Page 390 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 373

2 3
γ
ij
^ U p, ij tanh ^ =2 ^ U f, ij
4 + 5 ^ t ij ^ t p, ij
2 γ ^ =2 2
ij
2 3
γ
tanh ^
^ U p,i 1, j i 1, j =2 ^ U f,i 1, j
+ 4 + 5 ^ t i 1, j ^ t p, ij
^
2 γ i 1, j =2 2
^ U f, ij ^ U f,i 1, j
+ ^ t p,i +1, j ^ t p, ij + ^ t p,i 1, j ^ t p, ij ¼ 0 ð7:237Þ
γ
γ
γ
^
^ sinh^ ij γ i 1, j sinh^ i 1, j
ij
Defining the following parameters,
1
^
^
heat transfer parameter ^ ψ ¼ U p, ij + U f, ij ^ η f , ij (7.238)
i, j
2
1
^
^
bypass heat transfer parameter ϕ ¼ U f , ij ^μ f, ij (7.239)
ij
2

γ
tanh ^ =2
ij
fin efficiency ^ η ¼ (7.240)
f , ij
γ ^ =2
ij
2
fin bypass efficiency ^μ ¼ (7.241)
f, ij
^ sinh^γ
γ
ij ij
we can express Eq. (7.237) as

^
^
^
^
ϕ i 1, j p,i 1, j + ^ ψ ^ t p, ij + ^ ψ i 1, j p, ij + ϕ + ϕ i 1, j ^ t p, ij ϕ ^ t p,i +1, j
^ t
^ t
ij
ij
i, j
¼ ^ ψ ^ t ij + ^ ψ i 1, j i 1, j
^ t
i, j
(7.242)
or in a matrix form as
^ ^
^ ^
QT p ¼ CT (7.243)
^
^
where Q and C are the M p M p and M p M matrix, respectively, and their
nonzero elements are given in Table 7.1.
Eq. (7.243) can be expressed as
^
^ ^
T p ¼ PT (7.244)
where
1
^
^
^
P ¼ Q C (7.245)