Page 394 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 377

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
q qk k ql l
k¼1 l¼1
(7.261)
N 0 M
X X 1
0 00 0
+ g ðÞ^ t + e sðÞ^ t l x ^ t q x
e 0
s
g
qk k ql l s q
k¼1 l¼1
N 0 M N 0
X X 00 X
00
e 00 000 Δτ s e 0 00 Δτ s 00 000 0
lk θ sðÞ +
θ q x , s +
s
θ sðÞ ¼ g e g e lq e e g ðÞ^ t
l lk k lq q lk k
k¼1 q¼1 k¼1
M
X 1
+ g ðÞ^ t q x 00 t ^ 00 ð l ¼ 1, 2, …, N Þ (7.262)
00
s
e 00
lq q
q¼1 s

2 e
∂ θ f , ij 1
e
e
e
e
sC f, ij θ f, ij ¼ K ij 2 + U f , ij θ ij θ f, ij + U f, ij ^ U f, ij ^ t ij ^ t f, ij
∂y s
(7.263)
y ¼ 0 : θ f, ij ¼ θ p, ij ; y ¼ 1 : θ f , ij ¼ θ p,i +1, j (7.264)
e
e
e
e
7.4.4 Dynamic response to inlet fluid temperature
disturbances
If the disturbances occur only in the inlet fluid temperatures, all other param-
eters such as thermal flow rates and heat transfer parameters are kept con-
^
_
^
0
^
_
_
0
0
stant, that is, C ¼ C ¼ C, U ¼ U ¼ ^ U, G ¼ G ¼ G, G ¼ G ¼ G ,
^
^
000
00
00
000
000
00
G ¼ G ¼ G , and G ¼ G ¼ G , then, Eqs. (7.259)–(7.264) reduce to
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
(7.265)
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
∂θ f, ij ∂θ f ,i 1, j
+ K ij K i 1, j (7.266)
∂y
∂y
y¼0 y¼1