Numerical Techniques for Estimating the Surface Areas of Ellipsoids Representing Food Materials

Abstract

Abstract Food materials (seeds, grains, fruits and vegetables) resembling the shape of a general ellipsoid were modelled for accurate determination of surface area from measurements of their three principal dimensions. The process of surface area estimation involved partitioning the ellipsoid into an appropriate number of elliptical discs, determination of the surface areas of the edges of the discs and summing them to obtain the total surface area. Two types of perimeters were used to determine the edge surface areas: (1) the average perimeter; and (2) the root mean square perimeter. For edge length, three measures were considered: (1) parallel to the axis; (2) inclined to the axis; and (3) length of the elliptical arc. Six different surface area models were developed by combining the two perimeters with the three edge lengths and these were validated against exact analytical solutions for a sphere, prolate spheroid and oblate spheroid. The models with parallel edge length always produced the greatest underestimate of surface area and they cannot, in fact, give accurate results, even if a large number of discs is used. The models with inclined and elliptic arc edge length performed well in the entire range of width/length of prolate spheroids and thickness/length of oblate spheroids. For practical computations involving the general ellipsoid, a minimum of 100 discs was found to be sufficient, as it produced a maximum deviation of less than ±0·1% from 1000 discs for all the models and geometries considered. The performances of two approximate formulae for the surface area of the general ellipsoid were also studied for comparison but they were suitable only in the near-spherical range. Predictions were made and compared with the reported surface areas for a few paddy varieties, which were experimentally determined by previous researchers, and were found to deviate within +7·0 to +30·0%.