Dynamics of Core–Shell-Structured Sorbents for Enhanced Adsorptive Separation of Carbon Dioxide

Abstract

One of the key environmental problems underlying climate change and global warming is the persistent increase in atmospheric carbon dioxide concentration. Carbon capture and storage (CCS) systems can be based on, among others, solid porous sorbents (e.g., zeolites). A promising alternative to traditionally used sorbents may be appropriately structured hybrid adsorbents. With the proper geometry and synergistic combination of the sorbent with another material, e.g., a catalyst or a substance with certain useful physical features, they can gain new properties. The present study examined the dynamics of CO₂ sorption in core–shell particles and, as a reference, in particles with a uniform structure. It was assumed that the sorbent (zeolite 5A) incorporated in a single particle had the form of microcrystals, which implies a bidisperse particle structure. As a second particle-forming material, a nickel catalyst (behaving as an inert) was adopted. The computational results confirmed that particle structure can provide an additional design parameter for adsorption columns and adsorptive reactors. The sorption-inactive shell proved to play a protective role when thermal waves moved through the bed. In addition, an important element determining sorption dynamics in core–shell particles was revealed to be the structure (e.g., mean pore diameter) controlling intraparticle mass transport.

1. Introduction

Continuing global warming is one of the fundamental challenges facing modern industry and society. The leading contributor to global warming is undoubtedly anthropogenic carbon dioxide. Its concentration in the atmosphere increased by about 50% since the beginning of the Industrial Revolution in the 17th century [1]. In view of continuing population growth and technological progress, a significant reduction in CO₂ emissions is essential to prevent the consequences of global warming. Large industrial sites such as fossil fuel or biomass-based power plants, refineries, cement plants, steel or iron smelters are among the principal emitters of CO₂ to the atmosphere. Given that fossil-based and energy-intensive technologies will be exploited for many years to come, it is crucial to develop and implement mid-term preventive measures prior to the eventual attainment of energy transformation goals. In this regard, a very attractive solution to reduce the environmental impact of sectors that are difficult to decarbonize is the use of carbon capture and storage (CCS) and carbon capture and utilization (CCU) technologies [2,3].

The most important technologies for CO₂ capture include physical and chemical absorption, adsorption, membrane and cryogenic separation [3,4,5]. Separation of CO₂ involving solid adsorbents can be used both for industrial flue gas treatment and in the direct air capture (DAC) process [2], and it has attracted a lot of attention recently as a promising direction towards the advancement of CCS and CCU technologies [6]. Adsorption processes are also widely employed in biogas upgrading, typically containing several tens of percent of CO₂ [7].

Various types of adsorbents can be utilized for CO₂ separation, ranging from activated carbon, zeolites, silica, polymers, alumina, metal oxides to metal–organic structures (MOFs) and sorbents derived from industrial residues [6]. The most important criteria for selecting a successful candidate for CO₂ capture include selectivity towards CO₂, equilibrium adsorption capacity, kinetics of adsorption and desorption, pore structure, and ease and low cost of regeneration [6]. Developing an adsorbent with the desired properties poses many challenges. For instance, both the performance of zeolites and MOFs is negatively affected by the presence of H₂O, NOx and SOx [3]. In fact, in the case of zeolites, it is well known that those with a low Si/Al ratio are hydrophilic, so H₂O competes with CO₂ for active sites. In industrial practice, to overcome this problem, biogas and flue gas are usually pre-dried prior to sorption-based decarbonization, which is reflected in increased capital and energy costs. Therefore, an attractive alternative being explored is the design of hybrid sorbents with hydrophobic features. Recently, hybrid zeolite W-ZSM-5 and W-silicalite-1 with tungsten embedded in the framework were proposed to tackle the competitive adsorption of CO₂ and H₂O [8]. The presence of tungsten resulted in three times higher CO₂/H₂O selectivity for W-silicalite-1 (with silicalite-1 without tungsten being a reference). For the hybrid zeolite, the increase in this selectivity compared to zeolite without tungsten was not significant but still noticeable [8]. Another recent study [9] showed that core–shell particles with ZSM-5 forming the core and coated with silicalite-1, and additionally characterized by hierarchical porosity (i.e., zeolite micropores are accessible via larger pores) exhibited enhanced hydrophobicity. The authors demonstrated that the proposed hybrid structure allowed to reduce H₂O adsorption capacity by about 40% (with respect to ZSM-5) at 20% relative humidity [9].

