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)

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

εg = *f(AF)*/*g(AF) * 1 .

Taking the first derivatives of *f(AF) *and *g(AF), *we have

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*1/ 2 /[0.5(*AF *+ 1)]) = 0

limit

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

20