MATHEMATICAL MODELLING OF PNEUMATIC CLEANING OF COWPEA (Vigna unguiculata (L).Walp)

Abstract

The presence of impurities in cowpea affects its economic value as well as industrial and domestic utilisation. This problem has persisted despite the use of various manual and mechanical cleaning methods. There exists a need to improve the efficiencies of existing cowpea cleaning devices. Literature is sparse on modelling of pneumatic separation of cowpea. This study was therefore designed to develop and validate a mathematical model for the efficient separation of impurities from cowpea in a pneumatic cleaner. Some physical and aerodynamic properties of four cultivars of cowpea (Ife 98-12, IT90K-277-2, Ife Brown and Drum) and impurities (chaff, insect infested grains and immature grains) were determined at harvesting and at storage moisture contents range (9-22% dry basis) using standard methods. A two-dimensional mathematical model for separating cowpea and impurities in a vertical flow airstream was developed using the force and acceleration components of each particle in the horizontal and vertical directions. The resultant equations were solved numerically for displacements using a proprietary software at injection angles of 15, 30, 45 and 60o which are greater than the coefficient of friction for cowpea on mild steel and air velocities of 4.0, 6.0, and 8.0 m/s which are also greater than the impurities’ terminal velocities. The displacements were used to predict the trajectories of cowpea, impurities and their separation. The model was validated using a pneumatic cleaner and the separation efficiencies were evaluated at the angles of injection and air velocities. A paired t-test was used to compare the experimental and predicted efficiencies. The axial dimensions, geometric mean diameter, thousand grain mass, projected area and terminal velocity of the cultivars ranged from 5.0-10.3 mm, 6.1-7.0 mm, 147.5-252.8 g, 28.0-43.6 mm2 and 13.4-14.5 m/s respectively. The mass and terminal velocity of the impurities ranged from 8.2 x 10-2-15.0 x 10-2 g and 1.5-3.5 m/s respectively. Vertical displacements of the cultivars below the point of injection ranged from 3.9-4.8 m. The vertical displacements of the impurities above and below points of injection varied from 3.4-72.1 m and 0.70-3.2 m respectively. The predicted separation distances between cultivars and impurities ranged from 5.0 x 10-2 - 897.0 x 10-2 m. The predicted separation efficiency ranged from 30.0-45.8%, 59.3-70.6%, 76.5-83.9% and 55.2-61.6%, 77.0-79.0%, 85.0-86.9% at 4.0, 6.0, and 8.0 m/s for injection angles of 15-30o and 45-60o respectively. Separation efficiencies of the machine ranged from 27.7- 44.6%, 61.4-71.9% and 76.6-80.4% for angles of 15-30o at 4.0, 6.0, and 8.0 m/s respectively. For angles of 45-60o, it ranged from 59.2-60.9%, 75.4-76.0% and 87.2-88.8% at 4.0, 6.0, and 8.0 m/s respectively. The model predicted significantly (p≤ 0.05) the separation efficiency within the experimental range. A mathematical model for the separation of cowpea and impurities in a pneumatic cleaner was developed. It was established that separation of impurities from cowpea was best at 45o angle of injection at 4.0-6.0 m/s. Therefore, it is recommended that the hopper should be inclined to the airstream at this angle.