Page 365 - Design and Operation of Heat Exchangers and their Networks
P. 365
Dynamic analysis of heat exchangers and their networks 351
M w M w
∂θ i _ ∂θ i X _ dt i X
ð
ð
C i + C i + U ik θ i θ w,k Þ + ΔC i + ΔU ik t i t w,k Þ
∂τ ∂x dx
k¼1 k¼1
M w
X
_ d^ t i
^ ð
ð
+ C i + U ik t i ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, MÞ (7.139)
dx
k¼1
Eliminating the first derivatives of t i and ^ t i in Eq. (7.139) with
Eqs. (7.119), (7.147), we can express Eq. (7.139) as
!
M w M w _
X X
∂θ i _ ∂θ i ΔC i
ð
C i + C i + U ik θ i θ w,k Þ + ΔU ik U ik ð t i t w,k Þ
∂τ ∂x _
k¼1 k¼1 C i
!
M w _ ^
X
U ik C i U ik
+ ð ^ t i ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, Mð Þ (7.140)
^ _
k¼1 C i
With the same method, Eqs. (7.114)–(7.116) can be linearized as follows:
M M
∂θ w,k X X
C w,k + U jk θ w,k θ j + ΔU jk t w,k t j
∂τ
j¼1 j¼1
M
X
^
+ U jk t w,k ^ t j ¼ 0 k ¼ 1, 2, …, M w Þ (7.141)
ð
j¼1
N 0 M N 0
X X X
0 g θ τ Δτ 0 00 0 0
0
0
θ i τ, x g θ j τ Δτ ij , x Δg τðÞt
i ik k ik ij j ik k
k¼1 j¼1 k¼1
M N 0
X X
0
0
Δg ij τðÞt j x 00 +^ t i x g ^ t 0
j i ik k
j¼1 k¼1
M
X
g ^ t j x 00 ¼ 0 i ¼ 1, 2, …, MÞ (7.142)
ð
ij j
j¼1
N 0 M N 0
X X X
00 g θ τ Δτ 000 00 00 00 000 τ 0
000 0
θ τðÞ g θ i τ Δτ , x Δg ðÞt
l lk k lk li li i lk k
k¼1 i¼1 k¼1
M N 0
X X
000 0
00
00
Δg τðÞt i x 00 + ^ t g ^ t
li i l lk k
i¼1 k¼1
M
X
00 00 00
ð
g ^ t i xðÞ ¼ 0 l ¼ 1, 2, …, N Þ
li
i¼1
(7.143)