How much can novel solid sorbents reduce the cost of post-combustion CO2 capture? A techno-economic investigation on the cost limits of pressure-vacuum swing adsorption

This paper focuses on identifying the cost limits of two single-stage pressure-vacuum swing adsorption (PVSA) cycles for post-combustion CO2 capture if the “ideal” zerocost adsorbent can be discovered. Through an integrated techno-economic optimisation, we simultaneously optimise the adsorbent properties (adsorption isotherms and particle morphology) and process design variables to determine the lowest possible cost of CO2 avoided (excluding the CO2 conditioning, transport and storage) for different industrial flue gas CO2 compositions and flow rates. The CO2 avoided cost for PVSA ranges from 87.1 to 10.4 ¤ per tonne of CO2 avoided, corresponding to CO2 feed compositions of 3.5 mol% to 30 mol %, respectively. The corresponding costs for a monoethanolamine based absorption process, using heat from a natural gas plant, are 76.8 and 54.8 ¤ per tonne of CO2 avoided, respectively showing that PVSA can be attractive for flue gas streams with high CO2 compositions. The “ideal” adsorbents needed to attain the lowest possible CO2 avoided costs have a range of CO2 affinities with close to zero N2 adsorption, demonstrating promise for adsorbent discovery and development. The need for simultaneously optimizing the particle morphology and the process conditions are emphasized.


Introduction
Carbon dioxide capture and storage (CCS) from point sources are expected to play a key role in decarbonising the global energy and industry sectors. 1,2 The feasibility of implementing CCS may vary from one industry to the other since several factors such as CO 2 composition, pressure, the flow rate of the flue gas, system-level integration aspects etc. affect the attractiveness of CCS. 3 In the context of system-level integration, post-combustion CO 2 capture can be retrofitted into existing chemical/power plants in a rather straightforward manner without restructuring the plant layout and has been identified as one of the viable technologies in short-to medium-term. 4 While the majority of industrial sources emit CO 2 containing gases at atmospheric pressure, the variation in CO 2 composition is typically in the range of 3.5-30% across industries. 5 Although there are several CO 2 capture technologies considered for post-combustion CO 2 capture, the associated energy penalty and cost expenditure remains a barrier for the large-scale implementation. 6 Among all, solvent-based CO 2 capture is at the forefront owing to its technological maturity and commercial implementation. Adsorption-based processes are proposed as an alternative to the traditional solvents for their ability to lower the energy penalty and the costs related to the capture. 6 One of the main drivers determining the performance of adsorption processes is the choice of adsorbents. Recent developments in material science have facilitated material chemists to discover several new classes of adsorbents, such as metal-organic frameworks (MOFs), covalent-organic frameworks (COFs), etc., that can be highly tuned for CO 2 capture applications. 4 Since each class can typically consist of hundreds of thousands of materials, including both real and hypothetical structures, the selection of suitable adsorbents remains crucial for assessing the potential of adsorptive CO 2 capture. The quest for the best performing adsorbents has resulted in several in-silico material screening studies that use various performance metrics to rank materials. [7][8][9][10][11][12][13][14][15][16] While the initial focus primarily relied on simplified process metrics (derived under equilibrium conditions) as means to evaluate the performance of adsorbents in the real process, integration of dynamic process modelling and optimisation into adsorbent screening was later identified as a reliable tool to evaluate the realistic performance of adsorbents. 10,12,13,[16][17][18][19] As a result, studies focusing on the multiscale screening of known material databases have emerged wherein the adsorbent properties determined through molecular simulations are later incorporated into the process simulation and optimisation routines to identify/rank top material performers based on key processoriented metrics such as the CO 2 purity, CO 2 recovery, energy penalty, productivity or cost of the CO 2 capture. 13,[16][17][18] Alternatively, the problem of identifying the desirable adsorbent properties in processes was also approached through process inversion 20,21 . The process inversion approach focuses on determining the "ideal" adsorbent properties that result in the best process performance through an integrated adsorbent-process optimisation. In other words, the adsorbent properties are simultaneously optimised along with the process variables in the optimisation. This approach helps in identifying the best performance limits of adsorptive CO 2 capture. 20 Both simplified and detailed process models have been used in this approach. For instance, feature spaces of adsorbent properties such as CO 2 and N 2 adsorption isotherms, heat of adsorption, Henry's constant, etc., were probed using process inversion approach in order to determine the lowest energy penalty 20,22,23 and CO 2 capture costs 15,21 for post-combustion adsorptive CO 2 capture. More recently, Pai et al. explored adsorbent properties such CO 2 and N 2 adsorption equilibria that minimise the energy penalty and maximise the productivity for different flue gas compositions using a machine learning model. 24 In our earlier publication, 25 we demonstrated that the realistic performance of adsorbents should be assessed by incorporating a comprehensive techno-economic analysis framework with detailed process modelling and optimisation. This is primarily because the cost assessment captures the inherent complexities associated with the scale-up of the processes for industrial applications, which otherwise are not quantified when using process performance metrics such as energy penalty or productivity.
