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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/| f^ Qf tbe inniverett/of Midconsin I HEAT AND THERMODYNAMICS IViblUKed b^ the McGraw-Hill Book. Company SucccAsors to tKe Rook Departments of the McGraw Publbhing Company Hill Publishing' Companyr I\ibljaher5 of books for Electrical World The Engineering and Mining Journal Engineering Record American Machinist Electric Railway «Joumal Coal Age Metallurgical and Chemical Engineering R>wer l> WWWWW W WWWW W W W 9 W 9tr W WWWWWW W WW t9WWWWtW9f99WWWW91WWWW9WWWW9WWWWWWWrwmWwii HEAT AND THERMODYNAMICS BY F. M. HAETMANN In dkorff^ of th€ Depcurtmeni of Phynctt and Eledrieal and Meehanieal Eni/inoenng. Cooper Union Day and Night SchooU McGRAW-HILL BOOK COMPANY 239 WEST 39TH STREET, NEW YORK 6 BOUVERIE STREET, LONDON, E.C. 1911 COPYBIOHT, 1911 BY MoGRAW-HILL BOOK COMPANY 163128 TGr H^5 ERRATA Page 45, after the equation, p,«o(l+&T) = po«^(l+ax), delete remainder of page and read : The foregoing equation follows immediately from Boyle^s LaWy and by solving for t, we find ^ Pr—po poa-PrP Page 239, multiply right hand member of equation (b) by J. Page 283, first line after equation (28), in place of, '' equation (27)," read "equation (28)." HartmAnD's Heat and Thermodynainicw ^gS^giK^gS^gSC)g5HfeS^;SS-^!- 3128 Ca 163128 W\AR TGr PREFACE With so many good works in existence, both on Heat and Thermodynamics, it may perhaps appear presimiptuous to publish the following text. The author, however, has long felt the need of a text, in teaching the subject of thermodynamics, which properly covers, without introducing too much material, the fundamental principles of heat measurements. To expect an average student to cull from his text book on physics, or some treatise on heat, no matter how well the subject may have been taught, an introduction to thermodynamics is, in general, expecting somewhat more of him than he can accomplish. But it has been found, by experience, that a short cour^ on the fundamental principles of heat, given as an introduction to the subject of thermodynamics, greatly reduces the difficulties, experienced by most students, in pursuing this subject. Since it is almost impossible for a student to understand a complex piece of apparatus, imless he can actually examine it, long and tedious descriptions have been purposely avoided. Likewise, for the reason that photographs are seldom, if ever, of any value, all pictorial illustrations are diagrammatic. It is, of course, impossible to teach the subject of thermo- dynamics without the application of differential and integral calculus; but the aim has been throughout to keep within the bounds of elementary mathematics. However, a fair knowledge of the calculus, on the part of the reader, has been assumed. Very few teachers, if any, can present an unbiassed view of a speculative theory; furthermore, before a student has thor- ou^ly mastered the groundwork of any subject, he is not in a position to properly discriminate between the various arguments that may be advanced, either for or against a speculative theory. VI PREFACE It must also be remembered that the average student looks upon his instructor as an infallible authority; and that he accepts a theory on the mere say so of his instructor, no matter how flimsy the arguments upon which it may be based. How fre- quently one meets those who are in a condition so deplorable that they can talk very glibly about electrons, ionization, etc., and are driven helplessly into a comer by one or two weU directed questions. Whether there is or is not such a thing as an atom has nothing to do with the law of definite propor- tion. Facts will always remain and theories change to fit them. It is for .these various reasons that speculative discussions, such as that of the kinetic theory of gases, have been avoided, and that very hypothetical mediiun — ^the ether — has found no place in this text. It cannot be too strongly emphasized that before we. teach metaphysics to a student we must first give him a thor- ough training in mathematics and physics. It is the author's opinion that the best that can be done in any technical course is to thoroughly teach the fundamental principles underlying the subject, and that it is impossible to ^ve a training which makes the student a practical engineer. This part must be learned in practice, and the engineer must keep up to date and in proper touch with his profession by reading the current engineering literature, and by studying individual problems as they arise. It must not be understood that the author expects this text to supersede such admirable works as Peabody's treatise on " The Thermodynamics of the Steam Engine," Zeuner's " Tech- nische Thermodynamik," etc.; but rather as a proper prepara- tion for the reading of such works. Finally, the author cannot express his feelings too strongly in regard to the pleasure he experienced, as a student, while reading Tyndall's "Heat a Mode of Motion," and Ewing's "The Steam Engjie and other Heat Engines." Thanks are hereby expressed to Mr. Albert Goertz for the care with which he read the manuscript. F. M. H. CoopiBB Union, July, 1911. TABLE OF CONTENTS HEAT GHAPTZR PAGB I. Temperature and Thermal Unitb 1 II. Calorimetrt 11 III. Production op and Eppects op Heat 21 rV. Expansion op Solids and Liquids 34 V. Fundamental Equations op Gases 41 • VI. Elasticities and Thermal Capacities op Gases 68 Vn. Propagation op Heat 80 THERMODYNAMICS VIII. Fundamental Principles 93 IX. Steam and Steam Engines 116 X. Entropy 128 XI. Applications of Temperature-Entropy Diagrams 137 XII. Elementary Steam and Engine Tests 151 XIII. Compound Engines 180 XrV. Internal Combustion Engines and Fuels 192 XV. Ideal Coefpicient op Conversion and Elemejttary Tests . . . 210 XVI. Compressed Air and Compressors 230 XVII. REPRIGERATIOif 265 XVin. Steam Turbines 289 vu HEAT CHAPTER I TEMPERATURE AND THERMAL UNITS 1. The fundamental conception of hotness or coldness is one of bodily sensation. That is, an object is said to be hot or cold depending upon whether it gives us one sensation or another when we are near it or in contact with-it; and the more intense the sensation, the hotter or colder the object is said to be. Exper- ience, however, teaches us that the estimates so formed are not accurate; since the intensity of the sensation experienced, in any given case, depends not only upon the condition of the body under consideration, but very largely upon our experience imme- diately preceding. Observation shows that, in general, as bodies are heated or cooled they change in volume; and in most cases, other things being equal, bodies increase in volume when heated and decrease in volume when cooled. Observation fiulher shows that there is a continual exchange of heat among bodies; i.e., if in any S3rstem of bodies some are gaining heat, others are losing heat. Or, in other words, there is a continual tendency toward equilibrium. 2. Temperature. Assimie now, that we are dealing with two bodies, A and B, and that it is desired to determine which is the hotter. To do this, some test specimen, upon which 2 HEAT previous observation has shown a continuous expansion with continued application of heat, may be used in the following manner: The test specimen is put into contact with A, and after a suitable interval of time its length, say, is accurately measured; it is then put into contact with B, and its length is again measured. If the length is now greater than it was before, i.e., if the test specimen expanded when put into contact with S, after having been in contact with A, B is hotter than A; for, by previous observation it was found that the test specimen expanded con- tinually as it became hotter. The body S, therefore, was capable of imparting more heat to the test specimen than the body A could impart to it, and B is said to be at a higher temperature than A. The difference of temperature, then, between two bodies may be measured by the amount of change in dimensions which a test specimen imdergoes when, after having been in contact with one of the bodies, it is brought into contact with the other body. Such a test specimen is called a thermometer; and, for accurate measurements, the nature of the thermometer must be such that no appreciable change is brought about in the body whose temperature is sought. If the two bodies, A and B, of the previous discussion, are now brought into contact, and after a suitable interval of time the test specimen is put into contact with A and then with B, there will be no change in its dimensions; i.e., the two bodies are in thermal equilibrium, or, in other words, at the same temperature* But, the body B will have lost heat and the body A will have gained heat; hence, when a body has the capability of imparting heat to another body, it is said to be at a higher temperature. Solids, liquids, or gases may be employed in the construction of thermometers; but it is important that the substance used does not change its state during the change of temperature. For, the rate at which a body changes in volume, with respect to change in temperature, depends upon its physical state; i.e., though a substance may exist in the three different states, its rate TEMPERATURE AND THERMAL UNITS 3 of expansion will, in general, be entirely different in the various states; being usually the greatest for gases and the least for solids. The rate of expansion, in general, changes abruptly in passing from the solid to the liquid, and from the liquid to the gaseous state. The three substances most generally employed in the con- struction of thermometers are: Mercury, alcohol, and dry air] the most convenient and most commonly used being mercury. There are two standard temi)eratiu'es, arbitrarily chosen, upon which all thermometric scales are based; the one is that of melting ice, and the other* that of the vapor of boiling water under a pres- sure of one standard atmosphere. The pressure of one atmosphere being taken equal to that of a column of mercury, at the tempera- ture of melting ice, whose height is 76 cm., at 45° latitude and sea level, where the acceleration of gravity is 980.60 cm. per sec. per sec; or, in c.g.s. units, a pressure of 1.01325X10^ dynes per square centimeter. Thebmometric Scales and Thermometers 3. There are Oiree thermometric scales in use: The centigrade scale, the zero of which is the melting-point of ice, and the point corresponding to the temperature of the vapor from boiling water under standard conditions, called the boiling-point, is marked 100. Hence there are 100 units, called degrees, for the interval between the melting-point and the boiling-point. The Fahrenheit scale is marked 32 for the melting-point and 212 for the boiling-point; hence, there is an interval of 180 degrees between the two fixed points. The Riaumur scale is marked for the melting-point and 80 for the boiling-point; hence, there is an interval of 80 degrees between the two fixed points. Fig. 1 is a diagrammatic representation of the relation of the three scales. From the foregoing it is obvious that the value of a degree on the Fahrenheit scale is 5/9 of a degree on the centi- 4 HEAT grade scale, and the value of a degree on the Reaumur scale is 5/4 of a degree on the centigrade scale. Since both the centi- grade and Fahrenheit scales are in common use, it is convenient to have a simple method of conversion from one scale to the other. Let it be desired to convert the temperature 6 on the centigrade scale to the Fahrenheit scale. Since one degree centigrade equals 9/5 degree Fahrenheit, it follows that an interval of 6*^C. is equal to an FlG.1. g interval of — 6®F.; but, since the 5 melting-point on the Fahrenheit scale is marked 32, we must add 32 to give the Fahrenheit reading. Therefore, to convert a temperature on the centigrade scale to the Fahrenheit scale, we must multiply the reading by 9/5 and add 32. And similarly, to convert Fahrenheit to centigrade we must subtract 32 from the reading and multiply by 5/9. A method of conversion, which is simpler, is to solve for that temperature for which both scales read the same. This temperature is obviously below zero, and is therefore negative. g Let 6 be that temperature on the centigrade scale; then, —6+32 5 is the reading on the Fahrenheit scale. But these two readings are by condition equal. Hence, from which 6=|e+32; e=-40. Therefore, to convert from one scale to the other, the most con- venient way is to add 40 to the reading, multiply this by the con- TEMPERATURE AND THERMAL UNITS 5 version factor and deduct 40. To convert from centigrade to Fahrenheit; add 40 to the reading, multiply by 9/5 and deduct 40. To convert from Fahrenheit to centigrade, add 40, multiply by 6/9 and deduct 40. 4. Mercurial Thermometer. The mercurial thermometer con- sists of a capillary tube of, as nearly as obtainable, uniform bore, on one end of which is blown a bulb. The bulb is filled with mercury and heated so as to drive out all the air. When this has been satisfactorily performed, so that nothing but merciu'y remains in the bulb and tube, the tube is hermetically sealed. The bulb and tube are then immersed in ice from which the water is allowed to drain away, and the point to which the mercury falls marked 'on the stem. Next, the thermometer is immersed in saturated steam, under standard preasure, and the point to which the mercury rises marked on the stem. It is, however, necessary to allow considerable time to elapse between the seal- ing of the tube and the determination of the fixed points; since glass, after having been heated, does not, upon being cooled, immediately return to its original volume. If the bulb contracts after the fixed points have been placed on the stem, the ther- mometer will read too high. Joule foimd that the bulb of a certain thermometer, upon which he had taken observations for twenty years, was still changing slightly at the end of that time. After the fixed points are determined, a thread of mercury is detached, and, by means of it, the tube calibrated to the desired scale. In this way the units on the scale represent equal voliunes, and not necessarily equal lengths. But in good thermometers the tube is of so nearly uniform bore that the lengths of a degree do not differ by any considerable amount over different parts of the scale. Since glass changes in volume when its temperature changes, it is obvious that the indications of the mercurial thermometer are proportional to the relative changes between the mercury and glass, and do not necessarily indicate the true changes in 6 HEAT temperature; i.e., the indicated temperatures between the two fixed points depend upon the substances used in construction. It is readily seen that two mercurial thermometers, if constructed of different qualities of glass, which have not the same rates of expansion with respect to mercury throughout the entire scale, will differ slightly in their readings for some parts of the scale, even though they read alike for the fixed points. 6. Alcohol Thermometer. The alcohol thermometer is con- structed in a manner similar to the mercurial thermometer. Its chief advantage lies in the fact that it may be used for temperatures below the melting-point of mercury, which is — 39®C. However, on account of its low boiling-point, which is 78.2®C., alcohol cannot be employed for high temx)eratures; on the other hand, the boiling-point of mercury is 357®C. The discussion of the air thermometer will be deferred imtil after the discussion of the laws of gases. 6. Thermo Couple. When the junction of two dissimilar metals is heated, an e,m,f. is developed; and since this e.m.f. is a fimction of the temperature, such a combination, called a thermoelectric couple, or simply thermo couple, may be employed to indicate temperatures. The thermo couple is, in many cases, where a bulb thermometer cannot be employed, a very conven- ient device for measuring changes of temperature; and it is par- ticularly valuable in enabling us to estimate changes of temper- ature above the boiling-point of mercury. By employing proper metals, such as platiniun and iridium, very large ranges of tem- perature can be measured; the meltmg-pomt of iridium bemg about 2500'*C. and that of platmum about ITTS^'C. 7. Resistance Thermometer. The fact that the ohmic resist- ance of a metal is a function of its temperature makes it possible to estimate changes in temperatures, by noting the changes in resistance of a particular conductor employed for the purpose. From the foregoing, it is obvious that if any body or combina- tion of bodies manifest some change, which is a function of the TEMPERATURE AND THERMAL UNITS 7 temperature and readily measurable, such body or combination of bodies may be employed to indicate temperatures. But it is to be carefully noted that the device employed must be such that the temperature of the body, upon which the measurements are made, is not appreciably altered by the test body; and that, in any case, the changes produced in the test body are peculiar to it, and not necessarily proportional to the changes that would be produced in some other instrument of a different tjrpe. There- fore, temperatures must alwajrs be referred to some scale chosen as a standard. This will be dealt with more fully in the discussion of thermodynamics. Heat as a Measurable Quantity 8. If a quantity of water at a temperature ti, be mixed with an equal quantity of water at a temperature t2, the resulting temperature of the mixture is very nearly the mean between the two initial temperatures. If it requires a certain quantity of heat, g, to raise n grams of water through a given temperature interval, then it obviously requires a quantity of heat, mg, to raise mn grams of water through the same temperature interval. If the water be cooled through the same temperature interval, then there is imparted to the surrounding bodies a quantity of heat which is equal to that absorbed by the water while being raised through that temperature interval. If the quantities of heat, required to raise a given mass of water through equal temperature intervals throughout the chosen thermometric scale, were all equal, then the resulting temperature obtained when mixing equal masses of water would be the exact mean between the two initial temperatures. This is shown by experiment to be very nearly, but not quite true. Hence, there is not strict proportionality, in the case of water, between change of temperature, according to our thermometric scale, and change of heat. 8 HEAT If equal masses of water and some other substance, say, mercury, be mbced, the resulting temperature will differ considerably from the mean between the two initial temperatures. Experiment shows that if mercury at a temperature ti, be mixed with water at a temperature T2, the mass of the mercury must be 29.85 times the mass of the water so as to give a resulting temperature equal , T1+T2 to-g-. 9. Thermal Capacity. The thermal capacity of a body is numerically equal to the ratio of change in heat to the corresponding change in temperature produced by it. The preceding paragraph states that water has, mass for mass, a greater thermal capacity than mercury in the ratio of 29.85 : 1; or the thermal capacities of equal masses of mercury and water are to each other as 0.0335 : 1. If copper be compared with water the ratio is found to be as 0.0933 : 1. In general, the ratio is less than unity, has different values for different substances, and varies somewhat with change of temperature. One notable exception is hydrogen gas, where the ratio is found to be, at constant pressure, as 3.409 : 1; and at constant volume, as 2.42 : 1. To compare different quantities of heat, it is necessary to choose some substance as a standard. On account of convenience, water has been so chosen. 10. The Calorie. The quantity of heat required to raise the temperature of 1 kilogram of water through 1 degree centi- grade, is called a calorie^ and is the unit adopted for heat measure- ments. But, since this quantity varies slightly for different temperatures it becomes necessary, in making exact measure- ments, to specify some particular quantity. There are three different definitions for the calorie: (1) The quantity of heat required to raise the temperature 6f 1 kilogram of water from 0®C. to 1**C., called the zero calorie. (2) One hundredth part of the heat required to raise the tem- TEMPERA.TURE AND THERMAL UNITS 9 perature of 1 kilograin of water from 0°C. to 100°C., called the mean calorie. (3) The quantity of heat* required to raise 1 kilogram of water from 15®C. to 16®C., called the common calorie. Since it is impossible to realize accurately the first or second of these, on account of the difficulty experienced in working with water at 0°C., the last, or common calorie, is the one most gen- erally used. Furthermore, due to the fact that a great many heat measurements are made in the range between 15^C. and 25^ C, no large corrections for change in thermal capacity, due to change in temperature, when the common calorie is employed, are necessitated; hence, this unit is more convenient than the others. Since, in ordinary heat measurements, masses are generally specified in grams, a secondary unit, called gram calorie, having the gram instead of the kilogram for the unit mass, is usually found more convenient than the calorie. In what follows, unless otherwise specified, by calorie is to be imderstood the quantity of heat required to raise the temperature of 1 kilogram of water from 15**C. to 16®C., and by gram calorie, one thousandth part of the calorie. 11. British Thermal Unit. The thermal unit most commonly employed in engineering practice, in England and America, is the British Thermal Unit or B.T.U.; it is the quantity of heat required, at ordinary temperatures, to raise the temperature of 1 pound of water through 1 degree Fahrenheit. 12. Thennal Capacity per Unit Mass and Specific Heat. The ratio of the quantity of heat required to raise the temperature of a given mass of a substance through a given temperature interval, to the quantity of heat required to raise an equal mass of water through the same temperature interval, is called the specific heai of the substance. And, since the unit of heat — ^the gram calorie — ^is the quantity of heat required to raise the temperature of 1 gram of water through 1 degree centigrade, it follows 10 HEAT that the quantity of heat, measured m gram calories, required to raise the temperature of 1 gram of a substance 1 degree centigrade, is numerically equal to the specific heat of the sub- stance; or, in other words, the specific heat of a substance is numerically equal to its thermal capacity per unit rruiss. 13. Water Equivalent. By the water equivalent of a body is imderstood the mass of water which has a thermal capacity equal to that of the given body, and is numerically equal to the mass of the body multiplied by its thermal capacity per unit mass. 14. Method of Mixtures. Assume a mass of water mi, at a temperature xi, to be mixed with a mass m2, of some other sub- stance, at a higher temperature t2, yielding for the mixture a resulting temperature of 0. Then, if no heat is lost to or gained from the surfoundings, during the operation, and the thermal capacities of the water and substance are sensibly constant for the temperature ranges experienced, it follows that, since the heat gained by the water is equal to that lost by the substance, we must have mi(0-Ti)=m2c(T2-0); (1) where c is the thermal capacity per unit mass of the substance, amd m2C is its water equivalent. To take a numerical example, assume 268 grams of water at a temperature 10°C. to be mixed with 1000 grams of mercury at a temperature 100°C., giving a temperature of 20°C. for the mix- ture. Substituting in equation (1), we have 268(20 - 10) = 1000c(100 - 20) ; from which, the thermal capacity of mercury, per unit mass, is _ 268(20-10 ) _ ''" 1000(100-20) "•"'*^^- CHAPTER II CALORIMETRY 16. As thermometry has for its object measurement of tem- peratures, so has calorimetry for its object the measiu'ement of quantities of heat. In equation (1), Art. 14, it was shown what must be the relation between the masses involved, the changes in temperature, and the thermal capacity per unit mass of a substance, when two substances are mixed and assume a common temperature. The actual determination of thermal capacities is, however, not so simple. Since, in general, the vessel in which the mixing takes place suffers a change in temperature, its thermal capacity must be taken into consideration. Furthermore, there is usually an exchange of heat between the vessel, in which the experiment is performed, and the surrounding medium during the progress of the experiment. The vessel, specially designed, in which the mixing takes place, is called a calorimeter. It therefore follows, from what has just been said, that in making heat measurements it is necessary to know not only the thermal capacity of the calo- rimeter, but also, the rate at which, for a given difference of tem- perature, exchange of heat takes place between the calorimeter and the surrounding medium. 16. Law of Cooling. The rate at which a body loses heat to surroimding bodies ia independent of its thermal capacity, and depends upon the nature and area of its exposed surface, and the difference in temperature between it and the surroundings. For small differences of temperature, i.e., up to a difference of about 11 12 HEAT 15°C., from ordinary room temperature, the rate of cooling is very nearly proportional to the difference of temperature. This is known as Newion^s Law of Cooling. The foregoing then states that the rate at which a body loses heat at any instant is a func- tion of its surface, and of the diflFerence of temperature between it and surrounding bodies. Newton's law may be stated as follows: f-'^'^ <') where Q is quantity of heat, t time, K some constant, depending upon the surface of the body, and t the difference of temperature. If m is the mass of the body, and c the thermal capacity per unit mass, then equation (1) may be written "4'=-^^ ^2) If c is constant, then equation (2) becomes dt There will always be an exchange of heat between the calo- rimeter and surroimdings; but this can be reduced to a small quantity by choosing the masses of water such that the result- ing temperature of the calorimeter is as much below the room temperature as was its initial temperature above the room tem- perature. When it is possible, large differences of tempera- ture, between the calorimeter and room, should be avoided. 18. Cooling Constant of a Calorimeter. When it is impossible to have the initial and final temperatures differ by equal amounts from the room temperature — one, of course, being above and the other below — ^then, to obtain accurate results, correction must be made, as the case may be, for loss or gain in heat. To do this, the calorimeter is filled with water, at about 15®C. or 16®C. above room temperature, to the same height as it will be when the experiment proper is performed. The temperature b then CALORIMETRY 15 noted at short intervals of time, the water being continuously stirred to insure a uniform temperature throughout at any instant. If the room temperature has remained constant during the progress of the experiment, then, obviously, equation (4), Art. 16, applies and the constant k of this equation is determined. In general, however, better results are obtained by plotting the observations; using times as abscissas and differences in temper- atures, between the calorimeter and the room, as ordinates, and passing a smooth curve through the points so found. The slope of the tangent then, to the curve at any point, is the rate of change of temperature at that point; and this slope, divided by the dif- ference of temperature, or in other words, by the ordinate of the point, is, according to Newton's law of cooling, a constant for any point on the curve. Drawing a number of tangents and divid- ing the slope of each by its ordinate, will give quotients nearly equal; and the mean of these quotients will be a fair value for the rate of change of temperature for imit difference of temper- ature. The rate so obtained multiplied by the thermal capacity of the calorimeter and contents is numerically equal to the quan- tity of heat lost, by the calorimeter, per unit time per imit differ- ence of temperature. 19. Determination of Thermal Capacities. Thermal capacities may be determined in various ways; the simplest, though not necessarily the most accurate, and not applicable in all cases, is the method of mixtures. The substance, whose thermal capacity is sought, is heated to a temperature ti, which is noted. It is then inmiediately transferred to a calorimeter of thermal capacity C, containing a mass of water m2, at a temperature T2. If the result- ing temperature is 6, and no heat has been lost to or gained from the surroundings during the operation, then the heat lost by the one side must be equal to the heat gained by the other; hence we have mic(Ti-e) = (m2+C)(e-T2); (6) 16 HEAT where mi is the mass of the substance, and c its thermal capacity per miit mass. From equation (6), we find ^, (m.+C)(e-T2) wi(Ti-e) which determines the thermal capacity per unit mass of the substance. If the resulting temperature differs materially from the room temperature, then corrections will have to be made, from the curve of cooling of the calorimeter. 20. Method of Cooling. If heat is generated or absorbed when two substances are mixed, the thermal capacity of a sub- stance cannot be determined by the method of mixtures. As an example, if sulphuric acid is mixed with water, considerable heat is evolved; hence, recourse must be had to some method, other than the method of mixtures, in determining the thermal capacity of sulphuric acid. This may conveniently be done by what is known as the method of cooling. Assume that we have a calorimeter of known thermal capacity, C Let the calorimeter be filled to a definite height with water, and the time noted which is required for the calorimeter and contents to cool, from a temperature ti to a temperature T2, when exposed to a definite and constant room temperatiu'e. Next, the calorimeter is filled to the same height with the liquid, whose thermal capacity is sought, and the time which is required to cool from ti to T2, when the calorimeter and contents are sub- jected to precisely the same conditions as when filled with water, is again noted. Then, since the average difference of temperature between calorimeter and contents and the surroundings is the same in both cases, it follows that the thermal capacities, in the two cases, are to each other directly as the times required in cooling through the same temperature intervals. Therefore, if ti is the time required for the calorimeter and water to cool from ti to T2, and t2 is the time required for the calorimeter and the sub- CALORIMETRY 17 stance whose thennal capacity is sought, to cool through the same temperature interval, it follows that C+nii __ C+m2C. rgv ^1 fa ' where tni is the mass of water, m2 the mass of the liquid, and c its thermal capacity per imit mass. From equation (8), since all quantities excepting c are known, the thermal capacity per unit mass is determined. It is, however, not necessary to know the thermal capacity of the calorimeter. For, if the calorimeter be first cooled through the temperature interval, ti— T2, when empty, then when filled with water, and again when filled with the liquid whose thermal capacity is sought, and the times t, hy and fa are noted, we shall have C C+mi C+m2C t ti fa (9) where t is the time required when empty, h when filled with water, and fa when filled with the liquid whose thermal capacity is sought, 4 and the other symbols having the same significance as before. Eliminating C, from equation (9), we find mi(fa— . . '^m(h^) ^^^^ A convenient form of apparatus, for the method of cooling, is an alcohol thermometer with its bulb greatly enlarged and having the form of a hollow cylinder. The substance is then placed directly inside of the bulb and the whole thermometer, while cooling, exposed to a constant temperature; this may readily be brought about by placing the thermometer inside of a vessel which is surrounded by melting ice. 21. Mechanical Equivalent of Heat From mechanics we have the following statement: '' The change in kinetic energy 18 HEAT that a body undergoes in passing over a given path is equal to the work done in traversing that path." The foregoing statement is very simple, and readily comprehended when considered in the purely mechanical sense. That is, it is merely stated that when- ever a given amount of kinetic energy has been destroyed or developed in a system, an equivalent amount of work has been done by the system or, as the case may be, on the system. The statement, however, does not concern itself with the transforma- tion of one form of energy into another. The most general experience, common to all, is that of the destruction of energy, in the form of mechanical motion, by fric- tion or impact and the simultaneous evolution of heat. However, it was not until 1842 that a clearly formulated statement was made, by J. R. Mayer, to the eflfect, that when heaJt is corwerted into work, or vice versa, the ratio of the numbers representing the two quardities involved is constant. Unfortunately, the figures upon which Mayer based his calculations were in error, and consequently the value obtained for the mechanical equivalent was also in error. Shortly subsequent to Mayer's enunciation. Joule began his series of experiments to determine the mechanical equivalent of heat by direct measurement. Joule's method was essentially as follows: A vessel having fixed vanes, was filled with water, and a paddle-wheel was caused, by means of falling weights, to rotate in the water. The fixed vanes prevented the water from assimiing a rotary motion. The heat developed manifested itself by a rise in temperature, and the work done was measured by the weights and distance fallen. A series of experiments was then made, using mercury instead of water. Another series of experiments was made by causing one iron plate to rotate with friction over another iron plate under water. It is, of course, understood that experiments like these are attended by great difficulties, and various precautions must be taken and corrections made which cannot here be enumerated. CALORIMETRY 19 Still, Joule obtained fairly consistent results; and, the figures he finally published were not greatly in error. The results were expressed in meter-kilograms of work per calorie; i.e., Joule foimd when using water that it required 423.9 m.kg8. of work to develop one calorie, 424.7 m.kg8. of work per calorie when using mercury, and for the experiment with iron, the number was found to be 425.2. It must be remembered that, if comparisons are to be made between results obtained by experiments, which have been performed in different localities, corrections will have to be made for variations in the value of g. Rowland varied Joule's method by using a motor drive instead of falling weights. The vessel was suspended and the torque required to prevent rotation measured. Thi3 enabled a much more rapid expenditure of energy, and a consequent rapid rise in temperature; thus making the correction due to cooling much smaller. Rowland's experiment covered the range from 5°C. to 36**C. Since the thermal capacity of water varies for different temperatures, a variation was found for the mechanical equivalent of heat. Rowland found, at Baltimore, where g = 980.0, for the mechanical equivalent of heat, of the common calorie, 427.3 m.kgs. This may be taken as being substantially correct. Anthony modified Rowland's method by having a continuous flow of water through the calorimeter, the temperature of the inflowing water was constant, and its rate of flow was so regulated that the vessel was always at room temperature. The mass of wajer flowing for a given time was determined by weighing, the number of rotations made by the paddle was recorded on a speed counter, and the torque was noted, together with the tem- perature of the inflowing and outflowing water. By this method, fifince the vessel is always at room temperature, no corrections for cooling are required; and furthermore, since the vessel suffers no change in temperature, its thermal capacity need not be known. The values obtained by this method were in practical concordance with those foimd by Rowland. 20 HEAT In the English system of units, the value for the mechanical equivalent of heat, usually employed, is 778 ft. -lbs. per B.T.U. Various experiments have been performed, by sending an elec- tric current through a conductor woimd upon an insulating support and submerged in water. By noting the current, the applied 6.m./., and the time, during which the current has been flowing, the energy input is readily computed. Results obtained by this method agree almost precisely with those obtained by the methods previously described. A safe value to use, which if in error is only slightly so, and which is correct for all practical purposes, is 4.195X10^ ergs per common gram calorie. It will now readily be seen that, since we can measure electrical quantities with great precision, the best method for obtaining a definite quantity of heat is by sending a steady current for a given time through a given resistance; it being remembered that there are alloys whose resistances are practically independent of temperature. CHAPTER III PRODUCTION OF AND EFFECTS OF HEAT 22. Since, in a great many cases, those changes which evolve heat during their progress, require the application of heat to bring about a change in the reverse order, it is inadvisable to consider the production of heat and the effects of heat independently. Whenever a change of a chemical nature takes place there is either an absorption or a liberation of heat. In general, heat is absorbed when a compound is split up into the elements com- posing it; and heat is liberated when the elements recombine to form the compoimd. Those compoimds which evolve heat, during their formation, are called exothermic compounds; and those rare compoimds which absorb heat, during their formation, are called endoihermic compounds. It will now be instructive to consider some particular substance and the various changes which take place with the continuous application of heat. Suppose that we are dealing with a definite mass of ice, under a given pressure, whose temperature is below that of its melting-point for the applied pressure. (The melting- point of ice changes slightly with change of pressure; i.e., the melting-point is lowered about 0.0075°C. for each increment in pressure equal to one atmosphere.) A definite amoimt of heat then, must be applied to raise the temperature of the ice to the melting-point. If heat be then further applied the temperature will no longer change; but, a change of physical stale takes place together with a continuous absorption of heat until all the ice is melted. 21 22 HEAT 23. Heat of Fusion. The quantity of heat required to convert unit mass of a solid into the liquid state withotU change of tem- perature is called the heat of fusion of the substance. In the case of ice, the heat of fusion is approximately 80 gram calories per gram. If now, after all the ice has been converted into water, heat be continuously applied, the temperature will rise progress- ively until, if the liquid be under a pressure equal to one stand- ard atmosphere, the temperature of 100*^C. is reached. The temperature will then cease to rise, provided the pressure be maintained constant, and a change of physical state, namely, vaporization at constant temperature, takes place progressively with the continuous application of heat until all the water is evaporated. The heat of fusion of ice may be determined as follows: Let L, m, and ti, respectively, represent the heat of fusion, the mass, and the initial temperature of the ice; and M, and T2, respectively, represent the water equivalent of calorimeter and contents, and the initial temperature of calorimeter. Let the ice now be sub- merged in the water in the calorimeter imtil it is all melted, and the calorimeter and total contents assume a common temper- ature 6. The total heat, then, consumed by the ice in having its temperature raised from ti to 0, being converted into water at this temperature, and in raising the temperature of the liquid from to 0, is mcTi+m(L+6); where c is the thermal capacity per unit mass of ice. But this quantity of heat must be equal to hence, From which ilf(T2-e); ?ncTi+w(L+e) =M (T2-e). L=-(T2-e)-.(CTi-he). PRODUCTION OF AND EFFECTS OF HEAT 23 24. Heat of Vaporization. The quantity of heat required to convert unit mass of a liquid into a vapor mthaui change of temperature is called the heat of vaporization of the substance. In the case of water, at lOO^C, the heat of vaporization is approx- imately 537 gram calories per gram. The heat of vaporization for most substances becomes less as the temperature rises. The heat of vaporization of water may be determined by the method of mixtures as follows: Let superheated steam, at a tem- perature Ti, be passed into a calorimeter containing water at a temperature T2. This is continued until a convenient rise of temperature is obtained in the calorimeter, and a mass of steam m has been condensed. The quantity of heat, given up by the steam, is mc(Ti — T)+wr+w(T— 6) ; where t is the temperature at which the condensation takes place, r the heat of vaporization, c the thermal capacity per unit mass for superheated steam, and the resulting temperature. But this quantity of heat must be equal to M(6-T2); where M is the thermal capacity of the calorimeter and water initially contained in it. Hence, mr+m{c(Ti-T)+(T-e)}=M(e-T2); from which r=^(0-':2)-{c(Ti-T)+(T-e)}. To obtain accurate results by means of calorimetric methods, as previously stated, it is always necessary to take certain pre- cautions and apply proper corrections. 26. Sublimation. Under certain conditions a substance may pass directly from the solid to the gaseous state without passing 24 HEAT through the liquid state. Such a change is called syblimaMon. Substances such as camphor and iodine, when gently heated, pass readily from the solid to the gaseous state without melting. Ice, under normal pressure, also sublimes at temperatures lower than the melting-point. 26. Superheating. If now, after all of the liquid has been converted into vapor, heat be further applied, the temperature will rise progressively with continued application of heat, and the vapor will become superheated. All of the foregoing may, instructively, be represented diagram- matically; bearing in mind that the thermal capacities per unit mass of water for the three states are: Solid 0.504, liquid 1, and g(i8e(m8 0.481. Fig. 2. Let, as in Fig. 2, the temperatures be taken as ordinates and quantities of heat as abscissas. Then, if we assume some arbitrary zero, such as 0, for the initial condition of the ice, we have for the application of the quantity of heat Oe, the increment of tempera- ture ea, to the point of fusion. The application of the quantity of heat ef then brings about the conversion from the solid to the liquid state at constant temperature. The further application of the quantity of heat fg brings about th^ elevation of temper- ature, from the temperature of fusion, to that of vaporization. The application of the quantity of heat gh brings about complete PRODUCTION OF AND EFFECTS OF HEAT 25 vapoTziation at constant temperature. The further application of heat brings about superheating, as shown by the line di. In general, the physical history of substances with continued application of heat will be similar to the process just discussed; but, the ratios of the quantities involved will be entirely dififerent for each substance. Assume now, that the process takes place in the reverse order, step by step; i.e., the steam is first cooled to the point of conden- sation, then condensation takes place at constant temperature, and so on, step by step, until the initial condition is reached. Then during each change, heat is liberated precisely equal in amoimt to that which was absorbed when the change was taking place in the opposite direction. 27. Reversible Processes. The changes just described and depicted in Fig. 2 are, however, not the only changes involved. Assmne the applied pressure to be maintained constant throughout the entire change, the volume then of the substance will be chang- ing continually; and, in general, will be increasing with the tem- perature. When the volume is increasing, work is being done by the substance in overcoming the applied pressure. When the volume is decreasing, work is being done an the substance by the applied pressure. From the initial condition up to 4°C., this being the temperature ccM*responding to the maximum density of water, work is being done on the substance. For all tem- peratures higher than this, the substance expands continuously with the continued application of heat; and work is being done by the substance in overcoming the applied pressure. When the process takes place in the reverse order, then, wherever heat was absorbed and work done by the substance during the direct process, heat will be liberated by, and work will be done on the substance during the reverse process. And if these quantities be mutually equal, and no permanent changes have been made to take place in the surroimding bodies, by this cycle of operations, either process is said to be a reversible process. That is, when a system 26 HEAT undergoes a change, or a series of changes, the process is said to be reversible if, after it has taken place, a second process can be made to take place, in a manner, such that when the system is again in its initial condition, there remain, due to these various changes, no changes oviside of the system. A little consideration will show that vaporization at constant pressure, and hence at constant temperature, provided we could have perfect insidation and no friction, would be a reversible process. For, under the assiuned conditions, the amount of work done by the vapor, during its formation, in overcoming the external pressure, is precisely equal to the amount of work done on the vapor during its condensation. Furthermore, the quantity of heat absorbed, during vaporization, from a reservoir of heat at constant temperature, is precisely equal in amount to the quantity of heat rejected, to the reservoir, during condensation. Hence, since all quantities involved balance each other, and no changes have been brought about in the surroimdings, the process is reversible. The cooling and heating of a substance at constant pressure, together with its consequent changes in volume, can be made a reversible process only by the aid of a perfect regenerator. The following discussion will make this clear. Assume that a body at a temperature Tn, which is also the temperatm-e of the first reservoir, cools to a temperature ti, by being put, successively, into contact with n reservoirs, perfectly insulated from each other, and each reservoir differing in temperature from the one adjacent to it by an amount equal to (xn— Ti)/(n— 1). The body then, in cooling, gives up to each reservoir, excepting the first, a definite quantity of heat, and has done upon it, by the constant external pressiu-e, a definite amount of work. Let the process now take place in the reverse order, i.e., the body at a temperature ti is put into contact with the reservoir at a temper- ature T2; a definite quantity of heat will be absorbed, which will be precisely equal to that rejected when put into contact with PRODUCTION OF AND EFFECTS OF HEAT 27 the reservoir at a temperature ti, after having been in contact with the reservoir at a temperature t2. Let this be continued until the temperature Tn is again reached. Now, the work done by the external pressure on the body, while cooling from the tem- perature Tn to the temperature ti, is precisely equal in amount to the work done by the body, in overcoming the external pressm-e, while being heated from the temperature ti to the temperature Tn. This process, however, is not perfectly reversible; since the reservoir at the temperature Tn has given up heat and received none, and the reservou: at the temperature ti has received heat and given up none. In the limit, however, as the fraction (Tn--Ti)/(n— 1), approaches zero for its value, the process becomes perfectly reversible. But this implies a perfect regenerator; i.e., a series of reservoirs which are perfectly insulated from each other, and still have a continuous variation in temperatmre throughout the series; but this is practically impossible. Hence, it is obvious that, since in the case of vaporization, we must assume no radiation and no friction to make the process reversible, and in the case of cooling and heating of a body, we must assmne a perfect regenerator and no friction and radiation to make the process reversible, the process as described and represented diagranmiatically in Fig. 2 is reversible only in an ideal sense; i.e., an ideally reversible process, 28. Irreversible Processes. If a constant e.m,f. be applied to the terminals of a homogeneous conductor, a current will flow which is directly proportional to the applied e.m.f., and inversely to the resistance of the conductor. After a time the conductor will reach a constant temperature; i.e., the rate at which the energy is being converted into heat by the conductor, due to its resistance, will be equal to the rate at which heat, expressed in the same imits, is given to the surroundings by the conductor. This, however, does not mean that there is thermal equilibriimi. In this case thermal equilibrium can only be brought about by dis- connecting the applied e.m./., and consequently, discontinuing the 28 HEAT dissipation of energy. This being done, the conductor will finally assume the temperature of the surroimdings, and thermal equilib- rium will have been established. This process, viz, the con- version of energy, in the form of an electric cxirrent, into energy, in the form of heat, differs essentially in one particular feature from the process discussed in Art 27. The former process, which we termed an ideally reversible procesSj can be made to take place, barring various losses, in the reverse order. The latter process, however, cannot be made to take place in the reverse order; i.e., it is absolutely impossible to cause a current to flow in a homogeneous conductor by applying heat to it. Such a process is called an irreversible process. Another example of an irreversible process is that of the con- version of energy, in the form of mechanical motion, into heat by friction. For it is impossible to restore a system of bodies to their initial positions by the application of a quantity of heat to the sm*faces, equal to that which was evolved, due to friction, during their displacements. The same is true for the case of impact. When impact takes place between two or more bodies, a certain amount of kinetic energy is always converted into heat. But it is absolutely impossible, by the direct application of heat, to restore the. kinetic energy which was destroyed during impact. Also, when thermal equilibrium is established by mixing sub- stances, initially at different temperatures, the process is abso- lutely irreversible. 