Computing Particle Temperature
Select the Compute Particle Temperature check box in the physics interface Advanced Settings section to define an auxiliary dependent variable for particle temperature. The particle temperature Tp (SI unit: K) is then computed along each particle trajectory by integrating the first-order equation
(5-16)
where
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mp (SI unit: kg) is the particle mass,
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Cp (SI unit: J/(kg·K)) is the particle 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 particle.
Validity of the Particle Temperature Calculation
The particle temperature is treated as a single value for each particle, not as a temperature distribution throughout the particle’s volume. Therefore, the temperature computation is only valid when the temperature throughout the particle can be considered uniform; that is, the heat transfer resistance within the particle is negligibly small compared to the heat transfer resistance at the surface of the particle. This is typically true for small particles with high thermal conductivity.
The Biot number Bi (dimensionless) can be used to determine whether the particle 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 particle volume to particle surface area, and kp (SI unit: W/(m·K)) is the particle thermal conductivity. If the Biot number is very small, much less than unity, then the conductive heat transfer within the particle takes place on a much shorter time scale than convective heat transfer at the surface of the particle, so the particle temperature can be treated as a uniform value.
Convective Heat Losses
Use the Convective Heat Losses feature to apply convective heat transfer at the surface of the particles. This feature adds the following contribution to the total heat source Qt in Equation 5-16:
(5-17)
where
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h (SI unit: W/(m2·K)) is the heat transfer coefficient,
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Ap (SI unit: m2) is the particle surface area, and
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T (SI unit: K) is the temperature of the surrounding fluid at the particle’s position.
Strictly speaking, T is the temperature that the surrounding fluid would have at the particle’s position, if the particle were not there; the fluid very close to the surface of a warmer or cooler particle will show a temperature gradient. Assuming that the fluid temperature stays relatively constant over length scales comparable to the particle diameter, we can think of T as the ambient or free-stream temperature at a large distance from the particle surface.
The heat transfer coefficient h can be specified directly or by entering the Nusselt number Nu (dimensionless),
(5-18)
where k (SI unit: W/(m·K)) is the thermal conductivity of the fluid (assumed to be isotropic) and dp (SI unit: m) is the particle diameter. In the following section it will be shown that Nu = 2 is the appropriate value for a particle moving at the same velocity as the surrounding fluid; some more general expressions for the particle Nusselt number are also given in Ref. 34.
Heat Transfer in a Stationary Fluid Surrounding a Sphere
Neglecting viscous dissipation, thermal expansion, and any other external heat sources, the heat equation for a fluid surrounding a particle is
where
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T (SI unit: K) is the fluid temperature,
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t (SI unit: s) is time,
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u (SI unit: m/s) is the fluid velocity relative to the particle,
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k (SI unit: W/(m·K)) is the thermal conductivity of the fluid,
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ρ (SI unit: kg/m3) is the mass density of the fluid, and
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Cp,f (SI unit: J/(kg·K)) is the specific heat capacity of the fluid at constant pressure.
Using the following assumptions:
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The particle is spherical,
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The particle thermal conductivity is homogeneous and not temperature-dependent,
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The fluid is stagnant (u = 0) in the particle frame of reference, and
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There is no heat source or sink within the particle.
The heat equation is simplified to
This may also be written in terms of the thermal diffusivity κ (SI unit: m2/s),
The heat equation can be further simplified using the quasi-steady approximation. The particle thermal conductivity is assumed to be large enough that the particle temperature is spatially uniform. The time scale for the transient behavior in the heat equation is also assumed to be very small, so that the time derivative terms can be removed, yielding
(5-19)
with boundary conditions
where Tp (SI unit: K) the temperature at the particle surface and Ta (SI unit: K) is the ambient or free-stream temperature at a large distance from the particle surface.
The solution to Equation 5-19 satisfying these boundary conditions is
From Fourier’s law, the heat flux is
so the inward radial component of the heat flux at the particle surface is
Assuming there is no evaporation or condensation, and negligible radiative heat transfer, so that all of the inward heat flux is used to raise the particle temperature, the heat source (or sink) is the product of the inward heat flux with the particle surface area Ap (SI unit: m2),
Comparison with Equation 5-17 shows that
Or, recalling the definition of the Nusselt number from Equation 5-18,
Radiative Heat Losses
Use the Radiative Heat Losses feature to make the particles undergo radiative heat exchange with their surroundings. This feature creates the following contribution to the total heat source in Equation 5-16:
where
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εp (dimensionless) is the particle emissivity,
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σ = 5.670373 × 10−8 W/(m2·K4) is the Stefan-Boltzmann constant,
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Ap (SI unit: m2) is the particle surface area, and
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T (SI unit: K) is the temperature of the enclosure or ambient surroundings.
User-defined Heat Source
Use the Heat Source feature to create a user-defined heat source for the particles. This feature creates a user-defined contribution to the total heat source Q in Equation 5-16.