While previous studies provide some insights into understanding the underlying relation-4 ships between the adsorbent properties and the process performance, in this study, we pose the following key questions: 1. If "ideal" adsorbent(s) were discovered, what are the cost limits of adsorptive postcombustion CO 2 capture from industrial flue gases? 2. How do the costs compare with the benchmark technology, i.e. absorption?
Addressing these questions is critical to understand the true potential of adsorption processes and thus allow for advances in both material discovery and process design. This paper aims to answer the questions mentioned above by employing a process inversion approach. We restrict our analysis to single-stage pressure-vacuum swing adsorption (PVSA) technology, a widely studied class of adsorption processes for CO 2 capture applications. In this study, we define the cost limits as the lowest possible achievable costs for capturing CO 2 from post-combustion industrial flue gases using "ideally" desired adsorbent features in the PVSA process considered. Moreover, we also use this opportunity to show the impact of parameters such as the vacuum level required in the process, pellet morphology and adsorbent costs on PVSA costs. Further, we carry out a one-to-one comparison with benchmark monoethanolamine (MEA) solvent cases for various industrial applications. Finally, the cost performance of two "real" adsorbents are evaluated and compared with the limits to identify the potential for "material innovation". This is the first study to report such a comprehensive analysis to the best of the authors' knowledge.
The current article is organised as follows: the next section summarises the different cases we considered to encompass the wide range of industrial applications. In the computational details section, the process inversion approach through integrated techno-economic optimisation framework is explained, and details of adsorbent properties, process model, scale-up and economic assessment are provided. The results and discussion section reports the findings obtained from optimisations and compares them with benchmark MEA-based CO 2 capture case. The merits and demerits of PVSA for post-combustion CO 2 capture are discussed in the concluding remarks, along with some perspectives towards the advancement of adsorptive CO 2 capture.

Case study
For this study, a case matrix comprising a wide range of CO 2 compositions at different flue gas flow rates is considered to represent various industrial post-combustion flue gas sources adequately. 5 Under dry conditions, the flue gas consists of CO 2 /N 2 binary mixture and the CO 2 molar compositions in the flue gas are varied between 3.5% and 30%. This range corresponds to flue gas sources from simple cycle gas turbine plants, natural gas combined cycle power plants, coal-fired power plants, cement and steel industries. Further, the analysis is extended to different flue gas flow rates ranging from 303 tonnes/h to 3696 tonnes/h to account for the effect of the scale of operation. Table 1 illustrates the case matrix used in this study. In all cases, the flue gas is available at 1 bar and 35 • C for post-combustion CO 2 capture.
The system under consideration includes CO 2 capture from dry flue gas. We exclude the following while estimating the costs: the process that emits CO 2 containing flue gas, CO 2 conditioning, CO 2 transport and CO 2 storage. The process layout of adsorptive CO 2 capture is provided in Fig. S1 in the ESI †. The dry flue gas further undergoes compression followed by cooling to 25 • C. Multiple adsorption process trains with each N columns are employed to treat the dry flue gas. The CO 2 rich product and N 2 are collected separately using separate vacuum pumps.
Further, adsorptive CO 2 capture is benchmarked against the baseline monoethanolaminebased (MEA) technology to fully understand the potential of adsorption process technology for various industrial applications. To be consistent, the system boundaries for both PVSAand MEA-based CO 2 capture were kept the same. The MEA-based CO 2 capture performances based on Fu et al 5 are summarised in the ESI †. 6

Computational details
The cost limits of adsorptive CO 2 /N 2 binary mixture separations are determined using a recently developed integrated techno-economic optimisation model. 25 Our computational framework integrates adsorbent, process and economic aspects to determine the cost-optimal performance of adsorptive post-combustion CO 2 capture on industrial scales, as shown in Fig.   S2 in the ESI †. Both adsorbent properties and process design parameters are simultaneously optimised herein to determine the lowest possible costs.

