Influence of the main operating parameters on the DRPSA process design based on the equilibrium theory

Among the adsorption-based separation processes for gaseous mixtures, those exploiting pressure variations, so-called Pressure Swing Adsorption (PSA) processes, are the most popular. In this work, we focus on the specific PSA configuration known as Dual Reflux-Pressure Swing Adsorption (DR-PSA) given its ability to achieve sharp separations. In the case of binary mixtures, an analytical approach based on Equilibrium Theory has been proposed to identify the operating conditions for complete separation under the assumption of linear isotherms. This same approach is not available when the separation is not complete. Accordingly, in this work we study the features of non-complete separations by solving numerically a general DR-PSA model with parameter values suitable to approach equilibrium conditions (no mass transport resistances, no axial mixing, isothermal conditions and no pressure drop), thus reproducing the analytical solution when complete separations are examined. Even for non-complete separations, triangularly shaped regions at constant purity can be identified on a plane whose axes correspond to suitable design parameters. Moreover, we found a general indication on how to select the lateral feed injection position to limit the loss in product purities when complete separation is not established, whatever is the composition of the feeding mixture. Finally, a sensitivity analysis with respect to pressure ratio, light reflux ratio and heavy product flowrate is proposed in order to assess how to recover product purities according to the specific degrees of freedom of a DR-PSA apparatus.


D bed
Diameter of the adsorption bed, m Dynamic viscosity, Pa ⋅ s High pressure to low pressure ratio,

Introduction
In recent years, economical and energy-efficient separation processes are more and more welcome since they represent a large part of the costs in chemical, petrochemical and related industry (Sholl and Lively 2016). In this frame, adsorptionbased processes are an attractive option and the development of novel solid materials could further foster the profitability of this type of separations when compared to more traditional gas separation techniques (Agarwal et al. 2008). In the case of gaseous mixtures, great attention is currently paid to processes applying adsorbent regeneration by changing the operating pressure, so-called Pressure Swing Adsorption (PSA) processes. It has been demonstrated that the PSA technology can be successfully applied to several separations of industrial interest, such as hydrogen purification, air separation, CO 2 capture, and CH 4 upgrading (Grande 2012).
Despite the large potential of PSA processes, a major thermodynamic limitation adversely affects the performances of their traditional configurations (Skarstrom 1959;Yoshida et al. 2003). Indeed, with reference to binary mixtures, usually only the pure light component (i.e., the least adsorbable, in the following referred to as B ) or the heavy one (i.e., the most adsorbable, in the following referred to as A ) can be produced.
To overcome this limitation, the alternative process version called Dual Reflux PSA (DR-PSA) has been proposed (Leavitt 1992). While relying on the basic features of PSA, complete separations are enabled by a lateral feed injection and two recycles (one of heavy and the other of light component). Following the first proposal, several papers explored different experimental applications of DR-PSA for separating different mixtures, namely: mixtures carrying CO 2 (Wawrzynczak et al. 2019;Shen 2017;Li et al. 2016;Kim et al. 2016;Sivakumar et al. 2011a, b;Takamura et al. 2001;Diagne et al. 1995Diagne et al. , 1994), N 2 /O 2 (Wang et al. 2019;Tian et al. 2017), N 2 /CH 4 (Weh et al. 2020;Xiao et al. 2019;Zhang et al. 2016;Salemann et al. 2015), and N 2 /C 2 H 6 (Mc Intyre 2002, 2010. This body of work built the basic understanding of the main features and capabilities of DR-PSA. The empirical tuning of some operating parameters has been proposed Zhang et al. 2016;Saleman et al. 2015) to improve the separation performances, while innovative process configurations have been designed (cf. Tian et al. 2017;Sivakumar et al. 2011a, b) again with the final aim to maximize purity and recovery.
