TITLE : Radiating particles in a flame systems

BY : Dr S V Zhubrin, CHAM Ltd

DATE : March 2002

FOR : Multi-physics demonstration case

INTRODUCTION

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.

MODELLING CONSIDERATIONS

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, R_I in W m^-2, accounting for radiating particles and gases together can be derived as:

div ( G_radgradR_I ) + a_g ( 4sT_g⁴ - R_I) + a_p ( 4sT_p⁴ - R_I) = 0

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

The exchange coefficient, G_rad, is expressed by:

G_rad = ( 3(a_g+s_g) + 3(a_p+s_p) - C_gs_g )^-1

where

• a_g is the gas absorption coefficient, m^-1
• s_g is the gas scattering coefficient, m^-1
• a_p is the equivalent particle absorption coefficient, m^-1
• s_p is the equivalent particle scattering coefficient, m^-1 and
• C_g is the symmetry factor of a scattering phase function.

a_p = e_pA_p

where e_p is the emmisivity of particle and A_p is the volumetric particle projected area, m^-1.

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

A_p = 1.5r₂d_p^-1

The equivalent particle scattering coefficient is defined as:

s_p = (1 - s_p) (1 - e_p) A_p

where s_p stands for particle scattering factor.

The s_p and e_p are related by the "incident-radiation-sharing" equation:

s_p + e_p = 1

The symmetry factor, C_g, is used to model anisotroping scattering by means of a linear-anisotropic scattering phase function. C_g 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 C_g=1 corresponding to complete forward scattering.

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

A zero value of C_g 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:

S_H1,rad = a_g (R_I - 4sT_g⁴) + a_p (R_I - 4sT_p⁴)

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

S_H2,rad = e_pA_s (0.25R_I - sT_p⁴)

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

A_s = 6r₂d_p^-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:

S_{R, wall} = 0.5 e_w (4sT_w⁴ - R_I)(2 - e_w)^-1

where e_w 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 IMPLEMENTATION

The radiation model is implemented for the combustion of pulverized coal in a wall-fired furnace as described in details here.

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