Page 434 - Design and Operation of Heat Exchangers and their Networks

Experimental methods for thermal performance of heat exchangers 417

If the axial dispersion coefficient approaches infinitely large values, that
is, Pe¼0, the fluid temperature approaches a uniform distribution. This is
the case of stirrers, for example. It is easy to obtain the real-time fluid tem-
perature response:
ð 1+ Ba 1 Þe a 2 τ 1+ Ba 2 Þe a 1 τ
ð
θ Pe¼0 x, τð Þ ¼ 1 (8.122)
ð
Ba 1 a 2 Þ
in which a 1 and a 2 are the eigenvalues of the governing equations for Pe¼0
1
ð
a 1,2 ¼ ½ NTU 1 + BÞ +1Š
2B
1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
ð
NTU 1+2Bð Þ + 2NTU + 1 NTUBÞ (8.123)
2B
For B¼0, Eq. (8.122) reduces to
NTU NTU τ

θ Pe¼0,B¼0 x, τð Þ ¼ 1 e NTU + 1 (8.124)
NTU+1
The effect of the axial fluid dispersion on the outlet fluid temperature
dynamics is shown in Fig. 8.6. It illustrates that the initial temperature rise
at the outlet of the heat exchanger is more sensitive to the axial dispersion.
For the case that the value of NTU is about 2, the enlargement of the initial
temperature rise due to the axial dispersion is the most significant, as is
shown in Fig. 8.7. Only for small values of NTU, the axial dispersion effect
can be neglected. If the working fluid is a liquid, the effect of the axial dis-
persion becomes more significant. The outlet fluid temperature responses of
the fluid with B>0 are given in Fig. 8.8.
The effect of axial heat conduction in the wall on the outlet fluid tem-
perature dynamics is similar to that of the axial fluid dispersion. The differ-
ence is that it does not change the initial temperature rise if the thermal
capacity of the fluid is negligible, as is shown in Fig. 8.9.
The earlier solutions are the temperature responses to a unit step change
in inlet fluid temperature. The same method can be used to obtain the
dynamic responses to other inlet temperature variations such as the expo-
0
0
nential variation θ ¼ 1 e τ=τ ∗ and cosine variation θ ¼ 1 cos 2πτ=pÞ
ð
(Luo, 1998). The temperature response to an arbitrary inlet fluid tempera-
ture variation can be calculated by means of the Duhamel theorem (Grigull
and Sandner, 1990), which yields
τ
ð
0
θ x, τð Þ ¼ θ U x, τ τÞf τðÞdτ + θ U x, τÞθ 0ðÞ (8.125)
ð
ð
0