Iterative Standard Strategy for Non-Linear Optimization in Melting Furnace Charge Calculations

Abstract

This paper describes an approach to furnace charge calculation for melting processes, to achieve a specified target melt in crucible or induction furnaces with minimum material cost. There has been a challenge with the problem regarding material loss, contaminations, and unwanted physiochemical reactions during the process. Those put a high degree of complexity, non-linearity, and uncertainty on the calculations. The current study presents a model that takes three important complexities of the problem into account, including non-homogeneous element loss during melting, non-metal contaminations in scraps/charges, and correction of possible initial melt in the furnace. The model was based on the mass balance of chemical elements along with the optimization of weights of charge materials. It presents a re-arrangement of non-linear mass balance formulations into an iterative standard linear-programming framework. To evaluate the performance of the model, an industrial-scale test case was introduced. The test problem was to find an optimum combination of charge materials to achieve target brass alloy C47940 in a 10-ton induction furnace. Eight different types of charge material were introduced to the model, with different amounts of non-metallic contaminations and different ratios of element loss in each. Also, 7-ton out-of-range initial melt was considered in the furnace to be corrected. The matrix of coefficients was build according to the numerical algorithm of the model. Optimizations were successfully performed in 3–5 iterations with Excel solver. For the test case, the calculations showed the optimum mass fractions of charge burdens and predicted to give 9649 kg melt with ~ 262 kg of total materials loss. An optimality analysis was conducted and showed that the solution has reached the minimum possible cost. The non-linear iterative algorithm revealed a convergent and fast performance. This feature provides a backbone for reliable and fast optimization in melting operations which is of significant benefit for industrial automation.