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/q*0 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 *T*s = 85C and *q/q*0 = 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.

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 *A*1 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

27

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