Analytical solution of the three-pass counter-crossflow heat exchanger with both fluids unmixed throughout

This paper presents an analytical-algebraic solution procedure for a three-pass counter-crossflow heat exchanger in which both fluids remain unmixed throughout the entire exchanger. The mathematical model is derived under the standard assumptions of classical heat exchanger theory and extends the framework previously developed for two-pass configurations. The governing differential equations describing the temperature fields in each pass are solved analytically, while the coupling between successive passes is resolved through an algebraic closure procedure. Specifically, the inlet temperature distributions for subsequent passes are represented by assumed functional forms, and the unknown coefficients are determined from a linear algebraic system obtained by collocation at enough points. This approach yields explicit functional expressions for the temperature fields, mean outlet temperatures, and exchanger effectiveness. Unlike purely numerical or matrix-based methods, the present formulation preserves the analytical structure of the solution while restricting numerical intervention to the determination of a finite set of algebraic coefficients. The procedure is demonstrated in detail for the three-pass configuration, but its structure is directly extendable to two-pass and higher-order multipass arrangements. The resulting analytical-algebraic framework provides a transparent and systematic basis for performance prediction and design calculations for complex multipass crossflow heat exchangers.