Page 367 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 353

The solutions of Eqs. (7.151)–(7.153) for the steady-state fluid and wall
temperatures at the new mean operating conditions can be obtained by
replacing “^” in Eqs. (7.134), (7.124) with “¯” as follows:

T xðÞ ¼ V xðÞD (7.155)

M M
X X
t w,k ¼ U ik t i = U ik ð k ¼ 1, 2, …, M w Þ (7.156)
i¼1 i¼1
where

V xðÞ ¼ He Rx ¼ h ij e r j x i (7.157)
M M

0 00 1 0 0
D ¼ V GV G T (7.158)
It should be pointed out that the governing equation system,
_
Eqs. (7.140)–(7.143), is an approximate model if C, U, G, G , G , and
0
00
G are functions of time for τ>0; therefore, the linearized model is only
000
0 _
00
valid for small disturbances in T , C, U, G, G , G , and G .
000
0
7.3.4 Analytical solution with numerical inverse algorithm
The governing equation systems for linear problems and linearized non-
linear problems are solved by means of the Laplace transform. Applying
the Laplace transform to Eqs. (7.140)–(7.143), the governing equations
become a set of ordinary differential equations:

!
M w
M w X e _
X
e
_ dθ i + U ik θ i θ w,k + ΔC i ð t i t w,k Þ
e
e
e
sC i θ i + C i Δ e U ik U ik
dx _
k¼1 k¼1 C i
!
_
M w
1 X C i
+ U ik ^ U ik ð ^ t i ^ t w,k Þ ¼ 0 ð i ¼ 1, 2, …, MÞ (7.159)
s ^ _
k¼1 C i
M M
X X
sC w,k θ w,k + U jk θ w,k θ j + ΔU jk t w,k t j
e
e
e
e
j¼1 j¼1
M
1 X
^
+ U jk t w,k ^ t j ¼ 0 k ¼ 1, 2, …, M w Þ (7.160)
ð
s
j¼1