2.1. Selection of Earth system boundaries

The first step in the ERA method is to select an environmental sustainability objective and a set of suitable boundaries describing the objective. The boundaries can be global or regional, interdependent or independent, and be either driver, pressure, state or impact in the DPSIR frameworkFootnote ¹ (Niemeijer & de Groot, Reference Niemeijer and de Groot2006, Reference Niemeijer and de Groot2008). The condition for selecting a boundary is that it needs to be measurable using the methods chosen to allocate the safe operating space (SOS; step 3) and quantify the impacts (step 4). In general, each boundary demarcates the maximum value for an indicator that can be tolerated by the Earth system before it collapses or changes irreversibly (Catton, Reference Catton1986; Rees, Reference Rees1996). The uncertainty range of the boundary can reflect the parameter uncertainty and/or increased risk to society associated with the transgression of the boundary value (Steffen et al., Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015).

The ERA method is based on the precautionary principle, which means that despite the often large uncertainties in both boundary values and impact measurements, the resulting ERA budgets needs to be viable with high confidence (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019, Reference Desing, Brunner, Takacs, Nahrath, Frankenberger and Hischier2020). Therefore, an acceptable probability of violating the defined boundaries P _v is required as an input parameter. This value is a normative choice, reflecting the acceptance of risk in society.

For the ERA calculations, the boundary values are stored in a three-dimensional matrix (n _ESB × 1 × n _runs), with the third dimension containing random values for each element in the other two dimensions according to the probability distribution of the respective boundary.

(1)$$\eqalign{& ESB\quad = \quad \matrix{ {\vskip-5pt {\rm boundaries}\downarrow } \cr \matrix{\hskip -1.5pt{\displaystyle 1} \hfill \cr \vdots \hfill \cr \hskip -9pt n_{ESB} \hfill} \cr } \quad \mathop {\left[{\matrix{ {} \cr {ESB_i} \cr {} \cr } } \right]}\limits^ {\displaystyle 1} \matrix{ {} \cr {} \cr {} \cr {\vskip4pt\hskip-6pt{\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} n_{{\rm runs}}} \cr }} $$

2.2. Resource segment definition

The second step in the ERA method is to select the resource segment k to be investigated. Such a segment comprises one or several resources (e.g., metals). This choice may be limited depending on the data source used for allocation. For example, the Exiobase database (Stadler et al., Reference Stadler, Wood, Bulavskaya, Södersten, Simas, Schmidt and Tukker2018; Tukker et al., Reference Tukker, Poliakov, Heijungs, Hawkins, Neuwahl, Rueda-Cantuche and Bouwmeester2009) provides information on the aggregated industry level (e.g., casting of metals). In this case, the resource segment needs to comprise at least all (bulk) metals. When a resource segment comprises more than one resource, the relative SoP of each resource needs to be specified. The SoP expresses the fraction that a single resource contributes to the total mass flow of the segment. The SoP data are stored in a vector (n _resources × 1).

(2)$$\eqalignb{& \matrix{ {\quad \quad \quad {\rm resources}{\mkern 1mu} \downarrow } \cr } \matrix{ {\quad \quad \,\,\,\,\,\,\quad 1} \cr } \cr & SoP\quad = \quad \matrix{ A \cr \vdots \cr X \cr } \quad \quad \quad \quad \left[{\matrix{ {} \cr {\displaystyle{{m_j} \over {\sum\nolimits_{i = A}^X {m_i} }}} \cr {} \cr } } \right]} $$

2.3. Allocation of safe operating space

The ESBs define the SOS (Rockström et al., Reference Rockström, Steffen, Noone, Persson, Chapin, Lambin and Foley2009a), within which collective human activities can be considered environmentally sustainable. This space needs to be allocated to specific activities in order to set boundaries for those activities. In the ERA method, the ESBs are allocated to the resource segment under investigation, resulting in segment boundaries SB _k. This allocation is performed by assigning a share of the safe operating space (SoSOS) (Ryberg et al., Reference Ryberg, Owsianiak, Richardson and Hauschild2018b) to the segment k for each boundary i. There are many different possibilities to define the SoSOS (EEA & FOEN, 2020) (e.g., economic value of a resource – Ryberg et al., Reference Ryberg, Owsianiak, Clavreul, Mueller, Sim, King and Hauschild2018a, 2018b; grandfathering approach – Hollberg et al., Reference Hollberg, Lützkendorf and Habert2019; or future emission scenarios – Pineda et al., Reference Pineda, Tornay, Huusko, Delgado Luna, Aden, Labutong and Faria2015). In general, any allocation principle can be applied in the ERA method.

