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3. Radiative-heat-transfer models in PHOENICS

PHOENICS is supplied with five models of radiation, namely:-

  1. The composite-flux model of Schuster and Hamaker, as formulated by Spalding [1980]; this is also known as the six-flux model;
  2. The composite-radiosity model of Spalding [1994]; this is similar to the P-1 spherical-harmonic model ( Ozisik [1973]).
  3. The Rosseland [1936] diffusion model, which in PHOENICS is derived from the radiosity model.
  4. The IMMERSOL model, which is a more complete version of (b).
  5. The surface-to-surface radiation model.
  6. The P-1 T3 radiation model, which is derived from IMMERSOL to produce a composite-radiosity model with radiant temperature as dependent variable; this is similar to the P-1 model.

Preliminary notes on each

Models a, b, c, d and f use the "radiative-conductivity" concept, whereas model e allows fully for angular effects.

Of these, only d and f can also handle conjugate heat transfer (i.e. heat conduction within large immersed solids) and two-phase flow (i.e. additional suspended solids within the flowing medium).

Model a is restricted to Cartesian and cylindrical-polar grids, whereas models b, c, d and f are applicable to BFC grids also.

The PHOENICS implementation of all models is restricted to "gray" radiation, i.e. to that in which the influence of wave-length can be neglected.

Models d and e can handle radiation between solids separated by non-absorbing media, whereas the others cannot. Model e is, in principle, the more accurate; model d is the more economical.

Structure of this Encyclopaedia article

Because of its novelty and wide applicability, model d (IMMERSOL) is presented first, in section 3.

Sections 4, 5 and 6 are devoted to the older models a, b and c. Model d is the only one to combine universal applicability with economic practicability for complex geometries.

Model e is not described in this article; but information about it can be found in the lecture entitled "Surface to Surface Radiation" in the Lectures on PHOENICS section of POLIS.

Model f is described in Section 7.

A remark about other models of radiative heat transfer

Other models of radiation are known, for example:-

PHOENICS implementations of the discrete-transfer and discrete- ordinates methods have been reported by Kjaldman [1993] and Muller et al [1994], respectively.


L.Kjaldman, 'Numerical simulation of combustion and nitrogen pollutants in furnaces', TRC Finland, VTT Publications 159, (1993).

J.Muller, C.Gevers and M.Brunet, 'Three dimensional radiative heat transfer modelization using the discrete-ordinates method', EUROTHERM Seminar 36, Advanced Concepts and Techniques in Thermal Modelling, Sept. 21-23, ENSMA, Poitiers, France, (1994).

M.N.Ozisik, 'Radiative heat transfer', John Wiley, New York, (1973).

S.Rosseland, 'Theoretical astrophysics', Oxford Univ. Press, Clarendon, London and New York, (1936).

D.B.Spalding, 'Idealisations of radiation', In Mathematical Modelling of Fluid-Mechanics, Heat-Transfer and Chemical-Reaction Processes, Lecture 9, HTS/80/1, Imperial College, Mech. Engng., Dept., London, (1980)

D.B.Spalding, 'Proposal for a diffusional radiation model', Unpublished technical memorandum, CHAM, London, (1994).