29. Dissociation. Under Art. 26 we discussed the physical history of water with the continued application of heat up to the point of its superheated vapor. If heat be still further applied to the superheated vapor its temperature will continue to rise, up to some very high temperature, when complete dissociation takes place; i.e., the vapor splits up into its two constituent elements, viz, hydrogen and oxygen. Such a change is called a chemical change. If, while the pressure is maintained constant, heat be further applied, the gaseous mixture will rise in temper- PRODUCTION OF AND EFFECTS OF HEAT 29 atiire and increase in volume progressively with continued appli- cation of heat. If now, the process be reversed, i.e., the mixture be cooled, the temperature and volume will diminish until the temperature of dissociation is reached. When this point is reached the two gases recombine to form steam, and precisely the same amount of heat is evolved as was absorbed to bring about decom- position. The quantities of heat, however, which are involved in chemical decomposition and recomposition are very large in comparision with those quantities involved during changes in temperature and changes of physical state. Indeed, our greatest source of supply of energy, in the form of heat, is that due to chemical combination; viz, the combination of carbon, in the form of coal, with oxygen. It is true that the amount of dissociation is a function of the temperature; i.e., even water at ordinary temperatures has a small percentage of dissociation. This, however, does not invalidate the statement that, the energy absorbed during dis- sociation is equal to that liberated upon recombination. To give an illustration of the quantities of heat evolved during chemical combination, we may take as examples the combination of hydrogen and oxygen to form steam, and the combination of carbon and oxygen to form carbon-dioxide. In the former case, 1 gram of hydrogen combining with oxygen to form steam (H2O), about 34,000 gram calories are evolved, or expressed in mechanical units, 1.43X10^^ ergs. In the latter case, i.e., when 1 gram of carbon combines with oxygen to form carbon-dioxide (CO2), about 8000 gram calories are evolved, or, expressed in mechanical units, 3.36X10^^ ergs. '30. Electrolysis. Dissociation may also, in general, be brought about by electrolysia. That is, if an electric current be passed through a chemical compound, in the form of a solution, the com- poimd will be split up into its constituents. ' Suppose that we are dealing with a solution of copper sulphate (CUSO4), and that the two electrodes are absolutely inert as regards 30 HEAT chemical reactions. Then if an electric current be passed through the solution^ copper will be deposited on the negative electrode {cathode), and the radical SO4 will be liberated at the positive electrode {anode). The SO4 thus liberated will combine with hydrogen of the solvent to form sulphuric acid (H2SO4) ; and at the same time oxygen will be liberated. The equation, repre- senting this reaction, is CUSO4+H2O = H2SO4+CU+O. K an electric current be passed through water, then the water will be split up into its two elements, hydrogen and oxygen; hydrogen being given off at the cathode and oxygen at the anode. In the case of dissociation by heat, a definite quantity of heat disappears for a given amount of dissociation; and an evolution of an equal quantity of heat upon recombination. But, since a given quantity of heat represents a definite amount of energy, it follows that dissociation involves storing of energy. Likewise, when dissociation is brought about by electrolysis a definite amount of energy is consumed for a given amoimt of dissociation, which must necessarily be equal to that consumed when the same amount of dissociation is brought about by the application of heat; since the energy stored is the same in amount for both cases. It is true that a solution becomes heated when conveying a current; but, this has nothing to do with the dissociation. The development of heat being merely due to the resistance of the solution, the same as when any other conductor is conveying a current. It is immaterial, so far as the foregoing argument is concerned, whether we consider the solution initially partly ionized and the current merely a icarrier of the ions, or that the current actually splits up the compound. 31. Faraday's Discoveries. Faraday showed experimentally that the amoimt of dissociation is directly proportional to the PRODUCTION OF AND EFFECTS OF HEAT 31 time and the intensity of the current; and furthermore, that the amoimt of chemical action is the same for all parts of the circuit. The latter part may perhaps be best illustrated as follows : Assmne that there are two voltameters connected in series; the first containing a solution of copper-sulphate and the second water. Then upon passing a steady current through the circuit a definite amount of copper will be deposited on the cathode of the first voltameter, for a given interval of time, and a definite amount of hydrogen liberated at the cathode of the second voltameter, during the same interval of time; and these two quantities mil be in the same raiio as their chemical combining numbers. That is, for every gram of hydrogen set free at the cathode of the second voltameter, 31.59 grams of copper will be deposited on the cathode of the first voltameter; where, if hydrogen be taken as unity, 31.59 is the chemical equivalent of copper in copper-sul- phate. During the same time that the 31.59 grams of copper are being deposited on the cathode of the first voltameter, 1 gram of hydrogen must be liberated to combine with the sulphion (SO4), set free to form H2SO4. 32. Counter Electromotive Force. Since the amount of dis- sociation, other things being equal, varies directly as the cur- rent, and the amount of energy stored during dissociation depends upon the compound dissociated, it follows that every compound offers a definite counter e.m.f, to dissociation. And any applied e.m.f. less than this cannot bring about dissociation. To make this clear, the e.m./. necessary to dissociate water will here be cal- culated. The amount of hydrogen set free, per coulomb of elec- tricity conveyed, is 0.000010357 grams. The number 0.000010357 is called the electro chemical equivalent of hydrogen. Let, in the cg.s. system of units, z be the electro chemical equivalent of hydrogen, I the current, and t the time, then the mass of hydrogen liberated, during the time t, is m^Izt (1) 32 HEAT Let h be the heat, expressed in mechanical units, required to dissociate 1 gram of hydrogen, then mh=IEt; (2) where E is the applied e.m./. Substituting in equation (2), the value of w as given in equation (1), we find Izth-^IEt; from which zh^E (3) If now, in equation (3), we substitute for z and h their values, remembering that for 1 gram of hydrogen, combining with oxygen to form steam, A = 1.43 X 10^^ ergs; and that z, in the cg.s, system of units, equals 0.00010357 grams per imit quantity of electricity, we find or, £?= 1.43X1012x0.00010357 c.g.8. units e.m/., „ 1.43X10^2x0.00010357 , ^^ u E = r>.^ = 1.48 volts. 10** It must, however, be emph&sized that, in general, the e.m/. for most cells is a function of the temperature; and, therefore, to calculate the e.m.f, this function must be known. We are not prepared, here, to take up this matter. The coimter e.m.f. is readily determined by experiment. 33. Junction of Dissimilar Metals. If the junction of two dissimilar metals be heated an e.m.f. is developed which is a func- tion of the temperature, and the metals which form the junction. To take a concrete example, assimie a junction of antimony and bismvih. Such a junction, if heat be applied to it, develops an e.m.f. which tends to send a current from bismvih to antimony; and if a current be sent from bismuth to antimony, by the applica- PRODUCTION OF AND EFFECTS OF HEAT 33 tion of an external e.m.f.y there will be a tendency to reduce the temperature of the junction. On the other hand, if a current be sent through the junction from' antimony to biamvih, heat will be developed. 31. As a r^mn^, we may then state that the most general effects of heat are: To change the volumes of bodies; to bring about physical changes of state; in general, to promote chemical dissociation; to develop an e,m,f. at the junction of dissimilar metals. For the prodiiction of heat we may state the following examples: The production of heat by the mechanical compression of bodies which expand upon the application of heat; when the physical state of a body changes in the reverse order from the change when heat is applied; in general, by chemical combination; at the junc- tion of two dissimilar metals when a current is passed in a direc- tion opposite to that of the developed e.m.f. when heat is applied. Also, the production of heat when an electric current is conveyed by a homogeneous conductor; and, in general, when friction is being overcome, and mechanical motion destroyed by impact. 36. The Principle of Energy. The various relations discussed in this chapter may now be summed up and stated quantitatively in a very simple manner. This generalization, known as the principle of energy, or conservation of energy, is one of the most extensive of generalizations, and may be stated in substance as follows: If in any system, from which no energy escapes and into which no energy enters, account be taken of aU forms of energy, then no maUer what transformations take place within the system, the sum total is a constant quantity. So far as experience goes, the foregoing statement is consistent with all phenomena; and hence, in all subsequent demonstrations its truth will be assiuned. CHAPTER IV EXPANSION OF SOLIDS AND LIQUIDS 36. Linear Expansion. It has been previously stated that, in general, bodies expand when the temperature is augmented. It is found by experiment that a body, such as a metal rod, increases in length by approximately equal amounts, between 0°C. and lOO^C, for equal increments of temperature. But, even though there is an approximate proportionality between change in length and change in temperature for moderate ranges, such as just specified, it must not be inferred that this is generally true for large ranges or high temperatures. If a body of unit length expand in length by an amoimt a for unit increment of temperatiu"e, then a body of length I will expand in length by an amount Za for an increment of 1 degree; and for an increment of t**, between 0°C. and lOO^C, the body will expand in length, approximately, by an amount Zocr. Hence, if the length of the body at 0** be denoted by lo, and at t** by Z„ we have from which ^ = fo(l+flpr) (1) The quantity a is called the coefficient of linear expansion, and may be defined as the ratio of the change in length, per unit change in temperature, to the length at zero. The quantity in the parenthesis, viz, l+or, is called the /ocfor of linear expansion. 34 EXPANSION OF SOLIDS AND LIQUIDS 35 37. Voltuninal Expansion. Homogeneous isotropic bodies will change in amount by like fractional parts of their original dimensions in all directions when the temperature changes. Assume that we are dealing with a rectangular parallelopiped whose three edges at zero temperature are oo, bo, and co, its volume then, at zero, is »o=ao&oco. . (2) If the temperature, now, be changed to t®, the three edges become: ao(l+ore), 6o(l+ore), and co(1+oct); from which we have, for the volume at t°, ^=ao6oco(l+opc)3 (3) Substituting in equation (3), for ooioco the value vo, as given by equation (2), we have t;,=«o(l+(rr)3 (4) Expanding equation (4), we find v,=vo(l+S(rz+3a?z^+Q?'^) (5) Now, a is a very small quantity in ^comparison with the dimen- sions of most bodies; hence, the two terms containing a^ and a^ may be neglected; and we have v,=vo(l+Sopz) (6) Equation (6) is, of course, only approximately true; but, that it is correct for most practical purposes will be evident from an inspection of 'the following table, which gives the coefficients of linear expansion, per degree centigrade, for a few solids. It must be remembered that for substances like brass and glass the nmn- bers are only approximate; and in any case the value found for the coefficient of expansion will depend somewhat upon the treat- ment to which the specimen was subjected in its preparation. 36 HEAT Coefficients of Lineab Expansion Platinum CSSOXIO"*^ Copper 1.678X10"* Steel (annealed) 1.095X10"* Zinc 2.918X10"* Brass 0.187X10"* Glass 0.083X10"* Invar (steel containing 36% nickel) 0.087X10"* It is only necessary to substitute the values of a, as given in the foregoing table, in equation (5) and it becomes evident that equation (6) is approximately true. Hence, we may say that-, the coefficient of voluminal expansion is practically equal to three times the coefficient of linear expansion. 38. Non-Isotropic Bodies. There are certain bodies which have different physical properties in diflferent directions. Such bodies are termed non48otropic. A notable example is that of Iceland spar, in which it is found that the coefficient of linear expansion in one direction is 2.63X10"*; whereas, in a direction normal to this it is found to be only 0.544 X 10"*. It is also interesting to note that Iceland spar manifests different optical properties in different direltions, 39. Expansion of Liquids. Liquids, in general, change more . rapidly in volume than do solids for equal changes in temperature. However, since, in the case of liquids, the term linear expansion is meaningless, we deal only with voluminal expansion. The determination of the coefficient of linear expansion is quite simple; hence, the description of the methods employed in its determination was omitted. The determination of the coef- Jtdent of voluminal expansion of a liquid is attended by various difficulties; and, since the discussion of the principles involved will proves instructive, a few methods will be described. EXPANSION OF SOLIDS AND LIQUIDS 37 Assume that we have a glass flask tennmating in a tube of capillary bore, and that the mass of the flask when empty is known. The flask is then filled, to a definite mark on the capillary tube, with the liquid at a temperature ti, whose coefficient of voluminal expansion is sought. The mass of the vessel and contents is now determined. The difference between this mass and the mass of the flask gives us the mass of the liquid at the temperature ti. If Di is the density of the liquid at the temperature ti, and V the volmne of the liquid in the flask at the same temperature, then J5i=-pr; (7) where Mi is the mass of the liquid in the flask at the temperature Ti. Let, now, M2 be the mass of the liquid in the flask, when its temperature is T2, the flask being filled to precisely the same mark on the tube as when the temperature was ti. The volume of the flask will now be r = F{l + P(T2-Ti)}; where ^. is the coefficient of voluminal expansion of tfie glass of which the flask is composed. ^ may be computed from the coef- ficient of linear expansion, or determined xlirectly by experiment, as will be shown subsequently. The density of the liquid at T2, will now be ^^-l^~Fil + &(T2-.l)} ^^^ Dividing equation (7) by equation (8), we find 38 HEAT If the coefficient of voluminal expansion of the liquid, between 0° and the other two temperatures under consideration, is approx- imately constant, we have where Do is the density of the liquid at 0** and a the coefficient of voluminal expansion. Similarly, we find Z)2=j^; (11) l+aT2 from which, by dividing equation (10) by equation (11), we find Di l+aT2 D2 1+aTi (12) Finally, by equating the right-hand members of equations (12) and (9), we find •±S,^.|,+M.-,.)l (.3, In equation (13), all quantities excepting a are known; hence, its value is determinate. Another method is as follows: A solid, whose coefficient of voluminal expansion is accurately known, and which does not react chemically with the liquid whose coefficient of voluminal expansion is sought, is weighed in the liquid, first at temperature ti, and second at temperature T2. The weight of the solid being known, its loss of weight for the two temperatures is known; and, smce the volume of the solid for any temperature may, from its known coefficient of voluminal expansion, be computed, the densities of the liquid for the two temperatures ti and t2 are readily found, and from these, as previously shown, the coefficient of voluminal expansion. EXPANSION OF SOLIDS AND LIQUIDS 39 ■10 'a^ ? i-_ 40. Direct Measurement of CoeflSicient of Voluminal Expan- sion. The method about to be described, and the one by which Regnault determined the coefficient of expansion of mercm-y, depends upon the principle that when commimicating columns of liquids are in equilibrium their heights are inversely as their densities. In Fig. 3, AB and CD are two vertical iron tubes, cross connected by the horizontal tube BD. The two horizontal tubes, AE and CF, terminate in the vertical tubes EO and FI, which are connected by the inverted glass U-tube GJI, The tube containing the stop cock s, is connected to the receiver of a compression pump; and after the apparatus has been filled with mercury, air is forced in from the compressor until the mercury in the tubes EG and FI is at a convenient height. The stop cock B is then closed. If, now, the temperature, and consequently the density, of the mercury in the tubes AB and CD be the same, the columns in EG and FI will be at the same level. On the other hand, if the temperatures of the two columns AB and CD be not the same, then the colmnns in EG and FI will not be at the same level. For, since there is free communication between B and Z), the pressure must be the same at these two points. Further- more, the pressure of the air inside the U-tube being everjrwhere the same, and, since this pressure plus the pressure, due to the column in FI, balances the pressure due to the column DC, and the same pressure plus the pressure, due to the column in EG, balances that of the column BA, it follows that if the pressures, due to the two columns AB and CD, are not the same, the two columns in EG and FI cannot be at the same level. ■I' ■d I E Fig. 3. 1 40 HEAT Suppose, now, that the column in EO, FI, and CD be main- tained at 0°C. and the column AB at t°C., th«i the column in FI must exceed the column in EG jin height, by an amount A, such that (i?-A)(l+orr)=ff; (14) where a is the coefficient of voluminal expansion of mercury. From equation (14) we find 41. If, now, it be desired to determine the coefficient of expan- sion of some other liquid, it becomes only necessary to take a flask and determine its mass when empty, then the mass of flask and contents when filled to a given mark at various temperatiu*es, first with the liquid whose coefficient is sought, and then when filled with mercury to the same mark for the various temperatures. From the mass of mercury required to fill the flask at various temperatures, and from the known density of mercury for these temperatures, the volume of the flask is readily computed. From the known volume of the flask and the mass of liquid required to fill it at the various temperatures, the densities corresponding to those temperatures are found. CHAPTER V •FUNDAMENTAL EQUATIONS OF GASES . 42. Isofhermal Equation. Experiment shows that, between certain limits, for the so-called permanent gases, such as hydrogen, oxygen, nitrogen, etc., the product of pressure and volume is a constant, for constant temperature. Expressed symbolically ?w=*; (1) where p is the applied pressiu^ per unit area,* v the corresponding volume of the gas, and k some constant whose value depends upon the units chosen. Equation (1) is usually designated as Boyle's Law. But, in any case, the equation which expresses the relation between the pressure and volume of a gas, at constant temperature, is its isothermal equation. 43. Gay-Lussac's Law. As a fiuther result of experiment it is foimd that all gases which obey Boyle's Law have the same constant temperaiure coefficient; i.e., all gases, under constant pressure, expand by the same fractional part of their voliunes at zero temperature, for equal increments of temperature. This is known as Oay-Lussac's Law. If we denote by a the increment in volume, for a imit volmne of a gas, under a constant pressure PO; when its temperature changes from zero to imity, then the volume of a gas at t degrees is v^=vo+VQ(xz) (2) * In all subsequent equations, unless otherwise stated, p will be used to denote pressure per unit area. 41 42 HEAT where «o is the volume of the gas at zero temperature, and v^ the volume, under the same pressure, at t degrees. Writing equation (2) in another form, we have t;, = t;o(l+opr) (3) If, after the temperature t, and the corresponding volume Vtf under the constant pressure po; has been attained, the pressure is augmented, the temperature being maintained constant, until the gas assumes its original volume vo, we must, from Boyle's law, have the following relation: p,=po(l+oeT) •. . . (4) As a matter of fact the experiment is most conveniently per- formed, by varying the pressure, so as to maintain the volume constant, as the temperature is varied. Equation (4) may there- fore be considered as being the expression of experimental results. Multiplying both sides of equation (4), by vo, we obtain Pt«^o=Pot^(l+aT) (6) If, now, while the temperature is maintained constant, the pressure be varied, the volume will vary according to BoyWs Law; i.e., pt;=p,vo=po«o(l+aT); (6) where p is any pressure and v the corresponding volume at the temperature t. Since Gay-Lussac^s Law also holds for temperatures below zero, a, of course, being a decrement, we have pv=povo{l — (xx) (7) Equation (7) reduces to zero when ore equals unity; hence T = l/a is the temperature below zero for which pv=0. FUNDAMENTAL EQUATIONS OF GASES 43 For the centigrade scale a =0.003665, very nearly; hence, T = 1/0.003665 = 273, very nearly. Therefore, if there were no deviation from the relation expressed by equation (7), then at — 273®C. the product of pressure and volume would become zero. As a matter of fact, all gases liquefy at temperatures above -273°C. This point, 273°C. below zero, is called the ''absolute zero "; and temperatures measured from this zero are called tem- peratures on the " absolute scale," Substituting now, in equation (6), for a its value, viz, 1/273, we have ?w=Pot;o(l+273)=|^'(273+T). ... (8) But, in equation (8), pofo/273 is a constant for any particular mass of a gas, and 273 +t is the temperature as measured on the " absolute scale" Replacing the former by R and the latter by r, equation (8) becomes VV=RT (9) The numerical value of R depends, of course, upon the units chosen. Equation (9) is called the characteristic equation of a gas, and shows that the product of pressure and volume is directly propor- tional to the temperature as measured on the " absolute scale." * 44. Departure from Boyle's Law. For ordinary pressures,^ Boyle's law is approximately true; i.e., for air the ratio of the » product of pressure and volmne, when the pressure is one atmos- phere, to the product of pressure and volume, when the pressure is two atmospheres, is about 1.002. The ratio of the initial value of p« to the final value of pv becomes greater as the final pressure becomes greater, imtil certain very high pressures, which are dif- ferent for the various gases, are reached; after which the value * Since aU knowledge is relative, the expressions ''absolute zero^' and ''absolute scale" are not well chosen. 44 HEAT of pv increases rapidly with increase of pressure. The values of the pressures for which, at ordinary temperature, the product pv is a minimum, are as follows : 100 meters of mercury for oxygen, 65 meters for air, and 50 meters for both nitrogen and carbon- dioxide. For hydrogen the deviations from Boyle's Law are less and in the opposite direction. Since Boyle's Law is not rigidly true, it follows that all equations, which have been based on it, are not rigidly true. However, for ordinary ranges of pressure and temperature the character- istic equaiion is very nearly true. And, for the mathematical discussion of gases, it is very convenient to assume that we are dealing with a gas for which the characteristic equation is rigidly true. Such a gas is called a perfect or ideal gas. 46. Gas Thermometer.* If a gas be confined in such a manner that its volume is maintained constant while the tem- perature varies, then it follows, from equation (9), that between those limits for which the equation is approximately true, the pressure varies directly as the temperature. All that is necessary then, to enable us to measure temperatures accurately, is a glass bulb of convenient size, filled with a gas, the most convenient being dry air, and some device by means of which the pressure can be regulated and measured. Such an arrangement constitutes a gas thermometer. Fig. 4 is a diagrammatic representation of the arrangement. A is a glass bulb, filled with gas, and connected by an inverted capillary U-tube, to the U-tube (Jbc. S is a vessel, open at the top, partially filled with mercury, and communi- cates, by means of a flexible tubing, with the U-tube at 6. The tube 6c being open at the top, the mercury in B and he must be at the same level. Assume that, when the temperature of A is at zero, the mercury in S, in 6c, and in 6a is at the same level; then, if the temperatiu-e of the bulb A, and consequently that of * In all subsequent demonstrations, unless otherwise specified, the symbol T will be employed to designate temperatures as measured by the ideal gaa thennameter, the zero of which is about — 273°C. FUNDAMENTAL EQUATIONS OF GASES 45 o the gas in it, be increased, the gas will expand if the pressure be constant. Hence, to maintain the gas at its original volume, the vessel B must be raised so as to increase the column be, in a manner, such that the column ba, and hence the volume of the gas. is maintained constant. The difference in the height of the colmnns be and ba measures the increment in pressure; and hence, the increment in tem- perature may be calculated. A correction must, however, be made for the change in volume, due to change in temperature, for the bulb A. This correction is readily applied, provided the coefBcient of voluminal expan- sion for the glass, confining the gas, be known. Let po and ro, respectively, rep- resent the pressure and volume of the gas for the temperature at zero, p, the observed pressure for the temperature t, a the coefficient of expansion for the gas, and P the coefficient of voluminal expansion for the glass. The new volimie, then, for the gas at the temperature t, will be wo(l + ^t) ; hence Px«o(l + &t) = pot^(l + ax) . Fig. 4. Now, pT==Po(l+y^); where a' is the /Apparent expansion of the gas. Making this ysubstitutio) eliminating, w/ find (l-Ha'T)(l + M/l+aT; from whicl / ifficienlf oj for p/ axu J'etf Eifratt "doyJe-'J Law, miJ :s<>lvinj ;f'>y'x', "^^-find T- f*r-p /v*-/v^* 46 HEAT When extreme precision is sought, corrections must also be made for changes in volume of the bulb, due to changes in pressure. Gas thermometers are used only for purposes of standardization. 46. Expansion without Doing External Work. Experiment has shown that there is no energy consumed in the simple expan- sion of a gas; i.e., when a gas expands in such a manner that no external pressure is overcome, and hence, no external work is being done, no energy is consumed by it. This experiment was first performed by Gay-Lussac (who apparently did not realize its full significance) in the following manner: Two vessels, one of which was exhausted and the other filled with air under a pres- sure, were placed in a calorimeter and surrounded by water. When the stop cock in the tube, connecting the two vessels, was opened, the air from the one vessel expanded into the other, bringing about an equalization of pressures. During this process the gas increased in volume without doing any work external to the system. That is, work was done by the gas under a high pressure, in the one vessel, in expanding against the increasing pressure of the gas in the other vessel. But, since the temperature of the system after the completion of the process was found to be precisely equal to that of the system before the process *began, it follows that the total energy of the system is unchanged by the expansion. If energy were required to bring about an increase in volume, the temperature of the system at the end of the process would necessarily be less than at the beginning of the process. Or, to express the result in another way, since the temperature of the system is unchanged by the change in volume, the work done by the gas in the vessel, initially under the higher pressure, is pre- cisely equal in amount to the work done on the gas in the other vessel. The results just deduced may be embodied in a simple statement; i.e., the intrinsic energy of a perfect gas is a function * only of the temperature. Or, to put it still in another way, the heat of disgregation of a perfect gas, when no external work is being done, is zero. FUNDAMENTAL EQUATIONS OF GASES 47 The foregoing experiment was subsequently repeated, in the most careful manner, by Joule and found to be approximately, though not rigidly, true. 47. Thermal Capacities of Gases. If a gas, under a pressure p, expand by an amount in volume dv, the external work done will be numerically equal to pdv; where p is the pressure per unit area. This is shown as follows: From the definition of work, we have dw-^Fds] (10) where F is the applied force and ds the displacement. But, since Fy the applied force, is nimierically equal to the product of p, the pressure per imit area, and the area A, we have dw=pAd8 (11) But Ad8=dv) hence, by substituting in equation (11), we have dw=pdv (12) Since, according to the experiments of Gay-Lussac and Joule, when the temperature of a gas is augmented, that part of the heat which is required to elevate the temperature of the gas is practically the same for equal ranges of temperature, no matter whether the volume be varied or maintained constant, it follows that the quan- tity of heat required to bring about a given elevation of temper- ature, when the pressure is maintained constant, is greater than the quantity of heat required to bring about an equal elevation of temperature, when the volume is maintained constant. For, in the former case, heat is required, not only to elevate the temper- ature of the gas, but also to do the external work due to the expan- sion of the gas; whereas, in the latter case, the heat consimied is only that required to elevate the temperature of the gas. 48 HEAT The ratio of change in heat to the corresponding change in temperature, in a unit mass of gas, when the volume is maintained constant, is a measure of the thermal capacity per unit mass at constant volume, and is denoted by Ct. Hence, for a unit mass of gas, we have (§).--' (13) In a like manner (14) where C, represents, for a gas, the thermal capacity per unit nuiss, under a constant pressure. The ratio Cp/Cv=n is practically a constant for all the per- manent gases; and, in the case of air, is approximately 1.405. The quantity (Cp—Cv), is evidently a measure of the external work done by the xmit mass of gas in expanding against the pressiu*e p, while it is being heated through a range of 1 degree. Assiune that we have a given mass of gas m, whose volume is V at 0°C., confined in a cylinder by a piston of area A, imder a pressure p, per unit area. If the piston be perfectly free to move, and heat be applied bringing about an elevation of temperature T, the pressure being maintained constant during the process, then the volume will be increased by an amount vaT. The dis- tance through which the piston moves during the expansion is vox/ A ; and the external work done, which is numerically equal to the product of force and displacement, is W^pA-j-^pvan (16) The heat consumed in doing the external work, expressed in mechanical imits, is H=Jm{Cj,-C,)'z; , . (16) FUNDAMENTAL EQUATIONS OF GASES 49 where J is the mechanical equivalent of heat. But, imder the assumed conditions, H and W are nimierically equal; hence, from equations (15) and (16), we have Jfn{Cp — C,)t = pv(xt ; from which Equation (17) * enables us to compute the mechanical equiv- alent of heat from the known constants of a gas. The constants of dry air are as follows: Cp = 0.2376, C,= 0.1690, a =0.003665, and the mass of 1 c.c. of air at 0°C., under a pressure of 1.01325 XlO^' dynes per sq.cm., is 0.001293 grams*. Substituting these values in equation (17), and assuming 1 c.c. for the initial voliune, we find . 1.0132 X10^X0.003665 . mos^iA? i • ■^= 0.001293(0.2375-0.1690) ^^-^^^^^^ ^'^ ^' ^"^ "^^™' The value of J thus obtained differs by a small percentage from that obtained by the direct conversion of mechanical work into heat. We have here then a complete verification of the numer- ical relation between heat and work. That is, in the one case, mechanical work is directly converted into energy, in the form of heat, and the numerical ratio of the two quantities involved is determined. In the other case, heat is converted into work, and the numerical relation between the two quantities is again determined; and the difference, between the two values so deter- mined, is well within the limits of observational error. * It was by this method that J. R. Mayer first computed the mechanical equivalent of heat. Various writers have attempted to take some of the credit from Mayer, by asserting that it was not then known that no energy is required for the simple expansion of a gas. But Gay-Lussac had per- formed this experiment, and anyone reading Mayer's original papers will see that he was aware of this and interpreted the experiment properly. 50 HEAT 48. Adiabatic Equation. If a gas is compressed, work is being done on it and heat is necessarily developed; and since the pres- sure of a gas, other things being equal, rises with the temperature, it follows that, unless the heat developed by the compression is abstracted from the gas as rapidly as it is developed, the pressure must rise more rapidly, with respect to the amoimt of compres- sion, than it would during isothermal compression. There are certain processes where compression and dilatation take place ■so rapidly that there are practically no exchanges of heat between the various parts of the system. Changes, during which no heat enters or escapes, are called adiabatic changes. A good example of adiabatic changes are the compressions and rarefactions which take place in a medium when a soimd wave exists in it; the time of compression, or rarefaction, being so small that practically no heat is transferred from particle to particle. In general, if we are dealing with a unit mass of a perfect gas, we may write dQ^CdT+pdv] (18) where all quantities, of coiu*se, are expressed in the same units. dQ, expressed in mechanical units, is the quantity of heat absorbed by the gas, or else abstracted from it, C^dT is the quantity of heat involved in bringing about the change of temperature dr, and pdv represents work done either by the gas or on the gas. As a matter of illustration, assume work is being done on the gas in a manner such that its temperature rises and that there is also heat given to the surroundings. If we consider work done by the gas and heat absorbed by the gas positive, then, in equation (18), if applied to this case, both dQ and pdv become negative; on the other hand, Cj,dT remains positive. If, now, during the change in volume, no heat enters or escapes, the process will be adiabatic; and equation (18) becomes C,dr+pdt;=0 (19) FUNDAMENTAL EQUATIONS OF GASES 61 In equation (19), if dv is positive, dT must be negative; i.e., if external work is done by the gas it is done at the expense of the intrinsic energy of the gas, and the temperature must fall. Like- wise, if work is done on the gas, since according to our assumption no heat escapes, its temperature must rise. To solve equation (19) we will substitute for dT its value, as found from equation (9), Art 43; i.e., by differentiating pv=RT, we find dT=^P^^ (20) Now, R can be expressed in terms of the two thermal capacities. To find this relation, assume that we are dealing with a imit mass of the gas and allow it to expand under a constant pressure p, while its temperature is increased by unity. If vi is the initial and V2 the final volume, the external work done is p(«2— t;i) = Cp— Cf. It being understood that all quantities are measured in mechan- ical units. But, from equation (9), it follows directly, that p{v2-vi)=R(T2-Ti); and, since by the conditions we have T2'-Ti equal to unity, it follows that Substituting this value of i2 in equation (20), and the value of dT so obtained in equation (19), we find CPp±^+pdv=^0; from which t) V+f -O- m 62 HEAT But, as previoiisly explained, CplCt is practically a constant. Representing this constant by n and substituting in equation (21), we have dv .dp ^ n — h— =0; from which, by integration, logv'^+logp^fci; where ki is the constant of integration. Or, expressed in another form, logp»"=fci; from which pv^'^k; (22) where X; is a constant depending upon the units chosen. Equation (22) gives the relation of pressure and volume for a gas, during adiabatic changes, and is known as the adiabatic equalion. General Equations of Gases 49. The change of heat involved when a gas suffers a change is a fimction of the temperature, pressure, and volume; i.e., Q=f{T,p,v). But, since any two of these quantities may vary independently of the third, we may write Q=/(r,p), Q=f{T,v\ and Q^rip.v). By partial differentiation of these functions, we obtain «-{ii).^''+(f) *■ • •. • <^) «-(w)5+©,*- • • • • <«' and '^^=(if).'^+(S),*'-- • • • • (26) FUNDAMENTAL EQUATIONS OF GASES 53 If it be assumed that we are dealing with a iinit mass of gas^ then in equation (23), we have m since in this case, (^) is the thermal capacity per unit mass at constant pressure. In a like manner, ( IV/T is also a imit thermal capacity, and is the ratio of change in heat to change in pressure at constant temperature. Substituting, in equation (23), we have dQ=Cj,dT+mdp (26) In equation (24) (g)dr=WT; since (■^) is the thermal capacity per imit mass of the gas at constant volimie. In a like manner. ( IV/T is a unit thermal capacity, and is the ratio of change in heat to change in volume at constant temperature; hence, by substi- tuting in equation (24), we find dQ^C,dT+ldv (27) Again, in equation (25) the two quantities, viz, (t" ) > ^^^ ratio of change in heat to change in pressure at constant volume, and ( ^) , the ratio of change in heat to change in volume at constant 64 HEAT pressure, are both unit thermal capacities. If we represent the former by j, and the latter by o, then equation (25) becomes dQ=-jdp+odv (28) Since all of these equations must be true for the particular case when the left-hand members are equal, the right-hand mem- bers of equations (26) and (27) may be equated, and we have Cj,dT+mdp = CpdT+ldv (29) From the fundamental statement v^fiPy r), we have *-(|)*+(r.).^^- Substituting this value of dv, in equation (29), we have and C,dT+mdp=lc,+l(^^\dT+l(^^Jlp. . . (30) Since this equation is true when the corresponding changes in the two members are equal, it follows that and ^°it\ (32) Again from which P=/(f, T); *-© *+(ll)5- (»> FUNDAMENTAL EQUATIONS OF GASES 55 Substituting the value of dp as given in equation (33), in equation (29), we have from which Cp+m (i).}'^^+'"(S)r*'=^'^^+^' hence, by equating like coefficients, we find C,-a=-m(||)^ (34) By equating the right-hand members of equations (26) and (28), we find CpdT+mdp=jdp+odv (35) From the fundamental statement T=f(v, V), we have '''-(l7)*+(lf)*^ and substituting this value of dT, in equation (35), we obtain hence, by equating like coefficients, we find From equations (27) and (28), we have CvdT+ldv =jdp + odv ; 56 HEAT and, by substituting for dT its value, we find from which We then have the following values: and •'(f).--' From the characteristic equation . ■pv=RT, we find hence (lv\ ^R i=|(C,-C.) (38) Again hence m=~(C,-C,). " (39) FUNDAMENTAL EQUATIONS OF GASES 57 Also .J hence R Finally i=^C^ (40) (ir\ =e. ,Bo/, R' and o=|Cp (41) Substituting in equations (26), (27), and (28) the values of I, m, j, and as just determined, we find dQ = CpdT--iC„-C,)dp, (42) dQ = C,dT+^{Cp-C.)dv, (43) and dQ=^C.dp+|c,(fo. (44) These equations, viz, (42), (43), and (44), may be put into different forms; since, from the characteristic equation pv=-RT, we have T/p=v/R, and p/R=-T/v. From the assumption, then, that in the fimdamental statement (3=/(r, p, t;), 58 HEAT any two of the variables may vary independently, while the third is maintained constant, we have obtained three distinct equa- tions, viz, equations (42), (43), and (44). It is of further interest to note that if in any one of these three equations the right-hand member is equated to zero, the adiabatic equation is obtained. Assuming that the change in equation (44) is adiabatic, then dQ = 0, and we have Ctvdp^—Cppdv; from which and Cpdv dp^ »/*= -If' from which n log v= —log p+ki. Or, expressed in another form and, finally pV'^k (46) This may also be obtained from the other equations, as may be readily shown. From we have ^m_ pdv+vdp Substituting this value of dT in equation (43) and equating to zero, we find Ct{pdv+vdp) = '-p(Cp—Cv)dv; from which Cvvdp^—Cppdv; FUNDAMENTAL EQUATIONS OF GASES 59 which is identical with the result obtained from equation (44) under the same assumption. Again, substituting in equation (42), for iT its value and equating to zero, we find from which which is again identical with that previously obtained imder the same assimiption. If it be desired to find the temperature of a gas, corresponding to a given pressure and volimie, during an adiabatic change, in terms of the initial temperature and pressure and the given pres- sure, or in terms of the initial temperature and volume and the given volume, we proceed as follows: Let Ti, pi, and v\ be, re- spectively, the initial temperature, pressure, and volume, p and r, respectively^ the pressure and volume, for which the corre- sponding temperature, T, is sought. Then, since the two points are on the same adiabatic, we have Pivi''=7w'*; (46) and from the characteristic equation, we have pit;i=ii!ri, (47) and 3W=i2r (48) Substituting in equation (46), the values of v\ and v, as found from equations (47) and (48), we find n-l 2'=ri(^) " (49) 60 HEAT In a similar manner, by substituting in equation (46), the values of pi and p, as found from equations (47) and (48), we obtain t-tM:-! (50) Vapors 50. Vaporization. The gaseous states of bodies, which under ordinary conditions of temperature and pressure are either liquids or solids, are called vapors; and the process by which the vapor is formed is called vaporization. In general, vaporization takes place in two distinct ways. In the one process, called evaporation, vapor is continually formed at the exposed surfaces of liquids; and in the other process, called ebvUition, bubbles of vapor are formed in the body of the liquid or at the heated surfaces. 61. Evaporation. If a liquid be enclosed in a space, only part of which is occupied by the liquid, then vapor immediately forms and occupies the space above the liquid. This continues imtil the vapor has reached a certain density which depends upon the temperature, and is greater as the temperature is higher, but is always the same for the same temperature. In other words, for any given temperature there is a maximum density and hence, a maximum pressure, which the vapor is capable of exerting. When this state is reached the vapor is said to be saturated. That is, for the given temperature the space contains the maximiun possible amount of vapor. If, after this state has been reached, the temperature be maintained constant and an attempt be made to increase the pressure by the application of an external force the result will be, not an increment in pressure, hut a diminvJtion in vdlume, at constant pressure, and a corre- sponding amount of condensation. In other words, the pressure for a saiuraied vapor ai constant temperature is constant. Or, FUNDAMENTAL EQUATIONS OF GASES 61 to put it in still another way, the temperature of a saturated vapor is uniquely defined by its pressure. 62. Addition of Vapor Pressures. The rate of evaporation depends, of course, upon the rate at which heat is being supplied; but, as has just been stated, the final pressiu*e reached depends merely upon the temperature. Furthermore, evaporation takes place more rapidly in a vacuum than in a space occupied by the vapor of some other substance; however, the final pressure reached by the vapor will be almost, though not quite, as high, when the space is partially occupied by some other gas or vapor, than it would be were the space originally a vacuum, provided always, that the temperature be the same and that there be no chemical action between the vapors. This statement was first made by Dalton, viz, when evaporation takes place in a space filled by another gas, which has no action on the vapor, the final pressure reached by the mixture is equal to the sum of the pressures of its , constituents. Careful experiment shows Dalton's statement to be approximately, though not rigidly true. 63. EbuUition. As has been previously stated, when heat is applied to a liquid, the temperature rises progressively with continued application of heat until a certain point, which depends upon the pressure, is reached, when the temperature remains constant. This is the boiling-point for the given pressure; and is that temperature for which the pressure of the vapor is equal to the superimposed pressure. Since the pressure at any point in the liquid, is equal to the pressure at the surface plus the pres- sure due to the liquid, from the surface to the point under con- sideration, it follows that, the temperature varies for different depths below the surface of the liquid. Hence, the temperature of the boiling liquid is not a constant throughout; but increases slightly with the depth. When equilibrium has been attained, i.e., the temperature becomes constant, then all the energy that is supplied, in the form of heat, is consimied in converting the liquid into a vapor. This 62 HEAT energy oonsists of two parts, viz, one part being that energy which is required to overcome the inherent forces, that is, to separate the particles so as to form vapor, and the other part to overcome the external pressure during the augmentation of volume. The former is called the heat of disgregation and the latter the heat of expansion. I Pressure, however, is not the only factor that fixes the boiling- point of a liquid. As examples, the following may be cited: The nature of the material of the containing vessel has some influ- ence. If the liquid be first carefully freed from the imprisoned air, the temperature may be raised considerably above the tem- perature at which ebullition ordinarily takes place. Impurities in the liquid influence the boiling-point. And finally, salts dis- solved in ajiquid always raise the boiling-point. As an example, the boiling-point of a saturated solution of water with conmion salt is about 109°C. But the temperature of the saturated vapor of a liquid is always the same for the same pressure, no matter what the temperature of the liquid. It is for this reason tha.t the temperature of steam, rather than that of water, imder a pressure of one standard atmosphere haa been chosen as the boiling-point. 64. Critical Temperature. When a liquid is heated in a closed vessel the vapor accumulates above the liquid and augments the pressure. Up to a certain point, differing for different liquids, there is a sharp definition between the liquid and vapor; but, for every liquid there is reached, finally, a temperature when this definition ceases, and the liquid disappears and is completely converted into vapor, even though the volume occupied by the vapor is but little greater than that occupied by the liquid. The temperature at which this takes place is called the critical tem- perature for the substance. And, it appears that, for temper aiures, higher than this, no meatier what the applied pressure, the substance can exist only in the gaseous state. The following table gives a few substances together with their approximate critical temperatures, and the corresponding pressures: FUNDAMENTAL ^EQUATIONS OP GASES 63 Subfltanoe. Temperature in Degrees, C. Pressure in Atmospheres. Carbon-dioxide 31 156 194 365 -118 -146 -234 77 79 36 195 50 33 20 Sulphur-dioxide Ether Water OxvKen Nitrofcen HvdroKen **rf *** v©**** ••• 66. It is interesting to note that there apparently is a relation between heat of disgregation of a substance and its critical tem- perature. Let ff be the specific volume of the liquid; i.e., the volume occupied by unit mass of the liquid, and 8 the specific volimie of the dry saturated vapor, then the increment in volume, when a unit mass of a liquid is converted into vapor, is /i=«-ff; (51) where fi is the increment in volume. The external work, or the heat of expafision, is Tr=p/t=p(s-yi0 7 =40.3 gram calories per gram. This gives for the heat of disgregation, which is the difference between the heat of vaporization and the heat of expansion, 536.5-40.3=496.2. Zeimer gives an empirical equation, for the heat of disgregation for water, which gives results very close to those obtained by equation (55) ; this equation is p = 575.4 -0.791t; (56) where p is in gram calories per gram and t in degrees centigrade. In general, the heat of disgregation becomes less as the tem- perature becomes higher; and the critical temperature appears to be that for which the heat of disgregation becomes zero. In the case of some liquids, very close agreement is found between the values for the critical temperatures, as found by direct experiment, and those values calculated from the empirical equations for the heat of disgregation. In other cases, again, there are discrepancies of considerable magnitude. A notable case is that of water; but, it must be remembered that the critical temperature of water is very high, and therefore, its determination is attended by dif- *8 not being known definitely to the fourth significant figure, it is immaterial whether we use 1670 or 1669; or, in other words, the volume of the liquid, in this case, is negligibly small in comparison with that of its vapor. FUNDAMENTAL EQUATIONS OF GASES 65 ficulties. Furthermore, the heat of vaporization for water has not been determined for such high temperatures; hence, the empir- ical equation for such ranges is doubtful. 