Page 388 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 371

∂t p, ij U p, ij U p,i 1, j ∂t f, ij
C p, ij ¼ t ij t p, ij + t i 1, j t p, ij + K ij
∂τ 2 2 ∂y
y¼0

∂t f,i 1, j
K i 1, j (7.219)
∂y
y¼1
N 0 M

X X
0
0
t q τ, x 0 ¼ g t τ Δτ 0 + g ql t l τ Δτ ql , x 00 (7.220)
q qk k qk l
k¼1 l¼1
N 0 M
X X
000 0
00 g t τ Δτ 00 00 00 00 00
t τðÞ ¼ + g t q τ Δτ , x ð l ¼ 1, 2, …, N Þ
l lk k lk lq lq q
k¼1 q¼1
(7.221)
τ ¼ 0 : t ij ¼ ^ t ij , t p, ij ¼ ^ t p, ij (7.222)
2
∂t f, ij ∂ t f, ij
C f, ij ¼ K ij + U f, ij t ij t f, ij (7.223)
∂τ ∂y 2
y ¼ 0 : t f , ij ¼ t p, ij ; y ¼ 1 : t f, ij ¼ t p,i +1, j (7.224)
τ ¼ 0 : t f, ij ¼^ t f , ij (7.225)
_
in which C ij and C f,ij are the thermal capacities of the fluid and the fins, C ij is
the thermal capacity flow rate of the fluid, U p,ij ¼(αA) p,ij is the heat transfer
parameter between the fluid and the separating plate where A p is the primary
surface area, U f,ij ¼(αA) f,ij is the heat transfer parameter between the fluid
and the fins where A f is the secondary surface area, K ij ¼(λ f A fc /h f L) ij ,
x ¼ x=L ij , and y ¼ y=h f , ij . All these parameters refer to the parameters of
channel ij. C p,ij is the thermal capacities of the separating plate ij.
7.4.2 Steady-state temperature distribution in plate-fin heat
exchangers
For the initial stead state, the governing equation system (7.218)–(7.225) for
section q (q¼1, 2, …, M p ; i¼1, 2, …, n; j¼1, 2, …, m) reduces to
0 1 1
^ ð
^ _ d^ t ij U p, ij
C ij ¼ ^ t p, ij +^ t p,i +1, j 2^ t ij + ^ U f, ij @ ^ t f , ij dy ^ t ij A (7.226)
dx 2
0
^ U p, ij ^ U p,i 1, j
^ t ij ^ t p, ij + ^ t i 1, j ^ t p, ij
2 2

∂^ t f, ij ∂^ t f ,i 1, j
¼ 0 (7.227)
+ K ij K i 1, j
∂y ∂y
y¼0 y¼1