Page 399 - Design and Operation of Heat Exchangers and their Networks

382 Design and operation of heat exchangers and their networks

The linearization and Laplace transform of the governing equation sys-
tem yield
0 1
^ ψ
e
dθ ij
e
e
e
e
sC ij θ ij + _ C ij ¼ ψ θ p, ij + θ p,i +1, j 2θ ij + @ e ψ ij e A ^ t p, ij +^ t p,i +1, j 2^ t ij
_ C ij
ij
ij
dx ^ _
C ij
(7.296)

θ
sC p, ij p, ij ¼ ψ θ ij θ p, ij + ψ i 1, j e e ij e e
e
e
∗ e
θ i 1, j θ p, ij + ϕ θ p,i +1, j θ p, ij
ij

+ ϕ i 1, j θ p,i 1, j θ p, ij + e ψ ^ t ij ^ t p, ij + e ψ i 1, j ^ t i 1, j ^ t p, ij
e
e
ij

+ ϕ ^ t p,i +1, j ^ t p, ij + ϕ i 1, j ^ t p,i 1, j ^ t p, ij
e
e
ij
(7.297)
where
1
e ψ ¼ U p, ij + U f, ij η ij (7.298)
e
e
ij
2
1
ψ ¼ U p, ij + U f, ij η ij (7.299)
ij
2
1
ϕ ¼ e U f, ij μ (7.300)
e
ij ij
2
1
ϕ ¼ U f, ij μ (7.301)
ij
2 ij
Eq. (7.297) can be expressed in the matrix form as
e ^
e ^
e ^
e ^

QΘ p ¼ CΘ QT p CT ¼ CΘ QP C T (7.302)
e
e
e
which yields the excess plate temperature vector in the Laplace domain as
e ^
Θ p ¼ PΘ + PT (7.303)
e
e
where The M p M coefficient matrices P and P are calculated from
e
1
P ¼ Q C (7.304)

e ^
1
P ¼ Q C QP (7.305)
e
e
in which the nonzero elements of Q, Q, C,and C for different block arrange-
e
e
^
mentsaregiveninTable7.3,andPiscalculatedfromEq.(7.245)andTable7.1.
By eliminating the plate temperatures with Eqs. (7.303), (7.244),
Eq. (7.296) can be expressed in the matrix form as
dΘ
e
^
¼ AΘ + BT (7.306)
e
dx