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

arXiv: https://arxiv.org/abs/2209.15051 · HTML: https://mlsanity.com/paper/2209.15051/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Synchronized Object Detection for Autonomous Sorting, Mapping, and Quantification of Materials in Circular Healthcare](https://mlsanity.com/paper/2405.06821/) (2024)
- [Circular Microalgae-Based Carbon Control for Net Zero](https://mlsanity.com/paper/2502.02382/) (2025)
- [CiRL: Open-Source Environments for Reinforcement Learning in Circular Economy and Net Zero](https://mlsanity.com/paper/2505.21536/) (2025)

## Abstract (arXiv)

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) [1] have been proposed as an approach to design material flows based on the idea that any supply chain can be seen as a set of thermodynamic compartments that can be added, removed, modified or connected differently. Compared to the well-established material flow analysis (MFA), TMNs leverage dynamical energy balances and ordinary differential equations along with the usual mass balances, thus tackling circular economy as a material network design problem analogous to traditional engineering design approaches (e.g., design of thermodynamic cycles, electrical and hydraulic networks) rather than as an analysis of stock-and-flow data. Hence, TMNs allow the depiction of highly dynamic material stocks and flows whose variations can occur in less than 1 minute; achieving such modelling accuracy with MFA would be more data intensive. In this paper, we first develop several circularity indicators of TMNs using a graph-based formalism. Then, we illustrate their calculation using two numerical examples for the case of fluid materials and one numerical example for the case of solid materials, for which the detailed hybrid dynamical equations and simulation outputs are provided. The paper source code is publicly available.