In this study, the heat transfer analysis of a conceptual Stirling engine heater has been carried out. Basically the heater is a parallel flow heat exchanger consisting of a square duct and a circular duct. The circular duct is concentrically situated inside the square duct. A hot gas emanating from a burner passes through the circular duct, while the working fluid of the engine flows through the gaps between the square and circular ducts in laminar regime. The flow of the working gas through the gap is assumed to be a fully developed steady-state flow, and the working fluid side Nusselt number has been numerically investigated. The problem was treated in two different manners. In the first manner, the cross-cut of the flow passage was divided into two symmetrical sections and the field equations ( momentum and convective energy) were solved on one of the symmetrical sections aft er transforming into a boundary fitted domain and then the Nusselt number was determined. In this treatment, at the symmetry line of the flow passage, the symmetry condition was used as a boundary condition for both velocity and temperature fields. In second manner, the field equations were solved for the whole of the flow passage aft er again transforming into a boundary fitted domain. The Nusselt number has been investigated for both uniform heat flux and uniform wall temperature. In the case of uniform heat flux, the convective energy equation transforms to an equation with a source term. In the case of uniform wall temperature, the energy equation transforms to an equation that has been usually treated as an eigenvalue problem. In this study, both equations were solved by a finite difference method. For solution of the uniform wall temperature equation, a numerical algorithm was presented. For a uniform heat flux, the hydraulic diameter-based Nusselt number has been determined as 1.128. For the case of uniform wall temperature, the Nusselt number appears to be the function of the Peclet number ( Re x Pr) but, except for very low values of the Peclet number such as 10, the Nusselt number is a constant of 0.894.