Similar efforts towards the design of the best process conditions have been made on the modelling side. Different models have been proposed, ranging from detailed models requiring numerical solutions (Rossi et al. 2019a, b;Zou et al. 2017;May et al. 2017;Bhatt et al. 2014;Thakur et al. 2011) to simplified semi-analytical models based on the so called Equilibrium Theory (Bhatt et al. 2018(Bhatt et al. , 2017(Bhatt et al. , 2013Kearns and Webley, 2006a, b). In the latter case, the design conditions which allow the complete separation of binary mixtures in the case of linear isotherms have been investigated (Bhatt et al. 2013(Bhatt et al. , 2015(Bhatt et al. , 2017(Bhatt et al. , 2018 and two key process parameters have been identified: the capacity ratio C , proportional to the ratio between the number of moles injected during the adsorption step and the volume of the column, and the dimensionless feed injection position z feed . In the plane defined by these two parameters, a Triangular Operating Zone (TOZ) is established inside which complete separation is guaranteed (Bhatt et al. 2013). The TOZ is characterized by a single, constant value of a third parameter G (the reflux ratio, proportional to the ratio between the light reflux and the feed flowrate). Such an approach represents a first step towards the predictions of optimal separation conditions, but its validity is significantly limited by the assumptions of the Equilibrium Theory, with particular reference to the strong requirement of complete separation. However, in many practical processes complete separation is not required since purity values larger than a given target value usually suffice. Therefore, the main aim of this work is to explore the role of the different process conditions on the separation performances retaining the main assumption behind the equilibrium model but under conditions of non-complete separation. In other words, based on the main results provided by the Equilibrium Theory (i.e., the identification of the TOZ), we analysed the process behaviour outside such a region through a detailed mathematical model. Extensive simulations with different operating conditions allowed identifying new triangular operating regions on the C − z feed plane at constant purity and non-complete separation. These results enable to understand the effects of different key process parameters (i.e., heavy product flowrate, light recycle ratio, and pressure ratio) in a quantitative way, thus providing effective guidelines to tune the separation performances also in the case of non-complete separations.

DR-PSA configuration
The DR-PSA process considered involves four steps performed by two identical beds working simultaneously. These steps are: (i) feed (FE), during which the gas mixture to be separated is fed to an adsorption bed at constant pressure; (ii) blowdown (BD), where the pressure is reduced from the highest value P H to the lowest one P L ; (iii) purge (PU), in which the bed which was previously in FE is regenerated at constant pressure; and (iv) pressurization (PR), during which the pressure is raised again from P L to P H . Four different configurations of the process can be applied, based on the pressure of the FE step ( P H or P L ) and on the stream which is used for pressurization ( A-rich or B-rich) (cf. Kearns and Webley 2006b). In this work, we focus on the configuration named DR-PH-A, where pressure switching is performed with the A-rich stream and the feed stream is injected at P H . The corresponding half-cycle is sketched in Fig. 1, where the A-rich stream is indicated as "heavy product stream" and the B-rich stream as "light product stream". The lateral feed injection is represented in the figure by the molar flowrate ̇n feed and it is injected into the column at an intermediate position z = z feed , where z is the dimensionless axial coordinate. As further detailed in (Rossi et al. 2019a), two tanks are involved in the specific process under examination, 1 and 2 . The first tank, 1 , is used to collect the B-rich stream leaving column 1 during FE and the two leaving streams are the light product and the light recycle to column 2 during PU, respectively. The second tank, 2 , collects the A-rich stream during step PU of column 2 and step BD of column 1, Fig. 1 Sketch of the first half-cycle of the process. During the second half, the two columns are exchanged while the leaving streams are heavy product during PU and heavy recycle during BD, respectively. The molar fractions of the light product, y i,L , and those of the heavy product, y i,H , are computed as cycle average values of the molar fractions entering 1 and 2 , respectively. The presence of the two tanks, with a large enough volume, is to tone down possible irregularities in the streams flowrates (Sivakumar and Rao 2011a;Rossi et al. 2019a).

Mathematical model, numerical approach and solving procedure
A detailed model has been developed and numerically solved as discussed elsewhere (Rossi et al. 2019a). A singlebed approach was applied, i.e., one single column is simulated all along the entire cycle (FE, BD, PU, and PR) while assuming that the two beds are identical. Notably, the simulation is carried out from an arbitrary initial condition until Cyclic Steady State (CSS) conditions are established, i.e., the model results of two subsequent cycles are identical in terms of integration variables according to a given tolerance.
The main model assumptions are: • Ideal gas mixtures; • Isothermal conditions; • Mass transport between gas and solid phase described by the Linear Driving Force (LDF) model (Farooq et al. 1989); • Pressure drops evaluated by the Blake-Kozeny equation (Thakur et al. 2011); • Negligible axial dispersion, as expected in gas phase adsorption processes (Liao and Shiau 2000).
