Analytical solutions to counter-flow heat exchanger subjected to external heat flux and axial conduction

A counter-flow heat exchanger with longitudinal conduction in separating wall is analyzed along with external heat flux. The inhomogeneous system of model equations is solved with the method of variation of parameters. Closed-form expressions are obtained for balanced flow with three types of wall end-conditions and the zero heat in-leak limit verified against reported expressions. In addition, analytical solution is derived for the general unbalanced flow model with non-adiabatic wall ends that is validated against the limiting case of no axial conduction. It enables examining property variations and arbitrary external load by dividing the system into smaller segments and applying the derived expressions to each element. Heat in-leak has higher impact on deterioration in the low NTU region studied when directed toward the hot fluid. As the external parasitic enhances with NTU, a critical NTU is noted in the range 2.8-5.1 for the various cases analyzed. At this optimum condition, the net energy reaching colder stages is between 27% and 55%.