Adsorbent features
Adsorbent properties that are required for process modelling include CO 2 and N 2 adsorption isotherms, crystal density, isosteric heats of adsorption, pellet porosity, pellet diameter and specific heat capacity. Physicochemical properties such as CO 2 and N 2 adsorption isotherms, crystal density and isosteric heats of adsorption are inherent crystal properties, while pellet porosity, pellet diameter and specific heat capacity are properties of adsorbent within the adsorption column.
A practically deployable sorbent for CO 2 capture should have several critical features, e.g., low cost, scalability, stability, etc. However, the ability to separate CO 2 and N 2 , i.e., the two key components of flue gas, is arguably the most important feature. Most practical adsorption-based CO 2 and N 2 separations exploit the differences in affinity between CO 2 and N 2 on a specific sorbent. The affinity is expressed in the form of an adsorption isotherm that relates the fluid and solid phase concentrations at equilibrium. The hypothetical CO 2 and N 2 adsorption isotherms were expressed in terms of the competitive dual-site Langmuir (DSL) isotherm model. The advantages of using the competitive DSL isotherm model involve computational simplicity (because of the explicit formulation), and also the ability to adequately represent the mixture equilibrium predictions from single component parameters for many practical systems 10,26 . The competitive DSL isotherm model (for component i) is given by, In Eq. 1, c i is the fluid phase concentration of component i, q * i is the equilibrium solid-phase loading of the component i, q sb,i and q sd,i represent saturation capacities for the two sites and, b i and d i are the temperature dependent adsorption equilibrium constants defined as: where ∆U b,i and ∆U d,i are the internal energies of the two sites. Different hypothetical CO 2 and N 2 adsorption isotherms can be generated by varying the parameters q sb , q sd , b 0 , d 0 , ∆U b and ∆U d . For many known adsorbents, CO 2 adsorption is heterogeneous and the DSL isotherm model can reasonably describe the equilibrium while N 2 adsorption is homogeneous. 20 Consequently, the DSL isotherm parameter variation was constrained such that both thermodynamic consistency and homogeneity of N 2 adsorption are maintained.
This can be accomplished by describing the competition between CO 2 and N 2 between the two sites using equal energy site (EES) formulation. 18 Here, the saturation capacity of each site remains the same for both components, i.e., q sb,CO2 = q sb,N2 and q sd,CO2 = q sd,,N2 .
Also, the internal energy of adsorption and constants b 0 and d 0 for N 2 are kept identical between the two sites, i.e., ∆U b,N2 = ∆U d,N2 and b 0,N2 = d 0,N2 . Experimental evidence also supported this type of formalism for Zeolite 13X. 27 The physicochemical properties for achieving lowest costs are examined by considering the following parameters q sb,CO2 , q sd,CO2 , In a recent study, Farhamini et al 16 showed that both pellet porosity ( p ) and pellet diameter (d p ) can significantly affect the process performance. Hence, the variation of p and d p was also considered in the optimisation. Other properties such as crystal density and the specific heat capacity of the adsorbent are held constant to that of Zeolite 13X. While specific heat capacity can potentially impact the process performance, 16 especially under adiabatic conditions, it is, however, held constant because of lack of data.  Owing to its simple features, this cycle has been benchmarked by various studies 16,18,20,21 and was also demonstrated at a pilot plant facility. 28 We considered a more complex six-step PVSA cycle with dual reflux (DR) as the second cycle. 20,21,29 The cycle schematic is shown in Fig. S4 in the ESI †. In addition to the four steps above, this cycle comprises two reflux steps:

Process model and economic analysis
5. Light reflux (LR) step after the evacuation at P L where the light product from the adsorption column is used as reflux to purge the column in the LR step.
6. Heavy reflux (HR) step after the adsorption step at P H by using the product from the LR step in order to increase the CO 2 partial pressure in the column.
