,n,. J. Heat Mass Transfer. Vol. 27, No. II, pp. 1953-1962, 1984 0017 zyxwvutsrqponmlk Printed in Great Britain Y3lu,x4$3.w +o.oo Pergamon Press Ltd. Heat and momentum transfer around a pa i r of spheres in viscous flow REUVEN TAL (THAU),t DAH NAIN LEES and WILLIAM A. SIRIGNANOt t Mechanical Engineering Department and $ Mathematics Department, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. (Received 8 July 1982 zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIH and in revisedform 11 July 1983) Abstract-The Navier-Stokes and energy equations have been solved numerically for a pair of spheres in tandem at Re = 40 for two different spacings using bispherical coordinates. The solutions compare favorably with experimental results. The drag coefficient and the average Nusselt number of either sphere is always less than that of a single isolated sphere, the effect being much stronger on the downstream sphere. zyxwvutsrqponmlkji 1. INTRODUCTION THERE has been recent interest in the problem of interaction between two burning droplets. Twardus and Brzustowski [l] solved Laplace’s equation in bispherical coordinates for the diffusion field around two stationary droplets, as a function of the distance between the droplets, obtaining criteria for the flame shape as a function of droplet size, spacing and stoichiometry. Umemura et al. [2, 31 addressed the same problem, for droplets of equal or non-equal sizes and included Stefan flow as well as the energy equation in their model. Using the SchvabZeldovich transform- ation, Umemura et al. [2,3] obtained the burning rate and the flame surface shape for pairs of burning droplets. Labowsky [4] used the method of images to solve the Laplace equations in the interdroplet field for two and more droplets and concluded that droplet lifetimes are longer as the spacing is reduced. Forced convection was excluded from the above-mentioned studies. Rex et al. [S] studied experimentally the burning of pairs offuel droplets. It was observed that as the spacing is reduced from larger values with negligible droplet interference, the burning rate increases, presumably because heat losses from the flame surface are reduced by the proximity of a second heat source. As the spacing is reduced, the burning rate reaches a maximum and then decreases as the spacing is further decreased. This decrease in the burning rate could be produced by the ‘competition’ for oxygen of the two droplets surrounded by one flame sheet. Sangiovanni and Kesten [6] studied experimentally the burning oflinear arrays of droplets and found that the burning rate decreases monotonically as the spacing is reduced. The apparent discrepancy between the experimental works mentioned above could be caused by the presence of free convection in the experiments performed by Rex et al. [S] and of forced convection in the experiments performed by Sangiovanni and Kesten 161. In real combustors, the Reynolds number based on fuel droplet diameter can be as high as 200. As indicated by Prakash and Sirignano [7, 81 and Lara-Urbaneja and Sirignano [9], the most important effect of forced convection on a droplet is creation of internal cir- culation within the droplet. The internal circulation changes the characteristic heating times of the droplets by an order of magnitude. Subsequently, the droplet vaporization characteristic significantly differs from that given by the d2-law for stationary droplets. For that reason, the existing combustion analyses for pairs of droplets in the absence of convection are limited in their applications. As a first step in the study of the combustion of a pair of fuel droplets in the presence of forced convection, we address the problem of heat and momentum transfer around a pair of solid spheres in viscous flow, at intermediate Reynolds numbers. This problem has not been solved theoretically although a considerable amount of experimental work has been performed on the subject by Rowe and Henwood [lo] and Tsuji et al. [ 111. The only theoretical work on a pair of spheres in tandem is limited to Re << 1 [12]. In the above-mentioned works it was found that the drag coefficient ofeither sphere was always less than that ofa single isolated sphere. Rowe and Henwood [lo] performed measurements of drag forces on two spheres in line as well as with the centerline inclined to the flow direction, as part of a broad study of a hydraulic model of a fluidized bed. The Reynolds number range considered was 32-96. Although this range is different from that considered by Tsuji et al. [I 11, the drag results of the two works are in reasonable agreement. In fact, Rowe and Henwood [lo] indicate that the Reynolds number is not a criti- cal parameter in a fluidized bed. The experimental apparatus used by Rowe and Henwood [lo] was as follows : l/2 in. (12.7 mm) spheres were made of polyethene. A small hole was drilled in the sphere and this was plugged with solder to make the overall density just greater than that of water. The sphere was supported by a 4 in. (101.6 mm) length of 0.007 in. (0.18 mm) Nichrome wire which was pushed into it. Most of the wire was supported by a water-tilled glass tube. Distilled water was circulated through an open-topped Perspex tank. The water stream caused deflection which was measured by sighting a cathe- 1953