Exact solution for the conjugate fluid–fluid problem in the thermal entrance region of laminar counterflow heat exchangers

A generalized Lévêque solution is presented for the conjugate fluid–fluid problem that arises in the thermal entrance region of laminar counterflow heat exchangers. The analysis, carried out for constant property fluids, assumes that the Prandtl and Peclet numbers are both large compared to unity, and neglects axial conduction both in the fluids and in the plate, assumed to be thermally thin. Under these conditions, the thermal entrance region admits an asymptotic self-similar description where the temperature varies as a power j of the axial distance, with the particularity that the self-similarity exponent must be determined as an eigenvalue by solving a transcendental equation arising from the requirement of continuity of heat fluxes at the heat conducting wall. Specifically, the analysis reveals that j   depends only on the lumped parameter kˆ=(A2/A1)1/3(α1/α2)1/3(k2/k1), defined in terms of the ratios of the wall velocity gradients, AiAi, thermal diffusivities, αi, and thermal conductivities, ki, of the fluids entering, 1, and exiting, 2, the heat exchanger. Moreover, it is shown that for large (small) values of kˆ the solution reduces to the classical first (second) Lévêque solution. Closed-form analytical expressions for the asymptotic temperature distributions and local heat-transfer rate in the thermal entrance region are given and compared with numerical results in the counterflow parallel-plate configuration, showing very good agreement in all cases.