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

= *AF*1/2 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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