Chemical Engineering Science, Vol. 47, No. 17/l& pp. 42954303, 1992. 0009-2509/92 $5.00 + om Printed in Great Britain. 0 1992 Pergamon Press Ltd AN EXPERIMENTAL CLUE TO THE IMPORTANCE OF DILATION IN DETERMINING THE FLOW RATE OF A GRANULAR MATERIAL FROM A HOPPER OR BIN R. B. THORPE University of Cambridge, Shell Department of Chemical Engineering, Pembroke Street, Cambridge CB2 3RA, U.K. (First received 19 February 1992; accepted in revised form 30 April 1992) Abstract-In this paper the experiments of Fickie et al. (1989) on the density of a granular material flowing through a wedge-shaped hopper are discussed with respect to their impact on the theoretical modelling of the flow from a hopper or bin under the influence of gravity. It is concluded that the degree of dilation found in these experiments is most significant. Furthermore, the experiments suggest that the material is less dense at the outlet than might be expected from the current theory. This in turn suggeststhat no discontinuity in the stress and velocity fields is needed to model the flow. The dilation is capable of reducing the flow so much that the challenge to the theoretician is no longer to tind ways of increasing the dissipation of energy in their models but to decreaseit. INTRODUCTION The flow of a granular material through a converging duct has long fascinated mankind. The most obvious early scientific use was that of the hourglass and many an ancient must have wondered at the regularity of the flow which caused the device to be the most accurate timepiece of its day. However, the theoretical principles which determine this most regular of flows have only been developed in the second half of the twentieth century. The current and consistent ap- proach to the prediction of the flow rate in an hour- glass or from a bin or hopper is to couple the stress field in the granular material to its strain field and, hence, its acceleration. The earliest attempt is due to Savage (1965), who considered the flow of a cohesion- less material from an acute smooth-walled hopper. The assumption of an acute half-angle and smooth walls ensures that the stress-strain coupling can be expressed in terms of a single ordinary differential equation; the problem becomes one-dimensional. The flow field is radial and the stress field has one of its principal directions in the radial direction. The solu- tion of the resulting ODE, subject to suitable bound- ary conditions, gives the flow rate of material from the hopper and, hence, the stress field. This simple but ideal&d solution was independently derived by both Sullivan (1972) and Davidson and Nedderman (1973); the latter workers elaborated the solution in some detail. The purpose of this paper is to discuss the signific- ant improvements which have been made in the pre- diction of the flow rates form hoppers based on this continuum approach. This will be done by reference to the relative improvement in the predicted discharge rate with reference to the value which is measured in experiments. The smooth-walled model being the least complex, and first in the series, predicts flow rates which are nearly twice those observed_ There is, however, a complication in small hoppers in that the flow rate is further reduced by a boundary layer at the wall of the hopper which is of a thickness of three or four particle diameters (Hagen, 1852; Beverloo et al., 1961; Nedderman and Laohakul, 1980). This bound- ary layer cannot be predicted by continuum theories and, in any comparison, this effect must, therefore, be factored out. According to Nedderman ef al. (1982), the experimental rate of discharge of a coarse granular material from a wedge-shaped hopper can be ad- equately correlated by the following equation which is based on the work of Beverloo et al. (1961), Rose and Tanaka (1959) and Myers and Sellers (1977): IV= 1.03pg” .5(1 - kd)(b - kd)1.5 tan-‘.’ &, (6, < 45”). (1) The first significant improvement to the smooth- walled solution was to include the effect of wall friction and is due to Savage (1967). There was a later more successful attempt by Brennen and Pearce (1978) for a wedge-shaped (two-dimensional) hopper. The three-dimensional homologue is the conical hopper and this was analysed by Ngyuen et al. (1979). These sets of authors sought a perturbation solution to the governing set of partial differential equations. The variables were perturbed in 0, the angle from the axis of symmetry. The resulting solution could be shown to be only approximate, but gave an idea of the importance of wall friction on the Row rate: the rate of discharge predicted by the second-order perturbation was significantly (about 50%) less than the rate pre- dicted by the smooth-walled theory and a mere 10% above the experimentally measured values. Chen et al. (1984) and Thorpe (1984) independently pointed out that by using only one term (i.e. up to first-order) in the expansion about the axis of symmetry, it is pos- sible to reduce the equations to ordinary differential form whilst retaining most of the effect of wall friction. The flow is radiai but the principal directions of the CES 47:17/18-F 4295