A study is made of turbulent heat transfer in a vertical circular tube under conditions of combined forced and free convection with uniform heat flux at the wall and volume heat sources. The basic conservation laws are reduced to three coupled, linear integrodifferential equations which are numerically solved by means of a digital computer. An improved expression for eddy viscosity variation with radial distance and degree of turbulence is used in the solution, as is an expression for the ratio of eddy diffusivity of heat to momentum. The parameters in the integrodifferential equations are: Prandtl number, Pr; Rayleigh number, Ra; friction Reynolds number, Re*; and a volume heat source parameter, F. For fixed values of these parameters, the solution results in: the velocity and temperature profiles; the pressure drop parameter, C; mixed-mean-to-wall temperature difference, φm; Nusselt number, Nu; and Reynolds number, Re. The solutions to this fairly general problem compare very closely to solutions of the following special cases (with no volume heat source): pure forced convection, laminar flow; pure forced convection, turbulent flow; and combined forced and free-convection, laminar flow.