Hybrid particles featuring a core–shell structure also receive considerable attention in the field of catalysis, including catalytic processes of CO₂ conversion that fall into the aforementioned CCU technologies [1,10,11]. They belong to the broader family of dual-function materials (DFMs). The latter usually consist of an adsorption and a catalytic component, and thus, in the case of CO₂, DFMs allow both its capture and further conversion to valuable chemical compounds, such as methane [1,10,11].

While the main goal behind the formulation of DFMs both with a core–shell structure and any other structures (e.g., particles with a uniform distribution of individual components throughout the volume or hollow yolk–shell structures) is process integration, their advantages and features go well beyond. In the case of thermocatalysis, a particle with a catalytically active core may be encapsulated within an inactive (from both a sorption and reaction point of view) shell with protective properties, e.g., to prevent sintering or poisoning of the catalyst [11]. The core–shell structure can also be used to create a micro- or nano-sized confined reaction environment to stabilize and enhance the process. Some examples of the latter are Ni nanoparticles embedded in silicalite-1 shell which were used in dry methane reforming [12] or Co₃O₄@SiO₂ “nanorattles” synthesized for catalytic combustion of toluene [13]. It is worth emphasizing at this point that improving the effectiveness of syngas production is also an extremely important problem to be solved in the context of the greenhouse effect. The same applies to volatile organic compounds (VOCs) treatment processes, given that practically all VOCs contribute to global warming directly by absorbing infrared radiation.

Core–shell particles can also be utilized to run so-called tandem reactions, such as the process of direct synthesis of dimethyl ether (DME) from synthesis gas. For two-step catalytic processes, with DME synthesis being an example, the active centers catalyzing the first step are usually incorporated in the core, while the second step is accomplished when the intermediate product diffuses through the porous shell [11,14]. Another possibility is to encapsulate the active core in a shell with appropriately sized pores to increase the catalytic process selectivity [11].

With regard to CCS and CCU technologies, perhaps one of the issues of greater interest is the development and implementation of DFMs that can be used in the cyclic process consisting of CO₂ adsorption, followed by its methanation. The combined CO₂ capture and methanation (CCCM) process can generally be performed in two connected units, i.e., adsorber and reactor, or in a single unit, i.e., adsorptive reactor, the latter with a bed made of a mixture of sorbent and methanation catalysts or of DFM [15]. In all cases, the process is carried out cyclically. For two units, after the sorbent bed breakthrough occurs the process of its regeneration is initiated, and the desorbed CO₂ is transported to the reactor connected in series, where methanation is carried out. When using the adsorptive reactor, the adsorption stage and reactive regeneration are carried out in the same apparatus. An extensive literature review of CCCM, including the use of DFMs in this process, can be found in reference [15]. However, there is little work on the effect of the structure of the hybrid particle itself on the performance of the two stages. Recently, the encapsulation of a nickel catalyst in a zeolite shell to realize the methanation process was proposed [16], but the main purpose of forming such a shell was a protective function during methanation step, and the possibility of utilizing such DFM in CCCM was not analyzed.

Taking into account the relevance of the topic and the literature gaps, a simulation study focused on analyzing the dynamics of hybrid particles composed of zeolite and nickel catalyst at the stage of CO₂ adsorption itself was undertaken. Following the arrangement of individual functionalities in the case of hybrid particles utilized for tandem reactions, it was hypothesized that from a process point of view it may be advantageous to locate the sorbent in the core and the catalyst in the shell. This is because such an arrangement can have a protective function at the adsorption stage as thermal waves travel through the apparatus [17]. In addition, it is known that for capture-reactive regeneration systems, especially at the beginning of regeneration, the release rate of the adsorbed reactant can be very high [18], and therefore such a particle structure may also be advantageous when running the second step of the CCCM process.

