The problem of non-isothermal quasi-steady state evaporation and condensation of aerosol spheres is examined to determine the rates of simultaneous heat and mass transport in the Knudsen (transition) regime. Now expressions for the mass and heat fluxes are obtained that show explicitly the dependence of the rate processes on the Knudsen number, the accommodation coefficients for mass and energy transport and on the molecular weight ratio of the vapor and gas molecules. The analysis, based on the solution of the Boltzmann equation by the method of Grad for Maxwellian molecules, is shown to yield results in the continuum regime (Kn much less than 1) in reasonable agreement with classical methods based on continuum theory and with measured water droplet evaporation rates in dry air. Computations of heat and mass transport rates for ice sublimation for upper tropospheric and stratospheric conditions for sizes that correspond to the continuum and transition regimes show that the process is very nearly isothermal. Parametric studies explore the effects of temperature, humidity and accommodation coefficients on the heat and mass transport processes. Although the method of Grad is known to fail in the free-molecule regime, the results agree with more rigorous theoretical solutions for isothermal processes in the near-continuum regime and with an earlier solution for hard sphere molecules in the near-continuum regime. It is shown that flux-matching or resistance models used for the transition regime do not show the correct dependence on the Knudsen number and other parameters.