When the heating intensity is below the critical threshold required for the onset of Rayleigh-Benard (RB) convection, heat transport across a horizontal fluid layer uniformly heated from below is driven by conduction, placing a limit on its magnitude. A methodology to increase the heat flow through the use of spatial heating non-uniformities is proposed. The non-uniformities create convection whose pattern is dictated by the pattern of the heating, and this convection supplements the conductive heat transport. It has been shown that the effectiveness of this convection, the primary convection, is a strong function of the heating wave number and it may increase the heat flux by up to ten times when compared with the conductive state if the most effective heating wave number is used. The primary convection is subject to transitions to various secondary states and the critical conditions for these transitions have been determined using linear stability theory. Three modes of secondary convection have been identified giving rise either to rolls parallel to the primary rolls, to rolls orthogonal to the primary rolls, or to oblique rolls. Conditions leading to their onset define the limits on the heat transfer predictions based on the analysis of the primary convection. In general, transition to secondary convection occurs at smaller Rayleigh numbers in the presence of heating non-uniformities than those required for the onset of RB convection, however, under certain conditions these non-uniformities increase the critical Rayleigh number. (C) 2017 Elsevier Ltd. All rights reserved.