In particular, in this study, the dynamics of CO₂ sorption in core–shell particles (Figure 1a,b) and in a particle with a uniform structure (Figure 1c) were analyzed via numerical simulations. The latter, namely the uniform structure, was examined as a reference for comparison purposes. In the case of core–shell particles, both sorbent in core and sorbent in shell arrangements were analyzed. It was assumed that the sorbent (zeolite 5A) incorporated in a single hybrid spherical particle is in the form of microcrystals, which is reflected in the bidisperse structure of the particle, characterized by a bimodal pore size distribution. It was further assumed that the second material composing the particle is a nickel catalyst, which behaves as an inert, i.e., sorption-inactive material, during the adsorption step.

2. Materials and Methods

2.1. Mathematical Model of a Single Hybrid Particle

As stated earlier, one of the key assumptions of the model was that the hybrid spherical particle has a bidisperse structure, which is due to the fact that the sorbent (here, zeolite 5A) is in the form of microcrystals. Figure 2 graphically shows a cross section of such a particle for a core–shell configuration, in which the sorbent is located in the core. It is evident that the sorption-active region is characterized by the presence of pores of different diameters, namely those resulting from the voids between crystals and the pores in the zeolite microcrystals, where the actual adsorption process takes place. The region possessing the second functionality (in this case, a nickel catalyst that behaves like inert at the sorption stage and in Figure 2 located in the shell) can have both a compact but porous structure or can also be constituted of microparticles. In the present study, given that the chemical reaction process is not analyzed, it was assumed for the catalyst (or inert), that the mass transport occurs only in its macropores.

Other main assumptions of the mathematical model of the hybrid spherical particle are as follows:

• The analyzed process of physical adsorption of CO₂ in the hybrid particle occurs under non-isothermal conditions;
• The particle is spherical and symmetrical, which leads to a one-dimensional description of the concentrations of components and temperature along its radius;
• The gas mixture within the particle and its surroundings contains only CO₂ and N₂;
• The gas mixture follows the ideal gas law;
• The main mechanism of mass transport is diffusion, and therefore viscous flow is neglected;
• The mass transport in macropores follows a molecular and Knudsen diffusion mechanism, whereas in the micropores of the crystals, configurational (intracrystalline) diffusion mechanism takes place [19];
• CO₂ is the only component that is subjected to adsorption which is motivated by the results reported in [20];
• The adsorption equilibrium of CO₂ on zeolite 5A is given by Thoth isotherm [21], and the mass transfer from the gas phase in the micropores to the solid microcrystals is described by the linear driving force (LDF) model [22,23];
• The chemisorption of CO₂ on zeolite 5A is neglected, which is motivated by the fact that in the absence of H₂O its physisorption strongly dominates over chemisorption [24];
• During the adsorption process examined in this study, the nickel catalyst behaves like inert, which was motivated by the findings provided in [25];
• Physical, thermal and transport properties are independent of temperature.

Given the above formulated assumptions the mass balance of an arbitrary component i in a hybrid spherical particle can be written as follows:

$${\Gamma }_m\frac{\partial C_i}{\partial t}=D_{eff}\left(\frac{{\partial }^2{C}_i}{\partial r^2}+\frac{2}{r}\frac{\partial C_i}{\partial r}\right){-\rho }_{p,ads}{f}_{ads}\frac{\partial q_i}{\partial t},$$

(1)

with the mass transfer (in case of CO₂) from the gas in the micropores to the solid microcrystals described by the LDF model as follows:

$$\frac{\partial q_{\mathrm{C}{\mathrm{O}}_2}}{\partial t}=k_{LDF}\left(q_{\mathrm{C}{\mathrm{O}}_2}^{*}-q_{\mathrm{C}{\mathrm{O}}_2}\right),$$

(2)

where

$${\Gamma }_m={\epsilon }_{p,ads}{f}_{ads}+{\epsilon }_{p,cat}{f}_{cat}, k_{LDF}=\frac{15D_c}{R_c} \mathrm{a}\mathrm{n}\mathrm{d} D_{eff}=\frac{{\epsilon }_p}{{\tau }_p}\frac{D_K\times D_m}{D_K+D_m}.$$

(3)

In Equations (1) and (3) f_ads and f_cat = 1 − f_ads denote, respectively, the volume fraction of adsorbent and catalyst in the hybrid particle (referred to the total volume of the solid) being a function of the radial coordinate, r. In all cases studied in this paper, it was assumed that each type of functionality, i.e., adsorbent and catalyst, occupies 0.5 volume fraction of the solid within the particle. Thus, for both core–shell arrangements examined, the radius of the core was set to R_sc = 0.7937 × R_p, with R_p being the radius of the hybrid particle.

