Page 142 - Design and Operation of Heat Exchangers and their Networks

130 Design and operation of heat exchangers and their networks

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If C and U L are constant in each channel (they may vary from channel to
channel), the previous ordinary differential equation system is linear and can
be solved analytically. According to the theory of linear algebra, the general
solution of Eq. (3.309) is obtained in the matrix form as

Rx
T xðÞ ¼ He D (3.311)
or

M
X
r j x
t i xðÞ ¼ h ij e d j i ¼ 1, 2, …, MÞ (3.312)
ð
j¼1
Rx r i x
in which e ¼diag{e } is a diagonal matrix and r i (i¼1, 2, …, M) are the
eigenvalues of matrix A. H is an M M square matrix whose columns are
the eigenvectors of the corresponding eigenvalues.
Eq. (3.311) or (3.312) is valid only if the eigenvalues differ from each
other. It has been proved that all eigenvalues of matrix A are real; however,
Eq. (3.311) or (3.312) might have multiple eigenvalues (Zaleski and
Jarzebski, 1973, 1974; Malinowski, 1983). A practical method to avoid mul-
tiple eigenvalues is to add very small random deviations to the input param-
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eters such as C ij and U L,ij . Such small deviations have almost no effect on the
results.
The coefficient vector D in Eq. (3.311) is determined by the boundary
conditions. In order to model the flow arrangement inside the multistream
heat exchanger, we define the mixing nodes in the multistream heat
exchanger. At a mixing node, several fluid streams are mixed and then
splitted again, such as manifolds of plate heat exchangers with complex flow
arrangement, headers of shell-and-tube heat exchangers between flow
passes, and internal manifolds of plate-fin heat exchangers. Each mixing
node is dealt as a channel. The number of the mixing nodes is denoted
by M M . The temperatures of the mixing nodes do not depend on the coor-
dinate x; therefore, they can be expressed as
t i xðÞ ¼ d i i ¼ M +1, M +2, …, M + M M Þ (3.313)
ð
Now, we extend the temperature vector to include the temperatures of
the mixing nodes by
T
½ Š (3.314)
Θ x ðÞ ¼ t 1 x ðÞ, t 2 x ðÞ, ⋯, t M x ðÞ, t M +1 , t M +2 , ⋯, t M + M M
The solution, Eqs. (3.311), (3.313), can be expressed as

T xðÞ ¼ V xðÞD (3.315)