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<p>The transition towards a circular economy has gained importance over the last years since the traditional linear take-make-dispose paradigm is not sustainable in the long term. Recently, thermodynamical material networks (TMNs) have been proposed to approach the creation of circular material flows as a design problem. Material flow analysis (MFA) is effective when many stocks and flows data are available, but it lacks accuracy in the frequent case in which data are missing. In contrast, TMNs are robust to a lack of data because they leverage ordinary differential/difference/hybrid equations derived from thermodynamics laws. In this paper, we develop several circularity indicators of TMNs using graph theory, detail the mathematical formulation of the models for repeatability, report the results of numerical simulations, provide the algorithms to compute the circularity indicators and made them publicly available (<a href="https://github.com/fedezocco/TMNcircularity-MATLAB">https://github.com/fedezocco/TMNcircularity-MATLAB</a>). By providing thermodynamics-grounded dynamical-systems-based circularity indicators, we contribute to the literature of circularity indicators since the literature lacks physics-based circularity measures and mainly consists of static snapshots of material use. In contrast, our dynamical systems capture both slow and fast flow variations that occur in less than 1 minute, e.g., the sudden variation that occurs in the stock of a building when a batch of material enters it. Specifically, first we considered the case of fluid materials, which are well-described by ordinary differential equations. Then, we addressed the case of materials transported in batches such as critical raw materials (CRMs) and plastics; this case required a combination of continuous- and discrete-time terms. The proposed circularity indicators capture properties of TMNs to be used for what-if analysis in the design of circular material flows.</p>

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Circularity of Thermodynamical Material Networks: Indicators, Examples, and Algorithms

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Federico Zocco

摘要

The transition towards a circular economy has gained importance over the last years since the traditional linear take-make-dispose paradigm is not sustainable in the long term. Recently, thermodynamical material networks (TMNs) have been proposed to approach the creation of circular material flows as a design problem. Material flow analysis (MFA) is effective when many stocks and flows data are available, but it lacks accuracy in the frequent case in which data are missing. In contrast, TMNs are robust to a lack of data because they leverage ordinary differential/difference/hybrid equations derived from thermodynamics laws. In this paper, we develop several circularity indicators of TMNs using graph theory, detail the mathematical formulation of the models for repeatability, report the results of numerical simulations, provide the algorithms to compute the circularity indicators and made them publicly available (https://github.com/fedezocco/TMNcircularity-MATLAB). By providing thermodynamics-grounded dynamical-systems-based circularity indicators, we contribute to the literature of circularity indicators since the literature lacks physics-based circularity measures and mainly consists of static snapshots of material use. In contrast, our dynamical systems capture both slow and fast flow variations that occur in less than 1 minute, e.g., the sudden variation that occurs in the stock of a building when a batch of material enters it. Specifically, first we considered the case of fluid materials, which are well-described by ordinary differential equations. Then, we addressed the case of materials transported in batches such as critical raw materials (CRMs) and plastics; this case required a combination of continuous- and discrete-time terms. The proposed circularity indicators capture properties of TMNs to be used for what-if analysis in the design of circular material flows.