The allocated segment boundary (SB _k), which is the absolute share of the total boundary available for the respective resource segment, is calculated by multiplying the diagonal of SoSOS with ESB for each Monte Carlo simulation run (third dimension). It has the same matrix dimension as ESB.

(3)$$SB_k = diag\lpar {SoSOS_k} \rpar \times ESB$$

2.4. Environmental impacts of resource production

After specifying the segment boundaries, the impacts on these boundaries need to be determined for one unit of resource provision. Impacts not only arise from extraction and primary material production, but also from EoL treatment. Every material introduced into the socioeconomic system eventually finds its way back to the environment, be it through unintentional dispersion (e.g., abrasion) or intentional discharge (e.g., landfill) (Desing et al., Reference Desing, Brunner, Takacs, Nahrath, Frankenberger and Hischier2020). Following the responsibility principle,Footnote ² environmental impacts for final disposal have to be added to primary production. Impacts associated with the use of the material, such as product manufacturing, product use and cycling strategies, are not considered in the ERA method, as these are covered in the respective sectors in the economy. Impacts resulting from the input of energy and materials in the use phase of the material are considered in the respective ERA budgets. Direct use impacts are to be considered in the relevant segments. Similarly, secondary material production can be considered as a separate segment with its own boundary, and thus it is assumed not to affect the ERA of primary materials. Cycling strategies do increase the resource base available to the economy, but only delay and do not prevent materials from entering final sinks.

Four principal disposal options can be distinguished: open dump (= dispersion), landfilling, incineration and sewage sludge. If the disposal routes are known (e.g., for plastics in Switzerland, see Kawecki & Nowack Reference Kawecki and Nowack2019), an overall waste process can be created as a combination of the four options. Otherwise, impacts can be modelled with an uncertainty range spanning the best and worst option.

The application of the precautionary approach necessitates modelling of uncertainties for unit impacts. For example, uncertainties in life cycle assessment (LCA) stem from the inventory uncertainties (e.g., measurement errors, spatial and temporal variability; Ecoinvent, 2013; Muller et al., Reference Muller, Lesage, Ciroth, Mutel, Weidema and Samson2014) and different life cycle impact assessment methods (e.g., different time horizons for global warming potentials). Data for unit impacts (UI) are stored in a three-dimensional matrix (n _resources × n _ESB × n _runs):

(4)$$\eqalignb{& \matrix{ {\quad \quad \quad {\rm resources} {\mkern 1mu} \! \downarrow } \cr } \matrix{ {\quad \,\,\,\,\,\,\quad \ 1\ \quad \ldots\quad n_{ESB}} \cr } \cr & UI\quad = \quad \matrix{ A \cr \vdots \cr X \cr } \quad \quad \quad \quad \left[{\matrix{ {\displaystyle{\vskip10pt{m_{CO_2}} \over {m_A}}} \,\,\,\quad \,\,\,\,\,\,\quad \cr {}{}{} \cr\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, {}{}{\displaystyle{{E_{el}} \over {m_X}}} \cr } } \right]}\matrix{ {} \cr {} \cr {}\cr {} \cr {{\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} n_{{\rm runs}}} \cr } $$

As a consequence of the global scope of the ERA method, the processes modelled to calculate the unit impacts have to be representative of the global average for primary production (e.g., global datasets in ecoinvent). Attention has to be paid to ensure that secondary materials are not included in these global averages.