Model constitutive equations are summarized below for a binary mixture (i = 1, 2): They are total (1) and component (2)  (1) Blake-Kozeny Eq. (4). The evaluation of the solid phase concentration at equilibrium, q * i , is carried out through a suitable equilibrium isotherm. In this work, since we aim to extend the results of the Equilibrium Theory, linear adsorption isotherms (5) are considered: The resulting system of Partial Differential Equations (PDEs) has been discretised in the spatial domain applying the Finite Volume Method (FVM) with first order upwind interpolations, Lagrange extrapolations, and Van Leer's Flux Limiters. This approach has been reported to efficiently reproduce the complex dynamics typical of transient adsorption processes (Casas et al. 2013;Haghpanah et al. 2013;LeVeque 2002;Webley and He 2000;Rossi et al. 2019a). Once discretised in space, the system of PDEs reduces to a set of Ordinary Differential Equations (ODEs) which has been integrated in time using Matlab® routine ode15s. A comprehensive summary of boundary and initial conditions is presented in Table S1 of the Supplementary Information. In particular, an air-like composition of the feed stream (i.e., y A,feed = 0.79 ) has been considered.
All the reported simulations have been performed using a spatial grid made of 300 nodes. The grid independence of the results has been checked by comparison with the results corresponding to finer grid simulations (500 nodes). Moreover, the achievement of CSS conditions has been checked both in terms of accuracy of material balance closure and constant purity values of both the product streams.

Non-complete separation study
The separation of a binary mixture of oxygen and nitrogen using zeolite 5A as adsorbent is considered as case study. Realistic values of the process parameters have been selected, with the exception of those required to approach equilibrium conditions. Namely, very large values of transport coefficients and particle size have been applied in order to have negligible mass transport resistances and pressure drops, respectively.
In Table 1, the operating and process parameters for the case study with y A,feed = 0.79 are reported.
Moreover, the definitions of the four main variables discussed in the following are summarized below: Taking advantage of the semi-analytical solution based on Equilibrium Theory (Ebner and Ritter, 2004;Bhatt et al. 2013), the region of complete separation in the C − z feed plane (TOZ) is first evaluated, as shown in Fig. 2. At the chosen values of adsorption selectivity of each component , pressure ratio ( ) , and feed molar fraction (y A,feed ) , any pair of values of the two operating parameters C and z feed inside such a region results in complete separation (pure heavy and light components in each product stream). Note that this picture corresponds to a single value of the reflux ratio G , a quantity proportional to the ratio between the light reflux, ̇n LR , and the feed flow rate, ̇n feed .
The reliability of the detailed model for the conditions here investigated has been verified by simulating the process with the demanding (from a numerical point of view) operating conditions inside the TOZ indicated by the empty symbol in Fig. 2. The expected complete separation is correctly predicted by the detailed model, as clearly shown by the concentration profiles at CSS in Fig. 3. About 250 cycles are required to achieve CSS conditions from a column filled with the feeding mixture as initial condition. The material balances are also fulfilled with a residual error (relative to the fed mixture) for each component below 0.3%.
Let us now explore sets of operating conditions not corresponding to complete separation, that is, any pair of ( C, z feed ) values outside the TOZ. In order to evaluate the sensitivity of the process performances to changes in the two key parameters ( C, z feed ), all the conditions indicated by filled symbols in Fig. 2 have been simulated by keeping unchanged all the operating parameter values apart from C and z feed (see Table 1; note that C values have been modified by changing only the duration of the feed step). Since FE time and distance from the column end become so small to be impractical for C values smaller than 0.0015 and for z feed values smaller than 0.05, these regions were not investigated.
Note that the variation of C may be obtained also through the variation of other process parameters (cf. Eq. 7). However, once selected the solid adsorbent (and the corresponding properties), the value of the pressure ratio, and the unique G value estimated through the analytical solution based on Equilibrium Theory, we verified that the same results as those proposed in the following are obtained varying C through the variation of V bed . Therefore, we consider these results meaningful whatever is the parameter used to change the C value, feed duration t FE or bed volume V bed . The CSS purities of all the performed simulations are summarized in Table S2 of the Supplementary Information, while, in Table 2 the specific parameter values for the two other mixtures investigated are reported. Note that the parameters not reported in Table 2 have been set equal to those reported in Table 1.
