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2 Theoretical background 17 (LDF) equation. Therefore, both approaches are presented in order to illustrate the performed strategy. Moreover, the pressure dependence of mass transfer kinetics is considered through the implementation of the Darken correction due to utilisation of highly microporous CMS adsorbent in high-pressure PSA processes. 2.2.1 LDF model The linear driving force model of adsorption kinetics was first introduced by Gluekauf for adsorption phenomena in chromatography [68]. It assumes that (1) the temperature and loading of the adsorbent are uniform and do not depend on the particle radius, and (2) the entire mass transport resistance is transferred to the boundary layer of the solid particle [37,51]. The graphical representation of concentration gradients within the adsorbent according to LDF model is presented in Fig. 2.2.1-1. Thus, the mass transfer rate of a single adsorbate k into a spherical adsorbent particle is quantified as shown in Eq. 2.2.1-1 [68]. The equation is applied when the diffusion process in the solid is the controlling stage, i.e. intra-particle mass transfer resistances dominate. wk (t) = MTC (w* (t) − w (t))= 15D (w* (t) − w (t)) (Eq. 2.2.1-1) t LDF k k r2 k k 0 where: w(t) is the average adsorbate concentration in the adsorbent particle at time t, and w*(t) is the adsorbate concentration in the adsorbent particle that would be in equilibrium with the instantaneous gas phase partial pressure of the adsorbate and the adsorbent temperature at time t. MTCLDF is the effective mass transfer coefficient at adsorbate loading w and the adsorbent temperature, which can be estimated based on intra-particle diffusivity D and adsorbent particle radius r0. Fig. 2.2.1-1 Concentration gradients within the adsorbent particle according to LDF model [51] Therefore, all mass transfer resistances occurring within the porous system of the adsorbent are lumped together and characterised by MTCLDF. The equation was originally derived from a series expansion of the diffusion problem in a single particle subjected to a step change in external concentration, which is valid for long times of approximation, thereby for a long-term adsorption [69,70]. However, the equivalent equation can be derived considering that thePDF Image | Modelling and Simulation of Twin-Bed Pressure Swing Adsorption Plants
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