A set of principles for applying Circular Economy to the PV industry: Modeling a closed-loop material cycle system for crystalline photovoltaic panels

AuthorAcceptedManuscript(AAM) Thismanuscriptistheauthor-acceptedversionofthearticle: "AsetofprinciplesforapplyingCircularEconomytothePVindustry:Modelingaclosed-loopmaterialcycle systemforcrystallinephotovoltaicpanels" DOI:10.1016/j.spc.2021.03.033 ThisAAMismadeavailableunderthelicense:CCBY-NC-ND4.0 1 A set of principles for applying Circular Economy to the PV industry: Modeling a ClosedLoop Material Cycle system for Crystalline Photovoltaic Panels Rubén Contreras Lisperguer1, E.Muñoz-Cerón2 and J.Aguilera3 *, J. de la Casa3 1 Natural Resources Division, Energy and Water Unit United Nations Economic Commission for Latin America and the Caribbean *Corresponding author E-mail: [email protected]; 2 Department of Graphic Engineering, Design and Project. University of Jaen. IDEA Research Group (Research and Development in Solar Energy) E-mail: [email protected] ∗Corresponding author 3 Electronics and Automation Engineering Department. University of Jaen IDEA Research Group (Research and Development in Solar Energy) E-mail: [email protected] E-mail: [email protected] Keywords: Photovoltaic; Cradle-to-cradle; Closed-loop-cycle; Upcycling; Recycling; Circular Economy Abstract The growing popularity of crystalline silicon photovoltaic (C-Si PV) panels will generate a massive amount of waste when these reach their end-of-life (EoL). To assure the promise of sustainable energy technology, an effective solar photovoltaic (PV) material recovery system must be implemented. In this paper, we assess a Circular Economy (CE) system based on Cradleto-Cradle (C2C) philosophy as an alternative to tackle the daunting challenge of this waste, and exploring its impacts in the form of a Closed-Loop Material Cycle (CLMC). The novel concept of circular time is introduced, and the material separation of C-Si PV materials is also discussed by theoretically introducing a novel framework. Further, a numerical simulation experiment has been performed in order to evaluate the performance of the proposed theoretical model, on a CLMC to assess the flow of silicon used in C-Si PV, aiding in future planning and logistics for a CE in the PV industry. The initial results obtained in the simulation show that it is not possible to close the material flow without introducing new raw material into the system unless the operational lifespan of the PV panels is reduced. This research seeks to fill a gap in CE theoretical literature by introducing a new framework and identifying some of the challenges and limits of implementing a CE system in the form of a CLMC system based on C2C principles (C2C-CLMC). 1. Introduction The world is undergoing a massive transition from a predominantly carbon-based energy system dependent on fossil fuels towards a low-carbon energy system dominated by renewable energy (RE) technologies. Solar photovoltaic (PV) panels are one of the most widely used RE technologies with a global record of 98 GW of installed capacity and connected to the grid in 2019(IRENA, 2020), representing approximately 23 percent of all globally installed RE power 2 Nomenclature A Helmholtz free energy 𝐴𝐴𝑖𝑖𝑜𝑜 Molar Helmholtz free energy 𝐴𝐴𝑝𝑝𝑖𝑖 𝑜𝑜 Molar Gibbs energy dNi Set of moles of i components 𝛿𝛿𝛿𝛿𝑏𝑏 Boundary work 𝛿𝛿𝛿𝛿𝑟𝑟𝑟𝑟𝑟𝑟 Reversible work 𝛿𝛿𝛿𝛿𝑠𝑠 Shaft work N𝑚𝑚𝑖𝑖 Total number of moles mixture N𝑝𝑝𝑖𝑖 Total number of moles pure components Am Total Helmholtz free energy P Pressure µpi Chemical potential pure materials Tc Circular time 𝑇𝑇 Temperature Tind Industrial time Td Dwelling time Trr Removal and recovery Trs Reservoir time 𝑅𝑅 Constant gas 𝛿𝛿𝑚𝑚 Minimum work separation random mix 𝜕𝜕 𝜕𝜕𝜕𝜕 Space derivative 𝜕𝜕 𝜕𝜕𝜕𝜕 Time derivative 𝜌𝜌� Average material density 𝑢𝑢�𝚤𝚤 Average processing velocity at 𝚤𝚤 𝜌𝜌 �𝑖𝑖 𝑛𝑛+1−𝜌𝜌 �𝑖𝑖 𝑛𝑛 ∆𝜕𝜕 Numerical approximation for 𝜕𝜕𝜌𝜌 � 𝜕𝜕𝜕𝜕 𝜌𝜌 �𝑖𝑖𝑛𝑛𝑢𝑢 �𝑖𝑖−𝜌𝜌 �𝑖𝑖−1 𝑛𝑛 𝑢𝑢 �𝑖𝑖−1 ∆𝜕𝜕 Numerical approximation for 𝑢𝑢�𝚤𝚤𝜕𝜕𝜌𝜌 � 𝜕𝜕𝜕𝜕 𝜌𝜌�2 𝑛𝑛+1 Joint point of the system 𝜉𝜉󰆻𝑛𝑛 Average processing rate of 𝜌𝜌� 𝜌𝜌�𝑖𝑖𝑢𝑢�𝑖𝑖 Flux of Si capacity in 2019 (IRENA, 2020). The global cumulative capacity installed until 2019 was about 635 GW (ISE, 2020) and it is projected to increase approximately to 1337 GW by 2025 (IEA, 2020). In this context, silicon (Si) wafer PV technology represents about 95 percent of the total PV technology production in 2019 (ISE, 2020). Renewable energy and solar PV technology particularly have been growing exponentially, mostly because of climate change and new policies adopted by many countries to cut carbon emissions, after the Paris Climate Accords in 2015. High demand on materials needed to manufacture renewable energy technologies is already compromised by dramatic rates of PV panels and other RE technologies production, constraining the possibility of achieving the clean energy system ambitioned by 2050(Moreau et al., 2019). The increasing demand for PV technology described above, with panels that have an average operational life of 20 to 30-years (Deng et al., 2019), will result in large amounts of waste when panels reach the end-of-life (EoL) phase, expecting around 78 Mt of solar PV waste by 2050 (Chowdhury et al., 2020). Moreover, C-Si PV panels are a highly competitive renewable energy technology that has rapidly evolving in the last decade, showing an exponential cost reduction with a fifteen times reduction cost only from 2008 to 2019 (Green, 2019) and increasing efficiencies from 5% in 1950 to the record lab efficiency of 26.7% in 2020 (Green et al., 2021). Furthermore, the global energy transition towards renewables comes with an increased demand for mineral production by around 200–900% from the electricity sector between 2015 to 2050 (Watari et al., 2019), highlighting the importance and the urgency for improving the material efficiency and management and move towards a circular paradigm, otherwise the energy transition may become unsustainable. 3 C-Si PV panel technology depends on the availability of valuable metals, including Si. For instance, demand estimations for Si in Europe will grow on a high-PV deployment scenario, from 42,854 t in 2020 to 234,962 t in 2030 (European Commission, 2018). Hence, improving Si’s efficient management throughout its EoL may help to ensure its supply to the PV industry. Moreover, the Waste Electrical and Electronic Equipment (WEEE) Directive 2012/19/EU has signaled a way in that sense, and has pushed for the development of PV recycling facilities in the EU (EC, 2012). Researchers have already recognized the importance and need of a comprehensive recycling system (i.e. logistics, recycling, storage, and sustainable waste treatment) for all materials within PV modules (Xu et al., 2018), achieving economic feasibility and reducing environmental impacts (Deng et al., 2019), while promoting a PV closed material cycle may be able to improve resource efficiency and waste reduction (Sica et al., 2018). Moreover, the manufacturing of a crystalline solar cell entails several steps (Woodhouse et al., 2019), where, after a multistage distillation process (Siemens process), the polysilicon is transformed into cylindrical ingots of monocrystalline Si, which are then shaped and sliced into thin wafers (Ferroni and Hopkirk, 2016). Hence, the high-purity and energy-intensive Si used in the PV industry is a valuable material (Deng et al., 2019), and this is why the main focus of our research is about the EoL of the Si confined in a crystalline PV panel. Contreras-Lisperguer highlighted the importance to move from recycling, that has become a key stage at the EoL of the linear paradigm Cradle-to-Grave (C2G), where C2G is