Page 361 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 347

channel i and those from the outlet of channel j to the inlet of channel i,respec-
tively. Similarly, we have the energy balance at the exit of each stream as
N 0 M
X X
00 τ g t τ Δτ 00 00 00 00 00
000 0
t ðÞ ¼ + g t i τ Δτ , x ð l ¼ 1, 2, …, N Þ (7.112)
l lk k lk li li
k¼1 i¼1
000
00
where Δτ li and Δτ lk are the time delays from the outlet of channel i to exit l
and those from entrance k to exit l, respectively.
Introducing the excess temperature θ ¼ t ^ t and the dimensionless spa-
tial coordinate x ¼ x=L and substituting them into Eqs. (7.108)–(7.112) with
the assumption that the heat exchanger runs at first at a steady state denoted
with “^,” we can express the governing equation system as follows:

M w
X
∂θ i _ ∂θ i _ d^ t i
ð
C i + C i + U ik θ i θ w,k Þ + C i
∂τ ∂x dx
k¼1
M w
X
+ U ik t i ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, MÞ (7.113)
ð
^ ð
k¼1
M
∂θ w,k X
C w,k U jk θ j θ w,k
∂τ
j¼1
M
X
^
U jk t j ^ t w,k ¼ 0 k ¼ 1, 2, …, M w Þ (7.114)
ð
j¼1
N 0 M
X X

0
0 g θ τ Δτ 0 00 0
0
θ i τ, x g ij θ j τ Δτ ij , x +^ t i x
i ik k ik i i
k¼1 j¼1
N 0 M
X X
0 0 ^ 00
g ^ t g ij t j x i ¼ 0 τ > 0; i ¼ 1, 2, …, MÞ (7.115)
ð
ik k
k¼1 j¼1
N 0 M
X X
00 000 0 000 00 00 00 00
θ τðÞ g θ τ Δτ Þ g θ i τ Δτ , x + ^ t
ð
l lk k li li i l
k¼1 i¼1
N 0 M
X X
000 00 00 00 00
g ^ t g ^ t i x ¼ 0 τ > 0; l ¼ 1, 2, …, N Þ (7.116)
ð
lk k li i
k¼1 i¼1
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 (7.117)
ð
where x and x are the dimensionless coordinate vectors of channel inlets
00
0
i i
and outlets

_
_
0, C i > 0 1, C i > 0
0 00
x ¼ , x ¼ ð τ > 0; i ¼ 1, 2, …, MÞ (7.118)
i _ i _
1, C i < 0 0, C i < 0