Page 350 - Design and Operation of Heat Exchangers and their Networks

336 Design and operation of heat exchangers and their networks

By eliminating θ w with Eq. (7.72), Eqs. (7.70)–(7.71) can be written in the
e
matrix form
dΘ ^ Rx
e
^
^
^
¼ AΘ + BT ¼ AΘ + BHe D (7.74)
e
e
dx
in which the elements of the coefficient matrices A and B are

1 sC w + U 2 U 1 U 2
a 11 ¼ sC 1 + U 1 , a 12 ¼ ,
_ sC w + U 1 + U 2 _
ð
C 1 C 1 sC w + U 1 + U 2 Þ
n n
ð 1Þ U 1 U 2 ð 1Þ sC w + U 1
a 21 ¼ , a 22 ¼ sC 2 + U 2 ,

_ C 2 sC w + U 1 + U 2 _ C 2 sC w + U 1 + U 2
" #

^
^
^
1 sC w + U 2 U 2 U 1 U 1 ^ U 1 U 2 e
b 12 ¼ b 11 ¼ e U 1 + e U 2 _ C 1 ,
^ ^
_ C 1 U 1 + ^ U 2 sC w + U 1 + U 2 sC w + U 1 + U 2 _ C 1
2 3
n n
ð 1Þ U ^ U sC w + U 1 ^ U 1 ð 1Þ ^ U ^ U
2 2
b 21 ¼ b 22 ¼ 4 e U + e U 1 2 e _ C 2 5
1
2
^ n ^
_ C ^ U + ^ U sC w + U + U 2 sC w + U + U 2 1Þ _ C
1
1
2 1 2 ð 2
The solution of Eq. (7.74) can expressed as
ð
Rx R x ξÞ 1 ^ Rξ ^ Rx ^ Rx ^
ð
Θ ¼ He D + He H B ^ He Ddξ ¼ He D + He D (7.75)
e
in which R is the eigenvalue matrix of the coefficient matrix A,

r 1 0
R ¼ , where
0 r 2
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2
r 1,2 ¼ a 11 + a 12 ð a 11 a 12 Þ +4a 12 a 21 (7.76)
2
H is the corresponding eigenvector matrix:

h 11 h 12 1 1
H ¼ ¼ (7.77)
h 21 h 22 ð r 1 a 11 Þ=a 12 ð r 2 a 11 Þ=a 12
H ¼ HQ (7.78)

ð
q 11 = ^r 1 r 1 Þ q 12 = ^r 2 r 1 Þ
ð
Q ¼ (7.79)
q 21 = ^r 1 r 2 Þ q 22 = ^r 2 r 2 Þ
ð
ð

q 11 q 12 1
Q ¼ ¼ H B ^ H (7.80)
q 21 q 22