Page 391 - Design and Operation of Heat Exchangers and their Networks

374 Design and operation of heat exchangers and their networks

^
^
Table 7.1 Nonzero elements of Q and C in Eq. (7.243) for different block
arrangements.
Block arrangement Nonzero elements
^
^
^ ¼ ^ ψ + ^ ψ + ϕ + ϕ
q
1<i n j ij ij i 1, j ij i 1, j
ðÞ, ijðÞ
^
^
q ^ ¼ ϕ , ^ ¼ ϕ
q
ij
ij
ðÞ, i 1, jÞ i 1, j ðÞ, i +1, jÞ ij
ð
ð
c c Þ ¼ ^ ψ
ij i 1, j
^ ijðÞ, ijðÞ ¼ ^ ψ , ^ ijðÞ, i 1, jð
^
^
q
i¼1 Sequential ^ ¼ ^ ψ 1, j + ^ ψ n j , j + ϕ 1, j + ϕ n j , j
ð
ð 1, jÞ,1, jÞ
^
^
^ ¼ ϕ n j , j , ^ ¼ ϕ 1, j
q
q
ð
ð
ð 1, jÞ, n j, jÞ ð 1, jÞ,2, jÞ
c Þ,1, jÞ ¼ ^ ψ , ^c ¼ ^ ψ
^ 1, jð ð 1, j ð 1, jÞ, n j, jÞ n j , j
ð
^
q
Symmetry I and II ^ ¼ ^ ψ + ϕ
ð
ð 1, jÞ,1, jÞ 1, j 1, j
^ ^ c Þ,1, jÞ ¼ ^ ψ
q
1, j
ð 1, jÞ,2, jÞ ¼ ϕ , ^ 1, jð ð 1, j
ð
q
q
Symmetry III ^ ¼ 1, ^ ¼ 1
ð
ð
ð 1, jÞ,1, jÞ ð 1, jÞ,2, jÞ
q
q
Sequential ^ ¼ 1, ^ ¼ 1
ð
ð n j +1, jÞ, n j +1, jÞ ð n j +1, jÞ,1, jÞ
ð
^
i¼n j +1 Symmetry I ^ ¼ ^ ψ + ϕ
q
ð
ð n j +1, jÞ, n j +1, jÞ n j , j n j , j
^
^ ¼ ϕ ,
q
ð n j +1, jÞ, n j, jÞ n j , j
ð
^ c ¼ ^ ψ
ð
ð n j +1, jÞ, n j, jÞ n j , j
q
q
Symmetry II and III ^ ¼ 1, ^ ¼ 1
ð
ð n j +1, jÞ, n j +1, jÞ ð n j +1, jÞ, n j, jÞ
ð
c
q
n j ¼1 ^ ¼ 1, ^ 1, jð Þ,1, jÞ ¼ 1
ð 1, jÞ,1, jÞ ð
ð
By eliminating the fin temperature ^ t f and plate temperature ^ t p in
Eq. (7.226) with Eqs. (7.233), (7.244), we obtain the equation for the fluid
temperature:
^ _ d^ t ij
C ij ¼ ^ ψ ^ t p, ij + ^ t p,i +1, j 2^ t ij (7.246)
ij
dx
or in the matrix form as
^
dT ^ ^
¼ AT (7.247)
dx
^
where the elements of the M M coefficient matrix A are
^ ψ ij
a ^ p + ^ 2δ ij,q
p
^ ij,q ¼
^ _ ij,q ð i +1, jÞ,q
C ij

i ¼ 1, 2, …, n j ; j ¼ 1, 2, …, m; q ¼ 1, 2, …, M (7.248)