Page 395 - Design and Operation of Heat Exchangers and their Networks

378 Design and operation of heat exchangers and their networks

N 0 M

X 0 X
0 0 Δτ s e 0 00
e qk Δτ ql s e
θ q x , s ¼ g e θ sðÞ + g e θ l x , s (7.267)
q qk k ql l
k¼1 l¼1
N 0 M
X X 00
00
e 00 000 Δτ s e 0 00 Δτ s 00 00
lk θ sðÞ +
θ q x , s
θ sðÞ ¼ g e g e lq e ð l ¼ 1, 2, …, N Þ
l lk k lq q
k¼1 q¼1
(7.268)

2 e
∂ θ f , ij
e
e
e
sC f, ij θ f, ij ¼ K ij 2 + U f , ij θ ij θ f, ij (7.269)
∂y
y ¼ 0 : θ f , ij ¼ θ p, ij ; y ¼ 1 : θ f, ij ¼ θ p,i +1, j (7.270)
e
e
e
e
Although the excess temperature in the Laplace domain θ and θ p are the
e
e
functions of x, they can be treated as parameters in Eqs. (7.269), (7.270);
therefore, they can be solved alone, which yields
!
coshγ 1
U f , ij ij
e 1+ e
θ f, ij ¼ sinhγ y coshγ y θ ij
ij
ij
sC f, ij + U f, ij sinhγ
ij
(7.271)
!
coshγ sinhγ y
ij ij
e
sinhγ y coshγ y θ p, ij + e
θ p,i +1, j
ij
ij
sinhγ sinhγ
ij ij
where
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

γ ¼ sC f, ij + U f , ij =K ij (7.272)
ij
The integral and two derivatives of the fin temperature in the Laplace
domain appearing in Eqs. (7.265), (7.266) are obtained as

ð U f, ij 1 η
1 η
f, ij f, ij
e θ ij + e e (7.273)
θ f, ij dy ¼ θ p, ij + θ p,i +1, j
e
0 sC f, ij + U f, ij 2

∂θ e 1 1 i
h
η θ
f , ij
e
θ p, ij μ θ p,i +1, j
K ij ¼ U f , ij ij ij sC f , ij + U f , ij η + μ ij e ij e (7.274)
ij
∂y 2 2
y¼0

∂θ 1
e
θ
η
f ,i 1, j
e
K i 1, j ¼ U f ,i 1, j f ,i 1, j i 1, j
∂y 2
y¼1
1 h i
θ
e
sC f ,i 1, j + U f ,i 1, j μ f ,i 1, j p,i 1, j η f ,i 1, j + μ f ,i 1, j θ p, ij (7.275)
e
2