56. Total Heat of Steam. Before leaving the subject of satu- rated vapors, we will give, on account of the importance in steam calculations, the empirical equations for the total heat of satu- rated steam, and the heat of vaporization. By total heat of steam is meant the quantity of heat required to raise the temperature of unit mass of water, from the melting-point of ice, to the tem- perature imder consideration and convert it into saturated steam at that temperature. In the French system the total heat is given by the equation fl = 605+0.305 T gram calories per gram. . . . (57) In the English system the total heat is given by the equation ff = 1082 +0.305 T B.T.U. per pound (58) In equation (57), H equals the quantity of heat, in gram calories, required to raise 1 gram of water from 0°C., to the temperature T°C., and convert it into a saturated vapor at that temperature. In equation (58), -ff equals the quantity of heat, in British thermal units, required to raise 1 pound of water from 32°F. to t^F., and convert it into a saturated vapor at that temperature. 67. Heat of Vaporization for Water. The empirical formula, which gives the heat of vaporization of water, for the various temperatures, is r=1114-0.7TB.T.U. per pound. . . . (59) Equation (59) is not quite as accurate as equation (58) ; but for most steam calculations it is sufficiently precise; since, by means of it the heat of vaporization is foimd, with an error of less than 1 per cent, between lOO'^F. and 400**F. 66 HEAT 68. Superheated Vapors. Any vapor which, for a given pressure, is at a temperature higher than that corresponding to saturation, for the given pressure, is said to be superheated. This is only possible when the vapor is not in contact with its own liquid. Vapors which have been superheated obey Boyle's law approximately; and furthermore, for adiabatic changes, the equation holds; n having different values, depending upon the vapors with which we are dealing. 69. Hygrometry. Hygrometry has for its object the deter- mination of the state of the atmosphere with respect to the aqueous vapor present. The amount of aqueous vapor present in the atmosphere is a very variable quantity. The effect of the vapor, however, depends not only upon the quantity present, but also upon the temperature. These two facts are included in the single statement, that the effect of the vapor depends upon the relative humidity. By the expression relative humidity, is meant the ratio of the actual density of the vapor, contained by the air, to the density which it would have if there were saturation for the given tem- perature. Or, if expressed as a percentage, the relative humidity is the percentage of saturation for the given temperature. 60. Dew Point. The dew point is that temperature at which the vapor present in the atmosphere, begins to condense; i.e., the point of saturation. Assume that a certain portion of the atmosphere is cooled until the vapor present begins to condense. The temperature at which this takes place is readily found by experiment. By referring to a curve giving the relation of temperature and pressure of saturated steam, we can readily find the pressure corresponding to the dew point. Designating this pressure by pi, and by p2 the pressure of saturated vapor corresponding to the temperature of the atmosphere, then since FUNDAMENTAL EQUATIONS OP GASES 67 non-saturated vapors approximately obey Boyle's law, the density of the vapor present in the atmosphere is to the density the vapor would have were there saturation, very nearly, as pi is to p2. Hence, we have approximately, for the relative humidity A = 100^ per cent. To illustrate further, we will consider a concrete case. Assimie the temperature of the atmosphere to be 25°C., and that of the dew point 15°C. From the steam curve we find pi = 1.278 cm. of mercury, and p2= 2.369 cm. of mercury. From this, the rela- tive hmnidity, expressed as a percentage, is foimd to be 1 278 A = 100X^|g=53.9 per cent. 61. Absolute Humidity. The amoimt of moisture, expressed in grams, contained by a cubic meter of air is called the absolute humidity. This is found in a very simple manner. The dew point is determined, givmg the temperature of the saturated vapor, and from this, by referring to steam tables, the mass per unit volume is foimd. CHAPTER VI ELASTICITIES AND THERMAL CAPACITIES OF GASES 62. The adiabatic equation, for gases and vapors, being so frequently employed in the discussion of the theory of heat motors, it is important to say a few words about the determination of the ratio Cp/Cv=^n. The determination of thermal capacities of gases is attended by far greater difficulties, than those found in the determination of thermal capacities of liquids and solids. This is due to the fact that the thermal capacity of a gas is always very small in comparison with that of the containing vessel. 63. Thermal Capacity at Constant Pressure. Regnault was the first to determine accurately the thermal capacities of gases under constant pressure.* The method is essentially as follows: The gas, whose thermal capacity is sought, is contained in a large reservoir, imder a high pressure, from which it is passed through a spiral tube immersed in a bath, the temperature of which is maintained constant. The spiral tube is of sufficient length to insure the gas leaving it to be at the same temperature as the bath. In the tube, connecting the reservoir with the spiral tube, is a valve, by means of which the pressure of the gas is maintained constant. A second spiral tube, through which the gas must pass, is immersed in a calorimeter filled with water; the thermal capacity of the calorimeter and contents being known. The spiral tube, immersed in the calorimeter, is of such length that the tem- perature of the gas, throughout the progress of the experiment, is reduced to that of the calorimeter before being discharged into * For a full description, see Preston's "The Theory of Heat," Chapter IV, Section VI. 68 ELASTICITIES AND THERMAL CAPACITIES OF GASES 69 the atmosphere. From the initial and final pressure of the gas in the reservoir, together with its temperature, which is main- tained constant, by means of a suitable bath, the mass of gas pass- ing through the calorimeter, during the progress of the experiment, is readily found. Furthermore, from the thermal capacity of the calorimeter and contents, together with the initial and final temperatures, the quantity of heat absorbed by the calorimeter during the progress of the experiment, proper corrections being made for losses, is determined. And, since the temperature of the gas before entering the calorimeter, as well as the average final temperature, is known, and also the mass of gas which has passed through the calorimeter, the thermal capacity per imit mass is determinate. 64. Thennal Capacity at Constant Volume. In the experi- ment just described the quantity of gas employed is not limited by any containing vessel; for, the reservoir in which the gas is contained may be of any size whatsoever, without having any influence on the result. Therefore, a large quantity of gas may be used, and consequently, a considerable range of temperature may be obtained in the calorimeter. However, when it is desired to determine the thermal capacity of a gas at constant volume, the quantity of gas upon which we are experimenting, is limited by the containing vessel; and the thermal capacity of the con- taining vessel is always large in comparison with the thermal capacity of the enclosed gas. 66. Joly's Steam Calorimeter.* In the most primitive form, the steam calorimeter consists of the pan of one side of a beam balance placed in an enclosure with the specimen, whose thermal capacity is sought, supported by the pan. When steam is admit- ted into the enclosure, condensation takes place until the temper- ature of the test specimen is equal to that of the steam. The steam then passes through the enclosure without further conden- For a complete description of this apparatiis, see Preston's "The Theory of Heat," Chapter IV, Section V. 70 HEAT sation. When this condition has been reached, the balance is counterpoised and the mass of condensed steam, which has been collected by the pan, is noted. From the initial and final temper- ature of the test specimen, together with the quantity of water collected by the pan, and the heat of vaporization for this par- ticular temperature, the thermal capacity of the test specimen is readily found. 66. Differential Steam Calorimeter. In this form, both pans of the balance, which are made so that they have equal thermal capacities, are suspended in the enclosure. On the one pan is placed a spherical vessel, which has been exhausted, and on the other pan a spherical vessel of like dimensions and equal thermal capacity, filled with the gas whose thermal capacity is sought. When steam is now admitted into the encloito'e, the quantity of water which collects jn the pan, supporting the vessel containing the gas, is greater than that which is collected in the pan supporting the exhausted vessel. This is necessarily so; since the vessel, together with the contained gas, has a thermal capacity greater than the exhausted vessel. From the excess of condensation in the'one pan over that in the other, which is obtained directly by weighing, together with the initial and final temperatures of the enclosure and the mass of gas contained by the one vessel, the thermal capacity per unit mass of the gas at constant volume is readily foimd. Correction, of course, being made for change in volume of the containing vessel for change in temperature.* It must, of course, always be remembered that it is impossible to obtain absolutely accurate results by this method; since the thermal capacity of the gas is always small in comparison with that of the containing vessel. Still, Dr. Joly, who is the inventor of this method, has obtained fairly good results. 67. Method of Clement and Desoimes. In this method the gas, whose thermal capacity is sought, is contained in a large vessel For a complete description, see "The Theory of Heat," by Preston, Chapter IV, Section V. ELASTICITIES AND THERMAL CAPACITIES OF GASES 71 provided with a delicate manometer. When the contained gas has assumed the temperature of the surroundings, its pressure, which must diiBfer from the atmospheric pressm'e, is carefully ascertained. When this has been done, a stop cock, having a large orifice, is opened and then closed after a very short interval of time. The time which elapses between the opening and closing of the stop cock must be so small that the change in the gas may be assumed adiabatic. During this change the temperature changes; i.e., there will be, either an elevation of temperature, if the pressure in the flask was initially less than the atmospheric pressure, or else, a diminution of temperature, if the pressiu*e in the flask was initially greater than the atmospheric pressure. After the vessel and contents have again assumed the initial temperature, viz, the temperature of the surroimdings, the pres- sure is again carefully noted. If, now, we denote by pi the initial pressure of the gas, by vi the corresponding volume per unit mass, by p the atmospheric pressure, which is also the pressure of the gas when the stop cock is open, and by V2 the volume per unit w>ass after the stop cock is closed, then since the change is assumed adiabatic, we may write PlVi'*=pV2'' (1) Also, since the initial and final temperatures are the same, we have PlVl=P2V2; (2) where p2 is the pressure in the vessel after the temperature of the surroundings has again been assumed. From equation (1), we find \W Pi' from which ^^log (p/pi) . log (t;i/t;2) 72 HEAT From equation (2) we find from which log^=Iog2?. V2 Pl Substituting in equation (3) for log (viM), its value, log (P2/P1), we obtain ^^log( p/pi) log (P2/P1) Or, expressing this in another form, we have ^^ logp-logpi .g. logp2-logpi Since p, pi, and p2 are known, n is determinate. In this manner. Roentgen found for dry air the value n = 1.405. This method is open to criticism, in so far that when the stop cock is opened, oscillations occur; and it does not necessarily follow that, at the instant of closing, the pressure in the vessel is equal to that obtaining outside. 68. Isothermal and Adiabatic Elasticities. The ratio of the two thermal capacities of a gas is most accurately found by determining the speed of propagation of a disturbance through the gas. We will first show that the ratio of the two thermal capacities is numerically equal to the ratio of the two elasticities. From the statement of Boyle, we have, the temperature being maintained constant, pv=fci; (6) where fci is a constant, depending upon the units chosen. By differentiation, we find immediately pdv+vdp=0; ELASTICITIES AND THERMAL CAPACITIES OF GASES 73 from which -^-p; (7) -(t) c the minus sign denoting merely that the volume decreases as the pressure increases. Now, the left-hand member of equation ^ (7), is numerically equal to the ratio of change in imit stress to the corresponding change per unit volume; and is, therefore, by definition, the expression for the modulus of elasticity. Hence,/ for a gas obeying Boyl^a law, the elasticity is numerically equal to the pressure. If now, we take the adiabatic equation, viz, pt;**=fc2, (8) where k2 is again a constant depending upon the units chosen, and differentiate, we find »-i>^w7..^n. from which and v^dp+nv^ " pdv = 0; vdp+npdv=0; ^ np (9) -(?) The left-hand member of equation (9) again expresses, accord- ing to definition, the modulus of elasticity. Hence, the modulus of elasticity when no heat is allowed to enter or escape, i.e., for adiabatic changes, is numerically equal to the product of the ratio of the two thermal capacities and the pressure. Denoting the isothermal elasticity by Et, and the adiabatic elasticity by Eh, we have Eh_np_ . . Erj"" ^^°^ 74 HEAT From equation (10) we see that the ratio of the two principal elasticities is the same as the ratio of the two principal thermal capacities. 69. Propagation of Wave Motion in an Elastic Medium. To properly appreciate how the ratio of the two elasticities, and hence the ratio of the two thermal capacities, of a gas is fomid from the speed of propagation of sound in the gas, it is essential to study the character of the motion by means of which sound is propagated in an elastic medium. Let ABf of Fig. 5, be a prism, of indefinite length and constant cross-sectional area, filled with a homogeneous elastic medium; and let the piston P have impressed upon it a constant acceler- ation toward the right. If the medium had absolutely no inertia, or were perfectly rigid, then the whole substance, between A and — IT 1 i| 1 1 1 ^1 1 1 1 ^1 1 1 1 i| 1 1 1 — MIL Fig. 5. Bj would suffer precisely the same displacement in a given interval of time. Due, however, to the inertia, the layer next to the piston will be compressed; the pressure of this layer now being greater than that of the medium in the imdisturbed condition, it will react upon the second layer and compress this, which again in turn compresses the third layer, etc. Finally, when every layer throughout the prism, has been compressed by an amount such that its internal pressure is precisely equal to the applied pressure, then the acceleration of each layer will be the same and equal to that of the piston. . Assume now, that the piston P is caused to vibrate periodically, with a small amplitude s. In tracing out a vibration we will begin by assuming the piston in the neutral position, moving toward the right, and the pressure of the medium, throughout, ELASTICITIES AND THERMAL CAPACITIES OF GASES 75 the same as that m the undisturbed condition. As the piston moves toward the right, condensation takes place in the medium; the condensation being greatest for the layer in contact with the piston and becoming less as the distance from the piston increases. Suppose now, that when the piston has reached its maximum difh placement 8 toward the right, the wave of condensation has reached the section represented by a; i.e., the pressure of the medimn at the section a is the same as that in the undisturbed condition, and greater for all portions to the left of a. As the piston now begins to move toward the left, the wave of condensation continues moving toward the right; but, the pressure behind the piston begins to decrease, and by the time the piston has again reached its neutral position, the pressure of the medium directly in contact with the piston is the same as that in the undisturbed condition. The wave of condensation will, in the meantime, have traveled to the section h ; the distance ab being equal to Aa. The maximum condensation is now at a, and tapers off to zero from a to & and from a to A. As the piston now continues moving toward the left, the medimn behind it becomes rarefied; and a waoe ofrar^ faction travels toward the right. By the time the piston has reached its extreme left-hand position, the wave of rarefaction will have reached the section a, and the wave of condensation the section c; where the distances he and ab are equal. The maximum condensation now exists at b, and the maximum rarefaction at the piston. As the piston now begins its journey toward the right, the pressure behind it begins to rise, imtil it reaches the neutral position, when the pressure at the piston is equal to that of the medium in the undisturbed condition. In the meantime, the wave of condensation has traveled to the section d, where the distances cd and he are equal. Hence, during the time required by the piston to complete a period, the disturbance has traveled from A to d; and the conditions now existing are: Maximum condensation at c, maximum rarefaction at a, and at A, b, and d, the pressure is equal to that of the medium in the imdisturbed condition. 76 HEAT As the piston now continues moving toward the right, a wave of condensation moves toward the right from Aj and also from d; and by the time the pistoB has completed its second cycle, the disturbance will have traveled to h; where the distances dh and Ad are equal. The condition of the medium between A and d is now at every section precisely the same as it was at the end of the first cycle. Also, the condition of the medium between d and h is precisely the same as it is between A and d; i.e., the condition at e is the same as at a, at / the same as at b, at g the same as at c, etc. At the end of the third cycle the disturbance will have traveled to the right of A, a distance equal to Ad=dh; and the condition of this portion will also be the same as the con- dition of the portions from A to d and from d to h. At the end of N cycles, the disturbance will have traveled through a distance equal to the product of N and Ad. The distance through which the disturbance travels while the piston goes through one cycle is called a wave length, and repre- sented by the letter X. Or, in other words, this is the distance a disturbance travels before conditions are beginning to be exactly reproduced. From the previous discussion, it is obvious that if there be performed N vibrations per imit time, and X is the wave length, then the speed of propagation is given by S^N-k (11) It is important to note that the distance through which the disturbance travels during a cycle depends upon the time consimied in performing that cycle; i.e., if the frequency — the number of vibrations per unit of time — ^be increased, then according to the discussion, the wave length will be proportionately less, such that the product of wave length and frequency is constant. This is fully verified by experiment for the speed of propagation of soimd in gases. ELASTICITIES AND THERMAL CAPACITIES OF GASES 77 70. Speed of Propagation in Terms of Elasticity and Density. Assume, as in Fig. 6, a cylinder of indefinite length and constant cross-sectional area A, filled with a homogeneous elastic medium whose density is p, and pressure per unit area in the undisturbed condition p. Assume further the frictionless piston P, having applied per unit area a pressure p+Jp; where Jp is a small frac- tional part of p. Now, as a matter of convenience, assume that the prism, repre- sented in Fig. 6, is divided into unit lengths, 1, 2, 3, etc., up to N; where N represents the distance the disturbance travels in a time t. The effect of the application of a pressure to the piston, in excess of the pressure of the medium, will be twofold; i.e., the medium will be compressed and also set in motion. It is evident that when any element of the medium in the prism has reached p I P+AP 2 8 N As Fig. 6. a pressure per unit area equal to p+Jp, it cannot be further com- pressed, but will merely serve to transmit the applied stress to the next element. Let Ja be the amount of shortening a unit length undergoes while its pressure rises from p to p+Jp. The total shortening then, that the prism of length N undergoes in being compressed from the pressure p to a pressure p+Jp, and consequently the distance through which the piston moves during the time this change takes place, is and, since the time consumed to bring about this change is t, the speed with which the piston has been moving is d NJs t t (12) 78 HEAT But, at the instant that the pulse has passed through the distance iV^all that portion of matter included in the length iV,of the prism, is moving with a speed the same as that of the piston as given by equation (12); hence, its kinetic energy is m' »'> Since the total change in volume is AN As, and the average resisting pressmre per unit area is p+ Jp/2, the work, due to com- pression, is W2^{ANA8){v+^^ (14) But the total work done on the system must be equal to w\-\'W2\ hence; from which m-y Ap (15) Now, N/t^Sf the speed of propagation; and Jp/Js, in the limit, represents the ratio of unit stress to unit strain, and hence, is equal to (x, the modulus of elasticity. Substituting, in equation (15), we have finally s=^/^; (16) i.e., the speed of propagation of a disturbance through an elastic medium is numerically equal to the square root of the ratio of elasticity to density. The speed of propagation of soimd in air is very readily deter- mined by experiment; and is found to be, at 0°C., very nearly 332 meters per sec. The vibrations in a sound wave take place ELASTICITIES AND THERMAL CAPACITIES OF GASES 79 so rapidly that the changes are sensibly adiabatic; hence (i, in equation (16), will be replaced by the adiabatic elasticity, and we have from which .-fS (17) Substituting numerical values, in equation (17), we find _ (33,200 )2 X0.001293 _ ^" 1.0132X10« -A.4U0i-. The value 1.405 is generally used for dry air; but, for most practical purposes, 1.4 is sufficiently close. CHAPTER VII PROPAGATION OF HEAT 71. Heat is transferred from one place to another in three distinct ways, viz, by radiation^ by corwection, and by conduction. 72. Radiation. In Chapter II, we dealt with Newton's law of cooling, without considering in what manner the cooling takes place. As a matter of fact, in the cases considered, the pooling was due to two distinct phenomena. To illustrate this, we will consider a concrete case, viz, an incandescent lamp, which con- sists of a filament inside of a glass bulb; the bulb having been exhausted, so that the filament is practically in a vacuum. The propagation of heat from the filament to the glass bulb, that is, through a vacumn, is called radiation; or, in other words, radiation is the propagation of heat through space without the aid of any maierial substance. On the other hand, the dissipation of heat from the surface of the bulb is due, not only to radiation, but also convection; and the propagation of heat from the inner siuf ace of the bulb to the outer surface is due to conduction. The propagation by convection and conduction will be considered later. 73. Theory of Ezchanges. Provost, in 1792, promulgated the theory that there is a continual exchange of heat between bodies, even when they are at the same temperature. Provost's theory may, perhaps, be best explained by means of the following illustration: Suppose a body suspended in a vessel, which has been completely exhausted. Assume further, that the walls of the enclo- sure are maintained at a constant temperature, and that the body, when first placed in the enclosure, has a temperature higher than 80 PROPAGATION OF HEAT 81 this. The temperature of the body will immediately begin to fall, due to its radiating heat to the walls of the enclosure; and this will continue until the temperature of the body is the same as that of the walls, when it becomes constant, and the body has apparently ceased radiating. If the temperature of the walls be now reduced, by immersing the vessel in a bath of lower tem- perature, the temperature of the body will fall and it will again be radiating heat. The body then, apparently, ceases to radiate heat when its temperature has fallen to that of the walls of the enclosure, and again begins to radiate heat when the walls are lowered in temperature; and again ceases to radiate heat when its temperature has fallen to that of the walls, and so on indefinitely. If, initially, the temperature of the walls had been higher than that of the body, heat would have been radiated from the walls to the body. Now, according to the theory of exchanges, the body does not cease to radiate when its temperature has fallen to that of the walls; but, the body and the walls are continually radiating and absorbing heat. That is, it is assumed that, when the body is at a higher temperature than the walls, it is radiating heat more rapidly than it is absorbing heat, when at a lower temperature than the walls, it is gaining heat more rapidly by absorption than it is losing heat by radiation, and when at the same temperature the rates of radiating and absorbing heat are the same. Without being committed to this theory, it will be interesting to note certain conclusions which must necessarily follow from it. 74. Emissivity* Experiments on radiant heat show that some bodies emit heat, other things being equal, more copiously than others. It is also foimd that bodies which are good radiators are also good absorbers. The capability which a body has for emitting heat is called its emissivity. Suppose now, that we have two bodies, placed in a space, imper- vious to heat. Then, according to Provost's theory of exchanges, they will both radiate and absorb heat, even though they be at 82 HEAT the same temperature. If now, one of the bodies absorbs heat more readily than it emits heat, its temperature will rise; this, however, is contradictory to experience. If, on the other hand, one of the bodies radiates heat more readily than it absorbs heat, its temperature will fall; which again contradicts experience. It therefore follows, if Provost's theory of exchanges holds, that bodies have precisely the same capability for radiating heat that they have for absorbing heat. This appears to be in concordance with experiment. 75. Stefan's Fonmila. We know that Newton's law of cooling |s very limited in its application; i.e., it does not hold when the difference of temperature between the body under consideration and the surroimding mediiun exceeds 15°C. to 20°C. In other words, it is only an approximate statement. Dulong and Petit performed a nmnber of classical experiments by means of which they endeavored to determine the law of cooling. Their exper- iments were, however, limited in range of temperature; since, the maximmn temperature reached was only about 240^C. From their experiments, they deduced the equation, for the quantity of heat lost per unit time, Q=mk\k^-1); (1) where Q is the quantity of heat, t the temperature of the enclosure, the difference in temperature between the enclosure and the radiating body, both measured on the centigrade scale, and m a constant, depending upon the substance and the nature of its surface. For k the value of 1.0077 was found. Stefan, from an examination of the results obtained by Dulong and Petit, deduced a formula for the loss of heat by radiation; i.e., = fc(ri*-r2*); (2) where k is constant, and Ti and T2 are the temperatures, as meas- ured on the ideal gas thermometer, respectively, of the radiating body and the enclosure. PEOPAGATION OF HEAT 83 Equation (2) appears to give results, in accordance with exper- iments, up to temperatures of about 1700°C. to 1800°C. It, however, appears from subsequent experiments, that Stefan's formula is not rigidly true; and consequently will require modi- fication. But, for practical purposes, Stefan's formula may be considered correct for the limits of temperature as stated. Considerable research work is still being done in regard to radiation at high temperatures; and whether, or not, a simple expression will finally be found which will be true for all temper- atures, is an open question. 76. Convection. Referring again to the incandescent lamp, and considering the dissipation of heat from the surface of the bulb, we find that part of the heat is absorbed by the atmosphere surrounding the bulb, and the remainder is transferred by radia- tion. Due to the absorption of heat, the gases in contact with the bulb become heated and therefore change in density. This change in density destroys the equilibrium, in regard to pressure; and hence, currents are established, called convection currents, tending to restore equilibrium. In this manner, heat is conveyed from one portion of space to another by currents in the atmos- phere; i.e., the particles in contact with the bulb become heated and are replaced by particles at a lower temperature. These particles, in turn, become heated and are replaced by other particles; each particle carrying away a certain amount of heat. Since, in general, the density of liquids changes with change of temperature, it follows, that when a liquid is not of a uniform temperature throughout, convection currents will be established; and these will, of course, tend to bring about equilibrimn. Thus, if a vessel containing a liquid, be heated at the bottom, the liquid in contact with the heated surface becomes less dense, rises, and is replaced by a portion of the liquid of higher density, which in turn becomes heated, is replaced by a denser portion, and so forth. 77. Conduction. If a body, such as a inetal rod, be heated at one end, then it is foimd that the temperature along the rod 84 HEAT gradually rises; i.e., heat is transferred wiihont the displacement of matter. Or, to put it in another way, heat is transferred from particle to particle, in a manner such that the particles maintain their relative positions. The propagation of heat through a solid is called candtLction. Assmne that we are dealing with a homogeneous body, bounded by two parallel plane surfaces, indefinite in extent, and that one surface is maintained at a temperature ti, and the other at some lower temperature T2. Then, after a certain time, steady condi- tions will be established. Consider now, the simplest case pos- sible, viz, a prism of constant cross-sectional area, normal to the two surfaces, and extending from one surface to the other. Now, since the two surfaces of the body are indefinite in area, we are justified in assuming that there is no lateral flow of heat; i.e., the heat flows through the prism in parallel stream lines, and the quantity of heat absorbed by the surface at a temperature Ti, for a given interval of time, will be precisely equal in amount to the quantity of heat given oflF by the other surface, at a temperature T2, during the same interval of time. Or, in other words, the flow of heat through the prism will have become uniform; and the quantity of heat passing any section, parallel to the two surfaces, will be the same throughout. Then, as a fundamental principle, verified by experi- FiG. 7. ment, the temperature slope, or the rate of fall of temperature €dong the prism, is con- stant. Hence, if the distance between the two surfaces is represented by s, the temperature slope is r='^'; (3) where r is the temperature slope, or the rate of fall of temperature. The temperature at any point may be found as follows: Let, as in Fig. 7, a be the surface at a temperature of ti; b the surface at I I I I |r I I ^.__ I I I I I I PROPAGATION OF HEAT 85 a temperature of ^2, and a the distance between the two surfaces. The rate of fall of temperature between the two surfaces is given by equation (3); and the fall of temperature from the surface a to the plane x, parallel to the two surfaces, is X T; = -(ti-T2); o hence, the temperature of the plane x, is Tx = Tl (ti — T2) (4) S Theory indicates and experiment verifies that the quantity of heat which is transferred by a prism, such as has just been discussed, is proportional to the area of the exposed surfaces, to the time, and to the temperature slope. Stated symbolically QozAtr] (5) where Q is the quantity of heat transferred, A the cross-sectional area of the prism, t the time, and r the temperature slope. To make statement (5) an equality, we must introduce a proportion- ality factor; i.e., Q=KAtr; from which ^=1 («) K is the ratio of the quantity of heat, passing any section, to the product of the area of the section, the time, and the temperature slope at that section. This ratio is called the coefficient of con- ductivity of the substance; and, of course, diflfers for different substances. From equation (6) it follows that the coefficient of conductivity K, of a substance, is numerically equal to the quantity of heat which flows across a section of unit area, in unit time, when the tempera- ture slope is unity. In the c.g.s. system, and using the centi- grade scale, the coefficient of conductivity of a substance is numer- 86 HEAT ically equal to the quantity of heat, measured in gram calories, which flows across a section 1 sq.cm. in area, in 1 second, when the temperature slope at the section is l^C. per centimeter. 78. Flow of Heat along a Bar. If a bar be maintained at a constant temperature at one end, and the remainder of the bar be exposed to a space of lower temperature, which is also main- tained constant, the fall of temperature along the bar will not be the same as that of the prism previously discussed. For, since the bar is at a higher temperature than the enclosure, it will con- tinually give up heat to the surroundings by radiation and con- vection. Eventually, heat will be supplied to every portion of the bar, by conduction, as rapidly as it is dissipated by radiation and convection. That is, the temperatures along the bar will finally assmne steady values. But, as previously stated, the tem- peratiu'e slope along the bar will not be constant. For since, when a steady condition has been assumed by the bar, the quantity of heat which passes any section, for a given interval of time, is neces- sarily equal to the quantity of heal which is dissipated from the bar beyond that section, for the same interval of tims, it follows that the quantity of heat which passes a section of the bar becomes less as the distance from the end, which is maintained at a con- stant temperature by the application of heat, increases. Hence since, other things being equal, the quantity of heat, which passes any section of the bar, is directly proportional to the temperature slope at that section, it follows that the temperature slope decreases with increase of distance from the heated end. 79. Determination of Coefficient of Conductivity. Since, it is impossible to realize in practice those ideal conditions which were assumed in the discussion of the flow of heat between two parallel walls having areas of indefinite extent, recourse must be had to other methods. A bar maintained at a constant tem- perature at one end, and having the remainder exposed to a space of constant temperature, furnishes a convenient means for deter- mining the coeflScient of conductivity. PROPAGATION OF HEAT 87 To do this, we proceed as follows: After the bar has assumed a constant condition throughout, its temperature is ascertained at a niunber of definite points along it; this is most conveniently done by means of a thermo couple, which is calibrated by com- paring with a standard thermometer. The results are then plotted, differences of temperature between the bar and its enclosure as ordinates and distances along the bar as abscissas. The curve passed through the points so found, shows the difference of tem- perature between the bar and the enclosure, throughout the length of the bar; and the slope of the tangent, drawn to any point of this curve is numerically equal to the temperature slope at that section. This gives us r for equation (6) ; and A of this equation, viz, the area of the section, is determined directly from the dimen- sions of the bar. It now remains to determine Q/t, i.e., the quan- tity of heat which passes a section per unit time. To do this, a second experiment is necessary. The bar is now heated imtil its temperature is uniform throughout and sUghtly higher than the highest temperature on the curve for the rate of fall of temperature along the bar. The bar is then placed in the enclosure, under precisely the same conditions as obtained when the curve for the rate of fall of temperature was determined, and its temperature noted at definite intervals of time. From the data so obtained, a second curve is plotted, differences of temperature between the bar and the enclosure as ordinates and times as abscissas. The curve so obtained is the curve of cooling; and the slope of the tangent, drawn to any point of this curve, is numerically equal to the rate of change of temperature of the bar, with respect to time, for the particular difference of temperature between the bar and its enclosure at that time. Let it now be desired to determine the quantity of heat, which passes in a unit of time, some particular section of the bar, repre- sented by the point a, on the curve A, of Fig. 8. Curve A is the curve representing the temperatures along the bar, and curve B, the curve of cooling. If now, that part of the bar to the right 88 HEAT of a be divided into elements, such as oft, so short, that without appreciable error, the fall of temperature along the element may be considered constant, then the temperature of the element may be taken as the mean of the two temperatures at the points a and 6. If this mean temperature be then projected across to the curve of cooling B, and at the point c, so f oimd, a tangent be drawn, then the slope of this tangent is niunerically equal to the rate of change of temperature with respect to time, for a difference of temperature equal in amount to the difference between that of the mean temperature of the element ab and its enclosure. If Fig. 8. now, we take the product of the thermal capacity of iJie element ab, and the rate of change of temperature just found, we obtain q/t, the quantity of heat lost, per unit of time, by the element ab at the instant when its temperature is defined by the point c. But, since the temperatures of the various parts of the element ab are constant, the mean temperature is a constant, and differs con- tinually from the temperatm-e of the enclosure by an amount pre- cisely equal to the difference of temperature as found from the curve of cooling for the instant when the temperature is repre- sented by the point c. Therefore, the element ab is continuously losing heat, at a constant rate, equal in amount to the quantity just found from the curve of cooling for the temperature represented PROPAGATION OF HEAT 89 by the point c. In a similar manner, the quantities of heat, escap- ing per unit of time, from the various elements to the right of the element ab, are found. Taking the sum of the quantities of heat so found, for all the elements to the right of ab, the quantity of heat Q/t, of equation (6), which passes the section a, in a unit of time, is found. From ^hich, by substitution, K is found. Experiment shows, that, in general, the conductivity of solids decreases slightly with increase of temperature. 80. Conductivity in Non-isotropic Substances. If there be a source of heat at a point in an isotropic substance, i.e., a substance having like physical properties in all directions, then other things being equal, heat will be propagated with equal speeds in all directions; and the temperatures at equal distances in all directions from the source of heat, at any instant, will be foimd the same. Or, in other words, the source of heat will be the center of spherical isothermal surfaces. On the other hand, substances which are non-isotropic do not conduct heat with equal speeds in all directions. As an example, the conductivity of Ice- land spar is greatest in the direction of the axis of symmetry, and equal in all] directions at right angles to this axis. It will be remembered that the coefficient of expansion for Iceland spar is also greatest in the direction of the axis of symmetry, and equal in all directions at right angles to this axis. 81. Non-homogeneous Solids. Tyndall found, by experiment- ing with cubes of wood, that the speed of propagation of heat is greatest, in the direction of the fibers; i.e., parallel to the length of the tree, and least, parallel to the annual layers. And in a direction normal to both the fibers and the annual layers, i.e., radial to a section of a tree, a value was foimd for the conductivity slightly greater than that parallel to the annual layers; but, con- siderably less than that parallel to the fibers. Wood, however, on the whole is a very poor conductor in comparison with metals. It will be of interest here to note that the speed of propagation of soimd through wood is different for the three directions; i.e., 90 HEAT the speed of propagation is greatest, parallel to the fibers, least, parallel to the annual layers, and radial to a section of the tree, it is somewhat greater than it is parallel to the annual layers, but considerably less than that parallel to the fibers. 82. Conductivity of Liquids. The determination of the co- efficient of conductivity of a liquid is attended by difficulties which are not experienced when dealing with solids. For, in the case of liquids, if we wish to determine the true conductivity, convection currents must be avoided. It is, therefore, necessary to heat the column of liquid from the top. It is impossible here, to consider all the necessary precautions which must be taken to insure accurate results. The principle involved, however, is precisely the same as for solids. That is, to determine accurately the temperature slope along the colimm and the quantity of heat passing a given section for a definite interval of time. 83. Conductivity of Gases. The determination of the coef- ficient of conductivity of a gas is still more difficult than is the determination of the coefficient of conductivity of a liquid. For, in a case of a gas, not only must convection currents be eliminated, but radiation must also be taken into account. This makes it extremely difficult to obtain even fairly accurate results. As a matter of interest, the following coefficients of conductivity for a few substances are given. They are all expressed in the c.g.s. system with the gram calorie as the unit quantity of heat. That is, the numbers in the table represent in each case, the quantity of heat, in gram calories, which passes a section 1 sq.cm. in area, in 1 second, when the temperature slope is l^C. per cm. Silver 1.01 Glass 0.002 Copper 0.891 Firebrick 0.0017 Aluminum 0.344 Cork 0.0007 Zinc 0.265 Paraffine 0.0002 Iron 0.167 Water 0.0014 Mercury 0.0152 Ether 0.0003 Ice 0.0057 Hydrogen 0.0004 Granite 0.005 Air 0.000066 PROPAGATION OF HEAT 91 The student must always remember that the results given in the tables for the coefficients of expansion and conductivity must be taken as being only approximate. For, the physical properties of a substance depend very largely upon its chemical purity; and, furthermore, the properties any substance may manifest, will depend very largely upon its physical history and composition. This is especially true for alloys, such as brass, organic growths, such as cork, and complex compositions and mixtures, such as glass. It is interesting to note that, for metals, the order is the same for electrical conductivity as it is for thermal conductivity; i.e., good conductors of heat are also good conductors of electricity, and vice versa. However, there is not, as was at one time sup- posed, strict proportionality. THERMODYNAMICS CHAPTER VIII FUNDAMENTAL PRINCIPLES 81. First Principle of Thermodynamics. The first principle of thermodynamics is merely the application of the principle of energy to the special case of mechanical work and heat; and may be stated as follows: When heat is converted into work, or work into heat, the ratio of the numbers representing the two quardities involved is a constant. The foregoing statement is, of com^e, the result of direct experiment. 86. Second Principle of Thermodynamics. The second pririr ciple of thermodynamics is stated variously by different authors. Indeed, in some cases, the statement is preceded by discussions which involve almost the whole theory of heat. For our purposes, however, the statement first enimciated by Clausius will suffice. This statement is essentially as follows: Heat cannot pass from a body of lower temperature to one of higher temperature urUhovt the aid of some external agent. This statement, though not the result of direct experiment, is in conformity with our conunon experience. As an example, we know from experience that heat passes by conduction and radiation from regions of higher tem- perature to regions of lower temperature. To illustrate further, ^assume that we are dealing with two bodies A and B, and that the temperature of the former is lower than that of the latter; then 03 94 THERMODYNAMICS heat may be made to pass from A to B, by applying heat to A, until its temperature is the same as that of B, bringing the two bodies into contact, and by the further application of heat to A, heat will pass from it to B. Heat may also be made to pass from A to B, it work be first done on the former, such as compressing it, until its temperature is the same as that of B; then by bringing the two bodies in o contact and developing, by a further expendi- ture of work, more heat in A, heat will pass from it to B. But, until there is a tendency to liaise the temperature of A above that of B, no heat will pass from the former to the latter. Assume, now, a third body, C, imder compression and at the same temperature as -A. By bringing the two bodies A and C into contact, and allow- ing C to expand against the external pressure, thus performing work, its temperature will fall and a certain quantity of heat will flow from A into C. The body C may now be removed from A, and compressed adiabatically until its temperature is equal to that of B, and then by bringing C into contact with B, and by a further expenditure of work on C, heat will flow from it to B. At the end of this process, C may be removed from B and allowed to expand adiabatically, and, if the various ranges have been properly chosen, it will at the end of this cycle of operations be in precisely the same condition as it was at the beginning. But, A now contains less heat, and B contains more heat than it did when the process began; and since C is in the same condition as it was at the beginning, heat has been transferred from a body of lower temperature to one of higher temperature by the aid of an external agent. 86. Heat Motors. Heat Motors , or Heat Engines, are devices by means of which energy in the form of heat, is converted into energy, in the form of mechanical motion. All heai motors consist of three parts; viz, a source of heai, a working substance, and a refrigerator. Furthermore, all actual heat motors act periodically; i.e., operate on cycles; and for each cycle a certain quantity of heat is abstracted from the source by IW^^Stt?f'£r4inMRfi' ^^^*"^ makiDg a general kM'tCBii«IEtiBiiS7a|'&ive, as an illustration, a 'j|^'U,^i||/,**)'^]^g^Qonucal, method for the 'V V(»*Ht^t7l(8fS^6i^oir of boiling water ||rgf<^l^<*^#g|[Si[{^t 0°C., and the working jeaJSit^'^^-fl^'a^*^*"^!^^ rod ab. One end of gg^ converted into work, ititti, etc., is rejected to the M l^f^d the other end f^ainst on an axis through 0. A engages one of the teeth drum, having wound over llppose now, that the lei^h t to its coefficient of Unear IB raised from 0°C. to 100° J;the disk turns through an ^f ■;^^|^eg:2|^G|^^^:|:tooth. The resistance R, '" '"^ i«SEil£3i|^«|^^Jci^a^ the pawl will engage the 96 THERMODYNAMICS next tooth. If the rod be now surrounded by a bath of melting ice, it will contract and engage the next tooth; the pawl, in the meantime, holding the disk in position. This process may be repeated indefinitely, until the resistance has been displaced through any desired distance. The cyde is then as follows : The rod at the temperature of melting ice is put into position, and sur- roimded by a i>ath of boiling water at a temperature of 100**C. In consequence of this elevation of temperature, the rod expands and turns the disk through a certain angle and in this manner does work in overcoming the resistance R. The heat taken from the source consists of two parts: One part being consumed in elevating the temperature of the rod, and is numerically equal to the prod- uct of the mass of the rod, its thermal capacity per unit mass, and the elevation of temperature. The other part consists of the heat equivalent of the work done in displacing the resistance R. The rod, now being disconnected and surrounded by melting ice, gives up to the refrigerator, in cooling from 100°C. to O^C, an amount of heat precisely equal to that absorbed in being heated, without any external work being done, from 0°C. to 100** C. Therefore, the dififerenoe between the heat taken from the source and that given to the refrigerator is equivalent to the work done in displacing the resistance R. Since this completes a cycle it may be repeated indefinitely without any change in the relation of the quantities involved. If, now, we represent by Qi, the quantity of heat absorbed from the source, during a cycle, and by Q2, the heat rejected to the refrig- erator, then the external work done is Tr=J(Qi-Q2); (1) where W is the external work done, and J the mechanical equiv- alent of heat. Since for every cycle the quantity of heat Qi has forever disappeared from the source, and only the part Qi— Q2 has been converted into work, it follows that, with the contrivance FUNDAMENTAL PRINCIPLES 97 just described, it is impossible to convert all the heat, taken from the source, into external work. 88. Heat of Expansion. As explained in Arts. 46 and 47, when a gas is heated and expands against an external pressure, the heat required is practically equal to that required to elevate the temperature of the gas, plus the heat equivalent of the exter- nal work done; i.e., the external work done, expressed in heat units, is practically equal to the heat of expansion. This, however, is by no means the case when a metal rod is heated and expands against an external pressure; for, in this ca3e, the ?ieat of expansion consists of two parts, viz, the heat equivalent of the external work done, and the heat required to expand the rod against its own inherent forces. The former may be called the external heat of expansion, and the latter the internal heat of expansion. In the present state of our knowledge we are unable to assign the proper relative values for the heat consumed in elevating the temperature of a substance and the internal heat of expansion; but; for most substances, the latter is a relatively large quantity. Since, now, in the cycle discussed in Art. 87, the internal heat of expansion is not recovered as work, but is rejected to the refriger- ator, it follows that such a contrivance cannot use heat eco- nomically. 