a linear lifetime for PV panels, from the extraction of material used in manufacturing them (cradle) until the EoL, when they are considered waste (grave) and only a fraction of the materials reach the recycling phase, towards up-cycling based on the circular paradigm Cradle-to-Cradle (C2C), where C2C implies a circular system, where all materials are used indefinitely and at EoL they can become primary resource to manufacture another panel (up-cycling) (Contreras-Lisperguer et al., 2017), more details about C2C and C2G principles are available at the Appendix A. Consequently, recycling is not an appropriate solution to solve the waste accumulation and environmental challenges ahead since recycled materials were already considered waste, therefore, this option is conceptually flawed from the outset. Furtheremore, Non-RenewableAbiotic-Primary-Resources (NRAPR) have a lengthened natural renewal cycle and are viewed as finite (e.g. minerals can takes millions of years) (EEA, 2005), therefore, it is important to minimize our reliance on limited resources by recovering them at the EoL. Here, NRAPR has been defined as natural sources (including energy sources), such as minerals, metals, iron ore and 4 crude oil, which are considered stock in the environment that is “depleted through physical extraction of resources (mining and subsequent processing) from the environment into the economy” (Schulze et al., 2020). CE, particularly in the form of a Closed-Loop Material Cycle (CLMC), has the potential to reduce the impacts and pressure over these resources, but also make a better use of it and the key role of redesign and design for recovering and recycling in a sustainable CE system (Velenturf and Purnell, 2021). However, given the complexity of creating, and implementing a C2C-CLMC system, together with the impracticability of conducting experiments on the supply chain and logistic systems at global or regional scale, it is required the development of a credible scientific framework for system modeling, based on thermodynamics and mathematics, to stimulate research, and recognition in the academic community. In this article, we introduce a new theoretical framework that stably simulate and forecast the material flow of Si at a global/hemispheric scale for a C2C-CLMC system. We hope our study leads to a wider use of inductive methods in this transition from a linear to a circular economical system. The article is divided as follows. First, in section 2 a literature review is presented providing a context for our research. Then, at section 3 the methodologies used in this research are introduced In section 3.1, the energetic and thermodynamic implications of material separation in a system are introduced. In section 3.2 we propose a new procedure to comprehend material separation for a C2C-product. Afterward, at section 3.3 we address the challenge of time and space scales interactions by defining a suitable one and at section 3.4 the concept of circular time (Tc) and its components are introduced. A description of Dwelling time (Td) is provided in the Appendix B. Dwelling time estimation. The mathematical part at section 3.5 is devoted to contemplating the right set of equations that described a C2C-CLMC system. A detailed description of the conservation of mass equation is provided in Appendix C, and the stability analysis of our model is provided in Appendix D. Then, in section 3.6 we incorporate a numerical simulation to test our model at hemispheric scale. Finally, the preliminary assumptions and boundary conditions are introduced at section 3.7, while a detailed calculation of Td for crystalline silicon photovoltaic PV panel is available in Appendix E. In section 4, a discussion of the numerical simulation test results is included. As an alternative to the classic sensitivity analysis a brief note using a simplified ensemble analysis on the variations of Td is available in the section 4.1. Finally, conclusions, and recommendations are presented in the final section 5. 5 2. Literature review Today’s PV industry is consuming Earth’s NRAPR generating massive amounts of waste at the EoL of manufactured PV panels. This waste provokes several wide-range environmental impacts: eutrophication, mineral depletion, loss of biodiversity, and CO2 emissions (Contreras Lisperguer et al., 2020). Consequently, sustainable recycling methodologies for C-Si PV panels are essential to move towards a sustainable Circular Economy (CE). Briefly, a C-Si panel structure is composed of many electrically connected solar cells with Ag contacts on the front and back, coated with an ethylene vinyl acetate (EVA) layer, which acts as electrical insulation, providing to the solar cell with protection from the environment and increasing its lifespan, then, this structure is heated until is sealed, then it is covered by glass and finally this laminated structure is encased in an aluminum frame (Gorjian and Ashish, 2020). This kind of sandwich structure, and particularly the use of EVA poses a major challenge when it comes to recovering and recycling the materials used in the C-Si PV panel manufacture because it can be removed only by thermal, chemical process and a combination of both (Farrell et al., 2020). Thus, the generation and accumulation of waste coming from PV panels and other RE technologies is set to become a global challenge in the decades to come. A typical mass composition of 1000 kg of C-Si PV panel material is approximately 70% glass, 18.5% aluminum, 5.1% EVA, 3.65% Silicon, 1.5% Back foil sheet, 0.11% Cooper, 0.053% Silver, 0.035% Led, and 0.018% Tin (Contreras Lisperguer et al., 2020). Recovering and recycling all those materials embedded in a panel may have different alternatives pathways that can be categorized into physical, thermal, and chemical treatments and/or a combination of those treatments (Chowdhury et al., 2020). Thus, the efficiency of the recycling process at EoL will dependent on the effectiveness of the chosen treatment for recycling, and this is specially challenging due to the “sandwich” enclosed nature of the C-Si PV panel structure (Farrell et al., 2020). However, despite the importance of recycling methodologies at the EoL, especially for a CE system, there is very limited data available (Ardente et al., 2019), nonetheless, the recycling technologies are still in infancy, and many PV panel recycling facilities are still in their pilot stage and very little is known about their environmental impacts (Contreras Lisperguer et al., 2020). The physical treatment is a mechanical process that usually considered crushing and grinding after the aluminum frame is manually removed along with cables and electric box (Chowdhury et al., 2020). Hence, the physical treatment is rather limited for the recovery of valuable materials, allowing only the recovery of primarily (Padoan et al., 2019). The thermal treatment uses high temperatures to remove the EVA to access the PV cell (Contreras Lisperguer 6 et al., 2020). This treatment is typically carried out by pyrolysis, nonetheless, this process is highly energy-intensive limiting it use from an economic and environmentally sustainable point of view (Padoan et al., 2019). The chemical treatment entails another option to remove the EVA and to recover valuable metals. This approach usually requires the use of different organic solvents to remove the EVA (Chowdhury et al., 2020). Nevertheless, most of the recycling pathways treatments for C-Si PV panels are a combination of the physical, thermal, and chemical treatments briefly described above (Padoan et al., 2019). Moreover, it is essential to assess the related environmental impacts of all the different