Computing Grain Temperature
Select the Compute Grain Temperature checkbox in the physics interface’s Grain Temperature section to define an auxiliary dependent variable for the grain temperature. The grain temperature Tg (SI unit: K) is then computed along each grain trajectory by integrating the first-order equation
(3-26)
where
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mg (SI unit: kg) is the grain mass,
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Cp,g (SI unit: J/(kg·K)) is the grain specific heat capacity,
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t (SI unit: s) is the time, and
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Qt (SI unit: W) is the sum of all heat sources and sinks affecting the grain.
Validity of the Grain Temperature Calculation
The grain temperature is treated as a single value for each grain, not as a temperature distribution throughout the grain’s volume. Therefore, the temperature computation is only valid when the temperature throughout the grain can be considered uniform; that is, the heat transfer resistance within the grain is negligibly small compared to the heat transfer resistance at the surface of the grain. This is typically true for small grains with high thermal conductivity.
The Biot number Bi (dimensionless) can be used to determine whether the grain temperature can be treated as a uniform value. The Biot number is defined as
where LC (SI unit: m) is a characteristic length, typically the ratio of grain volume to grain surface area, and kg (SI unit: W/(m·K)) is the grain thermal conductivity. If the Biot number is very small, much less than unity, then the conductive heat transfer within the grain takes place on a much shorter time scale than convective heat transfer at the surface of the grain, so the grain temperature can be treated as a uniform value.
Contact Heat Transfer
Use the Compute conductive heat transfer checkbox in the physics interface Additional Variables section to apply the heat flux across the boundary between grains in contact and between grains and walls. Compute conductive heat transfer checkbox is only available when
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the Hertz–MD or the Hertz–MD with adhesion is selected from the Contact force model list in the physics interface Force section and
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the Compute grain temperature checkbox is selected in the physics interface Additional Variables section.
This heat flux contribution is added to total heat source Qt in Equation 3-26 and is defined as (Ref. 6)
(3-27)
where the contact radius is calculated as
(3-28)
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Fn is the normal component of the contact force; see Contact Force: Hertz–MD (Mindlin and Deresiewicz) Model, Contact Force: Hertz–MD with Adhesion Model, or Contact Force: van der Waals Force for more details.
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Req is calculated using Equation 3-13.
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Eeq is given by Equation 3-12.
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Cr is the temperature correction factor for contact radius.
The temperature correction factor, Cr equals to 1 when no correction is needed. It is a common practice in granular flow to use smaller value of Young’s modulus, E, compared to the actual value, E0, to simulate the bulk flow behavior. The main advantage is to be able to use larger time step without compromising the accuracy in bulk behavior of a granular system. However, taking smaller value of E might result in overestimation of heat flux as the contact patch between grains or between grain and surface increases. To fix this, usually a correction factor with following expression is used:
where Eeq,0 is the equivalent Young’s modulus calculated using real values of the Young’s modulus of grains and walls.
Conductive heat transfer between grain and wall is calculated assuming the wall to have infinite radius and conductivity. Consequently, Equation 3-27 and Equation 3-28 become
(3-29)
(3-30)
where Eeq is given by Equation 3-15.
Convective Heat Transfer
Use the Convective Heat Transfer feature to apply convective heat transfer at the surface of the grains. This feature adds the following contribution to the total heat source Qt in Equation 3-26:
where
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h (SI unit: W/(m2·K)) is the heat transfer coefficient,
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Ag (SI unit: m2) is the grain surface area, and
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T (SI unit: K) is the temperature of the surrounding fluid at the grain’s position.
Strictly speaking, T is the temperature that the surrounding fluid would have at the grain’s position, if the grain were not there; the fluid very close to the surface of a warmer or cooler grain will show a temperature gradient. Assuming that the fluid temperature stays relatively constant over length scales comparable to the grain diameter, we can think of T as the ambient or free-stream temperature at a large distance from the grain surface. The Granular Flow interface does not support the effect of fluid flow on a grain’s motion, so the Convective Heat Transfer feature should be used assuming there is fluid in the system, and that the fluid has zero effect on grain dynamics.
The heat transfer coefficient h can be specified directly or by entering the Nusselt number Nu (dimensionless),
where k (SI unit: W/(m·K)) is the thermal conductivity of the fluid (assumed to be isotropic) and dg (SI unit: m) is the grain diameter.