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.
AF1/2 ln(AF) - AF + 1
[(AF + 1) ln(AF)/ 2] - AF + 1
εg = f(AF) = AF1/2ln(AF) AF + 1
εa = g(AF) = [(AF + 1) ln(AF)/2] AF + 1
f ′(AF) = [AF1/2ln(AF)/2] + AF 1/2 1
g′ (AF) = ln(AF)/2 + 1/2AF 1/2 .
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
f ′′(AF) = [AF 3/2ln(AF)]/4
g′′(AF) = (AF 1 AF 2)/2 .
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
f ′′′(AF) = [(3AF 5/2ln(AF))/8] AF 5/2/4
g′′′(AF) = AF 3 AF 2/2
limit εg + εa = limit ( f ′′′(AF)/ g′′′(AF)) = (-1/ 4)/(1/ 2) = -1/ 2
AF → 1
AF → 1
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 n1 and n2. Here, we will
only address the simpler case where n1 = n2. This is the result that occurs for "high
radiators" according to Bhm (1988), in which case n1 = n2 = 1.3. In that case eq 3-1
q2 / q0 = (Tm1)2 /(Tm1)0