(3-4)

In the case where the above criterion for the approach factor is met, the error of

approximation is always less than 4%. The arithmetic mean always overestimates

the logarithmic mean and thus any estimates of the heat transfer based on the

arithmetic mean will overestimate the actual heat transfer that will be achieved. This

could result in undersized heat exchangers, assuming that no other margin of safety

is included, which is of course seldom the case.

As an alternative approximation to the logarithmic mean temperature difference,

the use of the geometric mean temperature difference is proposed. The geometric

mean temperature difference for this type of heat exchanger is defined as

2

(3-5)

The geometric mean temperature difference, like the arithmetic mean tempera-

ture difference, has the advantage that an explicit expression for the supply or return

temperature can be obtained from it. The geometric mean temperature difference,

however, is a much better approximation of the logarithmic mean temperature

difference than is the arithmetic mean temperature difference, as will be shown

below.

To simplify the analysis, we introduce the following expressions

∆*T*sa = *T*s *T*a

(3-6)

∆*T*ra = *T*r *T*a

(3-7)

(3-8)

where ∆*T*sa = greatest temperature difference between fluids (C)

∆*T*ra = smallest temperature difference between fluids (C)

Two limiting cases of heat transfer set the range of values possible for the

approach factor *AF*. The first case is the case where no heat transfer takes place in the

heat exchanger. In this case the temperature of the water flowing through the

radiator will not decrease, and thus the supply and return water temperatures will

be equal and the approach factor becomes unity. The other limiting case occurs when

the maximum amount of heat transfer occurs in the heat exchanger, in which case

the return temperature equals the air temperature and approach factor becomes

zero. Thus, we have the following range of values for the approach factor *AF*

0 ≤ *AF *≤ 1 .

(3-9)

Now we can examine the errors that can result from each of the approximations

presented above over the entire range of possible approach factors. First, we define

the relative error of each of the approximations

εa = (*T*ma/*T*ml) 1

(3-10)

εg = (*T*mg/*T*ml) 1

(3-11)

where εa is a relative approximation error for the arithmetic mean temperature

difference (dimensionless) and εg is a relative approximation error for the geometric

mean temperature difference (dimensionless). Then, by combining eq 3-2, 3-3, 3-6,

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