Even though incomplete separation is predicted in all cases outside of the TOZ, the sensitivity of the purity of each component when changing C or z feed is quite different. This is shown in Figs. 4 and 5 for light and heavy component where, taking advantage of the usual C − z feed representation, curves at constant purity are visualized. Note that, since discrete values are typically calculated, a proper interpolation of such data is required, as detailed in the Supplementary Information. Interestingly, these curves identify again triangular regions inside which the product purity is larger than the purity specific of each curve. The vertex of the new triangles moves up and to the right at decreasing purity, i.e., higher productivity and z feed values become feasible when the separation target is reduced. On the one hand, this behaviour results in a significant increase of the range of operating conditions which can be applied to achieve the selected separation performance. On the other hand, the variation of the purity of each component is much sharper at z feed values smaller than that corresponding to the vertex than at larger values. Finally, the behaviour is quite asymmetric for the two  This asymmetric behaviour can be easily explained through the overall material balances. Given the selected operating conditions (which involve product flowrates values of the light and heavy component suitable to achieve complete separation), the product flowrates are equal to ̇n HP = y A,feed ⋅̇n feed and ̇n LP = y B,feed ⋅ṅ feed = (1 − y A,feed ) ⋅̇n feed . Introducing these relations into the overall material balance of component B on a complete cycle, the following equation can be obtained: which, given the constraint y B,feed = 1 − y A,feed , can be readily rearranged as: The last equation provides a linear relationship between the purity of the light component, y B,L , and that of the heavy one, y A,H , as a function of the specific feed composition.  Such relationship is shown in Fig. 6 for typical values of y A,feed . It is evident that the purities of light and heavy components are equal only when y A,feed = 0.5 , while they must be different for non-equimolar feed composition. More specifically, the variation of the light product purity must be larger than that of the heavy component when the feed mixture is richer in the heavy component, as in Figs. 4 and 5. Moreover, given the constraint represented by Eq. (11), the two figures are not independent. Accordingly, in the following we discuss the process performances with respect to the light component, with the exception of the cases in which ̇n HP is varied, that is, when Eq. (11) is not valid. Figure 4 also shows a different sensitivity of the purity values depending on the feed positions being smaller or larger than that of the TOZ vertex. This behaviour has some practical implications. Even if the same purity can be achieved at feed positions larger and smaller than that of the TOZ vertex, larger feed positions should be highly preferred given their weaker sensitivity of the product purity to the variation of the process operating conditions (that is, in the position of the operating point in the C − z feed plane).
The same analysis has been repeated with different feed mixture composition. In particular, the two mixtures, y A,feed = 0.21 and 0.50 , have been investigated. TOZs have been identified (see Table 2 and Fig. S2 of the Supplementary Information) and several simulations have been performed to span the corresponding C − z feed plane. The obtained results fully confirm the general behaviour discussed above as well as the corresponding conclusions.

Process performance improvement
In the previous section, process conditions ensuring a given extent of separation have been identified starting from operating conditions provided by the simplified solution based on Equilibrium Theory (e.g., feed position, pressure ratio, recycle ratio, and adsorbent amount at given feed flowrate). In the following we discuss how to improve the separation performance of a given unit when keeping most of the design parameter values as predicted by the Equilibrium Theory but selecting operating conditions outside the TOZ. In the literature, the heavy product flowrate ( ̇n HP in Fig. 1; Saleman et al. 2015;Zhang et al. 2016;May et al. 2017) and the light recycle ( ̇n LR in Fig. 1; Bhatt et al. 2014;Shen et al. 2017;Zhang et al. 2016;Tian et al. 2017;Li et al. 2016) have been identified as major operating parameters to tune the product purity. On the other hand, the effect of changing the pressure ratio, , has not been systematically investigated, even though it can be expected to affect significantly the DR-PSA process performances.
Three dimensionless variables, accounting for the change of the three process parameters mentioned above with respect to their values provided by the Equilibrium Theory, are defined as follows: The first dimensionless parameter, R , involves the heavy product flowrate; the second one, G R , the light reflux, and the last one, R , the pressure ratio. In all cases, the superscript ET indicates the reference value provided by the Equilibrium Theory for complete separation, i.e., the values considered in the previous sections (see Table 1 of the Supplementary  Information). The impact of changing such values on the separation performances has been quantified considering three specific operating points in the usual C − z feed plane, as shown in Fig. 7.
The set of operating conditions labelled Run1 in Fig. 7 is located inside the TOZ, while that labelled Run2 corresponds to a lateral feed injected at z feed larger than the optimal one, and Run3 corresponds to column shorter than the minimum value predicted by the Equilibrium Theory. While complete separation is achieved only in Run 1, the other two sets of conditions correspond to non-complete separation, as shown in Figs. 4 and 5. In the following, the impact of changes of the three dimensionless parameters Fig. 6 Product purity values at different feed composition. The lighter grey surface refers to y A,feed < 0.5 , while the other to y A,feed > 0.5 (Eqs. 12 to 14) will be explored: using Run 1 as reference, Runs 2 and 3 allow to check whether a partial separation recovery can be easily achieved outside the TOZ by changing only one of the parameters values provided by the Equilibrium Theory.
