Temperature distributions recorded by thermocouples in a solid body (slab) subject to surface heating are used in a mathematical model of 2-D heat conduction. The corresponding Dirichlet problem for a holomorphic function (complex potential), involving temperature and heat stream function, is solved in a strip. The Zhukovskii function is reconstructed through singular integrals, involving an auxiliary complex variable. The complex potential is mapped by the Schwartz-Christoffel formula onto an auxiliary half-plane. The final heat conduction flow net (orthogonal isotherms and heat lines) is compared with the known Carslaw-Jaeger solution and shows a puzzling topology of energy fluxes for simple temperature-boundary conditions.