Page 493 - Advanced thermodynamics for engineers

20.7 DIFFUSION AND HEAT TRANSFER 485

The system is isolated, hence variations DU and Dm in part I give rise to variations DU and Dm
in part II. The variation in entropy due to these changes may be found from Taylor’s series.

2 2 2
vS vS 1 v S 2 v S 1 v S 2
DS I ¼ DU þ D m þ 2 DU þ DUDm þ 2 Dm
vU vm 2 vU vU vm 2 vm (20.70)
m U
þhigher order terms
DS II may be found by a similar expansion, but the linear terms in DU and Dm are negative.
Thus, the change of entropy of the universe is

DS ¼ DS I þ DS II
2 2 2 (20.71)
v S 2 v S v S 2
¼ 2 DU þ 2 DUDm þ 2 Dm þ higher order terms.
vU m vU vm vm U
The time rate of change of DS, i.e. the rate of generation of entropy, is given by
2
2
2
2
d v S v S v S v S
_
_
ðDSÞ¼ ð2DUÞDU þ 2 DmDU þ DUD _ m þ ð2DmÞD _ m
dt vU 2 vU vm vU vm vm 2
2
2
2
2
v S v S v S v S
¼ 2DU _ DU þ Dm þ 2D _ m Dm þ DU
vU 2 vU vm vm 2 vU vm
vS vS
_
¼ 2DU D þ 2D _ m D
vU m vm U
(20.72)
and the rate of generation of entropy per unit volume is
d 1 d _ vS vS
ðDsÞ¼ ðDSÞ¼ DU D þ D _ m D ; (20.73)
dt 2 dt vU vm
m U
because the combined volume of systems I and II is 2V, and the original terms were defined in relation
to a single part of the system.
It was stated in Eqn (20.12), from Onsager’s Relationship, that for a two-component system
Tq ¼ J 1 X 1 þ J 2 X 2
which may be written, for this system, as
d
Tq ¼ ðDsÞ¼ J U X U þ J m X m (20.74)
dt
Comparison of Eqns (20.73) and (20.74) enables the forces and fluxes to be defined, giving
vS 9
J U ¼ DU _ X U ¼ D >
>
vU m >
=
(20.75)
vS >

J m ¼ D _ m X m ¼ D >
>
vm
;
U