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Many of the heat transfer processes
encountered in nuclear facilities involve a combination of
both conduction and convection.
For example, heat transfer in a steam generator involves
convection from the bulk of the
reactor coolant to the steam generator inner tube surface,
conduction through the tube
wall, and convection from the outer tube surface to the
secondary side fluid.
In cases of combined heat
transfer for a heat exchanger, there are two values for h.
There is the convective
heat transfer coefficient (h) for the fluid film inside the
tubes and a convective heat transfer
coefficient for the fluid film outside the tubes. The thermal
conductivity (k) and thickness
(Dx) of the tube wall must also be
accounted for. An additional term (Uo),
called the overall heat transfer coefficient, must be used
instead. It is common practice to relate the total rate
of heat transfer ( ) to the cross-sectional area for heat
transfer (Ao)
and the overall heat ˙Q transfer
coefficient (Uo).
The relationship of the overall heat transfer coefficient to
the individual conduction
and convection terms is shown in Figure 6.
Recalling Equation 2-3:
˙Q =UoAoDTo
where Uo
is defined in Figure 6.
An example of this concept applied to
cylindrical geometry is illustrated by Figure 7, which shows a typical combined heat
transfer situation.
Using the figure representing flow in a pipe,
heat transfer by convection occurs between temperatures
T1 and T2;
heat transfer by conduction occurs between temperatures T2
and T3;
and heat transfer occurs
by convection between temperatures T3
and T4.
Thus, there are three processes
involved. Each has an associated heat transfer coefficient,
cross-sectional area for heat transfer,
and temperature difference. The basic relationships for these
three processes can be expressed
using Equations 2-5 and 2-9.
˙Q =h1
A1 (T1-
T2)
DTo
can be expressed as the sum of the DT
of the three individual processes.
If the basic relationship for each process is
solved for its associated temperature difference and substituted
into the expression for DTo
above, the following relationship results.
This relationship can be modified by
selecting a reference cross-sectional area Ao.
Solving for Q results in an equation in the
form 
.
Equation 2-10 for the overall heat transfer
coefficient in cylindrical geometry is relatively difficult
to work with. The equation can be simplified without losing
much accuracy if the tube that
is being analyzed is thin-walled, that is the tube wall
thickness is small compared to the tube diameter.
For a thin-walled tube, the inner surface area (A1),
outer surface area (A2),
and log mean surface area
(A1m), are all
very close to being equal. Assuming that A1,
A2, and A1m
are equal to each
other and also equal to Ao
allows us to cancel out all the area terms in the
denominator of Equation 2-11.
This results in a much simpler expression
that is similar to the one developed for a flat plate heat
exchanger in Figure 6.
The convection heat transfer process is
strongly dependent upon the properties of the fluid being
considered. Correspondingly, the
convective heat transfer coefficient (h), the overall
coefficient (Uo),
and the other fluid properties may vary substantially for the
fluid if it experiences a large temperature
change during its path through the convective heat transfer
device. This is especially true
if the fluid’s properties are strongly temperature
dependent. Under such circumstances, the temperature
at which the properties are "looked-up" must be
some type of average value, rather than
using either the inlet or outlet temperature value. For
internal flow, the bulk or average value of temperature is
obtained analytically through the use
of conservation of energy. For external flow, an average film
temperature is normally calculated,
which is an average of the free stream temperature and the
solid surface temperature. In
any case, an average value of temperature is used to obtain
the fluid properties to be used in the
heat transfer problem. The following example shows the use of
such principles by solving a
convective heat transfer problem in which the bulk
temperature is calculated.
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