The value of the configurational diffusion coefficient, D_c, often referred to as the intracrystalline diffusion coefficient, was calculated using the following formula [19,23]:

$$D_c=D_0\mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{E_{diff}}{RT}\right),$$

(4)

while the value of the molecular (binary) diffusion coefficient, D_m, was calculated using a formula developed by Chapman and Enskog [26], and the value of the Knudsen diffusion coefficient, D_K, was determined using a classical formula originating from the kinetic theory of gases [27], namely

$$D_m\equiv D_{ij}=1.883\times {10}^{-22}\frac{T^{1.5}}{p{\sigma }_{ij}^2{\mathsf{\Omega }}_D}{\left(\frac{1}{M_i}+\frac{1}{M_j}\right)}^{0.5} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} i={\mathrm{C}\mathrm{O}}_2 \mathrm{a}\mathrm{n}\mathrm{d} j={\mathrm{N}}_2,$$

(5)

$$D_K=48.5d_{pore}\sqrt{T/M},$$

(6)

The equilibrium concentration of CO₂ in the solid phase was described using the isotherm of Thoth for zeolite 5A [21]:

$$q_{{CO}_2}^{*}=\frac{a{p}_{C{O}_2}}{{\left[1+{\left(b{p}_{C{O}_2}\right)}^n\right]}^{1/n}},$$

(7)

where

$$a=a_{0}exp\left(\frac{E_{ads}}{T}\right), b=b_{0}exp\left(\frac{E_{ads}}{T}\right), n=n_0+\frac{c}{T}.$$

(8)

The energy balance for the gas and solid phases within the particle is expressed by the following equation:

$${\Gamma }_h\frac{\partial T}{\partial t}={\lambda }_s\left(\frac{{\partial }^{2}T}{\partial r^2}+\frac{2}{r}\frac{\partial T}{\partial r}\right){+\rho }_{p,ads}{f}_{ads}\left(-\mathsf{\Delta }{H}_{ads,\mathrm{C}{\mathrm{O}}_2}\right)\frac{\partial q_{\mathrm{C}{\mathrm{O}}_2}}{\partial t},$$

(9)

where

$${\Gamma }_h=\left({\epsilon }_{p,ads}{f}_{ads}+{\epsilon }_{p,cat}{f}_{cat}\right){\rho }_g{c}_g+{\rho }_{p,ads}{f}_{ads}\left(c_{s,ads}+c_{g,ads}\right)+{\rho }_{p,cat}{f}_{cat}{c}_{s,cat}.$$

(10)

Following the model assumption, in the calculations, constant mean values of ρ_g and c_g were adopted. The same assumption was made for the heat capacity of the adsorbed phase, c_g,ads, and the adsorption enthalpy, ΔH_ads,CO2, which was calculated from Clausius–Clapeyron equation [27], namely:

$$-\mathsf{\Delta }{H}_{ads,\mathrm{C}{\mathrm{O}}_2}=\frac{R{T}^2}{p_{\mathrm{C}{\mathrm{O}}_2}}{\left(\frac{\partial p_{\mathrm{C}{\mathrm{O}}_2}}{\partial T}\right)}_{q_{{CO}_2}^{*}} \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} p_{C{O}_2}=\frac{q_{{\mathrm{C}\mathrm{O}}_2}^{*}}{{\left[a^n-{\left(q_{{\mathrm{C}\mathrm{O}}_2}^{*}\right)}^n\right]}^{1/n}}.$$

(11)

Keeping in mind the spherical symmetry of the hybrid particle and in order to take into account the external resistances to mass and heat transport, the following boundary conditions were associated with Equations (1) and (9):

$${\left.\frac{\partial C_i}{\partial r}\right|}_{r=0}=0 \mathrm{a}\mathrm{n}\mathrm{d} {\left.D_{eff,i}\frac{\partial C_i}{\partial r}\right|}_{r=R_p}=k_{m,i}\left(C_{bulk,i}-C_i\left(R_p,t\right)\right),$$

