Page 345 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 331

x ¼ 0 : t 1 ¼ t 1,in τðÞ (7.29)
x ¼ L : t 2 ¼ t 2,in τðÞ (7.30)

τ ¼ 0 : t 1 ¼^ t 1 xðÞ, t 2 ¼^ t 2 xðÞ, t w ¼^ t w xðÞ (7.31)
7.1.2.3 Crossflow

C 1 ∂t 1 _ ∂t 1 ð αAÞ 1
+ C 1 ¼ ð t w t 1 Þ (7.32)
L x ∂τ ∂x L x
C 2 ∂t 2 _ ∂t 2 ¼ ð αAÞ 2 ð t w t 2 Þ (7.33)
L y ∂τ C 2 ∂y L y

∂t w
ð
ð
ð
C w ¼ αAÞ t 1 t w Þ + αAð Þ t 2 t w Þ (7.34)
∂τ
1
2
x ¼ 0 : t 1 ¼ t 1,in τ, yð Þ (7.35)
y ¼ 0 : t 2 ¼ t 2,in τ, xð Þ (7.36)
ð
ð
τ ¼ 0 : t 1 ¼^ t 1 x, yð Þ, t 2 ¼^ t 2 x, yÞ, t w ¼^ t w x, yÞ (7.37)
in which L x and L y are the length of the heat exchanger in the flow direction
of fluid 1 and that in the flow direction of fluid 2, respectively.

7.1.2.4 Parallel-flow and counterflow heat exchangers
Consider a parallel-flow or counterflow heat exchanger consisting of fluid 1,
fluid 2, and the wall separating these two fluids. We combine these two types
together with an index n to indicate the flow direction of fluid 2. For a parallel-
flow heat exchanger, n¼0, and for a counterflow heat exchanger, n¼1:

C 1 ∂t 1 _ ∂t 1 ð αAÞ 1 (7.38)
L ∂τ + C 1 ∂x ¼ L ð t w t 1 Þ

C 2 ∂t 2 n _ ∂t 2 ð αAÞ 2 (7.39)
ð
L ∂τ + 1Þ 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 (7.40)
∂τ
2
1
x ¼ 0 : t 1 ¼ t 1,in τðÞ (7.41)
x ¼ nL : t 2 ¼ t 2,in τðÞ (7.42)
τ ¼ 0 : t 1 ¼ ^ t 1 x ðÞ, t 2 ¼ ^ t 2 x ðÞ, t w ¼ ^ t w x ðÞ (7.43)
in which 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.