89« Camot's Cycle. The first scientific discussion of a peri- odically acting thermodynamic engine is that due to Sadi Camot, published in 1824. In this discussion, ideal conditions are assumed ; i.e., it is assumed that there are no losses due to radiation and friction. In other words, it was Camot's object to show that under certain given conditions, assmning ideal processes, a definite fractional part of the heat taken from the source, by a periodically acting engine, is converted into work; and that, for the given conditions, this is the maximal amount of work that may be realized. The following demonstration will make this clear. Assume that we are dealing with any working substance whatso- ever, confined in such a manner that it may be put into contact pinr:'M^VS'H*W'on the p-ti (pressure-volume) 9>KtS^B?ML@^ volume of the working sub- t?|SriBi(]Bil»«^|it is removed from the refrig- iilWf^a^'0'Mp^!ltStwt4i''3'' ordinates and volumes by is now insulated and com- Lce of which its temperature i^Qied until the temperature of at of the source, and its con- i:^^t:^lume, is represented by the tbstance during this compres- der the curve; i.e., the area now put into contact with isothemally, by any desired _ Is pressure and volume, at the |;|C||^^^h^^J*^t^A||^bythe point C. During this - ;. . j^. . j^. .^. .^. .g. .^. ..-!. FUNDAMENTAL PRINCIPLES 99 isothermal expansion a certain quantity of heat Qi, has been abstracted from the source, and work has been done by the working substance, represented by the area BCFK, The working sub- stance is now again insulated and allowed to expand adidbaticoMy, in consequence of which its temperature falls. This expansion is continued until the temperature of the working substance has fallen to that of the refrigerator, and the work done by it, during this expansion, is represented by the area CDEF. The working substance is now put into contact with the refrigerator and com- pressed isotherrrMUy until its condition, as regards pressure and volume, is again represented by the point A. During this isother- mal compression, toork was done on the working substance repre- sented by the area DEGA ; and, a quantity of heat Q2, was rejected to the refrigerator. Since now, the working substance, as regards pressure, volume, and temperature, is in precisely the same condition as it was at the beginning of the cycle, its irUrinsic energy is also the same; it therefore follows, from the first principle of thermodynamics, that the difference between the heat abstracted from the source and that rejected to the refrigerator, expressed in mechanical units, is equal to the net work done. By an inspection of Fig. 10 it is obvious that the net work done is represented by the area ABCD; and, from equation (1), we have Wi=J{Qi-Q2); where Wi is the net work done. But the heat, expressed in mechanical units, abstracted from the source is W2^JQi; therefore, the ideal coefficient ofcorwersion^OT the maximal fractional part of the heat, abstracted from the source, which in an ideal process can be converted into work, is ^ — jQi — ~Qr ^^^ 100 THERMODYNAMICS The result, just found, has been deduced without making any assumption in regard to th^ nature of the working substance; it is therefore perfectly general. Suppose that the working substance suffers a physical change of state during the cycle; the foregoing demonstration still holds. For, since the working substance is in precisely the same condition as regards pressure, volume, and temperature at the end of the cycle as it was at the beginning, it follows that whatever physical changes of state have taken place during any part of the cycle, changes of a like kind must have taken place in the reverse order during some other part of the cycle; and hence, are balanced. Therefore, the difference between the heat taken from the source and that rejected to the refrigerator, expressed in mechanical units, is equal to the external work done. 90. Since the relation, expressed in equation (2), was deduced without considering the properties of the working substance, it must be independent of those properties. There being, however, no other quantities involved in the right-hand member of this equation, excepting quantities of heat, and since these do not depend upon the properties of the working substance, they must be functions of the two temperatures. That is, the quantity of heat taken from the source must be some function of the tempera- ture of the source, and the quantity of heat rejected to the refrigerator must be some function of the temperatxure of the refrigerator. Just what values are to be assigned to these func- tions must be determined for some specific case, which is con- sistent with the demonstration. 91. Camot's Cycle a Reversible Process. The ideal cycle just described is a reversible process. For, if the working substance, at that part of its cycle when its condition, as regards pres- sure and volmne, is represented by the point A, Fig. 10, and its temperature is the 'same as that of the refrigerator, is put into contact with the refrigerator and allowed to expand isoOiermaUy to the point D, it will abstract from the refrigerator, a quantity FUNDAMENTAL PRINCIPLES 101 of heat Q2, and do an amount of external work, represented by the area ADEG. The working substance is then insulated and com- pressed adiabatically , in consequence of which its temperature will rise; let this be continued until its temperature is the same as that of the source, and its condition, as regards pressure and vol- ume, is represented by the point C, and an amount of work, represented by the area FEDC, has been done on the working substance. The working substance is now put into contact with the source, and compressed isothermaMy until its condition, as regards pressiu-e and volume, is represented by the point B, a quantity of heat Qi being rejected to the source, and an aziiount of work, represented by the areaiPCSX ,has been done on the work- ing substance. The working substance is now insulated and allowed to expand adiabaticaUy imtil its temperature has fallen to that of the refrigerator; its pressure and volume being the same as at the beginning, and the external work, represented by the area BAGKy having been done by it. Taking the smn, we find that the work done by the working substance is represented by the area KBADE; and the work done on the working substance is repre- sented by the area BKEDC. Finally, the net work done on the working substance is represented by the area ADCB. But, during this process, the quantity of heat Q2 has been taken from the refrigerator, and the quantity of heat Qi has been transferred to the source. Since now, the working substance is in precisely the same condition as regards temperature, pressure, and volume, as it was initially, it follows that the difference between the heat rejected to the source and that taken from the refrigerator, expressed in mechanical imits, is equal to the net work done on the working substance; i.e., TF=J(Qi-02). 92. It will now be shown that, for a given source and refrig- erator, an engine operating on the Camot cycle, that is, a reversible 102 THERMODYNAMICS engine, converts into work as large a fractional part of the heat taken from the source as is possible under the assumed conditions. To do this, we assmne that we have two engines A and B, operating between the same source and refrigerator, the former acting direct and driving the latter, which is nmning reversed. Let Ha and Ha' be, respectively, the heat taken from the source and that rejected to the refrigerator by the engine A during a given interval of time; and likewise, let Ht and Hi/' be, respectively, the heat transferred to the source and that abstracted from the refrigerator by the engine B, during the same interval of time. Assimie,now, that the engine A, which is nonreversible, can convert a larger fractional part of the heat taken from the source into work than could the engine B if it were running direct. We then have Ha'^Hg' Hb—Hb' ^rtv Ha' ^ Hb' ^^^ Also, the work done hy the engine A must be equal to the work done on the engine B, since the former is driving the latter; hence, we have W=J{Ha'-Ha")=J{m'-Hn (4) From equation (4) it follows that the numerators of the inequality, expressed by statement (3), are equal; hence Ha K,Hb • Also, from equation (4), we find Hb — Ha ^^Hb — Ha ; hence, since Hb is greater than Ha, Hb" must be greater than Ha'. But, with a reversible engine, operating on the Camot cycle between a certain source and refrigerator, the quantities of heat involved are always the same for a definite amount of work, no matter whether the engine is acting direct or reversed; FUNDAMENTAL PRINCIPLES 103 it therefore follows that under the assumed conditions, the source must be gaining heat, since the quantity of heat Ua taken from it by the non-reversible engine is less than the quantity of heat Uh rejected to it by the reversible engine. Likewise, since the quantity of heat Ka' rejected to the refrigerator, by the non- reversible engine, is less than J?/', that taken from it by the revers- ible engine, it follows that the refrigerator is continually losing heat. Hence, under the assumed conditions, we have a system in which heat is being transferred from a body of lower temper- ature to one of higher temperature, without the aid of an agent external to the system. Since this, however, contradicts the second principle of thermodynamics, we must conclude that the assumption made, viz, that any engine can convert into work a larger fractional part of the heat taken from the source than is possible by means of a reversible engine, operating on a Camot cycle between the same source and refrigerator, is in error. There- fore, a reversible engine converts into work as large a fractional part of the heat taken from the source as is possible under the given conditions. We may, however, consider this in another manner. Assume that there is a third engine, operating between the same source and refrigerator, abstracting heat from the source, and rejecting heat to the refrigerator at a rate such that both the source and refrigerator are maintained at a constant temperature. This third engine may then be employed in doing external work, and we have a system which is doing work without the expenditure of energy. This, however, contradicts the principle of energy; hence we must again conclude that the original assumption is in error. Hence the conclusion that, /or any given conditions, no engine can convert into work a larger fractional part of the heat taken from the source than that converted into work by a reversible engine. 93. Reversible Engine as a Standard. It must be remembered that all processes so far discussed, in this chapter, are ideal proc- esses and cannot be realized in practice; in other words, since the 104 THERMODYNAMICS cycles of an actual thermodynamic engine, are attended by friction and radiation, they are necessarily irreversible. As a matter of fact, as stated in Art. 27, all processes are irreversible. Reversible processes are merely ideal; i.e., conceptions of perfect operations. When we speak of the efficiency of a mechanical contrivance, as being the fraction p, we simply mean that the output is the fractional part p of the input. And the closer p approaches unity, the nearer the machine is considered to be to perfection. But here again, our standard is an ideal one; i.e., we are comparing our actual machine with one that is ideally perfect. Since it has been shown that a reversible engine, operating on the Camot cycle, between a given source and refrigerator, converts into work as large a fractional part of heat taken from the source as possibly can be converted into work under the given conditions, we are justified in taking this engine as a standard with which to compare the performance of actual engines. 94. Camof s Cycle with a Perfect Gas as a Working Sub- stance. Assume that we have confined in a cylinder, by means of a frictionless piston, a perfect gas, and that there is a source of heat at a temperature Ti, and a refrigerator at a temperature T2. Let the condition of the gas, as regards pressure and volume, be represented by the point Ay Fig. 11, when put into contact with the source; pressures being represented by ordinates and volmnes by abscissas. We will, furthermore, assume ideal conditions; i.e., perfect conduction for the isothermal processes, perfect insulation so that adiabatic processes may take place, and no losses. Consider now, the four processes as follows : (1) The cylinder containing the gas at the pressure pi, volume vi, and temperature Ti, is put into contact with the source of heat at temperature 7^i, and the gas is allowed to expand isother- mdHy by any desired amount, say to the point B\ its pressure now being p2 and volume V2- During this expansion work is done on the piston by the gas, measured by the area imder the curve AB, and a quantity of heat Q\ is taken from the source. It being FUNDAMENTAL PRINCIPLES 105 assumed that the temperature of the source during the abstraction of the heat Qi, remains constant; this may be brought about by supplying heat to it at a proper rate. (2) The cylinder is now removed from the source, is perfectly insulated, and the gas is allowed to expand adiabatically, in con- sequence of which its temperature falls, due to the fact that the external work is done at the expense of the intrinsic energy of the gas. This expansion is continued until the temperature of the gas has fallen to T2, that of the refrigerator; in the meantime, (P..V V FlQ. 11. work has been done on the piston by the gas, measiu*ed by the area under the curve BC. The pressure is now pz and the volume ^3. (3) ThQ cylinder is now put into contact with the refrigerator and the gas is compressed isothermcdly, until its pressure is p4, and volume »4, as represented by the point D. During this com- pression a quantity of heat Q2 is developed, and work is done by the piston on the gas, measured by the area under the curve DC. It being assmned that the temperature of the refrigerator, during the absorption of the heat Q2, remains constant; this may be brought about by abstracting heat from it at the proper rate. (4) The cylinder is now removed from the refrigerator, is perfectly insulated, and the gas is compressed adiabatically until 106 THERMODYNAMICS its temperature is Ti, that of the source, and its pressure and vol- ume are, respectively, pi and vi. During this compression work was done, by the piston on the gas, measured by the area under the curve AD. The gas being now in precisely the same condition as it was initially, its intrinsic energy must also be the same. Since, now, A and B are on the same isoth&rm, and likewise, C and D are on the same isotherm, we have, from the character- istic equation, Pit'i=P2V2=-Bri, (5) and P3f3=P4V4=-Kr2 (6) Also, since B and C are on the sam^ adiabatic, and likewise, A and D are on the same adiabatic, we have P2V2** = P3t'3^ (7) and pit;i»=p4t;4" (8) From equation (5) we find and From equation (6) we find and Pi = -r-, (9) P2 = ^ '. (10) P3 = -— , (11) B>T2 /-rt\ P4=— (12) Substituting the value of p2 as given in equation (10), and that of p3 as given in equation (11), in equation (7) we find ^^1 , RT2 „ — -V2''=' va'*; V2 Vz FUNDAMENTAL PRINCIPLES 107 from which V2 \ 1 Again, substituting in equation (8) the values of pi and p4>as given by equations (9) and (12), we find RTi „ RT2 „ Vl ^4 from which (14) From equations (13) and (14) it follows that from which V3 V2 (15) va Vl Equation (15) shows that the volume at C must be to the volume at D, as the volume at JS is to the volume at A, so that when the gas is compressed adiabatically from Z), it will come to the point A. Since we are dealing with a perfect gas, its intrinsic energy is a function of the temperature only, and is, therefore, independent of pressure and volume. Therefore, the work done by the gas in going along the adiabatic from £ to C, is exactly equal to the work done an the gas in going along the adiabatic from Z> to il. Hence to obtain the net work done during the cycle, and the quan- tities of heat involved, it is only necessary to consider the two isothermal processes; viz, the heat abstracted from the source, and the external work done, by the gas in going from A to B, along the isotherm Ti, and the heat rejected to the refrigerator, 108 THERMODYNAMICS m and work done, on the gas, in going along the isotherm T2 from CtoZ). Since the temperature, during the isothermal expansion, from A to £ is constant, it follows that the heat abstracted from the source is directly proportional to the external work done. Hence we have Qi^A { pdv; where A is the heat equivalent of a unit of work. But p, for any part of this process is equal to RTi/v;- hence dv Jvi V =ARTi\og^ (16) By similar reasoning we find, that the heat rejected to the refrigerator, during the isothermal compression, in going from C to D, is Q2=A I pdv. But, for any part of this process p is equal to RT2/V; hence Q2=ART2r- =ART2log^ (17) The net work done, measured in heat units is, by condition, proportional to the difference between the heat taken from the source and that rejected to the refrigerator; hence iiTr=Qi-02=Ai2^riiog^-r2iog^); FUNDAMENTAL PRINCIPLES 109 and the ideal coefficient of conversion, since Qi has forever dis- appeared from the source, is ^ ^ Wrilog^-7'2log^^ «' ^Bniog? Vl from which, since by equation (15), v^hx—vz/v^^ we find ^ — qT" — rr ^^^^ Equation (18) shows that, for a perfect gas operating on a Camot cycle, the xdM coefficient of conversion is the ratio of the difference in temperature of source and refrigerator, to the tem- perature of the source, as measured on the ideal gas thermometer. Equation (18) is usually written in the following form: S — XL ,^-. •n — -g-; (19) where S is the temperature of the source and R that of the refrig- erator, both being measured by means of the ideal gas thermometer. 96. A little consideration will showthat the foregoing discussion, and result obtained, is perfectly consistent in everyway with that of the Camot cycle using any working substance. We are there- fore justified {Art. 90) in assuming that even under ideal conditions the maximum quantity of work that can be realized from an engine working between a given source and refrigerator and ab- sorbing the quantity of heat H from the source, is W=Jh{^) (20) Writing equation (19) in another form, we have 110 THERMODYNAMICS from which it is obvious that, for y) to approach unity, R must either approach zero, or 8 must approach infinity. Experience, however, shows that it is not economical to attempt to maintain the refrigerator at a temperature lower than that of the surround- ings. Also, as the temperature of the source is increased, a point is soon reached for which radiation and pressures become excess- ive, and lubrication becomes difficult. It therefore follows, with conditions such as obtain on the earth's surface, that even a perfect engine can convert only a small fractional part of the heat, taken from a source, into work. 96. Reversible Engine and Refrigeration. An engine operating in a reverse order, i.e., one that is taking heat from a body of lower temperature, transferring heat to a body of higher temper- ature, and absorbing external work, constitutes a refrigerating machine. Let, for any given time. Hi be the quantity of heat transferred to a body of higher temperature, and H2 the quantity of heat abstracted from a body of lower temperature, by a per- fectly reversible engine; i.e., a perfect refrigerating machine. We will then have the following relation: H1—H2 __ S—R .^i\ Hi ~ S ^^^^ From equation (20) we have, for the amount of work that must be done, to transfer the quantity of heat Hi, to the body of higher temperature, W==JHi^^ (22) In general, however, in the case of refrigerating machines, we are concerned principally with the work that must be done to bring about a certain absorption from the body of lower temperature; i.e., the amount of refrigeration. It is therefore advisable to deduce an expression for the amount of work that must be done in terms of H2, the quantity of heat taken from the body of lower FUNDAMENTAL PRINCIPLES 111 temperature, instead of the quantity of heat Hi, rejected to the body of higher temperature. From equation (21) we find H2 __R Hi~P from which Hi=H2^ (23) Substituting in equation (22) the value of Hi, as given by equation (23), we find W^JH^^; (24) which gives the desired relation. 97. In Art. 95, it was stated that it is not economical to attempt to maintain the temperature of the refrigerator lower than that of the surrounding medium. We are now prepared to demon- strate this mathematically. Let Hi be the heat taken from the source, at a temperature S, and let Ri be the temperature of the surroundings. If then the temperature of the refrigerator be also Ri, the work that, under perfect conditions, may be realized is Wi=JHi^^ (25) Assume now, that the refrigerator, by means of a reversible engine, is maintained at some temperature R2, lower than Ri. The work that can now be realized, by means of a perfect engine, is W2=JHi^^ (26) Subtracting equation (25) from equation (26), member by mem- ber, we obtain, due to lowering the temperature of the refriger- ator, for the gain in work, W2-Wi^J^{Ri-R2) (27) 112 THERMODYNAMICS To maintain the temperature 722 we must, by means of a revers- ible engine^ abstract heat from the refrigerator at the same rate that the direct engine is rejecting heat to it, and transfer heat to the surroundings. The heat rejected by the direct engine is H2^Hi-AW2=Hi'-Hi^~^=Hi^. . . (28) The work that must be expended in transferring this quantity of heat from the body of temperature i22, to the surroundings at a temperatiu*e Ri, is =J^{Ri-R2) (29) By comparing equations (29) and (27), it is obvious that, even under ideal conditions, the amount of work that must be done by the reversible engine, to maintain the temperatiu^ of the refrigerator, below that of the surroundings, is equal to the gain in work by the direct engine, due to the lower temperature of the refrigerator. It therefore follows that, even without consid- ering losses, there can be nothing gained by attempting to have the temperature of the refrigerator lower than that of the earth's sur- face. As a matter of fact, if the' temperature of the refrigerator is lower than that of the surroimdings, heat will continually pass from the surroundings to the refrigerator, and the reversible engine must do an amount of work greater than that given by equation (29). Furthermore, due to imperfections of the engines, the gain in work realized by the direct engine will be less than that specified by equation (27), and the work that must be done on the reversible engine will be greater than that specified by equation (29) ; hence, there is a decided loss when the refrigerator is main- tained at a temperature lower than that of the surrounding media. FUNDAMENTAL PRINCIPLES 113 98. Thermod]niamic Scale of Temperatures. The thermo- dynamic scale of temperatures^ which was first proposed by Lord Kelvin, will be made clear by the following considerations. Assimie a series of n perfect heat engines arranged in such a manner that the refrigerator of the first engine is the source of the second engine, the refrigerator of the second engine is the source of the third engine, etc., and furthermore, that the beat rejected by any engine is absorbed by the engine next lower in the scale. To show that, if the difference in temperature between soiLrce and refrig- erator for the various engines is the same, they are all doing the same amount of work. Let, as in Fig. 12, the two adiabatics, AB and CD, be cut by the isotherms Ti, T2, Tz, etc., such that the temperature intervals are all equal, and each equal to t; i.e., ri-r2=r2-r3=rn-rn-Hi=T. The ideal coeflScients of conversion for the various engines, begin- ning with the first, then are ri' 2^' ¥3' ' "fn' (30) 114 THERMODYNAMICS If H is the quantity of heat absorbed by the first engine from its source, during a given interval of time, then n—ii — y Ziy- is the heat rejected to its refrigerator, and absorbed by the second engine, during the same interval of time. In a similar manner, the quantity of heat supplied to the third engine is „i2 _ „i^ i 2 — is __ Tr£3 Tl ^Tl T2 " T{ The quantities of heat supplied to the various engines, beginning with the first, then are "^7 "^jT* -"jTj • • • -"~y — } ^jT' ' ' ^ ^ Since, now, the work done by any engine of the series is equal to the product of its ideal coefficient of conversion and quantity of heat, expressed in mechanical units, absorbed by it, it follows from expressions (30) and (31), that all the engines are doing the sameamount of work; i.e., is the work done by each engine of the series. The results just deduced, being independent of the properties of any substance, a thermodynamic scale of temperature may be established in the following manner: Assume a series of n heat engines, working between a given source of temperature Ti, and refrigerator of temperature Tn+u ^^ such a manner that the n engines are all doing the same amount of work, and each engine is absorbing the heat rejected by the engine next higher on the scale. If we then designate the difference of temperature between FUNDAMENTAL PRINCIPLES 115 the source and refrigerator of any one of these ideal engines, as a unit of temperature, we will have a scale of temperatures inde- pendent of any substance, and depending only upon the perform- ance of a perfect engine. But, from the discussion just given, we found that by assuming the temperature intervals, as measured on the ideal gas thermometer, equal, the engines were all doing the same amount of work; hence, the thermodynamic scale is identical with that of an ideal gas thermometer; and differs but slightly, for temperatures not exceeding 500°C.; from those as f oimd by means of the ordinary gas thermometer. CHAPTER IX STEAM AND STEAM ENGINES m 99. The proper design of a heat engine presupposes, on the part of the designer, a knowledge of the construction of mechan- ical contrivances; i.e., how to construct a machine which shall withstand the stresses imposed upon it in the performance of its duties, with the lowest cost. The expression, lowest cost, must not be interpreted as meaning lowest first cost; but it must be understood to mean that the interest^on the capital invested, for both machinery and ground rent, plus depreciation, plus cost of power lost, must be a minimum. This part of the subject comes under the heading of machine design; and, properly speak- ing, has nothing to do, except in so far cw fuel economy is affected by the design, with the subject of thermodynamics. But, a thorough knowledge of the characteristics of the working substance and the changes it undergoes, during its various stages, is fully as impor- tant, if not more so, in the designing of an engine, as is a knowledge of machine design. It is for this reason, since steam is so widely used as a working substance, that so much research work has been done, to accurately determine its characteristics. 100. Steam Operating on Camot's Cycle. Assume that we are dealing with a unit mass of water, at a temperature T2, which corresponds to that of the refrigerator, and let its condition, as regards pressure and volume, be represented by the point D of Fig. 13. The water is compressed adiabatically until its tem- perature is Ti, that of the source, and its condition, as regards pressure and volume, is represented by the point A. If the water 116 STEAM AND STEAM ENGINES 117 is now placed into contact with the source, and the pressure is maintained constant, vaporization will take place. Assume this to be continued until all the water has been converted into satu- rated steam, whose condition, as regards pressure and volume, is represented by the point B, The steam is now allowed to expand adiabatically imtil its temperature has fallen to T2, that of the refrigerator; its pressure and volume being now represented by the point C During this adiabatic expansion a certain amount of condensation, which will be discussed later, has taken place. The mixture of steam and water is now put into contact with the Ar ,Tl B D^ ^ Arf V Fig. 13. refrigerator and compressed isothermally until complete conden- sation has taken place, and its condition, as regards pressure and volume, is again represented by the point D, Since, now, the condition of the working substance, as regards temperature, pressure, and volume, is precisely the same as it was initially, its intrinsic energy is also the same. Therefore, the net work done during the cycle is measured by the area DABC. Further- more, since the process is ideally reversible, the ideal coefficient of conversion is the same as previously deduced for any working substance. 118 THERMODYNAMICS 101. Relation of Temperature and Density of Saturated Steam. It is frequently of prime importance to know the density of satu- rated steam for a given temperature; and it being difficult to determine this relation by direct experiment, it will be shown how it is found from the relation of pressure and temperature of a saturated vapor, this being easily determined by direct experiment. To show how to deterinine the relation of temperature and density of the saturated vapor of a substance, it will be assumed that we are dealing with a unit mass operating on a Camot cycle, as just described, and an indefinitely small difference of temperature, AT J between source and refrigerator. This is represented dia- gramjnatically in Fig. 14, where T+dTis the temperature of the A T+^T ^ B V Fig. 14. source, and T the temperature of the refrigerator. This being a reversible process, the ideal coefficient of conversion is TQ = AT T+AT' which, in the limit, becomes TQ = dT T' If the quantity of heat taken from the source, in going from A to By is Q, then the work done is W^Jd dT (1) STEAM AND STEAM ENGINES 119 aao ^ 1 ^^ ^ ^^ "■ ^^^^ ^■" ■^ ■■" ■^ ■"■ ■^ ^" ^~ ^ ■^ ^ ■" ■" 1 ' 1 1 ' 1 1 1 1 p ' / S5 / RELAT ON r 1 1 OF TEMPERATURE AND PRESSURE OF SATURATED STEAM I 1 1 200 / / / / / 176 / / / / 1 ^.150 ii / o z / ^ 1 , §126 / / fi i CO / / 100 ' i / / V / 76 i r / J / y > z' 50 / y / / / 25 / / ^ y __^ X - -^ 1^ — — U T Ef lip, ,1S IC >EC iHl IE SF Al HR • ac ^ too ] 20 THERMODYNAMICS The work done during the cycle may also be expressed in terms of the initial and final volimies and the change in pressure dp, corresponding to the change in temperature dT. That is, if a is the volume of unit mass of the liquid, and 8 the volume of unit mass of satiu-ated vapor, then the work done, during the cycle, is W={8-a)dp (2) Now the right-hand members of equations (1) and (2) must be equal; since they are expressions for the same amount of work, hence dT from which and JQ^dT -^^'4-% (3) The quantity Q, in equation (3), represents the quantity of heat required to convert unit mass of the liquid into a saturated vapor at the temperature T, and may be replaced by r, the heat of vaporization; hence, equation (3) becomes In equation (4), a, the voliune of unit mass of the liquid, for the temperature T, is readily found by experiment; and likewise r, the heat of vaporization. dT/dp is found from the curve giving the relation of temperature and pressure of the saturated vapor. Hence, since J, the mechanical equivalent of heat, is known, 8 is determinate; and the reciprocal of this gives the density of the saturated vapor. STEAM AND STEAM ENGINES 121 As a matter of interest, the curve showing the relation of temperature and pressure, for saturated steam, is given on page 119. 102. Perfect Steam Engine and Boiler. In the previous dis- cussions it has been assumed that all of the heat is taken in at the highest temperature. This, however, is by no means the case, even under perfect conditions, with a steam engine and boiler. For the present, we will confine ourselves to the operation of a reciprocating engine, which has supplied to it saturated steam from a boiler. The reciprocating engine consists essentially of the following parts: A source of heat, the boiler y where steam is generated under a constant pressure, and hence, at a constant temperature, a cylinder and piston, and a refrigerator, or condenser, at constant temperature, by means of which the steam, after expanding and doing work against the piston, is converted into water and returned to the boiler. The cycle of operations is as follows: The piston P is at the position as represented in the diagram. Fig. 15, and the condition of the steam, as regards pres- sure and volume, is represented by the point A, the point of admission. That is, at this point, the valve in the pipe connect- ing the boiler with the cylinder is opened, and steam is freely admitted. The piston advances to the point B, while vapori- zation takes place at the temperature Ti. To simplify matters, we will assume that we are dealing with unit mass of water and that complete evaporation has taken place when the volume is represented by the point B. The quantity of heat then, taken from the boiler, is ri, the heat of vaporization at the temperature Ti. The line of admission, AB, is a straight line and parallel to the axis of volumes, since vaporization has taken place at constant temperature; and hence, at constant pressure. The external work done, during this advance of the piston, is measured by the BiieskABFE. The point B is the point of cut-off; i.e., the admission- valve is closed, and the steam is allowed to expand adiabatically until its temperature has. fallen to T2, that of the condenser. \ 122 THERMODYNAMICS In the meantime, the external work, measured by the area BCGF^ has been done. The exhatLst-valve now opens, and the steam remaining in the cylinder, is compressed isothermally, in contact with the condenser, until complete condensation has taken place, and the work represented by the area DCGE, has been done by the piston. The condensed steam, at the temperature T2, is returned to the boiler and heated from the temperature T2 to that of Ti, thus completing the cycle. The net work done during A Ti B JG Fig. 15. this cycle is evidently measured by the area ABCD, and is neces- sarily less, as will now be shown, dvs to not taking in all the heat at the maximum temperature^ than that which could be realized by a Camot cycle. The maximum amount of work that could be realized from an engine taking in an elementary quantity of heat dQ, at the temperature T, between the temperatures T2 and Ti, operating on a Camot cycle, and rejecting heat to the condenser at the tem- perature T21 is T-r2 dW^JdQ T (5) STEAM AND STEAM ENGINES 123 But, dQ is equal to cdT; where c is the thermal capacity and dT the change in temperature. And, since we are dealing with imit mass of water, dQ is practically equal to dT; since for water c is almost constant and equal to unity. Therefore, equation (5) may be written dW^J^^dT; (6) from which we obtain, for the total work that could be realized, under ideal conditions, from the heat required to elevate the tem- perature of xmit mass of water from T2toTi, W'=Jl ^-J^dT =j(ri-r2-r2iog^j).. . . (?) The work that, under ideal conditions, could be realized from the heat taken from the source during vaporization, since this is absorbed at constant temperature, is v/_ r.- T1 — T2 W^'^Jn^^; (8) where ri is the heat of vaporization at the temperature Ti, Adding equations (7) and (8), we obtain for the total work that may be realized, for the given conditions, W, = W' + W''=j(Ti-T2-T2log^+ri'^^^ . (9) Since the total heat, abstracted from the soiu'ce, expressed in mechanical units, is J(Ti-T2+ri), 124 THERMODYNAMICS the maximum work which would have been realized, had the operation been on a Carnot cycle, is W2=J{Ti-T2+ri)^^^^ (10) Dividing equation (9) by equation (10), we find W2 Ti+n-Tz ^ '■^ If now, in equation (11), 7r^|rlog^>r2, (12) then W1/W2 is less than unity. To prove that the expression, given by the inequality (12), holds, we assume that T2, the tem- perature of the condenser, is fixed, and that Ti is a variable, which may be represented by T; remembering that T is always greater than T2f and that both are positive. Expression (12) may then be written y'^lo5^=ftr2; (13) where A: is a proportionality factor. From this, we find log^=fc(l-f) (14) Substituting, in equation (14), for T/T2, a new variable, x, we have logx=fc(l-i); and, differentiating with respect to x, we obtain T *=a; = -jr; (15) STEAM AND STEAM ENGINES 125 from which, if TJT^ equals unity, fc equals unity; and, if TJT^ becomes greater than unity, M, must be greater than unity or equation (15) cannot hold. But this means that the left-hand member of equation (13) must be greater than T^'y and hence W1/W2 is less than unity. It therefore follows that a steam engine, which rejects condensed steam to a boiler cannot, even under perfect conditions, convert into work as large a fractional part of the heat taken from the source as can an engine operating on a Camot cycle, between the same limits of temperature. To illustrate the foregoing, we will deal with a concrete case; i.e., assiune the temperature of the entering steam, and of the con- denser, respectively, 356°F. and 140^F. This gives: Ti = 816,* 72 = 600, and ri=865. Substituting these values, in equation (11), we find QiA.QAR 816X600 , 816 W^ ^^^+^^^"816:^600^^^600 ^,^ W2^ 816+865-600 =^^-^ ^^^ "^^*' giving a loss of about 9 per cent due to not taking in all the heat at the maximum temperature. 103. Unresisted Adiabatic Expansion of Steam. If dry saturated steam is allowed to expand adiabatically, from a chamber of given pressure to one of lower pressm-e, without doing work, the steam becomes superheated. This is due to the fact that, when the steam enters the chamber of lower pressure, eddy cur- rents are developed; and as they subside, the kinetic energy, possessed by them, is converted into heat. Since the process is adiabatic, and no external work is done, the total heat cantentf i.e., the total quantity of heat contained by the steam, will be the same at the end of the process as it was at the beginning. But since, the total heat of steam decreases as the pressure is * According to recenl; experiments, the zero for the thermod3maniic scale is 491. 65 ^F. below the melting-point of ice; but, in general, 492 is sufficiently accurate. 126 THERMODYNAMICS decreased, and the final pressure is lower than the initial pressure, the steam must become superheated. If the steam is not initially dry, then it will become drier by unresisted adiabatic expansion. Assume that we are dealing with a unit mass of a mixture of steam and water under a pressure pi, for which the heat of the water and the heat of vaporization are, respectively, hi and ri. The total heat of the mixture, then is H=hi+qiri; (16) where gi is the dryness; i.e., the fractional part of the liquid which is present as steam. After expansion, since the total heat content remains the same, we have H=h2+q2r2; (17) where A2, q2, and r2 are, respectively, the heat of the liquid, the dryness, and the heat of vaporization for the final pressure p2. Equating the right-hand members of equations (16) and (17), we obtain hi+qiri = h2+q2r2 (18) By priming is meant the percentage of moisture present; and if this is low, the steam may become superheated by the unresisted adiabatic expansion, and q2j in equation (18), becomes unity. It will be shown later how, under certain conditions, advantage may be taken of this, and the initial priming determined exper- imentally. 104. Resisted Adiabatic Expansion. If steam, the initial priming of which is low, expands adiabatically in such a manner that external work is done, it will become wetter. If in equation (9), Art. 102, it is assumed that complete evaporation has not taken place before the adiabatic expansion begins, then the work, expressed in heat units, yielded per cycle, is W=Ti-'T2-T2\ogp+qiri'^p; . . (19) STEAM AND STEAM ENGINES 127 where qi is the dryness. The total heat absorbed, in elevating the temperature of the water from T2 to Ti, and evaporating it to the dryness qi, is Hi==Ti-T2+qiri (20) And since, imder the assmned conditions, the difference between the heat abstracted from the source and that converted into work, must be equal to H', the heat rejected to the condenser, we find, by subtracting equation (19) from equation (20), ff' = (Zin^'+r2log^ (21) But, the heat rejected to the condenser, after adiaoatic expansion to the temperature T2, must be equal to the heat liberated during condensation, i.e., H' = q^2; (22) where 52 is the dryness and r2 the heat of vaporization corre- sponding to the temperature T2' Equating the right-hand mem- bers of equations (21) and (22), we obtain q2r2 = qiri jT + ^2 log ^r ; from which «-5fe+'»«S) (») Equation (23) enables us to compute the dryness, during resisted adiabatic expansion, provided the initial dryness be known. In the next chapter, the relation expressed in equation (23) will be deduced by a much simpler and shorter method. CHAPTER X ENTROPY 106. It is obvious that a substance, in going from one isotherm to another, always suffers the same definite change in temperature; and furthermore, that this change in temperature is independent of changes in pressure and volimie. That is, the change in tem- perature in going from one isotherm to another is independent of the path pursued during the change. A good analogue of this is the change in potential a body undergoes in going from a sur- face of potential Vi, to a surface of potential F2, the change in potential, F2— Vi, being independent of the path pursued in bringing about the change. It will now be shown that, in going from one curve to another, both curves representing reversible adiabatic processes, there is some definite constant change. Equation (19) of Art. 48 specifies for reversible adiabatic processes C,dT+pdv=0; (1) from which, by substituting for p its value as obtained from the characteristic equation, pv^RTy and separating the variables, we find C,f+R^=0 (2) By integrating equation (2), between limits, we obtain C.log|-+Blogf=0; (3) i 1 Vl 128 ENTROPY 129 where Ti and vi are, respectively, the temperature and volume before the change, and T and v, respectively, the temperature and volume after the change. Since equation (3) is equal to zero, no matter what the limits of integration, it follows that there is something which does not change during a reversible adiabatic process. Integrating equation (2) for the primitive, we find C.logr+iilogt;=A;; (4) where A; is a constant of integration. Equation (4) shows that the fundamental differential equation for a perfect gas, yields upon integration for a reversible adiabatic process a constant. But since T and v, in equation (4), may have any values whatsoever, pro- vided, always, they are so related that the process is adiabatic, it follows that, no matter what the range, there is some function which remains constant: which conclusion is the same as that drawn from equation (3). Hence, since there is some function which remains constant during a reversible adiabatic change, there must be some definite constant change in going from one curve, repre- senting a reversible adiabatic process, to another curve, represent- ing a reversible adiabatic process. In equation (4), the constant k evidently represents some particular condition for the gas, which remains constant, during an adiabatic process; and its value depends upon the unit of measure and zero chosen. The condition of a gas, as expressed by equation (4), was called by Clausius the entropy of the gas; and, as just stated, the numerical value of the entropy depends upon the units chosen and the arbitrary zero from which it is measured. Equation (4) may now be stated as follows: The entropy of a substance during a reversible adiabatic change remains constant. 106. Change of Entropy. If the left-hand member of equation (1) is not equal to zero, i.e., heat is either added or abstracted 130 THERMODYNAMICS while the gas changes in volume and temperature, the process is no longer adiabatic, and the equation becomes dQ=C,dT+pdv; from which, by substituting for p its value as obtained from the characteristic equation, we have dQ = C.dT+RT~ (5) V Dividing equation (5) by T, we obtain f-cf +4 m If we represent the entropy of the gas by q), equation (4) becomes *, ^ ■^ ^ w;.- kJ w £ "** >.^ - "^ ■*-« - ■^ ■*«, p • _J ■^ ^ - oo n . g 00 i'5 m z ID ■" > o z o ■0 •< o PI — < + z 30 r gfe •< h^ 1^ ^^ — ■*■ ^ — bi ^ -- ^ - -r '^ ^- :i !? 3 u- , oc ^ "^ ^ ■^ ^ •^ g ^ ' /^ 1 — ^ •'' il - 1 ^ 142 THERMODYNAMICS where r is the heat of vaporization corresponding to the temper- ature T, and q the increment in entropy. Let, now, the initial drjmess, at the temperature Ti, be gi, such that the entropy is represented by the point E, Since the quantity of liquid evaporated, at a given temperature, is directly propor- tional to the quantity of heat added, and the increment in entropy is also directly proportional to the quantity of heat added, it follows that the increment in entropy is directly proportional to the amount of evaporation. Hence, the initial dryness is AE If, now, adiabatic expansion take place, the entropy remains constant, and the vertical line EF represents the relation of tem- perature and entropy. Hence, the dryness corresponding to the temperature T is given by 5=^ (8) ^ mo In a similar manner, provided always the initial dryness be known, the dryness corresponding to any temperature during an adiabatic change, may be found. And it makes no difference whether we are dealing with an expansion or a compression. It is obvious that if a T- 9 curve be plotted to a convenient scale, for water, together with the corresponding saturation curve, in a manner as has just been described, between such temperature limits as are likely to occm: in practice, we may at once, from such a sheet, provided always the initial dryness be known, determine the amount of dryness at any temperature for adiabatic changes. As a matter of convenience, such a sheet is given on page 141. 113. Zero Curve. Assume that various horizontal distances between the curves AD and BC are all divided into the same number of parts, and each part is the same fractional part of the APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 143 total distance. If the points, so found, are connected by smooth curves, as shown in Fig. 20, then the dryness along any particular curve is a constant. For, the horizontal distance between any curve, such as mn, and the curve AD, no matter at what temperature the distance be measured, is always the same fractional part of the total increment in entropy, at that temperature, due to vapor- ization; and therefore, represents the same fractional part of vaporization. If, now, an adiabatic, such as ab, be drawn, it is found to cut the curve mn; i.e., the curve mn passes to the right of the adiabatic ab, as the expansion progresses, and shows that Fig. 20. the steam becomes wetter during adiabatic expansion and drier during adiabatic compression. If, on the other hand, an adiabatic, such as cdj be drawn, it is found that the curve of equal dryness op, as the expansion progresses, passes to the left of it; and shows that the vapor becomes drier during adiabatic expansion and wetter during adiabatic compression. It is thus seen that if the vapor be initially quite dry, it becomes wetter during adiabatic expansion, and if the dryness is very low, it becomes drier when expanded adiabatically. If the adiabatic at any point becomes tangent to the curve of constant dryness, then there is at that point, no change in dryness during adiabatic changes. By finding a nimiber of points on the various curves where the tangent is vertical, and joining these 144 THERMODYNAMICS points by a smooth curve, then at any point to the right of this curve the dryness is decreased by adiabatic expansion, and at any point to the left of this curve, adiabatic expansion increases the dryness. Such a curve is known as the **zero curve"; and for temperatures such as are common in practice, does not lie very far from 50 per cent dryness. 114. Loss of Work Due to Using Steam Non-ezpansively. Assume, in the first place, that no expansion whatsoever is allowed; • but that at the instant of cut-off, the steam is put into contact with the refrigerator, and condensation takes place at constant voliune. Then the work lost, due to using the steam non-expan- sively, is represented by the area BCK, of Fig. 15, for the ideal case discussed in Art. 102. As a matter of fact, during the greater part of the eighteenth century, steam was used in this manner; i.e., the steam was not worked expansively, but immediately after cut-oflF, the steam was condensed by a jet of water, either in the cylinder, or in an adjoining condenser. The result, however, is the same whether the steam be condensed by a jet of water, inmiediately after cut-off, or allowed to escape to a space of lower pressure; for in either case, there is the same gradual diminution of pressure in the cylinder. The conditions which obtain, when steam is used non-expan- sively, are best studied by the aid of the T-(f diagram. In Fig. 21, DA is the T-^ curve for the heating of unit mass of water, AB the curve for evaporation at the temperature Ti, BC the saturation curve, BE the curve for adiabatic expansion, BnF the curve of condensation at constant volume, and FD the condensation curve at constant temperature. The condensation curve BnF is determined as follows: For any temperature T, we have to determine a point n such that mn Q= — ; mp where q is the dryness corresponding to the temperature T. APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 146 The volume of the steam in the cylmder, remaining sensibly constant, since the volume of liquid present is practically negli- gible, we have ««=«i; (9) where a is the volume of unit mass of saturated steam at the tem- perature r, and si the volume originally occupied at the temper- ature T\, 8 and «i, are found from steam tables. From equation (9) we find ?