(12)

$${\left.\frac{\partial T}{\partial r}\right|}_{r=0}=0 \mathrm{a}\mathrm{n}\mathrm{d} {\left.{\lambda }_{eff}\frac{\partial T}{\partial r}\right|}_{r=R_p}={\alpha }_q\left(T_{bulk}-T\left(R_p,t\right)\right),$$

(13)

with the convective mass, k_m, and heat transfer coefficient, α_q, determined from the relations:

$$k_m=\frac{\mathrm{S}\mathrm{h}{D}_m}{2R_p} \mathrm{a}\mathrm{n}\mathrm{d} {\alpha }_q=\frac{\mathrm{N}\mathrm{u}{\lambda }_g}{2R_p},$$

(14)

where the Sherwood, Sh, and Nusselt, Nu, numbers were determined from the correlations reported, respectively, in [28,29]:

$$\mathrm{S}\mathrm{h}=2+1.1{\mathrm{S}\mathrm{c}}^{0.33}{\mathrm{R}\mathrm{e}}^{0.6} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{N}\mathrm{u}=2+1.1{\mathrm{P}\mathrm{r}}^{0.33}{\mathrm{R}\mathrm{e}}^{0.6}.$$

(15)

The viscosity of the gaseous mixture was computed using Wilke’s formula, and its thermal conductivity from Wassiljewa’s formula [26].

The initial conditions for equations Equations (1) and (9) were defined as follows:

$$C_i\left(r,0\right)=0,T\left(r,0\right)=T_0 \mathrm{a}\mathrm{n}\mathrm{d} q_{C{O}_2}\left(r,0\right)=0 \mathrm{f}\mathrm{o}\mathrm{r} 0\le r\le R_p.$$

(16)

2.2. Numerical Solution and Model Parameters

The model equations with associated boundary and initial conditions were solved numerically using the method of lines [30], which involves approximation of spatial derivatives using finite differences. First, the spatial domain was discretized using N = 251 equidistant nodes, and then the spatial derivatives of state variables were approximated at the discrete nodes using central differential quotients. The resulting system of ordinary differential equations was then integrated in time using a solver ode23tb from Matlab 2022b software.

The following issues were analyzed via numerical simulations:

• The effect of particle structure on the dynamics of CO₂ adsorption, including the time required to saturate the particle with adsorbate;
• The effect of particle structure on sorption dynamics in particles subjected to temperature perturbations in a bulk gas;
• The effect of different temperature perturbations on sorption dynamics in core–shell particles with adsorbent located in the core;
• The effect of pore diameter on sorption dynamics in core–shell particles with adsorbent located in the core.

As mentioned earlier, due to the relatively large number of model parameters, in this study the analysis was limited to the case where the volume ratio of adsorbent to catalyst in the hybrid particle is 1:1. For a particle with a uniform distribution of sorbent and catalyst throughout the volume, this implies that f_ads(r) = f_cat(r) = 0.5. On the other hand, for core–shell particles, to avoid numerical difficulties, the following S-shaped logistic function was used to describe the radial distribution of sorbent:

$$f_{ads}\left(r\right)=\frac{1}{1+\mathrm{e}\mathrm{x}\mathrm{p}\left[\pm A\left(r/R_p-R_{cs}/R_p\right)\right]}.$$

(17)

The sign before the expression under the exponent in Equation (17) depends on whether the sorbent is located in the core (+) or in the shell (−), whereas A denotes the steepness parameter and ratio R_cs/R_p is the function midpoint. As mentioned earlier, the value of R_sc was set to 0.7937, whereas the value of A was set to 1000. The function so defined approaches the Heaviside step function yet is still a continuous function and it takes values from 0 (in catalytic zone) to 1 (in sorbent zone). In addition to eliminating numerical problems, such a function allows for a more realistic description of the core–shell structure, since it is obvious that in the real particle at the boundary between the different functionalities there will be narrow transition zone containing both materials. Its thickness can be specified by the parameter A.

The values of key model parameters employed in the numerical simulations are reported in

Table 1. Values of key model parameters used in numerical simulations.

