We consider conjugate forced-convection heat transfer in a rectangular duct. Heat is exchanged through the isothermal base of the duct, i.e., the area comprised of the wetted portion of its base and the roots of its two side walls, which are extended surfaces within which conduction is three-dimensional. The opposite side of the duct is covered by an adiabatic shroud, and the external faces of the side walls are adiabatic. The flow is steady, laminar, and simultaneously developing, and the fluid and extended surfaces have constant thermophysical properties. Prescribed are the width of the wetted portion of the base, the length of the duct, and the thickness of the extended surfaces, all three of them nondimensionalized by the hydraulic diameter of the duct, and, additionally, the Reynolds number of the flow, the Prandtl number of the fluid, and the fluid-to-extended surface thermal conductivity ratio. Our conjugate Nusselt number results provide the local one along the extended surfaces, the local transversely averaged one over the isothermal base of the duct, the average of the latter in the streamwise direction as a function of distance from the inlet of the domain, and the average one over the whole area of the isothermal base. The results show that for prescribed thermal conductivity ratio and Reynolds and Prandtl numbers, there exists an optimal combination of the dimensionless width of the wetted portion of the base, duct length, and extended surface thickness that maximize the heat transfer per unit area from the isothermal base.

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