Our model of the consumer's heat exchanger should accept as input the supply
temperature to the radiators and the heat load q2. The values of the operating
parameters at the design "0" condition will also be needed. The output from the
model will be the return temperature. We would like this model to be as accurate as
possible while being in a simple form, thus allowing ease in both analytical and
numerical procedures involving it. As stated earlier, it is not possible to obtain an
explicit model if the log mean temperature difference is used. Thus, we will proceed
using our geometric mean approximation for the log mean temperature difference.
Making this substitution, we have
n1
( )( )
q2 / q0 = Tmg / Tmg
.
(3-16)
0
2
Note that the log mean temperature difference at the design 0 condition is also being
approximated with the geometric mean temperature difference. Since all of the
temperatures are known at the design condition, we could have evaluated the log
mean temperature difference and used the result here and still achieved an explicit
result. However, to ensure that no error occurs in the resultant model at the design
condition, we have used the geometric mean approximation. This will also reduce
the errors at the "off-design" (2) condition. The same procedure has been adopted
for the model using the arithmetic mean temperature difference.
To obtain our model for the return temperature as a function of the load and
supply temperature, we simply solve eq 3-16 for the return temperature at the 2
condition (actual load). The result is
2/n1
-
(Tr )2 = Ta + (Ts - Ta )21(Tmg )0 (q2 / q0 )
2
.
(3-17)
We can also obtain a model for the return temperature using the arithmetic mean
approximation to the log mean temperature difference. It is
{ [
]} - (T ) .
(Tr )2 = 2 Ta + (Tma )0 (q2 / q0 )1/n1
(3-18)
s 2
To evaluate the performance of our models that use approximations to the log mean
temperature difference, we need a model that uses the log mean temperature
difference. As noted earlier, this model will be implicit and thus will require solution
by an iterative numerical method of some type. The model can be arranged in several
forms for numerical solution, one being
-1/n1
(Tr )2 = Ta + (Ts - Ta )2 / exp (q2 / q0 )
(Ts - Tr )2 /(Tm1)0 .
(3-19)
A number of iterative methods can be used to solve this implicit equation for the
return temperature (Tr)2. Most iterative methods are very sensitive to the quality of
the initial estimate. Here, we are fortunate to have the geometric mean temperature
difference approximation that can be used to obtain the initial estimate. Figure 5
below shows some of the results for the three models developed. In addition more
detail as well as numerical values are given in Table 3. It is clear from Figure 5 that
the model using the arithmetic mean temperature difference is unacceptable for
most values of the load ratio q/q0, while the model using the geometric mean
temperature difference is acceptable over the entire range of values given for q/q0.
The results for the model using the log mean temperature difference were obtained
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