Chemical drive pressure coefficient

The chemical potential // and drive (afhnity) X are not considered independent concepts in the conventional thermodynamic formalism. Therefore, one does not directly define the temperature and pressure coefficients a and f), as well as oc and Ji, rather, they are always expressed in terms of different quantities ... 

From Fig. 19.3a-c, and as opposed to purely sorption controlled processes, it can be seen that during pervaporation both sorption and diffusion control the process performance because the membrane is a transport barrier. As a consequence, the flux 7i of solute i across the membrane is expressed as the product of both the sorption (partition) coefficient S, and the membrane diffusion coefficient Di, the so-called membrane permeability U, divided by the membrane thickness f and times the driving force, which maybe expressed as a gradient of partial pressures in place of chemical potentials [6] ... 

Conversely, the correct approach to formulate the diffusion of a single component in a zeolite membrane is to use the MaxweU-Stefan (M-S) framework for diffusion in a nonideal binary fluid mixture made up of species 1 and 2 where 1 and 2 stands for the gas and the zeohtic material, respectively. In the M-S theory it is recognized that to effect relative motions between the species 1 and 2 in a fluid mixture, a force must be exerted on each species. This driving force is the chemical potential gradient, determined at constant temperature and pressure conditions [68]. The M-S diffiisivity depends on coverage and fugacity, and, therefore, is referred to as the corrected diffiisivity because the coefficient is corrected by a thermodynamic correction factor, which can be determined from the sorption isotherm. 

The three main driving forces which have been used within diffusion models (moisture content, partial pressure of water vapor, and chemical potential) will now be discussed. Attempts to predict diffusion coefficients theoretically will also be reviewed, together with experimental data for fitted diffusion coefficients and their dependence on temperature and moisture content. 

Performances of PV membranes are represented by parameters such as separation factor, flux of permeates, and service life. The separation factor of a membrane is a measure of its permeation selectivity (permselectivity) and is defined as the ratio of the concentration of components in the permeate mixture to that in the feed mixture. The component flux is the amount of a component permeating per unit time and unit area, and is given by the product of the permeability coefficient of the membrane and the driving force. The driving force is the gradient in the chemical potential of the components between the feed and the permeate side of the membrane. These values are influenced by operating variables such as temperature, composition of each component in the feed mixture, and permeate side pressures (see Fig. 107). 

An approach that is conceptually simpler and does not require the prescription of transport to hydraulic or diffusion mechanisms was proposed by Janssen [47], and Thampan et al. [22] (hereafter TMT) based on the use of chemical potential gradients in the membrane. More recently, Weber and Newman [27] developed a novel model where the driving force for vapour-equilibrated membranes is the chemical potential gradient, and for liquid-equilibrated membranes it is the hydraulic pressure gradient. A continuous transition is assumed between vapour- and liquid-equilibrated regimes with corresponding transition from 1 to 2.5 for the electro-osmotic drag coefficient. 

For ideal gases, the driving force on the left side of these equations reduces to (l/P)dp/dz, which for constant pressure systems is equivalent to dy/dz, and Eqs. f 15-541 sinplify to Eqs. fl5-52a.bl. For nonideal systems, the chemical potential can be written in terms of the activity coefficients Ya nd Yb liquids or the fugacity coefficients for gases. The results for conponent A for liquids are... 

A2. In the second interpretation, Eq. (10.27), the driving force dc/dx is identical to a gradient of chemical potential hence, the equation for D involves the activity coefficient y of the polymer. The equation fits into the analysis of an interacting multicomponent system. Of course, it must be remembered that osmotic pressure and chemical potential are closely related, as discussed in Chapter 9. 

It might at first seem strange that we can have a number of different mass transfer coefficients in contrast to essentially one heat transfer coefficient. However, consider the situation for chemical equilibrium constants where we also have a number of different constants (based on activities, fugacities, partial pressures, concentrations, etc). In dealing with the mass transfer coefficient it becomes very important to clearly know the characteristic driving force used for the system. While the units of the coefficients are helpful, they are not foolproof example both ky and k have same units). 

Most of the multicomponent systems are non-ideal. From thermodynamic viewpoint, the transfer of mass species i at constant temperature and pressure from one phase to the other in a two-phase system is due to existing the difference of chemical potential 7t, x p between phases, in which /t,- p =p + T Fln where y, is the activity coefficient of component i is p at standard state. In other words, for a gas (vapor)-liquid system, the driving force of component i transferred from gas phase to the adjacent liquid phase along direction z is the... 

The Flory-Huggins expression can be used to evaluate the dpldC term at the upstream pressure for conditions where Henry s law does not apply however, only the region of Henry s law is discussed here. Figure 10 shows the local diffusion coefficient for the various penetrants as a function of local fugacity in silicone rubber (45). This can be easily converted to a plot ofZ)Ap(C ) vs C using the sorption isotherm data if desired. For carbon dioxide, the fugacity differs by as much as 22% from the pressure at the maximum pressure studied. The chemical potential difference is of course the true thermodynamic driving force for diffusion and should be used in mixed-gas calculations. 

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