The process simulations were carried out using a non-isothermal, one-dimensional math-  30 . The resulting ordinary differential equations (ODEs) were then integrated using ode23tb, a stiff ODE solver in MATLAB. The cycle simulations were carried out based on a standard uni-bed approach where a single column undergoes all cycle steps in a sequence. The column was initialised with a feed mixture at P L and simulated until the process reached cyclic steady state (CSS). When the mass balance error for the PVSA process equals 1% or less was observed for five consecutive cycles, the process was considered to attain CSS. A minimum number of 50 cycles were simulated to ensure that the CSS criterion was adequate. For simulations where the system fails to achieve CSS, a maximum number of 500 cycles were simulated, after which it was assumed that the system attained CSS. At CSS, state variables such as composition, pressure and temperature profiles were determined to calculate key performance indicators. The process model was previously demonstrated to reproduce both lab-scale and pilot-scale experiments. 27,28 The column scheduling was carried out based on the method proposed by Khurana and Adsorbent decision variables: CO 2 DSL isotherm parameters (q sb,CO2 , q sd,CO2 , ∆U b,CO2 , and pellet diameter (d p ).
Most of the process decision variables were kept the same as that of the previous study 25 .
Additionally, the high pressure (P H ) in the adsorption step was also varied. Although feed velocity in the adsorption step can be explicitly varied in the optimisation, it was not considered as a decision variable in the present work. This was because the optimiser in an earlier study always approached the upper bounds in the techno-economic optimisations in order to reduce the total number of trains required for the separation 25

Results and Discussion
Cost limits of four-step PVSA cycle Unique optimisations were carried out to determine the minimum CO 2 avoided cost for each of the cases specified in Table 1. The optimisation was performed by varying both adsorbent and process design variables simultaneously in the optimisations. Note that the requirements of minimum 95% CO 2 purity and 90% CO 2 recovery were considered as constraints for all optimisations in this paper. The costs of adsorbents have been set to zero in these optimisations, which were based on the assumption that the "ideal" hypothetical adsorbent(s) identified using process inversion can be synthesised or available for zero-cost. Although not plausible practically, this assumption will determine the absolute minimum costs of building and operating simply the PVSA process alone without additional expenditures related to the adsorbent.   related to the cost limits of the six-step DR cycle. 14 the minimum CO 2 avoided cost obtained for 30% CO 2 composition is 12.2 per tonne of CO 2 avoided and the CO 2 avoided cost increases 1135% when the CO 2 composition is reduced from 30% to 3.5%. In Fig. 1 can be attributed to the P H and P L required in the process and due to low efficiency of the vacuum pumps at low pressures. Figure 2 shows the optimal values of P H and P L obtained over a range of CO 2 compositions. The shaded region around the optimal values represent the upper and lower limits of the solutions obtained within the 5% vicinity of the minimum CO 2 avoided costs. The rationale behind this is to account for the variation of P H and P L on the minimum avoided cost. As expected, P L decreases from ≈ 0.11 bar to 0.01 bar with lowering CO 2 composition from 30% to 3.5%, respectively; contrarily, P H increases from 1.8 bar to 2.9 bar. This trend is observed because the process requires a certain amount of working capacity from the adsorbent to meet the 95% CO 2 purity and 90% CO 2 recovery constraints. Hence, a higher pressure ratio, P H /P L is required.
We now turn our attention to the adsorbent properties that link to the cost limits. This is a key observation that points to the possibility that multiple adsorbents may be able to provide the comparable (low) cost of CO 2 capture. However, they may display widely varying CO 2 isotherms. This also highlights why the interplay between material property and process performance should be studied together. The optimal N 2 isotherms are shown in Fig. 3(f). Since the N 2 affinity for all cases was almost zero, the isotherms around the optimum were not considered. This again confirms that low N 2 adsorption is a very desirable property of an ideal adsorbent. compositions. For 3.5% and 7.5% CO 2 compositions, CO 2 purity-recovery constraints were not met in the optimisations and hence, we did not consider these compositions for the discussion. The cost limits decreased with an increase in CO 2 compositions. The difference between two cases remains minor (≤6%) for CO 2 compositions from 20% to 30% while at 13% CO 2 composition, a difference of 14% was observed. This indicates that the four-step PVSA process can still be operated at higher P L (≥0.1 bar) for higher CO 2 compositions but requires ultra-deep vacuum for lower CO 2 compositions in order to meet purity-recovery requirements.  Effect of pellet porosity and pellet size. One set of adsorbent decision variables in the cost limit optimisations relate to adsorbent properties in the pelletised form, namely, pellet porosity and pellet diameter. 16 Here, we seek to investigate the influence of pellet properties towards achieving the cost limits at different CO 2 compositions. We conducted this study by comparing two optimisation cases: in the first case, pellet porosity and pellet diameter were treated as decision variables along with other adsorbent and process decision variables in the optimisations, while the second case involves keeping pellet porosity and pellet diameter as fixed values. The cost limits discussed earlier represents the first case, meanwhile unique optimisations were carried out for the second case using fixed values of pellet porosity ( p =0.37) and pellet diameter (d p =1.5 mm) from the previous publication 25 which represents typical experimental values. 27,28 Figure 4(a) illustrates the comparison between the two cases. For the entire range of CO 2 compositions considered, the difference between the minimum CO 2 avoided costs for the two cases varies between 9 to 22%. At higher CO 2 compositions (i.e. ≥13%), the cost limits for fixed pellet properties are about 9-11% higher than the cost limits where pellet properties were varied. This difference increases to 19-22% at lower CO 2 compositions (i.e. <13%). The improvement in CO 2 avoided costs through optimisation of pellet morphology can be attributed to the increased values of optimal p and d p as shown in Fig. 3

(g)-(h).
