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A compressible lattice Boltzmann-finite difference method is extended by the phase-field approach into a monolithic scheme to study fluid flow and heat transfer through regular arrangements of solid bodies of circular, elliptical and irregular shapes. The advantage of using the phase-field method is demonstrated both in its simplicity of accounting for flow and thermal boundary conditions at solid surfaces with irregular shapes and in the capability of generating such complex-shaped objects. For an array of discs, numerical results for the overall solid-to-gas heat transfer rate are validated via experiments on flow through arrays of hot cylinders. The thus validated compressible LB-FD-PF hybrid scheme is used to study the dependence of heat transfer on flow and thermal boundary conditions (Reynolds number, temperature difference between the hot solid bodies and the inlet gas), porosity as well as on the shape of solid objects. Results are rationalized in terms of the residence time of the gas close to the solid body and downstream variations of gas velocity and temperature. Perspective for further applications of the proposed methodology are also discussed.