logarithmic mean temperature difference are always less than the values for an
approach factor of zero.
The other limiting value for the approach factor is unity. In this case no heat
transfer occurs. First, let's examine what happens to the error from the arithmetic
mean approximation εa. We must again use l'Hpital's rule, proceeding as before
f(AF) = (AF 1) ln(AF)
g(AF) = 2(AF 1)
f′ (AF) = 1 + ln(AF) + 1/AF
g′ (AF) = 2 .
In the limit as AF approaches unity, we have
(
)
( f (AF)/ g(AF)) =
( f ′(AF)/ g′(AF)) =
limit εg + 1 = limit
limit
AF → 1
AF → 1
AF → 1
((1 + ln(AF) + 1/ AF)/ 2) = 1 .
limit
AF → 1
So, we see that the error induced by using the arithmetic mean temperature
difference as an approximation for the logarithmic mean temperature difference
approaches zero as AF approaches unity. Now let's look at what happens to the error
for the geometric mean temperature difference as AF approaches unity. Again we
see that l'Hpital's rule is needed and we proceed as follows
= AF1/2 ln(AF)
f (AF)
g (AF)
= AF 1
f′ (AF)
= AF 1/2 [(ln(AF)/2) + 1]
g′ (AF)
=1.
In the limit as AF approaches unity we have
(εg + 1) = AlFmit1 ( f (AF)/ g(AF)) = AlFmit1 ( f ′(AF)/ g′(AF)) =
limit
i
i
AF → 1
→
→
(AF
[(ln(AF))/ 2) + 1]) = 1 .
-1/ 2
limit
→1
AF
Thus, we find that the error for the geometric mean temperature difference
approximation to the logarithmic mean temperature difference also approaches
zero as AF approaches unity. The errors resulting from using the arithmetic and
geometric mean temperature differences as approximations for the logarithmic
mean temperature difference are shown in Figure 4. Some numerical values for these
errors are also given in Table 2.
Several important observations can be made by studying Table 2. First, we note
that the error from approximating the logarithmic mean temperature difference
with the arithmetic mean temperature difference is always positive. Since the heat
transfer is proportional to the logarithmic mean temperature raised to some positive
power, using the arithmetic mean temperature difference as an approximation will
always over-predict the actual heat transfer. Also note that, as we have shown
analytically, the arithmetic mean temperature difference approaches infinity as the
AF goes to zero and approaches zero as AF goes to unity. For the geometric mean
temperature difference, the error resulting from using it as an approximation for the
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