treatments pathways, and the Life Cycle Assessment (LCA) methodology is considered to be an effective tool to assess such impacts generated during the recycling the C-Si PV panels used by researchers and academia (Ardente et al., 2019). In a recent study (Contreras Lisperguer et al., 2020), we investigated the alternatives recycling pathways at EoL and its environmental impacts, furtheremore, we provided a thorough estimation of the environmental impacts and potential alternatives to reduce the environmental impacts of the recycling process. We highlighted the potential benefits of a C2C-CLMC scenario as a way to move towards a CE, however, it was highlighted the urgency to minimize impacts coming from current energy-intensive thermal treatments and toxic chemical treatments, which are mainly generated due to the designed enclosed sandwich nature of current silicon PV modules. For instance, in terms of environmental impacts, the recovery of PV materials in a C2C-CLMC scenario results in reduction of the Climate Change impact factor (kg CO2 eq) by 74%, compared with an Open scenario (Contreras Lisperguer et al., 2020). However, it is necessary to consider new design pathways that facilitate disassembly, recovery of materials and the entire recycling process supported by improved standards and regulations. Other studies using LCA models along with material flow analysis assessed the efficiency related to the recycling process for c-Si PV waste. Its results illustrate the material flows for PV recycling process, based on the average practices in Western European WEEE recycling plants that the overall recycling rate achieved by such is around 24%, which is well below the current minimum target of 80% (in mass) of reuse and recycling set by the WEEE Directive (Ardente et al., 2019). While a research project achieved an estimated recycling rate of 83%, using an novel recycling process for c-Si PV panels (Ardente et al., 2019). The importance to increase the recycling performance for C-Si PV panels at the commercial level is paramount to successfully implement a C2C-CLMC into the PV industry. This becomes even more important when considering that recent research indicates that, for 7 example, the highest amount of PV waste in Australia will be crystalline silicon-based PV panel, reaching around 80% by 2047 (Mahmoudi et al., 2019), while for the OECD countries it is estimated about 29 million tons PV waste by 2058 (Mahmoudi et al., 2021). The potential today to upcycle C-Si PV in a C2C-CLMC system is still limited because PV panels have not been designed to be recovered and reused at EoL (Contreras-Lisperguer et al., 2017), however, silicon-based PV panels are manufactured mostly with highly pure (i.e. Si) and valuable materials (e.g. copper, aluminum, silver, etc.) (Chowdhury et al., 2020), therefore, the potential reuse of those materials can be technically feasible, however, there is very limited research conducted on the resource efficiency in the recycling of such valuable metals (Ardente et al., 2019). Furthermore, in a C2C-CLMC system such as a CE is key to weigh the physical availability of all materials, particularly for recycling, recovery, and re-use at EoL, therefore, modeling material cycles and flows are important tools for resource management and sustainability (Balanay and Halog, 2018), but also key to achieve resource efficiency and to reduce competition on access to raw materials and its supply (Penaherrera and Pehlken, 2020). However, methodologies such as material flow analysis, and life cycle assessment are inefficient to foreseeing the temporal variations of material stocks (Penaherrera and Pehlken, 2020). On the other hand, a growing body of literature examines the use of dynamics models based on partial differential equations (PDE). These models have mainly been used to describe material flow in manufacturing facilities (Glatt et al., 2018), supply chains models (Yuan et al., 2020), game theory approach (Chen et al., 2021), and optimal control of supply chains models (Rarità et al., 2021), while other studies provides some generic considerations for modeling and simulation for an end-to-end supply chain system (Chilmon and Tipi, 2020) or the theoretical stability of the conservation laws based on PDE in numerical simulations (Chiarello et al., 2019). Previous work in material flow using dynamics models based on PDE indicated above has mostly focused on modeling material flow at the theoretical level in small facilities at local space-scales (i.e. factories, conveyor belts, manufacturing processes) and at local time-scales (e.g. seconds, hours and days of some manufacturing process). Furthermore, the latest research available on the study of closed-loop systems has mostly focused on the use of mathematical-statistics, such as multi-objective evolutionary algorithm (Cheraghalipour et al., 2018), optimization models (Accorsi et al., 2020), network models (Fu et al., 2021), and game theory (Shekarian and Flapper, 2021). Despite the interest, no one to the best of our knowledge has studied the topic of dynamics simulations using PDE of material flow for products with a long useful life on a large-scale (i.e. 8 thousand kilometers), like PV panels. We have yet to consider a holistic analysis of the thermodynamic implications in material separations under the C2C framework. Incapability to perform experiments with real-time material flow in the real world warrants a need to develop a deeper understanding of this issue. In this scenario, material flow simulation methods are essential for the understanding and implementation of a complex C2CCLMC system at a global scale. Further, such simulations are also needed for strategic decisionmaking purposes, since material flow patterns provide vital information for logistics, planning, and material control. Modeling systems can help to improve the knowledge, without disturbing the PV industry, about the limitations and implications of applying new paradigms such as CE in our industrialeconomic systems before implementing them (e.g. measurements, storage, balance of the material, time, and space scales, etc.). A proper transition from a linear to a circular economical system requires developing a proper theoretical model based on mathematics and thermodynamics (Contreras-Lisperguer et al., 2017). A first step towards the implementation of a C2C-CLMC system involves laying out the necessary theoretical framework to do so, therefore, a sound C2C-CLMC system design requires first a robust theory based on mathematic and thermodynamic principles little is known about the implications of a C2C-CLMC system at a global/hemispheric. 3. Methods It is expected that if well-designed under C2C principles, a C-Si PV panel at EoL will contain high-quality primary resources. The C2C approach promotes a circular industrial symbiosis based on the central equation “waste = food”, whereby all materials are used indefinitely (Braungart et al., 2007). A CLMC system based on the C2C philosophy (C2C-CLMC) promises to reduce or even eliminate the very concept of waste. Toxic residues would also be completely replaced by nontoxic alternatives and/or effectively encapsulated to prevent any interaction with the environment if the manufacturing sector can implement proper design and values based on a consistent theory. Despite the clearness and optimism behind that assertion, there are serious obstacles other than just planning the implementation of a Closed-Loop supply chain for PV that includes recycling. A fundamental question still needs to be answered. Do we actually know the implications of the laws governing the material flow in a C2C-CLMC system principles? This question can be answered only by deducing consequences from hypothesis and subjecting them to experimental verification. Since it is practically impossible to experiment with the material flow in a supply 15 𝜌𝜌� 𝑖𝑖 𝑛𝑛+1−𝜌𝜌� 𝑖𝑖 𝑛𝑛 ∆𝜕𝜕 + ℱ 𝑖𝑖 𝑛𝑛−ℱ 𝑖𝑖−1 𝑛𝑛 ∆𝜕𝜕 = 0 (16) In equation (16) x and t are discretized with uniform values of ∆𝜕𝜕 and ∆𝜕𝜕, respectively. For brevity, we use the notation e.g. 