Parameter	Value	Parameter	Value
cg	1.014 × 103 J/kg·K−1	τp,ads = τp,cat	3
cg,ads	97.461 J/kg·K−1	a0	9.875 × 10−7 mol·kg−1·kPa−1
cs,ads	1000 J × kg−1·K−1	b0	6.761 × 10−8 kPa−1
cs,cat	1107 J × kg−1·K−1	c	−2.002 × 10 K
dpore	1.7 × 10−7 m	Eads	5.625 × 103 K
Rc	5 × 10−7 m	n0	2.7 × 10−1
Rp	5 × 10−3 m	ΔHads,CO2	−3.555 × 104 J·mol−1
εp,ads = εp,cat	0.4	D0	5.9 × 10−11 m2·s−1
λs,ads	0.5 W·m−1·K−1	Ediff	2.633 × 104 J·mol−1
λs,cat	0.84 W·m−1·K−1	p	101,325 Pa
ρg	1.117 kg·m−3	Tbulk	323 K
ρp,ads	1087 kg·m−3	yCO2,bulk	0.2
ρp,cat	3532 kg·m−3

. In particular, the parameters of the temperature-dependent Thoth isotherm (Equations (7) and (8)) describing the equilibrium of CO₂ adsorption on zeolite 5A were taken from the work [21], while the preexponential coefficient and diffusional activation energy that appear in the expression for the intracrystalline diffusion coefficient (Equation (4)) were taken from the work [19].

Figure 3a shows adsorption isotherms of CO₂ on zeolite 5A employed in this work, while the impact of mean pore diameter on diffusion coefficients is illustrated in Figure 3b. Note that for the baseline average pore diameter of macropores, i.e., pores between microparticles forming a single hybrid pellet, of d_p = 1.7 × 10⁻⁷ m and at T_bulk = 323 K the value of effective diffusion coefficient calculated from Equation (3) is D_eff = 1.3237 × 10⁻⁶ m²·s⁻¹, while at the same temperature the value of the configurational (intracrystalline) diffusion coefficient determined from Equation (4) is as low as D_c = 3.2512 × 10⁻¹⁵ m²·s⁻¹.

3. Results and Discussion

Prior to a comparative analysis of the different hybrid particle configurations (Figure 1), the dynamics of the adsorption process of CO₂ in a particle with a core–shell structure, and with a sorbent situated in the core (referred to as core(sorbent)–shell) was analyzed. In the calculations, the initial temperature of the particle was assumed to be equal to the temperature of the bulk gas (T₀ = T_bulk) and the concentration of carbon dioxide in the pores and in the solid sorbent was set to zero (Equation (16)). Although the model used in this study is one-dimensional, in order to better illustrate what is happening within the particle, in Figure 4 the results of simulations of adsorption dynamics are presented in a polar coordinate system. The contour plots shown in Figure 4 display the evolution over time of the distributions of each state variable in the particle cross section.

The symbol t_end in the last row of the Figure 4 corresponds to the particle saturation time, which in the analyzed case amounted to t_end = 794 s. It was assumed that the adsorption process stops when the norm from the difference between the values of all discrete state variables in consecutive time instants is lower than 10⁻⁴.

As can be observed in Figure 4a–c, despite the relatively large size of the hybrid particle (R_p = 5 × 10⁻³ m), after just 10 s CO₂ is present in the entire volume of the particle. Although the concentration of CO₂ in the core macropores is still very low at this particular time (Figure 4a), the slight increase in temperature (Figure 4b) as well as solid-phase concentrations of CO₂ (Figure 4c) confirm that the adsorption process takes place throughout the entire core. In the first instance of the process (Figure 4a–i), the distributions of all state variables are non-monotonic, due to the overlapping of mechanisms of diffusion, adsorption and heat conduction that takes place within the particle. As can be clearly observed, heat transport in the particle by conduction also causes an increase in temperature in the inactive shell. Once the sorbent embedded in the hybrid particle is saturated, the concentration of CO₂ in the pores becomes uniform (Figure 4j), the temperature reaches back the value equal to the bulk gas temperature (Figure 4k), and the solid-phase concentration of CO₂ in the entire volume of the core (Figure 4l) becomes equal to the equilibrium value for T = 323 K, i.e., $q_i^{*}$ = 2.506 mol/kg.

