Page 382 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 365

Since the problem is linear, an easy way to obtain the system responses to
arbitrary inlet temperature disturbances is to use Duhamel’s theorem
(Grigull and Sandner, 1990):
ð τ
2
X U U
00 θ 00 j 0 1 ðÞ τ zÞdz + θ 00 j 0
ð
θ τðÞ ¼ z ðÞθ ð τ ðÞθ 0ðÞ i ¼ 1, 2Þ (7.211)
i i j i j
j¼1
0
00U j is the response of the outlet temperature of stream i to a unit
in which θ i
0(1)
step change in the inlet temperature of stream j and θ j (τ)¼dθ j (τ)/dτ
0
(i, j¼1, 2).
If the mass flow rates are disturbed, the heat transfer coefficients will also
vary with time. By assuming that the properties of the fluids are constant and
using the empirical correlation for the Nusselt number,
m
Nu ¼ CRe Pr n (7.212)
the heat transfer parameter disturbances can be approximately expressed as

m 1 ^ U
½
U τðÞ ¼ 1+ στðފ (7.213)
where
^
στðÞ ¼ _m τðÞ= _m 1 (7.214)

For a step change in one of the mass flow rates at τ¼0, σ is a constant.
_ ^ _ m 1 _
Thus, we have C ¼ 1+ σð ÞC, U ¼ 1+ σð Þ ^ U, ΔC τðÞ ¼ 0, and ΔU(τ)¼
0. In this case, the problem is linear. The temperature responses of the outlet
stream temperatures to the step disturbances in mass flow rates are shown in
Fig. 7.8 and Fig. 7.9 with solid lines.
To check the validity of the linearized model, the problem is resolved by
_ ^ _ m 1 _ ^ _
ð
setting ΔC τðÞ ¼ σC, ΔU τðÞ ¼ 1+ σÞ 1 ^ U, C ¼ C, and U ¼ ^ U,
which yields a nonlinear problem. The temperature responses obtained
with this linearized model are shown in Fig. 7.8 and Fig. 7.9 with dotted
lines. The deviations between the linear model and linearized model are
<0.7% of the maximum temperature difference. This error is caused due
_
to omitting the nonlinear terms ΔC τðÞθτðÞ and ΔU(τ)θ(τ), that is, there
_
are constant deviations between C τðÞ and its mean value in the new
_
operation state, C, and between U(τ) and U. For small disturbances in
_
ΔC τðÞ and ΔU, the linearized model is a good approach.