Page 362 - Design and Operation of Heat Exchangers and their Networks

348 Design and operation of heat exchangers and their networks

7.3.2 Initial steady-state solution
For the initial steady state, all the excess temperatures are zero, and the
governing equation system reduces to

M w
^ _ d^ t i X
^ ð
C i + ^ U ik t i ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, Mð Þ (7.119)
dx
k¼1
M
X
^
ð
^ U jk t j ^ t w,k ¼ 0 k ¼ 1, 2, …, M w Þ (7.120)
j¼1
N 0 M
X X

0 0 0 ^ ^ t j ^ 00
^ t i ^x g ^ ^ t g x ¼ 0 i ¼ 1, 2, …, MÞ (7.121)
ð
i ik k ij j
k¼1 j¼1
N 0 M
X X
00 000 0 00 00 00
x
^ t ^ g ^ t g ^ ^ t i ^ ¼ 0 l ¼ 1, 2, …, N Þ (7.122)
ð
l lk k li i
k¼1 i¼1
T T
00
0
00
00
0
x
^ 0
where x ¼ ^x , ^x , …, ^x 0 and ^x ¼ ^ , ^ , …, ^ 00 are the initial
x
x
1 2 M 1 2 M
dimensionless coordinate vectors of channel inlets and outlets at τ¼0:
8 8
^ ^
0,C i > 0 1,C i > 0
< _ < _
0 00
x
^ x ¼ , ^ ¼ ð i ¼ 1, 2, …, MÞ (7.123)
i i
: ^ _ : ^ _
1,C i < 0 0,C i < 0
Eq. (7.120) can be expressed as
M
X
U jk
^ ^ t j
j¼1
^ t w,k ¼ ð k ¼ 1, 2, …, M w Þ (7.124)
M
X
^ U mk
m¼1
Eliminating ^ t w,k in Eq. (7.119) with Eq. (7.124) yields
0 1
M
X
U jk
^ ^ t jC
B
M w
^ _ d^ t i X B j¼1 C
^
ð
^ B
C i + U ik t i C ¼ 0 i ¼ 1, 2, …, MÞ (7.125)
dx B X C
M
k¼1 @ A
^ U mk
m¼1
which can be written in the matrix form as
^
dT
^ ^
¼ AT (7.126)
dx