As expected, the results for the normalized mass flow rates in Table 3, which use
the geometric mean approximation for the log mean temperature difference, are
much better than those obtained using the arithmetic mean approximation. Clearly,
the errors induced by the arithmetic mean approximation are unacceptable for any
loadsupply temperature condition that deviated significantly from the design
condition for the radiators. Over the range of supply temperatures and heat loads
given in the Table 3, the average error in the return temperature obtained using the
geometric mean approximation is only 7% compared to 69% for the results obtained
using the arithmetic mean approximation. Also, note that the errors in approxima-
tion of the mass flow rate average only 3.6% for the model using the geometric mean
approximation, while the average error for the model using the arithmetic mean
approximation is about 19%.
In addition, it should be noted that the model based on the arithmetic mean
approximation in a number of instances at lower loads results in physically im-
possible return temperatures, i.e., ones lower than the room air temperature of 20C.
Note that at lower loads (lower q/q0 values), the errors of approximation tend to be
larger. This is predicted by our error analysis carried out earlier, since the approach
factor is lower in these cases. Also note that, like our basic geometric mean
approximation, our model based on it is conservative and under-predicts the heat
transfer on the average. An exception sometimes occurs at supply temperatures
lower than the design condition of 90C. Since our model for the radiator predicts
the return temperature based on the return temperature at the design 0 condition,
the error in approximation of the return temperature at the design condition has an
effect on the error in our model at conditions other than the design condition. At
lower supply temperatures and high loads, for example Ts = 85C and q/q0 = 1.0, the
approach factor is 0.83 for the geometric mean approximation and is thus higher
than the approach factor of 0.71 encountered at the design condition. Because the
error in the geometric mean approximation decreases with increasing approach
factor, our model for the prediction of the return temperature actually under-
predicts it slightly at that point. This under-prediction is not, however, a cause for
concern, since it is so slight and in addition it exists at a loadtemperature condition
that would not normally be encountered because it would require mass flow rates
greater than the design condition.
DESIGN OF A SINGLE PIPE SEGMENT WITH A CONSUMER MODEL
In Chapter 2 we developed a methodology to determine the optimal pipe
diameter for a single pipe segment. In the example given, it was assumed that both
the supply temperature and return temperature were constant over the entire yearly
cycle. This is of course not the case, and now that we have a simple model for the
consumer's space heating substation, we can examine what the effect is of coupling
this model with our design methodology.
First, let's consider what the effect is of assuming a constant supply temperature,
as we had done earlier, but rather than assuming a constant return temperature as
well, let this be determined by our consumer model. The varying return temperature
will affect the heat losses by altering the A1 parameter (eq 2-5) to some degree; this
will be addressed later. The primary effect, however, will be on the mass flow rate.
The heat load will be assumed to vary sinusoidally, as before, except now the
variation in the mass flow rate will not be sinusoidal itself, but will be determined
by the load and the return temperature from the consumer model. The relationship
between the mass flow rate, load, design supplyreturn temperatures and the actual
supplyreturn temperatures was given by eq 3-21. If we substitute our expression
for the return temperature as determined using the geometric mean approximation
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