A multi-dimensional indicator for material and energy circularity: Proof-of-concept of exentropy in Li-ion battery recycling

Summary Recycling processes are an important stage in the raw material life cycle, as it enables the transition from a linear economy into a circular one. However, the currently available indicators of productivity in recycling technologies respond to the needs of a linear economy. In this work, a parameter called “exentropy” is proposed, offering the possibility to simultaneously account for mass preservation and the energy efficiency of transformative stages. As a proof-of-concept of this indicator, the analysis of a lithium-ion battery recycling process under various concentrations of a leaching reagent (i.e., 0.1M, 1M, and 2M) is presented. It is shown that, when the energy or mass dimensions are considered independently, the processes considered optimal may have conflicting characteristics. In contrast, the multi-dimensional analysis identified the process option offering the best compromise for both material and energy preservation, an aspect closer to the goals of the circular economy.


INTRODUCTION
5][6] While the currently dominant linear economy model gives an erroneous assumption of unlimited resource stocks, 4 immediate action is needed to prevent the depletion of natural resources in reality. 7One approach to address this challenge is that of the ''circular economy'' (CE), which fundamental principle is to replace the end-of-life (EOL) stage and keep value longer through the different value retention stages, such as reusing, refurbishing, or recycling. 8The idea of a CE model was proposed in the mid-20 th century, and it has gained attraction in recent years due to the risks of resource scarcity as population grows along with an increased demand of products and services. 9While various definitions are found in the literature, [9][10][11][12] in general, the CE represents a regenerative system in which waste is minimized by closing, narrowing, and slowing materials and energy loops during a life cycle of a product.In this context, ''closing'' loops represents the recovery of materials into a value chain, ''narrowing'' loops refers to avoiding overly complicated routes for the circulation of materials in a useful form, while ''slowing'' represents the extension of the useful lifetime of a product, all of which results in a decreased consumption of resources within a given period.
Therefore, to close the loop and to transform the linear economy into circular one, EOL products are considered as resources, thus reducing the need for primary material inputs.In an ideally circular scenario, waste would not be generated at all, 4 so the utility and value of materials would be preserved, effectively reducing the need for virgin raw materials. 5As the aim in CE is also to reduce energy losses in addition to material losses, circulation of energy and energy degradation should also be considered. 10Thus, the implementation of CE practices can further help to deal with problems indirectly associated with virgin materials production.These problems might include poor energy and materials intensity, dependency on foreign countries for critical materials, increased use of fossil energy, and even climate change. 6,9,13Regarding the use of energy, in CE systems the use of renewable energy sources should be prioritized to reduce emissions, and energy degradation should be studied more as we want to avoid any unnecessary energy losses. 12,14espite the interest gained in the CE, there is a lack of consensus regarding the right indicators to analyze and compare technological solutions required for its implementation. 2,15Without indicators accounting for the CE goals by definition, there is a risk that processes aimed at recovering economically valuable materials are energy intensive, consume hazardous reagents, or produce undesired emissions.
Therefore, objective, scientific, and systemic indicators of materials circularity are needed for the design and evaluation of industrial ecosystems. 8,10Circularity parameters are fundamental, both to support decision making and to serve as a framework for technological innovation. 1 To shift the recycling principles toward CE, a wide variety of recyclates should be concentrated to a degree where they can be reincorporated into the value chain with minimum losses to the environment.To study the concentrating action of material processing systems, the concept of statistical entropy (SE) can be applied to describe the distribution pattern of an element or substance through a system. 16The concept of SE was created by Shannon and Boltzmann, 17 who borrowed the concept of thermodynamical entropy, introduced by R. Clausius in 1865, 16 and applied it to information theory.SE aims to represent the loss or gain of information in a system, whereas thermodynamical entropy, which is based on the 2 nd law of thermodynamics, represents the creation of disorder, or randomness of energy in microsystems.Even though SE and thermodynamical entropy follow a similar mathematical form, SE is measured in information bits [bit], and therefore, no physical relationship exists between the two entropy terms. 