Particle velocities in vertical pneumatic conveying systems

Abstract

The calculation of pressure gradients in solid particle conveying systems requires a knowledge of particle velocities. In the following discussion it is shown that, in the case of large particles in vertical pipes, a general equation for particle velocities can be written in terms of the known particle and system properties and the empirical coefficients, Co, Co, and As, which relate the drag force and the solids force to the flow parameters. It is further shown that, for a given rate of solids flow, the excess pressure gradient due to the presence of the solids passes through a minimum as the airflow is changed and that at this minimum the particle velocity is independent of the solids factor A,. Moreover, if the multi-particle drag coefficient CD is approximately equal to the single particle drag coefficient Co,, then at the minimum, the relative velocity between the air and the solids is approximately 41% greater than the single particle terminal velocity in air. The flow of dilute suspensions of solid particles in air has been studied extensively and satisfactory explanations of the effect of a change in the airflow on the total pressure gradient and the stability of the suspension have been given (Zenz & Othmer 1960; Wen & Galli 1971; Kunii & Levenspiel 1969). In these texts the phenomenon of saltation in horizontal flow is described together with the sudden increase in the system pressure gradient which occurs when, at constant solids flow, the airflow falls below the saltation value. The equivalent choking velocity in vertical flow has also been described and the more gradual increase in pressure gradient as the airflow reduces beyond a pressure gradient minimum is recognised as being due to a fall in particle friction and to a rise in static head as the mixture density increases at lower air and particle velocities. A knowledge of particle velocities is necessary for predicting pressure drops accurately in such systems but no generally accepted analytical method exists which enables these velocities to be calculated. Some investigators have concluded that for vertical flow the relative velocity between the air and the solids is equal to the single particle terminal velocity. Others have refuted this assumption and produced data to show that the relative velocity is not a constant but increases with increasing air velocity. Basic analyses relating air/particle drag forces and frictional forces due to particle collisions have been published by Hinkle (1953) and by Wen & Galli (1971). In the Hinkle thesis the equation developed requires experimental data relating air and particle velocities before a solution for pressure drop can be found. The relationship which was found to correlate the measured particle and gas velocities is of an empirical nature and is of uncertain value outside the conditions in the test. Wen & Galli derive a general equation for horizontal flow which relates particle and gas velocities but its solution requires data on the solids frictions factor As together with an iterative or graphical solution for the solid particle velocity in terms of the air velocity. In the following analysis it is shown that in the case of vertical flow a general solution for particle velocities can be written in terms which include the solids factor As and the multiparticle drag coefficient CD. By considering only the excess pressure gradient due to the