Page 194 - Entrophy Analysis in Thermal Engineering Systems

Appendices 191

" ! !#
n + a j ξ
k i
X 0
G ξðÞ ¼ n + a j ξ h j TðÞ T s s + s T, j R ln j R lnp
f
i
m j j n + aξ
i
j¼1
(C.4)
At the minimum G m , one must solve dG m /dξ¼0 to get
f
f
( ! )
n + a j ξ a j
k h i i
0 j
X
+ T s R ln a j ¼ 0
a j h j TðÞ T s s + s T, j
j n + aξ + T s R lnp
i
j¼1
(C.5)
Upon simplification and rearrangement, we obtain
! ( )
Y n + a j ξ a j 1 X n h io
i
k
k
j 0
¼ exp
n + aξ a j h j TðÞ T s s + s T, j a lnp
j
i
j¼1 T s R j¼1
(C.6)
For a reactive system with a known initial state and composition that inter-
acts with its surrounding that is at temperature T s , the only unknown in
Eq. (C.6) is ξ.If ξ at minimum G m happens to be close to ξ eq (as was seen
f
in Fig. 10.6) the composition of the mixture at the final state may then be
readily obtained from Eq. (C.3) and using ξ determined from Eq. (C.6).
Reference
[1] J.H. Cotterill, The Steam Engine Considered as a Thermodynamic Machine, A Treatise
on the Thermodynamic Efficiency of Steam Engines, second ed., E & F. N. Spon,
London, 1890.