HEAT-ENGINES.] S T E A M-E N G I N E 481 expansion the external work is done entirely at the expense of the substance's stock of internal energy. Hence in the adiabatic expan- sion of a gas the temperature falls, and in adiabatic compression it rises. To find the change of temperature in a gas when expanded or compressed adiabatically we have only to combine the equations and we find y _ l It is clear from the above that if, during expansion, n is less than 7 the fluid is taking in heat, and if n is greater than 7 the fluid is rejecting heat. Iso- 39. Another very important mode of expansion or compression Jhermal is that called isothermal, in which the temperature of the working expau- substance is kept constant during the process. siou. In the case of a gas the curve of isothermal expansion is a rectangular hyperbola, having the equation PV = constant = cr . When a gas expands (or is compressed) isothermally at tempera- ture T from Vj to V. 2 the work done by (or upon) it (per tt>) is where r is the ratio V^/A r 1 as before. 1 During isothermal expansion or compression a gas suffers no change of internal energy (by 34, since T is constant). Hence during isothermal expansion a gas must take in an amount of heat just equal to the work it does, and during isothermal compression it must reject an amount of heat just equal to the work spent upon it. The expression CTlog e ?- con- sequently measures, not only the work- done by or upon the gas, but also the heat taken in during isothermal expan- sion or given out during isothermal compression. In the diagram, fig. 11, fig- li- the Hue AB is an example of a curve of isothermal expansion for a perfect gas, called for brevity an isothermal line, while AC is an adiabatic line starting from the same point A. Carnot's 40. Ve shall now consider the action of an ideal engine in cycle of which the working substance is a perfect gas, and is caused to opera- pass through a cycle of changes tions. each of which is either isother- mal or adiabatic. The cycle to be described was first exa- mined by Caruot, and is spoken of as Carnot's cycle of opera- tions. Imagine a cylinder and piston composed of a perfectly non - conducting material, *> except as regards the bottom *| of the cylinder, which is a T/ 1 conductor. Imagine also a - hot body or indefinitely ca- g I pacious source of heat A, kept always at a tempera- * FIG. 12. Carnot's Cycle, with a gas for working substance. ture T 1; a perfectly non- conducting cover B, and a cold body or indefinitely capacious receiver of heat C, kept always at a temperature T 2 , which is lower than r l . It is supposed that A, B, or C can be applied to the bottom of the cylinder. Let the cylinder contain 1 Ib of a perfect gas, at temperature T ]( volume V a , and pressure Pto begin with. The suffixes refer to the points on the indicator diagram, fig. 12. (1) Apply A, and allow the piston to rise. The gas expands isothermally at TJ, taking heat from A and doing work. The pressure changes to P& and the volume toV&. (2) Remove A and apply B. Allow the piston to go on rising. The gas expands adiabatically, doing work at the expense of its internal energy, and the temperature falls. Let this go on until the temperature is T 2 . The pressure is then P and the volume V c . (3) Remove B and apply C. Force the piston down. The gas is compressed isothermally at T 2 , since the smallest increase of temperature above T. 2 causes heat to pass into C. Work is spent upon the gas, and heat is rejected to C. Let this be continued until a certain point d (fig. 12) is reached, such that the fourth operation will complete the cycle. (4) Remove C and apply B. Continue the compression, which is now adiabatic. The pressure and temperature rise, and if the point d has been properly chosen, when the pressure is restored to its original value P a , the temperature will also have risen to its 1 In calculations where this expression is involved it is convenient to remem- ber that log e , the hyperbolic logarithm, of any number is 2'3026 times the common logarithm of the number. original value TJ. [In other words, the third operation must be stopped when a point d is reached such that an adiabatic lino drawn through d will pass through .] This completes the cycle. To find the proper place at which to stop the third operation, we have, by 38, T 1 /T 2 = (V c /V(,)y- 1 in the second operation, and again T]/T 2 = ( V d /y a )> -Mn the fourth operation. Hence V c /V 6 = Vjj V a , and V&/V a , the ratio of isothermal expansion, is equal to V c /Vd, the ratio of isothermal compression. For brevity we shall denote either of these last ratios by r. 41. The following are the transfers of heat to and from the working fluid, in successive stages of the cycle : (1) Heat taken in from A = c Tl log e r (by 39). (2) No heat taken in or rejected. (3) Heat rejected to C=CT 2 log e r (by 39). (4) No heat taken in or rejected. Hen-ce, by the first law of thermodynamics, the net external work done by the gas is C(TJ-T,) log e r; and the efficiency of the engine ( 23) is This is the fraction of the whole heat given to it which an engine following Carnot's cycle converts into work. The engine takes in an amount of heat, at the temperature of the source, pro- portional to TJ ; it rejects an amount of heat, at the temperature of the receiver, proportional to T 2 . It works within a range of tem- perature extending from TJ to r. 2 , by letting down heat from TJ to TO ( 24), and in the process it converts into work a fraction of that heat, which fraction will be greater the lower the temperature T 2 at which heat is rejected is below the temperature T a at which heat is received. 42. Next let us consider what will happen if we reverse Carnot's Carnot's cycle, that is to say, if we force this engine to act so that the same cycle indicator diagram as before is traced out, but in the direction reversed. opposite to that followed in 40. Starting as before from the point a and with the gas at r lt we shall require the following four operations : (1) Apply B and allow the piston to rise. The gas expands adiabatically, the curve traced is ad, and when d is reached the temperature has fallen to r v (2) Remove B and apply C. Allow the piston to go on rising. The gas expands isothermally at T 2 , taking heat from C, and the curve dc is traced. (3) Remove C and apply B. Compress the gas. The process is adiabatic. The curve traced is cb, and when b is reached the temperature has risen to T t . (4) Remove B and apply A. Continue the compression, which is now isothermal, at TJ. Heat is now rejected to A, and the cycle is completed by the curve ba. In this process the engine is not doing work ; on the contrary, work is spent upon it equal to the area of the diagram, or c(T 1 -T 2 )log e r. Heat is taken in from C in the first operation, to the amount CT 2 log e r. Heat is rejected to A in the fourth operation, to the amount CT 1 log e r. In the first and third opera- tions there is no transfer of heat. The action is now in every respect the reverse of what it was before. The same work is now spent upon the engine as was formerly done by it. The same amount of heat is now given to the hot body A as was formerly taken from it. The same amount of heat is now taken from the cold body C as was formerly given to it, as will be seen by the following scheme : Carnot's Cycle, Direct. Work done by the engine = C(T I - T 2 ) log e r ; Heat taken from A = CTjlog e r ; Heat rejected to C=CT 2 log e r. Carnot's Cycle, Reversed. Work spent upon the engine = c (T X - T.,) Iog 6 r ; Heat rejected to A^cr^og^- ; Heat taken from C=CT 2 log e r. The reversal of the work has been accompanied by an exact reversal of each of the transfers of heat. 43. An engine in which this is possible is called, from the Revers- thermodynamic point of view, a reversible engine. In other words, ible a reversible heat-engine is one which, if forced to trace out its engine. indicator diagram reversed in direction, so that the work which would be done by the engine, when running direct, is actually spent upon it, will reject to the source of heat the same quantity of heat as, when running direct, it would take from the source, and will take from the receiver of heat the same quantity as, when running direct, it would reject to the receiver. By " the source of heat " is meant the hot body which acts as source, and by the receiver " is meant the cold body which acts as receiver, when the XXII 6 1
