9-17
9-24 An air-standard cycle executed in a piston-cylinder system is composed of three specified processes. The cycle is to be
sketcehed on the P-v and T-s diagrams; the heat and work interactions and the thermal efficiency of the cycle are to be
determined; and an expression for thermal efficiency as functions of compression ratio and specific heat ratio is to be
obtained.
Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is
an ideal gas with constant specific heats.
Properties The properties of air are given as R = 0.3 kJ/kg·K and cv = 0.3 kJ/kg·K.
Analysis (a) The P-v and T-s diagrams of the cycle are shown in the figures.
(b) Noting that
c p  cv  R  0.7  0.3  1.0 kJ/kg  K
k
cp
cv
P
1.0
 1.429
0.7

3
2
Process 1-2: Isentropic compression
v
T2  T1  1
v2



k 1
 T1 r
k 1
1
 (293 K )(5)
0.429
 584.4 K
v
w1 2,in  cv (T2  T1 )  (0.7 kJ/kg  K )(584.4  293) K  204.0 kJ/kg
q12  0
T
From ideal gas relation,
T3 v 3 v 1


r
 T3  (584.4)(5)  2922
T2 v 2 v 2
Process 2-3: Constant pressure heat addition
3
3
2
1
s

w2 3,out  Pdv  P2 (v 3  v 2 )  R(T3  T2 )
2
 (0.3 kJ/kg  K )(2922  584.4) K  701.3 kJ/kg
q 23,in  w2 3,out  u 23  h23
 c p (T3  T2 )  (1 kJ/kg  K )(2922  584.4) K  2338 kJ/kg
Process 3-1: Constant volume heat rejection
q 31,out  u13  c v (T3  T1 )  (0.7 kJ/kg  K)(2922 - 293) K  1840.3 kJ/kg
w31  0
(c) Net work is
wnet  w2 3,out  w1 2,in  701.3  204.0  497.3 kJ/kg  K
The thermal efficiency is then
 th 
wnet 497.3 kJ

 0.213  21.3%
2338 kJ
q in
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
9-18
(d) The expression for the cycle thermal efficiency is obtained as follows:
 th 
wnet w23,out  w1 2,in

q in
q in

R (T3  T2 )  c v (T2  T1 )
c p (T3  T2 )

c v (T1 r k 1  T1 )
R

c p c p (rT1 r k 1  T1 r k 1 )

T1 
c v T1 r k 1 1 
 T r k 1 
R
1




cp
c p T1 r k 1 (r  1)

T1 
1 
R
1


c p k (r  1)  T1 r k 1 

1 
1 
R

1 

c p k (r  1)  r k 1 
1 
1 
 1
 1   
1  k 1 
(
1
)

k
k
r


 r

since
c
R c p  cv
1

 1 v  1
cp
cp
cp
k
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.