The optimal values of p vary between 0.42 and 0.76, whereas the optimal d p lies in between 3.0 and 5.0 mm, which are greater than the typical pellet sizes used in PVSA operations.
It is worth noting that Farmahini et al also report similar ranges for p and d p in their energy-productivity optimisations. 16 Figure 4: (a) Impact of different process parameters on the cost limits of the four-step PVSA cycle. (b) Comparison between the cost limits of the four-step PVSA cycle with minimum CO 2 avoided costs obtained for real adsorbents (Zeolite 13X and IISERP MOF2). Note that the cost ratio was defined as the ratio between minimum CO 2 avoided costs obtained for the examined cases and the cost limits reported in Fig. 1 (a) at each CO 2 composition.
Influence of adsorbent costs. While an adsorbent cost of zero is interesting to understand the cost limit of PVSA, adsorbent costs cannot realistically be expected to be zero.
Although the adsorbent costs are dependant on the raw materials used to synthesise them, scale-up methods, etc., the question we asked is: if hypothetical adsorbents could be synthesised at similar costs as that of commercial adsorbents, what would be their contribution in bringing down CO 2 avoided costs? Hence, we studied the influence of adsorbent costs on the cost limits by considering three different adsorbent costs: 1) zero; 2) 1500 and; 4500 per tonne of adsorbent. To provide context, commercial adsorbent like Zeolite 13X costs about 1500 per tonne. 15,25 Like previous case studies, we carried out unique optimisations for each case where we minimised the CO 2 avoided cost by varying both adsorbent and process decision variables. The cost limits at zero adsorbent cost serve as a reference. As can be seen from Fig. 4(a), the adsorbent costs we considered have a marginal effect on the cost limits for all CO 2 compositions. For the case of 1500 per tonne, the cost limits obtained are <10% higher than the cost limits with zero adsorbent cost, whereas 7-13% higher when the adsorbent costs are increased three times the cost of Zeolite 13X.

Comparison with real adsorbents
The cost limits obtained previously are compared to the performance of real adsorbents. The rationale behind this case study was to understand the performance gaps between "ideal" and "real" adsorbents, hence the theoretical room for improvement by developing novel adsorbent materials. Two real adsorbents are considered: Zeolite 13X, the current benchmark material for post-combustion CO 2 capture 30,33 and; IISERP MOF2, a novel MOF that was found to be promising in a recent screening study 17 and in a techno-economic assessment. 25 Figure 4(b) shows the comparison of minimum CO 2 avoided costs obtained for Zeolite 13X (black line) and IISERP MOF2 (green line) and cost limits. Of the two "real" adsorbents, Zeolite 13X always resulted in higher CO 2 avoided costs than IISERP MOF2. At 30% CO 2 composition, the minimum CO 2 avoided cost obtained for Zeolite 13X was 18.7 per tonne of CO 2 avoided (see Table S18 in the ESI †) which is about 53% higher than the cost limit at the same CO 2 composition. The gap monotonically increases with reducing the CO 2 composition. For instance, the difference in CO 2 avoided costs between the Zeolite 13X and the cost limits at 3.5% CO 2 composition is approximately 175%. Higher CO 2 avoided costs for Zeolite 13X can be attributed to its non-linear CO 2 and high capacity N 2 isotherms, as shown in Fig. 3. While the band of hypothetical CO 2 isotherms from the cost limit optimisations are fairly linear, the non-linearity of the CO 2 isotherm for Zeolite 13X results in long blowdown and evacuation steps and, consequently, the capital costs. 25 On the other hand, previous studies have consistently shown that lower N 2 affinity significantly reduces electricity consumption. 17,23 Since the Zeolite 13X higher N 2 affinity as compared to hypothetical N 2 isotherms, higher electricity costs are incurred compared to the "ideal" adsorbents (see Table S18 in the ESI †).
