Page 161 - Entrophy Analysis in Thermal Engineering Systems

156 Entropy Analysis in Thermal Engineering Systems

!
1 X X
k
k
f i ΔG m
Φ ¼ ¼ (10.10)
G
m, j G
m, j
j¼1 j¼1 T s
T s
where G m, j ¼n j g m, j , and ΔG m denotes the change in G m due to the reaction.
10.4.2 Endothermic reaction
An endothermic reaction requires heat to be supplied from an external
source. The amount of heat Q that should be transferred to the system from
a heat source that is at temperature T s to maintain an isothermal reaction can
be determined using the first law.

k k
X
X
i f
n j h j TðÞ + Q ¼ n j h j TðÞ (10.11)
j¼1 j¼1
The total entropy generation associated with the isothermal reaction is
obtained by

k k
X
X
Q
f i
Φ ¼ n j s j T, pÞ n j s j T, pÞ (10.12)
ð
ð
j¼1 j¼1 T s
The negative sign of Q/T s denotes the reduction in the entropy of the heat
source. Combining Eqs. (10.11) and (10.12) to eliminate Q would again lead
to Eq. (10.9). So, Eqs. (10.9) and (10.10) are valid for both endothermic and
exothermic reactions. It must be remembered that the formulation rests on
the key assumption of constant T s .
10.4.3 Gibbs function
An essential element in the arguments of Gibbs is the condition of revers-
ibility without altering temperature. For a system undergoing a non-
isothermal process, he postulated that “it is not necessary … that the
temperature of the system should remain constant during the reversible
process, … provided that the only source of heat or cold used has the same
temperature as the system in its initial or final state. Any external bodies may
be used in the process in any way not affecting the condition of
reversibility.” He then elucidated “uniformity of temperature and pressure
are always necessary for equilibrium, and the remaining conditions, when
these are satisfied, may be conveniently expressed by means of the function
ζ” [18], where ζ is indeed the Gibbs function G.