Packed Bed of Pellets
For the modeling of packed beds of pellets, the The Heat Transfer in Packed Beds Interface interface implements an hybrid multiscale model of local thermal nonequilibrium.
As in the standard local thermal nonequilibrium model, two temperatures, one for the fluid phase and one for the pellets, are solved for. By assuming that the thermal conductivity of the pellets is much smaller than the one of the fluid phase, heat conduction among different pellets is neglected, and a 1D microscale equation for temperature conduction along the radial coordinate in the pellets is defined to replace the macroscale heat transfer equation. It is coupled to the macroscale heat transfer equation in the fluid phase, either by assuming temperature continuity, or a convective heat flux, at the outer surface of the pellets.
This model provides higher accuracy than the standard local thermal nonequilibrium model for highly nonlinear problems, in particular when combustion occurs within the pellets.
Spheres, cylinders, flakes, or any shape of pellets may be considered, as long as the specific surface area for heat exchange between the pellets and fluid phase can be determined. Porosity within the pellets themselves may be considered as well, in addition to the bed porosity.
Equations for Heat Transfer in a Packed Bed of Pellets
Macroscopic Heat Transfer in the Fluid
For the fluid phase, the heat equation is similar to the one solved in the standard local thermal nonequilibrium model:
(4-45)
In these expressions:
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 εp is the packed bed porosity (dimensionless)
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 ρf is the fluid density (SI unit: kg/m3)
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 Cp, f is the fluid heat capacity at constant pressure (SI unit: J/(kg·K))
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 qf is the fluid conductive heat flux (SI unit: W/m2)
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 kf is the fluid thermal conductivity (SI unit: W/(m·K))
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 Qpe,f is the pellet-fluid heat exchange term (SI unit: W/(m3·K))
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 Qf is any fluid heat source (SI unit: W/m3)
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 u is the fluid velocity vector (SI unit: m/s). It should be interpreted as the Darcy velocity, that is, the volume flow rate per unit cross sectional area. The average linear velocity (the velocity within the pores) can be calculated as uf = u/εp, where εp is the fluid’s volume fraction, or equivalently the porosity.
Microscopic Heat Transfer in the Pellets
In the pellets, the heat balance equation at normalized radius r is
(4-46)
where
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(ρCp)pe,eff is the effective volumetric heat capacity at constant pressure inside the pellet filled with fluid (SI unit: J/(kg·K)), defined by
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 ρpe is the density of the porous matrix within the pellet (SI unit: kg/m3)
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 Cp,pe is the volumetric heat capacity at constant pressure of the porous matrix within the pellet (SI unit: J/(kg·K))
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εpe is the porosity within the pellet (dimensionless)
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Tpe is the temperature along the radial coordinate of the pellet (SI unit: K)
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rpe is outer radius of the pellet (SI unit: m)
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kpe,eff is the effective thermal conductivity of the pellet filled with fluid (SI unit: W/(m·K)), defined by
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kpe is the thermal conductivity of the porous matrix within the pellet (SI unit: W/(m·K))
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Qpe is any heat source applied to the pellet (SI unit: W/m3)
Coupling Condition at Pellet-Fluid Interface
As in the local thermal nonequilibrium model, the interstitial heat exchange between the fluid and the outer surface of the pellets is accounted for.
The heat flux at the outer surface of the pellet, applied in the microscale pellet equation, is
with hpe,f the interstitial heat transfer coefficient (SI unit: W/(m2·K)), defined as in the standard local thermal nonequilibrium model.
The corresponding heat exchange term applied in the macroscale heat transfer equation of the fluid is:
with Sb the specific surface area of the pellets. For a packed bed of spherical pellets, it is defined as
with dpe the diameter of the pellet (SI unit: m).
When the pellets are not spherical, a correction factor is applied to the radius, to extend this formula.
When continuity of temperatures at the fluid-pellet interface can be assumed, that is, when
then the interstitial heat flux can be rewritten as
Average Temperature in the Packed Bed
The porous temperature, T, has the following definition (Ref. 35):