As can be observed from the Fig. 4(b), the marginal gap between the green line and the reference value 1.0 indicates the superior performance of IISERP MOF2, and this can be attributed to features of CO 2 and N 2 isotherms illustrated in Fig. 3. For all CO 2 compositions, the CO 2 isotherms of IISERP MOF2 are within the band of hypothetical CO 2 isotherms from the cost limits case. In addition, lower N 2 affinity similar to hypothetical N 2 isotherms contributed to lower electricity costs. 23,25 Further when the CO 2 compositions are lowered, the difference between the minimum CO 2 avoided costs of IISERP MOF2 and the cost limits also decreases non-monotonically from 20% to 9%.
Based on these results, it can be inferred that the dual-site Langmuirian-type "real" adsorbents can achieve relatively low CO 2 avoided costs at high CO 2 compositions, while their performances are discouraging at low CO 2 compositions. Conversely, the performances obtained by IISERP MOF2, an adsorbent approaching those "ideal" features previously outlined, are consistently close to the cost limits.

Cost limits of six-step DR cycle
As discussed previously, the four-step PVSA cycle relies strongly on the deep vacuum (< 0.1 bar) to meet CO 2 purity-recovery constraints for CO 2 compositions lower than 30%.
Although limiting the lower limit of P L to 0.1 bar in the optimisations resulted in minimum CO 2 avoided costs slightly higher (≤ 14%) than the cost limits for CO 2 compositions ≥ 13%, the CO 2 purity-recovery constraints were, however, not met for lower CO 2 compositions, i.e < 13%. Hence, we investigated a more complex six-step cycle with DR 20,21,29 by carrying out unique optimisations to determine if this cycle can yield lower cost limits than the four-step PVSA cycle while facilitating process operation at industrially feasible vacuum levels over a range of CO 2 compositions. Figure 1(b) shows the comparison of cost limits between the two PVSA cycles. As can be observed that the cost limits obtained for the six-step DR cycle are lower than the four-step cycle. For 30% CO 2 composition, the difference between the cost limits is 1.8 per tonne of CO 2 avoided (i.e. ≈ 15% lower for six-step DR cycle). When the CO 2 composition is lowered to 20%, the cost limits of the six-step DR cycle were found to be 24% lower than the cost limits achieved for the four-step cycle. The cost reduction (≈ 42%) is more significant as the CO 2 composition is lowered from 20% to 3.5%. As can be seen from Figs. 1(d) and (f), the decrease in capital and, more significantly, operating costs have contributed to the cost reductions of the six-step DR cycle. As compared to the four-step cycle, the electricity costs have significantly dropped, especially at lower CO 2 compositions. This can be attributed to the optimal P H and P L (shown in Fig. 2) required to achieve the lowest CO 2 avoided costs. Optimal P H for the six-step DR cycle always remained lower than that of the four-step cycle over a range of CO 2 compositions which indicate lower compression costs. Another interesting aspect remains that the six-step DR cycle can be operated with P L ≥ 0.1 bar over the entire range of CO 2 compositions. This is a significant result because the industrially used vacuum pumps can now be employed. The ability to operate vacuum pumps at milder vacuum levels further entails lower electricity consumption, not only connected to the higher P L but also, the higher vacuum pump efficiencies. The better performance of the six-step DR cycle over the four-step cycle can be attributed to the dual reflux steps, i.e. the HR and LR steps. The LR step in the six-step DR cycle helped recover the residual CO 2 from the column after the evacuation step, and the effluent of this step was used as the heavy reflux before the depressurisation steps. The HR step increased the overall CO 2 partial pressure in the column. Hence, the CO 2 purity-recovery targets can be achieved without depressurising the column to deep vacuum levels. 29 The optimal adsorbent properties linked to the cost limits of the six-step DR cycle are shown in Fig. S6 in the ESI †. Similar to the four-step cycle, we also noticed a huge variation of CO 2 isotherms and pellet properties within a 5% range from the minimum avoided costs for the six-step DR cycle. The CO 2 isotherms for the six-step DR cycle were also found to be fairly linear with almost zero N 2 adsorption (see Fig. S6 in the ESI †). For CO 2 compositions of 3.5%, 7.5% and 13%, the CO 2 isotherms of the six-step DR cycle showed a huge variation with higher loadings as compared to the four-step cycle. The band of CO 2 isotherms for both the cycles were comparable for 20% and 30% CO 2 compositions. On the other hand, the pellet diameters were closer to the upper limit of 5 mm, whereas the pellet porosity lies in the range of 0.36-0.63.

