This report contains the detailed derivation of the conservation equations of mass, momentum, and energy for a fluidized bed. The development of the chemical production term is included in the Appendix. The equations contain momentum exchange terms, viscosity and thermal conductivity transport coefficients for the fluid and solid, and other flow quantities which require constitutive relations before a solution is attempted. An iterative procedure, involving fluidized bed test data and the solution of these flow equations, could be employed to model these constitutive relations. The equations are then specialized to the one dimensional, steady flow for a high-temperature two-phase chemically reacting multispecies fluid and a single species solid. Solutions to this system are discussed.