High-resolution observations of sunspots have revealed the existence of dark cores inside the bright filaments of the penumbra. Here we present the stationary solution of the heat transfer equation in a stratified penumbra consisting of nearly horizontal magnetic flux tubes embedded in a stronger and more vertical field. The tubes and the external medium are in horizontal mechanical equilibrium. This model produces bright filaments with dark cores as a consequence of the higher density of the plasma inside the flux tube, which shifts the surface of optical depth unity toward higher (cooler) layers. Our results suggest that the surplus brightness of the penumbra is a natural consequence of the Evershed flow, and that magnetic flux tubes about 250 km in diameter can explain the morphology of sunspot penumbra.