Comparison with MEA absorption
Here, the competitiveness of PVSA for post-combustion CO 2 capture is analysed by comparing its cost limits with current benchmark MEA absorption. The CO 2 avoided costs for MEA obtained from two scenarios are considered: in the first scenario, the source of steam supply for MEA-based capture comes from a natural gas (NG) boiler, whereas in the second scenario, the steam is considered to be generated through heat recovery from the industrial facility. While the first scenario serves as a more general representation of standard MEAbased capture, the second scenario is highly site-specific, i.e. depends on the availability of suitable process waste heat in the industrial facility or nearby industries. The choice of these MEA scenarios comes from the fact that the source of steam supply strongly affects the overall CO 2 avoided costs obtained using the MEA solvent 5 , and such variations must be considered when assessing the techno-economic performance of PVSA for a fair comparison.  Table 1, five different flue gas flow rates spanning the entire spectrum of various post-combustion industrial point sources are considered. The two MEA scenarios mentioned above are used for comparison. Given the inherent way in which the PVSA operates as multiple modules, it is expected that the overall CO 2 avoided costs will not be influenced by the plant size (or flue gas flow rate). To corroborate this assumption, we carried out optimisations to determine the cost limits of the four-step PVSA cycle at different flue gas flow rates from Table 1 for a fixed CO 2 composition of 20%. We present the results in Fig. S8 in the ESI †. As expected, the minimum CO 2 avoided costs obtained at different flue gas flow rates from unique optimisations are almost identical (<2% difference).
Based on these results, we extended the same values of the cost limits obtained for both PVSA cycles in Fig.1(b) at different CO 2 compositions over a range of flue gas flow rates considered without re-running the optimisations for each case. Figure Fig. 6(a). In addition, the range within the 5% vicinity of cost limits is also shown. As can be seen from the figure, the overall trend, considering also the ranges for the 5% vicinity, is that botht the number of trains and the column footprint remains fairly constant. This trend is consistent as the amount of flue gas to be treated remains the same, immaterial of the CO 2 composition. The required number of PVSA trains for a fixed CO 2 composition linearly increases with the flue gas flow rates as shown in Fig. 6(b) (also see  Fig. 6(a). As can be seen from the figure, the column footprint ranges between ≈1000-2200 m 2 . Over a range of flue gas flow rates at 20% CO 2 composition, the column footprint, as illustrated in Fig. 6(b), varies almost linearly from 209 m 2 to 2234 m 2 when the flow rate changes from 313 to 3696 tonne h −1 , respectively. It is to be noted that the total footprint of the plant will be higher than the values reported after adding the area occupied by compressors, vacuum pumps and piping. Electricity scenarios. As previously described, the electricity demand remains the significant factor towards achieving the minimum CO 2 avoided costs for PVSA. In this study, we used the standard European electricity price of 58.1 per MWh. 25,34 For the electricity consumed, we also accounted for the specific direct emissions of 38 kg CO 2 per MWh based on the assumption that the electricity consumed by the PVSA is supplied through a deeply decarbonised power system based on a fossil-based power plant with CCS and renewables. 25 As indirect CO 2 emissions associated with electricity consumption increase the CO 2 avoidance cost 35 , the premise of a deeply decarbonised power system is consistent with the search for the cost limit. Since the source of electricity generation and its characteristics depends on several parameters such as plant location and electricity mix, we conducted an optimisation study with alternative scenarios to investigate the impact on PVSA cost limits. For this analysis, the electricity price of 58.1 per MWh and the specific direct emissions of 38 kg CO 2 per MWh remain as the base case. In the first scenario, we reduced the cost of electricity to 50% of the base case while the specific direct emissions were kept the same as the base case. This scenario is representative of cases in which the PVSA facilities access low-cost renewable electricity production with preferential industrial tariffs excluding transmission costs as can happen, for example, in Norway. 36 The second scenario considers the electricity generation with higher CO 2 intensity, i.e., the cost of electricity remains the same as that of the base case while the specific direct emissions are increased to 262 kg CO 2 per MWh corresponding to the CO 2 -intensity of the European average electricity mix 34,37 . The motivation for the second scenario comes from the existing power production systems that are significantly based on fossil-fuel power plants without CCS.

