Dispersion in curved tubes and channels is treated analytically, using the velocity distribution of Topakoglu (1967) for tubes and that of Goldstein (1965) for curved channels. The result for curved tubes is compared with that obtained previously by Erdogan & Chatwin (1967) and it is found that the presentdispersion coefficient contains the Erdogan & Chatwin result as a limiting case.The most striking difference between the results is that Erdogan & Chatwin predict that the dispersion coefficient is always decreased by curvature if the Schmidt number exceeds 0.124, which is the ease for essentially all systems of practical interest. In contrast, the present result, equation (76), predicts that the dispersion coefficient may be increased substantially by curvature in low Reynolds number flows, particularly in liquid systems which would be of interest in biological systems.Two competing mechanisms of dispersion are present in curved systems. Curvature increases the variation in residence time across the flow in comparison with straight systems and this in turn increases the dispersion coefficient. The secondary flow which occurs in curved tubes creates a transverse mixing which decreases the dispersion coefficient. The results demonstrate that the relative importance of these two effects changes with the Reynolds number, since the dispersion coefficient first increases and then decreases as the Reynolds number increases. Since secondary flows are not present in curved channels the dispersion coefficient is increased over that in straight channels for all cases.

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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