Modern Engineering Thermodynamics

10.11 Energy Efficiency Based on the Second Law 349
b. The second law availability efficiency of the preheater is given by Eq. (10.36), where the flow availability is given by
Eq. (10.20):
a f = h − h 0 − T 0 ðs − s 0 Þ + V2 + gZ
2g c g c
and since both flow streams are constant specific heat ideal gases, this equation reduces to
a f = c p ðT − T 0 Þ − T 0 c p ln T T 0
− R ln p p 0
+ V2
2g c
+ gZ
g c
We neglect kinetic and potential energy terms here, so we can evaluate this equation for each of the four flow
streams as
kJ
a f = 0:990 ð500: − 20:0KÞ
in-exh
kg.K
− ð20:0 + 273:15 KÞ 0:990 kJ
kg .K
+ 0 + 0 = 201 kJ/kg
kJ
a f = 0:990 ð400: − 20:0KÞ
out-exh
kg.K
− ð20:0 + 273:15 KÞ 0:990 kJ
kg .K
+ 0 + 0 = 135 kJ/kg
kJ
a f = 1:004 ð20:0 − 20:0KÞ
in-air
kg .K
− ð20:0 + 273:15 KÞ 1:004 kJ
kg.K
+ 0 + 0 = 34:0 kJ/kg
kJ
a f = 1:004 ð118:6 − 20:0KÞ
out-air
kg .K
− ð20:0 + 273:15 KÞ 1:004 kJ
kg .K
+ 0 + 0 = 41:9 kJ/kg
Then Eq. (10.36) gives the second law availability efficiency as
ln
ln
500: + 273:15
20:0 + 273:15
400: + 273:15
20:0 + 273:15
ln
ln
20:0 + 273:15
20:0 + 273:15
118:6 + 273:15
20:0 + 273:15
− 0:272 kJ
kg.K
− 0:272 kJ
kg.K
− 0:286 kJ
kg.K
− 0:286 kJ
kg .K
ln 1:10
1:00
ln 1:00
1:00
ln 1:50
1:00
ln 1:40
1:00
ε nonmixing HX
=
_m air½ða f Þ out air
− ða f Þ in air
Š ð0:800 kg/sÞð41:9 − 34:0 kJ/kgÞ
= = 0:119 = 11:9%
_m exh ½ða f Þ in exh
− ða f Þ out exh
Š ð0:800 kg/sÞð201 − 135 kJ/kgÞ
Exercises
44. Determine the second law efficiency of the air preheater in Example 10.15 when the exhaust inlet temperature
isincreasedfrom500.°C to 750.°C. Assume all the other variables (except T 4 ) remain unchanged.
Answer: ε HX
= 48.0%.
45. Calculate the second law efficiency of the air preheater in Example 10.15 when the exhaust exit temperature is decreased
from 400.°C to 350.°C. Assume all the other variables (except T 4 ) remain unchanged. Answer: ε HX
= 24.0%.
46. The mass flow rates in Example 10.15 are changed from 0.800 kg/s to _m exhaust = 0:900 kg=s and _m air = 0:700 kg=s:
Assuming all the other variables remain unchanged, determine the new second law thermal efficiency of the air
preheater. Answer: ε HX
= 18.4%.
If the hot and cold fluids mix inside the heat exchanger (see Figure 10.20b), then the second law availability
efficiency becomes
Second law efficiency of a heat exchanger in which the hot and cold fluids mix
ε mixing HX
= ð1 − yÞða f 3 − a f 2 Þ
yða f 1 − a f 3 Þ
(10.37)
where y is the hot mass fraction defined as y = _m H / _m m = _m 1 / _m 3 :