16Statistical entropy analysis (SEA) was proposed by Rechberger and Brunner, 16 who combined material flow analysis (MFA) with Shannon's SE function.According to MFA, a system consists of stages (q), transformative processes, i.e., units (u), and streams (s).A visualization of this concept can be found in supplemental information (Figure S1).The purpose of MFA is to balance all material flows in a process by systematically accounting for all input and output streams feeding and resulting from the transformative processes at every stage.The SE in each stage is determined by the concentration of each component (i) in each stream where it is present.Accordingly, an increase of SE represents the dilution of substances while the SE value of highly concentrated materials approaches to zero.Therefore, SEA can provide a systemic view of all concentration changes occurring in a system.
To compare different elements inside a system, we can use an indicator known as relative statistical entropy (RSE), which is derived from SEA.With RSE it is possible to make clearer comparisons between different elements and components in the same system, as it standardizes the entropy value of each component using their theoretical maximum as a benchmark. 18Furthermore, RSE can be used to calculate the substance concentrating efficiency (SCE), an indicator utilized to show the ability of a system to concentrate substances.Whereas the values of RSE goes from 1 to 0, where 1 indicates a diluted and 0 a concentrated material, the values of SCE are from 0% to 100%, where 0% indicates a diluted and 100% a concentrated material.
The aim in recycling processes is to obtain the lowest possible values of RSE, as this represents an efficient separation of the mixed materials feed. 19More concentrated resources are easier to control, manage, and utilize as valuable products, thus avoiding their downcycling or irreversible losses, 20 which supports the concept of closed loop recycling.In the production stages, highly concentrated raw materials with a low entropy content are combined to create functional products.The resulting mixed substances in manufactured goods have a higher entropy, in addition to the intrinsic entropy generated by the inefficiencies of manufacturing and distribution stages.By the EOL stage, these products are randomly mixed with other products, resulting in the stage with highest entropy of the product life cycle.Without further action, this status will remain unchanged and thus, a resource efficient economy is only possible by applying efficient waste management strategies.Under this perspective, recycling processes have the aim of transforming these high-entropy mixtures into low-entropy recycled materials. 20hile the study of material flows is fundamental for the analysis of processes from the CE perspective, energy preservation is widely acknowledged to be another relevant dimension for the analysis of materials circularity. 10Indeed, the recovery of valuable materials should consider the energy consumption and losses associated with each transformative process.A suitable methodology that accounts for the preservation of energy is exergy analysis (ExA).Unlike measurements of energy consumption, exergy is a property that helps quantifying the irreversible losses of useful energy. 21,22In the context of CE, it is important to identify whether the useful energy introduced in the system can be further utilized or whether transformative process dissipate it permanently. 21The ExA helps to identify and quantify the sources of exergy destruction (Ex D ), which should be minimized for a more efficient use of resources. 23,24Ex D can also be evaluated using an indicator called relative exergy content (REX), which is a fractional value representing the exergy content of a stage relative to the total exergy content of the system input.When exergy is being destroyed, the exergy content of a stage is being reduced.Therefore, the value of REX at the final stage can also be represented as exergy conservation efficiency (ExCE), an indicator to present how much exergy has been preserved during the process.Analogous to RSE and SCE for, the value of REX goes from 1 to 0 and values of ExCE from 0% to 100%, representing no exergy destruction at all and a completely exergy destructive system, respectively.
