Dynamic Models for Metal Flows

Abstract

Metal flow analysis, a discipline in industrial ecology, describes the manner in which metals propagate throughout a coupled ecological and industrial system. The current models in industrial ecology are static representations that provide analysts with only a qualitative understanding of the dynamics at play. As such, many industry metrics are difficult to ascertain, such as recyclability and loss rates, future metal demand, and sustainability thresholds. The discipline can be expanded for its current state to a more quantitative framework where reliance on mathematical modeling is central. The intent of this thesis is to provide a fundamental framework for the mathematical modeling of metal flows. A seamless transition is made from the static models to mathematical representations by incorporating dependence (independent or random variables) to the existing models and studying the rate at which metal flows throughout a system. Due to the composition of the static metal flow models (disjoint domains and adherence to conservation of mass), compartmental modeling that has been used for over a century in epidemiology to study the spread of infectious diseases is utilized. Deterministic models consisting of ordinary differential equations are constructed and analyzed. Parameter optimization techniques using the Levenberg-Marquardt algorithm and numerical schemes for the acquisition of approximation solutions to the models are constructed and programmed in the Matlab language. In attempts to provide users with the ability to interface with the models—analyze and solve the models based on various parameters— standalone graphical user interfaces written in Matlab are constructed. To provide insight into the modification and improvement of the models, economic principles are introduced into the ordinary differential equation models, a pricing independent variable is included into the models producing systems of reaction-diffusion equations, and economic fluctuations and volatility are modeled using stochastic differential equations.