3-7, 3-8 and 3-10, we can arrive at the following expression for the relative error of
the arithmetic mean temperature difference εa in terms of the approach factor AF
εa = { [(AF + 1) ln(AF)]/[2(AF 1)] } 1.
(3-12)
Similarly, we combine eq 3-2, 3-5, 3-6, 3-7, 3-8 and 3-11 to arrive at an expression for
the relative error of the geometric mean temperature difference εg in terms of the
approach factor AF
εg = {[(AF)1/2 ln(AF)]/[AF 1]} 1.
(3-13)
Now we can study the approximation errors for the arithmetic and geometric
mean temperature differences over the range of possible approach factors by
examining eq 3-12 and 3-13 respectively. It is immediately obvious that the error
from the arithmetic mean temperature difference approximation εa becomes infinite
as AF approaches zero. However, it is not clear what the error from the geometric
mean temperature difference approximation becomes as AF approaches zero. To
determine what value εg approaches as AF approaches zero, we use l'Hpital's rule.
It states that
limit ( f (x)/ g(x)) = limit ( f ′(x)/ g′(x))
(3-14)
x→λ
x→λ
where f(x) and g(x) are some functions of x that both approach either zero or infinity
when x approaches the value λ. To apply this to the error expression for the
geometric mean temperature difference, we let
x = AF
f(x) = f(AF) = ln(AF)
g(x) = g(AF) = (AF1)/AF1/2 = AF1/2 AF 1/2
εg = f(AF)/g(AF) 1 .
Taking the first derivatives of f(AF) and g(AF), we have
f ′(AF) = 1/AF
g′ (AF) = 0.5 AF 1/2 + 0.5 AF 3/2 .
Now we can determine the value that εg approaches as AF → 0 from
(εg + 1) = AlFmit0
( f (AF)/ g(AF)) =
( f ′(AF)/ g′(AF)) =
limit
i
limit
AF → 0
→
AF → 0
(AF1/ 2 /[0.5(AF + 1)]) = 0
limit
AF → 0
Thus, we find that the error from approximating the logarithmic mean temperature
difference with the geometric mean temperature difference εg reaches 100% as the
approach factor AF goes to zero. Although this is a very high relative error, it is still
much better than that of the arithmetic mean temperature difference approximation,
which becomes infinite at the same condition. Of course, in reality this limiting case,
where heat transfer is at its maximum value and the approach factor becomes zero,
is never achieved. As we will now show, the errors attributable to using the
geometric and arithmetic mean temperature difference approximations for the
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