In fluid descriptions of the solar wind the heat conductive flux is usually determined by the use of the classical Spitzer-H¿rm expression. This expression for the heat flux is derived assuming the gas to be static and collision-dominated and is therefore strictly not valid in the solar wind. In an effort to improve the treatment of the heat conductive flux and thereby fluid models of the solar wind, we study an eight-moment approximation two-fluid model of the corona-solar wind system. We assume that an energy flux from the Sun heats the coronal plasma, and we solve the conservation equations for mass and momentum, the equations for electron and proton temperature, as well as the equations for heat flux density in the electron and proton fluid. The results are compared with the results of a ''classical'' model featuring the Spitzer-H¿rm expression for the heat conductive flux in the electron and proton gas. In the present study we discuss models with heating of the coronal protons; the electrons are only heated by collisional coupling to the protons. The electron temperature and heat flux are small in these cases. The proton temperature is large. In the classical model the transfer of thermal energy into flow energy is gradual, and the proton heat flux in the solar wind acceleration region is often too large to be carried by a reasonable proton velocity distribution function. In the eight-moment model we find a higher proton temperature and a more rapid transfer of thermal energy flux into flow energy. The heat fluxes from the corona are small, and the velocity distribution functions, for both the electrons and protons, remain close to shifted Maxwellians in the acceleration region of the solar wind. ¿ American Geophysical Union 1996