AF = (Tr Ta)/(Ts Ta) .
(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
Tmg = (Ts Ta)1/2 (Tr Ta)1/2 = ( Ta + TsTr TaTs TaTr)1/2.
(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
∆Tsa = Ts Ta
(3-6)
∆Tra = Tr Ta
(3-7)
AF = ∆Tra/∆Tsa
(3-8)
where ∆Tsa = greatest temperature difference between fluids (C)
∆Tra = smallest temperature difference between fluids (C)
AF = approach factor for the heat exchanger (dimensionless).
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 = (Tma/Tml) 1
(3-10)
εg = (Tmg/Tml) 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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