2.5. Upscaling of resource production

The last step of the ERA method increases the production of a resource (and thus its environmental impacts) until the probability density functions (PDFs) of both the boundaries and environmental impacts overlap. As long as all possible impacts are smaller than all possible boundary thresholds, the system can be considered fully sustainable. However, as soon as some possible impacts are larger than some possible boundaries (i.e., when two PDFs start to overlap), there is an increasing probability that the system will violate the environmental sustainability criterion and thus be not sustainable (see Figure 2). If all impacts are larger than their respective boundaries, then the system is certainly not sustainable.

Fig. 2. Schematic representation of the concept of probability of boundary violation, which results from the overlap from the probability distribution of the environmental impacts with the distribution of the respective boundary.

The UI of the individual resources within one segment are combined to an overall unit impact for the segment, UI _k (i.e., the impact caused by the production of one unit of the resource mix as specified by the SoP; see Section 2.2).

(5)$$UI_k = SoP^T \times UI$$

The production volume of the resource mix in the segment ERA _k is then increased stepwise, until one segment boundary SB is violated with the chosen probability of violation P _v. As the calculation of P _v,i in each step depends on the shape of the distributions of both impacts and boundaries, a numerical approach is taken. The initial ERA _k,i is calculated as the $1-{{P_v} \over 2}$-quantile of the UI _k PDF and the ${{P_v} \over 2}$-quantile of the SB _k PDF. In an iteration loop, the P _v is calculated and the ERA _k,i+1 increased or decreased until P _v,i equals the required value P _v.

(6)$$ERA_{k\comma i} = \displaystyle{{quantile\,\left({SB_k{\rm \vert }\displaystyle{{P_v} \over 2}} \right)} \over {quantile\,\left({UI_k{\rm \vert }1-\displaystyle{{P_v} \over 2}} \right)}}$$

(7)$$ERA_{k\comma i + 1} = \left\{{\matrix{ {ERA_{k\comma i}\cdot \lpar {1-5\cdot P_v\lpar {1 + P_{v\comma i}} \rpar } \rpar } \hfill & {P_{v\comma i} \gt P_v} \hfill \cr {ERA_{k\comma i}\cdot \lpar {1 + P_v-P_{v\comma i}} \rpar } \hfill & {P_{v\comma i} \lt P_v} \hfill \cr } } \right.$$

The resource budget ERA _k is calculated for each boundary separately. As all SBs need to be respected, the smallest ERA _k defines the budget for the resource segment. For each resource in the resource segment k, the resource budget is calculated through multiplying the smallest ERA _k with the SoP. ERA is a vector (n _resources × 1) and contains the ERA budget for each of the resources in the segment with the chosen confidence 1 − P _v.

(8)$$ERA = SoP\cdot min\lpar {ERA_k} \rpar $$

3.1. Selection of Earth system boundaries: adaptation of planetary boundaries

As the sustainability objective, we choose the protection of the Holocene-like state of the Earth system and use the PB framework as a set of ESBs (Rockström et al., Reference Rockström, Steffen, Noone, Persson, Chapin, Lambin and Foley2009b; Steffen et al., Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015). The adaptation of the boundaries is based on the literature and for the purpose of demonstrating the ERA method. Furthermore, the probability of violating the boundaries is set to P _v = 0.01 (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019) to illustrate the method. This is in between the probability of failure tolerated for critical technical systems (usually <0.001; see Table S2) and deemed acceptable in current Earth system governance (e.g., P _v = 0.33 for reaching the 2.0°C target (IPCC, 2013); P _v = 0.5 for reaching the 1.5°C target (IPCC, 2018), despite potentially catastrophic effects (Lenton et al., Reference Lenton, Rockstrom, Gaffney, Rahmstorf, Richardson, Steffen and Schellnhuber2019)).