𝜌𝜌�𝑖𝑖𝑛𝑛= 𝜌𝜌� (𝑖𝑖∆𝜕𝜕,𝑙𝑙∆𝜕𝜕) with 𝑖𝑖,𝑙𝑙 ∈ {0,1,2, … . } If we say 𝛽𝛽=∆𝜕𝜕 ∆𝜕𝜕, then we can rewrite (16) as 𝜌𝜌� 𝑖𝑖 𝑛𝑛+1=𝜌𝜌� 𝑖𝑖 𝑛𝑛−𝛽𝛽(ℱ 𝑖𝑖 𝑛𝑛−ℱ 𝑖𝑖−1 𝑛𝑛) (17) The resulting numerical scheme (17) serves to model the flow of an imaginary and arbitrary volume of Si in a C2C-CLMC system. However, in order to complete our numerical scheme, we must close the material flow. The numerical solution proposed to close the loop for our C2C-CLMC model has been solved by introducing a joint point. The joint point represents in the numerical scheme the transportation process of the recovered materials from the reservoir to the industrial facility for reuse. Therefore, the strategy here is to connect the EoL phase with the recovered material processing facility in terms of material flow. The joint point has the following form 𝜌𝜌� 2 𝑛𝑛+1=𝜌𝜌� 2 𝑛𝑛−𝛽𝛽[𝜌𝜌� 2 𝑛𝑛𝑢𝑢� 2 −(𝜌𝜌� 1 𝑛𝑛𝑢𝑢� 1 +𝜌𝜌� 𝑟𝑟𝑛𝑛𝑖𝑖 𝑛𝑛 𝑢𝑢� 𝑟𝑟𝑛𝑛𝑖𝑖 )] (18) Equations (17) and (18) are the numerical solution of the governing equation of our C2CCLMC system. Its solution may be found by numerically solving for 𝜕𝜕𝜌𝜌 � 𝜕𝜕𝜕𝜕 and extrapolating the motion forward in time of the material flow, but the initial and boundary conditions must first be established. In the following section we introduce the initial and a numerical experiment that will help to test the reliability and stability of our model at a global/hemispheric performance scale. 3.6 Numerical simulation tests In order to assess the stability and convergency of our proposed numerical method described in the previous section, we have introduced assumptions made to test our model in the form of a numerical experiment. However, it would be very demanding to simulate and render the material flow and processing facilities of all C-Si PV panels produced in the world using high-resolution three-dimensional structural data, and appropriate machine representations. This is due to the limited computing power that will require such modeling, therefore, the boundary conditions and initial assumptions imposed in this numerical test are made in order to facilitate the processing of the numerical discretization solution and numerical calculations. The simplified assumptions and boundary conditions will give us the possibility to confirm if the proposed model is stable and its solutions converges for a complex large-scale C2C-CLMC 16 system, aiming to verify if the system is able to close the material cycle and to assess how long it would take to do so. In this simplified numerical test, performed in one-dimension (1D) for a C2C-CLMC system, we have simulated a case in which PV material (Si) is added into a closed system, in a single year, for any given single year, in order to simulate its entire life-cycle of material flow for Si, from the input of material extracted from a mining process into the manufacturing facility, PV manufacturing, use, decommissioning, recovering, reservoir, and reuse as a raw material in the manufacturing sector (upcycling) in a C2C-CLMC system . Furthermore, the PV panels adhere to the thermodynamic implications of material separation in a C2C-product introduced in section 3.2. This numerical experiment was implemented with a computer running MATLAB® (The MathWorks Inc., 2013). Figure 1: A simplified closed-loop material cycle system (Adapted from Contreras-Lisperguer et al. 2017) 3.7 Preliminary assumptions and boundary conditions In this section we have established the conditions that define the main characteristics of the C2C-CLMC system for this numerical experiment. The boundary conditions are key to the functionality of this model. Since in a C2C-CLMC system we are dealing with boundaries that are not naturally given by geometric motion of the volume of Si and it is not immediately obvious what these conditions are. Actual boundary conditions are more subtle. First, let R be region of the planet bounded by a closed curve with finite length (i.e. rectifiable curve) RC. If ∆𝜕𝜕 is known in R and x is given on RC, then the boundary conditions are prescribed in such a way that x is always known on RC, and ∆𝜕𝜕 is known in R, the solution of equation (16) (see section 3) for the region R will be possible to determine. Therefore, since x is a function of time in RC, velocity is known in the C2C-CLMC system. Consequently, the distribution of 𝜌𝜌�(𝜕𝜕,𝜕𝜕) after a ∂t will be 17 known everywhere in R. Because of this, we may assert that the flow of Si is determined by the specification of x on the boundary, and the 𝜌𝜌� on that part of the boundary where the flow of Si is initially entering the interior region of R. Here, it is important to define the individual averaged units of 𝜌𝜌� through defining the quantities in the system. Density (𝜌𝜌�𝑖𝑖) is defined as the averaged units of 𝜌𝜌� per length at 𝜕𝜕𝑖𝑖 and the flux (𝜌𝜌�𝑖𝑖𝑢𝑢�𝑖𝑖) is defined as averaged units of 𝜌𝜌� per time at 𝜕𝜕𝑖𝑖. To facilitate the numerical test some preliminary assumptions and boundary conditions have been imposed. This approach reduces computational burden of the simulation, time and work efforts for large-scale system modeling, allowing a better performance of the entire numerical test. The preliminary assumptions and boundary conditions are introduced at Table 1. Table 1 Summary of the assumptions and boundary conditions imposed in the numerical simulation Assumptions and boundary conditions 1 In order to evaluate numerically the density flow of Si in a C2C-CLMC system at hemispheric scale, we have defined the following initial conditions: to reduce computational cost, we are only considering Si used in PV panels as a single NRAPR in 1-D. 2 All nodes are in a continuous processing system (24/7) with a 100% efficiency for each node 3 The manufacturing plant and distributors are subsidiaries of the same business group. The recycling yield is imposed at 99% 4 The PV panels are all manufactured according to C2C philosophy sufficient for the highest possible C2C certification and based on the elementary principles introduced in section 3.2 to facilitate the disassembly phase. Hence, the PV panels in question were designed to eventually be upcycled. 5 All returned Si is accepted for inventory either actively (moving directly back to the manufacturing process after the EoL) or passively (at the reservoir, waiting to return to the manufacturing plant –depending on demand). Figure 1 shows the corresponding nodes. 6 The entire PV panel supply chain system is a vertically integrated and automated manufacturing facility. Since this numerical experiment is to test, for numerical simplicity we have imposed an input of 180,000 C-Si PV panels/250 W, assuming a 4.31 g of Si per watt (Castellanos et al., 2018). 7 For the sake of simplicity, a maximal density function for Reservoir has been defined given by ∑𝜌𝜌� 𝑖𝑖 (𝜕𝜕) [0, 𝜕𝜕]= 𝑛𝑛 𝑖𝑖 𝜕𝜕 [0, max{𝜌𝜌�𝑖𝑖}], which is maximum capacity allowed at the reservoir. 8 The length (L) of the C2C-CLMC system for a hemispheric scale has a typical value of about L=4000 km (ContrerasLisperguer et al., 2017). For this numerical experiment the distance between each node is assumed to be 1000 km. 9 The initial boundary condition (IBC) for Si density is defined when the system is initially empty at t=0 on the node-1, therefore, 𝜌𝜌� (x,0)=0. 10 An averaged unit of 𝜌𝜌� is has been imposed as 1.5 kg of silicon per PV panel at 𝜕𝜕 𝑖𝑖 . 