P-1 Radiation Model Theory

The P-1 radiation model is the simplest case of the more general P-N model, which is based on the expansion of the radiation intensity into an orthogonal series of spherical harmonics [70] (p. 762), [424] (p. 782). This section provides details about the equations used in the P-1 model. For information about setting up the model, see Using the Radiation Models in the User's Guide.

5.3.3.1. The P-1 Model Equations

As mentioned above, the P-1 radiation model is the simplest case of the P-N model. When modeling gray radiation, the following equation is obtained for the radiation flux if only four terms in the series are used:

(5.18) Heat Transfer

where is the absorption coefficient, is the scattering coefficient, is the incident radiation, and is the linear-anisotropic phase function coefficient, described below. After introducing the parameter

(5.19)

Equation 5.18 (p. 146) simplifies to

(5.20) The transport equation for is

(5.21) where is the refractive index of the medium, is the Stefan-Boltzmann constant and is a

user-defined radiation source. ANSYS Fluent solves this equation to determine the local incident radiation when the P-1 model is active.

Combining Equation 5.20 (p. 147) and Equation 5.21 (p. 147) yields the following equation:

(5.22) The expression for can be directly substituted into the energy equation to account for heat

sources (or sinks) due to radiation.

ANSYS Fluent also allows for the modeling of gray radiation using a gray-band model. For non-gray radiation,Equation 5.21 (p. 147) is rewritten as:

(5.23) where is the spectral incident radiation, is a spectral absorption coefficient, is the refractive index of the medium, is a user-defined source term, and is the Stefan-Boltzmann constant. is defined as:

(5.24)

where is a spectral scattering coefficient and is the linear anisotropic phase function coefficient.

The spectral black body emission ( ) between wavelength and is given as

(5.25) where is the fraction of radiant energy emitted by a black body in the wavelength interval from 0 to at temperature in a medium of refractive index . and are the wavelength bound-aries of the band.

The spectral radiative flux is calculated as

(5.26) The radiation source term in the energy equation is given as

(5.27) Modeling Radiation

5.3.3.2. Anisotropic Scattering

Included in the P-1 radiation model is the capability for modeling anisotropic scattering. ANSYS Flu-ent models anisotropic scattering by means of a linear-anisotropic scattering phase function:

(5.28) where

= unit vector in the direction of scattering

= unit vector in the direction of the incident radiation

= linear-anisotropic phase function coefficient, which is a property of the fluid

Value of ranges from –1 to 1. A positive value indicates that more radiant energy is scattered forward than backward, and a negative value means that more radiant energy is scattered backward than forward.

A zero value defines isotropic scattering (that is, scattering that is equally likely in all directions), which is the default in ANSYS Fluent. You should modify the default value only if you are certain of the aniso-tropic scattering behavior of the material in your problem.

5.3.3.3. Particulate Effects in the P-1 Model

When your ANSYS Fluent model includes a dispersed second phase of particles, you can include the effect of particles in the P-1 radiation model. Note that when particles are present, ANSYS Fluent ignores scattering in the gas phase. That is,Equation 5.29 (p. 148) assumes that all scattering is due to particles.

For a gray, absorbing, emitting, and scattering medium containing absorbing, emitting, and scattering particles, the transport equation for the incident radiation can be written as

(5.29) where is the equivalent emission of the particles, is the equivalent absorption coefficient, and is the refractive index of the medium. These are defined as follows:

(5.30)

and

(5.31)

In Equation 5.30 (p. 148) and Equation 5.31 (p. 148), , , and are the emissivity, projected area, and temperature of particle . The summation is over particles in volume . These quantities are computed during particle tracking in ANSYS Fluent.

The projected area of particle is defined as

(5.32)

where is the diameter of the ^th particle.

The quantity in Equation 5.29 (p. 148) is defined as Heat Transfer

(5.33)

where the equivalent particle scattering factor is computed during particle tracking and is defined as (5.34)

Here, is the scattering factor associated with the ^th particle.

Heat sources (sinks) due to particle radiation are included in the energy equation as follows:

(5.35)

5.3.3.4. Boundary Condition Treatment for the P-1 Model at Walls

To get the boundary condition for the incident radiation equation when modeling gray radiation, the dot product of the outward normal vector and Equation 5.20 (p. 147) is computed:

(5.36) (5.37) Thus the flux of the incident radiation, , at a wall is . The wall radiative heat flux is computed

using the following boundary condition:

(5.38) (5.39) where is the wall reflectivity. The Marshak boundary condition is then used to eliminate the angular dependence [353] (p. 778):

(5.40)

Substituting Equation 5.38 (p. 149) and Equation 5.39 (p. 149) into Equation 5.40 (p. 149) and performing the integrations yields

(5.41)

If it is assumed that the walls are diffuse gray surfaces, then , and Equation 5.41 (p. 149) becomes (5.42) Equation 5.42 (p. 149) is used to compute for the energy equation and for the incident radiation equation boundary conditions.

When using the gray-band model, the flux at the wall can be written as the following (assuming that the wall is a diffuse surface):

Modeling Radiation

(5.43)

where is the wall emissivity, is the spectral radiative flux at the wall, and is the spectral incident radiation at the wall.

5.3.3.5. Boundary Condition Treatment for the P-1 Model at Flow Inlets and Exits

The net radiative heat flux at flow inlets and outlets is computed in the same manner as at walls, as described above. ANSYS Fluent assumes that the emissivity of all flow inlets and outlets is 1.0 (black body absorption) unless you choose to redefine this boundary treatment.

ANSYS Fluent includes an option that allows you to use different temperatures for radiation and con-vection at inlets and outlets. This can be useful when the temperature outside the inlet or outlet differs considerably from the temperature in the enclosure. For details, see Defining Boundary Conditions for Radiation in the User's Guide.