BY : Dr S V Zhubrin, CHAM Ltd

DATE : March 2002

FOR : Multi-physics demonstration case

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.

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_{rad}gradR_{I} ) +
a_{g} ( 4sT_{g}^{4}
- R_{I}) +
a_{p} ( 4sT_{p}^{4}
- 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_{g}s_{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_{p}A_{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, r_{2} and
current particle diameter, d_{p} as follows:

A_{p} = 1.5r_{2}d_{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

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_{g}^{4})
+
a_{p} (R_{I} - 4sT_{p}^{4})

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

S_{H2,rad} =
e_{p}A_{s}
(0.25R_{I} - sT_{p}^{4})

where A_{s} is the volumetric particle surface area, m^{-1}
calculated as follows:

A_{s} = 6r_{2}d_{p}^{-1}

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_{w}^{4}
- 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 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