Page 431 - Design and Operation of Heat Exchangers and their Networks

414 Design and operation of heat exchangers and their networks

4
X
r i
r i c i e ¼ 0 (8.101)
i¼1
4 3
X r
2 i
r i
ð sB + NTUÞr i + r c i e ¼ 0 (8.102)
i Pe
i¼1
Then, we can solve the earlier equation system to obtain the coefficients c i .It
should be mentioned that the solutions (8.97) and (8.98) are valid only if the
eigenfunction (8.96) has no multiple roots. If there are any multiple eigen-
values, a simple treatment is to add a small deviation to a parameter (e.g.,
NTU) to avoid the multiple eigenvalues.
If the flow in the heat exchanger can be assumed as a plug flow, that is,
the axial dispersion can be neglected, Pe!∞, the eigenfunction reduces to

s + NTU
3 2 1 2
r + sB + NTUð Þr r s B + sBNTU + sNTU ¼ 0
K w K w
(8.103)
The Laplace transform of the fluid temperature becomes

3
X
e c i e r i x (8.104)
θ ¼
i¼1
in which r i (i¼1, 2, 3) are the three eigenvalues of Eq. (8.103) and c i (i¼1, 2,
3) are determined by solving the following equations:

3
X 0
c i ¼ θ (8.105)
e
i¼1
3
X 2
ð sB + NTUÞr i + r c i ¼ 0 (8.106)
i
i¼1
3
X 2
r i
ð sB + NTUÞr i + r c i e ¼ 0 (8.107)
i
i¼1
For the case that the axial heat conduction in the wall is negligible but the
axial dispersion in the fluid should be taken into account, we have

sNTU
2
r +Per +Pe sB + ¼ 0 (8.108)
s + NTU