Page 389 - Design and Operation of Heat Exchangers and their Networks

372 Design and operation of heat exchangers and their networks

N 0 M
X X
^
^
0
^ t q x 0 ¼ g ^ t + g ^ ^ t l x 00 (7.228)
^ 0
q qk k ql l
k¼1 l¼1
N 0 M
X X
00 0 ^ t ^ 00 00
^ t ¼ g ^ 000 ^ t + g ^ 00 lq q x ð l ¼ 1, 2, …, N Þ (7.229)
l lk k q
k¼1 q¼1
2
∂ ^ t f , ij
^
K ij + ^ U f, ij t ij ^ t f, ij ¼ 0 (7.230)
∂y 2
y ¼ 0 : ^ t f, ij ¼^ t p, ij ; y ¼ 1 : ^ t f, ij ¼^ t p,i +1, j (7.231)
Since the axial heat conduction in fins is neglected, the energy equation
for fins can be solved alone by taking the fluid temperature ^ t f and plate tem-
perature ^ t p as parameters, which yields

i sinh ^γ y
h ij

^ t f , ij ¼ ^ t ij + ^ t p,i +1, j ^ t ij ^ t p, ij ^ t ij cosh^γ
ij
γ
sinh^ ij (7.232)

+ ^ t p, ij ^ t ij cosh ^γ y
ij
1
ð γ
cosh^ 1
ij
^ t f , ij dy ¼ ^ t ij + ^ t p,i +1, j + ^ t p, ij 2^ t ij (7.233)
γ ij ij
^ sinh^γ
0
γ γ γ
ij
ij
∂^ t f , ij ^ ij ^ cosh^
¼ ^ t p,i +1, j ^ t ij ^ t p, ij ^ t ij (7.234)
γ
γ
∂y sinh^ sinh^
y¼0 ij ij
γ γ γ
∂^ t f ,i 1, j ^ i 1, j ^ i 1, j cosh^ i 1, j
¼ ^ t p,i 1, j ^ t i 1, j + ^ t p, ij ^ t i 1, j
γ
γ
∂y sinh^ sinh^
y¼1 i 1, j i 1, j
(7.235)
where
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ ^ U f, ij =K ij or K ij ¼ ^ U f , ij =^γ 2 (7.236)
^ ¼
ij ij
Eq. (7.232) shows that the fin temperature depends on the temperature
of the fluid flowing over the fin surface and the temperatures of the plates
above and below the fins. Substitution of Eqs. (7.234), (7.235) into
Eq. (7.227) yields a relationship between the fluid temperature and the plate
temperature, by the use of the relation (coshx 1)/sinhx¼tanh(x/2):