Page 366 - Design and Operation of Heat Exchangers and their Networks

352 Design and operation of heat exchangers and their networks

τ ¼ 0 : θ i ¼ 0 i ¼ 1, 2, …, MÞ, θ w,i ¼ 0 j ¼ 1, 2, …, M w Þ (7.144)
ð
ð
where
_
_
_
ΔC i τðÞ ¼ C i τðÞ C i , ΔU ik τðÞ ¼ U ik τðÞ U ik , Δg ij τðÞ ¼ g ij τðÞ g ,
ij
(7.145)
τ
τ
0
0
0
00
00
000
Δg τðÞ ¼ g τðÞ g , Δg τðÞ ¼ g τðÞ g , Δg ðÞ ¼ g ðÞ g 000
00
000
ik ik ik li li li lk lk lk
( (
_
_
0,C i > 0 1,C i > 0
0 00
x ¼ , x ¼ ð i ¼ 1, 2, …, MÞ (7.146)
i i
_
_
1,C i < 0 0,C i < 0
In the earlier equations, ^ t i and ^ t w,k are the steady-state solutions of the
fluid and wall temperatures at the initial operation conditions given by
Eqs. (7.134) and (7.124), respectively. t j and t w,k are the steady-state solu-
tions of the fluid and wall temperatures at the new mean operating condi-
tions described by the following governing equation system:
!
M M
M w
X X X
_ dt i
ð
C i + U ik t i U mk t m = U mk ¼ 0 i ¼ 1, 2, …, MÞ (7.147)
dx
k¼1 m¼1 m¼1
M M
X X
t w,k ¼ U jk t j = U mk ð k ¼ 1, 2, …, M w Þ (7.148)
j¼1 m¼1
N 0 M
X X

0 0 0 00
ð
t i x g t g t j x ¼ 0 i ¼ 1, 2, …, M Þ (7.149)
i ik k ij j
k¼1 j¼1
N 0 M
X X
00 000 0 00 00 00
ð
t g t g t i xðÞ ¼ 0 l ¼ 1, 2, …, N Þ (7.150)
l lk k li
k¼1 i¼1
or in the matrix form as
dT
¼ AT (7.151)
dx
0 0 ^ 00
T xðÞ ¼ G T + GT xðÞ (7.152)
00 000 0 00 00
T ¼ G T + G T xðÞ (7.153)
^
The elements of the coefficient matrix A are
0 1
B C
M w
1 X B U jn C
a ij ¼ U in B δ ij C ð i ¼ 1, 2, …, MÞ (7.154)
M
_ B X C
C i n¼1 @ A
U mn
m¼1