Page 374 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 357

0 00 000
e e
e
e
in which G, G , G , and G are the matching matrices with time delay,
0 Δτ s
000 Δτ s
00 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.
According to Eq. (2.140), the general solution of Eq. (7.165) can be
obtained by substituting the initial and new mean steady-state temperature
distributions into Eq. (2.140) and integrating the inhomogeneous term F xðÞ:
ð
1
∗
Rx
ð
Θ xðÞ ¼ He D + He R x ξÞ H F ξðÞdξ (7.171)
e
where
^
^
Rx
^ Rx
F xðÞ ¼ BT xðÞ + CT xðÞ ¼ BHe D + C ^ He D (7.172)
which yields
ð
Rξ
Rx
1
ð
Θ xðÞ ¼ He D ∗ + He R x ξÞ H BHe Ddξ
e
ð
^ Rξ
^
^
1
ð
+ He R x ξÞ H C ^ He Ddξ ¼ V xðÞD + K xðÞD + ^ K xðÞD (7.173)
r i x
Rx
where e ¼ diag e is a diagonal matrix, r i (i¼1, 2, …, M), and H¼
[h ij ] M M are the eigenvalues and eigenvectors of A in Eq. (7.165); r i (i¼1,
2, …, M)and H are the eigenvalues and eigenvectors of A in Eq. (7.151);
^
^
and ^ i (i¼1, 2, …, M)and H are the eigenvalues and eigenvectors of A
r
in Eqs. (7.126), (7.165).
^ Rx ^
Rx
The term Ke D + ^ Ke D is a special solution of the inhomogeneous
ordinary differential equation system (7.165), in which
r j x
Rx
V xðÞ ¼ He ¼ h ij e (7.174)
M M
K xðÞ ¼ HQe Rx (7.175)
hi h i

Q ¼ q ij ¼ p = r j r i (7.176)
ij
M M M M
hi
1
P ¼ p ¼ H BH (7.177)
ij
M M
^
^ K xðÞ ¼ HQe ^ Rx (7.178)
hi h i
^
Q ¼ ^ ij ¼ ^ = ^ j r i (7.179)
q
r
p
ij
M M M M
hi
^
1
P ¼ ^ ¼ H C ^ H (7.180)
p
ij
M M