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2 Theoretical background 19 The main advantage of this model is that a better fit of the experimental breakthrough curves than LDF for low adsorptive and adsorbate concentrations in the system can be obtained [37,73] due to amplified influence of adsorbent loading. However, when considering the dynamics of an adsorption column, the breakthrough curves can be described almost quantitatively by either the LDF or the QDF models for local adsorption kinetics, as presented in Fig. 2.2.2-1 [37]. Therefore, the selection between those mass transfer models should be based on developed concentration gradients within the adsorber mass transfer zone. 2.2.3 Darken correction The Darken correction of the intra-particle diffusivity shall be considered during characterisation of the mass transfer within highly microporous adsorbents. In the micropore (d < 2 nm), the diffusing molecules never escape from the force field of the pore wall; therefore, no bulk phase is recognised. In such circumstances, the strong concentration dependence of diffusivity occurs due to the effect of nonlinear adsorption equilibrium [32]. Since the driving force of mass transfer is always the gradient of chemical potential, the classic Fickian diffusion equation defining the surface diffusion flux can be modified, as presented in Eq. 2.2.3-1. J=−Ddcads=−D0 c d dz Rg T ads dz (Eq. 2.2.3-1) where: μ is the chemical potential, D is the diffusivity, and D0 is the corrected diffusivity. Assuming the equilibrium state, the chemical potential corresponds to that in the ideal gas phase, following Eq. 2.2.3-2. =0 +RTlnp (Eq.2.2.3-2) where: μ0 is the reference chemical potential. By combining Eq. 2.2.3-1 and Eq. 2.2.3-2, the surface diffusion flux can be expressed as shown in Eq. 2.2.3-3, which is valid at constant temperature and pressure conditions. J =−Ddcads =−D c dln p (Eq.2.2.3-3) dz 0adsdz By means of this strategy, the micropore diffusivity is defined as shown in Eq. 2.2.3-4, which depends on the equilibrium isotherm between gas and solid phases. The Darken correction states that surface diffusivity D at any loading is equal to the value at zero loading D0 multiplied by a thermodynamic correction factor dlnp/dlnw [34]. Therefore, the application of Darken correction requires very accurate equilibrium data to evaluate secant and tangent of adsorption isotherm, as presented in Fig. 2.2.3-1. D=D dlnp D dlnp=D wdp 0 dlnc 0 dlnw 0 pdw ads (Eq. 2.2.3-4)PDF Image | Modelling and Simulation of Twin-Bed Pressure Swing Adsorption Plants
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