Page 348 - Design and Operation of Heat Exchangers and their Networks

334 Design and operation of heat exchangers and their networks

^
^ H is the eigenvector matrix of A:

^
a a h 1i (7.59)
^ 12
^ 11 ^r i
^
a ^ 21 a r h 2i ¼ 0 ð i ¼ 1, 2Þ
^ 22 ^ i
which yields

1 1
^ H ¼ ^ n ^ _ ^ _ (7.60)
ð
1 1Þ C 1 =C 2
^
The coefficient matrix D should be determined by the boundary condi-
tions. Substitution of Eq. (7.53) into Eq. (7.56) yields
^
0
^
0
T ¼ ^ V D (7.61)
where
2 3
" # ^ 1 x ^ 2 x
^
^
r ^0
r ^0
^ v 0 11 ^ v 0 12 h 11 e 1 h 12 e 1
^ 0
V ¼ ¼ 4 5 (7.62)
^ v 0 ^ v 0 ^ ^ 1 x ^ ^ 2 x
r ^0
r ^0
21 22 h 21 e h 22 e
2 2
^
Then, D can be written as
^
^
D ¼ ^ V 0 1 T 0 (7.63)
The initial steady-state temperature distribution can be expressed in the
matrix form as
^
^
^
^ Rx
T ¼ ^ He V 0 1 T 0 (7.64)
^ ^
_
_
However, for the counterflow heat exchanger with C 1 ¼ C 2 , Eq. (7.57)
r
has a multiple root ^ 1 ¼^r 2 ¼ 0, and the solution (7.56) is not valid. As
has been mentioned in Chapter 2, a simple method to avoid the multiple
^ ^
_
_
eigenvalues is to add a small deviation to C 1 or C 2 , which has almost no
influence on the final results.
7.1.2.6 Linearization of the nonlinear problems
If the thermal flow rates or the heat transfer parameters vary with time,
Eqs. (7.38)–(7.40) represent a nonlinear problem, and the Laplace transform
cannot be applied to them. For small disturbances, the problem can be lin-
earized. We use the symbol “¯” to denote the time average values of thermal
flow rates and heat transfer parameters in the later period of the dynamic
process or at a new steady state. For small disturbances in thermal flow rates,
heat transfer parameters and inlet fluid temperatures, Eq. (2.174)