As indicators are needed to evaluate CE in different systems, the literature offers examples on the application of SEA to evaluate the recyclability of various materials and products, such as plastics, [25][26][27] construction materials, 28 electronic waste, 29,30 thermoelectric devices, 31 and lithium-ion batteries (LIBs). 18,32Furthermore, ExA has been used in the evaluation of power plants and processes and their internal recycling, [33][34][35] performance of recycling systems in general, 36 and recycling of specific materials and EOL products, such as coated magnesium, 37 battery recycling systems, 22,38,39 and the utilization of waste heat in hybrid and electric vehicles. 402][43] However, no studies have tried to combine both SEA and ExA in the field of recycling, to the best of the author's knowledge.Alternatively, SEA has been combined with energy in a recently published study by Moyaert et al. 44 to evaluate how different types of deli packaging material affect their recycling potential.Also, SE and exergy were combined with a third parameter, entransy, in a work by Vakalis et al. 42 to compare the efficiency of different gasification plants.Since our study will not focus on the possible utilization of heat in the process, the concept of entransy cannot be applied, as it is used to study heat transfer potential in a system.
In a previous study by our research group (Vela ´zquez-Martı ´nez et al. 32 ), a comparison between SEA and exergy efficiency analysis was presented to showcase the limitations of the ExA when dealing with separation processes with equal energy consumption but different materials concentration efficiency.Nevertheless, the objective of such study was to demonstrate the relevance of a systemic-level analysis of materials flow, rather than the combination of parameters for materials and energy preservation.Also, both material and energy balances obtained from a simulation of a PET recycling process were used in a study by Singh et al., 45 where they conducted a comprehensive analysis of PET recycling via a simulation study.They however did not combine these two parameters, but instead used material and energy balances to obtain information for the process analysis.
Evidently, indicators based solely on either material or energy balances can only capture a partial picture of the CE needs. 46A successful combination of the material-centric description with thermochemistry, and therefore exergy, is the key to gain a better understanding of circular systems. 24Exergy represents the maximum rate of work available when the system is brought into equilibrium with the surrounding system.From the perspective of raw materials, the analysis of circularity must have materials flow as basis, ideally regenerating itself with minimum losses (red arrows in Figure 1).The analysis of this circular model however should be enriched with the addition of other relevant dimensions following the definition of CE. Figure 1 shows, for example, how energy flows (represented in blue) and environmental footprints (represented in green) are intrinsically associated with the one-dimensional circularity of materials.As more dimensions of study are added in the analysis (e.g., time, quality, costs), a one-dimensional circular model will evolve into one evoking an n-dimensional hypersphere.
Therefore, the new concept introduced here is called ''exentropy'' (c), which offers a system-level analysis of material and energy preservation simultaneously, thus providing a more robust evaluation of circular solutions.The principle of c is that an irreversible loss of exergy, i.e., Ex D is only justified by the effective concentration of materials.Namely, the proper way to close the material loops is to direct energy resources in a useful form toward the recovery of materials.For instance, in systems where the design of products dies not consider recyclability, the recovery of materials will result in a significant use of energy and its consequent Ex D .As well, suboptimal recycling processes that consume excessive energy or generate significant material losses will likely result in unjustified Ex D to produce useful secondary raw materials.Thus, the concept of c hereby proposed represented the first hyperspherical engineering indicator ever proposed, as the concept combines circular material flows with energy flows.
To showcase the newly proposed methodology, the study of a hydrometallurgical LIB recycling process, based on the Retriev process 47-50 is hereby used as a proof-of-concept.The process was simulated using HSC Chemistry -software 51 to obtain the stream composition and energy values needed for the calculation of RSE, REX, and exentropy.With the aim of providing a useful proof-of-concept on exentropy analysis, the process simulation was performed under various concentrations of LiOH (namely, 0.1M, 1M, and 2M) used in the Li solubilization stage.As will be further explained in the Results and discussion section, this was a carefully chosen variable since it produces conflicting efficiency values between RSE and REX by affecting both values differently in each scenario.Thus, exentropy was applied to find optimal process conditions satisfying both materials and energy conservation.