In the next step, the analysis was made to determine which structure exhibits the best efficiency in the adsorption process. The results of the performed simulations are shown in Figure 5.

Due to the fact that in each configuration the mass of the adsorbent is the same, and thus each of them at a given temperature is able to adsorb the same amount of CO₂, the parameter against which the process was analyzed was the time required to saturate the particle. Figure 5a–c show the concentration of adsorbed carbon dioxide with respect to the particle radius for selected time instants, while Figure 5d illustrates a comparison of the temporal evolution of the total number of moles of adsorbed CO₂ during the process for the three analyzed structures. The moment of complete saturation of the sorbent incorporated into the hybrid particle is marked in Figure 5d using bullets. As expected, the structure with the sorbent located in the shell (referred to as core–shell(sorbent)) outperforms the other configurations when using process time as an evaluation criterion. This is caused by the shortest diffusion path required for the carbon dioxide molecules to reach the sorbent.

However, it turns out that the core(sorbent)–shell structure offers a number of desirable features that, in a global context, i.e., from the point of view of the dynamic process carried out in the fixed-bed adsorption column, can be exploited. Before addressing them, it is also worth looking at the temperature evolution in the particle. As is well known, physical adsorption is an exothermic process. Due to the resistance to mass transport in the particle, even in particles with a uniform structure, non-uniform concentration profiles and, as a result, non-uniform temperature profiles form at the stage of particle saturation (Figure 6a). In the case of core–shell particles, these distributions are non-monotonic at the beginning the process (Figure 6a). In addition, it turns out that for the core(sorbent)–shell configuration, the temperature in the center of the particle is (over time) always higher than the temperature on its surface. On the other hand, for the core–shell(sorbent) structure there is a kind of inversion, i.e., initially the temperature at the surface is higher, but as the adsorption zone advances deeper into the particle the temperature at the center becomes higher, since the heat is also transported to the inactive core positioned centrally.

These spatiotemporal peculiarities together with the different diffusion paths that the adsorbate has to follow in different configurations in combination with the different arrangement in the spherical geometry of the sorption centers make core(sorbent)–shell structure have important practical potential.

One of the phenomena occurring in the adsorption columns during the adsorption process is the formation of thermal waves that travel along the bed with the flowing gas. These waves can lead to local temperature disturbances and affect the local adsorption equilibrium negatively. Therefore, in the next step, simulations were carried out to evaluate the effect of temperature perturbations on adsorption dynamics.

Adsorbate-saturated particle being in thermal equilibrium with the bulk gas was adopted as the initial condition, that is, the final simulation results from Figure 5 were utilized. The implemented perturbation consisted in changing the temperature of the bulk gas, T_bulk, from 323 K to 350 K. Figure 7a–c show how the introduced perturbation affects the concentration of CO₂ in the solid phase, consecutively for the core(sorbent)–shell, core–shell(sorbent) and uniform structure, whereas Figure 7d illustrates the total amount of adsorbed CO₂ as a function of time, where t = 0 indicates the time of introduction of the perturbation. The new equilibrium ($q_i^{*}$ = 1.709 mol/kg for T = 350 K) is reached first in the core–shell(sorbent) configuration and is reached most slowly in the core(sorbent)-shell case. This means that the structure with the sorbent placed in the core is less sensitive to temperature perturbations. This property may have two particular applications. The sorption-inactive shell in the core(sorbent)–shell configuration can play a protective role as thermal waves travel through the bed. Indeed, an increase in temperature leads to a local shift of the sorption equilibrium, resulting in unwanted desorption of component being separated. In the case of a core(sorbent)–shell structure, this phenomenon is going to be hindered and slowed down, as evidenced by the analysis of the effect of temperature perturbation on particle behavior. Another case is when the adsorption step is followed by reactive regeneration. As a result of heating the bed, a huge amount of adsorbed component can be suddenly released, which can leave the column unreacted due to a shortage of a second reactant supplied with the gas stream. Thus, the use of a core(sorbent)–shell structure can enable a slower release of the component from the solid phase (both due to intraparticle mass and heat transfer resistances) and thus achievement of a higher conversion rate during the reactive regeneration step.