The results of this analysis are shown in Fig. 8, where the purities are reported as a function of the dimensionless operating parameters defined above.
For the cases of the variation of and G ( Fig. 8c and d) only the purity of B is reported. While, for the case of the variation of R , the purity of component A is reported too since, varying the heavy product molar flowrate, such a value cannot be computed through Eq. (11).
As expected, in the case of complete separation (Run1), the maximum purity value of y B,L = 1 is invariably found at unitary value of the corresponding dimensionless parameter, i.e. when the operating conditions predicted by the equilibrium model are used. Hence, by increasing , G , or , an equal, or at least for values not too far from that provided by the Equilibrium Theory in the case of , separation is obtained. This behaviour is imputable to the fact that Run1 lies in the TOZ region, which means complete separation when adopting ET , G ET and ̇n ET HP . For R < 1 , the molar flowrate extracted from tank 2 is inevitably lower than the amount of heavy component fed. Therefore, if a value smaller than ̇n HP T is collected, the remaining portion of the heavy component fed becomes part of the light product stream, thus reducing the light product purity. Analogously, if R > 1 , ̇n HP T is larger than the fed quantity of A and, therefore, part of the light component will be in the heavy product stream, reducing the heavy product purity. These behaviours are common to all the three runs considered, as shown by the three very similar results in Fig. 8a and b.
About the effect of R , a look at the concentration profiles inside the adsorption bed is helpful (cf. Fig. 9). In this work, the variation of R is obtained by changing the high pressure value of the cycle. Therefore, R smaller than 1 means lower P H , that is smaller adsorbed amounts of both A and B , leading to poorer process performances as shown in Fig. 8c. On the other hand, the amount adsorbed rises for R values larger than 1. However, the shape of the concentration profiles in the adsorption beds changes too. As shown in Fig. 9a, the plateau at constant molar fraction shifts towards the end of the column and settles at larger mole fraction with respect to the case R = 1.
For values of the light recycle ratio lower than that provided by the Equilibrium Theory ( i.e. , G R < 1 ), the molar flowrate of the light component recycled in the purging column does not allow proper desorption. As a consequence, a larger amount of light component will be present in the heavy product stream extracted from the bed during the PU step. On the contrary, when raising G , better purge occurs, as shown by the profiles in Fig. 8d. However, at the same time, the increase of the purge molar flowrate pushes the concentration profiles more and more towards the end of the adsorption bed, leading to their eventual breakthrough.

Conclusions
Through an efficient numerical strategy together with a detailed mathematical model, the performances of the Equilibrium Theory -based design of a DR-PSA separation • for non-complete separations, regions leading to purity values larger than a given value have triangular shape in the operating plane z feed − C , as those previously derived for complete separations, regardless the feed mixture composition; • the loose of purity outside the TOZ is much sharper on the left of the optimal feed position (i.e., that of the vertex of the TOZ) than on the right; this means that for a practical process design, the feed position located on the right of the optimal value predicted by the Equilibrium Theory makes the process more robust; • increasing the value of the heavy product flowrate allows for recovering (or at least for not losing, in case of operating conditions inside the TOZ) the light prod-uct purity; on the contrary, reducing the value of the heavy product flowrate enables the recovery (or at least not losing, in case of operating conditions inside the TOZ) the heavy product purity; this permits to tune the process performances to the desired target, also considering that the purity loss is larger for the component with lower feed concentration; • changing the pressure ratio from the value provided by the Equilibrium Theory never results in improving the process performances; this means that increasing the pressure ratio for improving the separation requires retuning all the other process parameters according to Equilibrium Theory; • similar conclusions arise for the light recycle ratio: changing its value from the one provided by the Equilibrium Theory does not allow for recovering better performance even in case of non-complete separation. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/. Fig. 9 Profiles of the heavy component gas mole fraction at the end of the feed step (Run1) when changing R (a) and G R (b). (--) parameter values smaller than the Equilibrium Theory value; (−) parameter values larger than the Equilibrium Theory value. Heavy lines refer to complete separation conditions (parameter value equal the Equilibrium Theory value)