Pergamon Press 1964.

Printed in Great Britain

HEAT TRANSFER IN AN ANNULUS WITH VARIABLE CIRCUMFERENTIAL HEAT FLUX? W. A. SUTHERLAND:

and W. M. KAYS§

(Received 10 March 1964)

Abstract-An analysis of heat transfer in a concentric circular tube annulus with an arbitrarily prescribed heat flux around the periphery of either wall, or both walls, is presented. Solutions have been obtained for the hydrodynamically and thermally fully developed condition for constant heat rate per unit of tube length, for both the laminar and turbulent flow regimes. With these results, the ensuing temperature variation around either wall may be predicted. Contrary to what might be expected, the waI1 temperature variation is very substantial in turbulent as well as laminar flow. An example shows the importance of this effect.

NOMENCLATURE a, b, A, B,

&

c nt

&A

Fourier series coefficients ; wall conduction parameter; hydraulic diameter; t

YY Y+,

a,

EH,

u.’

eigenfunction; Reynolds number; radius of zero shear; temperature; radial temperature function; velocity; y’@?C Q/P1 i dbn dum ;

Subscripts ave, in, a, 1, m, n, 0, W,

average; inlet; annulus ; inner waI1; mean ; harmonic index; outer wall; wall. INTRODUCTION

t This work was performed under U.S. Atomic Energy Commission Contract AT(O4-3)-189, Project Agreement 29. 2 Atomic Power Equipment Department, General Electric Company, San Jose, California. 5 Mechanical Engineering Department, Stanford University, Stanford, California.

THE CIRCULAR tube annulus is one of the more

important flow passage geometries for heattransfer systems, ranking closely behind the circular tube in engineering applications. It has 1187

1188

W. A.

SUTHERLAND

and W.

M.

KAYS

been shown by Reynolds et al. [l] that the thermal boundary condition for flow through an annulus can be reduced to four fundamental solutions. Of the large number of combinations of boundary conditions and flow regimes, the problem to be considered here is only that of fully developed laminar and turbulent flow with the heat flux specified at the wall and constant with tube length. For heat flux uniform around the tube periphery, this case has been treated analytically by Kays and Leung [2] for the concentric annulus for turbulent flow over an extensive range of Re, Pr, and r*, and has been substantiated by their experiments with air. The laminar flow counterpart of this problem is covered by Lundberg et af. [3] as a special case of the complete laminar flow concentric annulus problem. The additional condition to be considered here, however, concerns the variation of the heat flux around the periphery of the flow passage. This is of particular importance in a nuclear reactor, where the power distribution across a fuel element produces a variation of heat flux around the periphery of the flow passage. The importance of the variable circumferential heat flux problem has recently been shown by Reynolds [4] for the circular tube. This present paper extends the analysis of Reynolds [4] to include the annular geometry, building upon the annulus solution of Kays and Leung [2] for turbulent flow, and Lundberg et al. [3] for laminar flow. The technique used for the solution is to expand the known peripheral heat flux distribution in a Fourier series. Because the resulting temperature distribution can also be expressed as a Fourier series, the energy equation becomes a set of ordinary differential equations in terms of the eigenfunctions. These eigenfunctions are calculated herein, and may be used with the Fourier coefficients of any peripheral heat flux distribution to obtain the resulting wall temperature distribution. FORMULATION

perties. The system under consideration, along with some of the pertinent nomenclature, is illustrated in Fig. 1. Assuming equal eddy

Fig. I.

diffusivity for heat in the radial and circumferential...