11 To minimize mathematical complexity, we have assumed the curvature of Earth neglected. 12 In this simulation we have used a suitable discretization with a uniform grid (𝑖𝑖∆𝜕𝜕,𝑙𝑙∆𝜕𝜕) with 𝑖𝑖∆𝜕𝜕=10 𝑘𝑘𝑘𝑘 and with a total 404 space discretization points (101 per node). Thus, we have calculated the numerical stability of our model based on the selected ∆𝜕𝜕 and ∆𝜕𝜕. The calculated stability condition is 𝑢𝑢�𝑖𝑖Δ𝜕𝜕 Δ𝜕𝜕= 0.0003. This satisfies the Von Neumann stability requirement described in Appendix D. 13 We consider a finite number of 𝜌𝜌� of Si arriving in the system at 𝜌𝜌�(0, 𝜕𝜕) for t >0. Therefore, an arbitrary initial density inflow of Si entered the system. For this experiment a random initial input of 𝜌𝜌� was chosen, initially 5 units of 𝜌𝜌� entered the system, from a total of 180,000 units of 𝜌𝜌� that entered the system at an average rate of 1000 unit of 𝜌𝜌� per time step of the grid Δ𝜕𝜕=0.0003∙Δ𝜕𝜕 𝑢𝑢 �𝑖𝑖 (Δ𝜕𝜕 is estimated based on stability criteria described in Appendix D) . 14 The flux of Si is a function of x and t and is determined by the average density (𝜌𝜌�). Therefore, the flux can be defined as 𝜌𝜌�𝑖𝑖𝑢𝑢�𝑖𝑖, where 𝜌𝜌�𝑖𝑖 is known and 𝑢𝑢�𝑖𝑖 is a function of 𝜌𝜌� and represents the averaged velocity of Si particles, and i represents the cell within the computational domain 𝜖𝜖{1,2,...,i} for the periodic domain 𝜕𝜕∈[0, 𝐿𝐿], where L represents the total length of the system. Consequently, the average velocity vector can be defined as 𝑢𝑢�𝑖𝑖=𝑢𝑢�(𝜌𝜌 �𝑖𝑖)+𝑢𝑢�(𝜌𝜌 �𝑖𝑖+1). Here, 𝜌𝜌� propagates to the right on x. 15 The processing rate of averaged units of 𝜌𝜌� at each node (𝜉𝜉󰆻 𝑛𝑛 ) of the system are: 𝜉𝜉󰆻 1 ~1 · 10−2; 𝜉𝜉󰆻 2 ~1 · 10−9; 𝜉𝜉󰆻 3 ~18 ·10−3; 𝜉𝜉󰆻4~1 · 10−2 units of 𝜌𝜌� per second. 18 The processing rate of averaged units of 𝜌𝜌� at each node (𝜉𝜉󰆻𝑛𝑛) of the system (nodes 1, 2, 3, and 4 as shown in Figure 1), is represented as a constant rate that is specific for each node since the processing rate is highly dependent on the technology available at each node. Therefore, in this experiment, due to the high uncertainty of processing rates at each node a heuristic argument is used to show that the following is probably correct: where the unit of 𝜌𝜌� is entering the area enclosed by the node, the 𝜉𝜉󰆻𝑛𝑛 must be imposed. The motion at every node is determined by the specification of 𝜉𝜉󰆻𝑛𝑛 everywhere on the boundary at which the volume of Si is entering the node. Based on the time scale already defined in this article (Tc), we have then estimated the order of magnitude for the constant processing rate for each node. The imposed values for 𝜉𝜉󰆻𝑛𝑛 are the following: o For Node#1 and based on the definition of Tc, we have estimated Tind based on our imposed manufacturing rate of 180,000 in one arbitrary year, we have estimated a value of 𝜉𝜉󰆻1~1 · 10−2 units of 𝜌𝜌� per second. o For Node#3, based on the data available from the Full Recovery End of Life Photovoltaic (FRELP) project (FRELP Project, 2018), it is estimated that in a recycling PV plant about 38 kg of Si can be recovered per hour. Since 𝜌𝜌� is known, we estimated the order of magnitude as 𝜉𝜉󰆻3 ~18 ·10−3 units of 𝜌𝜌� per second. o For Node#4, there is no ‘reservoir’ or similar infrastructure to extrapolate information or empirical evidence, therefore, we have estimated the magnitude of 𝜉𝜉󰆻4 assuming that the magnitude of demand for recovered Si from the manufacturing solar and electronic industry is similar to the one from raw Si. We imposed the same order of magnitude at Node#1. Accordingly, 𝜉𝜉󰆻4~1 · 10−2 units of 𝜌𝜌� per second. o For Node#2, we have estimated Td. The estimation of Td is described at Appendix E. 4. Results and discussion Given that for this numerical experiment Td is approximately 30 years, as an initial test we performed a simulation of this C2C-CLMC system for 35 years (~1.1·109 s), illustrating a complete life-cycle of Si-flow described in Figure 1. The goal of this simulation was to confirm that after 35 years the initial input of units of 𝜌𝜌� returned to a manufacturing facility as raw material, fulfilling its purpose as a C2C-CLMC system. The resulting simulated profiles describing the flow of units of 𝜌𝜌� in percentage per time for each node for this C2C-CLMC system are illustrated in Figure 2. 19 Figure 2a Figure 2b Figure 2c Figure 2d Figure 2 Time series of the density percentage at each node from t=0 to t=35 years (on the top horizontal axis of the graph) and in seconds as standard international unit of time (SI) is on the bottom horizontal axis of the graph (𝜌𝜌�= average material density, s=seconds, yr=years). As shown in Figure 1, the flow of units of 𝜌𝜌� are assumed to be 1-D flow. This data is useful for studying the time response of the material flow at each node (Figure 2 - a, b, c and d) based on the conditions imposed in the numerical experiment. First, the raw material (initial input of Si units of 𝜌𝜌�) entered Node-1, where units of 𝜌𝜌� are used to manufacture solar cells and become an integral part of a PV panel. As expected, the percentage of units of 𝜌𝜌� decreased with time as manufactured PV panels left the manufacturing facility (see Figure 2a) to be used in a power PV plant at Node-2. Then, at Node-2 the units of 𝜌𝜌� increased up to the maximum 𝜌𝜌� because they will be operational for about 30 years in a single place generating electricity at the PV plant (see 20 Figure 2b). Once the PV panel ends its operative life, the PV are decommissioned and sent to a recycling facility (Node-3), where Si units of 𝜌𝜌� are recovered as raw material (see Figure 2c). The recovered material is stored in the “reservoir” (Node-4) and transported to the Node-1 as raw material (see Figure 2d). Based on data interpretation of Figure 2a, it is possible that the flow of Si units of 𝜌𝜌� was not able to return in its integrity to the manufacturing facility after the end of its operational life. We can see in Figure 2a that at approximately 1·109 s, the system had retrieved less than 10% of the total Si units of 𝜌𝜌� initially inserted in the system (see Figure 2a). Here, we can conclude that the system was unable to close the material loop after one life cycle of 35 years. Our initial hypothesis to explain this conundrum is to assume that the maximum 𝜌𝜌� at Node-2 (see Figure 2b) represents a critical convergence of units of 𝜌𝜌� attributed to a large Td. This triggered a shock wave that acted as a boundary between the high density observed at Node-2 and the low density observed at Node-3. The phenomenon is assumed to impact throughout the Node 3 and 4. This shock wave is reflected in a temporal variation of percentage of the units of 𝜌𝜌� between Node 2 (high percentage of units of 𝜌𝜌�) and 3 (low percentage of units of 𝜌𝜌�), therefore, temporal variation of units of 𝜌𝜌� requires spatial variation in the velocity 𝑢𝑢�𝑖𝑖, since continuity infers divergence of 𝑢𝑢�𝑖𝑖 must be 0. Moreover, due to the continuous nature of the PV material flow, the shock wave can be identified because of the change in densities between the nodes that describes the boundary between two PV material flow states that are characterized by different densities, and flow rates. Therefore, it is possible to conceive that the shock wave moves in time and space disturbing the entire system, and limiting the flow of units of 𝜌𝜌� to the Node-3, and restricting the capacity of the system to store Si units of 𝜌𝜌� at the reservoir in Node-4. Consequently, the PV material flow was unable to close at Node-1. Another interesting feature observed in Figure 2c and 2d is the “quasi-symmetry” in the distribution of 𝜌𝜌�. However, we consider this a trivial symmetry due to the restriction imposed by the shock wave and the geometrical symmetry of the system