RESULTS AND DISCUSSION
For the calculation of RSE, Figure 2 was drawn to show the schematic representation of the Retriev process according to MFA methodology.The mass flows of each stream can be found in the supplemental information (Tables S1-S4).
The values of relative statistical entropy were reduced in the system throughout all the scenarios The SEA results for the three different scenarios hereby explored are presented in Figure 3.
As seen, the values of RSE for all elements significantly decreased by the end of the process in every scenario.In all cases, Fe reached its lowest RSE value after the hammer mill, and the current collector materials (i.e., Al and Cu) were concentrated mainly during the screening step and slightly during carbon press filter.After leaching, Co (in the form of oxides) and graphite were separated as a mixture in the carbon press filtering step together with some remaining Al and Cu.The last change in RSE is found in the filtering step, where a concentrated Li stream in the form of Li 2 CO 3 is obtained as a final product.The behavior of RSE of all elements is similar in each scenario, but the values As mentioned earlier, the aim of recycling processes is to concentrate materials, so the values of RSE should be as close to zero as possible.As explained in ''Method details'' in Equation 8, a value of SCE can be calculated based on the value of RSE.Therefore, processes with a high value of SCE have a low value of RSE, as seen in Table 1, where the values of SCE were highest with 0.1M scenario.

The reducing relative statistical entropy values indicate the concentration of materials
The reduction in the RSE means that all materials were concentrated by the end of the process, even when accounting for potential losses, as expected in an efficient recycling process.Because Fe reached its lowest value after hammer mill, it means that it was recovered quite efficiently from the battery scrap as most of the steel casing remains as coarse particles that are easily removed after this stage.As explained more detailed in the ''Method details'' section, Fe is most likely recovered after the hammer mill and prior screening, and therefore in this study, it is considered to be separated from other materials during milling.The reduction of RSE of the current collector materials occurred mainly during the screening step as it is explicitly aimed at separating these coarse particles from fine electrode materials.The reductions in the RSE values for cathode metals, i.e., Co and Li, occurred mainly during hydrometallurgical processing step, as this part of the process was mainly for treating the black mass.There was no significant reduction in the RSE value during evaporation and Li 2 CO 3 formation, as the amount of water evaporated was relatively small compared to the mass of the mixture, and Li 2 CO 3 formation did not intend to concentrate materials, but to form the compound ''Li 2 CO 3 '', respectively.
Following the report by Bertuol et al., 52 it was assumed that some residues of the battery components were lost during shredding and milling due to the intrinsic separation inefficiencies of these processes.This affects the value of RSE by increasing it slightly, hence, n the elements reached an RSE value of zero.Fe was the only element considered to be in elemental form, but for the other elements, they were recovered either as mixtures (e.g., Cu and Al), or compounds (e.g., Co and Li as CoO and Li 2 CO 3 , respectively).This also affects the RSE values of the elements by increasing the values from zero.However, following the concentration efficiency as elements instead of different compounds is easier, as with pure elements only the element of interest can be followed instead of following several different compounds containing the same element.
The boundaries of the system hereby studied were set to consider process water and chemical reagents as part of the feed streams into the process.Since RSE accounts for a system-level concentration, the presence of large streams of process water will impacts its values.Indeed, the eventual separation of large streams of process water in the 0.1M LiOH scenario is computed as a successful concentration effort, resulting in a consistently lower value of RSE for all components.In reality, handling large water flows require additional efforts that are overlooked in the SEA.This exemplifies the need of additional dimensions for a robust analysis of circular systems, as will be further discussed in the subsequent Sections.
The more concentrated the materials are, the more suitable they are for production as secondary raw materials for similar applications.Therefore, to achieve a closed-loop recycling system and be more in accordance with CE, low values of RSE should be obtained at the end of a recycling process.On the contrary, in this case the value of SCE should be as high as possible, as it depends on the value of RSE in the final stage, and the lower the values of RSE are, the higher SCEs are obtained.A leakage of materials leads to material losses and therefore increased entropy in the system, thus resulting in high RSE values.Thus, RSE values should be low to obtain a more circular system.According to the results, the lowest values of RSE and therefore highest values of SCE were obtained with 0.1M scenario, so 0.1M LiOH would be the preferred option in terms of materials concentrating action.

