Gravity flow of bulk solids

There is hardly an industry in existence which does not use solid materials in bulk form. Where the volume of the solids is substantial, gravity is usually relied upon to cause the solids to flow. Such materials as ores, coal, cement, flour, cocoa, soil, to which the general term of bulk solids is applied, flow by gravity or are expected to flow by gravity in thousands of installations and by the billions of tons annually. Mining relies on gravity flow in block-caving and in ore passes; subsidence is a case of gravity flow of solids. Agriculture relies on gravity flow of its products in storage silos, in feed plants and on the farms. Every type of processing industry depends on gravity flow of some solid, often of several solids. Although vast quantities of bulk solids have been handled for many years, the author believes that this is the first comprehensive study of the subject. The fact that this work appears at this time is not accidental, but stems from the progress achieved during the past fifteen years in the mathematical theory of plasticity and in the techniques of numerical calculation. On the basis of recently developed and refined principles of plasticity, the problem of flow of bulk solids has been set up in mathematical terms. A few years ago, this would not have been possible; just as a few years ago the mathematically formulated problem would have been practically insoluble because there were no computers to carry out the necessary calculations. The careful reader of the author's previous reports and papers on the subject of flow of bulk solids will notice substantial modifications in the design formulae. No apology is offered for these seeming inconsistencies; the author has always approached the subject from the standpoint of the engineer who has had to provide definite recommendations on the basis of information at hand, at the time. Hence, as the volume of experience increased, the theory was developed, and the numerical data were computed, the design methods improved and changed - at times, radically* The work is presented in six parts. In Part I, the yield function applicable to bulk solids is described, and the flow properties of bulk solids are defined. The solids are assumed to be rigid-plastic, isotropic, frictional, and cohesive. During incipient failure, the solids expand (dilate), during steady state flow, they may expand or contract. The yield function is consistent with the principle of normality [7] which is specifically applied in incipient failure. Part II contains the theory of steady state gravity flow of solids in converging and vertical channels. The equations are first derived in a general form, applicable to problems of extrusion as well as gravity flow, in plane strain and in axial symmetry. Some of the derivations are more general than they need to be for this work. They will be referred to in other publications which are now in preparation [22, 23]. It is shown that, provided the slopes of the walls of a converging channel are sufficiently steep and mathematically continuous, the stress pattern in the neighborhood of the vertex of the channel is, primarily, a function of the slope and of the frictional conditions of the walls at the vertex, with the influence of the top boundary of the channel vanishing at the vertex . The particular stress field which develops at the vertex is called the radial stress field, because it is the field which can lead to a radial velocity field. Since the radial stress field is closely approached in the vicinity of the vertex, that field represents the stresses at the outlet of a channel. The region of the outlet of a channel is most important because it is there, that obstructions to flow originate. The radial stress field thus provides a basis for a general solution of flow in this important region of the channels. In Part III, the conditions leading to incipient failure are considered. General equations of stress are derived in plane strain and in axial symmetry. The conditions following incipient failure are discussed, and it is suggested that the velocity fields usually computed for conditions of failure are meaningless and that only initial acceleration fields can be computed. Two cases of incipient failure are analyzed: doming across a flow channel, and piping (which refers to a state of stress around a vertical, empty hole of circular cross-section). Part IV describes the flow criteria. The material developed in the previous three parts is brought together to relate the slopes of channels and the size of the outlets necessary to maintain the flow of a solid of given flowability on walls of given frictional properties. Part V describes the testing apparatus and the method which has been developed to measure the flowability of solids, their density, and the angle of friction between a solid and a wall. Finally, Part VI contains the application of the theory to the design of storage installations and flow channels, and discusses flow promoting devices, feeders, segregation, blending, structural problems, the flow of ore, as well as aspects of block-caving and miscellaneous items related to the gravity flow of solids. All these topics are approached from the standpoint of flow: their effect on flow and vice-versa. The reader will soon realize that many of the bins now in operation have been designed to fill out an available space at a minimum cost of the structure rather than to satisfy the conditions of flow. The result has been a booming business for manufacturers of flow promoting devices. While there are, and always will be, solids which are not suitable for gravity flow, the vast majority of them will flow if the bins and feeders are designed correctly. However, a correct bin will usually be taller and more expensive. It is up to the engineer to decide whether the additional cost of the correct bin will be balanced by savings in operation. This part is made as self-contained as possible to facilitate its reading to the engineer who has neither time nor inclination to study the theoretical parts. The reader versed in soil mechanics should note that the magnitude of the stresses discussed here is 100 to 1000 times smaller than that encountered in soil mechanics. Hence, some phenomena which may not even be observable in soil mechanics assume critical importance in the gravity flow of solids. For instance, the curvature of the yield loci (Mohr envelopes) in the ( cf, t ) coordinates is seldom detectable in soil mechanics, but in gravity flow the curvature assumes an important role in the determination of the flowability of a solid. By the terminology of soil mechanics, solids possessing a cohesion of 50 pounds per square foot are cohesionless: standard soil mechanics tests do not measure such low values. But a solid with that value of cohesion, an angle of internal friction of 30°, and a weight of 100 pounds per cubic foot can form a stable dome across a 3-foot-diameter channel and prevent flow from starting. In gravity flow, it is of interest to be able to predict whether or not flow will take place through a 6-inch-diameter orifice. This involves values of cohesion down to 8 pounds per square foot and even less for lighter solids.