A novel formulation is presented for solving the conjugate heat transfer problem that arises due to a thin flush heat source mounted on a conductive substrate. The geometry is a paradigm for direct air cooling of components on conducting boards. PCB thermal algorithms based on this approach are being developed for rapid estimation of the thermal field in a direct air cooled board. The algorithms are part of a suite of tools for integrated electronic packaging design being developed at the Center for Electronic Packaging Research (CEPR). This paper presents the formulation of the approach and demonstrates its utilization for parametric studies of board level thermal management, in particular for studying the effects of board conductivity. The unique formulation allows one to couple a wide variety of flow models to the solid conduction. The solid side is modelled with a Boundary Element Method (BEM). The temperature field in the fluid side is not explicitly solved, rather, analytical 'step temperature' solutions, relevant to the particular flow model, are used to express convective heat flux as a function of interface temperatures. A non-iterative solution for the conjugate problem is found by matching the temperatures and fluxes at the solid-fluid interface. Results of a parametric study of the effects of board conduction on component thermal performance are presented.