Page 494 - Advanced thermodynamics for engineers

486 CHAPTER 20 IRREVERSIBLE THERMODYNAMICS

From first and Second Laws, for a system at constant pressure
Tds ¼ du þ pdv mdm (20.76)
Hence

vs vS 1 DT

D ¼ D ¼ D ¼ (20.77)
vu vU T T 2
m m

vS m TDm mDT Dm DT
D ¼ D ¼ ¼ þ m (20.78)
vm T T 2 T T 2
U
Substituting for
Dm ¼ vDp sDT (20.79)
and
h ¼ m þ Ts ¼ u þ pv (20.80)
gives

vS Dp DT DT DT Dp DT
D ¼ v þ s þ h 2 Ts 2 ¼ v þ h 2 (20.81)
vm T T T T T T
U
Hence substituting these values in Eqn (20.8),

J U ¼ L 11 X U þ L 12 X m

J m ¼ L 21 X U þ L 22 X m
gives

DT Dp DT
J U ¼ L 11 2 þ L 12 v þ h 2
T T T
(20.82)

hL 12 L 11 L 12
¼ 2 DT v Dp
T T
and
L 21 DT Dp DT
J m ¼ 2 þ L 22 v þ h 2
T T T
(20.83)

hL 22 L 21 L 22
¼ 2 DT v Dp:
T T

If the flow of internal energy which occurs due to the diffusion is defined as J U ¼ U then

J m DT¼0

J U L 12 v Dp T
(20.84)
U ¼ ¼
J m DT¼0 T L 22 v Dp