Exergy analysis is used to study the useful energy inside a system
The study of exergy flows in the three systems under study are presented as Sankey diagrams in Figure 4.The Sankey diagram offers a clear visualization of all the input and output exergy streams, and the division of them into smaller streams as they are separated. 53The exergy values of material streams in the diagram are presented in supplemental information (Tables S1-S4).
As seen from Figure 4, the input exergy from LIBs remains generally in the process until the separation of metal oxides and graphite from the Li solution stream during carbon press filter.The biggest divisions of exergy between the input and carbon press filter occur during hammer mill and screening, as steel, Cu, and Al are recovered, respectively.After carbon press filter, the exergy streams of materials are much smaller compared to the LIB input stream, and no notable divisions of material exergy from the main stream cannot be observed.The exergy content of the final product, i.e., Li 2 CO 3 , is much smaller compared to the input exergy of LIBs.
The other thing that can be seen in Figure 4, is that the external resource inputs, i.e., reagents and electricity, can be traced, which was not evident with RSE.Resources are needed to recover materials efficiently, which inevitably leads to the destruction of exergy. 54This can also be observed in Figure 4, where all units lead to some Ex D .The most notable change in Ex D between the various scenarios occurs during the evaporation step (highlighted with a red circle) when Ex D decreases while the concentration of LiOH is increased.Ex D values at each stage are presented in Figure 5 to compare better Ex D between each scenario.
As seen, the total Ex D decreases when the concentration of LiOH is increased from 0.1M to 2M, so the scale of y axis is much smaller in 2M scenario compared to the 1M scenario.Ex D can also be characterized with fractional exergy destruction (FEx D ) curve, which visualizes more evidently the exergy distribution throughout the process.With the 0.1M scenario, there is a huge increase in FEx D during evaporation, a slight increase during shredding and milling, and almost no notable increases during other units.Conversely, scenarios with 1M and 2M LiOH had a similar Ex D throughout the whole process and evaporation is no longer the determinant Ex D unit.In these cases, shredding and milling contributed the most to Ex D due to the electricity consumption in these units. 54,55This can also be seen with FEx D , as its increase was steeper compared to the scenario with 0.1M LiOH.FEx D shows also that exergy was destroyed during Li leaching and Li 2 CO 3 formation, although this was not as significant compared with the previously mentioned stages.In conjunction, this results in ExCE values for the 0.1M, 1M, and 2M LiOH scenarios of 67%, 89%, and 90%, respectively.

Low exergy destruction indicates high exergy preservation in the system
The main exergy stream from LIB input to the carbon press filter is high, as this stream contains lot of elements with high elemental exergy, from which graphite has the highest molar fraction and a relatively high elemental exergy in that stream, it has the highest chemical exergy in the stream.This exergy is also much higher than the chemical exergy in the Li-containing stream, so the exergy content of the stream containing graphite is significantly higher.This is the reason also why the exergy content of the product is much smaller compared to the exergy of the input feed containing LIBs.Because of the notable increase in FEx D in the evaporation during 0.1M LiOH scenario, it clearly suggests that evaporation was the determinant Ex D unit.With the other two scenarios, the steeper increase in FEx D during the comminution steps indicated that comminution contributed more to Ex D than with 0.1M scenario.During Li leaching and Li 2 CO 3 formation the chemical compounds go through transformative processes, which may also cause exergy destruction together with electricity consumption. 21Operations such as sieving and filtering had low electricity consumption with no changes in chemical composition compared to other units, thus resulting in lower exergy destruction.
The amount of water also affects the results from the ExA.The 0.1M LiOH solution represents a comparatively large amount of water to be evaporated, associated with an extensive energy demand in comparison with the other scenarios.In exergy analysis, this is reflected by the high Ex D in the evaporation unit reported for the 0.1M scenario.In addition, since the mass of water used in the process decreases at higher concentrations of LiOH, the exergy losses in the form of water vapor is comparatively minor.Thus, the inefficiencies associated with highly diluted streams become evident with ExA, unlike the SEA described above.Nevertheless, both dimensions are needed for a proper analysis, as exergy is not an indication of materials concentration.
The results showed that much of the exergy is not being destroyed, especially with 1M and 2M scenarios.This is also according to CE principles, as the aim is to have as minor exergy degradation as possible in addition to the loss of materials.As the values of Ex D affects the values of REX (shown in Figure 7), Ex D should be minimized to have as small energy degradation throughout the process and therefore high values of ExCE.Therefore, based on the results, the optimum process would be the scenario with 2M LiOH.This nevertheless contradicts the outcome of RSE, where the diluted LiOH was the favored scenario.A method to integrate both the analysis of mass and energy conservation is thus necessary.

