Equations are derived for the prediction of radial velocity distributions for fully developed turbulent flow in smooth tubes both with and without heat transfer. The analysis results in an equation which represents both the conventional buffer layer and the laminar layer. In order to check the analysis and determine values for the constants appearing in the equations, tests were conducted to determine fully developed velocity distributions for air flowing without heat transfer in a smooth tube. The results are correlated by using conventional velocity and distance parameters, and agree closely with those of Nikuradse and other investigators. The analysis is extended to include flow with heat transfer, and equations for this case are obtained for Prandtl number of 1 in which the effects of the variation of fluid properties due to temperature variation across the tube were considered. By use of the equations for velocity and temperature distributions, relations are also obtained among Nusselt number, Reynolds number, and friction factor for a fluid with a Prandtl number of 1 for the case where the variation of fluid properties across the tube is large. The trends predicted analytically were found to be similar to those determined experimentally by measurement of average surface heat-transfer coefficients and friction factors of air flowing in tubes.