An explicit algebraic heat-flux (EAHF) model is derived by invoking the assumption of equilibrium turbulence for both the velocity and the thermal fields. Further modifications are achieved by applying an approximation technique to render the implicit heat flux vector relation explicit. Thus derived, the heat flux vector yields two terms. The first term is identical to that given by a simple thermal eddy diffusivity model, while the second term provides a correction to the streamwise heat flux. This second term is non-zero even when the streamwise mean temperature gradient is zero. It allows the modeling of the generation of a streamwise heat flux due to the interaction of the turbulent eddies with the mean temperature gradient normal to the streamwise direction. Previously derived near-wall corrections to the equations of the temperature variance and its dissipation rate are found to be equally valid for the EAHF model. The near-wall EAHF model is validated against direct numerical simulation data and experimental measurements. In addition, the calculations are also compared with those obtained from a second-order model and the simple thermal eddy diffusivity model. Two different near-wall Reynolds-stress models are used to calculate the velocity field and they are found to have little effect on the thermal field predicted by the EAHF model. In general, the results for the temperature field are in good agreement with data and are essentially unaffected by the second term in the EAHF model. On the other hand, the prediction of the streamwise heal flux is in good agreement with that given by a second-order model which correlates well with data.