While in the case of thermal waves, the local temperature rise in the bed is usually not very high, i.e., of a few or a few tens of degrees, reactive regeneration of the bed (e.g., CO₂ methanation reaction [15]) may be associated with the need for a significant increase in temperature. Of course, the higher the temperature, the more rapid is the initial phase of release of the adsorbed component from the solid, even for the core(sorbent)–shell configuration (Figure 8). However, the time required to reach a new equilibrium state for different temperatures is very similar, due to the significant slowdown in the process as the new equilibrium is approached.

To control this further, the next possible step is to modify the particle structure, for instance by modifying the macropores via which the adsorbate is transported to and from the surface of the sorbent microcrystals. Therefore, as a final step, it was examined how a geometric property, such as pore diameter d_pore, affects the process. In accordance with Figure 3b, this is a key factor affecting diffusion in the macropore of the particle. Numerical simulations of the effect of temperature perturbation (i.e., bulk gas temperature increase from 323 K to 383 K) on the core(sorbent)–shell particle dynamics were conducted for three values of pore diameter, namely 10, 100 and 170 nm.

Figure 9a–c show the radial distributions of the state variables 60 s after the introduction of the perturbation. Even a change by a factor of 17 in pore diameter does not significantly affect the temperature distribution within the particle (Figure 9b) and the time to establish a new adsorption equilibrium (Figure 9d, denoted with bullet), but it does have a crucial influence on the gas phase concentration. Indeed, a reduction in pore diameter increases the internal resistance to mass transport. Due to the small average pore diameter, carbon dioxide, which desorbs under the effect of temperature perturbation, is not able to diffuse and equalize the concentration inside the particle macropores relatively quickly. Based on these results, it can be assumed that the controlled release of the adsorbed component during reactive regeneration of the sorbent can be carried out by selecting an appropriate value of the pore diameter.

4. Conclusions

In this study, a mathematical model of a non-isothermal hybrid sorbent particle with a core–shell and uniform structure for carbon dioxide sequestration was developed and numerically resolved. It was assumed that the hybrid particle, in addition to the zeolite sorbent (zeolite 5A), contains a nickel catalyst, which can be exploited at the stage of reactive regeneration realized via methanation reaction. The performance of the hybrid particle was evaluated for three different configurations, that is core(sorbent)–shell, core–shell(sorbent) and a uniform mixture of both functionalities, i.e., adsorbent and catalyst. The obtained results were compared in terms of the time required to saturate the particle.

The core–shell(sorbent) configuration outperformed the remaining two cases, with the core(sorbent)–shell particle characterized by the highest value of the saturation time. Despite this, the core(sorbent)–shell arrangement still exhibits relevant properties due to its lower sensitivity to temperature perturbations, which are usually found in fixed-bed adsorption columns in the form of travelling thermal waves. Therefore, special attention was paid to the performance of the core(sorbent)–shell structure under the influence of perturbations, such as a step increase in the temperature of the gas surrounding the particle. It was shown that a protective and inactive (at the sorption stage) shell has the potential to prevent or slow down the undesired desorption of the component being separated and resulting from a shift in sorption equilibrium due to local temperature rise. Furthermore, it was analyzed how the intercrystalline pore diameter affects the sorption process. Adopting pores with smaller diameters increased the diffusion resistance and thus allowed for a more controlled release of the adsorbed substance from the particle.

Both the lower sensitivity of core(sorbent)–shell particles to temperature changes and the ability to control mass transport through appropriate selection of pores geometry are expected to play important role when using such particles in reactive regeneration stage of the process. Moreover, in reference to works related to thermocatalysis, where the catalyst is additionally surrounded by a protective shell (e.g., to avoid sintering) and given the results of the present work, an interesting design might be also a double-shell configuration, i.e., a core(sorbent)–shell(catalyst)–shell(sorbent) structure.

Analysis of the dynamics and optimization of the structure of core–shell particle for both CO₂ adsorption and methanation stages will be the subject of further studies. In addition, multicomponent adsorption is to be included in the model, in order to account for coadsorption of other species, and possibly also chemisorption of the species. Given the simplifying assumptions of the model provided in Section 2, which now restrict the extent of its applicability, this will require, among other things, accounting for the effect of temperature on physical, thermal and transport properties, considering viscous flow and, in the case of water coadsorption, taking into account the real nature of the gas phase.