Page 78 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 65

The temperature vector T consists of M temperatures of fluids and walls.
A is the M M coefficient matrix of the governing equations. The solution
of Eq. (2.174) with its initial condition (2.175) reads
τ
Z
Rτ 1 R τ τ Þ 1 0 0
0
ð
T ¼ He H T 0 + He H B τðÞdτ (2.176)
0
Rτ Rτ r 1 τ r 2 τ r M τ
in which e is a diagonal matrix, e ¼diag{e ,e ,⋯,e }, r i (i¼1, 2, …, M)
are the eigenvalues of the coefficient matrix A,and H is the eigenvector matrix
whose columns are the eigenvectors of the corresponding eigenvalues.
For the distributed parameter model, the Laplace transform can be
applied to eliminate the time variable. Then, the analytical solution in the
Laplace domain can be obtained. The real-time temperature responses
can be calculated by means of numerical inverse algorithm. It is suggested
to use the fast Fourier transform (FFT) algorithm to inverse the solution into
the real-time domain. The formula of Ichikawa and Kishima (1972)

" #
e aτ n M 1 1
X
f τ n ¼ Re fa + ikπ=τÞe 2iπnk=M faðÞ (2.177)
ðÞ
e
ð
e
τ 2
k¼0
can be adopted. With this algorithm, the temperature variation at all time
points τ n ¼2nτ/M in the time interval [0, τ] can be obtained simultaneously.
The value of a in Eq. (2.177) is taken as 4<aτ<5. M is an exponent of 2,
11
usually M¼2 ¼2048. The FFT algorithm has no special requirements for
the Laplace transforms to be inversed. However, if the function in the real-
time domain τ>0 has a discontinuity, the FFT algorithm will add an addi-
tional oscillation near this discontinuous point. If the value of M is large
enough, the oscillation becomes an additional sharp pulse. The maximum
value of the pulse amplitude is 8.949% of the step change at the discontin-
uous point (Anon., 1979).
In some cases, the inverse Laplace transform can be obtained analytically
by means of the residuum theorem (Anon., 1979):

Z σ + i∞ ∞
1 X
h i h i
sτ
L 1 e fsðÞe ds ¼ res fs j e s j τ (2.178)
e
fsðÞ ¼
e
2π i σ i∞ j¼0
in which s j is the jth singularity of function fsðÞ. For double Laplace trans-
e
form, we can get the analytical solution with Eq. (2.178) and then use
numerical inverse algorithm to get the real-time solution. This method
has been successfully used for predicting the transient behavior of crossflow
heat exchangers (Luo, 1998).