Mixture Theory for a Multi-component Solution

For a multi-component mixture with 1, 2, 3, At-species in a fundamentally miscible solution, which involves a process of chemical reactions, we establish a framework of continuum thermodynamics theory. The mixture might be a liquid or gas however, since we mainly use a mass-averaged velocity (defined later), the liquid solution might be more appropriate. The description is Eulerian because we are considering a fluid. 

For any solution we distinguish between solute and solvent. Here we can state that the first species is the solvent and the rest of the (2, 3, A )-species are solutes. We shall introduce a non-equilibrium thermodynamics theory for this solution that will treat the coupled problem of stress, diffusion with reactions and heat transfer. The field variables for the chemical field are listed in Table 3.1. 

We extend the theory of mixture given by Bowen (1976) (see also Drew and Passman 1998 Coussy 1995 Lewis and Schrefler 2000), though the notation is different. A continuum body with A-components is represented as B (a = 1,2, , N). It should be noted that due to the mixture theory the body B of the ath component defines the independent reference configuration for each component. The case of two components are schematically shown in Fig. 3.4. The motion of a spatial point x in the current body 2 c can be described by 

The particle velocity Va of a material point Xa at a time t is obtained as 

We here denote the material time-derivative d a)4 /dt of a function j x,t) with respect to the ath component as 

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A brief qualitative evaluation of CO2 adsorption selectivity in MOFs is the direct observation on differences in uptakes between separated gases under given conditions, usually based on the single-component adsorption isotherms. These isotherms can also be used to quantitatively estimate the adsorption selectivity. If adsorption species are presented at low loadings, namely within the Henry s regime, the adsorption selectivity for an equimolar mixture is close to the ratio of the Henry s constants for each species. At non-dilute loadings, however, more information is required to estimate multi-component adsorption. One common approach is to use ideal adsorbed solution theory (lAST) to predict multi-component adsorption isotherms and selectivity based on the single-component adsorption isotherms [60], This approximate theory is known to work accurately in many porous materials. 

In the thermodynamic model presented here, the Cubic-Plus-Association equation of state combined is used to model the fluid phases. The hydrate phase is modelled by the solid solution theory of van der Waals and Platteeuw. Good agreement between the model predictions and experimental data is observed, demonstrating the reliability and robustness of the developed model. The CPA EoS is shown to be a very successful model for multi-phase multi-component mixtures containing hydrocarbons, glycols and water. 

In this section the basic principles of membrane formation by phase inversion will be described in greater detail. All phase inversion processes are based on the same thermodynamic principles, since the starting point in all cases is a thermodynamically stable solution which is subjected to demixing. Special attention will be paid to the immersion precipitation process with the basic charaaeristic that at least three components are used a polymer, a solvent and a nonsolvent where the solvent and nonsolvent must be miscible with each other. In fact, most of the commercial phase inversion membranes are prepared from multi-component mixtures, but in order to understand the basic principles only three component systems will be considered. An introduction to the thermodynamics of. polymer solutions is first given, a qualitatively useful approach for describing polymer solubility or polymer-penetrant interaction is the solubility parameter theory. A more quantitative description is provided by the Flory-Huggins theory. Other more sophisticated theories have been developed but they will not be considered here. 

The present book has been written with the enthousiastic cooperation of my group of co-workers. Particularly important contributions are due to Dr. A. Bellemans and Dr. V. Mathot. Dr. Bellemans is associated with the recent developments of the theory of multi-component systems in the field of isotopes and pol3nner solutions as well as with the formulation of the average potential model. Moreover, he contributed actively to Ch. IX-XI on the average potential theory and to Ch. XVI-XVTI on Polymers. Dr. Mathot, who contributed to the section on Polymer Solutions in Ch. Ill, was associated with the extension of the cell model to multicomponent sterns. His main contribution in this field is his experimental work on the thermodynamical properties of simple mixtures. For these reasons I have invited Dr. Bellemans and Dr. Mathot to be co-authors. 

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