Exentropy analysis of a process combines materials concentrating action and exergy preservation
Before going to the actual results from exentropy analysis, it is important to understand what is happening behind the concept.For the sake of clarity, the Figure 6 shows the theoretical limits and potential interpretation of c q values.For a better understanding, the figure contains the life cycle of a product and a recycling process visualized according to the MFA methodology, and three exentropy scenarios for the exentropy analysis of a recycling process.The life cycle of a product in this case consists of a production stage, i.e., processing and refining, design and manufacture, and assembly, a consumption stage, i.e., distribution, use, and EOL, and a recycling stage. 32As recycling is applied to recover secondary raw materials from the EOL product and thus transform the linear economy into circular one, the concept of exentropy in this article is applied to evaluate the recycling step.Exentropy evaluates the efficiency of recycling systems based on both materials concentrating action and exergy preservation, and the values are presented for each stage according to the MFA methodology visualized in Figure S1.Both RSE and REX are calculated for each stage, and the combination of these two indicators gives the values of exentropy, as seen in the lower part of the Figure 6.
In cases where the RSE curve is above the REX curve, the difference between the curves is negative, and the process can be considered inefficient, as the exergy losses outweigh the expected benefits of materials concentration.In the best-case scenario (Figure 6A), REX would remain with a value of one (i.e., no exergy losses) and RSE would be zero (i.e., production of pure streams), resulting in c values equal to one in each stage.On the contrary, in the worst-case scenario (Figure 6B), RSE would be equal to 1 (i.e., no concentrating action occurring) and REX would be zero (i.e., all exergy is lost), and thus c values would be À1 in each stage.In reality, processes would likely be somewhere in between, as represented in Figure 6C, in which RSE decreases as the components get concentrated but at the expense of reducing REX.If the concentrating action is however dominant (i.e., positive c values), then the use of resources is justified, and the process can be considered efficient.

Exentropy requires both relative statistical entropy and relative exergy content values to be calculated
The results from the exentropy analysis hereby proposed is visualized in Figure 7.
In the first place, it is seen that the REX values remain above those of the RSE curve during most of the process stages.The only exception is seen during shredding, where negative values of c were obtained.After adding the c values in all stages, P c q values equal to 2.75, 2.88, and 2.80 were obtained for the scenarios with 0.1, 1, and 2M LiOH, respectively.Because all scenarios had the same number of stages, there is no need to calculate the c eff values, as they will give the same conclusion than c values alone.Positive exentropy values indicate that the process is efficient in terms of material concentrating action and energy efficiency Positive exentropy values suggest that all scenarios in this recycling process fulfill their purpose, as the materials concentration in the system (RSE) outweighs the irreversible exergy losses (REX).There is only one single stage with a negative exentropy value during the process, i.e., shredding.This is understandable since shredding is not an operation that directly results in any concentrating action, while its energy consumption inevitably leads to exergy losses.However, this is a stage required for subsequent processing, as materials liberation depends on it.Therefore, the impact of shredding on RSE reduction may become evident at a later stage of the process, illustrating the need of a systemlevel analysis of the process.As seen, the methodology hereby proposed provides an overview of the system highlighting potential points for redesign and optimization.In the analysis of each individual transformative process, negative values of c represent points where exergy destruction overshadows the benefits from the concentration of substances.Furthermore, the cumulative values of c reflect a system-level balance of energy invested into preserving material loops, making it multi-dimensional and objective parameter to define the efficiency of processes following the CE principles.The concept of exentropy offers a novel type of analysis that could further support other parameters associated with SE and exergy, such as the aforementioned methods by Nimmegeers and Billen, 25 Fernandes et al., 41 and Vakalis et al. 42 Based on RSE and REX, both are interesting methods to analyze the efficiency of materials concentration and energy conservation in a system, respectively.Nevertheless, these are independent parameters that only evaluate a single dimension relevant for circularity.To provide a more robust analysis, a new parameter hereby named ''exentropy'' was proposed to evaluate both material and energy circularity simultaneously.Since a high value of REX and a low value of RSE indicate an optimal system for exergy preservation and materials concentrating action, respectively, a higher c value represents an optimal process for materials and energy preservation.Therefore, the results suggest that it is in fact the scenario using 1M solution that is optimal, as it obtained the highest c values.Also, as low RSE and high REX values represent a circular system, the higher the c value is, the more circular the system is in terms of both materials and energy.This makes the method is suitable for evaluating which process is the most appropriate for closed-loop recycling, and whether the secondary materials of sufficient quality obtained could be used again in manufacturing or similar applications.

