is seemingly apparent from Table 2, we can prove this analytically by using

l'Hpital's rule. In this case it becomes necessary to take successive derivatives up

to the third derivatives in order to arrive at an expression that is not indeterminate.

This is an acceptable application of l'Hpital's rule, however. The analysis is

presented briefly below.

εg

=

[(*AF *+ 1) ln(*AF*)/ 2] - *AF *+ 1

εa

Let

εg = *f(AF) *= *AF*1/2ln(*AF*) *AF *+ 1

εa = *g(AF) *= [(*AF *+ 1) ln(*AF*)/2] AF + 1

In the limit as *AF *→ 1, we see that both *f *′*(AF) *and *g*′(*AF*) approach zero; thus, we still

have an indeterminate expression. Applying l'Hpital's rule to that expression

Again, we see that both *f *′′*(AF) *and *g*′′*(AF) *approach zero as *AF *approaches unity

and we are left with another indeterminate expression. Once again we take deriva-

tives so that we can apply l'Hpital's rule

(

)

limit εg + εa = limit ( f ′′′(*AF*)/ *g*′′′(*AF*)) = (-1/ 4)/(1/ 2) = -1/ 2

And we now have our desired result.

The significance of this result is that we now know that the error from using the

geometric mean temperature difference as an approximation to the logarithmic

mean temperature difference is always 50% or less of the error that would result

from using the arithmetic mean temperature difference. For applications where the

use of the logarithmic mean temperature difference is undesirable, use of the geo-

metric mean temperature difference will result in errors of less than 5% for values

of *AF *greater than 0.33. As Soumerai (1987) points out, given the other uncertainties

in heat exchanger design calculations, errors of this magnitude are certainly accept-

able. Most heat exchanger designs will have approach factors greater than 0.33 and

thus our findings here should be applicable in the majority of cases.

Now that we have this approximation for the logarithmic mean temperature

difference, we can construct a simple model for the consumer's heat exchanger using

it. Equation 3-1 will be used to construct our model. For our model two possible cases

exist, dependent on the values of the empirical parameters *n*1 and *n*2. Here, we will

only address the simpler case where *n*1 = *n*2. This is the result that occurs for "high

radiators" according to Bhm (1988), in which case *n*1 = *n*2 = 1.3. In that case eq 3-1

becomes

.

(3-15)

23

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