Thus  for  Carnot's  cycle,  represented  by  Fig.  20,  the  increase
of  entropy  during  isothermal  expansion  is


r-*'Sf-iPQ-t


because   the   temperature   is   constant.     In    like   manner    the
decrease  during  isothermal  compression  is


so


that  the  change  of  entropy  for  the  cycle  is

T       2'
But  from  the  efficiency  of  the  cycle  we  have

Q  T     '      "  Q      r '      "  T     r

A  complex  cycle  like  that  represented  by  Fig.  24  may  be
broken  up  into  two  simple  cycles  ABFG
and  CDFE,  for  each  of  which  individually
the  same  result  will  be  obtained  —  that  is,
the  increase  of  entropy  from  -4  to  5  is
equal  to  the  decrease  from  F  to  G,  and
the  increase  from  C  to  D  is  equal  to  the


Fig.  *4-

decrease   from  E  to  F,  so  that   the  sum-
mation of  changes  for  the  entire  cycle  gives  zero


MAXIMUM   EFFICIENCY


39


Fig.  25  represents  the  simplified  ideal  diagram  of  a  hot-air
engine,  in  which  by  the  aid  of  a  regenerator  the  adiabatic  lines
of  Camot's  cycle  are  replaced  by  p
vertical  lines  without  affecting  the
reversibility  or  the  eflSciency  of  the
cycle.  We  may  replace  the  actual
diagram  by  a  series  of  simple  cycles
made  up  of  isothermals  and  adia-
batics,  so  drawn  that  the  perimeter
of  the  complex  cycle  includes  the
same  area  and  corresponds  ap-
proximately with  that  of  the
actual  diagram.  The  summation
for  the  complex  cycle  is  clearly
drawing  the   adiabatic    lines    near


Fig.  35.
of  the  change  of  entropy
zero,  as  before.  But  by
enough  together  we  may
make  the  perimeter  approach  that  of  the  actual  diagram  as
nearly  as  we  please,  and  we  may  therefore  conclude  that  the
integration  for  the  changes  of  entropy  for  that  cycle  is  also  zero.