The PBs define boundary values for nine crucial Earth system processes.Footnote ³ When crossing the boundary values, fast and irreversible environmental change is expected to happen, leading to a new Earth system state being less hospitable for human civilizations and most other forms of life (Steffen et al., Reference Steffen, Rockstrom, Richardson, Lenton, Folke, Liverman and Schellnhuber2018). Respecting the PBs can be seen as necessary for reaching environmental sustainability, but not sufficient to ensure it (Chandrakumar & McLaren, Reference Chandrakumar and McLaren2018). The PB framework can be refined with more detailed control variable definitions (e.g., for biodiversity, see Alig et al., Reference Alig, Frischknecht, Nathani, Hellmüller and Stolz2019; Mace et al., Reference Mace, Reyers, Alkemade, Biggs, Chapin, Cornell and Woodward2014) or extended to other variables (e.g., net primary production O'Neill et al., Reference O'Neill, Fanning, Lamb and Steinberger2018; Running, Reference Running2012). To show this, we add a boundary on renewable energy potentials (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019), representing the amount of energy that can be appropriated by society without transgressing other critical PBs (e.g., land-system change) or compromising food supply. The global appropriable technical potential (ATP) for renewable energy is estimated to be $ATP_{el\comma p = 0.98} = \lpar {1.52_{{-}0.81}^{ + 1.24} } \rpar \times 10^{14}{\rm W}$ (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019). This added boundary is a global boundary for a driver of environmental pressure and is particularly relevant for a CE, as for increasingly closed material cycles, energy may become limiting (Ayres, Reference Ayres1999).

The PBs themselves need translation in order to be compatible with the measurement of impacts with units commonly used in LCA and environmentally extended input–output tables (EE-IOTs). This has been addressed by various authors using different approaches (e.g., impact reduction targets in LCA – Sandin et al., Reference Sandin, Peters and Svanström2015; deriving measurable boundaries – Dao et al., Reference Dao, Peduzzi, Chatenoux, De Bono, Schwarzer and Friot2015, Reference Dao, Peduzzi and Friot2018; Meyer & Newman, Reference Meyer and Newman2018; boundary characterization factors – Ryberg et al., Reference Ryberg, Owsianiak, Richardson and Hauschild2018b; per-capita impact allowance – Bjørn & Hauschild, Reference Bjørn and Hauschild2015; Doka, Reference Doka2016; combining footprints with PBs – Fang et al., Reference Fang, Heijungs and De Snoo2015). For the purpose of illustrating the method, we do not consolidate the various different approaches, which is a topic for further research. Meanwhile, we derive measurable boundaries and uncertainty ranges by using a combination of approaches in the literature (Table 1, Figure S1 and SM), except for the three boundaries described in the following. We adopt the biodiversity boundary from Doka (Reference Doka2016); however, we highlight the need to refine this boundary in the future. For example, up-to-date and United Nations Environment Programme (UNEP)–Society of Environmental Toxicology and Chemistry (SETAC) recommended methods to assess the global fraction of species loss exist (UNEP & SETAC, 2016) and can be converted into extinction rates (Alig et al., Reference Alig, Frischknecht, Nathani, Hellmüller and Stolz2019). The land appropriation boundary in Ryberg et al. (Reference Ryberg, Owsianiak, Richardson and Hauschild2018b) only considers forest area, which cannot be measured easily with LCA and EE-IOTs, whereas in Doka (Reference Doka2016), all biomes other than forest can be appropriated, endangering biodiversity in these biomes. Bjørn and Hauschild (Reference Bjørn and Hauschild2015) define the land boundary based on minimum biodiversity conservation targets; however, they do not consider the forest boundary set with climate considerations of the original PBs. We therefore adopt the land boundary setting in our earlier work (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019), which combines the remaining forest area boundary from Steffen et al. (Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015) with the ‘Nature Needs Half’ proposal (Dinerstein et al., Reference Dinerstein, Olson, Joshi, Vynne, Burgess, Wikramanayake and Saleem2017) for all other biomes.

Table 1. Translation of Earth system boundaries considered in this study (10 variables from the planetary boundaries framework (Rockström et al., Reference Rockström, Steffen, Noone, Persson, Chapin, Lambin and Foley2009b; Steffen et al., Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015) and complemented by the appropriable technical potential for renewable energy (Desing et al., Reference Desing, Widmer, Beloin-Saint-Pierre, Hischier and Wäger2019)) to annual flows compatible with environmentally extended input–output tables and life cycle assessment units. Intervals are expressed in this paper in their mathematical form (i.e., x = [x _min, x _max) meaning x _min ≤ x < x _max). This notation implies a level of confidence of the interval of p = 1. Uncertainty of values is reported for a specified level of confidence of the interval p, the 50% value and the lower and upper deviations that confine the interval.

