P1 Approximation Theory
The P1 approximation is available as a radiation discretization method in The Radiation in Participating Media Interface.
The P1 approximation is the simplest approximation provided by the method of spherical harmonics method (PN-method). This approximation provides additional accuracy compared to a Rosseland approximation even if it remains a very simple method. The P1 method relies on the following hypotheses:
•
The media is optically thick media: τ >>1, where τ is the optical thickness defined by the integral of absorption coefficient, κ, along a typical optical path:
•
The scattering is linear isotropic.
From a computational point of view this approximation has a limited impact because it introduces only one additional degree of freedom for the incident radiation G (SI unit: W/m2), which is a scalar quantity and adds a heat source or sink to the temperature equation to account for radiative heat transfer contributions. This method, however, fails to accurately represent cases where the radiative intensity propagation dominates over its diffusivity or where the scattering effects cannot be described by a linear isotropic phase function.
The P1 approximation accounts for the radiation transfer equation
by solving following equation for (Ref. 23):
(4-122)
where
•
DP1 is the P1 diffusion coefficient, defined as
•
a1 is the linear Legendre coefficient of the scattering phase function
•
Qr is the radiative heat source:
(4-123)
When scattering is modeled as isotropic, a1=0 and the P1 diffusion coefficient reduces to
The following boundary condition applies (Ref. 23):
where qr, net is the net radiative heat flux at the boundary.
Radiation in Participating Media
For the Participating Medium (Radiation in Participating Media Interface) feature node, the equation Equation 4-122 is implemented.
In addition Qr, defined by Equation 4-123, is added as an heat source in the heat transfer equation:
Opaque Surface
The Opaque Surface (Radiation in Participating Media and Radiation in Absorbing-Scattering Media Interfaces) boundary condition defines a boundary opaque to radiation and defines the incident intensity on a boundary:
The Opaque Surface feature accounts for the net radiative heat flux, qr, net, in the heat balance.
Two cases are considered, depending on surface emissivity value:
•
Gray wall: ε is a number between 0 and 1
•
Black wall: ε = 1
Gray Wall
The radiative heat flux at the boundary depends on the surface emissivity, ε:
with
Black Wall
The radiative heat flux at the boundary expression simplifies to
,
with
Semitransparent Surface
The Semitransparent Surface (Radiation in Participating Media and Radiation in Absorbing-Scattering Media Interfaces) boundary condition defines a boundary that emits radiation, and reflects one part of the incident intensity while the remaining is transmitted diffusively.
The net radiative heat flux, qr, net, accounted for in the heat balance, is defined as
where ρd is the surface diffuse reflectivity,  and τd is the surface diffuse transmissivity.
Incident Intensity
The Incident Intensity (Radiation in Participating Media and Radiation in Absorbing-Scattering Media Interfaces) node defines a boundary that receives incident radiative intensity Iext and that is transparent for outgoing intensity. On these boundaries, the relation between G (incident radiation), qr, net (net radiative heat flux) and Iext (incident radiative intensity) is
by defining
there is
which defines the heat radiative heat flux and also contributes to G boundary condition:
Radiative Source
The node Radiative Source accounts for a power density Q in the P1 approximation:
.