On the mathematical modelling and optimization of foil consumption for cylindrical bale wrapping

Aspects of mathematical modelling of wrapping round (i.e. cylindrical) bale related to the optimization of foil consumption are considered for individual bales of agricultural materials wrapped in plastic foil. The article details the mathematical description of bale wrapping, while also identifying both some solved problems and a few open ones, which can be solved to optimize the foil consumption. Mechanical properties of the sealing foil characterized by its Poisson ratio, arbitrary foil and bale size dimensions and overlapping width of subsequent wrapped foil strips were taken into account. After providing a basic understanding of how the foil is consumed in the course of wrapping process, the mathematical model for determining the number of entire foil wrappings is determined. Next the results of mathematical analysis are used for optimization purposes. The foil consumption per unit of the bale volume index is used as a measure of foil expenditure. Two problems of foil consumption optimization are stated and solved. The formulas for computing the optimal foil width and optimal bale size dimensions are derived and estimations of the solution errors are given. Computational results are presented and analyzed for exemplary bale silage.