modeled here. Additionally, the plots have different percentage of units of 𝜌𝜌� (see Figure 2c and 2d), which leads us to believe this similarity can only be attributed to this test. Applicability is clear from the previous results of our model since it provides a potentially useful approach to material flow simulation at hemispheric scale. Initial results have led us to question if the system will be able to close the material loop at any time in the future without introducing more new raw material into the system. In order to answer this question, and as a means of verifying initial findings and the consistency of the model, we ran a second simulation 21 with the same conditions but for three cycles totalizing 110 years (~3.5·109 s). These results are presented in Figure 3. Figure 3a Figure 3b Figure 3c Figure 3d Figure 3 Time series of the density percentage at each node from t=0 to t=110 years (on the top horizontal axis of the graph) and in seconds as standard international unit of time (SI) on the bottom horizontal axis of the graph (𝜌𝜌�= average material density, s=seconds, yr=years). The results observed in Figure 3 show the consistency of the model. The calculated flow of 𝜌𝜌� at each node (Figure 3 – a, b, c and d) appear consistent with the previous results presented in Figure 2. Once the initial input of raw material (Si units of 𝜌𝜌�) entered Node-1 and when PV panels leave the manufacturing site, the percentage of units of 𝜌𝜌� decreased to zero, detailed in Figure 3a. At ~1.6·109 s (~45 years) it is possible to observe an input of material, around 25% of the total 22 returned to the manufacturing facility (Node-1) as raw material. Finally, at ~2.6·109 s (~80 years) the units of 𝜌𝜌� are constantly returning to Node-1, reaching the maximum return around ~2.7·109 s (~85 years). Like in the previous simulation, Node-2, 𝜌𝜌� increased to a maximum accumulation between ~0.2·109 s (~5 years) and ~1·109 s (~30 years), and then descend (see Figure 3b). The high convergence of units supports our initial hypothesis confirms about the impact of Td on a closed system (Td ~29.949 years in a fixed location generating electricity). Thus, we initially suggest that a large Td creates an impact in the form of a shock wave that disturbs the C2C-CLMC system, causing it to reuse material ineffectively at EoL. This shock wave can be thought of analogously as a traffic jam, but instead of a red light or an accident that could last a couple of hours, PV panels are installed in a single place for 20-30 years or more with the sole purpose of generating electricity, causing an inertia that affects the entire closed system that can be seen in the results of our numerical experiments. Continuing analysis of Node-2, we observe that at ~1·109 s (~30 years, EoL for PV panel), the percentage of units of 𝜌𝜌� start diminishing due to the decommissioning process of PV panels. Following the trend observed in Node-3 (see Figure 3c), we estimate that at ~1·109 s (~30 years) the system was able to absorb around 25% of PV panels from Node-2, which is coherent with the amount of material reused at Node-1 (see Figure 3a). Finally, at 2.8·109 s (~90 years) the system was able to reuse the total of units of 𝜌𝜌� (Si material), gain stability, and close the material loop (see Figure 3a). For Nodes 3 and 4, we observed again a “quasi-symmetry” in the distribution of 𝜌𝜌�. However, the plots at Figures 3c and 3d are fairly different with a difference of several orders of magnitude, confirming our initial hypothesis. As was shown in the results of this numerical experiment, the temporal evolution of 𝜌𝜌� at each node initially suggests that a C2C-CLMC system at a global scale can be considered a system in steady-state, where the velocity of the material flow out of the Node regions can be faster than in it, which may be the result of high momentum flow. However, the small momentum flow is also transported by the material flow once it is out of the Nodes. This is an important attribute of a C2C-CLMC system, because recycling and upcycling cannot be instantaneous. Therefore, Node-4 plays a key role in helping the system reach stability after the shock wave triggered at Node-2, especially in cases where the recycling capabilities at Node-3 are limited. Consequently, Node-4 can provide a buffer capacity for the system, especially when non-pure or low-recyclable materials (e.g. highly random mixtures materials) are present in the system. 23 From the above results, some initial findings are drawn: 1. The numerical scheme used proves to be effective. The simulation is stable at Td and able to provide a convergent solution for this simulation of Si flow during its entire lifecycle at hemispheric scale. 2. The model was able to effectively predict 𝜕𝜕𝜌𝜌 � 𝜕𝜕𝜕𝜕 , the direction and magnitude of the material flow. 3. The model can reliably predict and identify a shock wave that affects the entire system. We initially suggest that Td is the main culprit of this phenomenon. 4. Results show that the density (averaged units of 𝜌𝜌� per length at 𝜕𝜕𝑖𝑖) behind the shock wave is different from the density ahead of the shock wave. This is why particles have enough time to experience delay under a steady process rate. Further research is needed to learn more about the shock wave propagation during material flow and the role of Node-4 (reservoir) in shock wave attenuation effects. 5. Although our simulations were originally implemented to evaluate the functionality and stability of our model, we initially believe our results can be considered a promising aspect for the implementation of a CLMC as a form of a CE system, contributing to the protection of the environment (Sauvé et al., 2016). Further analysis has to be done in terms of a real-time capable simulation. 6. The stability of the model and the successful numerical experiments performed show the encouraging performance of our model in capturing the dynamics of material flow at hemispheric scale. Promising initial results indicate the potential applicability of the proposed model to other complex physical systems at hemispheric and global scale at Td. Due to the non-linear nature of a CLMC system, as an alternative to the classic sensitivity analysis, a brief note using ensembles on the variations of Td is included in the following section. 4.1 Simplified ensemble analysis Though the uncertain aspects of our numerical model and its impact on a particular given set of assumptions have not been directly covered and are beyond the scope of this paper, our study does highlight the impact of Td on the entire system. A C2C-CLMC system is a non-linear system described by a set of differential equations. Since at least two first order systems exist in the series, two or more processes are equivalent to a second order system (Ingham et al., 2007). Therefore, our results agree with a higher-order 24 nonlinear system. Changing the parameters within this specific system of operations has little impact on the results of the system (Forrester, 1987; Tjøstheim, 2012). So, for high-order nonlinear systems, traditional linear sensitivity analysis may provide incorrect estimates (Saltelli, 1999). Instead, we try to understand how a given uncertainty in Td propagates throughout time in this dynamic system. In this research, we approach this question using the ensemble model framework. Ensemble model is an interesting statistical method that emerged in the last decade. It is used in many fields, such as: petroleum reservoir modeling (Jahanbakhshi et al., 2018), ocean dynamics (Nadiga et al., 2013), atmospheric science (Baran and Lerch, 2018), and hydrology (Seo et al., 2014), to name a few. In this section, we use a simplified ensemble analysis to assess the sensitivity of a C2CCLMC system at Td. An ensemble model was constructed by running formulations of our model with different values for Td (Fuentes and Foley, 2013). For this ensemble experiment, we ran a number of simulations for different Td [5, 10,15,20,25,30] (years) at Node-1, where Td=30 is our initial base-case scenario, then the ensemble results are compared to each other for Td=5,10,15,20,25. Figure 4 shows the results of the structure of the different profiles of 𝜌𝜌� with time for each Td. Figure 4: Ensemble results for different dwelling time (Td) (5, 10, 15, 20, 25, and 30 years), (𝝆𝝆 �=average material density, s=seconds, yr=years). 