Page 375 - Design and Operation of Heat Exchangers and their Networks

358 Design and operation of heat exchangers and their networks

The coefficient vector D in Eq. (7.173) is determined by substitution of
Eqs. (7.134), (7.155), (7.173) into Eq. (7.169):

1 0 0 0 1 0

^
00
0
0
0
0
D ¼ V GV 00 G Θ + ΔG T + G T 2 K 2GK ΔGV 00 D
e
e
e
e e
e
s

1
0 00 ^
^ K G ^ K + V GV D (7.181)
^0
^00
e
s
^
in which D and D are calculated by Eqs. (7.131), (7.158), respectively, and
the matrices
hi h i
V ¼ v 0 ij ¼ h ij e r j x 0 i (7.182)
0
M M M M
hi h 00 i
00
V ¼ v 00 ¼ h ij e r j x i (7.183)
ij
M M M M
0 0 00 00
K ¼ K xðÞ, K ¼ K xðÞ (7.184)
^ 0 ^ 0 ^ 00 ^ 00
K ¼ K ^xðÞ, K ¼ K ^xðÞ (7.185)
The excess exit temperature vector of the fluid streams is obtained from
Eq. (7.170):
00 000 0 000 0 1 000 00 00 00 00 00
Θ ¼ G Θ + Δ e G T + G G ^ 000 ^0 e 00 e D
T + G V D + G K + Δ e G V
e e
e
s

00 00 1 00
D
+ G ^ K + G G ^00 V ^00 ^ (7.186)
e
s
The temperature dynamics in the real-time domain can be obtained by
the use of the numerical inverse Laplace transform with FFT algorithm,
Eq. (2.178).
7.3.5 Dynamic model for startup problem
If the heat exchanger initially has a uniform temperature,
^ t i xðÞ ¼^ t w,m xðÞ ¼ t 0 ð i ¼ 1, 2, …, M; m ¼ 1, 2, …, M w Þ (7.187)
and then at τ¼0 undergoes sudden step changes in the inlet fluid temper-
_
0
atures ^ t , thermal capacity rates C i , heat transfer parameters U im , and flow
k
0
00
rate distributions—g ij , g ik , g li , and g lk —and then keep constant in τ>0,
000
except the inlet fluid temperatures. The inlet fluid temperatures can change
0 0
with time arbitrarily, ^ t ¼ ^ t τðÞ (τ>0; k¼1, 2, …, N ). The governing
0
k k
equation system (7.113)–() for the startup problem is expressed as