= 8 (10) A T. B L T n^x""'^ ^k 1 D f/ ^ _.__El„Ac T, ■•-— — *^^ Fig. 21. But, as has been previously shown, a relation must subsist, such that «= mp (11) hence, by combining equations (10) and (11), we find mn = m/p — . 8 (12) Finding a number of points in this manner, the curve BnF is determined; and the loss of work, due to using the steam non expansively, is obviously measured by the area BFE, 116. Loss of Work Due to Incomplete Expansion. If there is a partial adiabatic expansion before exhaust or condensation, 146 THERMODYNAMICS then the T-9 diagram takes the form as depicted in Fig. 22. DA is the T-q) curve for the heating of miit mass of water, AB the curve for complete evaporation, at the temi)erature Ti, BC the saturation curve, BG the curve for adiabatic expansion, to the temperature 7", GnF the curve of condensation at constant volimie, and FD the curve of condensation at the temperature 3^2. To determine the curve of condensation at constant voliune, we must find a point n for the temperature T, such that tnn Q= — ; mp (13) A T, E \ L- T' G \ - —I— ^y^ \. y^—Z^ir^ 1 r -^ ' \ 1 / ^ I \ / T, / -" ! \ / / . ^— _— _J A D F b EC Fig. 22. where g is the dryness corresponding to that temperature. Now, the volimie occupied at the point G, corresponding to the tempera- ture r, is gV; where g' is the dryness at the temperature 2", and s' the volume of unit mass of saturated vapor at 7". Also, for constant volume j gs=gV; (14) where q is the dryness at the temperature T, and » the correspond- ing volume for unit mass of saturated vapor. From equation (14), we have .« 3=vr 5 (15) APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 147 Finally, combining equations (13) and (15), we obtain mn=q'—mp (16) Finding a number of points in this manner, the curve OnF is determined. If the steam is initially not dry, the curve GnF is determined in precisely the same manner; but, the curve BG is shifted toward the left by a fractional part of the length AB, depending upon the amount of initial priming. The work lost, due to incomplete expansion, is measured by the area GFE; and, by an inspection of the figure, it becomes obvious that the loss of work decreases very rapidly as the expan- sion is increased. As an example, were the expansion continued up to the point a, the loss of work, due to incomplete expansion, would be measured by the small area abE. The greater the amount of expansion, after cut-off, the longer, necessarily, the stroke of the piston; but, the longer the stroke, other things being equal, the higher the first cost of the engine, and the greater the loss of work due to friction. Hence, there must be a point beyond which it is uneconomical to carry the expansion. Besides increasing the friction, there are still other losses introduced, by carrying the expansion too far; these will be considered later. Just how far to carry the expansion so as to give the best economy is a problem far too complex to be solved theo- retically. At best, theory can only serve as a guide, and the most economical expansion must be determined experimentally. For a simple engine, the point of cut-off may vary from about one-third to one-sixth of the total stroke; dependmg upon whether the engine is running non-condensing or condensing. But, it must always be remembered that the ratio of cut-off to length of stroke depends upon various conditions, which will be better understood after we have dealt with the actual behavior of the steam in passing through the cylinder. 148 THERMODYNAMICS 116. Gain of Work Due to Superheating. If the steam, after being completely evaporated, be superheated, the ideal coefficient of conversion is increased. But, this must not be under- stood to mean the same proportional gain in work; for, lubrica- tion and packing become more difficult as the temperature is increased; and when the temperature becomes very high, radiation becomes excessive. Let, in Fig. 23, DA be the T-(f curve for the heating of unit mass of water, AB the curve for evaporation, BE the curve for superheating, EI the adiabatic expansion curve, and ID the curve of condensation. The curve BE^ for superheating, is found as follows: The thermal capacity of superheated steam, for temperatures such as are found in practice, is approximately constant; hence, we have from which 9 T (p=ciog jt; (17) APPLICATIONS OF TEMPERATURE-ENTROPY DIAGRAMS 149 where 9 is the change m entropy m going from B to E, T^ the temperature to which the steam is superheated, and c the thermal capacity per imit mass of superheated steam. The thermal capacity per unit mass of superheated steam is, as has just been stated, practically constant, and is approximately equal to 0.480. With no superheating, and expansion along the adiabatic BJf the ideal coefficient of conversion is, ,_ ABJD ^ "ABGKD' and, with superheating and adiabatic expansion along £/, we find, for the ideal coefficient of conversion, „^ ABEID ^ ABEFKD' By an inspection of the figure, it becomes obvious that r{'>t{. If the steam be superheated to a temperature, such that the adiabatic EF passes through the point C, then the steam will be just saturated after it has been expanded to the temperat\ire 7^2. To find the amount of superheating that will bring about this condition, it is only necessary to equate entropies, for the points E and C. The change in entropy, in going from Z) to ^, is and the change in entropy, in going from D to C, is ^ '¥2 But, in order that the adiabatic EF pass through the point C, 'p' must equal 9"; hence. 150 THERMODYNAMICS from which log7'.=j^(^^-^^-log^^)+logTi. . . . (18) By means of equation (18), Tg is readily found 117. Double-acting Engine. Up to the present, we have been considering matters as though the engine were only single-acting; i.e., admission and exhaust take place only at one end of the cylinder. In general, however, this is not the case. By having proper valve arrangements, admission and expansion take place in one end of the cylinder while release and exhaust take place in the other end. And, the engine is double-acting; thus prac- tically doubling the capacity of the cylinder and giving a more uniform distribution of the work for a rotation of the fly-wheel. Hence, a single-acting engine, to carry its load properly, requires a fly-wheel of greater inertia than does a double-acting engine. 118. Condensing Engine. By a condensing engine is meant an engine which exhausts to a receptacle of some kind, called a condenser, where the steam is condensed at a comparatively low temperature, and the pressure in the condenser is maintained constant and lower than that of the atmosphere by means of a vacuum pump. In good condensers, the pressure is as low as the equivalent of one inch of mercury. By a non-condensing engine is meant an engine which exhausts directly to the atmos- phere at practically atmospheric pressure. Experience shows that, in general, when the supplied steam has a pressure of 100 lbs., or over, despite the fact that a certain amount of power is consumed in operating the vacumn pump, there is a decided gain in economy, when engines are operated condensing. Hence, in general, condensing engines are employed. CHAPTER XII ELEMENTARY STEAM AND ENGINE TESTS 119. Before proceeding to discuss the actual behavior of the steam as it passes through the cylinder of an engine, a brief description will be given of the methods pursued in determining the dryness of steam. It is obvious, from the discussions in the preceding chapter, that one of the essentials in studying the per- formance of a steam engine is a knowledge of the condition of the supplied steam. But, aside from the temperature-entropy diagram, if we wish to plot an adiabatic for steam on the p-v diagram, we must know the initial dryness. According to calculations by Zeuner, the equation ?w"=A:, (1) where A; is a constant, may be used for adiabatic changes. But the value of n depends upon the initial dryness. This value of n is given by the empirical equation w = 1.035+0.1g; (2) where q is the initial dr3mess. Equation (2) holds for all values between 70 per cent and 100 per cent dryness. Hence, if an adiabatic for steam is to be plotted, that value of n must be used in equation (1), which is found by means of equation (2), for the given initial dryness. 120. Throttling Calorimeter. It was shown, in Art. 103, that steam becomes drier, during unresisted adiabatic expansion; and furthermore, under proper conditions, if the initial priming be low, the steam may become superheated. Depending upon 151 152 THERMODYNAMICS J v\ 0O. this principle. Professor Peabody designed a calorimeter, by means of which the dr3mess of steam, provided the priming is low, may be determined. In Fig. 24, & represents the supply pipe, and A a vessel into which the steam expands; the rate of inflow being regulated by the valve Fi, and the outflow by the valve 72. G\ is a gauge, indicating the pressure of the steam in the supply pipe, G2 a second gauge, indicating the pressure of ^the steam in the vessel A, and m a thermometer, indicating its temperature. The valves Y\ and F2 are so regulated that the pressure in the vessel A is always con- m -\r siderably less than the pressure in the supply pipe. The vessel A is either well lagged with some non-conducting material, or else highly polished, to reduce radiation to a minimum. After the flow of steam has continued for some time, steady conditions will obtain; and, if the priming of the x| ^1 steam in the supply pipe is low, super- Pjq 24. heating will take place in the vessel A, and the thermometer m will register a temperature higher than that corresponding to saturated steam, imder a pressure as registered by the gauge G2. Let Ti be the temperature of saturated steam corresponding to the pressure indicated by the gauge G2, and T2 the tempera- ture registered by the thermometer; then T2— ti is the amoimt of superheating. The total heat of unit mass of steam then, in the vessel il, is H'=Ai+ri+c('C2-Ti); (3) where W is the total heat, Ai the heat of the liquid, corresponding to the temperature ti, r\ the heat of vaporization for this tern- ELEMENTARY STEAM AND ENGINE TESTS 153 perature, and c the thermal capacity per unit mass for superheated steam. The total heat, per miit mass, for the steam in the supply pipe, is fl"=A+gr; (4) where H" is the total heat, ft the heat of the liquid for the tem- peratiu'e t, which is the temperature of saturated steam for the pressure as registered by the gauge Gi, r the heat of vaporization for this temperature, and q the dryness of the steam in the supply pipe. For an adiabatic flow, however, the total heat for the two conditions is the same; hence, the right-hand mem- bers of equations (3) and (4) are equal, and we have A+gr=fti+ri+c(T2-Ti); from which ^ fti+ri+c(T2-Ti)~ft . . ff= (5) r In order to obtain reliable results, by the method just described, the two gauges Gi and (?2 must be accurately calibrated. On the other hand, a slight error in the thermometer does not appreciably alter the result; since the amount of superheating is necessarily small, the quantity of heat involved is small in comparison with the other quantities. But, (he thermometer mvM he of sufficient accuracy so that we may be assured that there is superheaiing. The amount of moisture that may be removed by throttling depends, of course, upon the difference between the pressure of the steam in the supply pipe and the pressure in the chamber into which it expands. If the pressure in the chamber is equal to that of the atmos|phere, and the pressure in the supply pipe is 100 lbs. per square inch, then the dryness must be about 96 per cent so that all the moisture may be removed by throttling. If the pressure in the chamber be reduced by means of a con- denser, a greater amount of moisture may be removed. If the 154 THERMODYNAMICS initial pressure of the steam be 150 lbs. per square inch, and the pressure in the chamber is atmospheric, then about 5 per cent priming may be removed by throttling. 121. Condensing Calorimeter. The dryness of steam may also be determined by condensation. There are various methods which may be pursued; one is to have a vessel partially filled with water at some low temperature, and passing steam into this until some convenient rise in temperature has been attained. The quantities of water and steam are determined directly by weighing, the initial and final temperatures of the vessel are read from a thermometer immersed in it, and the initial temper- ature of the supplied steam is determined from the pressure, as indicated by means of a gauge attached to the supply pipe. Let M be the water equivalent of the vessel and contents, m the mass of the condensed steam, and ti and T2, respectively, the initial and final temperatures of the vessel and contents, then if g is the initial dryness and t the corresponding temperature of the supplied steam, we have Af(T2— Ti)+mA2 = w{jr+mA; (6) where A2 is the heat of the liquid corresponding to the temperature T2, r and A, respectively, the heat of vaporization and the heat of the liquid, corresponding to the temperature t. From equa- tion (6) we find Jf(T 2-Ti)-m(ft-ft2) ,yv Equation (7) was deduced on the assumption that there is no radiation during the progress of the experiment. A correction for radiation may, however, be applied by taking a curve of cooling for the vessel. Due to the fact, that the mass of the condensed steam is determined by a difference in weighing, and that this mass is necessarily small in comparison with the mass of liquid initially contained in the vessel, a serious error may be introduced by an inaccuracy in weighing. ELEMENTARY STEAM AND ENGINE TESTS 155 A better method than the one just described, is that of passing the steam through a spiral tube, contained in a condenser through which there is maintained a continuous flow of water, in a manner such that the steam is completely condensed and reduced in temperature to that of the outflowing condensing water. After steady conditions obtain, the mass of steam con- densed, during a given interval of time, is determined by col- lecting it in a suitable vessel; and in a similar manner, by collecting in a separate vessel, the mass of water which passes through the condenser, during the same interval of time, is determined. Knowing the initial and final temperatures of the condensing water, together with its mass, the mass of the con- densed steam and its initial and final temperatures, then the dryness of the steam is determinate. Let M and m, respectively, be the mass of the condensing water and condensed steam for the same interval of time, ti and T2, respectively, the temperature of the condensing water for inflow and outflow, and t the initial temperature of the supplied steam, then, since the temperature of the outflow and that of the condensed steam is the safne, we have Af(T2--Ti)+m/i2=»w^+wiA; (8) where h2 is the heat of the liquid corresponding to the temper- ature T2, r and A, respectively, the heat of vaporization and the heat of the liquid corresponding to the temperature t, and q the dryness. From equation (8), we find M(T2~Tl)-in(ft-/t2 ) .... (9) mr Equation (9) was deduced on the assmnption that there is no radiation during the progress of the experiment. If the tem- perature of the vessel differs materially from that of the siu*- roimdings, corrections for radiation must be applied for this difference. Radiation may, however, be completely eliminated 156 THERMODYNAMICS by regulating the inflow such that the vessel and contents are continuously at room temperature. 122. Separating Calorimeter. Professor Carpenter devised an apparatus by means of which the moisture, present in the steam, is removed mechanically. The steam is passed from the supply pipe into a chamber, where it strikes against a convex surface, surrounded by a wire mesh, through which the escaping steam must pass. When the steam strikes the cup, the water present is separated, passes through the mesh, and is collected in the chamber; the dry satm^ated steam, meanwhile, passes into an outer jacket, which surroimds the chamber, and escapes from an orifice at the bottom, where it is condensed and collected. The quantity of water collected in the chamber is read directly from a glass gauge, which has been previously calibrated; and the quantity of steam which passes through the calorimeter is deter- mined by condensing and weighing. In making a determination, the valve in the pipe supplying steam to the chamber is opened, and when steady conditions obtain, a reading is taken on the glass gauge, and simultaneously, the exhaust pipe is passed into the condenser. When the oper- ation has been continued for a suflScient interval of time, the gauge is again read, and at the same instant, the exhaust pipe is removed from the condenser. The mass of steam, passing through the apparatus for the given interval of time, is found directly by the difference in weight of the condenser for final and initial conditions. And this mass compared with the sum of the two masses, i.e., the mass of the condensed steam and the mass of the water collected in the chamber, gives the dryness. One of the inherent difficulties common to all m£thod8, in deter- mining the dryness of steam^ lies in obtaining a sample which is a fair average of the steam supplied to an engine. 123. Clearance. The volimie swept out by the piston of an engine, during a stroke, is equal to the product of the area of the piston and length of its stroke. The voliune between the ELEMENTARY STEAM AND ENGINE TESTS 157 pLston and cylinder head at the end of the stroke, plus the \roliime of the supply and exhaust-passages leading to the admis- sion and exhaust-valves, is called the clearance. The clearance then is, that part of the volume through which the piston does not sweep, and is readily found by closing the valves and determining the volmne of water required to fill the space when the piston is at the end of its stroke. A convenient way of expressing the clearance of an engine is by a ratio; i.e., the ratio of the volume of the clearance, to the volume of piston displace- ment plus volume of clearance. The ratio of the clearance volume to the piston area gives the equivalent length of clearance. The clearance of different engines varies considerably, depend- ing upon the size of engine and type of valves used; and, other things being equal, the clearance for small engines is relatively larger than it is for large ones. In practice, depending on the type of engine, the clearance may vary from 2 per cent to 10 per cent. 124. Cushion Steam and Cylinder Feed. The mass of steam which remains in the clearance-space at the end of the exhaust- stroke, depends upon the time of closing of the exhaust-valve. Thus, if the exhaust-valve does not close xmtil the exhaust- stroke has been completed, then the pressure of the steam, remaining in the cylinder, is the same as that of the condenser, and the mass of the steam is equal to the product of the clearance- volume and the density of the steam. If, on the other hand, the exhaust-valve closes before the exhaust-stroke has been com- pleted, then the pressure of the steam, remaining in the cylinder at the end of the exhaust-stroke, will be higher than that existing in the condenser; hence, in this case, the mass of steam remaining in the cylinder is greater than that for a later closing of the exhaust-valve. The steam remainmg in the cylinder, at the end of the exhaust-stroke, is called the cushion steam; and the steam drawn from the boiler, per stroke, is called the cylinder feed. ' 158 . THERMODYNAMICS During expansion, both quantities are present, whereas, during compression, the cushion steam alone is present. 126. Wire Drawing. If the exhaust-valve closes late, the pressure of the cushion steam is less than that of the steam in the supply pipe, and a certain quantity of steam must pass into the cylinder, during each stroke, before the maximum pressure is reached. The entering steam, therefore, does not do as great an amount of work on the piston as it would do if the cushion steam had been compressed to the pressure of the incoming steam; in other words, it is a case of imperfectly resisted expansion. It is true that the incoming steam, if dry, becomes superheated, and if partially wet becomes drier, due to the partially unresisted expansion; but the pressure being lower, the heat which is evolved when the eddy currents subside, is applied at a lower temperature, and therefore, the imperfectly resisted expansion constitutes a thermodynamic drop. If the exhaust-valve closes at the proper time, then the pressure of the cushion steam is equal to the pressure of the incoming steam, and the thermodynamic drop, so f a(r as this part of the action of the engine is concerned, is avoided. Fur- thermore, the work which is done on the cushion steam, in compressing it from the condenser pressure to that of the incom- ing steam, is precisely equal to the work done by it, in expanding between the same limits of pressure; hence, there is no loss of work involved due to compression. There are, however, other unavoid- able losses. The piston advances rapidly, calling for a large sup- ply of steam, and the admission-valve does not open instantan- eously, but requires a definite time interval. Hence, due to the resistance offered to the flow of steam, by the supply passages and valves, there is a certain amoimt of throttling, the sameas when the cushion steam is at a pressure lower than that of the supplied steam; causing the pressure in the cylinder, during admission, to be less than that existing in the supply pip)e. And furthermore, in general, due to throttling, the pressure in the ELEMENTARY STEAM AND ENGINE TESTS 159 cylinder gradually decreases as the admission advances. The result of these combined causes, due to which the pressure in the cylinder during admission is lower than that of the supply pipe, or boiler, is known as mre dravnngy and constitutes a thermo- dynamic drop. There is also a loss of work during the exhaust-stroke, due to the fact that the exhaust-passages and valves offer a resistance to the flow of steam, which makes the pressure in the cylinder, during exhaust, always higher than that of the condenser. 126. The Indicator. One of the most important and, at the same time, one of the most delicate pieces of apparatus used in engine testing is the indicator. The indicator consists essentially of two parts; the first part being a small piston P fitted accurately into a cylinder and controlled by a helical spring S. The spring may be either inside of the cylinder or, as shown diagrammatically in Fig. 25, outside. The type of indicators having the spring above the cylinder are more convenient; and furthermore, since the springs, in this form, are not subjected to the same fluctua- tions of temperature, the results obtained are more satisfactory. The cylinder of the indicator, by means of a short supply pipe containing a cock, is tapped onto the cylinder of the engine, over the clearance space, in a manner such that the steam in the engine cylinder exerts its full pressure against the piston of the indicator throughout the entire cycle. If the indicator pis- ton moves freely, i.e., without appreciable friction, and the spring obeys Hooke's Law, then the movement of the piston will be proportional to the fluctuations of the pressure in the cylinder of the engine. To magnify the motion of the piston of the indi- cator, the end of its piston rod is connected, by means of a system of links, to a lever, in a manner such that a pencil point p, carried by the end of this lever describes, between the limits of travel, practically a right line. The springs are accurately calibrated to a definite scale with respect to the motion of the pencil point. Thus, if a spring is a 60 Jb, spring, it means that the pencil 160 THERMODYNAMICS point moves over a distance of 1 inch for a change in pressure, on the piston, of 60 lbs. per square inch; and a distance of 0.75 inches for a change in pressure of 45 lbs. per square inch, etc. The second part of the indicator consists of a drum JD, controlled by a spring, upon which the indicator card is wrapped. The drum has wrapped around its lower part a cord C, which in m^ m «a T> i^ ^1 00 Fig. 25. turn is connected by means of some mechanism, to the cross- head of the engine, in a manner such that the angular displace- ment of the drum is proportional to the linear displacement of the piston of the engine. The linear displacement of the surface of the drum, however, is less than that of the cross-head; i.e., the motion is reduced by the mechanism through which the ELEMENTARY STEAM AND ENGINE TESTS 161 cord, operating the drum, is connected to the cross-head. From the foregoing, it is obvious that, if the drum is stationary, and the cock in the supply pipe between the cylinder of the engine and the cylinder of the indicator is open, the pencil point traces a straight line on the indicator card. On the other hand, if the stop-cock is closed and the drum is in motion, the pencil traces a straight line at right angles to the former. This line is the atmospheric line, since the stop-cock is so arranged that when the steam is cut ofif from the indicator cylinder, a vent opens, allowing free access of the atmosphere to the space below the piston of the indicator. If, however, the stop-cock is open, and the driun is moving in unison with the piston of the engine, the position of the pencil point of the indicator, at any part of the cycle, is a measure of the pressure and volume, of the working substance, for that instant. Hence, during a cycle, the pencil point traces out a diagram, which shows to a reduced scale, as regards pressure and volume, the condition of the working sub- stance, for every part of the cycle. The diagram so traced, is the actual indicator diagram of the engine. 127. Indicator Diagram and Valve Adjustment. By means of the indicator diagram, the behavior of the working substance may be conveniently studied for the entire cycle; and further- more, we are enabled by it to judge, whether or not, the valves are properly adjusted, which is very important; since any faulty valve adjustment may seriously affect the eflSciency of the engine. Also, as will be shown in this chapter, by means of the indi- cator diagram, we are enabled to determine the power delivered, by the working substance, to the engine; hence, if the power delivered by the engine be known, the eflSciency of the engine, as a mechanical contrivance, is immediately determined. Fig. 26 is a reproduction of an indicator diagram taken from one end of the cylinder of a 40-H.P. engine, making 300 r.p.m.; the engine working non-condensing; i.e., exhausting to the atmos- phere. Fig. 27 is the indicator diagram for the same end of 162 THERMODYNAMICS the cylinder when the engme was exhausting to a surface con- denser; a partial vacuum being maintained by a pump. AB is the admission line, BC is the expansion line, C being the point where the exhaust-valve begins to open, and D the JH A I A .V- \ a ^^ E ~A. I » G Fig. 26. —J point where it is fully open; DE m the exhaust line, and at the point E compression begins. Just how far the compression will be carried before the admission-valve opens depends upon the set of the valves. In the diagrams here shown, the admission- FiG. 27. valve opened at the point a. The various points being illy de- fined is due to the time element involved in the opening and closing of the valves. The line OH is the line of zero volume, and is found by taking a distance, to the proj)er scale, to the left of FAf representing the equivalent length of the clearance. The ELEMENTARY STEAM AND ENGINE TESTS 163 line of zero pressure, or vacuum line OJy is found by measuring down from the atmospheric line FG, a distance representing the atmospheric pressure at the time the diagram was taken. Finally, HI shows, as registered by the gauge, the steam pressure in the supply pipe. 128. Comparison of Theoretical and Actual Curves. If it be desired to compare the expansion or compression curve, with an isotherm or adiabatic, a point on the curve is chosen, preferably about the middle, and the theoretical curve is made to pass through this point. As a matter of convenience, the following method for plotting curves is here given. To plot the curve whose equation is we proceed as follows: In Fig. 28, OA is the line of zero volmne, OB the line of zero pressure, and a a point on the curve. Lay off the line OC^ making an angle g with' the line OA, and the line ODy making an angle a with the line OB, such that l+tanp = (l+tana)" (10) Draw ad parallel to OB, and dh making an angle of 45° with OA. Now draw af parallel to OA, and through /, Jq making an angle of 45° with OB] then the point 6, which is the intersection of the line W), parallel to OS, with the line gh, parallel to OA, is a point on the curve. For, if we represent, for the point a, the pres- sure and volume respectively, by pi and v\, and similarly for the point 6, by p2 and V2, we have Pi=P2+P2tan p=P2(l+tan P); . • . . (11) and vi+vi tan a=r2; from which t;i"(l+tana)"=t;2'* (12) 164 THERMODYNAMICS Multiplying equations (11) and (12), member by member, we obtain PiVi'*(l+tana)^=P2i'2''(l+tanp); . . . (13) but, by construction, as stated by equation (10), (l+tana)" = l+tang; hence, equation (13) reduces to and 6 is a point on the curve. In a similar manner the points c and i are found, etc. Fig. 28. The value of n to be used, if we are dealing with steam, is found by means of Zeuner's equation, which is equation (2) of Art. 119. And if we wish to plot an isotherm, n in equation (10) is made unity, thus making the angles a and ^ equal. In drawing the lines OD and OC, some convenient value for a, say 15° to 20° is assumed, and the value of ^ is found by means of equation (10). In general, the expansion and compression curves, obtained by means of an indicator, for heat motors and compressors conform very closely to the equation ELEMENTARY STEAM AND ENGINE TESTS 165 where k and n are constants for a particular pro^ss. The value of 71; depending upon the nature of the working substance and the condition of operation, may lie anywhere between unity and 1.4. To find the value of n from the indicator diagram we may proceed as follows: The pressures and volumes corresponding to the points a and 6 (Fig. 28) are determined by means of a scale, and then, from the equation, a value for n is found. Similarly, values of n are found for a number of points along the curve; and these values will, in general, agree closely among themselves. And the mean of the values so found, compared with the ratio of Cp to Cv, for the given substance, is an indication of how closely the curve, under con- sideration, approaches an adiabatic. A method for finding the value of n from the indicator card, which has been found to give satisfactory results and is less labor- ious than the one just described, is as follows: Lay oflf the line OC (Fig. 28), making an angle ^ with the line OA; then choose a point on the curve, such as a, and draw the line ad parallel to OB, and through the point d, a line making an angle of 45** with the line OA, and cutting the line OC at the point h. Through A, now, a second line is drawn, parallel to the line OB, which cuts the curve at some point 6; two lines, now, parallel to the line OA, one through a and the other through 6, are drawn, and through g, where the line through 6 cuts the line OB, a line is drawn making an angle of 45° with the line OB and cutting the line, through a, at some point /. Now, this point / lies on a line OD, making some angle a with the line OB, Proceeding in this manner, a nmnber of points, for the line OD, are found which will lie very nearly in a straight line. Drawing a mean line through the points so found, the angle a is determined. And by substituting for ^ and a, in equation (10), n is found. 166 THERMODYNAMICS It will be noticed that in this case the angle a is determined from the angle ^ and the curve mider consideration; whereas, in the construction first ^ven in this article the curve is deter- mined by means of n and the angles a and ^. 129. Behavior of Steam throughout the Cycle. When an engine is first started, the cylinder walls are, of course, at a temperature much lower than that of the steam, and conden- sation takes place during admission, expansion, and exhaust. After a time, however, permanent cyclic conditions will obtain; i.e., regular periodical fluctuations will have been established, and each cycle, so far as practical conditions permit, will be an exact reproduction of the cycles preceding. When these permanent cyclic fluctuations have been estabUshed the incoming steam, during admission, comes into intimate contact with the cylinder walls, which are at a lower temperature, due to the cooling of the lower pressure exhaust steam which has been in contact, just immediately preceding, and condensation takes place. It is true, that, due to wire drawing, a certain amount of drying takes place; but, imless the supplied steam has been super- heated, considerable condensation will take place during admis- sion, and will continue during part of the expansion-stroke; and may, in some rare cases, continue throughout the whole of the expansion-stroke. In general, however, during expansion, some point is reached when the temperature of the steam falls below that of the cylinder walk, and reevaporation takes place; i.e., a certain quantity of heat is abstracted, from the steam, by the cylinder walls, during the earlier part of the stroke, and a certain quantity of heat is abstracted, from the cylinder walls, by the steam, during the latter part of the stroke. Even if the quantity of heat abstracted from the cylinder walls were equal to the quantity of heat given up to them, which is never the case, there would still be a thermodynamic loss; for the heat abstracted from the walls is applied at a lower temperature than that absorbed by the walls. The heat abstracted from ELEMENTARY STEAM AND ENGINE TESTS 167 the cylinder walls, to bring about reevaporation, during exhaust, is completely lost, since it is all rejected to the condenser.* It is found, by experiment, that the exchange of heat between metal surfaces and perfectly dry gases, is very small even for considerable differences of temperature; hence, the conclusion that, the rapid exchange of heat between the steam and the cylinder walls, in a steam engine, is due to a film of conducting moisture which collects on the surface of the cylinder walls. 130. Change of Dryness during Expansion. To determine the dryness during the expansion-stroke, it is necessary to know the cylinder feed and cushion steam. To determine the cyhnder feed, the exhaust steam, for a given interval of time, is condensed and weighed; and for the same interval of time, the number of working strokes made by the engine, is determined. From this, the mass of steam per stroke, i.e., the cylinder feed is foxmd. The cushion steam is found directly from the indicator diagram. Let, in Fig. 29, ABCD be the actual indicator diagram; OE and OF, respectively, the axes of zero volume and zero pressm-e. FiQ. 29. determined as described in Art. 127, and D the point where the exhaust-valve has been completely closed and compression begins. If the assmuption be now made that, at the point D, the steam is saturated, no serious error is introduced; for, since * For a comprehensive discussion of the influence of cylinder waUs, see " Thermodynamics of the Steam Engine," by C. H. Peabody. 168 THERMODYNAMICS the mass of the cushion steam is always small in comparison with the total mass of steam present, during the expansion, a small error made in determining the cushion steam, will not appreciably aflfect the saturation curve. On the assumption then, that at the point D the steam is saturated, the mass of cushion steam is readily found, by means of steam tables; since its volume and pressure are given by the diagram. Taking the sum of the cylinder feed and cushion steam, we have the total mass of steam and water present during expansion; and from this, the saturation curve GH may be plotted. That is, the curve GH gives the volumes, for the various pressures, the steam would have occupied had it been completely saturated. If then, at any pressure such as 01, the horizontal line IK be drawn, the dryness for that pressure is at once found by the rela- tion IJ 131. Exchange of Heat, during Expansion, between Steam and Cylinder Walls. If we plot, from the p-v diagram, a T- 9 diagram, which is easily done with the aid of steam tables, the transfer of heat, during expansion, between the steam and cylinder walls, is readily found. In the r-9 diagram. Fig. 30, CD is the saturation curve, AC is drawn at a temperature corresponding to the pressure at cut-oflf, and the point B is so located that the dryness g, for the point of cut-oflf, as found from the j>-v diagram (Fig. 29), is given by AB Taking in this manner the dryness, for various pressiu-es, on the p-v diagram, and transferring to the T- 9 diagram, the curve of dryness Bnu is foimd. The curve ux is the curve of conden- sation at constant volume, and is foimd as described in Art. 115. ELEMENTARY STEAM AND ENGINE TESTS 169 The point n, where the vertical line pn becomes tangent to the curve Bnu, is the point of mininiiini dr3niess, and is given by 9'= tnn mo' It is obvious from the diagram that, during the expansion, up to the point n, the steam is giving up heat to the cylinder walls; and during the remainder of the expansion-stroke, heat is abstracted by the steam, from the cylinder walls. Since the area under the curve is a measure of the heat abstracted, or rejected, it follows that, during the expansion, the heat given Fig. 30. to the cylinder walls is to that taken from them as the area pnBq is to the area pniU, As previously explained, the pressure in the cylinder during admission, due to wire drawing, is less than the pressure in the supply pipe; and as stated in Art. 125, superheating may occur. In general, however, on account of initial priming, even if there were no condensation during admission, due to the incoming steam coming into contact with the cylinder walls of lower temperature, there would still be present a certain amount of moisture. Most authors assume, in discussing the exchange of heat between the steam and cylinder walls, that the steam is 170 THERMODYNAMICS dry at cut-oflF. This assumption is neither justifiable nor nec- essary. No prediction can be made unless the dryness of the supplied steam is known. If, however, the .dryness of the steam in the supply pipe is known, together with its pressure and the pressure in the cylinder, during admission, the dryness of the steam in the cylinder, during admission, had there been no con- densationj is readily computed. Let this hypothetical dryness be represented on the T- (p diagram (Fig. 30) by Aa «' 'AC- Since, however, the actual dryness at cut-off, as found from the indicator diagram, is AB it follows that an amount of condensation, represented by the change in entropy Ba, has taken place during admission. Hence, the heat given up to the cylinder walls by the steam, during admis- sion, is measured by the area Babq, Heat is also given to the cylinder walls dining compression; this, however, is not entirely lost. Since, due to this, the tem- perature of the walls is raised, and the condensation during admission, is partially reduced. 132. Steam Jackets. The fluctuations in temperature of the cylinder walls, as described in Art. 129, are the more pronoimced the lower the speed of the engine. In other words, the higher the speed of the engine, the smaller the interval of time during which exchanges can take place between the cylinder walls and the steam, and as the speed becomes very high the exchange becomes very small. There is, however, another element to be considered, viz, the cooling of the cylinder, due to the fact that it is always at a higher temperature than the surroimdings. This loss of heat must continually be made up by the incoming ELEMENTARY STEAM AND ENGINE TESTS 171 steam; and hence, increases the condensation. This loss of heat is partially prevented by having the cylinder jacketed by some non-conducting material. In some cases, a steam-jacket is used, which is maintained full of live steam, taken directly from the supply pipe; and therefore, the pressiu-e of the steam, in the jacket is usually slightly higher than the pressure of the steam, during admission, in the cylinder. There is, therefore, less condensation, dining admission, than there would be were the steam jacket absent; and reevaporation begins earlier. On the other hand, the jacket increases the area of the exposed surface; hence, a greater loss of heat, due to radiation. If complete reevaporation takes place before the exhaust-valve opens, the steam during the exhaust-stroke is dry, and very little heat is absorbed by it from the steam in the jacket. The question then is, whether the thermodynamic gain, obtained by applying the heat at a higher temperature, to bring about reevaporation at the earlier part of the stroke, is greater than the energy lost, in the jacket steam, to bring about this reevaporation, plus the greater radiation and heat imparted to the exhaust steam. This question can be answered only by experiment. Experiments performed, on slow and moderate-speed engines, appear to indicate a decided gain in economy, by using a steam-jacket. In a great many cases, however, such discrepant results have been obtained, that it is extremely difficult to say imder just what conditions steam jackets are beneficial. 133. Brake Power. The output of an engine of low power, is most conveniently measured by a friction brake, which is a device by means of which the power, developed by the engine, is absorbed in overcoming the friction applied to the surface of its fly-wheel; the force required to prevent rotation of the brake, being measured by a balance. The most common form assumed by the friction* brake is depicted in Fig. 31. It consists of a number of wooden blocks fastened by means of bolts, to steel bands, wrapping, approx- 172 THERMODYNAMICS imately, two-thirds of the circumference of the fly-%heel. The wing-nut w on the bolt b makes it possible to vary the pressure to any desired value. The tie-rod t, going from the lower part of the bolt b to the lever, is merely to give rigidity to the brake. The rim of the fly-wheel is provided with flanges, so that water may be contained in it, to absorb the heat developed by the work done, in overcoming the friction. Assmne, now, that the fly-wheel is rotating in the direction as indicated by the arrow. Then, due to friction, the brake will tend to rotate in the same direction; and to prevent this, a certain force is applied to the lever, at the point p. This force Fio. 31. is most conveniently measured by a balance; which may be either a spring balance or a beam balance. Let the fly-wheel be making N r.p.m. (rotations per minute), the net weight registered by the balance, to prevent rotation, be W lbs., and d be the horizontal distance between the center of the shaft and point of contact p. Then, since power, is numerically equal to the product of angular velocity and torque, we have, employing the minute as the unit of time, P = 2T:NWd ft.-lbs. per min.; and since one horse-power is the equivalent of doing work at ELEMENTARY STEAM AND ENGINE TESTS 173 the rate of 33,000 ft.-lbs. per minute, we have, for the brake horse-power, ^•^•^•"337000" ^^^^ In the case of very small units, the torque is frequently meas- ured by wrapping a canvas belt aroimd the pulley, and applying tensions to its two free ends. The tensions are then varied, until the machine is loaded to the desired amount, and measured. The torque is then found by taking the product of the difference between the two tensions and the radius of pulley plus one-half the thickness of the belt. In this case, the heat developed by the work done, in overcoming the friction, is also absorbed by water contained in the pulley. When testing high-power machines, it is neither convenient nor desirable to make friction tests. One method used is that of connecting the engine under test to an electric generator, whose efficiency is known, and by means of an ammeter and voltmeter, or else by a wattmeter, determining its output. From the efficiency of the generator and the power delivered by it, the power delivered to it, by the engine, is readily foimd. Another method for determining the power delivered by an engine, is to make the shaft, through which the power is being transmitted, take the place of a transmission dynamometer. This is accomplished by determining the amount of twist, which a definite length of the shaft experiences, when transmitting the given power. Then, from the length and diameter of shaft, its modulus of rigidity, and the angle of torsion, the torque is readily foimd. 134. Indicated Power. The power expended on the piston of an engine, by the working substance, as found by means of the indicator diagram, is called the indicated power. During admis- sion and expansion, work is being done by the working substance on the piston; and during exhaust and compression, work is being done by the piston, on the working substance. Hence, the net work done by the working substance, during a cycle, is measured 174 THERMODYNAMICS by the area enclosed by the indicator diagram. If then, the area of the indicator diagram be determined and divided by the length of the stroke, reduced to the proper scale, the average ordinate is foimd. The average ordinate, so found, multiplied by the scale of the spring, used in taking the diagram, gives the mean effective pressure. The area of the diagram is most conveniently foxmd by means of a planimeter. There are certain types of planimeters, which are specially designed for determining the mean effective pressure from an indicator diagram. This type of planimeter is very convenient, inasmuch as it is only necessary to set it to the length of the diagram, employing a scale corresponding to the scale of the spring, used in taking the indicator diagram, and following the outline of the diagram with the tracing point of the instru- ment. The mean effective pressure is then given directly by the reading on the scale. The mean effective pressure is the average pressure on the piston, during admission and expansion, minus the average pressure during exhaust and compression; hence, it is the effective pressure, due to which external work is obtained. If the indica- tor spring has been calibrated to lbs. per square inch, then the mean effective pressure is also given in lbs. per square inch; and the total effective pressure on the piston is numerically equal to the product of the mean effective pressure and the area, expressed in square inches, of the piston. If we represent by P, the mean effective pressure, in lbs. per square inch, by A the area of the piston, in square inches, by L the length of the stroke in feet, and by N the number of cycles per minute, then the net work done on the piston, per minute, is W=PALN ft.-lbs.; and the indicated horse-power is ^•'^'^- " 33;000 ^^^^ ELEMENTARY STEAM AND ENGINE TESTS 175 136. Mechanical Efficiency. The indicated power of an engine is always greater than the power delivered by the engine, by an amount which is equal to the power consumed in overcoming the engine friction. The ratio of the brake horse-power, to the indi- cated horse-poiDer gives the mechanical efficiency; i.e., ^•»=m- (i«) 136. Thermal Efficiency. The thermal effi/Aency of an engine is given by the ratio of the power delivered by the engine to the power due to the heat taken from the source. As an example, assiune a steam engine to be taking M poimds of steam per minute from a boiler, the total heat of which, per poimd, is H, Let the heat of the water in the condenser be A, which we will assume is returned to the boiler without losses. Then the heat, expressed in mechan- ical units, which is taken per minute from the boiler, is ,JM{H-h) ft.-lbs.; and if W represents the number of ft.-lbs. of work delivered per mmute by the engine, then the thermal efficiency is W We will now illustrate equation (17) by a numerical example. Assume an engine making 300 r.p.m., doing work against a friction brake whose arm is 5 ft., and which requires a force of 135 lbs., applied at its end, to prevent rotation. If the engine is consiuning 16 poimds of saturated steam per minute, under a pressure of 80 lbs., and returns the water without losses directly to the boiler, from the condenser, where the pressure is 2 lbs., what is the thermal efficiency? Substituting, in equation (17), we find ^ 2xX300X 135X5 ^ ' . ^^ = 778X16(11 82^94:2) = ^'^ ^'' ^"^*' 176 THERMODYNAMICS where 1182* is the total heat of steam under a pressure of 80 lbs., and 94.2 the heat of the liquid, corresponding to the temper- ature of the steam, under 2 lbs. pressure. 137. Commercial Efficiency. The commercial efficiency of an engine is given by the ratio of the power delivered by the engine to the power which a perfect heat engine, working between the same temperature limits, would deliver. Let the symbols have the same significance as in Art 136, then the work, per minute, which a perfect heat engine would deliver, is JM{H-h)—^^ ft.-lbs.; and the commercial efficiency is W ^^= CO (18) JM{H-h)'^~~ Substituting in equation (18), the numerical data given as an illustration in the preceding article, we find ^ 2xX300Xl35X5 Ee = TK^ = «iy -U per cent. 778X16(1182-94.2)^ This is the proper method of comparison; i.e., comparing the actual performance of the engine with an ideally perfect engine, operating between the same temperature limits. When an engine exhausts to the atmosphere there is, of course, no heat returned to the boiler by means of the condensed steam, and the heat h, in equations (17) and (18), is lost. It is, however, not proper to charge this entire loss of heat against the engine; since, by proper arrangements part of the heat at least, contained by the liquid, can be returned to the boiler. There are other methods for rating the performance of engines, which are in certain cases, very convenient. One is, specifying * Taken from Peabody's Steam Tables. ELEMENTARY STEAM AND ENGINE TESTS 177 the number of pounds of steam per B.H.P. hour, consumed by the engine. Another is, specifying the number of B.T.U. per B.H.P. hxmry or the number of B.T.U. per K.W. hxmr of energy delivered to the bus-bar. The latter is especially expressive; giving, as it does, the rating of the power plant as a whole. 