We study radial flame ball solutions of a three-dimensional free boundary problem (FBP), which models combustion of a gaseous mixture with dust in a microgravity environment. The model combines diffusion of mass and temperature with reaction at the flame front, the reaction rate being temperature dependent. The radiative flux due to the presence of dust enters the equation for the temperature in the form of a divergence term. This flux is modeled by Eddington's radiative transfer equation. The main parameters are the dimensionless opacity and the ratio of radiative and thermal fluxes. We prove existence of spherical flame ball solutions for the FBP. Bifurcation diagrams are obtained, exhibiting the multiplicity of solutions. Singular limit cases of the parameter values are also discussed. © 2006 Society for Industrial and Applied Mathematics.