Page 396 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 379

Now, we can substitute Eqs. (7.274), (7.275) into Eq. (7.266) to get the
relationship between the fluid temperature and the plate temperature;

sC p, ij + ψ + ϕ + ψ i 1, j + ϕ i 1, j θ p, ij ϕ θ p,i +1, j ϕ i 1, j p,i 1, j
θ
e
e
e
ij
ij
ij
¼ ψ θ ij + ψ ∗ i 1, j i 1, j
θ
∗e
e
ij
(7.276)
where
1 h i
heat transfer parameter ψ ¼ U p, ij + sC f, ij + U f, ij η ,
ij
ij
2 (7.277)
∗ 1
ψ ¼ U p, ij + U f, ij η
ij ij
2
1
bypass heat transfer parameter ϕ ¼ sC f , ij + U f, ij μ ij (7.278)
ij
2

tanh γ =2
ij
fin efficiency η ¼ (7.279)
ij
γ =2
ij
2
fin bypass efficiency μ ¼ (7.280)
i, j
γ sinhγ
i, j i, j
The matrix form of Eq. (7.276) can be expressed as
QΘ p ¼ CΘ (7.281)
e
e
The excess plate temperature vector in the Laplace domain is presented
explicitly as
Θ p ¼ Q CΘ ¼ PΘ (7.282)
e
1 e
e
The nonzero elements of the M p M p matrix Q and M p M matrix C
for different block arrangements are given in Table 7.2.
Substitution of Eqs. (7.282), (7.273) into Eq. (7.265) yields
dΘ
e
¼ AΘ (7.283)
e
dx
where the elements of A are given as follows:
∗
ij
ψ
a ijðÞ,q ¼ p ijðÞ,q + p i +1, jð Þ,q
_
C ij

δ ijðÞ,q sC f, ij U f, ij ∗
sC ij + 1 η f , ij +2ψ ij ð q ¼ 1, 2, …, MÞ
_ sC f , ij + U f, ij
C ij
(7.284)