29
We optimised the PVSA cost limits based on the two alternative electricity scenarios. Figure 7 illustrates the cost limits obtained for two PVSA cycles under the two alternative scenarios. As can be seen from the figure, the cost limits of both PVSA cycles are lowered (≈23-32%) when the electricity prices dropped to 29.0 per MWh in the first scenario.
Under these circumstances, the six-step DR cycle outperforms MEA with NG boiler for all CO 2 compositions, whereas the four-step cycle gives lower costs for CO 2 compositions ≥7.5%.
If MEA-PWH is considered as reference, then the six-step DR and four-step cycles perform better than MEA for CO 2 compositions ≥7.5% and ≥13%, respectively. On the contrary, the PVSA cost limits have either increased or remained the same when the second scenario is considered. The high CO 2 intensity in the second scenario showed a substantial effect on two PVSA cycles at a 3.5% CO 2 composition where the CO 2 avoided cost increased to 246.4 per tonne of CO 2 avoided (64% higher than the base case) for the four-step cycle. When the six-step DR cycle is considered, the CO 2 avoided cost increased to 105.1 per tonne of CO 2 avoided, i.e. almost 21% higher than the base case. This is because the electricity consumption is significantly higher at 3.5% CO 2 composition (see Fig. 1(c)-(d)) than other higher CO 2 compositions. The four-step cycle, however, obtained 68.4 per tonne of CO 2 avoided, i.e. 17% higher costs compared to the base case at 7.5% CO 2 composition. For CO 2 compositions ≥13%, the CO 2 intensity has negligible effect on the PVSA cost limits.

Conclusions
Cost limits of two single-stage PVSA cycles for post-combustion CO 2 capture are investigated through techno-economic optimisations based on a process inversion approach. Using this approach, both adsorbent and process variables are simultaneously optimised based on NSGA-II algorithm to calculate the lowest possible cost of CO 2 avoided (excluding the costs of CO 2 conditioning, transport and storage) or cost limits at different flue gas flow rates and CO 2 compositions. The key results of this study can be summarised as follows: • We showed that the CO 2 composition in the flue gas significantly impacts the cost limits of PVSA, i.e., the lowest possible CO 2 avoided costs decrease with increase in CO 2 compositions. Between the two cycles considered, the six-step DR cycle achieved 15-42% lower costs compared to the four-step cycle, depending on the CO 2 composition.
• When compared with the established MEA solvent based on NG boiler as a steam source, the four-step PVSA cycle has at least 8% lower costs compared to the MEAbased CO 2 capture, whereas the six-step DR PVSA cycle has at least 35% lower costs for CO 2 compositions ≥7.5% over a range of flue gas flow rates.
• The optimisations indicated that the "ideal" adsorbents that facilitate lowest possible CO 2 avoided costs have fairly linear CO 2 adsorption isotherms and N 2 adsorption close to zero.
• It was found that modifying the pellet morphology can result in ≈9-22% lower CO 2 avoided costs based on a four-step PVSA cycle. The optimal pellet porosities and diameters were between 0.42-0.76 and 3-5 mm, respectively. Although the PVSA costs seem favourable, the practical implementation involves limitations due to the plant complexity in terms of the number of PVSA trains required to treat the flue gas, the footprint of the total plant and the associated complexities in plant integration. Some of the challenges can be offset by choosing horizontally-oriented columns instead of the vertically-oriented columns, and by potentially considering hybrid processes, e.g., PVSA+cryogenic 38,39 . Finally, the key outcome of the study is the demonstration that PVSA processes can be promising for treating flue gas streams with high CO 2 compositions, provided suitable low-cost adsorbents be developed. The fact that adsorbents with a variety of CO 2 isotherms can indeed yield similar costs is encouraging and provides the motivation of adsorbent discovery and development.

Conflicts of interest
There are no conflicts to declare.