Page 140 - Entrophy Analysis in Thermal Engineering Systems

134 Entropy Analysis in Thermal Engineering Systems

ð
ð
_ _ n H 2 O h H 2 O Þ h H 2 O Þ 0
P
ð
ð
0 s + _n O 2 O 2 R
Φ ¼ _n H 2 O s H 2 O Þ _n H 2 H 2 s Þ + (9.9)
T 0
Eliminating (h H 2 O ) P between Eqs. (9.4) and (9.9), we find
h Þ _n H 2 O h H 2 O Þ
_ ð _ n H 2 H 2 + _n O 2 O 2 R ð 0
h
Φ ¼
T 0
_ W FC
ð ð s Þ (9.10)
0 s + _n O 2 O 2 R
+ _n H 2 O s H 2 O Þ _n H 2 H 2
T 0
_
Setting Φ equal to zero leads to an expression for the maximum fuel cell
power. Hence,

ð ð s Þ (9.11)
0 s + _n O 2 O 2 R
¼ _ E in + T 0 _n H 2 O s H 2 O Þ _n H 2 H 2
_ W FC
max
where
_
ð
ð
0
E in ¼ _n H 2 H 2 h Þ _n H 2 O h H 2 O Þ (9.12)
h
+ _n O 2 O 2 R
The expression that we found for the maximum fuel cell power is neither the
change in Gibbs function at reaction temperature nor that at the surrounding
temperature T 0 . However, if the reactants enter the fuel cell at the surround-
ing temperature, i.e., T R ¼T 0 , Eq. (9.11) reduces to

ð G H 2 O Þ (9.13)
0
¼ G H 2 + G O 2
_ W FC
max
where G is the Gibbs function defined as
(9.14)
G ¼ H T S
In Eq. (9.13), G i (i:H 2 ,O 2 ,H 2 O) is calculated at the surrounding temper-
ature and pressure.
The maximum conversion efficiency of the fuel cell is obtained by

ð ð s s Þ
η ¼ max ¼ 1+ T 0 0 + _n O 2 O 2 R (9.15a)
_ n H 2 O s H 2 O Þ _n H 2 H 2
_ W FC
max _ E in _ E in
Or, per molar flowrate of hydrogen, one may write
ð s H 2 Þ s H 2 O Þ
ð
η ¼ 1 T 0 +0:5s O 2 R 0 (9.15b)
ð Þ h H 2 O Þ
max
ð
0
h H 2
+0:5h O 2 R
Also, the amount of heat dissipated to the surrounding at maximum effi-
ciency is
_
ð
ð
s Þ
0
0
s
Q ¼ T 0 _n H 2 O s H 2 O Þ _n H 2 H 2 + _n O 2 O 2 R (9.16)