31 Xu, Y., Li, J., Tan, Q., Peters, A.L., Yang, C., 2018. Global status of recycling waste solar panels: A review. Waste Manag. https://doi.org/10.1016/j.wasman.2018.01.036 Yuan, H., Bi, Y., Fu, H.C., Lam, A., 2020. Stability analysis of supply chain in evolutionary game based on stability theory of nonlinear differential equation. Alexandria Eng. J. 59, 2331–2337. https://doi.org/10.1016/j.aej.2020.02.025 Appendices A. Linear versus Circular: Cradle-to-Grave and Cradle-to-Cradle Cradle-to-Grave Cradle-to-Grave (C2G) is the current prevailing paradigm in the manufacturing sector and the entire supply chain system. The solar photovoltaic (PV) sector promotes inefficient recovery of residues and waste at End-of-Life (EoL) due to the linear lifetime of crystalline silicon photovoltaic (C-Si PV) panels. Since the extraction of material (cradle) is considered waste (grave) until EoL, only a small portion of the materials reach the “recycling phase” where most of the panels were not designed for recycling purposes or to be reused at the EoL (Figure 5). Here, the C2G process exemplifies “down-cycling”: Figure 5: Cradle-to-grave: Simplified linear flow of materials (adapted from Contreras-Lisperguer et al. 2017) Today, eco-efficiency principles provide a degree of environmental relief, along with certain economic benefits. Nevertheless, there is still a lack of a fundamental redesign of material flows and inadequately addresses the level of toxicity in some products (Braungart et al., 2007). Furthermore, there is no clear evidence, either historical or theoretical, that efficiency improvements will decrease the amount of resources used (Smil, 2008). Eco-efficiency can, however, be considered a link between current C2G paradigm to a new circular paradigm. 32 Cradle-to-Cradle The cradle-to-cradle (C2C) philosophy was introduced in the book Cradle-to-Cradle: Remaking the Way We Make Things (McDonough and Braungart, 2002). Products are designed and engineered to avoid toxicity to the environment, not just during their manufacture process, but over the full course of their useful life, while also making it possible to reutilize their materials at the EoL (McDonough and Braungart, 2002). The entire lifecycle of a C2C product mimics a circular industrial system, where all materials can be potentially used and reused indefinitely. Hence, the materials used to manufacture a PV panel or any other product at the EoL can be upcycled as a primary resource to manufacture the same product or a different one (Figure 6). The C2C is based on three main principles (McDonough and Braungart, 2002): • Waste equals food; • Use current solar income; • Celebrate diversity. These main principles can be divided more comprehensively based on the C2C certification program (McDonough Braungart Design Chemistry, 2012). The five principles are: a. Safe Materials: products are designed with non-toxic materials. b. Reutilization (upcycling): used in a closed-loop-cycle where all materials flow indefinitely in cycles. c. Water: Water control in the industrial process and the effective use of it. In-flow and out-flow of water is purer during the industrial process. d. Energy: Maximize use of renewable energy sources and technology. Energy is effectively used (Improve Energy Efficiency by Cascade Effect use on Industrial Systems). e. Social fairness: social impacts of C2C approach must be improved in terms of existing social conditions and future planning and design, always considering local conditions where industrial facilities operate. In a circular flow of mass, C2C defines two cycles or metabolisms depending on types of materials used in production: the biological and technical cycles. The biological cycle illustrates the life cycle of manufactured products that use exclusively biodegradable materials such as cotton, paper, or wood, among others, that can be considered biological nutrients (McDonough and Braungart, 2002). Since PV panels are not made with these materials, we do not discuss this cycle in the paper. The technical cycle (see Figure 6), as defined by the C2C philosophy, is the life cycle where material flows in circles, where all adequately selected materials are used and 33 then reused as raw material into the industrial system (McDonough and Braungart, 2002). The C2C philosophy promotes an eco-effectiveness by creating an industrial material flow like what happens in nature, where the use of materials is safe and healthy for the ecosystem, and there are materials flowing within different cycles, increasing the overall system efficiency (Braungart et al., 2007). Figure 6: Cradle-to-cradle: Technical cycle (adapted from Contreras-Lisperguer et al. 2017) C2C practitioners assert that the Life Cycle Assessment (LCA) tool is not suitable to assess the “C2C-ness” of a C2C-product (NL Agency, 2011), therefore, C2C CertificationTM (MBDC LLC., 2012) is the only available tool to assess a C2C-product and is implemented by the Cradle-to-Cradle Products Innovation Institute (Cradle to Cradle Products Innovation Institute, 2017). However, as others have mentioned, this exclusionary approach for the C2C Certification scheme is questionable (Llorach-Massana et al., 2015) an LCA tool may actually complement C2C certification (Toxopeus et al., 2015). The certification has five categories (Basic, Bronze, Silver, Gold, or Platinum), representing levels of accomplishment and adherence to prescribed requirements for each category. C2C philosophy has not only gained popularity in the manufacturing sector, but also garnered attention from critics. Based on the available scientific literature, most of the critics are focused on the C2C CertificationTM scheme. Llorach-Massana et al. 2015 and Bach et al. 2018 point out that the C2C CertificationTM does not consider the environmental impacts of a C2Cproduct from a life cycle approach and focuses mostly on the raw materials and end of life stages. Meanwhile, Toxopeus et al. 2015 highlights that the C2C CertificationTM mainly focuses on material health, with the category material reutilization a distant second, and the aspects of water 34 stewardship, renewable energy, and social fairness are implemented and assessed far less extensively. Other literature focuses on the benefits that LCA and eco-efficiency practitioners could gain from the C2C philosophy regarding the importance of resource efficiency and positive environmental effects from materials. Furthermore, de Pauw et al. 2015 stressed that LCA methodology must expand its boundaries in order to assess sustainability and main impacts from nature-inspired design products (i.e. C2C and biomimicry). Some authors underline the need to address conflict between sustainability, unlimited growth, and consumption by focusing on ecoeffectiveness along with the traditional focus on eco-efficiency in our industrial development (Hauschild, 2015). It is possible to conclude that the C2C philosophy has an important value by helping to the establish foundations under