Optimization and Analysis of Plastic Film Consumption for Wrapping Round Baled Silage Using Combined 3D Method Considering Effects of Bale Dimensions
Authors: Stankiewicz Anna 
Published: April 2021

Highlights

Solved the problem of optimal design, in the sense of minimal film usage, of round bale diameter and height.

Necessary and sufficient optimality conditions derived in the form of easy-to-solve cubic equations.

Bales of a maximum volume achievable with an actual wrapper and of optimal dimensions ensure minimal film usage.

Up to 10% savings in film usage if bale dimensions are optimally designed and wrapping parameters properly selected.

Abstract. The combined 3D method is used for wrapping cylindrical bales of agricultural materials based on biaxial rotation of the film applicators. The demand for minimization of plastic film consumption keeps increasing, with the goal to save the environment, reduce plastic costs, and minimize wrapping time. Consequently, methods have been reported to solve the problem of optimal wrapping parameters for the conventional wrapping method. In this article, a model-based problem of such a design based on round bale dimensions (diameter and height) that minimizes film consumption for the combined 3D method is mathematically formulated and analyzed. The film consumption per unit of bale volume is used as a measure of film usage. Generally, it is difficult to find the optimal bale dimensions that minimize the original film usage index, due to the discontinuity of the index. Thus, near-optimal parameters, being as important as optimal parameters for engineering applications, are looked for. The problem of selecting near-optimal bale dimensions was constructed by minimizing the continuous lower bound of the original film usage index. The necessary and sufficient optimality conditions for near-optimal bale dimensions were established in the form of standard cubic equations, which can easily be solved using both analytical and numerical methods. Based on the optimality conditions, analytical and numerical analyses were performed of the influence of film width, pre-assumed bale volume, and numbers of bottom and upper film layers on the near-optimal bale dimensions and film usage. The results indicated that the near-optimal bale diameter and height, hereinafter called optimal, monotonically increase, while the optimal film consumption monotonically decreases, with increasing pre-assumed bale volume. Therefore, it is recommended to use bales of a maximum volume achievable with an actual wrapper and of optimal dimensions, i.e., diameter and height. The film width also influences the optimal bale dimensions and film usage: the wider the film, the smaller the minimal film usage. To confirm the effect of near-optimal bale design on film usage, the errors of the near-optimality were examined for four to sixteen film layers. The results of the numerical experiments demonstrated that for four to sixteen layers of film, there are compositions of the bottom and upper film layers for which the relative near-optimality errors do not exceed 0.01% whenever the optimal bale dimensions are used. Simultaneously, inappropriate selection of wrapping parameters may result in increased film usage, measured by mean relative errors of 1% to 9.5%, which means up to 10% film cost savings when the bale dimensions are optimally designed according to the proposed approach, and the wrapping parameters are appropriately selected. Keywords: 3D bale wrapping, Mathematical model, Minimal film consumption, Round bales, Stretch film usage.