For the CO₂ boundary, we propose a different approach to be in line with the ERA method's scope of maintaining a sustainable state of the Earth system over time. The atmospheric CO₂ concentration needs to be constant at or below the PB. Without human influence, the CO₂ concentration in the atmosphere had been decreasing (Foster et al., Reference Foster, Royer and Lunt2017). Continuous CO₂ emissions are possible to the extent of continuous removal by natural processes. CO₂ removal by sedimentation and weathering has been relatively constant over the last 20 Ma (IPCC, 2013; Stein, Reference Stein1991), when atmospheric CO₂ concentration had been below 450 ppm (IPCC, 2013) and therefore within or below the PB, and so overcompensating for natural CO₂ release (e.g., volcanism). The condition of keeping the concentration constant allows continuous (but small) emissions of anthropogenic CO₂ of $\dot{m}_{{\rm C}{\rm O}_2} \lt [ {8.25\comma \;22} ] \times 10^{11}\,{{{\rm kg}} / {\rm a}}$. This boundary translation is in contrast to other studies, where the CO₂ emission boundary is set with a specific time horizon (e.g., 300 a – Ryberg et al., Reference Ryberg, Owsianiak, Richardson and Hauschild2018b; or for negative emissions required between 2050 and 2080 to reach the 350 ppm target before 2100, see Meyer & Newman, Reference Meyer and Newman2018).

3.3. Allocation of the safe operating space: grandfathering approach

The allocation principle has an important influence on the results (Bjørn et al., Reference Bjørn, Richardson and Hauschild2018; Ryberg et al., Reference Ryberg, Owsianiak, Clavreul, Mueller, Sim, King and Hauschild2018a; Sabag-Munoz & Gladek, Reference Sabag-Munoz and Gladek2017). To illustrate the ERA method, a grandfathering allocation approach (Sabag-Munoz & Gladek, Reference Sabag-Munoz and Gladek2017) is applied in this case study, where each resource receives a SoSOS equal to its historic impact share. For example, if the production of steel is responsible for 9% of the global CO₂ emissions today, steel receives the same share of the CO₂ boundary. This approach reflects the question: what if we rescale today's socioeconomic system to fit within ESBs? As today's world economy already transgresses six boundaries (see Figure 3), resource production as well as final demand would have to be downscaled significantly, leaving large parts of the global society without access to basic services. This scenario is therefore to be seen as indicative only.

Fig. 3. Panel (A) shows the relative impact contribution for nine resource segments (including supply chain impacts), which defines the share of the safe operating space (SoSOS) in the grandfathering approach. The contributions of the resource segment metals (5) are highlighted with red boxes. The rest of economy segment (10) represents all activities that are not part of the resource production chain (e.g., manufacturing of end-user devices), while the final demand segment (11) comprises the purchase and use of goods and services by the end consumer. In panel (B), the total global environmental impacts of the socioeconomic system in the year 2011 is compared to the global boundaries. Six out of the 11 boundaries are crossed with a probability greater than 1%. All values are scaled relative to the 0.5 percentile of the respective boundary distribution ($1\hat{ = }ESB_{p = 0.005}$).

