Page 393 - Design and Operation of Heat Exchangers and their Networks

376 Design and operation of heat exchangers and their networks

N 0 M

X X
0
0
θ q τ, x 0 ¼ g θ τ Δτ 0 + g θ l τ Δτ ql , x 00
q qk k qk ql l
k¼1 l¼1
(7.253)
N 0 M
X X
0
^
0
+ g ^ t + g ql t l x 00 ^ t q x 0
qk k l q
k¼1 l¼1
N 0 M
X X
00 g θ τ Δτ 00 00 00 00
000 0
θ τðÞ ¼ + g θ q τ Δτ , x
l lk k lk lq lq q
k¼1 q¼1
(7.254)
N 0 M
X X
000 0
00
00
+ g ^ t + g ^ t q x 00 ^ t 00 ð l ¼ 1, 2, …, N Þ
lk k lq q l
k¼1 q¼1
τ ¼ 0 : θ ij ¼ 0, θ p, ij ¼ 0 (7.255)
2
∂θ f, ij ∂ θ f , ij
^
C f, ij ¼ K ij + U f , ij θ ij θ f, ij + U f , ij ^ U f , ij t ij ^ t f, ij (7.256)
∂τ ∂y 2
y ¼ 0 : θ f, ij ¼ θ p, ij ; y ¼ 1 : θ f , ij ¼ θ p,i +1, j (7.257)
τ ¼ 0 : θ f, ij ¼ 0 (7.258)
The linearized model Eqs. (7.251)–(7.258) is solved by means of the
Laplace transform. In the Laplace domain, the equations earlier become a
set of ordinary differential equations:
0 1
1
ð

e
_ dθ ij U p, ij
e ¼ e e e @ e e A
sC ij θ ij + C ij θ p, ij + θ p,i +1, j 2θ ij + U f, ij θ f, ij dy θ ij
dx 2
0
!
e _
e
1 U p, ij C ij
^
+ U p, ij ^ t p, ij + ^ t p,i +1, j 2^ t ij
2 ^ _
C ij
0 1
! 1
e _ ð
e
U f, ij C ij
+ ^ U f, ij @ ^ t f, ij dy ^ t ij A (7.259)
^ _
C ij
0
1 1
e e e e e
sC p, ij θ p, ij ¼ U p, ij θ ij θ p, ij + U p,i 1, j θ i 1, j θ p, ij
2 2

e e 1 1
∂θ f, ij ∂θ f ,i 1, j
+ K ij K i 1, j + e U p, ij ^ U p, ij ^ t ij ^ t p, ij
∂y ∂y 2 s
y¼0 y¼1

1 1
+ e U p,i 1, j ^ U p,i 1, j ^ t i 1, j ^ t p, ij
2 s
(7.260)