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Processes 2022, 10, x FOR PEER REVIEW 4 of 19 Processes 2022, 10, 812 4 of 19 Figure 1. The CCA mathematical modeling pathway map cited from Li et al. [12]. Readapted with Figure 1. The CCA mathematical modeling pathway map cited from Li et al. [12]. Readapted with very small pores and that diffusion occurs in the internal void fraction [38]. A simplified In the aspect of kinetic models, three representative adsorption-kinetic models are permission from Ref. [12]. Copyright 2018 Elsevier. permission from Ref. [12]. Copyright 2018 Elsevier. 2.1. ATdhsoerptoiroen-dKiifnfuetsicosnMmooddelel further shows that the microstructure of particles consists of model was proposed to calculate the amount of the mass variation of the adsorbed gas taken into consideration. First, a homogeneous-solid or pore-diffusion model (HSDM) has component over time, as shown in Equation (2) [12]. Although the HSDM model is much been used to describe intraparticle mass transfer [36]. This model assumes that the adsor- closer to the real state, the complex solving process limits its practical applications. bent is homogeneous and that the adsorption process occurs at the external surface, fol- lowed by the diffusion of the adsorbate into the interior of the adsorbent particles [37]. dq ∞D D ∗c2c The mathematical expression for intraparticle diffusion is shown in Equation (1). dt=6q ∑r2 exp −(nπ) r2t (2) n=1 0 0 ∂𝑞 𝐷 ∂ ∂𝑞 = 𝑟 (1) ∂𝑡 𝑟∂𝑟 ∂𝑟 Second, the local equilibrium model is proposed for the strong-adsorbed component on the adsorbent material, which assumes that the mass transfer between solid adsorbent The pore-diffusion model further shows that the microstructure of particles consists particles and external gas instantaneously completes, regardless of the mass transfer resis- of very small pores and that diffusion occurs in the internal void fraction [38]. A simplified tance.Thus,thesecondmodelonlyworksforsomeidealstates,suchasCO/N separation model was proposed to calculate the amount of the mass variation of th2e ad2sorbed gas Finally, a first-order linear-driving-force (LDF) model was proposed by Lagergren [39], closer to the real state, the complex solving process limits its practical applications. which suggests that the uptake rate of a species into the adsorbent is proportional to the on activated carbon. component over time, as shown in Equation (2) [12]. Although the HSDM model is much 𝑑𝑞𝐷 𝐷 linear difference between the concentration of the species at the outer surface of the particle = 6𝑞∗ exp−𝑛π 𝑡 (2) (equilibrium adsorption capa𝑑ci𝑡ty) [40]. Th𝑟e average conc𝑟entration within the particle is expressed as: ∂t Second, the local equilibrium model is proposed for the strong-adsorbed component ∂q = k1(qe − q) (3) on the adsorbent material, which assumes that the mass transfer between solid adsorbent particles and external gas instantaneously completes, regardless of the mass transfer re- where k is the rate constant (1/s), which can be approximated by 15De/rc2, and De/rc2 is sistance. Thus, the second model only works for some ideal states, such as CO2/N2 sepa- the diffusion-time constant. The LDF mass transfer coefficient plays a considerable role in ration on activated carbon. the flow and adsorption process through a breakthrough experiment. Ma et al. [41] found Finally, a first-order linear-driving-force (LDF) model was proposed by Lagergren that a lower LDF mass transfer reduced the appearance time of the breakthrough point but [39], which suggests that the uptake rate of a species into the adsorbent is proportional to extended the completion time of the adsorption process. Moreover, the conventional LDF the linear difference between the concentration of the species at the outer surface of the equation is verified to underestimate the adsorption rate in the initial stage. Li et al. [40] particle (equilibrium adsorption capacity) [40]. The average concentration within the par- proposed an improved LDF equation derived from the parabolic concentration profile for ticle is expressed as:PDF Image | Numerical Research on the Pressure Swing Adsorption Process
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