138. Rankine's Cycle. Another important comparison may be made by the aid of Rankine's cycle, the indicator diagram of which is shown in Fig. 32. This indicator diagram is based on the assumption that the cylinder of the steam engine has no clearance and is perfectly insulated. AB represents the admis- sion at constant pressure pi, BC represents the adiabatic expan- sion to the pressure p2, and CD represents the exhaust, at constant pressure p2. Assume now, that we are dealing with a unit mass of liquid, whose specific volume is a, and that the dryness, during admission, is q\. If the specific volume of the steam, at the pressure pi, is «i, then the volume of the mixture, at the point of cut-oflf, is t;i = gisi+(l— gi)a=gi(si — a)+a=qiiJLi+(j; . (19) where {jli is the increment in volume due to complete evaporation at the pressure pi. Since the pressure, during admission, is con- stant, the work done by the steam, on the piston, is pit;i = pi(gi[J.i+a), (20) 178 THERMODYNAMICS The work done on the piston, by the steam, during the adiabatic expansion, must be equal to the difference between the intrinsic energy of the steam before and after expansion; i.e., £?i-£?2=J(Ai+gipi-A2-g2p2); .... (21) where Ei, hi, and pi are, respectively, the intrinsic energy, the heat of the liquid, and the heat of disgregation, corresponding to the pressure pi, and E2, A2, and p2 are, respectively, the intrinsic energy, the heat of the liquid, and the heat of disgregation, cor- responding to the pressure p2. Or, to put it in another way, the work done on the piston, by the steam, during the expansion, is the difference in the heat contentj expressed in mechanical units, before and after expansion. The work done on the piston, during exhaust, is -p2W2=-p2(g2iA2+cj); (22) • where 92 and (12 are, respectively, the dryness and increment in volume, due to complete vaporization, at the pressure p2. Taking the simi of the right-hand members of equations (20), (21), and (22), we find, for the net work done during the cycle, W=J{Apxqi\Li+hi''Ap2q2\>'2'-h2+qigi''q2?2)+(pi-V2), is Qi=MC.{T2-Ti)] (1) where M is the mass of the mixture, Cv the thermal capacity per unit mass at constant volume, and Qi the heat developed during combustion. The heat rejected at constant volmne, while the 212 THERMODYNAMICS pressure falls from that represented by the point E, to that repre- sented by the point B, is Q2=MC.{Tz-Ta)\ (2) where Q2 is the heat rejected by the working substance, while the temperature changes from Tz to T^. It is immaterial whether the change in temperature, from Tz to T^, since the final result is precisely the same, takes place inside or outside of the cylinder. The cofnditions are analogous to those discussed, in Art. 114, for condensation at constant volume in the steam cylinder. Since, now, there is no exchange of heat during the adiabatic expansion DE, and, likewise, during the adiabatic compression BCj and the thermal capacity of the mixture is approximately equal to that of the products of combustion, it follows, from equations (1) and (2), that the heat converted into work, during the cycle, is Qi-Q2^MCUT2-Ti)-{Tz-T^)]. ... (3) And, since the ratio of the heat converted into work to that abstracted from the source, is the ideal coefficient of conversion, we find _ Qi-Q2 _ MC,{(r2-ri)^(r3-r4)| . from which Since both DE and BC represent adiabatic changes for the same changes in volume, we have, from equation (60), Art. 49, (V2y^^ ^Tz ^Ta^ Tz-Ta \vj T2 Ti T2-T1' where vi is the volume of the mixture before compression, and V2 the volume after compression. Hence equation (4) becomes -©■"• (« IDEAL COEFFICIENT OF CONVERSION AND TESTS 213 Equation (5) shows that the ideal coefficient of conversion is a function of the ratio of the volume before compression to the volume after compression; and increases with the amount of precompression. 162. Theoretical Temperatures. The temperature which would obtain upon complete combustion, if there were no losses, is readily computed for any given case. That is, the theoretical rise in temperature, viz, T2—Ti,\3 numerically equal to the ratio of the heat of combustion to the thermal capacity of the products of combustion. However, it is found to be necessary, in order to have proper lubrication between the piston and cylinder walls, so as to prevent deterioration of material, to abstract heat from the cylinder walls, either by water jacketingf or else by air cooling. The former, that is water cooling j is brought about by having water, at a comparatively low temperatxire, circulate in a jacket surrounding the cylinder; and the latter, viz, air cooling, is brought about by increasing the surface of the exposed part of the cylinder by means of rihsy and having a stream of air plajdng over it continuously, by means of an air blower of some kind, or else, as is the case in some automobile engines, the circulation of air is brought about by the motion of the car. In any case, the heat abstracted, due to either water or air cooling ^ limits the rise in temperature. Therefore, the temperature found in the cylinder of an internal combustion engine, is always less than that pre- dicted from the heat of combustion and the thermal capacity of the products of combustion. Frequently, the actual temperature is found to be only 50 per cent of the theoretical temperature. 163. Standard Diagram. In deducing the expression for the ideal coefficierU of conversion for the internal combustion engine. Art, 161, certain assiunptions, in regard to thermal capacities and volumes before and after combustion, were made, which are only approximations. But the errors involved in these assmnp- tions are very small in comparison with the difference between the actual and theoretical temperatures obtaining in the cylinder. 214 THERMODYNAMICS However, the diagram described in Art 161, and the results deduced therefrom, though differing materially from what can be realized in practice, are very convenient as a basis for com- paring the performances of internal combustion engines. Elementary Engine Tests 164. Brake Power and Indicated Power. The power delivered by an internal combustion engine is determined in precisely the same manner as is that of a steam engine. This has been fully described in Art 133. Furthermore, the indicated power of an internal combustion engine is also found in the same manner as is that of a steam engine, as described in Art 134. But it must be emphasized that N, in equation (15) of Art 134, represents not the number of revolutions per minute of the fly-wheel, but the number of aydes per minvte in the cylinder under test. The ratio of Brake Power to Indicated Power is, of course, in the case of an internal combustion engine, as well as in the case of a steam engine, a measure of the mechanical efficiency. It is found, however, that the mechanical efficiency of an internal combustion engine, other things being equal, is always less than the mechanical efficiency of a steam engine. This is principally due to the fact that, owing to the high temperatiu*es existing in the cylinders of internal combustion engines, the lubrication is not as good as that obtained in steam cylinders. 166. Thermal Efficiency of Internal Combustion Engine. The thermal effi^dency of an internal combustion engine is, of course, the ratio of the power delivered by the engine, to the power due to the fuel consumed. In making a test, the engine is loaded by means of a brake, or some other contrivance, to the desired amount. Then, in the case of a gaseous fuel, the voliune of gas consmned is measured by means of a meter. Simultaneously with this, as described in Art. 158, the calorific value of the gas is determined. The best results are obtained if continuous tests IDEAL COEFFICIENT OF CONVERSION AND TESTS 215 are made for the calorific value of the fuel; that is, if the supply to the fuel calorimeter is tapped directly onto the main, supplying fuel to the engine, and samples of the fuel are tested, for calorific values, throughout the entire run. The ratio, then, of the work done by the engine, during the test, to the work, expressed in the same imits, due to the fuel consumed, which is equal to the product of the volume of gas consumed during the run and the mean calorific value of the gas, as found by means of the gas calorimeter, is a measure of the thermal efficiency. Or, if a liquid fuel be used, the work due to the fuel consumed, is found from the product of the mass of liquid consumed, during the run, and the mean calorific value per imit mass. The calorific value, per imit mass of the liquid, is determined as described in Art. 159. 166. Actual Indicator Diagram of Intemal Combustion Engine. By means of the indicator diagram, taken from an intemal com- bustion engine, the behavior of the working substance may be conveniently studied. The actual indicator diagram differs, of course, from the ideal indicator diagram, as depicted in Fig. 40. Whereas, in the ideal indicator diagram, the line representing the aspirating stroke, is parallel to the axis of zero pressure, in the actual indicator diagram the line representing the aspirating stroke approaches the axis of zero pressure, as represented by AB in Fig. 41. This is due to the throttling eflfect of the inlet- valve, on account of which, the pressure in the cylinder decreases as the piston advances. In Fig. 41, 01 and OH are, respectively, the axes of zero pressure and of zero volmne, and A A' is the atmos- pheric line. The curve AB, as just stated, represents the aspira- ting stroke; and shows the presswe in the cylinder at the end of this stroke, less than the atmospheric pressure, by an amount A'B. The compression of the mixture, which is approximately adiabatic, is represented by the curve BC. The combustion of the mixture, and consequent rise of pressure in the cylinder, is represented by the curve CD, which is, if the ignition has been properly timed, practically parallel to the axis OH. DE is the 216 THERMODYNAMICS curve representing the expansion of the products of combustion. At E the exhaust-valve begins to open, and the pressure decreases rapidly to the end of the stroke F. The expulsion stroke then begins, and the pressure continues to decrease rapidly up to the point G. At this point, the exhaust-valve is fully open and the pressiu'e decreases gradually as represented by the curve OA, to the end of the stroke, where the pressure is practically atmos- pheric, and the cycle has been completed. The press\u*e in the cylinder, during expulsion, is higher than that of the atmosphere due to the resistance offered by the exhaust-valve, to the outflow of the products of combustion. H That part of the diagram, which represents the effects due to throttling, has been purposely exaggerated. The curve DE is usually, more or less, wavy; this may be due to various causes. If the vibrations appear to be regular and of decreasing amplitude, they are principally due to the inertia of moving parts of the indicator. On the other hand, if the pressure is apparently constant for a time, then suddenly decreases, etc., the waves are due to friction between the piston and cylinder of the indicator. This trouble is readily removed by proper cleaning and lubrication of the piston and cylinder. Furthermore, waves may be established in the mixture in a manner similar to that described in Art. 69; i.e., as the piston begins to compress the IDEAL COEFFICIENT OF CONVERSION AND TESTS 217 mixture a wave of compression travels through the medium to the other end of the cylinder, where it is reflected, with change of sign. This reflected wave then travels toward the piston; and when it meets the piston reflection again takes place, etc. In this manner, inequalities in pressure may be established, which under certain conditions may persist throughout the compression and expansion strokes. However, in general, these inequalities will not be manifested to any marked degree on the indicator diagram, since the inertia, of the moving parts of the indicator, will tend to suppress them. By an inspection of Fig. 41, it is obvious that the work done on the piston during the aspirating stroke is measured by the area JABK; and, likewise, the work done by the piston during the com- pression stroke is measured by the area KBCJ, Diuing combus- tion, since there is no displacement of the piston, the work done is zero. During expansion the work done on the piston is measured by the area JDEFK. And, during expulsion, the work done by the piston is measured by the area KFGAJ. By taking the algebraic sum, we find that the net work done by the working sub- m stance, during the cycle, is measured by the difference between the areas CDEFGi and AiB. Hence, if the mean effective pressure is determined by means of a planimeter, the tracing point of the planimeter, in tracing the area AiB must travel in a sense opposite to that pursued in tracing the area CDEFGi, That is, if i be the starting point, then, to find the difference between the two areas, the tracing point of the planimeter must follow, in order, the path i, C, JD, J5, F, (?, I, A, B, i. The area AiB represents the work lost, due to valve throttling, and is, in well designed engines, small in comparison with the area CDEFGi. If the power lost, due to valve throttling, is large in comparison with the total indicated power, the valves must be readjusted. In general, the spring which gives good results for measuring the indicated power, has a modulus so high that the 218 THERMODYNAMICS part of the diagram, representing the power lost, due to valve throttling,, is too small to be accurately measured. But, by using a stop, so as not to injure the spring, a much lower scale spring may be employed. In this manner the power lost, due to throt- tling, and also the amount of precompression may be accurately determined. 167. Efficiency and Precompression. In Art. 161, equation (5), it was shown from theoretical considerations that, other things being equal, the thermal efficiency increases with the amoimt of precompression. This is foimd to be so in practice. There are, however, limits, beyond which the precompression may not be carried, due to the severe strains to which the engine is sub- jected diu'ing the explosion of the mixture. Tests made, in the Cooper Union Laboratories, on a Fair- banks 8 H.P. gas engine, gave the following results : R.P.M. B.H.P. Efficiencien. VjVi Thermal, Per Cent. Mechanical, Per Cent. 4.72 4.96 6.09 5.31 368 395 414 478 6.54 7.60 7.92 9.14 13.9 15.8 16.1 19.8 74.0 72.0 68.0 67.0 The value given for the B.H.P. is, in each case, the maximum load the engine would carry for the given precompression. On attempting to carry the precompression higher than that given by t;i/v2 = 5.31, it was found that the vibrations set up in the engine were so violent that satisfactory operation could not be obtained. From the table it is seen that the thermal efficiency increases rapidly with increased precompression. The mechanical efficiency, however, is considerably reduced. The fuel used during these tests was illuminating gas having a calorific value of about 590 B.T.U. per cubic foot. The amount of precompression which, in any case, gives the best results depends, of course, upon the quality of the fuel used. IDEAL COEFFICIENT OF CONVERSION AND TESTS 219 It must, however, be emphasized that in any case, without considering the severe strains to which the engine is subjected, the amount of allowable precompression depends upon the tem- perature of ignition for the fuel used. For, if the temperature of the mixture due to the heat developed during the compression, becomes higher than that of ignition, premature explosions will occur, and the engine will not operate successfully. 168. T-(p Diagrams and Internal Combustion Engines. The T- (p diagram, very frequently is a material aid in studying the Fig. 42. effect produced by a change in the cycle upon which an internal combustion engine operates. As an example, if we plot the T- 9 diagram for a foiu'-phase cycle, the efifect produced by changing the amount of precompression is obvious from an inspection of the figure. Let A, in the T-? diagram. Fig. 42, represent the condition of the mixture, as regards temperature and entropy, at the end of the aspirating stroke. Then, since the compression is assumed adiabatic, the entropy remains constant while the temperature rises from T^, that before compression, to Ti, that 220 THERMODYNAMICS after compression; the line representing this being parallel to the T axis, and the condition of the mixture; as regards temperature and entropy, is given by the point B. During combustion there is a rise in temperature, and also, an increment in entropy. The increment in entropy is given by 91 r^« dT where M is the mass of the gas present, Cv the thermal capacity per unit mass at constant volume, and T2 the temperature, when complete combustion has taken place. Though the thermal capapities of gases vary somewhat, for the ranges of temperature obtaining in an internal combustion engine, the vanations are probably not very large. Hence, so far as the present discussion is concerned, no serious error is introduced by assuming C» constant, and equation (6) becomes 9i=Maiog^ (7) The curve BCj therefore, representing the relation of temper- ature and entropy, during combustion, is logarithmic. During the expansion, which is assiuned adiabatic, the entropy is con- stant, while the temperature falls from T2 to Tz, Hence the curve, CD, representing this change, is parallel to the T axis. Finally, heat is rejected, the temperature falls from Ta to 74, and the relation of change in temperature to change in entropy is again logarithmic, as represented by the curve DA, By assmning the thermal capacity of the products of combustion constant, while the temperature falls from T^ to T4, we find for the change in entropy, 92 = Maiog^ (8) IDEAL COEFFICIENT OF CONVERSION AND TESTS 221 From Art 161 we have hence, smce T2 is greater than Ti, T2 — Tz must be greater than Ti — T^, and CD on the diagram, must be greater than BA, Since, by Art. 109, the area under the curve, BC, is propor- tional to the heat absorbed, and the area under the cm^e, AD^ is proportional to the heat rejected, it follows that the ideal coeflS- cient of conversion is ^ Area FBCE-Areo, FADE _ Area ABCD . . ^ AresiFBCE Atqel FBCE' ' ' ^^ From equation (9), and by an inspection of Fig. 42, it is obvious that the ideal coefficient of conversion increases with increased precompression. Thus, if the precompression had been such that the temperatiu'e at the end of the compression were T2, as repre- sented by the point G, such that combustion takes place at the constant temperature 7^2, the ideal coefficient of conversion would be Area AGCD if) = Area FGCE ' which is obviously greater than that specified by equation (9). This is the principle upon which the Diesel motor operates; i.e., an attempt is made to bring about the application of heat at constant temperature. Again, if after complete combustion has taken place, the expansion be continued until the temperature, as represented by the point K, has been reached, the ideal coef- ficient of conversion is still further increased, and is given by the relation Area AGCK TQ = Area FGCE which brings us back to the Camot cycle. 222 THERMODYNAMICS But, as has been previously explained, in Art 115, this requires a stroke of greater length than is consistent with economy. 169. Actual p-v and T-9 Diagrams of Internal Combustion Engine. The quantity of heat which a gas absorbs, or liberates, during a given temperature change, depends upon whether the change takes place at constant pressure or at constant volume. The change, however, usually takes place in a manner such that neither the pressure nor the volume remains constant. When both pressmre and volume vary, the change in entropy is readily found from equations (42), (43), and (44), of Art. 49. Equation (42) states that dQ=^Cj4T~{Cp-C,)dp. Assuming the process reversible, then, dividing through by T, we have and from which 92-(pi = Cplog^-(C,-Qlog^^ . . . (10) i 1 Pi where 2\ since, for any particular case, p2 is the desired final pressure, pi, however, is a variable; and the value of W obviously depends upon the value chosen for pi. Since, the only variable in the right-hand member of equation (19) is pi, the value found for TF is a minimum when the expression, is a minimum. Differentiating this expression, with respect to pi, equating to zero, and solving for pi, we find Pi = Pa*P2*. (20) 244 THERIVIODYNAMICS Substituting the value of pi as given by equation (20), in equations (16) and (18), we find and. n-l --;^-|fer-'l (22) From equations (21) and (22) it is seen that, if the work done, during a cycle, by a two-stage compressor, is to be a minimum, it must be equally divided between the two cylinders. Taking the sum of the right-hand members of equations (21) and (22), we find the net work done, when employing the most efficient compression possible, by a two-stage compressor, in taking air under a pressure pa and expelling to a receiver, under a pressure P2, IS w 2n V2 n-l 2n -1 ft.-lbs. per pound. (23) If the compression is brought about by three stages, the final pressure being pa, and the pressures of the intermediate receivers, respectively, pi and p2, then, on the assumption that, in the two intermediate receivers, the temperature is reduced to that of the atmosphere, the work done in the first, second, and third cylinders is given, respectively, by Wi = n TF2 = n-l _n_ n-l (24) . . (25) and n-l w^-;hH^^ ' -' (26) COMPRESSED AIR AND COMPRESSORS 245 Taking the sum of Wi, W2, and Wz we find, for a cycle, the net work done by the three-stage compressor, is The right-hand member of equation (27) is a minimum when the expression included in the brace is a minimum. Differentiating this expression, first, assimiing pi variable, and pa, P2, and ps constant, equating to zero, and solving for pi, we find pi = Vp^2 (28) Equation (28) gives the relation of pi to pa and p2 such that the process in going from pa to p2 shall involve a minimum amoimt of work. Differentiating again, this time, however, assuming Pi and p3 constant, and p2 variable, in order to obtain the relation P2 must bear to pi and pz such that a minimum amount of work is involved, while the process takes place from pi to pz, we find P2 = Vpipz (29) By elimination we find, from equations (28) and (29), and Pi = ^Pa^PZj P2 = i^ VaPz' (30) (31) Substituting, in equations (24), (25), and (26), the values of pi and p2, as given by equations (30) and (31), we find '^-»-v-£ n-l 3n -1 , • • • • \ The effective piston displacement not being equal to the actual piston displacement, does not affect the expression deduced for the work done on an air compressor; for, the air remaining in the cylinder, at the end of the expulsion stroke, does an amount of work * K usually has a value of about 50. COMPEESSED AIR AND COMPRESSORS 249 on the piston, in expanding, which is practically equal to that which was done on it while being compressed. The efifect of the clearance, then, is merely to reduce the capacity of the cylinder. 180. Throttling and Other Imperfections. The capacity of a cylinder of an air compressor is very frequently more seriously affected by other causes than it is by clearance. In the first place there is always, due to valve friction, a certain amount of throt- tling, which causes the pressure in the cylinder, during the aspira- ting stroke, to be less than atmospheric. Further, due to imper- fect valve action, i.e., the valves not opening or closing at the proper time, the capacity is reduced. And, finally, the temperature of the cylinder walls is usually higher than that of the incoming air, which again tends to reduce the capacity of the cylinder. These combined causes may reduce the apparent capacity, depending upon the speed of the machine, from 5 to 20 per cent. 181. Adiabatic Expansion in Motor. The cycle of an air motor is practically the reverse of that of an air compressor. The admission-valve opens and air from the mains, under a practically constant pressiu-e, forces the piston forward to the point of cut- off, and the work done on the piston, per pound of air is Wi = pivi; (43) where pi is the pressure in the main, and vi the volume of one pound of air at cut-off. The expansion is then practically adia- batic, and the work done in expanding from the pressure pi, to Pa, that of the atmosphere, is n-l The exhaust-valve then opens, the air is expelled imder a pressure Pa, and the work done, by the air, is L n-l Wz= —PaVa— —pV^ViPa » (45) 260 THERMODYNAMICS Taking the sum of the right-hand members of equations (43), (44), and (45), we find, for the net work done on the motor, per poimd of air, n-l n-l n-l n n-l plVi \ 1 -fe)" ft.-lbs. (46) Since the temperature in the mains is practically atmospheric, piVi is the product of pressure and volume, for one pound of air under ordinary conditions, and may be replaced by the constant 27,800. Hence, equation (46) becomes Tr=27,800-^|l-(^)"" n-i[ \pi/ (47) The indicator diagram for the preceding discussion is shown in Fig. 49, in which AB represents the admission, BC the expansion. V Fig. 49. and CD the expulsion. The work done on the piston during admission and expansion, is measured by the area ABCOF; and the work done by the piston, during exhaust, is measured by the area CGFD. Hence, the net work done, by the air, is measured by the area A BCD. COMPRESSED AIR AND COMPRESSORS 251 Equation (46) may be put into another form, by substituting n-l for pivi, its value RTi, and for {jpJvi) " t its value Ta /Ti. Making these substitutions, we find '^'^h'^H'-r)' from which, for adiabatic processes, since in that case n=Cp/Cv, we obtain And, since we have finally 72 = J\Cp — Cv) y W^JCp{Ti-Ta) (48) It must be noted that, in equation (48), Ti is the temperature in the mains, which is practically that of the atmosphere, and Ta, the temperature of the air after expanding adiabatically from the pressure pi, that existing in the mains, to pa, that of the atmos- phere. 182. Reheating. When air at atmospheric temperature, and under a high pressure pi, expands to atmospheric pressure pa, the corresponding temperature, Ta, will be very low. As an exam- ple, if air under a pressure of five atmospheres, and at atmospheric temperature Ti, expands adiabatically to a pressure of one atmos- phere, its temperature becomes approximately, n-l Ta = Ti(^) "" =5^^(iy =330=-130°F.; n having been assumed to have the value 1.4. Temperatures as low as this, due to the fact that the moisture present in the air freezes, makes lubrication difficult and clogs the valves, are undesirable at the exhaust of an air motor. To prevent this, 252 THERMODYNAMICS recourse must be had to reheating; i.e., the air from the mains is passed through a heater before being admitted to the motor. In being heated at constant pressure, the volume of the air is increased, and the ratio of the two volumes is given by T/Ti) where T is the temperature of the air after heating. Hence, equation (46), giving the work done, per pound of air, on the air motor, becomes '^'-i;x;^^'"'{'-fer} (*»> If, now, Ta is the temperature at the end of the adiabatic expansion, to the pressiu*e Pa, and since n — Cp/Cvy piVi=RTi =J{C^—C^Ti, n-l and {pa/pi)~^ = Ta/Tiy we find, by substituting in equation (49), TFi=|]x^^J(C,-c.)ri(i-^'). . . (50) And further, since the ratio of final to initial pressure is the same whether there be reheating or not, the ratio of final to initial temperature must also be the same for both cases; hence, T a ~ ■* /T7 I where Ta is the finaL temperature when there is no reheating. Substituting this value of Ta in equation (50), and simpUfying, we find W^^JC,^{T^^Ta) (51) Equation (51) is the expression, in terms of the three temper- atures, with reheating, for the work done per poimd of air, on the air motor. Equation (48) is the expr^sion for the work done per poimd of air without reheating. Taking the difference between equations (51) and (48), we find, due to reheating, for the gain in work 1^' - JCvY^Tx - Ta) -JCj^iTi - Ta) = JC^iTi - Ta)~~^. (52) COMPRESSED AIR AND COMPRESSORS 263 The heat consumed, expressed in mechanical imits, in raising the temperature of 1 pound of air at constant pressure, from Ti to r,is jCp(r-ri); and the work which could be realized from this quantity of heat, by means of a Camot cycle, is Tr"=JCp(r-ri)?^ (53) Taking the ratio of TT', as given by equation (52), to T7", as given by equation (53), we find In equation (54), the first factor, viz, T/Ti, is always greater than unity, and for any given case, T\ — Ta is a constant. Hence the ratio, W'/W', is greater than unity until the air is reheated to a temperature such that T T-Ti Ti Ti-T, (55) And, for reheating to a temperature higher than this, the ratio becomes less than unity. Solving equation (55), for T, we find JT 2 r=^ (56) Hence, for reheating to temperatures lower than that given by equation (56), there is a thermodynamic gain; i.e., the gain in work, due to the heat applied in reheating the air, is greater than that which could be realized if an equal quantity of heat were utilized on a Camot cycle for the same limits of temperature. To illustrate, we will assmne a particular case and solve for T. Let pi, the pressure in the mains, be six atmospheres, Ti be 522, and pa, the final pressure, be one atmosphere; then 254 THERMODYNAMICS From equation (66), we find r.^'- where / is an experimental constant depending upon the nature of the liquid and inner surface of pipe, and H, s, L, P, and A are, respectively, the loss in head, speed of flow, length of pipe, wetted perimeter, and area of stream. For any particular cross-section the ratio of -4. to P is a constant; which is termed the hydraulic radius, and may be replaced by the symbol K. Hence, equation (57) may be written Since the temperature of air, flowing in a pipe of any consider- able length, is sensibly constant, the product of pressure and volume is also practically constant; and hence, as the pressure falls the speed of flow must increase. Therefore, since equation (58) assumes a constant speed, it is not directly applicable to the flow of air, or any other gas. In the limit, however, we have dH=f^dL (59) And, since the loss of head is numerically equal to the work done by a unit mass of the substance, we have dH = pdv; (60) where p is the pressure, and dv the change in volume, per unit mass, for the section under consideration. From equations (59) and (60), we find pdv=f£^dL (61) 256 THERMODYNAMICS There is, of course, due to change in speed, also a change in kinetic energy; but, in general, this is so small in comparison with the total loss of head that it may be neglected. Substituting, in equation (61), for dv its value as obtained from the equation pv=RT, we find RT s2 ^p—^m^'^ (62) Under steady flow the mass of air passing any section, for a given interval of time, is a constant throughout the entire length of pipe. Hence, we have, for the speed of flow, -f-f/: m where M is the mass passing any section per unit time, v the volume per unit mass, and A the cross-sectional area of the pipe. Sub- stituting the value of «, as given by equation (63), in equation (62), we find from which X^PdP—f2^KA^j^ dL; .... (64) where pi and p2 are, respectively, the initial and final pressures, and L the length of the pipe. Finally, integrating, as indicated in equation (64), we find From equation (63) we have Pi*=— 2.42-; (66) COMPRESSED AIR AND COMPRESSORS 257 wbere «i is the initial speed. Dividing equation (65) by equation (66), member by member, we find pi2 '^gKRT ^^^^ Solving equation (67), respectively, for p2, «i, and/, we find ^M'-^y <^> «,=(£i Pi 2 ^ fj. f > . • . • (69) and By means of equations (65), (66), (68), and (69), the necessary calculations, for any given case, may be made; and, by means of equation (70), the coefficient / may be foimd for a given set of observations. The ratio A/P is, for cylindrical pipes, a function of the diam- eter only; i.e.. We may substitute, then, in equation (68), the following con- stants: g=32.2, X=Z)/4, and iJ = 53.3, and find P2=Pi/l ^ nrV^Pi(^-7^k^V' (71) .3x-r) ^ ^^^^^^ ' 32.2 X 53.3 X -7 4 In a similar manner, the various equations may be simplified. Equation (71) is, perhaps, best illustrated by assiuning a concrete case, and solving for the terminal pressure. As an example, let it be required to find the final pressiu'e, for the case when the initial pressure is six atmospheres, the temperature 258 THERMODYNAMICS 62*^F., the quantity of air required 1200 cu.ft. per minute, the length of pipe 5 miles, and the diameter of the pipe is 1 ft. First of all, from equation (71), it is obvious that the pressure may be specified in any units whatsoever. From a series of observations made by Riedler and Gutter- muth upon the compressed air system of Paris, extending over a distance of about 10 miles, the diameter of the cast-iron pipe being very nearly 1 ft. (exactly 300 mm.), Professor Unwin deduced for the coefficient /, in this particular case, the value of 0.0029.* It must be remembered that this is not the coefficient for a straight piece of cast-iron piping; including as it does, bends and joints, and also a small amount of leakage. Though transmissions to such distances are unusual, the value just quoted for the coefficient is probably a good average value to use for a practical case for the same diameter of piping. That is, in any practical case, for piping of an equal diameter, we should probably find the average losses per given length, approximately the same. From the conditions we have pi=88.2 lbs. per square inch, T=522, L=26,400ft., D = l ft., and si = (1200/60)/ =25.5 4 ft. per second. For /, we will use the value 0.0029. Substituting these values, in equation (71), we find cQo/i 0.0029X25.5 X26,400 \i .^ ^ „ P2 = S8.2\1 429X522 ) ^^ ^' ^^' Thus giving a loss in pressure of about 11.8 per cent. It must, however, not be understood from this, that the percentage loss of power in transmission is also 11.8 per cent. The efficiency of transmission is found by taking the ratio of the work which the air motor can do in expanding adiabatically from the pressure P2 to that of the atmosphere, to that which would have been obtained had adiabatic expansion taken place before transmission. * "On the Development and Transmission of Power," by W. C. Unwin. COMPRESSED AIR AND COMPRESSORS 259 That is, the efficiency of transmission is _ n-V \ \p2/ J _ \P2/ _ ,^2) Substituting in equation (72), for pa, pi, and p2, respectively, 14.7, 88.2, and 77.8, we find, for the efficiency of transmission. i'(—Y \77.8/ ^ ^ . . 1? = ^=0.945; 1 /14.7\7 V88.2/ where n is assumed equal to 1.4. It is thus seen that, though the loss in pressure is about 11.8 per cent, the loss in power, due to this loss in pressure, is only about 5.5 per cent. The efficiency of transmission may also be defined, depending upon the point of view, as the ratio of the work that could be realized, before transmission, by allowing the air to expand isothermally, to that which would be realized by means of isother- mal expansion after transmission. In any case, for pressures such as are ordinarily employed, the value found, for the efficiency of transmission, by this comparison will not differ materially from that found by means of equation (72). If we make the computa- tion for this particular case, we find, by assuming isothermal processes, , 77.8 ^""^li^ ^'=— 88:2 = 0-«30; which differs approximately, only 1.5 per cent from the value found by comparing adiabatic processes. 260 THERMODYNAMICS In the case of water, the coefficient /, other things being equal, is a constant for all diameters. This, however, is not the case for gases. In the case of air, the coefficient / is some function of the diameter. Various empirical formulsB have been proposed, for cast-iron piping, by means of which / is f oimd, in terms of the diameter. None of them, however, are true for all diameters. As an example, the following formula, proposed by Professor XJnwin, may be cited. According to this formula, the coefficient is /=0.0027(l+j|g). However, by computing the coefficient, for various diameters, by means of this formula, and comparing with the values, as found by actual experiments, it is foimd that there is considerable discrepancy, as the following table will show: D. In Feet. Coefficient. By Experiment. By Formxila. 0.492 0.656 0.980 0.00449 0.00377 0.0029 0.00435 0.00393 0.00351 For the two smaller diameters there is very close agreement; but, in the case of the one of 0.98 ft. diameter, the discrepancy is considerable. Professor Unwin has proposed the value 0.003 for all diameters of 1 ft. or over. 184. Composite Diagram. We are now prepared to show, by means of the p-v diagram, the losses for the compressor, the line, and the motor. Let, in Fig. 50, 01 and OH represent, respect- ively, the axes of zero pressure and zero voliune; and the line AB the aspirating stroke. Assume further that the compressor is one working on two stages, compressing first adiabatically, in the low-pressure cylinder, from the pressure pa, as represented COMPRESSED AIR AND COMPRESSORS 261 by the point S, to a pressure p, as represented by the point C. At the point C, the exhaust-valve of the low-pressure cylinder opens and the air is expelled to the receiver at the constant pres- sure p.. In the receiver, the temperature falls to its initial value, and \he volume shrinks by an amoimt represented by CD; the point D being on the isotherm BF. The condition of the same mass of air now, as regards pressm-e and volimae, at the end of the aspirating stroke, in the high-pressure cylinder, is represented by the point D. Compression now takes place adiabatically from the point D, to the point S, to a pressure pi. When the pres- H sure pi has been attained, the exhaust-valve opens and expulsion takes place under constant pressure, as represented by the line EG. In the reservoir, the temperature of the air falls to its initial value, and the volume shrinks by an amount EF; the point F being on the isotherm BDF. During isothermal transmission, the pressure falls by an amoimt represented by GK; and at the end of admission in the air motor, i.e., at the point of cut-off, the condition of the air, as regards pressure and volume, is represented by the point L. The point L is again on the isotherm BDF, From the point of cut-off, L, the air expands adiabatically, as represented by the curve LM. At the point M release occurs; and the expulsion stroke is represented by the line MA» From an inspection of the figure it is evident that the net work done, per cycle, by the compressor, is measured by the area 262 THERMODYNAMICS ABC DEO; and, the net work recovered by the air motor, per cycle, is measured by the area KLMA. Hence, the total loss of work is measured by the area BCDEOKLM. The work lost, per cycle, due to the compression and expansion not being isother- mal, i.e., in the compressor and motor, is measured by the area BCDEFLM; and, the work lost in transmission is measured by the area FGKL. In general, the thermodynamic loss in the com- pressor and motor is large, in comparison with the loss, due to friction, during transmission. 186. Theoretical Efficiency of System. It will now prove instructive to assume a concrete case and make, without consider- ing other losses, a comparison between the three losses; that is, the thermodynamic loss, due to the compression, in the com- pressor, being adiabatic in place of isothermal, the loss in trans- mission, and the thermodynamic loss, due to the expansion in the motor, being adiabatic in place of isothermal. Let it be required to compress the air to a pressure of six atmos- pheres by means of a two-stage compressor. The work required, per pound of air, according to equation (23), will be n-l -"^Mifr-^] =^X27,800(6^-1) = 56,800 ft.-lbs. per pound. If the expansion now take place isothermally, after cooling, the work recovered will be W2 = PaVa log ^- = 27,800 log 6 = 49,800 ft.-lbs. per pound. Pa This is a loss of about 12.3 per cent. Assimie, now, that the transmission line has the same constants as that discussed in Art. 183. Then the pressure at the end of COMPRESSED AIR AND COMPRESSORS 263 the line will be 77.8 lbs. per square inch; and the work that can now be recovered, due to isothermal expansion, will be 1^8 = 27,800 log j^= 46,300 ft.-lbs. per pound. This is a loss of about 6.2 per cent of the total work. If the expansion now takes place adiabatically, the work done on the air motor is Tr3 = 27,800xJ^{ l-(^y } =36,900 ft.-lbs. per pound. This gives a loss in the air motor of about 16.6 per cent of the total work done. We have then, the following : Per Cent. Loss in compressor 12.3 Loss in transmission 6.2 Loss in motor 16.6 Efficiency of system. . . . , 64.9 Total 100 From the foregoing computations, it is obvious that the efficiency of the system is low, not due to the loss in transmission; but on account of the combined losses in the compressor and motor. Assume, now, that the air is reheated to a temperature 275 °F. above the smroundings. The work which the air will now do on the motor is T 7Q7 Tr4=—Tr3=i|ix 36,900 =56,300 ft.-lbs. per pound; where T is the temperature to which the air is heated before bdng admitted to the motor. This gives, for the gain in work, for the same quantity of air consumed, by the motor, approxi- mately 52.6 per cent. To make a comparison now, between the work done on the motor and that done on the compressor, it will be necessary to 264 THERMODYNAMICS add to the work done on the compressor, the work due to the heat consumed in reheating the air. The heat consumed in elevating the temperature of 1 pound of air from the temperature Ti to the temperature T, is Q=Cp(r-ri)B.T.U. And the work which would be realized on a Camot cycle, is Substituting the various values, we find Tr6=778X0.238X275X^= 17,600 ft.-lbs. Taking the ratio now, of Wa to the sum of Wi and W5, we find, for the efficiency of the system, 75.7 per cent, as against 64.9 i)er cent, obtained without reheating. Commercially, however, the gain is greater than that indicated by the computations. For, as previously stated, a low-grade fuel may be employed, and the motor operates better, especially so if a small percentage of water is injected into the heater. This water is evaporated in going through the heater, and condensed in going through the motor. There is involved m this operation a small thermodynamic loss; but otherwise, the effect is good, since the water present helps to prevent leakage. Finally, reheating has the effect of increasing the capacity of both the compressor and line. It must be emphasized that in no case are efficiencies obtained as high as those indicated by the foregoing computations. Due to imperfect valve action, leakage, and mechanical losses, in both the compressor and motor, the efficiency of the system may be reduced by 10 to 15 per cent below that indicated by the com- putations.* *For actual tests on air transmission BystemBy see Unwin, "On the Development and Transmission of Power." CHAPTER XVII REFRIGERATION 186. The object of refrigeration is to maintain the temperature of some body; or aggregation of bodies, at some point lower than that of the smroundings. This may be done in two ways. One method is to abstract heat directly, by means of a refriger- ating machine, from the medium surrounding the bodies. The other method is to bring about the desired lowering of temperature by means of ice. The ice employed, to bring about the desired refrigeration, may be harvested, during the cold season, from rivers and lakes, or else, the so-called " artificial ice," produced by means of refrigerating machines, may be used. Since the putrefaction of organic growths, such as foodstuffs, is retarded with lowering of temperature, and, in general com- pletely prevented when the temperature becomes sufficiently low, the prime object of refrigeration is not the maintaining of low temperatiu'es, but rather the effects due to such low temper- atures; i.e., the preservation of foodstuffs during storage and shipment, and, in general, the promotion of health and comfort. 187. Commercial Refrigerating Machines. Refrigeration may be brought about in various ways. But commercially successful refrigerating machines are restricted to two types; viz, refriger- ating machines in which air is the working substance, and machines in which some volatile liquid, such as anunonia, or carbon-dioxide, is employed as a working substance. For ammonia machines, there are again two distinct methods of operation; viz, com- pressor machines, and absorption machines. These various types will be discussed subsequently under separate headings. 265 266 THERMODYNAMICS All commercial refrigerating machines operate as reversed engines; but, it must not be understood from this that the machine is reversible. The working substance abstracts heat from a body of, relatively, low temperature, called the refrigeratory con- sumes energy either in the form of mechanical work or heat, and rejects heat to a condenser or cooler. The heat rejected to the cooler, barring various losses, is equal to the heat taken from the refrigerator plus the heat equivalent of the energy consumed in bringing about the transfer. Equation (24), of Chapter VIII, states that, for an engine operating reversed, on a Camot cycle, W^JH2^; (1) where W is the energy consumed in bringing about the transfer, H2 the heat abstracted from the refrigerator, S the temperature of the source, and R the temperature of the refrigerator. The source, in the case of a reversible engine, corresponds to the cooler of a refrigerating machine. In discussing the Camot cycle, it was foimd that, other things being equal, the greater the range in temperature, the greater the amount of work realized for a given quantity of heat abstracted from the source. On the other hand, equation (1) clearly indi- cates that, other things being equal, for a given quantity of heat H2i abstracted ftom the refrigerator, the wo k which must be done by the compressor decreases as the difference of temperature between the cooler and refrigerator is decreased. Hence, the range in temperature between refrigerator and cooler, for refrigerating machines, should be as small as possible. 188. Air Refrigerating Machine. The air refrigerating system consists essentially of four parts; viz, a cold storage room, a compression cylinder, an expansion cylinder, and a cooler. The cycle is as follows : During the aspirating stroke, of the compressor piston, air flows into the cylinder, from the cold storage room, REFRIGERATION 267 which during the return stroke is compressed, practically adia- batically, to the desired pressure, and expelled to the cooler. The cooler, usually, consists of a series of pipes in which the air is cooled by water circulating through the tank in which the pipes are placed. From the cooler the air passes into the expansion cylinder, where it does work on the piston, expanding practically adiabatically, and is finally exhausted, at a low temperature, to the cold storage room. The work done in the expansion cylinder is utilized in helping to drive the compressor. Hence the work, barring mechanical losses, which must be supplied to the com- pressor by means of some motor, is the difference between that done in the compression cylinder and that done in the expansion cylinder. 189. Ideal Coefficient of Performance. To make a mathe- matical discussion, of the cycle just described, it will be necessary to assume ideal conditions. Let T2 be the temperature of the air entering the cooler, at the end of the adiabatic compression, Ta its temperature as it leaves the cooler and is admitted to the expansion cylinder, To its temperature at the end of the adiabatic expansion as it enters the cold storage room, and Ti its temper- ature as it leaves the cold storage room and enters the compressor. It will now be assumed that the pressures in both the cooling pipes and cold storage room are constant throughout the cycle, and the machine is mechanically perfect. By equation (15), Art. 177, we have for the work done per pound of air, on the piston of the compressor, Wi=JCp(T2-Ti) (2) By equation (48), Art. 181, we have for the work done per pound of air, on the piston, in the expansion cylinder, W2 = JCp{Ta-To) (3) 268 THERMODYNAMICS The work which must be supplied, to make the process take place, is the difference between Wi and W2; i.e., Wz=JCp{{T2'-Ti)'-{Ta-To)] (4) Since, according to the assumed conditions, the ratio of the ranges in pressures for the two cylinders are the same, we find r.4: (« Substituting the value of To, as given by equation (5) in equation (4) we find Wz=JCAT2-Ti)^^^ (6) The heat per pound of air, expressed in mechanical units, taken from the refrigerator, is W^^JCpiTi-To). Substituting again, for To, its value, we find Tr4=JC,^^(r2-ra) (7) Taking the ratio of W^ to Wz we find, for the ideal coefficient of performance, ^ Wz T2-T1 ^^^ Equation (8) again shows that, the smaller the difference in temperature between refrigerator and cooler, the larger will become the ratio of the work equivalent of the heat abstracted from the refrigerator, to the work supplied. It is, of course, obvious that, due to the fact that it is practically uneconomical to construct cooling pipes of sufficient volume, such that the pres- sure throughout the cycle in the cooler is constant, and further, since the pressure in the cold storage room varies, the ratio, REFRIGERATION 269 as expressed by equation (8), cannot be realized in practice. Furthermore, due to various losses, which must be experienced, in the case of an actual refrigerating machine, this ratio is still further reduced. Solving equation (8), for the work that must be supplied to a perfect machme, we find Ws^^wJ^^^ (9) The commercial efficiency of a refrigerating machine may be defined as the ratio of the work which would have to be done, for the given range of temperature and given quantity of heat removed from the refrigerator, on a perfect machine, to that actually required. If Ws is the work actually required, then the conmiercial efficiency is Ws'Ws^ Ti ^^"^ The heat which must be carried away by the circulating water, in the cooler, per pound of air, is Hl^C^{T2-Ta) (11) The cycle of an air refrigerating machine may be conveniently represented by means of the T-? diagram. By equation (12), Art. 169, the change in entropy, when both the pressure and volume vary, is 92-9i = C.log2?