Page 139 - Advanced thermodynamics for engineers

6.2 EFFICIENCY AT MAXIMUM POWER 125

Substituting in Eqn (6.8) gives

U H A H T H ðs T C =T H Þ
T 2 ¼ þ T C ;
U H A H
U C A C 1 þ
U C A C

U H A H T H s s 2 (6.16)
¼ þ T C ;
ðU C A C þ U H A H Þ
( 1=2 1=2 )
1=2 U H A H T H þ U C A C T C
¼ T C
U H A H þ U C A C
Similarly,
1=2 1=2
( )
1=2 U H A H T H þ U C A C T C
T 1 ¼ T : (6.17)
H
U H A H þ U C A C
To be able to compare the effect of varying the resistances, it is necessary to maintain the total
resistance to heat transfer the same. For example, let
U H A H þ U C A C ¼ 2:
U H A H
Then, if ¼ 1 (as in the previous example), U H A H ¼ 1.
U C A C
Consider the effect of having a high resistance to the high temperature reservoir, e.g. U H A H /
U C A C ¼ 1/2.
This gives
2 2
U H A H ¼ ¼ :
U C A C 3
1 þ
U H A H
Then,

W _ ð1=2 1=4Þ
_
¼ 1600 ð1 1=2Þ¼ 267 units; giving W ¼ 177:8 units:
U H A H ð1 þ 1=2Þ=2
_
Q _ H W
_
¼ ¼ 534 units and Q ¼ 355:6 units:
H
U H A H U H A H ð1 sÞ
Then,
Q _ C Q _ C 1 1
_
_
_
¼ $ ¼ Q W ¼ 133:5 units; giving Q ¼ 177:8 units
H
C
U C A C U H A H 2 2U H A H
Hence,
1
T 1 ¼ ð133:5 þ 400Þ¼ 2 533:5 ¼ 1067 K; and T 2 ¼ 533:5K:
s