Page 263 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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CONVECTION IN POROUS MEDIA
8.5 Forced Convection
Flow through packed beds are important in many chemical engineering applications. Gen-
erally, the grain size in the packed beds will vary depending on the application. As the
particle size increases, the packing close to the walls will become non-uniform, thereby
creating a channelling effect close to the solid walls. In such cases, the porosity value
can be close to unity near the walls, but will decrease to a free stream value away from
the walls.
In such situations, the ability to vary the porosity within the domain itself is essential
in order to obtain a correct solution. Although the theoretical determination of the near
wall porosity variation is difficult, there are some experimental correlations available to
tackle this issue. One such widely employed correlation, given by Berenati and Brosilow
(Berenati and Brosilow 1962), will be used, that is,
cx

= e 1 + exp − (8.73)
d p
where e is the free stream bed porosity taken to be equal to 0.39, and c is an empirical
constant (c = 2for d p = 5 mm). In general, the problem in this case is formulated on the
basis of particle size d p , that is, the Reynolds number is based on the particle size.
Figure 8.4 shows the problem definition of forced flow through a packed bed. The inlet
channel width is 10 times the size of the grain. The length of the channel is 6 times that of

10

p = 0

u = u = 0
2
1
T = 1
60

Parabolic inlet
and u = 0
profile for u 1 2
T = 0

Figure 8.4 Forced convection in a channel filled with a variable porosity medium. Geom-
etry and boundary conditions