Particulate effects in P-1 radiation model

Appendix 2

Radiating particles in a flame systems

BY : Dr S V Zhubrin, CHAM Ltd

DATE : January 2002

FOR : Demonstration case for V3.4


The implementation in PHOENICS of a radiation model which is based on P-1 approximation and accounts for particulate effects is presented. The P-1 is the simplest case of the more general P-N modelling approach based on the first order spherical harmonic expansion of the radiation intensity.

The current model is an extension of P-1 approximation suitably generalized to handle the effects of absorbing, emmiting, and scattering particles on anisotropic scattering in absorbing, emmiting, and scattering media.

The model is easy to implement and solve with little computing efforts. It is readily applied to complex BFC geometries.

The model offers the advantages of simplicity, high computational efficiency and relatively good accuracy, if the optical thickness is not too small.

The implementation of the radiation model for the pulverized coal-combustion in a wall-fired furnace is demonstrated.


Thermal radiation is modelled by the expanding the radiation intensity in terms of first order spherical harmonics.

Model equations

Assuming that only four terms representing the moments of the intensity are used, the conservation equation of incident radiation, RI in W m-2, accounting for radiating particles and gases together can be derived as:

div ( GradgradRI ) + ag ( 4sTg4 - RI) + ap ( 4sTp4 - RI) = 0

where s = 5.68 10-8, is Steffan-Boltzman constant, W m-2 K-4, Tp and Tg are the gas and particle temperatures, K.

The exchange coefficient, Grad, is expressed by:

Grad = ( 3(ag+sg) + 3(ap+sp) - Cgsg )-1


The equivalent particle absorption coefficient is defined as:

ap = epAp

where ep is the emmisivity of particle and Ap is the volumetric particle projected area, m-1.

The latter is calculated from particulate volume fraction, r2 and current particle diameter, dp as follows:

Ap = 1.5r2dp-1

The equivalent particle scattering coefficient is defined as:

sp = (1 - sp) (1 - ep) Ap

where sp stands for particle scattering factor.

The sp and ep are related by the "incident-radiation-sharing" equation:

sp + ep = 1

The symmetry factor, Cg, is used to model anisotroping scattering by means of a linear-anisotropic scattering phase function. Cg ranges from -1 to +1 and represents the amount of radiation scattered in forward direction.

A positive value indicates that more radiant energy is scattered forward than backward with Cg=1 corresponding to complete forward scattering.

A negative value means that more radiant energy is scattered backward than forward with Cg=-1 standing for complete backward scattering.

A zero value of Cg defines the scattering that is equally likely in all directions, i.e. isotropic scattering

Phase-energy source terms

The volumetric source term, in W m-3, for the gas mixture enthalpy, due to radiation, is given by:

SH1,rad = ag (RI - 4sTg4) + ap (RI - 4sTp4)

The volumetric heat source, due to particle radiation, included in the particulate phase energy equations is as follows:

SH2,rad = epAs (0.25RI - sTp4)

where As is the volumetric particle surface area, m-1 calculated as follows:

As = 6r2dp-1

Boundary conditions

For symmetry planes and perfectly-reflecting boundaries, the radiation boundary conditions are assumed to be zero-flux type.

For the incident radiation equation, the following boundary sources per unit area are used at the walls:

SR, wall = 0.5 ew (4sTw4 - RI)(2 - ew)-1

where ew is the wall emmisivity.

The sources for incident radiation at the inlets and outlets are computed in the manner similar to the walls.

Often, it is safe to assume that the emmisivity of all flow inlets and outlets is unity (black body absorption). If the temperature outside the inlet or outlet considerably differs from that in the enclosure, the different temperatures should be used for radiation and convection fluxes at inlet and outlets.


The radiation model is implemented for the combustion of pulverized coal in a wall-fired furnace as described in detail here. The composite-radiosity model of the original simulation is replaced by P-1 formulations detailed above.

All model settings have been made by PIL commands and PLANT settings of PHOENICS 3.4