Page 397 - Design and Operation of Heat Exchangers and their Networks

380 Design and operation of heat exchangers and their networks

Table 7.2 Nonzero elements of Q and C in Eq. (7.281) for different block arrangements.
Block arrangement Nonzero elements
q ijðÞ, ijðÞ ¼ sC p, ij + ψ + ψ + ϕ + ϕ
1<i n j ij i 1, j ij i 1, j
Þ ¼ ϕ Þ ¼ ϕ
q ijðÞ, i 1, jð i 1, j , q ijðÞ, i +1, jð ij
∗ ∗
ij
c ijðÞ, ijðÞ ¼ ψ , c ijðÞ, i 1, jð Þ ¼ ψ i 1, j
i¼1 Sequential q 1, jð Þ,1, jÞ ¼ sC p,1, j + ψ 1, j + ψ n j , j + ϕ 1, j + ϕ n j , j
ð
q ¼ ϕ n j , j , q 1, jð Þ,2, jÞ ¼ ϕ 1, j
ð
ð 1, jÞ, n j, jÞ ð
∗
Þ,1, jÞ ¼ ψ , c ¼ ψ ∗
c 1, jð ð 1, j ð 1, jÞ, n j, jÞ n j , j
ð
1
Symmetry I and II q 1, jð Þ,1, jÞ ¼ sC p,1, j + ψ 1, j + ϕ 1, j
ð
2
Þ,2, jÞ ¼ ϕ , c 1, jÞ,1, jÞ ¼ ψ ∗
q 1, jð ð 1, j ð ð 1, j
Symmetry III q (1,j), (1,j) ¼1, q (1,j), (2,j) ¼ 1
Sequential q (n j +1,j), (nj+1,j) ¼1, q (n j +1,j), (1,j) ¼ 1
1
i¼n j +1 Symmetry I q ¼ sC p, n j , j + ψ n j , j + ϕ n j , j
ð
ð n j +1, jÞ, n j +1, jÞ 2
q ¼ ϕ , c ¼ ψ ∗
ð
ð
ð n j +1, jÞ, n j, jÞ n j , j ð n j +1, jÞ, n j, jÞ n j , j
Symmetry II and III q (n j +1,j), (nj+1,j) ¼1, q (n j +1,j), (nj,j) ¼ 1
1
n j ¼1 q 1, jð Þ,1, jÞ ¼ sC p,1, j + ψ , c 1, jÞ,1, jÞ ¼ ψ ∗ 1, j
1, j
ð
ð
2
ð
The boundary conditions (7.267) and (7.268) can be expressed in the
matrix form as
0
0 e 0 e e e 00
Θ xðÞ ¼ G Θ + GΘ xðÞ (7.285)
e
000 00
00 0
00
Θ ¼ G Θ + G Θ xðÞ (7.286)
e
e e
e e
0 00 000
where G, G , G , and G are the matching matrices with time delay,
e
e e
e
000 Δτ s
00 Δτ s
0 Δτ s
whose elements are given by g e Δτ ij s , g e 0 ik , g e 00 li , and g e 000 lk ,
ij ik li lk
respectively.
With the similar method aforementioned, the solution in the Laplace
domain can be expressed as
1 0 0
0
e
Θ xðÞ ¼ V xðÞ V GV 00 G Θ (7.287)
e
e e

1
00 000 00 0 0
0
Θ ¼ G + G V 00 V GV 00 G Θ (7.288)
e
e
e
e
e
e
h i h i
r q x 0 r q x 00
00
with V xðÞ ¼ h pq e r q x p , V ¼ h pq e p , and V ¼ h pq e p ,
0
M M
M M M M
where r q (q¼1, 2, …, M) and h pq (p, q¼1, 2, …, M) are the eigenvalues