An analytical study of laminar counterflow double‐pipe heat exchangers

Abstract

An orthogonol expansion technique for solving a new class of counterflow heat transfer problems is developed and applied to the detailed study of laminar flow concentric tube heat exchangers. The exchanger problem is solved for fully developed laminar velocity profiles, negligible longitudinal conduction in the fluid streams and in the exchanger walls, and with fluid properties which are independent of the temperature.A description of the variation of the local Nusselt numbers and the temperature a t the wall between the two streams is given. Also reported are bulk temperature changes in the two streams and mean overall Nusselt numbers. It is shown that for long exchangers, which are of some industrial importance, asymptotic Nusselt numbers exist in counterflow as in single-phase and cocurrent systems. Numeric01 volues of asymptotic Nusselt numbers are reported for a wide range of parameters. Comparisons are made with single-stream solutions such as the Graetz problem, with empirical correlations of experimental data, and with cocurrent flow exchangers.To solve this problem it was necessary to derive new orthogonality relations, and also expressions for determining positive and negative sets of eigenvalues and eigenvectors. Satisfoction of inlet boundary conditions a t both ends of counterflow exchangers requires a complete set of eigenfunctions and thus one must use both the positive and negative sets.Forced convection heat transfer in bounded conduits has received and continues to receive a great deal of attention in the literature. Theoretical studies are generally extensions of the classical Graetz problem of a fluid in fully developed laminar flow in a circular duct with a uniform wall temperature boundary condition. Extensions have been made to different geometries and to a variety of boundary conditions (5, 14,16) in an attempt to simulate physical situations more closely.Beyond the initial concern for the single-stream problem, it is natural to speculate on the applicability of these results to double-pipe heat exchangers which are used extensively in industrial applications. However, the question remains as to what type of condition exists at a wall between two streams in a practical heat exchanger. Stein tems are open-ended with respect to the specified boundary conditions, but the counterflow system is neither openended nor completely bounded, as shown in Figure 2c.As a consequence of these differences the counterftow problem has at least two unique aspects:1. It requires the development of new orthogonality relations for the sets of positive and negative eigenfunctions associated with positive and negative sets of eigenvalues. ___.___Richard New York. J.Nunge is with Clarkson College of Technology, Potsdam, 2. Complete orthogonal expansions utilizing both sets of eigenfunctions must be written at both ends of the counterflow problem to satisfy inlet conditions. Therefore, it is necessary to find a new procedure for evaluating the eigenvectors, or expansion coefficients, associated with the complete se...

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