Conclusions
A proper analysis of the circularity of process should consider the impact of transformative stages on various dimensions such as materials and energy preservation.As shown in this work, processes can be evaluated either on a material-basis using RSE or on an energy-basis using ExA, but there might be limitations when analyzing these two dimensions independently.Using these methods, it is difficult to objectively assess whether it is preferable to obtain purer material streams at the expense of higher exergy destruction or if it is better to obtain less pure streams but with a higher exergy preservation.To address the limitation of the current methodologies, a new method called exentropy was introduced to simultaneously account for both dimensions of material concentration and energy preservation.To provide a proof-of-concept of this indicator, a recycling process was studied under various concentrations of LiOH used as an extractive reagent.The results from SEA and ExA on the recycling process case study offered different conclusions for the optimal concentration of LiOH to be used.Accordingly, it was possible to obtain low values of RSE with a high cost of exergy destruction and vice versa.In this manner, it was illustrated how c can be used to identify the process conditions offering the optimal trade-off between materials concentration and energy preservation.Admittedly, true process optimization would require a larger number of scenarios, but this was considered beyond the aim of this proof-of-concept study.
Exentropy can be used as a tool for the evaluation and optimization of processes by identifying transformation stages with unacceptably high exergy destruction.For instance, mechanical treatment methods such as shredding will likely result in a negative contribution to the system-level exentropy as they do not directly contribute to the concentration of species.Consequently, the purpose of such unit operations needs to be justified by their impact on c values at later stages of the system.Similarly, exentropy may also be suitable to compare different processes using a robust parameter that favors the efficient use of energy for materials recovery.This proof-of-concept represents only the first effort to provide an optimization method that accounts for the multi-dimensional nature of CE, offering the possibility to identify the most efficient process in terms of both material enrichment and energy conservation.

Outlook
In future studies, some limitations and assumptions made in this article must be reconsidered when approaching more realistic simulations.To have as detailed recycling processes as possible, the possibility of mixed battery chemistries in the feed should be considered, in line with the expectations in industrial recycling processes.Also, the process hereby presented considered only the battery materials flow excluding further treatment of side streams, such as off-gases.Furthermore, to have more accurate exergy calculations, mixing entropy could be considered in the calculations as well.To further expand on the analysis of a circular system, parameters covering other dimensions such as the environmental impact, and their economic implications should be considered.

Limitations of the study
The subsequent question is whether the result from the exentropy analysis in this article is a reasonable outcome, considering it once again conflicts with the independent suggestions from SEA and ExA.Upon critical analysis, it is firstly seen that the highest decrease in RSE was indeed obtained with 0.1M LiOH, but this scenario is burdened by a significant exergy destruction in the evaporation step.Admittedly, the simulation was carried out under the assumption that LiOH concentration did not significantly affect the productivity of the system or the quality of the products.Nevertheless, as a proof-of-concept, it helps exemplify that SEA can be favored under ideal circumstances that may overlook real implications in terms of energy consumption.The opposite could be argued with respect to the favorable exergy preservation with the 2M LiOH scenario.Certainly, the most efficient way to preserve energy is to prevent its utilization.Nevertheless, the extent of transformations needed to reintroduce materials to the value chain may not be achieved by such a stationary system.6) The value of RSE is dimensionless and allows for an easy interpretation of the entropic behavior of materials flowing in a system.It is also possible to describe the collective entropy of all components in the system with Equation 7. 7) where RSE total q is the total RSE in the process at the q th stage.The substance concentrating efficiency (SCE) 16,56 is calculated from the difference between input and output RSE values in a system, as described in Equation 8.

