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Consider a perforated plate which has 25% area open to flow. The pressure drop through the plate is known to be 0.5 times the dynamic head in the plate. The loss factor,
(7.19-11)
, defined as
is therefore 0.5, based on the actual fluid velocity in the plate, i.e., the velocity through the 25% open area. To compute an appropriate value for
, note that in the FLUENT model:
1. The velocity through the perforated plate assumes that the plate is 100% open.
2. The loss coefficient must be converted into dynamic head loss per unit length of the porous region.
Noting item 1, the first step is to compute an adjusted loss factor, which would be based on the velocity of a 100% open area:
(7.19-12)
,
or, noting that for the same flow rate,
,
(7.19-13)
The adjusted loss factor has a value of 8. Noting item 2, you must now convert this into a loss coefficient per unit thickness of the perforated plate. Assume that the plate has a thickness of 1.0 mm (10 factor would then be
m). The inertial loss
(7.19-14)
Note that, for anisotropic media, this information must be computed for each of the 2 (or 3) coordinate directions.
Using the Ergun Equation to Derive Porous Media Inputs for a Packed Bed
As a second example, consider the modeling of a packed bed. In turbulent flows, packed beds are modeled using both a permeability and an inertial loss coefficient. One technique for deriving the appropriate constants involves the use of the Ergun equation [ 98], a semi-empirical correlation applicable over a wide range of Reynolds numbers and for many types of packing:
(7.19-15)
When modeling laminar flow through a packed bed, the second term in the above equation may be dropped, resulting in the Blake-Kozeny equation [ 98]:
(7.19-16)
In these equations, is the viscosity, is the mean particle
diameter, is the bed depth, and is the void fraction, defined as the volume of voids divided by the volume of the packed bed region.
Comparing Equations 7.19-4 and 7.19-6 with 7.19-15, the permeability and inertial loss coefficient in each component direction may be identified as
(7.19-17)
and
(7.19-18)
Using an Empirical Equation to Derive Porous Media Inputs for Turbulent Flow Through a Perforated Plate
As a third example we will take the equation of Van Winkle et al. [ 279, 339] and show how porous media inputs can be calculated for pressure loss through a perforated plate with square-edged holes.
The expression, which is claimed by the authors to apply for turbulent flow through square-edged holes on an equilateral triangular spacing, is
(7.19-19)
where
= mass flow rate through the plate
= the free area or total area of the holes
= the area of the plate (solid and holes)
= a coefficient that has been tabulated for various Reynolds-number ranges
and for various
= the ratio of hole diameter to plate thickness
for and for the coefficient takes a value of
approximately 0.98, where the Reynolds number is based on hole diameter and velocity in the holes.
Rearranging Equation 7.19-19, making use of the relationship