Page 347 - Design and Operation of Heat Exchangers and their Networks
P. 347
Dynamic analysis of heat exchangers and their networks 333
^ _ d^ t 1
^ ð
C 1 + ^ U 1 t 1 ^ t w Þ ¼ 0 (7.50)
dx
n ^ _ d^ t 2
ð 1Þ C 2 + ^ U 2 t 2 ^ t w Þ ¼ 0 (7.51)
^ ð
dx
^ U 1 t 1 ^ t w Þ + ^ U 2 t 2 ^ t w Þ ¼ 0 (7.52)
^ ð
^ ð
^
^
0
^ t 1 x 0 ¼ ^ t , ^ t 2 x 0 ¼ ^ t 0 (7.53)
1 1 2 2
Substitution of Eq. (7.52) into Eqs. (7.50), (7.51) yields
2 3
d^ t 1
" #"#
^ 11 ^a 12
6 dx 7 a ^ t 1
6 7
¼ (7.54)
4 5
^ 21 ^ 22
d^ t 2 a a ^ t 2
dx
or
^
dT
^ ^
¼ AT (7.55)
dx
in which
^ ^
^
^
^ U 1 U 2 ^ U 1 U 2
a ^ 11 ¼ ¼ NTU 1 , ^a 12 ¼ ¼ NTU 1
^ _ ^ ^ ^ _ ^ ^
C 1 U 1 + U 2 C 1 U 1 + U 2
n ^ n ^
ð 1Þ ^ U 1 U 2 n ð 1Þ ^ U 1 U 2 n
^
^
a ^ 21 ¼ ¼ 1ð Þ NTU 2 , ^a 22 ¼ ¼ 1ð Þ NTU 2
^ _ ^ ^ _ ^
C 2 U 1 + ^ U 2 C 2 U 1 + ^ U 2
The general solution of Eq. (7.55) can be written as
^
^
^ Rx
T ¼ ^ He D (7.56)
^ 1 x
r
e 0
^ Rx
^ Rx
Here, we denote e as a diagonal matrix, e ¼ ^ 2 x , in which ^ 1 and
r
r
0 e
^
r ^ 2 are the eigenvalues of the coefficient matrix A:
a
^ 11 ^r a
^ 12
¼ 0 (7.57)
r
a
a ^ 22 ^
^ 21
Then, we have
" #
n ^
1 ð 1Þ ^ U 1 U 2 n
^
^
r
^ 1 ¼ 0, ^r 2 ¼ + ¼ NTU 1 1Þ NTU 2 (7.58)
ð
^ _ ^ _ ^ ^
C 1 C 2 U 1 + U 2