Page 141 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 129

Wolf (1964). Many significant discussions on the general solution were
made by Settari (1972), Zaleski and Jarzebski (1973, 1974), Malinowski
(1983), and Zaleski (1984).
Based on the pioneering research work of Kao and Wolf, Luo et al.
(2001) proposed a general analytical solution for the thermal performance
of parallel flow multistream plate-fin heat exchangers. By introducing three
matching matrices, a general form of the analytical solution for various types
of parallel flow multistream heat exchangers was explicitly expressed in the
matrix form by Roetzel and Luo (2001). The procedures will be introduced
in the following sections.

3.6.1 Multistream parallel channel heat exchangers without
interconnected wall
Consider a generalized multistream heat exchanger that consists of a bundle
of M parallel channels (M N f ) and contains N f fluid streams. The fluid flow-
ing through a channel exchanges heat with the fluids in all other channels. It
is assumed (1) the longitudinal heat conduction in the solid wall can be
neglected, (2) there is no heat loss to the environment, and (3) the heat trans-
fer coefficients and the properties of the fluids and wall materials can be con-
sidered constant along each channel. The general mathematical model can
be written as

M
X
_ dt i
C i ¼ U L, ij t j t i ð i ¼ 1, 2, …, MÞ (3.308)
dx
j¼1
_
with U L,ij ¼U L,ji and U L,ii ¼0, where C is a signed thermal capacity rate
(positive for the flow in the positive direction of the x-coordinate and neg-
_
ative for the counterflow), C ¼ _mc p , and U L is heat transfer parameter,
U¼kA/L. Eq. (3.308) can be rewritten into a matrix form as
dT xðÞ
¼ AT xðÞ (3.309)
dx
where A is an M M matrix:
2 M 3
1 X U L,12 U L,1M
U L,1l ⋯
6 _ _ _ 7
6 C 1 l¼1 C 1 C 1 7
6 M 7
6 1 X 7
U L,21 U L,2M
6 U L,2l ⋯ 7
A ¼ 6 _ _ _ 7 (3.310)
6 C 2 C 2 l¼1 C 2 7
⋮ ⋱
6 7
6 7
M
6 X 7
4 U L,M1 U L,M2 1 5
_ _ ⋯ _ U L,Ml
C M C M C M l¼1