SCE =
RSE i;input À RSE i;output RSE i;input Ã 100% (Equation 8) Where RSE i;input and RSE i;output are the input and output values of RSE of component i in the system, respectively.

Exergy analysis (ExA)
The total exergy of a system consists of its physical, chemical, kinetic, and potential exergy.Kinetic exergy is a result of the velocity with respect to system boundaries, and potential exergy is given by the velocity under any given body force field. 57However, kinetic and potential exergy are usually neglected since they are an order of magnitude lower compared to the contribution by chemical and physical exergy. 58In simple terms, physical exergy describes the maximum work obtained, when there is a difference in temperature or pressure between the system and the environment. 54,59Chemical exergy describes the exergy, or usable energy, intrinsically contained in an element or compound. 60The system-level exergy is the addition of exergy contributed by all elements and compounds within the system boundaries.The total exergy of a system thus depends on its composition, and it may change as its components undergo transformation processes.The total exergy of a stream can be presented as a sum of chemical and physical exergy of the components in that stream, as shown in Equation 9.
Ex tot;s z Ex ch;i + Ex ph;i (Equation 9) The process simulation software used in this work, i.e., HSC Chemistryâ calculates the total exergy for species using Equation 10: 51 Ex tot = X  10) where n k is the stoichiometric amount of element k in the species, b ref k is the elemental exergy of element k, DG 0 fð25 C;1barÞ is the standard formation of Gibbs energy of the species, N i is the total enthalpy of the species at the temperature of T, N ið25 C;1barÞ is the total enthalpy of the species in the standard state, S i is the total entropy of the species at a given temperature T, and S ið25 C;1barÞ is the total entropy of the species in the standard state.
Standard conditions in the calculations are usually 25 C and 1 bar, and the exergy calculations are based on Szargut theory of exergy.Solid and liquid streams are calculated in the standard pressure, and gaseous streams adds the possible pressure differing from the standard state to the exergy.In the software, elemental exergy is accounted as a part of the chemical exergy, and all values for each species are calculated separately and then added up to the total values of the streams.All the information for the calculations in this work was obtained from the HSC Chemistryâ, except for temperature, pressure, number of species and their masses in the stream.A challenge with exergy calculations in HSC is that the mixing entropy cannot be associated with specific species in a meaningful way, so entropy increase due to mixing of different materials is left out from the equation.Thus, in the calculations presented in this manuscript, only the stream exergy is accounted for.
The system may also use electricity or heat in the process.In HSCâ, electricity is calculated directly into the total exergy.For the heat input and losses, the software uses the Carnot cycle to convert heat to exergy, as presented in Equation 11.
Ex heat = q Ã 1 À T 0 T (Equation 11) where q is heat flow, T 0 is the temperature in the standard state, and T is the temperature of the heat source.Unlike energy, exergy does not satisfy the law of conservation.Thus, not all the input exergy is transferred from feed to products. 60The lefthand side of the exergy balance in Equation 12 considers the input exergy as the result of both the input exergy contained in the feed materials (Ex feed ) and the exergy provided by external energy inputs (Ex energy ).The terms related to the exergy output include product streams (Ex products ), waste streams (Ex waste ), the irreversible exergy destruction (Ex D ), and the exergy lost in the form of heat (Ex Q ).It is worth pointing

Figure 1 .
Figure 1.The concept of hyperspherical engineering: from circular to spherical and further to a hyperspherical solution

Figure 2 .
Figure2.The Retriev process for LIB recycling according to MFA methodology with different streams (s), stages (q) and units (u)

Figure 6 .Figure 7 .
Figure 6.Product life cycle, recycling process according to MFA methodology, and an exentropy analysis of a process including best-case scenario: (A), worst-case scenario (B), and regular case (C) k + DG 0 f ð25 C;1barÞ + N i À N ið25 C;1barÞ À T 25 C À S i À S ið25 C;1 barÞ Á (Equation

Table 1 .
Substance concentration efficiencies (SCEs) for different elements with different concentrations of LiOH in the Retriev process