+Cplog||? (12) In the cycle just discussed it was assumed that the pressure, during the absorption and rejection of heat, remains constant. Hence, equation (12) becomes 92-(pi = Cplogg=Cplog^^; .... (13) and the heating and cooling curves, on the T- 9 diagram, are loga- rithmic. Let, in Fig. 51, the point A represent the condition of 270 THERMODYNAMICS the air, as regards temperature and entropy, at the instant when it enters the compressor at the temperature Ti. During the adiabatic compression the entropy remains constant and the tem- perature changes from Ti to 72, as represented by the line AB. The cooling then takes place, as represented by the curve BC, at constant pressure, to the temperature Ta- The adiabatic expansion, from the temperature Ta to the temperature Tq, is represented by the constant entropy line CD. Finally, the rise in temperature, in the refrigerator, at constant pressure, from To to Ti, is represented by the curve DA ; and the cycle is completed. The heat abstracted from the refrigerator is measured by the area FADE, the heat rejected to the cooler is measured by the area FBCEj and the work done, on the compressor, is measured by the area A BCD. Finally, the ideal coefficient of performance is given by Aresi FADE V = Aresi ABCD' Since the rejection of heat to the cooler, and the abstraction of heat from the cold storage room, both take place at constant REFRIGERATION 271 pressure, equation (8) may be deduced in a very simple mann^. If the heat abstracted from the cold storage room, for a given interval of time, is ff2=Cp(ri-ro), then the heat rejected to the cooler for the same interval of time, is Hi = Cp(T2 — Ta) • Therefore, the ideal coefficient of performance is H2 C pCri-fo) ^ Ti . "^ H1-H2 C,{T2-Ta)-Cp{Ti-To) T2-T1' which is the same as previously foimd. It must be emphasized that the equations deduced, in this article, do not represent conditions as found in actual practice. For the pressure, in the cooling pipes, of any actual refrigerating machine will vary considerably throughout the cycle. Hence, the actual coefficient of performance, even when all other losses are neglected, will be less than that indicated by equation (8). Due to the fact that air has a low thermal capacity, air refrig- erating machines are necessarily bulky, and therefore, commer- cially imeconomical. However, there are certain places, as for example on board of ships, where it is inadvisable to use machines employing a volatile liquid. For, in the first place, there are possi- ble dangers from injurious escaping gases. But, even if the escap- ing gas is not injurious, there is always the possibility of a large leak, and consequently a total loss of the working substance, which cannot be replaced imtil the end of the trip. This, however, means a complete disablement of the plant. Hence, air machines are used only as a matter of expedience and not economy, in place of refrigerating machines employing a volatile liquid as a working substance. 190. Compression Machines Using Volatile Liquids. Com- pression refrigerating machines, using a volatile liquid for the 272 THERMODYNAMICS working substance, consist essentially of the parts as represented diagrammatically, in Fig. 52. A is the compression cylinder where the vapor is compressed, and then expelled into coils immersed in water in S; B being the condenser, or cooler. If the vapor is just saturated as it leaves the refrigerating coils, superheating may take place, during compression; this however is usually very small in comparison with the heat of condensation. Due to the high pressure in the condenser, and the low temper- ature, maintained by the circulating water, the vapor condenses, gives up the superheat and heat of condensation, which is carried -J Wl^J VJJIV/ rvMfVVyA, s*AAV» \ia^ \y «AX^1A«V* M^-t^ *T tj Aa^VXJ D 4 Va^ V tJ VV^A »«^^^ VI A p B B i: G ^ D Fig. 52. In the tank C, the liquid is under a pressure corresponding to that of its vapor, for the existing temperature; the temperature of the liquid in the storage tank, usually does not differ materially from that of the surroundings. As an example, if the liquid employed be ammonia and the temperature in the tank is 76**F., then the pressure of the vapor is approximately 140 lbs. per square inch. Due to this high pressure, under which the liquid is in C, it flows, through the expansion valve D, into the coils in the refrig- erator E, The pressure in the coils, due to the aspirating action of the compressor, is low. By regulating the expansion valve, or the speed of the compressor, or both, the pressure in the refriger- ator coils may be varied at pleasm*e. Since, when the liquid passes through the expansion valve, the process is adiabatic, and no work REFRIGERATION 273 is being done, the total heat content remains the same. There- fore, for thermal equilibrium to obtain, when the pressure falls from pi, that existing in the storage tank, to p2, that existing in the refrigerating coils, there must take place a certain amount of. evaporation, such that hi=h2+qr2; (14) where hi and A2, respectively, are the heats of the liquid corre- sponding to the pressures pi and p2, q the amount of dryness, and r2 the heat of vaporization at the pressure p2. From equation (14), we find «=^; (15) and the remainder of the liquid can, then, if completely vapor- ized, take from the surrounding medium the quantity of heat ff2 = (l-(?)r2 (16) In order that heat may flow from the medium in E, into the coils it is, of course, necessary that the temperature of the mediimi be higher than that of the liquid, in the coils, corresponding to the pressiu-e p2. If the difference of temperature is suflScient, the liquid will be completely vaporized; and the quantity of heat, as expressed by equation (16), will be removed from the refrigerator. If the difiference of temperature be greater than this, the vapor becomes superheated; and the quantity of heat removed from the refrigerator will be greater than that indicated by equation (16). The ideal p-v diagram, of the cycle just discussed, is represented in Fig. 53. The point A represents the condition, as regards pressure and volmne, of the vapor at the beginning of the aspira- ting stroke, and the point B represents the condition at the end of the aspirating stroke; the line AB, therefore, represents the volume, due to complete vaporization under constant pressure. The curve BC represents the compression, which is nearly adia- 274 THERMODYNAMICS batic, CD represents the expulsion, and also the condensation, under constant pressure, in the condenser, and DE the change in pressure, and consequent change in voliune, due to partial evaporation in passing through the expansion-valve. Therefore, the net work done, during the cycle, is measured by the area A BCD, Fina ly the ideal coefficient of performance is given by the ratio of the work equivalent of the heat removed from the refrigerator to the work equivalent of the area A BCD. The refrigerating coils, in which the vaporization takes place, may be placed directly in a cold storage room, in the form of pipes, or else placed in a tank containing a solution of some salt, called brine. The freezing point for the brine must, of course, be lower than the temperature in the coils. The brine may then be em- ployed, by circulating it through pipes, to bring about refrigeration in some place remote from the plant, or else, to produce ice, by abstracting heat from water, contained in tanks, immersed in the brine. The cycle of a compressor refrigerating plant, using a volatile liquid £us a working substance, is most instructively represented by the T— 9 diagram. However, before plotting the T— 9 dia- gram, it will be necessary to deduce an expression for the change REFRIGERATION 275 in entropy, for the substance, when passing through the expan- sion valve. To do this, let Ti be the temperature of the liquid in the storage tank, then by assuming some arbitrary temperature, say To, from which the entropy is measiu'ed, the entropy of a unit mass of the Uquid, before passing through the expansion valve, is 9i = cl -77=cIogjr; (17) where c is the thermal capacity, of the liquid, per unit mass. Assume some temperatiu*e T, in the refrigerating coils; T being, of course, less than Ti. The entropy, then of a unit mass of liquid and vapor, measured from the same zero, is 92 = cj y+^=clog jr+^; .... (18) where q is the amount of dryness, and r the heat of vaporization corresponding to the temperature T, Taking the difference between equations (18) and (17), we find, for the change in entropy in passing through the expansion valve, 9=92- 9i = c log ^-c log j^+^. . . . (19) But, gr=Ai-/i=c(Ti-r); substituting this value of gr in equation (19), and simplifying, we find 9=c(log|^^+y'-l) (20) Dififerentiating equation (20) with respect to Tj Ti being assumed constant, we find %--S-') <^'> Equation (21) shows, since T\>T, that as the temperature increases, the entropy decreases, and vice versa. Hence, the 276 THERMODYNAMICS entropy of the substance is increased by passing through the expansion valve. This is necessarily so, since the process is irre- versible. The T- 9 diagram, Fig. 54, indicates the various parts of the cycle. BC is the constant entropy line for the adiabatic com- pression of the vapor, from the temperature T2 to T,; if the vapor be just saturated, as shown, when the compression begins, it will become superheated during compression. CK represents the cooling of the vapor to the temperature of condensation, Ti; 7 C D Ti »/ A T, 1 \ \ \ * B F! Gj H 9 Fig. 54. and KD represents the condensation of the vapor, in the con- denser, at the constant temperature Ti. DA represents the tem- perature entropy curve for the cooling of the liquid, without expansion, from the temperature T\ to ^2. Had evaporation taken place, without expansion, after cooling along the curve DA, the quantity of heat removed from the refrigerator would be meas- ured by the area FABH. Due to expansion, however, through the expansion valve, the entropy of the substance increases in changing from the temperature Ti to ^2, as indicated by the curve DE. The curve DE is determined by assiuning various values of temperature, between Ti and T2, and solving, by means of equa- REFRIGERATION 277 tion (20), for the corresponding entropy. Hence the heat that it is now possible to remove from the refrigerator, in bringing about complete vaporization, is measured by the area 6EBH. Con- sequently, the amount of refrigeration that is lost, due to the change in entropy, in passing through the expansion valve, is measured by the area FAEG. Had there been superheating in the refrigerating coils, the quantity of heat removed from the refrigerator would be increased; but, due to this superheating, the vapor at the end of the compression will, likewise, be super- heated by an additional amount. Since the evaporation, which takes place while the liquid passes through the expansion valve, has no refrigerating value, but merely brings about thermal equilibrium, by reducing the temperatiu-e of the liquid to that existing in the coil, it follows that the change in entropy, along the curve DE^ depends upon the ratio of the heat of vaporization to the thermal capacity of the liquid. The higher the ratio of the heat of vaporization to thermal capacity of liquid, the smaUer the amount of vapomation required, for a given difference of temperatiu-e, to reduce the tem- peratiu*e of the liquid to that existing in the coil ; and conseiquently the smaller will be the area FAEG. Therefore, since the work done by the compressor is independent of the amoimt of vapor- ization that takes place, along the ciure DE, it follows that a liquid for which the ratio, of heat of vaporization to thermal capacity, is high, is best suited, from an economical standpoint, for refrigerating purposes. 191. Absorption Machines. The principle of operation of an absorption refrigerating machine is based on the fact that the volume of ammonia vapor that can be absorbed by a given volume of water, other things being equal, depends upon the temperature, and decreases rapidly as the temperature is increased. Hence if water, at a low temperature, is saturated with ammonia vapor, then, to drive off the vapor, heat must be absorbed by the water. Likewise, if ammonia vapor be passed into water at a low tern- 278 THERMODYNAMICS perature, absorption will take place with a consequent develop- ment of heat. An absorption refrigerating machine is represented, diagram- matically, in Fig. 55. A is a storage tank containing anmionia from which expansion takes place through the valve B, into refrigerating coils in C, where refrigeration takes place. From the coils in C, the ammonia vapor passes into the liquid in the absorber D. The liquid in D is a solution of ammonia in water, of slight concentration and, relatively, low temperature. ThQ H ^^9 :E = ' Bp FiQ. 55. liquid in the absorber being at a low temperature and only slightly concentrated, the incoming vapor is readily absorbed. The liquid of high concentration is removed from the bottom of D, by means of the pump P, and forced, at a, into the generator F. In the gener- ator is placed a heating coil H, by means of which heat is supplied to the highly concentrated solution, and raises its temperature. Due to the high temperature, part of the vapor is expelled from the solution, under a high pressure, and passes into the condenser O. The condenser is maintained at a, relatively, low temperature by means of circulating water. Due to this low temperature and the high pressure, the vapor condenses and flows into the storage tank A. By means of the valve /, the pressure in G and A is regu- REFRIGERATION 279 lated. Finally, the solution of low concentration, at the bottom of the generator F, is forced, due to the high pressure subsisting in the generator, into the absorber, at b. The pipes which convey the highly concentrated solution into the generator at a, and the solution of low concentration into the absorber at 6, both pass through the heat exchanger E. In the heat exchanger the solution at a low temperature, going from the absorber to the generator, takes up heat from the high temperature solution, going from the generator to the absorber. The cycle is, then, as follows : The absorption, in the absorber, corresponds to the aspirating stroke of the compressor, as repre- FiG. 66. sented by AB of Fig. 66. The change in pressure in going from the absorber through the generator is represented by the curve BCf and corresponds to the compression curve of the compressor. The line CD represents the condensation at constant pressure, the same as in Fig. 53. Finally, the curve DE represents the fall in pressure and consequent increment in volume, due to partial evaporation of the liquid, in passing through the expansion- valve B. There is, of course, in this cycle, as well as in the com- pressor cycle, due to a partial evaporation when the liquid passes through the expansion-valve, a loss in refrigeration. Furthermore, there is, due to the fact that the ammonia vapor when distilled in the generator, carries with it a certain amount of aqueous vapor, an unavoidable loss. 280 THERMODYNAMICS Thermodynamically speaking, the ideal coefficient of per- formance of the absorption machine, as well as that of any other refrigerating machme, is given by R where S is the temperature of the condenser, and R the tempera- ture in the refrigerator. On the other hand, the commercial efficiency is given by the ratio of the work which would have to be done on a perfect engine, to bring about the given transfer of heat, to the energy actually consmned. That is, if ^2 is the quan- tity of heat abstracted from the refrigerator, during a given interval of time. Hi the quantity of heat supplied to the generator, and W the work done on the pump, during the same interval of time, the commercial efficiency is given by JHi &= S-R R JHi+W (22) The following table, taken from " Landolt and Bdmstein," is given to show how the coefficient of absorption for ammonia vapor, under normal pressure, varies with the temperature: T K T K 98.7 15 60.6 1 92.7 16 59.1 2 87.7 17 57.6 3 83.6 18 56.1 4 79.9 19 54.7 6 • 77.3 20 53.5 6 75.6 21 51.9 7 73.9 22 50.6 8 72.3 23 49.6 9 70.6 24 48.6 10 68.9 25 47.6 11 67.2 26 46.5 12 65.5 27 45.5 13 63.7 28 44.4 14 62.1 29 43.4 REFRIGERATION 281 where t is the temperature in degrees centigrade, and K is the number of grams of ammonia vapor absorbed per 100 c.c. of water. By heai of dilviion of a substance is meant the quantity of heat which is evolved when a unit mass of the substance is diluted to an extent such that practically no more heat is evolved upon further d ution. According to experiments by Berthelot, when 1 gram of liquid anunonia has been dissolved in n grams of water, and this solution is then fully diluted, the heat evolved is as given in the following table: n Gram Calories. n Gram Calories. 1.04 1.06 1.13 1.98 75.6 74.4 68.8 40.0 3.18 3.76 6.11 10.1 22.6 18.8 12.3 0.12 The results given in the table were obtained from experiments conducted at temperatures of 14°C. By inspection it is seen that very little heat is evolved after the dilution is greater than 10 to 1. It has been proposed to employ the empirical equation n (23) for the heat of dilution when a solution, of 1 gram of ammonia dissolved in n grams of water, is fully diluted; H being the heat evolved, and h some constant. If equation (23) be applied to the values as given in the table, the value 78 be assigned to A, and the values of H computed and compared with the observed values, an idea will be obtained as to how closely the empirical equation conforms to the actual experimental results. 282 THERMODYNAMICS n Observed. Computed. n Observed. Computed. 1.04 1.06 1.13 1.98 75.6 74.4 68.8 40.0 75.0 73.6 69.0 39.4 1 3.18 3.76 6.11 10.1 22.6 18.8 12.3 0.12 24.5 20.7 12.8 7.7 The foregoing table shows that, when the mitial dilution is not greater than 6 to 1,. equation (23) gives fairly consistent results; but, for initial dilutions greater than this, the equation breaks down completely. Furthermore, since the lowest initial dilution in Berthelot's experiments was 1.04 to 1, equation (23) is necessarily doubtful for initial dilutions lower than this. Experiment shows that, if 1 gram of ammonia vapor is absorbed by water and completely diluted, there is evolved a quan- tity of heat equal to 496 gram calories; hence, if m grams of ammonia are absorbed, and complete dilution take place, there will be evolved 496m gram calories. Therefore, if we assume equation (23) to hold, there will be evolved, when m grams of ammonia vapor are absorbed by n grams of water, Hi^mQ-^ = m(Q--h); .... (24) n/m \ n / ^ where n/m is the number of grams of water per gram of ammonia, and Q the quantity of heat evolved when 1 gram of ammonia vapor is absorbed by water and completely diluted. If, now, m+k grams of ammonia be absorbed by n grams of water, the number of grams of water per gram of ammonia will be n/{m+k). Therefore, the quantity of heat ff2 = (m+A;)(Q-^XA), . . . . (25) will be evolved. Taking the difference between the right-hand members of equations (25) and (24), for the quantity of heat REFRIGERATION 283 evolved, when a solution containing m grams of ammonia to n grams of water, absorbs k grams of ammonia, we find =A;jQ-^(2m+fc)L . . (26) Substituting for Q and h the numerical values, we obtain H^k 78 496-— (2m+fc) gram calories. . . (27) Equation (27) may be reduced to English units as follows: £r=A;]893-^(2m+fc) B.T.U (28) That is, equation "^r^ is the expression for the heat, in B.T.U. , which is evolved when a solution containing m pounds of ammonia to n poimds of water, absorbs k pounds of ammonia vapor. As previously stated, the foregoing equations are empirical and are true only between certain limits of initial dilution; further- more, since the heat of absorption and dilution varies with the temperature, the results obtained by means of these equations are to some extent doubtful. The equations have been deduced merely to show the method of attack. Under ideal conditions the heat developed in the absorber is equal to that required in the generator; but, since the temperature of the generator must be higher than that of the absorber, it is impossible to utilize the heat developed in the absorber. Hence, to maintain the process, heat must be suppHed, by means of some independent source, to the high-temperature generator, and heat must be abstracted from the low-temperature absorber. In the case of a compression machine the energy consumed varies directly as the difference of temperature between the con- denser and refrigerator. This, however, is not so in the case of Cri^tm 284 THERMODYNAMICS an absorption machine; hence, for a wide range in temperature, the absorption machine is thermodynamically superior. A further advantage, which is mechanical, is that no compressor is required. In certain cases the heating in the generator is brought about by means of exhaust steam, from engines, which is again economical. Finally, the power consumed by the pump in an absorption machine is small in comparison with the other quan- tities involved.* 192. Comparison of Air and Axnmonia Machines. It was stated in Art. 189 that, due to the low thermal capacity of air, refrigerating machines employing air as a working substance are necessarily bulky. It is impossible to make a general comparison; but a rough estimate may be obtained by assumi!ng a concrete case. Let it be assumed that the temperature of the refrigerator is 32°F., and that the range in temperature of the air in passing through the refrigerator is lOO^F. One pound of air will then remove, from the refrigerator, CpCri-To) =0.238X100=23.8 B.T.U.; and to do this, the compressor must take in 12.4 cu.ft. The volume, per B.T.U. removed from the refrigerator, then is 12.4/23.8=0.521 cu.ft. Assume, now, an ammonia compression machine, with a temper- ature of 70°F. for the reservoir. The dryness after passing through the expansion valve will be ^ r2 540 ' and the quantity of heat that can be removed, by complete vaporization taking place at 32°F., is (l-g)r2 = 540(l-0.078)=498 B.T.U. per pound. *Fora comprehensive discussion of absorption machines see ''Modern Refrigerating Machinery'' by Hans Lorenz. f From Peabody's Steam Tables. REFRIGERATION 285 The specific volume of ammonia vapor, at 32®F., is approximately 4.74 cu.ft. per pound. Hence we find, for the volume, per B.T.U., 4.74/498=0.00952. Taking the ratio of the volimie for air, to that for ammonia, we find 0.521/0.00952=54.7. This shows that for the assumed conditions, other things being equal, the bulk of the compression cylinder of an air refrigerating machine is very large in comparison with that of an ammonia machine; but, further than this, the air machine must also have an expansion cylinder. For lower temperatures in the refriger- ator, the ratio of the two volumes becomes somewhat less. Assume the temperature of the refrigerator 15°F., then the dryness, after passing through the expansion-valve, is 61/554 = 0.110; and the quantity of heat that can be removed, by complete vaporization, is 554(1-0.110) =493 B.T.U. per pound. The specific volume for the vapor of ammonia at 15°F. is 6.68 cu.ft. per pound; hence, we find for the volume, per B.T.U., 6.68/493=0.0135 cu.ft. The volume of air that the compressor must now take in, at the temperature of 15°F., is approximately 12 cu.ft. ; hence the volmne of airperB.T.U., is 12/23.8=0.504 cu.ft. Taking the ratio of the volmne for air, to that for ammonia, we find 0.504/0.0135 = 37.3. The ammonia machine is also superior from the thermody- namic standpoint. By considering the two cycles, it is obvious 286 THERMODYNAMICS that the cycle of the ammonia refrigerating machine, approaches the Carnot cycle much more closely than does the cycle of an air machine. For, in the ammonia cycle, the greater part of the heat is abstracted and rejected, respectively, dm-ing vaporization and condensation; i.e., at constant temperature. On the other hand, in the case of the air cycle, both the abstraction and rejec- tion of heat take place with continuously varying temperature. The foregoing may be illustrated roughly as follows : As previously shown, the work done by a compressor per cycle, if the compres- sion is adiabatic, is Wi = JCp{T2 - Ti) per pound. The heat, expressed in mechanical imits, removed by 1 pound of ammonia, from the refrigerator, is W2=Jr(l-q); and the ideal coefficient of performance is r_W2_^ r(l-g) . . ^ Wi C^{T2-Ti) ^^^^ It was shown, in Art, 189, that the ideal coefficient of per- formance of an air refrigeration machine is . ^" = 7^ (30) In equations (29) and (30), Ti and T2 are, respectively, the temperatures before and after adiabatic compression. If we assume, now, that the ranges in temperature for the two machines are equal, we find, for the ratio of the performance for the two processes, y^ e« If we are dealing with ammonia, and conditions are such as ordinarily obtain in refrigerating plants, then, in equation (31), the numerator and Ti will be practically equal. But, Cp for REFRIGERATION 287 ammonia vapor is approximately 0.53; hence, the coefficient of performance for the ammonia machine, bb expressed by equation (29), is approximately double that for the air machine, as ex- pressed by equation (30). From the foregoing discussion it is obvious that, due to its enormous bulk, and consequent mechanical losses, together with its thermodynamic inferiority, the air refrigerating machine is very uneconomical, both from the standpoint of first cost and operation. 193* The Kelvin Heating Machine. Before leaving the sub- ject of refrigerating machines, it will be interesting to consider a heat engine running reversed as a warming machine. This was suggested as early as 1852 by Lord Kelvin. To illustrate this, let it be desired to maintain the temperature of a room higher than that of the surrounding atmosphere. This may be brought about by the direct application of heat, or else by a heat engine running reversed. Assume the heating to be brought about by an air refrigerating machine, such as discussed in Art. 188, then during the aspirating stroke a charge of air flows into the com- pression cylinder at a temperature Ta, This charge is now com- pressed to a temperature Ti and expelled into pipes, placed in the room which it is desired to heat, where heat is abstracted. After cooling, the air does work in the expansion cylinder and is expelled to the atmosphere. For a reversible engine, the heat rejected to the room is equal to the heat taken in from the atmosphere plus the heat equivalent of the work done on the air. If Hi is the heat rejected to the room, and Ha the heat taken in from the atmosphere, then Hi=Ha+AW: and W=JHi Ti-Ta If, now, Ti — To, the required range, be small, then the heat equivalent of W will be a small fractional part of Hi, 288 THERMODYNAMICS To illustrate further: Assume a situation where it is impos- sible to obtain fuel of any kind, but that there is available energy in the form of an electric current. Heating may then be brought about in two ways. That is, heat maiy be developed by passing the current through a suitable resistance, or else, the energy may be consumed in driving an electric motor, which in turn drives some form of reversed heat engine. In either case, the energy spent per unit time, due to the current consumed, is given by the product of e.m./. and current. To make a simple comparison it will be necessary to assume certain conditions. Let the tem- perature of the atmosphere be 0°F., and that required in the heat- ing coils, so as to maintain the room at a proper temperature, be 165°F. Now, to bring about equal heating effects, the heat dissipated per unit time must be the same in each case. Let H be the heat required per unit time, then AEIi=H] (32) where E is the applied e,m.f. and /i the current consumed, when the heating is brought about by means of resistances. Assume now, a perfect electric motor driving a perfect warming machine. The power consumed to bring about the same heating efifects, for the given temperatures, is AEh^H^^^^iyi, (33) where I2 is the current consumed by the motor. Solving by means of equations (32) and (33), for /2, we find showing that for a commercial eflSciency even as low as 26.4 per cent, the warming machine is thermodjniamically equal to the direct method. And for eflBciencics higher than 26.4 per cent, the warmmg machine is thermodynamicaUy superior. CHAPTER XVIII STEAM TURBINES 194. The detailed descriptions of the various types of steam turbines and the attendant mathematical discussions require an extended treatise. For such a treatise the reader is referred to Dr. A. Stodola's classical work, *' Die Dampf-turbinen." * No attempt will here be made to do anything further than lay down the most elementary principles, so as to enable the student to take up reading matter, on the subject, of an advanced nature. In steam turbines, as well as in water tiu^bines, there are impulse turbines and reaction turbines. However, in the case of water wheels, the types most frequently used are single stage; i.e., one stationary part, which carries the guides, by means of which the water is given the proper direction before entering the wheel, and one rotating part, carrying a number of curved blades, by means of which the energy stored in the water due to pressure and velocity, is absorbed. On the other hand steam turbines must be multi- stage, i.e., consist of a number of fixed parts called guides, and a nimiber of rotating wheels, called rotors; otherwise the speed would be impracticably high. Fig. 57 is a diagranmiatic repre- sentation for two stages of a multi-stage turbine. 196. Impact on Curved Surfaces. Before proceeding to make any mathematical discussions for steam turbines, it will be well to consider a few cases for a non-expansive fluid, such as water, impinging on curved surfaces. Let, in Fig. 58, abc represent the section of a curved blade, having impinging upon it a stream of water with a velocity, relative to the earth's surface, represented * Translated by L. C. Loewenstein. 289 290 THERMODYNAMICS in magnitude and direction by ka. This velocity is briefly designated as absolute velocity. The line ad represents in mag- nitude and direction the velocity of the blade. Now, while a GUIDE J J J J Fig. 57. particle of water starting from a moves to e, a distance equal to Fi, the tip of the blade a has suffered a displacement Vi, as repre- sented by ad. Hence the velocity of the water Yt^ relative to Fig. 58. the blade, is given by (fe. Therefore, the absolute velocity at entrance is equal to the vector sum of the velocity of the tip of the blade, at entrance, and the relative velocity. STEAM TURBINES 291 If the water is to glide onto the blade, so that there shall be no shock, the tangent, to the tip of the blade at a, must be parallel, as shown by ka, to the relative velocity Vr. If, now, there is experienced no friction by the water as it glides along the blade, and there is no sudden change in direction, then the magnitude of Vr will not change; and the water will leave the blade, parallel to the tip at exit, as shown by ci. If V2 is the velocity of the tip at exit, then the vector sum of Vr and V2, gives for the absolute velocity at exit, F2, as shown by cj. We then have, respectively, for the triangles of velocities at entrance and exit, ade and dj. If, from a we draw ag equal and parallel to V2, and close the tri- angle by eg, we find Vt, the vector difference between Vi and V2; i.e., the total change of absolute velocity. Resolving Vt into two components, one normal to the motion of the blade and the other parallel to the motion, we find Ve, represented by ef, the total change of absolute velocity in the direction of motion. Ve may be called the effective component, since this is the one producing the motion. From the diagram, it is obvious that the absolute velocity at entrance cannot be parallel to the motion of the blade, but must make some angle with it; otherwise entrance into the channel, included between abc and a'b'c', cannot take place. Similarly, at exit, there must be a normal component to carry the water away, so as not to interfere with the following blade. Fur- thermore, it is essential that the two surfaces of the blade tips, both at entrance and exit, come to a point and have, practically, a common tangent, parallel to the relative velocity, so that the following blade, a'b'c', may glide into the stream without shock. Unless this is so, there will be a loss in efficiency; for, whenever a stream of water impinges upon a surface with shock, there are developed eddy currents which, when subsiding, develop heat and the energy thus consiraied is dissipated to the surroundings. To find the theoretical efficiency of a system of blades, as depicted in Fig. 58, it is only necessary to take the ratio of change in kinetic energy, in passing through the channel, to the kinetic 292 THERMODYNAMICS energy at entrance. Let M be the mass of water, per unit time, passing through the channel; then^the kinetic energy, at entrance, is TFi = ^ (1) The kinetic energy at exit is W..«^; (2) and the energy given up to the sjrstem is Wi-W2=W3=^(yi^-V^) (3) Hence, the efficiency is given by Wi Vr'-Vi^ ... -"^Wz — F7~ ^^^ Equation (4) shows that V2 should be as small as possible; which means that it must be normal to the direction of motion of the blades and just sufficient to carry the required quantity of water away from the channel. We may consider this in another manner. Since force is numerically equal to rate of change of momentum, the eflfective force, in producing motion, is F=MV,] (5) where F* is the change of absolute velocity in the direction of motion. And the power developed, since power is numerically equal to the product of force and speed, is P2^MVeVi) (6) where P2 is the power developed. Equation (6) assumes vi and V2 nimierically equal. The power of the stream before imping- ing, since M is the mass of water which passes through the channel per unit time, is Pi=-^-; (7) STEAM TURBINES 2d3 where Pi is the power delivered by the stream. Taldng the ratio of Pa to Pi, we find for the eflSciency, 10 -pj- Vi^ (8) Equation (8) again shows, that if Vi and vi are fixed, the efficiency is a maximum when F« is a marimmn; i.e., the velocity com- ponent normal to the direction of motion is as small as possible. 196. The Pelton Cup. Pelton wheels may be taken as repre- senting impulse tiurbines in the case of hydraulic motors. One of the cups, as used in this tjrpe of wheel, is represented diagram- matically in Fig. 59. Vi is the absolute velocity of the entering Fig. 59. jet, and vi the velocity of the cup, whose section is represented by abc. The velocity of the water relative to the cup is given by Vr^Vl-Vl. (9) Since the direction of motion of the water at exit makes an angle 6 with the direction of motion of the cup, the component of the velocity of the stream at exit, parallel to the motion of the cup, is given by 7,= (Fi-t;i) cos e (10) 294 THERMODYNAMICS And the absolute velocity, parallel to the direction of motion of the cup, at exit, is F2'=t;i~(7i-t;i)cos e (11) The change in absolute velocity in the direction of motion, there- fore, is Fa = 7i-72' = Fi- {vi-(Fi-t;i) cos 61 = (7i-t;i)(l+cose). . . (12) If M is the mass of water, per unit of time, impinging on the cup, then, the force moving the cup is nmnerically equal to the product of change in velocity and mass; hence we find for the force acting, F=M(Fi-t;i)(l+cose) (13) Finally, since power is numerically equal to product of force and speed, we have, by multiplying both sides of equation (13) by vi, for the power developed by the cup, P2^Fvi^Mvi{Vi-vi)(l+Q0S%) (14) It is obvious, from Fig. 59, that the motion of the water cannot be completely reversed; i.e., the direction of motion of the water leaving the cup must be inclined to the direction of motion of the cup. For, otherwise, the stream at exit will interfere with the forward motion of the following cup. By assmning, in equation (14), the power and the velocity of the cup variable, and the other quantities constant, we find, by differentiating for a maximum, ^=M(Fi~2t;)(l+cos e) = 0; (W from which That is, for maximum power, the velocity of the cup must be one- half the velocity of the stream. STEAM TURBINES 295 Substituting in equation (14), this value for vi, we find, for the maximum power developed by the cup, P2'-M^(vi-^yi+cosii)=^^{l+ck, as represented in Fig. 60. •^■^^s impact, then, since for maxi- ^sup equals Vi/2, the absolute i^^^^^Dpifttifrifp^V is equal to Fi/V^. There- STEAM TURBINES 297 fore, since the efficiency is equal to the ratio of ihe kinetic energy absorbed by the cup to the kinetic energy of the stream, before impact, we find ^- 7i2 2 ^^^^ Hence, theoretically, the efficiency of an impulse turbine may vary between 50 and 100 per cent; depending upon the value of the angle 6. In practice it is attempted to have the angle 6 just sufficiently large so that the stream, leaving the cup, does not interfere with the following cup. In properly designed impulse wheels the actual efficiency may be, considering all losses, as high as 90 per cent. The results deduced clearly indicate, that in any case, it is essential, if a high efficiency is to be realized, to reduce the absolute velocity of the impinging stream, in going through the turbine, as nearly as ppssible, to zero. And this, in the case of steam turbines, is just as necessary a prerequisite for high efficiencies, as it is in the case of water turbines. The question, why is it possible, in the case of hydraulic motors, to convert so large a fractional part of the theoretical energy, due to the difference in topographical level, into actual work, and in the case of heat motors, so small a fractional part of the energy, due to the difference in " temperature level,'* naturally suggests itself. The answer is obvious. Every heat motor must act periodically. Even though the identical working substance is not used in the succeeding cycle, the result is just the same. For, the condition of the working substance, for the best results, must be at the beginning of each cycle the same as it was at the end of the preceding cycle. This is equivalent to cyclic operation. In the case of the hydraulic motor, however, the cycle is only partially completed. That is, the water, after having performed work, in falling through a certain height, is restored to its original condition by the action of the sun, which completes the cycle automatically. In other words, the water 298 THERMODYNAMICS at the height Hi, of the headrace, falls to the height H2, of the tailrace, and performs, theoretically, the amount of work w{Hi-H2); where Hi is the height of the headrace, H2 the height of the tail- race, and w the weight of water. But, to complete the cycle, the water must again be raised from the level H2 to Hi. This is done at the expense of the radiant heat from the sun, by means of which the water from streams, lakes, the oceans, etc., is evapo- rated and carried, by means of convection currents, to higher elevations, where condensation takes place, and the difference in elevation, Hi'-H2, is again established. 197. Flow of Fluids in Pipes of Varying Section (De Laval Nozzle). The flow of a gas or vapor, under steady conditions, Fig. 61. through a pipe of varying cross-section, is very similar to the flow of a liquid under similar conditions; but, the flow of a gas differs materially from that of a non-compressible liquid in two respects. That is, in general, the weight of the gas, or statical head, is negligibly small in comparison with the pressure head and velocity head; but, on the other hand, account must be taken of the expansion. Consider a pipe CC, such as is represented in Fig. 61, and assume a steady flow, i.e., the mass of gas entering the section at C, for a given interval of time, is equal to the mass leaving the section at C, during the same interval of time. And further, that the gas flows, without friction, in straight stream lines. Let m be the mass of gas that enters the channel at C, per unit STEAM TURBINES 299 time, with a speed Sa, and pressure pa, then an equal mass will leave the channel at C, during the same interval of time, with ' some speed St, and pressure p^. Assiune the pipe CC, to be divided into an indefinitely large number of sections, such that the thick- ness of each section is indefinitely small. Let the pressures on the left-hand side of the various sections, be, respectively, P«> P2, pa, . . . Pn-l, Pn] and on the right-hand side of the sections, P2, ps, P4, . . . Pn, Pb. Representing the respective cross-sectional areas by Ao} -^.2) -A3, . . . A-n) -Aft, and the corresponding thicknesses of the elements by dsa, ds2f dssy . . . dsn, then the work done by the positive pressiu-es, during the time that the displacement dsa takes place at C, and the displacement dsn takes place at C, is W = PaAad8a+P2A2d82+ ' ' ' +Pn^lAn-ldSn-l+PnAndSn. . (18) Likewise, the work done by the back pressures is W''=—p2A2ds2'-p^A^dsz—'''—pnAndsn—pbA^4sb^ . . (19) Taking the sum, of equations (18) and (19), we find, for the net work done, due to change in pressure, Wi = W'+W=PaAadsa-pi>Ai4si>. . . . (20) Changing, in equation (20), the subscripts a and 6 to 1 and 2, we have Wi=piAidsi'-p2A2d82 (21) 300 THERMODYNAMICS If dsi is the distance passed through^ at C, in the time dt^ and ds2 the distance passed through at C in the same interval of time, then dsi = sidty and ds2=S2dt; where si and 82 are, respectively, the initial and final speeds. Substituting, in equation (21), these values of dsi and d82, we obtain Wi = (piAi8i-p2A282)dt (22) Since, m is the mass of gas, flowing per unit time, we have, for steady flow, A181 A 282 where vi and V2 are, respectively, the volumes per unit mass of the gas corresponding to the pressures pi and p2. Substituting in equation (22), we obtain Wi = {viVi'-p2V2)mdt (23) The change in kinetic energy is and expressed in engineer's units, this becomes W2 = ^-^^mdt (24) The work due to expansion is, Wz = (mdt)]vdv] (25) and for the assumed conditions, Wi+W2+Wz-=0. STEAM TURBINES 301 Hence, S^ So C*^ (piwi —p2V2)mdt-\ H — mdt+ (rndt) I pdv=0; from which — 2^ — =PiVi-p2V2+j pdv (26) But, rrPi pdv= I vdp; hence, 82^ -Si^ f Pi , .^^. -"^^=Jp/^P (27) If, now, the process be such that the equation pv^=k L IL holds, then v=^k^p »; and by substitution, equation (27) becomes 52^- — = fcnj p ridp = —^kn{pi n -pg n ). . (28) 2g 1_ Substituting for k^ its value, we find 82^ — 81^ n L !L=i !Lii. n Or, since L L L i_ equation (28) may be simplified by dividing by i» , and multiply- ing the first term in the parenthesis by pi» vi, and the second term by ^2" f2. Performing this operation we find —2^ ;^(Pifi-P2r2) (30) 302 THERMODYNAMICS If the pipe is curved, then the pressure on the convex surface is less than on the concave surface. However, unless the change in direction is considerable, the difference in pressure is very small. The foregoing conclusions are the result of a modification of of Bemouilli's theorem; i.e., applying the theorem to a compress- ible fluid of negligible weight. If the initial speed is negUgible, as is the case when discharge takes place from a comparatively large vessel, then si, in equation (29), may be omitted, and we have n-l 82^ 2g Solving for 82, we find -„-V.{i-(i!) } (31) Now, the mass of fluid conveyed, per unit time, through any section is As m= — : V where A is the area of the section, s the speed, and v the volume per unit mass. But for steady flow this is a constant throughout the pipe. Hence we have ^^As^A^2 (3 V V2 where A 2 is the area corresponding to the pressure p2, and V2 the corresponding volume per unit mass. By combining equa- tions (32) and (33) we find n-l 1 A2S2_A2\ 2gn m V2 V2 [„^.''-{-(i:) " W- <3« Assuming again, the flow to be such that the equation pi>"=pi»i" (35) STEAJVI TURBINES 303 holds, we find Substituting this value of V2 in equation (34), we find Since, as previously stated, m is constant, equation (34) may be written 2 n+l 1 m -t-4»^F:{©-'-C-.)"}f^ <=" where A is any section, p the corresponding pressure, and v the corresponding volmne per unit mass. Since As/v is a constant, s/v must be a maximum, when A is a minimum. But, s/v becomes a maximum, and hence, A a minimum, when the value of the expression included in the bracket, of equation (37), becomes a maximmn. Hence, we may write and 2-n 1_ ^dx_2 p n n+l p^ _^ Pin pi n from which P M^+iY'^ (38> The value of p, as given by equation (38), is that value which makes s/v a maximmn, and, therefore, A a minimum. Substitut- ing this value of p, for p2, in equation (32), and reducing, we find 304 THERMODYNAMICS Finally, by means of equations (35), (38), and (39), we find ^=^«=d(i5Lp,„,)LA(^pi(4-y"^F. (40) V v\n+l I [n+lt;i\n+l/ J ^ ^ Equating the right-hand members of equations (36) and (40), we find 4? A n+l\n+l/ kg)" -©' (41) which gives the ratio of the area, for the pressure p2, to minimum area. Solving equation (38), on the assumption that the fluid is saturated steam, for which n equals 1.135, we find p=0.577pi. By means of equation (32), the final speed may be determined, when the initial and final pressures are known; and with the aid of equation (37) any one of the three quantities, viz, m, A, and p, may be found, if two of them are given. Since, in deducing the foregoing equations, we equated work, expressed by the product of pressure and volume, against energy, expressed by the product of mass and square of the speed, we must in substituting numerical values, in these equations, use the same system of units. As an example, if in equation (39), s is to be given in feet per second, pi must be given in lbs. per square foot, and vi in cubic feet. Reducing equation (39), so that pi is expressed in lbs. per square inch, and substituting for g and n the proper values, we find /2X32.2X1.135^-,. \2 s= — -. ,^- Xl44pii;i I =70.2(piri)J. \ Z.ioO / STEAM TURBINES 305 For steam under a pressure of 100 lbs. per square inch, the volume per pound is approximately 4.43 cu.ft.; hence, by substitution, we find 5 = 70.2(443)i = 1478 ft. per sec. If the pressure be 200 lbs. per square inch, for which the volume, per poimd, is approximately 2.29 cu.ft., we find 8 = 70.2(458)1 = 1503 ft. per sec. These two computations show that the variation in speed is small when compared with the variation in pressure. Reducing equation (40) in a similar manner, we find 2 1 A J 2X32.2X1.135 / 2 \o.i35 p, ^"144( 2.135 V2.135/ ^^ t;i 2 1 = 0.30A ( — j pounds per sec. ; where A is now in square inches. 198. Two Principal Types of Turbines. From the previous discussions on the flow of steam through pipes it is obvious thet the speed of flow, for any considerable difference in pressure, is very high; and that if any single-stage turbine, i.e., a turbine consisting of a set of nozzles and only one rotating part, were to « utilize practically all the kinetic energy of the steam, due to its speed at nozzle exit, the speed of the turbine would have to be abnormally high. As an example, some of the De Laval tur- bines, which were single stage, had speeds as high as 40,000 r.p.m. Though the efiiciencies of the De Laval turbines, from the stand- point of steam consumption, were not exceptionally low, the enormously high speeds were a serious disadvantage. For, in no other mechanical contrivance, not even dynamo electric machines, which are operated at relatively high speeds in com- parison with other machines, are such high speeds ever approached. 306 THERMODYNAMICS Therefore, to utilize the power developed by a single-stage turbine it is necessary to employ a reduction gear. But, a reduc- tion gear means an additional first cost, and a lowering of mechan- ical efl&ciency. Furthermore, proper lubrication becomes exceed- ingly difficult when machines are operated under speeds such as are attained by single-stage tm'bines. The difficulties, however, stated in the preceding paragraph, were overcome by the introduction of nudti-stage turbines.* That is, by allowing the steam to act successively upon the rotors of a multi-stage tiu-bine, its speed is gradually reduced, and the speed of the turbine need not be abnormally high. The turbine must of course be so designed that the steam expands, and the temperature is reduced continuously to the lowest possible value at exit. That is, the kinetic energy of the steam at exit must be, as nearly as possible, equal to zero. , There is then the choice of the following types of multi-stage turbines: Combined impulse and reaction, and impulse. 199. The Parsons Turbine. The Parsons turbine, at the pres- ent time, represents one of the commercially successful types of turbines; and may be considered a combined impulse and reaction turbine of the parallel-flow type. That is, the steam passes through the first set of guide blades approximately parallel to the shaft of the turbine, and has given to it the proper direction so that it may enter into the channels of the first rotor without shock. At exit from the first rotor, the steam enters a second set of guide blades, where it is again directed so as to properly enter the chan- nels of the second rotor, etc. In this way the steam reacts, expands, and falls in pressure continuously as it travels, from * There appears to be considerable confusion in regard to the meaning of the word '' stage." In some cases authors designate a turbine, as an n-6tage turbine when there are n rotors, which is consistent with the nomenclature employed in the case of hydraulic turbines. In other cases, however, namely the Curtis turbine, by number of stages is meant the number of sets of expanding nozzles. STEAM TURBINES 307 entrance to exit, through the turbine. Since the steam is con- tinually expanding, the length of the blades and spacing, for both guides and rotors, must be increased so that the ratio of steam speed and blade speed, upon which the efficiency of the tin-bine depends, is maintained constant. 200. The Curtis Turbine. The Curtis turbine is of the impulse type. The steam expands in a set of nozzles, where the pressure head is converted into velocity head, and then impinges on the ciUT^ed blades of a rotor. Part of the kinetic energy of the steam is absorbed by the first rotor; the steam then at reduced speed passes through a set of guide blades where it is directed so as to properly enter the channels of a second rotor, where the speed is still further reduced, etc., until, finally, the speed is very low. The steam is then expanded through a second set of nozzles, and again passes through a series .of rotors and guides, precisely as in the first stage. This is continued until the pressure of the steam has been reduced to the desired exhaust pressure. The niunber of stages, other things being equal, depends, of course, upon the range in pressure. Due to the fact that the speed of the steam is reduced in each rotor, the passages traversed by the steam must be continuously enlarged. This is brought about by reducing the curvature of the blades as well as lengthening them.* 20l. Comparison of Parsons and Curtis Turbines. Since the speed of the steam entering a Parsons turbine is moderately low, and for high efficiency its absolute velocity at exit must approach zero in value, it follows that the relative velocity must be high. That is, the relative velocity at exit being, approximately, the vector difference between the absolute velocity at entrance and the velocity of the wheel, it follows that, since the velocity of the steam at entrance is low, the relative velocity at exit will be high, and therefore, the velocity of the wheel must be high * For a comprehensive discussion on the design and testing of turbines see "The Marine St^am Turbine" by J. W. Sothern. 