which this CLMC system will be designed. B. Dwelling time estimation Dwelling time (Td) represents the time that materials remain in operation or with the consumer before being reintroduced as a primary resource into the manufacturing cycle. Td equals the timeframe between (a) the start of use and service and (b) removal or recovery. In reality, the operational lifespan of C-Si PV panels can only be approximated by mathematical models (e.g. trend analysis, probabilities, etc.). For C-Si PV panels, Td is assumed to be closely related to the operational lifespan until C-Si PV panel failure. This assumption facilitates estimation while fundamentally satisfying the definition of Td. With this said, Td can be represented as a continuous non-negative random variable that has a probability density function (pdf) denoted by f(t). The cumulative distribution function (cdf) of F(t) is then the integral of pdf. Here, F(t) represents the probability that a system will be functional in the interval from time 0 to time t, and can be written as: 𝐹𝐹(𝜕𝜕)=𝜕𝜕( 𝑇𝑇𝑖𝑖≤ 𝜕𝜕) = �𝑓𝑓(𝑠𝑠 t 0) 𝑑𝑑𝑠𝑠 𝑓𝑓𝑓𝑓𝑓𝑓 𝜕𝜕∈ℝ+ (B.1) In the case of C-Si PV panels, Td amounts to the time from when the typical C-Si PV panel starts operation -generating electricityuntil the same panel is no longer functional. Thus, using equation (B.1) it is possible to express the reliability function (Mishra and Joshi, 1996) as: 𝑅𝑅(𝜕𝜕)= 1 −𝐹𝐹(𝜕𝜕)=𝜕𝜕(𝑇𝑇 𝑖𝑖 >𝜕𝜕) 𝑓𝑓𝑓𝑓𝑓𝑓 𝜕𝜕> 0 (B.2) where 𝑅𝑅(𝜕𝜕) in equation (B.2) represents the probability that the PV panel is still working at time t and can be considered as a proxy of the average lifetime of a PV panel. Based on reliability theory (Ross and Ross, 2019) it can be written as follows: 35 𝑇𝑇𝑖𝑖=� 𝑅𝑅(𝜕𝜕) 𝑑𝑑𝜕𝜕 ∞ 0 (B.3) C. Conservation of mass equation and its linearization As an initial approximation, let us assume that the flow of mass of Non-RenewableAbiotic-Primary-Resources (NRAPR) at any time-space scale and at any manufacturing phase observed at 𝑇𝑇𝑐𝑐 will behave as a continuum rather than as discrete particles. Our initial assumption can be seen as an extension of symbiotic industrial material flows through the networks of different businesses at different time-space scales (Chertow, 2004). Let’s consider an imaginary finite NRAPR volume with dimensions 𝛿𝛿𝜕𝜕,𝛿𝛿𝑦𝑦,𝛿𝛿𝛿𝛿, containing a fixed collection of matter displaced across a field property 𝜀𝜀 (𝜕𝜕,𝑦𝑦,𝛿𝛿,𝜕𝜕) through any arbitrary manufacturing phase. As the system is comprised of a fixed collection of matter, the shape of the finite volume of material (NRAPR) will be distorted by the various forces applied during the manufacturing processes. However, since the density is unchanged over time, we can write the conservation of mass equation as: 𝜕𝜕𝜌𝜌 𝜕𝜕𝜕𝜕+∇∙(𝜌𝜌Υ)= 0 (C.1) Equation (C.1) is the Euler form of a first-order hyperbolic equation (Craik, 2013), which describes the mass-conservation equation for given bodies of NRAPR, where ρ represents the NRAPR or material density in terms of 𝜌𝜌 per units of length and Υ represents the threedimensional velocity vector. One more step is needed to comply with our first assumption that at any time and space scale, the flow of a mass of NRAPR observed at 𝑇𝑇𝑐𝑐 can be considered continuous. Since material flow happens at different time and space scales (Contreras-Lisperguer et al., 2017), it has both an average and a fluctuating component. Therefore, we shall now derive the conservation of mass equation in its averaged form. To avoid unnecessary complexities, we opt for the Favre approach over the Reynolds approach (Ruffin et al., 1997). After decomposing the variables into a mass-weighted mean and the fluctuating part (nonlinear), we apply Favre decomposition of the motion field in (1) and obtain the Favre time averaged conservation of mass equation: 𝜕𝜕𝜌𝜌� 𝜕𝜕𝜕𝜕+𝛶𝛶�𝛻𝛻∙𝜌𝜌�= 0 (C.2) Where 𝛶𝛶�=𝑢𝑢�𝚤𝚤+𝑣𝑣�𝚥𝚥+𝑤𝑤�𝑧𝑧 represents the average three-dimensional velocity vector at every stage of the life-cycle of a product. Equation (C.2) allows only for the determination of 36 mean quantities at a global/regional scale, which may differ from instantaneous ones. We shall now use a numerical scheme to solve this differential equation. D. Stability analysis of our model A stable numerical scheme is crucial to successfully modeling the C2C-CLMC system. It can be considered stable if any numerical errors generated during the solution of discretized equations are not amplified (LeVeque, 1992). Since equation (17) is a linear finite-differential equation with constant coefficients (Durran, 2010), we have used Von Newman Stability analysis to assess the stability of the scheme. We considered a single Fourier component in order to examine the growth of the error in time on an individual harmonic of the form 𝜓𝜓 𝑖𝑖 𝑛𝑛= 𝜓𝜓(𝜕𝜕 𝑖𝑖 ,𝜕𝜕 𝑛𝑛 )=𝛼𝛼𝑛𝑛𝑒𝑒iθ𝑖𝑖Δ𝜕𝜕 (D.1) where 𝛼𝛼 is the amplification factor. The numerical scheme is unstable if |𝛼𝛼|> 1. Hence, for stability we need to satisfy the requirement |𝛼𝛼|≤1 If we then introduce 𝜓𝜓𝑖𝑖𝑛𝑛 on our numerical scheme equation (D.1), then, we can rewrite (D.1) as (𝛼𝛼𝑛𝑛+1−𝛼𝛼𝑛𝑛)𝑒𝑒iθ𝑖𝑖Δ𝜕𝜕+ 𝜐𝜐𝛼𝛼𝑛𝑛(𝑒𝑒iθ𝑖𝑖Δ𝜕𝜕−𝑒𝑒iθ(𝑖𝑖−1)Δ𝜕𝜕) = 0 (D.2) where 𝜈𝜈=𝑢𝑢�Δ𝜕𝜕 Δ𝜕𝜕 Removing powers of α and using 𝑒𝑒iθ𝑖𝑖Δ𝜕𝜕 as a common factor gives 𝛼𝛼= 1 −𝜈𝜈+𝜈𝜈𝑒𝑒 iθΔ𝜕𝜕 (D.3) Then, by applying the exponential form identity we can rewrite 𝛼𝛼 as 𝛼𝛼= 1 −𝜈𝜈+𝜈𝜈(cos(𝜃𝜃𝜃𝜃𝜕𝜕)−isin(𝜃𝜃Δ𝜕𝜕)) (D.4) We can find the magnitude of 𝛼𝛼 by taking the square root of the multiplication of 𝛼𝛼 per its complex conjugate 𝛼𝛼∗ |𝛼𝛼|= √𝛼𝛼𝛼𝛼∗ 2 Consequently, |𝛼𝛼|= 1 −2𝜈𝜈+𝜈𝜈2+ 2𝑐𝑐𝑓𝑓𝑠𝑠𝜃𝜃−2𝜈𝜈2𝑐𝑐𝑓𝑓𝑠𝑠𝜃𝜃+𝜈𝜈2𝑐𝑐𝑓𝑓𝑠𝑠2+𝜈𝜈2𝑠𝑠𝑖𝑖𝑙𝑙2 (D.5) |𝛼𝛼|= 1 −2𝜈𝜈(1−𝑐𝑐𝑓𝑓𝑠𝑠𝜃𝜃)+ 2𝜈𝜈2(1−2𝑐𝑐𝑓𝑓𝑠𝑠𝜃𝜃)−1 + 2𝜈𝜈(1 −𝑐𝑐𝑓𝑓𝑠𝑠𝜃𝜃)(𝜈𝜈−1) (D.6) Simplifying the trigonometric expression (D.6) using trigonometric identities, we have |𝛼𝛼|= 1 + 4𝜈𝜈(𝜈𝜈−1) (D.7) Thus, equation (D.7) only satisfies the stability criteria, if and only if 𝜈𝜈≤1 then, |𝛼𝛼|≤1. |𝛼𝛼|= 1 + 4𝜈𝜈(𝜈𝜈−1)≤1 (D.8) 37 Since 𝜈𝜈≤1 is sufficient to demonstrate a condition of stability and 𝜈𝜈=𝑢𝑢�Δ𝜕𝜕 Δ𝜕𝜕, then, the Von Neumann stability condition is in the form of the Courant–Friedrichs–Lewy (CFL) condition, as 𝑢𝑢� Δ𝜕𝜕 Δ𝜕𝜕 ≤1 (D.9) and a non-negative Δ𝜕𝜕 and Δ𝜕𝜕 denotes 𝑢𝑢� Δ𝜕𝜕 Δ𝜕𝜕 ≥0 (D.10) in order to satisfy both conditions (D.9) and (D.10), we can rewrite the stability condition as 0≤𝑢𝑢� Δ𝜕𝜕 Δ𝜕𝜕 ≤1 (D.11) Our experiment must satisfy (D.11) to be considered stable and the satisfactory simulation. E. Estimating Dwelling time for a crystalline silicon photovoltaic panel To estimate Dwelling time (Td), we considered that the degradation and failure of a PV panel can be represented by a Weibull function (Charki et al., 2013; Pregelj et al., 2001). This function represents the Td distribution for C-Si PV panels in the form 𝑓𝑓(𝜕𝜕)=� 𝛼𝛼 𝛽𝛽�� 𝜕𝜕 𝛽𝛽� 𝛼𝛼−1 𝑒𝑒−( 𝜕𝜕 𝛽𝛽)𝛼𝛼 (E.1) where, 𝐹𝐹(𝜕𝜕)= P(𝑇𝑇𝑖𝑖≤𝜕𝜕)=�1−𝑒𝑒−(𝜕𝜕 𝛽𝛽)𝛼𝛼 𝑓𝑓𝑓𝑓𝑓𝑓 𝜕𝜕> 0 0 𝑓𝑓𝜕𝜕ℎ𝑒𝑒𝑓𝑓𝑤𝑤𝑖𝑖𝑠𝑠𝑒𝑒 then, 𝑅𝑅(𝜕𝜕)=𝑒𝑒−( 𝑡𝑡 𝛽𝛽)𝛼𝛼 𝜕𝜕> 0 (E.2) By applying equation (E.2) over (E.1), it is possible to estimate Td 𝑇𝑇𝑖𝑖=∫𝑅𝑅(𝜕𝜕) 𝜕𝜕 0=𝛽𝛽 Γ�1 + 1 𝛼𝛼� (E.3) Then the mean Td may then be written as follows 𝑇𝑇�𝑖𝑖= 𝛽𝛽(ln 2) 1 𝛼𝛼 (E.4) For this numerical experiment, the estimation of parameters 𝛼𝛼 and 𝛽𝛽 were made based on statistical methodologies and random failure sample (Pregelj et al., 2001; Ramadan, 2015) using the Weibull reliability toolkit for R (Weibull-based reliability toolkit for R, 2013), then, the parameters found for this particular experiment are 𝛼𝛼= 5.376 and 𝛽𝛽= 32.062. 38 Introducing 𝛼𝛼 and 𝛽𝛽 on equation (E.4), we obtain Td = 29.949 years ≈ 9·108 s, which matches the average lifecycle of a typical PV module. Since Td and 𝜌𝜌� are known, we estimated that the order of magnitude of 𝜉𝜉󰆻2~1 · 10−9 units of 𝜌𝜌� per second, which represents the decommissioning rate of units of 𝜌𝜌� once the PV panel is no longer functional.