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2 Theoretical background 13 equation. However, the larger this parameter is, the higher the degree of system heterogeneity, as shown in Fig. 2.1.1-1. The Sips equation circumvents the limitation of the rising adsorbate concentration associated with the Freundlich isotherm. At low adsorbate concentrations it reduces to the Freundlich isotherm; whereas at high concentrations it predicts a monolayer adsorption in a similar manner to the Langmuir isotherm. Namely, it predicts that the amount adsorbed reaches the saturation capacity, corresponding to complete coverage of all adsorption sites with adsorbate molecules. For this reason, the Sips isotherm should be used to describe only monolayer adsorption systems. However, as long as the equation is not used for equilibrium data extrapolation, it can be applied at higher pressures associated with multi-layer adsorption, but the parameter wsk* should not be considered as the ultimate saturation adsorption [58]. Moreover, since the Sips equation does not reduce to the linear isotherm at very low pressures, which is a constrain demanded by statistical thermodynamics, the application of the model in this range should be avoided. Fig. 2.1.1-1 Plots of the Sips equation as the function of bk(T)pk [34] 2.1.2 Isostere method The adsorption enthalpy is defined as the specific heat to be released/required when an adsorptive molecule binds/detaches to the solid surface of an adsorbent [59]. It remains a necessary parameter in the interpretation of adsorption and desorption processes, especially in the case of non-isothermal systems, since the temperature variation influences both adsorption equilibria and mass transfer kinetics. In gas-phase systems it can be determined experimentally via gas adsorption microcalorimetry technique [60]; or calculated straightforwardly based on analysis of adsorption isotherms via the so-called isostere method, which is simple and commonly practiced. The isostere method originates in the following argumentation [51]. Since the adsorption isotherm identifies the thermodynamic equilibrium state between gas and solid phases, the Gibbs free energy equals zero at its every point, according to Eq. 2.1.2-1. Subsequently, the entropy term can be substituted by the quotient of adsorption enthalpy and temperature, as shown in Eq. 2.1.2-2. Therefore, by assuming that the volume of the adsorbate phase is negligible compared to the volume of the gas phase (Eq. 2.1.2-3) and substituting the volume of the gas phase according to the ideal gas law (Eq. 2.1.2-4), the adsorption enthalpy can bePDF Image | Modelling and Simulation of Twin-Bed Pressure Swing Adsorption Plants
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