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

Federico Zocco

TL;DR

Several circularity indicators of TMNs are developed using a graph-based formalism and illustrate their calculation through examples and the paper source code is publicly available.

Abstract

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.

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

TL;DR

Several circularity indicators of TMNs are developed using a graph-based formalism and illustrate their calculation through examples and the paper source code is publicly available.

Abstract

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.
Paper Structure (15 sections, 91 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 91 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Circularity Indicators
  4. Graph Theory for Circular Economy
  5. Thermodynamical Material Networks
  6. Indicators and Algorithms
  7. Indicators
  8. Algorithms
  9. Examples for Fluid Materials
  10. Fluid Case and Nonphysical Dynamic Graph
  11. Fluid Case and Imposition of Mass Balance
  12. Example for Solid Materials
  13. Network Modeling and Simulation
  14. Indicators and Discussion
  15. Conclusion

Figures (6)

  • Figure 1: Graphical representation of the indicators in Table \ref{['tab:summaryIndicators']} considering a compartmental digraph with $n_c$ = 11, $n_v$= 5, $n_a$ = 6, and $n_\phi$ = 2. Legend: orange for flows (i.e., arcs) and stocks (i.e., nodes) that increase the indicator; light blue for arcs and nodes that decrease the indicator; green for arcs and nodes that could increase or decrease the indicator; black for arcs and nodes having no influence on the indicator.
  • Figure 2: Digraph and dynamics of the indicators in Example 1 for $t \in [0, 2]$.
  • Figure 3: Stocks and indicators in Example 2 for $[m_1^0, m_2^0, m_3^0, m_4^0]^\top = [10, 10, 10, 10]^\top$; the other indicators are unchanged and given in Example 1.
  • Figure 4: Compartmental digraph (top) and mass-flow digraph (bottom) of the network considered in the example for solid plastics. Dashed arrows are for time-dependent flows.
  • Figure 5: Dynamics of stocks and flows in the example for solid materials. The model equations are given in (\ref{['eq:cs-m1']})-(\ref{['eq:cs-constraints']}), while the values of the parameters are in Table \ref{['tab:valuesOfParam']}. Yellow is for the total mass/flow, blue is for PET, orange is for HDPE, and green is for PP.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6: Thermodynamical material network
  • Definition 7: Material flow network
  • Definition 8: Mass-flow matrix
  • Definition 9: Cycle geometric mean
  • Definition 10: Cycle harmonic mean
  • ...and 19 more