Page 439 - Design and Operation of Heat Exchangers and their Networks

422 Design and operation of heat exchangers and their networks

Introducing the following dimensionless variables and parameters,

C _ t t 0 α A p + A f
x ¼ x=L, y ¼ y=h, τ ¼ τ, θ ¼ , NTU ¼ ,
C p + C f t ref t 0 C _
C
C f A f A c, f λ f
ζ ¼ , ξ ¼ , B ¼ , K f ¼
hC _
C p + C f A p + A f C p + C f
we can express the governing equation system in dimensionless form and
then apply the Laplace transform to it, which yields
dθ ð 1
e
θ f dy θ
sBθ + ¼ NTU 1 ξÞθ p + ξ e e (8.131)
ð
e
e
dx 0

e
dθ f
e e
e
s 1 ζÞθ p ¼ 1 ξð ÞNTU θ θ p +2K f (8.132)
ð
dy
y¼0
0
e
e
x ¼ 0 : θ ¼ θ sðÞ (8.133)

2 e
d θ f
e e
e
sζθ f ¼ K f + ξNTU θ θ f (8.134)
dy 2
e
e
y ¼ 0 andy ¼ 1 : θ f ¼ θ p (8.135)
In the Laplace plane, the fin temperature can be solved alone by taking θ
and θ w as parameters, which yields

ξNTU ξNTU
θ f ¼ θ w
e e θ cosh γl tanh γ=2ð Þsinh γl + θ
e
e
sζ + ξNTU sζ + ξNTU
(8.136)
from which we further obtain
ð 1
ξNTU
θ
e e Þ e (8.137)
θ f dy ¼ η θ p +1 ηð
f f
0 sζ + ξNTU

h i
e
dθ f sζ + ξNTUÞθ p ξNTUθ
2K f ¼ η ð e e (8.138)
f
dy
y¼0
where the fin efficiency η f and complex eigenvalue γ are
η ¼ tanh γ=2ð Þ= γ=2Þ (8.139)
ð
f
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sζ + ξNTU
γ ¼ (8.140)
K f