Page 344 - Design and Operation of Heat Exchangers and their Networks

330 Design and operation of heat exchangers and their networks

(4) The cross-sectional areas of the flow passages are constant.
(5) The heat conduction resistance of the wall in the thickness direction is
negligible.
(6) The axial heat conduction of the wall along the flow direction can be
neglected.
(7) The heat loss to the environment is negligible.
(8) The heat exchanger runs at a steady state before the disturbances occur.
According to the earlier assumptions, we can express the energy equations as
follows.

7.1.2.1 Parallel flow

C 1 ∂t 1 _ ∂t 1 ð αAÞ 1 (7.21)
L ∂τ + C 1 ∂x ¼ L ð t w t 1 Þ

C 2 ∂t 2 _ ∂t 2 ð αAÞ 2 (7.22)
L ∂τ + C 2 ∂x ¼ L ð t w t 2 Þ

∂t w
ð
ð
ð
C w ¼ αAÞ t 1 t w Þ + αAð Þ t 2 t w Þ (7.23)
∂τ
2
1
x ¼ 0 : t 1 ¼ t 1,in τðÞ, t 2 ¼ t 2,in τðÞ (7.24)
τ ¼ 0 : t 1 ¼ ^ t 1 x ðÞ, t 2 ¼ ^ t 2 x ðÞ, t w ¼ ^ t w x ðÞ (7.25)
_
in which C is the thermal capacity, C¼Mc p ; C is the thermal capacity rate,
_
C ¼ _mc p ; L is the length of the heat exchanger; and ^ t 1 , ^ t 2 , and ^ t w are the tem-
perature distributions of fluid 1, fluid 2, and the wall at the initial steady state,
respectively.

7.1.2.2 Counterflow

C 1 ∂t 1 _ ∂t 1 ð αAÞ 1 (7.26)
L ∂τ + C 1 ∂x ¼ L ð t w t 1 Þ

C 2 ∂t 2 _ ∂t 2 ð αAÞ 2
L ∂τ C 2 ∂x ¼ L ð t w t 2 Þ (7.27)

∂t w
ð
ð
ð
C w ¼ αAÞ t 1 t w Þ + αAð Þ t 2 t w Þ (7.28)
2
1
∂τ