Page 433 - Design and Operation of Heat Exchangers and their Networks

416 Design and operation of heat exchangers and their networks

The singularities given by Eq. (8.114) are of the first order. The formula
for their residua are
h i
s j τ sτ
res θ f s j e ¼ lim s s j θ f sðÞe (8.116)
e
e
s!s j
Therefore, we have
h i
s 0 τ
e ðÞe ¼ 1 (8.117)
res θ f s 0
h i
2β β cos β x + asin β x
h i j j j j
s j τ ax + s j τ
res θ f s j e ¼ h ie
e
2
2
2
s j a + β +2a B + NTU = s j + NTU 2
j
(8.118)
Substituting Eqs. (8.117), (8.118) into Eq. (8.112), we obtain the real-
time solution:
h i
∞
X2β β cos β x + asin β x
j
j
j
j
θ xτðÞ ¼ 1+ 2 e ax
2
a + β +2a
j¼1 j
8 9
< s j τ s j τ =
e e
h i + h i
: 2 2 2 2 ;
s j B + NTU = s j + NTU s j B + NTU = s j + NTU
(8.119)
If thethermal capacity of thefluid in the testcoreis negligiblecompared
with the thermal capacity of the solid wall, that is, B¼0, Eq. (8.114) reduces to

2
NTU a + β 2
j
s j ¼ 2 ð j ¼ 1, 2, …, ∞Þ (8.120)
2
a + β +2aNTU
j
The real-time temperature response becomes
θ B¼0 x, τÞ
ð
h i 2 2
j ð
∞ 8NTUa β β cos β x + asin β x NTU β + a Þ
2
X j j j j ax β + a +2aNTU τ
2
2
¼ 1 e j
2
2
2
2
2
j¼1 β + a 2 β + a +2a β + a +2aNTU
j
j
j
(8.121)
The real-time solutions (8.119) and (8.121) converge well for Pe<20.
However, if Pe>50, they might not converge; therefore, a numerical
inverse algorithm is demanded.