Page 376 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 359

M w
∂θ i _ ∂θ i X
ð
ð
C i + C i + U ik θ i θ w,k Þ ¼ 0 i ¼ 1, 2, …, MÞ (7.188)
∂τ ∂x
k¼1
M
∂θ w,k X
ð
C w,k + U jk θ w,k θ j ¼ 0 k ¼ 1, 2, …, M w Þ (7.189)
∂τ
j¼1
N 0 M
X X

0
0
0 g θ τ Δτ 0 00
θ i τ, x g ij θ j τ Δτ ij , x ¼ 0
i ik k ik j
k¼1 j¼1
ð τ > 0; i ¼ 1, 2, …, MÞ (7.190)
N 0 M
X X
00 g θ τ Δτ 000 00 00 00
000 0
θ τðÞ g θ i τ Δτ , x ¼ 0
l lk k lk li li i
k¼1 i¼1
00
ð τ > 0; l ¼ 1, 2, …, N Þ (7.191)
ð
τ ¼ 0 : θ i ¼ 0 i ¼ 1, 2, …, Mð Þ, θ w,k ¼ 0 k ¼ 1, 2, …, M w Þ (7.192)
where
θ i ¼ t i t 0 ð i ¼ 1, 2, …, MÞ, θ w,k ¼ t w,k t 0 ð k ¼ 1, 2, …, M w Þ
(7.193)

_
_
0, C i > 0 1, C i > 0
0 00
x ¼ , x ¼ ð τ > 0; i ¼ 1, 2, …, M Þ (7.194)
i _ i _
1, C i < 0 0, C i < 0
The time delays Δτ ik and Δτ ij are the time intervals for the streams flowing
0
from entrance k to the inlet of channel i and those from the outlet of channel
j to the inlet of channel i, respectively.
After the Laplace transform, we obtain the governing equation system in
the Laplace domain as follows:
M w
X
e
_ dθ i
e
e
e
sC i θ i + C i + U ik θ i θ w,k ¼ 0 i ¼ 1, 2, …, Mð Þ (7.195)
dx
k¼1
M
X
e
sC w,k θ w,k U jk θ j θ w,k ¼ 0 k ¼ 1, 2, …, M w Þ (7.196)
e
ð
e
j¼1
N 0 M
X 0 X
0 0 Δτ s e 0 00
θ i x , s g e ik θ sðÞ g ij e θ j x , s ¼ 0
e Δτ ij s e
i ik k j
k¼1 j¼1
ð τ > 0; i ¼ 1, 2, …, MÞ (7.197)