The calculation of the relative historic impact of each resource segment is conducted with a top-down approach, using the Exiobase database v3.4 (Stadler et al., Reference Stadler, Wood, Bulavskaya, Södersten, Simas, Schmidt and Tukker2018; Tukker et al., Reference Tukker, Poliakov, Heijungs, Hawkins, Neuwahl, Rueda-Cantuche and Bouwmeester2009).Footnote ⁴ In Exiobase, the emissions for each industry are reported that are directly produced through the industry's activity itself. Industries concerned with the production or EoL treatment of materials are grouped into nine resource segments, and the remaining industries are aggregated into rest of economy. Impacts associated with consumption in households are contained in the final demand category. We use the approach from Cabernard et al. (Reference Cabernard, Pfister and Hellweg2019) to calculate the cumulative impacts for nine resource segments without double counting the impacts already included in the supply chain of another segment (e.g., metals necessary to produce plastics). A detailed description of the calculation, the impact characterization methods used and the uncertainty modelling can be found in the SM. The results of the SoSOS calculations are shown in Figure 3(A). The SoSOS is determined by the 50% value of the uncertainty distribution resulting from the Monte Carlo simulation. This set of values is chosen so that the total relative impacts add up to 1 and the full SOS can be utilized. The SoSOS has been determined with data from the year 2011 (the difference from the same procedure with data from 1995 is small, Person covariance r = 0.9954). The overall impact for the year 2011 on the global boundaries as calculated with Exiobase is qualitatively consistent with Steffen et al. (Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015), except for the two climate boundaries. In Steffen et al. (Reference Steffen, Richardson, Rockstrom, Cornell, Fetzer, Bennett and Sorlin2015), the impact on these boundaries is within the uncertainty range, whereas in our assessment, they exceed the boundaries by far (see Figure 3(B)). This is due to the translation of state variables (CO₂ concentration, irradiation imbalance) into pressure variables ($\dot{m}_{{\rm C}{\rm O}_2}$, $\dot{m}_{{\rm C}{\rm O}_2\comma {\rm eq}}$; see Section 3.1).

3.4. Environmental impacts of metals production

The impacts on the selected boundaries caused by the production and EoL treatment of one unit of material are calculated with a bottom-up approach using LCA and ecoinvent v3.5 (Wernet et al., Reference Wernet, Bauer, Steubing, Reinhard, Moreno-Ruiz and Weidema2016). This database is chosen because it aims to offer global coverage of the supply chains of materials in necessary detail. Hence, for each material, at least a global average dataset is available as required in the ERA method for calculating global resource budgets. The unit impacts are constructed from two processes: (1) primary material production (e.g., aluminium, primary, ingot); and (2) EoL treatment, which can consist of several processes (e.g., incineration, landfill). From ecoinvent, the cut-off (or recycled content) system model is chosen, as in this model the primary material production chains show the entire impacts of the production steps; no parts of these impacts are already allocated to secondary material cycles or to EoL treatment processes. At the same time, the EoL processes contain only the impacts related to the respective treatment processes. An overview of all of the processes considered and the impact characterization methods and uncertainty considerations for this analysis is provided in the SM.

3.5. Upscaling of resource production: ERA determination

Following the calculation steps (Section 2.5) and using n _runs = 10⁵ simulation runs leads to the results as depicted in Figure 4. The limiting SB for metals is CO₂ due to the large overshoot of global impacts on this boundary. The production in 2016 (USGS, 2016), ω, SoP and ERA budgets are given in Table 2 for the metal segment. The ERA for the whole segment is 3.5 × 10¹⁰ kg/a and thus is 40 times smaller than production output in the year 2016. In other words, the primary resource production must be reduced by a factor of 40 to be within the selected ESBs in the grandfathering approach. The CO₂ boundary is limiting the production of metals (see Figure 4), as it is the boundary that is crossed the most on the global scale (see Figure 3). Within the segment, steel is dominating the impact shares due to its large production share (SoP _steel = 0.9). The depletion time of extractable global resources (Henckens et al., Reference Henckens, Driessen and Worrell2014; USGS, 2016) at the rate of ERA is for all investigated metals >>1500 a (except 920 a for gold). This confirms the hypothesis that the depletion of physical resources is not the pressing constraint for a sustainable economy, but rather the environmental impacts caused.

Fig. 4. Panel (A) shows the relative contribution of each metal to the total environmental impacts of the resource segment metals. In panel (B), the cumulative impacts of metals (blue) are compared relative to the 0.5 percentile of the allocated boundary for the metal segment (green) ($1\hat{ = }SB_{p = 0.005}$). CO₂ is limiting the metal segment ($P_{v\comma {\rm C}{\rm O}_2} = 0.01$).