308 THERMODYNAMICS in order that the absolute velocity at exit may be low. On the other hand, in the case of an impulse turbine, the velocity of the blade, for the best efficiency, is approximately one-half that of the entering steam. Hence it is obvious that, other things being equal, the Parsons turbine is inherently a higher speed prime mover than is the Curtis turbine. 202. Turbines and Reciprocating Engines. No matter how operated the steam turbine is inherently a high-speed prime mover; and since power is proportional to the product of torque and angu- lar velocity, it follows that for equal output, the steam turbine, with its high rotative speed, will be of smaller dimensions than a reciprocating engine. Furthermore, where rotative motion is required, which is usually the case, the turbine needs no connect- ing rod and crank, as does the reciprocating engine. Again, where electric generators are direct connected, as in power plants, high speeds, up to a certain point, are desirable. Since, the power developed is equal to the product of e.m./. and current, high-speed generators, for equal output, will have a lower first cost and occupy less floor space than low-speed generators. Finally, the turbine has the further mechanical advantage of having a uniform turn- ing moment. On the other hand there are certain cases where low speeds are essential either to successful operation or economy; under such conditions the reciprocting engine is superior. As an illustration of this we may consider present conditions in marine engineering. As previously stated, the turbine, using high pres- sure steam is, for high efficiencies, inherently a high-speed prime mover; on the other hand the propeller of a ship, is, for high efficiencies, inherently a low-speed mechanism. On passenger liners, the increased rates, which passengers are ready to pay for a reduction of time in transit, more than pay for the increased cost of operation. However, on freight steamers, such is by no means the case; and it appears that for such steamers the recipro- cating engine combined 'with a low-pressure tm^bine, as regards economy, is at least equal if not superior to the turbine. STEAM TURBINES 309 Thermodynamically, the steam turbine is far superior to the reciprocating engine. For, in the turbine there is no alternate heating and cooling of the surfaces with which the steam comes into intimate contact. In other words, in the case of a turbine, very shortly after starting, steady conditions will prevail; and the incoming steam, therefore, does not come into contact with sur- faces which have been previously chilled by the low-temperature exhaust steam. That is, the steam changes gradually in pressure and temperature from admission to exhaust. And this means that w/yy///////7/^?77^>^ , V Fia. 62. there is no condensation excepting that due to radiation. Hence, in a turbine, condensation is largely eliminated in comparison with a reciprocating engine; and herein lies one of the great factors that makes the turbine thermodynamically superior to the reciprocating engine. Another important factor is the fact that in a turbine a good vacuum is utilized to much better advan- tage. This is illustrated by Fig. 62. Let, in the figure, ABODE be the indicator diagram of a reciprocating engine operating between the pressures as indicated by the points A and E, Then the net work done by the engine is measured by the area ABODE; and the net work realized by means of a turbine, for the same limits in pressure, is measured by the suresiABOFE. If, now, the back pressure be reduced, from that as represented by the line 310 THERMODYNAMICS ED, to that represented by the line HI, the net gain in work, by means of a reciprocating engine, is measured by the area EDIH, and that, in the case of a turbine, by the area EFGH, That is, due to mechanical considerations, the length of stroke of the reciprocating engine is fixed; and hence, full expansion cannot be realized. But, in the turbine full expansion is realized and the toe^ of the indicator diagram is utilized in doing useful work. This gain in work, in the case of low-pressure turbines, is quite appreciable. 203. •Turbine Tests. It has been found impossible, up to the present time, to devise any method by means of which to determine the indicated power of a tm-bine, in the same manner that the indicated power of a reciprocating engine is determined. There is, however, no difficulty experienced in determining the output, or brake power. The output is determined in precisely the same manner as described in Art 133; and the thermal effi- ciency is determined as described in Art, 136. If it be desired to determine the commercial efficiency, the method of procedure is precisely the same as that described in Art, 137. The com- parison frequently made is that between the actual output of the turbine, and that which would have been obtained on a Rankine cycle, as discussed in Art, 138. 204. Reciprocating Engines and Low-pressure Turbines. In 1906, H. G. Stott presented a paper* before the American Institute of Electrical Engineers on " Power Plant Economics," which gave a complete analysis of the various losses, from the coal bunkers to the bus-bars, of the Interborough Power Plant, located at Fifty-ninth Street and Eleventh Avenue, New York City. The prime movers employed at that time were of the Manhattan type compound Corliss engines; two engines being connected to one generator of 7500 K.W. maximum capacity. The following quotation is an extract from Mr. Stott's paper, "Power Plant Economics," Transactions of the A.I.E.E., Vol. XXV. STEAM TURBINES 311 which is one of the most complete and instructive analyses that has ever been made of a power plant: " Three years ago the steam-power plant for the generation of electricity had apparently settled down to an almost uniform arrangement of standard apparatus in which one power plant diflFered from another only in details of construction of engines, generators, and auxiliaries. As only about twenty years had then elapsed since the first central station was put in operation on a commercial basis, this uniformity of design seemed to indicate that in the near future it would only be necessary to purchase a standard set of power-plant drawings, and make the necessary changes in size of units in order to have a station of the best type known to the art. " The internal combustion or gas engine had from time to time been brought forward as a candidate for the position of prime mover, with every prospect of improved economy in fuel consumption; but with the exception of a few special instances it was not looked upon with favor, as shown by the almost universal use of the steam engine. " After a long period of development a new factor in power- plant design; namely, the steam turbine, was placed on the market in commercial sizes. It is safe to say that during the last three years no other piece of apparatus has had so stimulating an effect upon the power plant. Its effect upon the entire plant has been most beneficial, for it has revived the apparently moribund superheater. This has now been so developed and improved that superheat of 200° or 300° fahr. can be safely and economically obtained. With the development of the superheater further study of the problem of combustion has improved the efficiency of the furnace; and this most important subject is apparently susceptible of still further development. " One other important result of the steam-turbine develop- ment has been the development of condensing apparatus to such a point of efficiency that a vacuum within one inch of the simul- 312 THERMODYNAMICS taneous barometer reading can now be maintained without difficulty. *' Another change in the power plant has been the reversion to high-speed generators, resulting in decreased cost of the gen- erator and its f oimdations, as well as saving in floor space. " Last but not least the steam turbine has put the recipro- cating engine and the gas engine on the defensive and has actually been unkind enough to throw out hints in regard to the applica- tion of Dr. Osier's proposed methods to the treatment of older apparatus. " The reciprocating engine and internal combustion engine have not been slow in accepting this challenge; they have responded by showing so improved an economy (especially in the gas engine) that the situation has become most interesting to the power-plant designer. It is safe to say that the develop- ments of the next ten years will show very marked improvement in power plant efficiency. " In regard to this development the author wishes to direct attention to the basic fact that in power plants one should not look merely for increased efficiency in the prime mover, but should also investigate and analyze the entire plant from the coal to the bus-bars: first, in regard to efficiency; secondly, in regard to the effect of load-factor upon investment; and thirdly, the effect of the first and second upon the total cost of producing the kilowatt- hour, which is the ultimate test of the skill of the designer and operator. '* Efficiency. " In Table 1 will be found a complete analysis of the losses found in a year's operation of what is probably one of the most efficient plants in existence to-day and, therefore, typical of the present state of the art. STEAM TURBINES 313 (( Table No. 1 ANALYSIS OF THE AVERAGE LOSSES IN THE CONVERSION OF ONE POUND OF COAL INTO ELECTRICITY. 1. B.T.U. per pound of coal supplied . . 2. Loss in ashes 3. Loss to stack 4. Loss in boiler radiation and leakage . 5. Returned by feed-water heater 6. Returned by economizer 7. Loss in pipe radiation 8. Delivered to circulator 9. Delivered to feed-pump *10. Loss in leakage and high-pressure drips 11. Delivered to small auxiliaries 12. Heating 13. Loss in engine friction 14. Electrical losses 15. Engine radiation losses 16. Rejected to condenser 17. To house auxiliaries Delivered to bus-bar B.T.U. Per Cent. B.T.U. 14150 100.0 340 3212 1131 441 3.1 960 6.8 28 223 » 203 152 51 31 111 36 28 8524 29 15551 109.9 14099 14099 99.6 1452 10.3 Per Cent. 2.4 22.7 8.0 0.2 1.6 1.4 1.1 0.4 0.2 0.8 0.3 0.2 60.1 0.2 99.6 ti Discussion of Data in Table 1 " Item 1, Bi.u, per Pound of Coal Supplied, The thermal value of the coal used is evidently of prime importance, as it affects the cost efficiency of the entire plant. The method of purchasing coal used in the plant from which this heat balance is derived is that of paying for B.t.u. only, with suitable restrictions on the maximum permissible amount of volatile matter, ash, and sulphur. '' A small sample of coal is automatically taken from each filling of the weighing hoppers, so that the final sample represents a true average of a boat-load of coal. This final average sample is then pulverized and tested for heat value in a bomb calorim- eter, after which a proximate analysis is made of another por- 314 THERMODYNAMICS tion of the sample. This method of purchasing coal has been in use for two years, with highly satisfactory results. " Item 2, Loss in Ashes. It is doubtful whether a further saving in this item can be made, as the extra care and labor necessary to accomplish any improvement would in all probability oflFset the saving in coal. " Item 3. Loss to Stack. This is one of the most vulnerable points to attack, as the loss of 22.7 per cent, is very large. Recent investigations show that promising results may be obtained by the use of more scientific methods in the boiler room. In prac- tically all cases iUwill be found that this loss is due almost entirely to admitting too much air to the combustion chamber, result- ing in cooling of the furnace. This result is usually produced by '* holes " in the fire; these " holes '' may be due to several causes^ but usually are due to carelessness on the part of the fireman. " Fortunately, a very valuable piece of apparatus has been placed upon the market in the shape of a CO2 recording instru- ment. The results of a series of tests made with this instrument are shown in Figs. 63 to 66. " Fig. 63 shows the average condition of a furnace using small sizes of anthracite, with forced draught. The conditions are such that approximately 40 per cent, of the thermal value is being lost. " Fig. 64 shows what improvement may easily be obtained by watching the CO2 record, and indicates a saving of about 19 per cent, over the previous case. '* In the combustion of the small sizes of anthracite it is neces- sary to use a draught of not less than 1.5 in. of water; this breaks the crust of the fire in the thin spots, allowing the air to come through in such volumes that an enormous amount of heat is wasted in raising the temperaturie of the surplus air and at the same time causing inefficient combustion in the entire furnace. " Fig. 65 shows a record taken from a stoker boiler whilst the recorder was covered up to prevent the fireman from seeing the record. STEAM TURBINES 315 ^ _ V^ ^ - — ^■" - § ■^ "^ •^ 1 - 5 1 - ( 4 *=^ 1 1 p ?• ., ( ^ :^ ^ — * PER CENT. COs RECORD DEC. 18-19, 1905, BOILER NO. 29. AVERAGE PER CENT. CO, »4.7 HAND-FIRED, NO. 2 BUCK, UNDER FORCED DRAUGHT. - ^ « < — -^ ^N > _*i ^ ^ > 1 >, c. .^ 1. ^ ^ r > - J f 1 f X 4 ^ >• ^ L V ►^ 1 <:^ V % ' \ __ ^ i "^■— \ S _- -^ ► - * w f — — ' -^ 1 - < i; 3 rt - » - ©« a ^ CO UJ ^ - J^ = s - A - «D - lO - 09 M 0< oo aoviNaoaad !•• OO 316 THERMODYNAMICS 1 J L — ••^ " — ^. PER CENT. COf RECORD DEC. 19-20. 1905, BOILER NO. 29. AVERAGE PER CENT. COt =7.6 HAND-FIRED, FORCED DRAUGHT. CAREFUL OPERATION. - , ^ •^ ► - jt — 1 . < ^.^ " • s^ ^^^ =■ ^ — -— C ps - — • 1 — 5 ^. __ ^ ?• . ^ 5 " — 1 "^ ~ _ ^ — I^ c = :2* 1 — - "~ >* 1 > - 4 ?• = ^25, 1 " ( — ^ -^ 1 5 -^ - 1 1 ^ >. - ;?• ^ ^. ' ^ . "^ -^ * ^^^ p - 1 •< ^ 1 1 1 •^^ ■— - — — ' '^ ^9 -S - 1"- - eo - o» 3 111 ^ ^ rn -s - oo oo e« o« i« 00 a» 30VXN30U3d 9 » STEAM TURBINES 317 9 ^ - ^ ' PER CENT. CO,- RECORD NOV. 28-29, 1905, BOILER NO. 31.- AVERAGE PER CENT. COa->6.8 — ::J - ^^ N— o — r s < 1 « 0» ^^ "^ :> aa 00 ^ -T e- 2: 1 «o ' ^ c to **^ y 1 « - S ( S • 0» Zl _^ =- p 00 — -— ^~~ !>• • — > — "-i - < 1 «D PER CENT CO2 RECORD DEC. 22-23, 1905-BOILER NO. 31 AVERAGE PER CENT C02«11.4 STOKER OPERATED r "^ ^ i- ^ ^ ■^ •* 1 — - > eo iC M ~ - — — ? •• «H CO UJ ^ ^•— •^ 1 C -^ 1:3 1 k ;a r — S I a» I — 1 ' 00 1 > ti« — ' « = — — "^ ~ 5^ 10 • «£ "^ — =* •* — - - eo J -^ "^" OS "^ ^ 1- / I o 1 '/ 8 i / / / / / f

x^ / / 8 ^ X .-^ L^' ./ / -^ ^ ^ ^ X 0) ^ --" ^ ^ -— -^ ^^ .^ '^ f-> -<- _ -^ "^ 3 Ul o Q. CO S 6 3 C -I lij 3 U. 9* S CO ;s ?3 2 • aNnnoA 'jln30 uBd : *oo M STEAM TURBINES 321 " Mr. W. H. Patchell,* of London, who recently visited us, has introduced very large boilers, assembling two in one setting; each boiler has a normal evaporation of 33,000 lb. per hour and in this way has cut down to a minimum the radiating surface per square foot of heating surface. He has also introduced the iron case with magnesia lining, and with good results. " The question of boiler leakage is one in which the choice of the lesser of two evils is necessary; for in the tubular or cylindri- < o o > ce a 2 III Z < b. 10. 9.5 o ft 2 < > Ul z Id < > 5 a 111 "" ■" ^ "■ "" "™ ^^ ^^^ ■~ ■^ T" ■" ^ — — ' — r- - ~~ /" ^ J CURVES SHOWING RELATIVE ECONOMIES ATOIFFBRENT DRAUGHT IW THE COMBUSTION OF SEMI-BITUMINOUS COAL UNDEA A BABCOCK AND WILCOX BOILER.UPON A RONKY STOKER. / \ L / I — 1 \ — / V / \ / ft ^ 1 > ^ ~ ~ I > V ~ ^■^ ^^ > V ^ S - *■ ^ ~ '^ ^ ^-^ ~ ^ ^ ~ '^^ >- „--■ A n «^ ^ ^ "^^ ^ -f ^ ^ ^ ~ ^^ ^^ r- "^ •>i« ~" ?* ^ >--. [^ y ~ h- 4 ~ ^^~ ^^ J. \— ~~ h— J L ■~ "" / / V- ~ / X ^— ~ ■~ .1 0. 8 0. 3 1 DP 0. lAI 4 JG IK r: 0. IN 5 CI HE ( .sv ).6 VA1 rE R. 0. 7 0, .8 0. 9 1.0 400 s Ul <0 CC o X Ul -I O a Fig. 68. cal boiler the leakage will imdoubtedly be less than in the water- tube type, owing to the smaller number of joints in the water space. But these two advantages are offset by the increased difficulty of construction, and the danger of using large boilers of the tubular type, especially with high-pressure steam. " It is now generally admitted that there can be no more difference in the efficiency of different types of boilers under * See paper read December 7, 1905, before the Institution of Electrical Engineers, by W. H. Patchell. 322 THERMODYNAMICS similar conditions than there can be in electric heaters, press agents to the contrary notwithstanding. " Item 5. Returned by Feed-water Heater. ^ The importance of getting the feed water to the maximum temperature obtainable is generally recognized, and would seem to indicate that all auxili- HOUR - 23-HOUR RUN, JAN. t-2-06 : L 2 S 1 i i 5 6 1 r 8 { 1 10 11 12 13 14 15 16 17 18 19 20 21 2S 23 20 111 GRAPHICAL LOG OF AVERAGE READINGS TAKEN ON BOILER NO. 31, JAN. 1, 2, 1906. s 3l5 o > — 8io 1- ■^ \ — — . y \ - \ A / z UJ 2" tlJ ' \ / \ s_ / 0. \ ^ OlOO si 50 CO 0.25 j^0.20 5£o.l5 ,^_ <. J X \ /^ "-~ ■■"" \ y / \ J / ^ \ / s X / — \ \ """" "^ '^ i \ • <5 **•*" 3oi \ ^0,05 \ \ 1 J Fig. 69. aries should be steam driven so that their exhaust may be utilized in the feed-water heater; in this way the auxiliaries may operate at about 80 per cent, thermal efficiency. *' Item 6, Owing to the difficulty of pumping water at tem- peratures above 150 degrees fahr., when under pressure, it becomes necessary to install economizers for the purpose of increasing the feed-water temperature to 200 or 250 degrees fahr. STEAM TURBINES 323 As this increase of temperature is obtained from the waste gases at no expense for fuel, it only becomes necessary to consider the load-factor, as will be shown later, in order to decide whether economizers should be installed or not. In practically all cases where the load factor exceeds 25 per cent, the investment will be justified. " In deciding upon the size of economizer to be installed it is important to consider first, the influence of the economizer upon the available draught due to the obstruction it oflFers and also due to the reduced stack temperature; the second important consider- ation is to equate the interest and depreciation charges against the saving in fuel, and so determine the amount of investment justified in each particular case. " Item 7, Loss in Pipe Radiation, By the use of two-layer pipe covering, each layer being approximately 1.5-in. thick, and sections put on in such manner that all joints are broken, the radiation losses have become practically negligible. " Items 8 and 9. Heat Delivered to Circulating and Bailer-Feed Pumps. As these auxiliaries may be either electrically driven or steam driven it is interesting to note that the thermal efficiency of the electrically-driven pumps would be equal to the thermal efficiency of the plant, multiplied by both the efficiency of con- version from the alternating to direct current and by the motor efficiency. In this case, there would be a net thermal efficiency of 10.3X0.93X0.90=8.63 per cent., whereas the thermal efficiency of the steam-driven auxiliary discharging its exhaust into a feed- water heater at atmospheric pressure would be approximately 87 per cent. " Item 10, Loss in Leakage and High-Pressure Drips, The loss in leakage should be infinitesimal, and the high-pressure drips can be returned to the boilers, so that practically all the loss under this heading is recoverable. ^^ Items 11 y 12 y and 17 are probably unavoidable and of so small a magnitude as not to merit much consideration. 324 THERMODYNAMICS " Item IS. Loss in Engine Friction, Recent tests of a 7500- h.p. reciprocating engine show a mechanical efficiency of 93.65 per cent, at full load, or an engine friction of 6.35 per cent. As this forms only 0.8 per cent, of the total thermal losses it is relatively unimportant. Attention is called to the method of lubricating all the principal bearings by what is known as the flushing system, whereby a large quantity of oil is put through all the bearings by gravity feed from elevated oil reservoirs common to all the imits; after passing through the bearings the oil is returned by gravity to oil filters in the basement and then pumped up to the reservoir tanks again. About 200 gallons per hour are put through each engine, and of this quantity only about 0.5 per cent, is lost. This method of oiling undoubtedly contributes to the general results. " Item 14- As large electrical generators can now be obtained which give from 98 to 98.5 per cent, efficiency, it would seem as if the limit in design had been reached and that hereafter the ' problem of design is to be merely one of altering dimensions to suit varying sizes and speeds. While this is true as far as the efficiency is concerned, other problems are continually arising, such as the design of generators for an overload capacity of 100 per cent, to meet the demand for apparatus capable of taking care of great overloads economically for short periods, corresponding to peak loads of a railroad or lighting plant. *^ Item IS. Engine Radiation Losses. This source of loss has evidently been reduced to a negligible quantity by the use of improved material and methods of heat insulation. " Item 16. Rejected to CondenseTf 60.1 per cent. This imme- diately introduces the thermodynamics of the steam engine, a subject so broad that it will be impossible to do more than touch upon some of the most important points in considering steami- engine efficiency. " The efficiency * of any heat engine can be expressed by the • Defined as ideal coeflficient of conversion, Art. 89. — ^Author. STEAM TUBBINES 325 ratio of ^=— i~ — 2. where Ti is the absolute temperature of ii the steam entering the engine and T2 the absolute temperature of the steam leaving the engine. Thus in the engine whose steam-eonsmnption curve is given in Fig. 70, if the initial pressure is 175 lb. gauge and the vacuimi at the low-pressure exhaust 837-560 nozzle is 28 in., then the maximum thermal efficiency is 837 =33 per cent. This would be true for any form of engine or turbine working between the same temperature limits. ^ ^ ^ ^ imm ■^ ^ ■■" "" ^ ^ "" ^ — ■H ■" ~ " 4 T. l_ L \- t A. ECONOMY CURVE FOR 7 500 H.P. ENGINE LOAD EQUALLY DIVIDEO BETWEEN CYLINDERS. B. ECONOMY CURVE POR 7 600 H.P. ENGINE LOAD UNEQUALLY DIVIDED BETWEEN CYLINDERS, > / 20 1 t / oe J ^ T T / • / 1. / h / J r 1 J f — sc J r 0: J r Id J r (L y r / ^18 / r * J > / J f ^ J ^ J f > 0: ^^ ^ ,/ r :5 S. y ^ /" > i^ • y* < ^ X; s A. - - ^ •^ ^ -^ ii N s »■•■ V a " - 17 ^ — — -" ^ _ _ _ _ _ ^^ ^ ^ 4000 5000 6000 LOAD: KILOWATT- HOUP ( SWITCHBOARD READING) FiQ. 70. 7 000 " In Pig. 70, however, it is seen that the point of maximum economy shows a steam consumption of approximately 17 lb. per kilowatt-hour, which is equivalent to 20,349 B.t.u. per hour. One kilowatt-hour is equal to 3412 B.t.u. per hour, so that the actual efficiency of the steam engine and generator ig-Jil?. = 16.7 per cent. As the generator efficiency at this 20349 ^ 326 THERMODYNAMICS load is approximately 98 per cent, the net engine thennody* namic efficiency * is 7r^= 17 per cent. " The difference between the theoretical efficiency and the actual is then 33—17 = 16 per cent., of which 0.8 per cent, has already been accounted for in engine friction, so that the balance of 15.2 per cent, is due to cylinder condensation, incomplete expansion, and radiation. '' As the engine friction in a two-bearing engine with high- pressure poppet valves and low-pressure Corliss valves has by care- ful design been reduced to less than 0.8 per cent, gain cannot be expected here, so attention must be centered on the loss due to cylinder ccnJensation, etc., amounting to 15.2 per cent., in order to effect any improvement. '' Superheated steam is the only remedy at hand and with it we can probably effect an improvement of 5 or 6 per cent, by using such a degree of superheat in the boilers that dry steam will be had at the point of cut-off in the low-pressure cylinder. '' Any greater amoimt of superheat than this will merely result in loss to the condenser; for it should be remembered that the cylinder losses increase with the difference in temperature between the steam and exhaust portions of the cycle; in other words, the greater the thermal range of temperature the greater the condensation loss. This would seem to point to the use of more cylinders; but this involves additional first cost and fric- tion as well as more space and higher maintenance charges. " Fig. 71 shows what may be gained by reducing the temper- ature at the end of the cycle by means of increased vacumn, but in the case in point the maximiun vacuum obtainable in practice was used so that no additional economy can be expected in this way. * Defined as thermal efficiency under Art, 136. — ^Authob. .T Balance. ig reciprocating engines ;i|r.g;oiiS^j|f-S-. 12% ffl |i^jftJi&1Se2^S*ll^fce of superheat 6% ni^f* •]•"■»" -'2"§* ijji'.jjj. . j^. .j^. .^. .^. ^^: •*■ S^Cwi^t^^W^il^ efficiency of the entire 4|€n^;i^r|^e total thermal efficiency i«!apacity of the plant waa ■^ take care of the increas- Li^KcVraffic; and it was finally ■^I^£o operate on the exhaust 328 THEEM0DYNAMIC8 • The following quotation * indicates the all-around gain by this combination. " During the year 1908 it became apparent that owing to the ever-increasing traffic in the New York subway, it would be necessary to have additional power available for the winter of 1909-1910. " 2. The power plant of the Interborough Rapid Transit Com- pany, which supplies the subway, is located on the block boimded by 58th and_59th Streets, and by 11th and 12th Avenues, ad- jacent to the North River; it contains nine 7500-kw. (maximum rating) engine units, besides three 1250-kw. 60-cycle turbine units which are used exclusively for lighting and signal purposes. " 3. The 7500-kw. units consist of Manhattan-type com- pound Corliss engines, having two 42-in. horizontal high-pressure cylinders and two 86-in. vertical low-pressure cylinders. Each horizontal high-pressure cylinder and vertical low-pressure cylinder has its connecting rod attached to the same crank, so that the unit becomes a four-cylinder 60-in. stroke compound engine with an overhanging crank on each side of a 7500-kw. maximum rating 11,000-volt, three-phase, 25-cycle generator. The generator revolving field is built up of riveted steel plates of sufficient weight to act as a flywheel for the two engines con- nected to it. This arrangement gives a very compact two-bear- ing unit. The valve gear on the high-pressure cylinders is of the poppet type, and on the low-pressure of the Corliss double-ported type. '' 4. The condensing apparatus consists of barometric con- densers, arranged so as to be directly attached to the low-pressure exhaust nozzles, with the usual compound displacement circu- lating pump and simple dry-vacuiun pump. '' 5. These engine and generator units are in general probably the most satisfactory large units ever built, as five years' experience •"Testa of a 15,00(>-KW. Steam-Engine-Turbine Unit," by H. G. Stott and R. J. S. Pigott. Transactionfi of. the A.S.M.E., VoL XXXII. STEAM TURBINES 329 a with them has proy^; their uormal economic rating is 5000 kw., but they operate equally well (water rate excepted) on 8000 kw, continuously, '^ 6t In considering tjie problem of how to get an additional supply of power, every available source was considered, but by a process of elimination only two distinct plans were left in the field. " 7. The electric transmission of power from a hydraulic plant was first considered, but owing to the high cost of a double transmission line from the nearest available water power, and the impossibility of gett ng reliable service (that is, service having ft maximum total interruption of not more than ten minutes per annum) from such a line, further consideration of this plan was abandoned, " 8. The gas engine, while oflFering the highest thermo-dynamic eflBciency, at the same time required an investment of at least 35 per cent more than ordinary steam-turbine plant with a prob- able maintenance and operation account of from four to ten times that of the steam turbine, " 9, The reciprocating-engine unit of the same type as those already installed, was rejected in spite of its most satisfactory performance, on account of the high first cost and small range of economical operation. Reference to Fig, 72 will show that the economic limits of operation are between 3300 kw. and 6300 kw,; beyond these .imits the water rate rises so rapidly as to make operation undesirable under this condition, except for a short period during peak loads, " lOt The choice was thus narrowed dgwn to either the high- pressure steam turbine or the low-pressure steam turbine. There was sufficient space in the present building to accommodate three 7500^kw. units of the high-pressure type, or a low-pressure unit of the same size on each of the nine engines, so that the questions of real estate and building were eliminated from the problem, 330 THERMODYNAMICS "11. The first cost of a low-pressure turbine unit is slightly lower than that of a high-pressure imit, due to the omission of the high pressure stages and the hydraulic governing apparatus, but the cost of the condensing apparatus would be the same in both cases. The foundations and the steam piping in both cases 28 80 111 22 o25 ^ i ^ 21,x20 t g 19^10 o IS 5 17 16 < VARIATION OF WATER-RATE & BEST RECEIVER PRES8URE WITH LOAD . ORIGINAL ENQINE A:- WATER RATE B:- REC. PR. ^ > "^ ^ ^ y r^ y / ^ p X Ai-« . ^ \ 2000 8000 4000 5000 enqine load k.w. Fig. 72. 6000 7000 would not differ greatly. The economic results, so far as the first cost is concerned, would then be approximately the same, if we consider the general case only; but in this particular instance the installation of high-pressure turbines would have meant a much greater investment for foimdations, flooring, switchboard STEAM TURBINES 331 apparatus, steam piping and water tunnels, amounting to an addition of not less than 25 per cent to the first cost. '' 12. The general case of displacing reciprocating engines and installing steam-turbine units in their place was also con- sidered. The best type of high-pressure turbine plant has a thermal eflBiciency approximately 10 per cent better than the best reciprocating-engine plant, but the items of labor for operation and for maintenance, together with the saving of about 85 per cent of the water for boiler-feed purposes and the 10 per cent of coal, reduce the relative operating and maintenance charges for the steam-tiu-bine plant to 80 per cent, as compared to 100 per cent for the reciprocating-engine plant. ** 13. Assuming that the reciprocating engine plant is a first- class one and has been well maintained, about 20 per cent of its original cost (for engines, generators and condensers) may be real- ized on the old plant and so credited to the cost of the high-pres- sure turbine plant. But on the other hand, if the high-pressure turbine installation is to receive credit for the second-hand value of the engines, it must also have a debit charge for 100 per cent of the original reciprocatmg-engine plant which it displaced. The relative investments, therefore, upon this basis would be approximately equal for the high-pressure or the low-pressure turbine; but 80 per cent of the cost of the original engine plant would have to be charged against the high-pressure turbine plant, as against an actual increase in value (to the owner) of the engine by reason of its improved thermal efficiency, due to the addition of the low-pressure turbine. " 14. The preliminary calculations, based upon the manu- facturers' guarantees for the low-pressure and high-pressure turbines, showed that the combined engine-turbine unit would give at least 8 per cent better efficiency than the high-pressure turbine unit, so that it was finally decided to place an order for one 7500-kw. (maximum rating) unit, as by this means we would not only get an increase of 100 per cent in capacity, but 332 THERMODYNAMICS at the same time give the engines a new l^ase of life by bring- ing them up to a thermal efficiency higher than that attained by any other type of steam plant. " 16. The turbine installed is of the vertical three-^tage impulse type having six fixed nozzles and six which can be operated by hand, so as to control the back pressure on the engine, or the division of load between engine and turbine. An emergency overspeed governor, which trips a 40-in. butterfly valve on the steam pipe connecting the separator and the turbine and at the same time the 8-in. vacuum breaker on the condenser, is the only form of governor used. The footstep bearing, carrying the weight of the turbine and generator rotors, is of the usual design supplied with oil under a pressure of 600 lb. per sq. in. with the usual double system of supply and accumulator to regulate the pressure and speed of the oil pumps. " 16. The condenser contains approximately 25,000 sq.ft. of cooling surface arranged in the double two-pass system of water circulation with a 30-in. centrifugal circulating pump having a maximum capacity of 30,000 gal. per hr. The dry vacuum pimip is of the single-stage type, 12-in. and 29-in. by 24-in., fitted with Corliss valves on the air cylinder. The whole condensing plant is capable of maintaining a vacuum within 1.1 in. of the barometer when condensing 150,000 lb. of steam per hr. when supplied with circulating water at 70 deg. fahr. " 17. The electric generator is of the three-phase induction type, star-wound for 11,000 volts, 25 cycles and a speed of 750 r.p.m. The rotor is of the squirrel-cage type with bar winding connecting into common bus-bar straps at each end. This type of generator was chosen as being specially suited to the conditions obtaining in the plant. " 18. With nine imits operating in multiple, each one capable of giving out 15,000 kw. for a short time, operating in multiple with another plant of the same size, it is evident that it is quite possible to concentrate 270,000 kw. on a short circuit. If we STEAM TURBINES 333 proceed to add to this, synchronous turbine units of 7500-kw. capacity, which, owing to their inherently better regulation and enormous stored energy, are capable of giving out at least six times their maximum rated capacity, the situation might soon become dangerous to operate, as it would be impossible to design switching apparatus which could successfully handle this amount of energy. The induction generator, on the other hand, is entirely dependent upon the synchronous apparatus for its excitation, and in case of a short circuit on the bus-bars would automatically lose its excitation by the fall in potential on the synchronous apparatus. " 19. The absence of fields leads to the simplest possible switching apparatus, as the induction generator leads are tied in solidly through knife switches, whichare never opened, to the main generator leads. The switchboard operator has no control whatever over the induction generator, and only knows it is present by the increased output on the engine generator instru- ments. " 20. The method of starting is simplicity itself — the exciting current is put on the engine generator before starting the engine, and then the engine is started, brought up to speed and synchron- ized in exactly the same way as before. While starting in this way, the induction generator acts as a motor until sufficient steam passes through the engine to carry the turbine above synchronism, when it immediately becomes a generator and picks up the load. Three of these 7500-kw. low-pressure turbine units have been installed and tests run on Nos. 1 and 2. No. 3, having been just started, has not yet been tested. " 21. Instead of inserting in this paper the enormous accumu- lation of data incident to these tests, we have divided the paper into two parts in the hope that it would thus be more accessible for reference, the first part giving the reasons for adopting this particular type of apparatus, with a brief description of the plant and a smnmary of the results obtained, and the second part con- 334 THERMODYNAMICS taining all the principal data acquired during the tests, with sufficient explanation to make their meaning clear without refer- ence to the text." " 24. The net results obtained by the installation of low- pressure turbine units may be summarized as follows: ''a. An increase of 100 per cent in maximum capacity of plant. " 6. An increase of 146 per cent in economic capacity of plant. " c. A saving of approximately 85 per cent of the condensed steam for return to the boilers. " d. An average improvement in economy of 13 per cent over the best high-pressure turbine results. '' e. An average improvement in economy of 25 per cent (between the limits of 7000 kw. and 15,000 kw.) over the results obtained by the engine units alone. "/.An average unit thermal efficiency between the limits of 6500 kw. and 15,500 kw. of 20.6 per cent." 206. Summary. The two preceding quotations are self-explan- atory; hence no comment is necessary. But, before concluding, it must be remarked that the internal combustion engine and the steam turbine are still in the experimental stage; and that it is impossible to predict what the final adjustment will be. * It is true that the internal combustion engine has a higher thermal efficiency than has any other heat motor. But, due to complexity of construction, the internal combustion engine has a higher first cost; and furthermore, its regulation is inherently inferior to a reciprocating engine or turbine. Due to this, in spite of the fact that the reciprocating engine has a lower thermal efficiency, it still holds its place, on account of its simplicity and high over-load capacity; the latter bemg especially important in most power plants where it is necessary to take care of large "peak loads." It must be further remarked, that the installation of every power plant is finally affected by the economy of transmission. STEAM TURBINES 335 Whether power can be developed economically at any locality depends upon whether the cost of power for the particular locality is greater or less if developed at this particular point, or developed at some other point and transmitted to the point under considera- tion. This, of course, depends largely upon the economy of transmission. At the present time electrical engineers are giving consider- able attention to the subject of high-tension transmission. And if it develop that methods can be devised by means of which corona losses can be eliminated, or partially avoided, for potential differences far in excess of those employed at present, the subject of power plant economics will require revision. For, if corona losses can be avoided, the cost of power for any particular locality will be materially changed. And hence, the cost for the produc- tion of power will, likewise, be changed. To illustrate concretely: Assimie that it becomes possible to transmit with a potential difference of 300 kilo-volts instead of 125 or 150 kilo-volts. Under these conditions the economy of transmission is considerably increased; and the distances to which coal can be transported, to compete with the increased eflSciency of transmission, is considerably reduced. This, however, is not the only governing factor. Ground rent also influences the choice. That is, when the saving in transmission and the saving in groimd rent, by locating the plant at the coal fields, is balanced against the hauling of the coal, and the ground rent for a large city, it may develop that it is more economical to locate the power plant where the coal is mined. A similar argument, of course, applies to water-power plants. That is, the initial cost of a water-power plant is high, and therefore the distance, for a given potential difference, over which power can be profitably transmitted is limited; and, of course, the lower the cost of transmission, the greater the area over which profitable transmission may take place. Hence, as the potential difference, which may be employed in transmission, is increased, the smaller. 336 THERMODYNAMICS relatively, due to high ground rent, becomes the economy of a localized plant. Therefore, if it develop, that potential differences, far in excess of those employed at the present time, may be used, power plants in large cities, where ground rent is high, will dis- appear; and the future power plant will be located at the point where the raw material, for the development of power, is found. INDEX PAOB Absolute scale 43 Absolute zero 43 Adiabatic changes, change of dryness with 139 change of temperature with . .^ 59 Adiabatic equation for gases 50 Air, and adiabatic compression in compressor 235 and air compressors 230 composite diagram of compression, transmission, and expansion .... 260 compression and expansion 231 isothermal 233 constants of 230 loss of head in transmission pipes 254 Air compressor 230 throttling and other imperfections 249 Air motor, adiabatic expansion in 249 Air refrigerating machines 266 ideal coefficient of performance of 267 r-<|) diagram of 270 Air transmission system, theoretical efficiency of 262 Ammonia, coefficient of absorption 280 heat of dilution 281 Ammonia refrigerating machines, absorption 277 compression 271 Analysis of power plant losses 310 Anode 30 Boyle's law 41 departure from 43 Brake power 171 Brayton cycle 195 British thermal unit 9 Calorie 8 common 9 gram 9 mean 8 zero 8 337 338 INDEX PAOB Calorimeter 11 condensing 154 cooling constant of 14 Joly's differential steam 70 Joly's steam 69 separating 156 thermal capacity of 13 throttling 151 Calorimetry 11 Camot's cycle 97 a reversible process 100 coefficient of conversion 99, 109 steam operating on 116 with a perfect gas 104 Cathode 30 Centigrade scale 3 Characteristic equation 43 Chemical reactions of fuels 198 Clearance 156 equivalent length of ., 157 of air compressor 247 Commercial efficiency of steam engine 176 Compound engines, cross 189 tandem 183, 186, 188 Compressed air 230 Compressed air system, composite diagram of 260 theoretical efficiency of 262 Compression, adiabatic 235 isothermal 233 multi-stage 239 theoretical efficiency of 237 Condenser 121 Condensing calorimeter 154 Conduction 83 flow of heat along a bar 86 Conductivity, coefficient of 85 determination of coefficient of 86 in non-isotropic substances 89 non-homogeneous solids 89 of gases 90 of liquids 90 table of coefficients 90 Convection 83 Cooling, constant of calorimeter 14 Newton's law of 11 Stefan's law of 82 INDEX 339 PAGE Critical temperatures, table of 63 Cross-compound engine 189 Curtis turbine 307 Curves, theoretical and actual 163 Cushion steam 157 Cycle 76 Brayton 195 Camot's 97 with a perfect gas as a working substance 104 Diesel 196 four phase 193 of perfect steam engine and boiler represented by means of T-<|> diagram 137 Rankine's 177 reversible, as a standard 103 two phase 194 Cylinder feed 157 De Laval nozzle 298 Density and temperature of saturated steam 118 Dew point 66 Diesel cycle 196 Disgregation, heat of 62 Dissociation 28 Ebullition 61 Economics of power plants 310 and transmission 334 Efficiency and precompression 218 commercial 176 mechanical 175 of power plants and transmission 334 thermal 175 Elastic medium, propagation of wave motion 74 speed of 77 Elasticities, determination of ratio from speed of propagation of wave motion 77 isothermal and adiabatic and ratio of two thermal capacities . 72 Electro-chemical equivalent 31 Electrolysis 29 counter e.m.f. of 31 Faraday's statement 30 Emissivity 81 Endothermic 21 Energy, principle of 33 340 INDEX PAGI Engine and boiler, perfect) with steam 121 studied with aid of the T—^ diagram 137 dearanoe 166 internal combustion 192 actual indicator diagram of 215 actual p-v and T— diagrams of 222 double-acting cylinders 227 multi-cylinder 228 standard diagram 213 thermal efficiency of 214 !r-<|> diagram of 219 reversible 101 and refrigeration 110 as a standard 103 ideal coefficient of conversion 99, 109 simple thermodynamic 95 steam, condensing 150 double-acting 150 Elngines, compoimd 180 cross 189 double expansion 181 tandem, with large receiver 183 tandem, with small receiver 188 tandem, without receiver 186 triple expansion 190 heat 94 reciprocating and turbines 308 reciprocating, combined with low-pressure turbines 310, 328 Entropy 128 and temperature diagrams 134 change of 129 during reversible and irreversible processes 134 universal increment of 132 Evaporation 60 Exchanges, Provost's theory of 80 Exothermic 21 Expansion, adtabatic, in motor 249 change of drsmess during 167 double, in steam engine 181 external heat of 97 heat of 62 internal heat of 97 isothermal 233 triple, in steam engine 190 without doing external work 46 INDEX 341 PAGB ExpanBion, linear 34 coefficient of 34 factor of 34 table of coefficients of 36 voluminal 35 determination of coefficient of, for liquids 36 direct measurement of 39 Fahrenheit scale 3 Four-phase cycle 193 Frequency 76 Friction brake 171 Fuel, calorific value of 203 determination of 204 Fuels, and chemical reactions 198 and fuel tests 198 liquid 207 table of calorific values of 208 Fusion, heat of 22 I Gas, characteristic equation of 43 city 203 perfect or ideal 44 thermometer 44 Gases, adiabatic equation 50 change of temperature with adiabatic changes 59 conductivity of 90 elasticities of 72 expansion of, without doing external work 46 Gay-Lussac's law 41 general equations of 52 isothermal equation 41 table of constants 208 thermal capacities of 47 at constant pressure 48, 68 at constant volume 48, 69 determination of 68, 69, 70, 74 Gasoline 200 Gram calorie 9 Head, loss of, in transmission pipes 254 Heat, as a measiu'able quantity 7 content 125 effects of 33 engines or motors 94 flow along a bar 86 342 INDEX Heati mechanical equivalent of « 17 from constants of air 48 of dilution 281 of disgregation 62 of fusion 22 of vaporization 23 for water ; 65 production of 33 total, of steam 65 unit quantity 8 Heating machine 287 Humidity, absolute 67 relative 66 Hydraulic-radius 255 Hygrometry 66 Impact on curved surfaces 289 Indicated power 173 Indicator 169 Indicator diagram 161 and valve adjustment 161 comparison of theoretical and actual curves 163 ideal of internal combustion engine 210 standard of internal combustion engine 213 Internal combustion engine 192 actual indicator diagram of 215 actual p^ and T— 4> diagrams 222 and double-acting cylinders 227 efficiency and precompression 218 multi-cylinder 226 standard diagram of 213 thermal efficiency of 214 Irreversible processes 27 Isothermal equation 41 Isotropic 35 Kerosene 201 Mean effective pressure 174 Mechanical efficiency of steam engine 175 Mechanical equivalent of heat 17 from constants of air 48 Mixtures, method of 10 Motor, adiabatic expansion of air in 249 reheating air before expanding in 251 Motors, heat 94 INDEX 343 PAOB Multi-cylinder internal combustion engines 226 Multi-stage air compressors 239 Non-isotropic 36 Parsons turbine 306 Pelton cup 293 Power, brake ' 171, 214 indicated. 173, 214 Power plant economics 310 analysis of losses 310 efficiency and transmission^ 334 Precompression and efficiency 218 Pressure and temperature of saturated steam 119 Pressure, mean effective 174 Priming 126 Principle of energy 33 Radiation 80 Dulong and Petit's formula 82 Provost's theory of exchanges 80 Stefan's formula 82 Rankine's cycle 177 Reaumur scale 3 Reciprocating engines and turbines 308 engines exhausting to low-pressure turbines 310 Refrigerating machine 110 absorption 277 air 266 ideal coefficient of performance of 267 r-4) diagram of 270 commercial 266 commercial efficiency of 269 comparison of air and anunonia 284 compression, using volatile liquids 271 !r-4> diagram of 276 Kelvin heating 287 Refrigeration 265 and reversible engine 110 Refrigerator 94, 266 Resisted adiabatic expansion of steam 126 Reversible process, ideally 27 Reversible processes 25, 104 Separating calorimeter 156 Specific heat 9 344 INDEX PAcn Steam, behavior throughout the cycle 166 change of dryness during adiabatic expansion 126, 139 by means of T— 4» diagram 139 cushion 157 cylinder feed 157 double expansion engine 181 dryness during expansion in cylinder 167 engines, reciprocating and turbines 308 engines, reciprocating with low-pressure turbines 328 exchange of heat with cylinder walls 168 jackets 170 operating on Camot's cycle 116 relation of temperature and entropy 139, 141 resisted adiabatic expansion 126 saturated, density and temperature of 118 relation or pressure and temperature 119 total heat of 65 triple expansion engine 190 turbine 289 unresisted adiabatic expansion 125 wire drawing 158 with perfect engine and boiler 121 work and incomplete expansion 145 work and superheating 148 work without expansion 144 zero curve 142 Zeuner's equation for adiabatic changes 151 Stefan's formula 82 Sublimation 23 Superheated vapors 66 Superheating 24 Tandem compound engines 183, 186, 188 Temperature 1 critical 62 table of 63 difference of 2 Temperature and density of satiu'ated steam 118 Temperature and pressure of saturated steam 119 Temperature-entropy diagrams 134, 219 dryness by means of 139 of air refrigerating machines 270 Temperatures, theoretical in cylinder of internal combustion engine 213 thermodynamic scale of 113 Thermal capacities, determination of by method of cooling 16 INDEX 345 PAGB Thermal capacities, determination of by method of mixtures 15 Thermal capacities of gases 47 determination of, at constant pressure 68 at constant volume 69 ratio of, by method of Clement and De- sormes 70 ratio of, from speed of propagation of wave motion 77 Thermal capacity 8 by method of mixtures 10 per imit mass and specific heat 9 Thermal efficiency of internal combustion engine 214 of steam engine 176 Thermal equilibriimi 2 Thermo couple 6, 32 Thermodynamic drop 158 scale of temperatures 113 Thermodynamics, first principle of 93 second principle of 93 Thermometer 2 alcohol 6 gas 44 mercurial 5 resistance 6 Thermometric scales 3 Throttling and other imperfections of air compressor 249 calorimeter 151 Transmission of air and loss of head in pipes 254 of power and economy of plants 334 Turbine blades, impact on 289 Turbines, and reciprocating engines 308 comparison of Parsons and Curtis 307 Curtis 307 low pressure on exhaust of reciprocating engine 310, 328 Parsons 306 steam 289 tests 310 two principal types of 305 Two phase cycle 194 Unresisted adiabatic expansion of steam 125 Valve adjustment and indicator diagram 161 Vaporization 60 heat of 23 heat of, for water 65 346 INDEX PAOB Vapor pressures, addition of 61 Vapors 60 saturated 60 superheated 24, 66 Water equivalent 10 Water, relation of temperature and entropy 137, 141 Wave length 76 Wave motion, propagation of 74 speed of propagation 77 Wire drawing 158 Work, gain of, due to superheating steam 148 loss of, due to incomplete expansion of steam 145 loss of, due to using steam non-eacpaoaively 144 a*10A*)bb*1311 B89089669311A This book may be kept FOURTEEN DAYS A fine of TWO CENTS wfll becharged for each day the book is kept overtime. 9 No'4< 1 . • * No. 291-B I II i?- (: I. / • / /■ 3 ^ ^^ ^^^ ^ >