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Process Heat Transfer 165
where the logarithmic-mean diameter, D LM, is defined by
D LM = —————— (4.12)
In (D 0 /DO
Each term in the denominator of Equation 4.11 is the reciprocal of a heat-
transfer coefficient, and thus represents a resistance to heat transfer. The first term
in the denominator represents the resistance to heat conduction across a scale
formed on the inside surface of the tube, where the thickness and the thermal con-
ductivity of the scale is rarely known. The thermal conductivity and the thickness
of scale are not reported in the literature, but its reciprocal is designated by Rf j,
the resistance to heat transfer caused by the tube-side scale, where
R fi = x f i D 0 /k f i Di (4.13)
Also, Rf 0, the resistance to heat transfer (fouling resistance or fouling factor)
caused by the shell-side scale is equal to the last term in the denominator of Equa-
tion4.11.
R fo = x f o /k f o (4.14)
The scale thickness will vary with time. When a heat exchanger is first in-
stalled, it is clean. With use the scale thickness increases. If a fouling resistance is
specified, the time required to form the scale is indirectly specified, usually 1 to 1
!/ years [1]. When this period of time is reached, the heat exchanger must be
2
taken out of service and cleaned. The longer the time before cleaning (service
time), the greater the required heat-transfer area and cost of the heat exchanger and
hence capital cost, but the cost of cleaning and operating cost will be less. On the
other hand, if the service time is reduced, the heat-exchanger cost will decrease,
but the cleaning cost will increase. Therefore, there is an optimum service time
which minimizes the total cost. This optimization problem has been studied by
Crittenden and Khater [26].
The second term in the denominator of Equation 4.11 represents the convec-
tive resistance to heat transfer caused by the inside fluid film on the scale surface.
The third term is the conductive resistance caused by the tube wall, which is usu-
ally small, because the thermal conductivity of many metals is large. We will ne-
glect the conductive resistance to heat transfer, unless the thermal conductivity is
very small and tube wall thickness large. The fourth term is the convective resis-
tance to heat transfer of the outside fluid film on the scale surface. After substitut-
ing Equations 4.13 and 4.14 into Equation 4.11,
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