Membranes rate-transfer equation

Useful simplifications are often made in Equation 2.2. We will use gas-liquid contact as an example, and assume gas-filled homogeneous membrane of high porosity, thin wall, and low tortuosity. Since diffusion in gas phase is generally of three orders of magnitude faster than in liquid phase, one can show that and ka are quite high in this case compared to ki, and so the controlling resistance to mass transfer is in the liquid phase. This means A total is essentially the same as If is constant within the contactor the total mass transfer rate in Equation 2.4 can be approximated for the entire contactor as... 

Assuming that the rate of mass transfer is proportional to the concentration difference over the membrane according to Equation 12.1 and noting that in static extraction, the concentration in the donor phase Cd decreases as analyte is transferred over the membrane. We get the following differential equation ... 

Here, F, Zf and h are, respectively, the molar flow rate, mole fraction of component of i and total enthalpy, all in cell k their subscripts, ret and perm, refer to retentate and permeate streams. Equations (10.4) and (10.5) are mass balances and mass-transfer equations for each of the components present in the membrane feed. The cross-flow model [Equations (10.3)-(10.7)] was implemented in ACM v8.4 and validated against the experimental data in Pan (1986) and the predicted values of Davis (2002). The Joule-Thompson effect was validated by simulating adiabatic throttling of permeate gas through a valve in Aspen Hysys. Both these validations are described in detail in Appendix lOA. 

Equation Q7-6ai. which we will call the rate transfer or RT curve, relates the mole fracs on both sides of the membrane based on the RT parameters and the driving force. In deriving Eq. ri7-6ai we invoked the assumption that the membrane separator is perfectly mixed when we replaced y with yp. However, this equation is applicable to other flow configurations if written in terms of y and applied point-by-point on the membrane. Equation (17-f>a) is not an equilibrium expression since it was derived based on transfer rates, but as we will see, it can take the place of an equilibrium expression for binary systems. 

Ideally, it is necessary to use mass-transfer equations in the feed-gas phase and the product-gas phase along with the permeation equation (3.4.72). However, in general, the gas permeation rate through a nonporous membrane is so slow that mass-transfer equations in the feed-gas and product-gas phases are not needed. Note that, for species i transport through the membrane from the feed to the product gas, pif> pip, but fy need not be greater than Pp, although in practice it generally is. In such a case, flux expression (3.4.72) may also be expressed as... 

Membrane Reactors. Consider the two-phase stirred tank shown in Figure 11.1 but suppose there is a membrane separating the phases. The equilibrium relationship of Equation (11.4) no longer holds. Instead, the mass transfer rate across the interface is given by... 

These component balances are conceptually identical to a component balance written for a homogeneous system. Equation (1.6), but there is now a source term due to mass transfer across the interface. There are two equations (ODEs) and two primary unknowns, Og and a . The concentrations at the interface, a and a, are also unknown but can be found using the equilibrium relationship, Equation (11.4), and the equality of transfer rates. Equation (11.5). For membrane reactors. Equation (11.9) replaces Equation (11.4). Solution is possible whether or not Kjj is constant, but the case where it is constant allows a and a to be eliminated directly... 

Note that in the component mass balance the kinetic rate laws relating reaction rate to species concentrations become important and must be specified. As with the total mass balance, the specific form of each term will vary from one mass transfer problem to the next. A complete description of the behavior of a system with n components includes a total mass balance and n - 1 component mass balances, since the total mass balance is the sum of the individual component mass balances. The solution of this set of equations provides relationships between the dependent variables (usually masses or concentrations) and the independent variables (usually time and/or spatial position) in the particular problem. Further manipulation of the results may also be necessary, since the natural dependent variable in the problem is not always of the greatest interest. For example, in describing drug diffusion in polymer membranes, the concentration of the drug within the membrane is the natural dependent variable, while the cumulative mass transported across the membrane is often of greater interest and can be derived from the concentration. 

For undersaturated ([M] < Km) systems with relatively fast internalisation kinetics (kmt > k, ), the uptake of trace metals may be limited by their adsorption. Because the transfer of metal across the biological membrane is often quite slow, adsorption limitation would be predicted to occur for strong surface ligands (small values of k ) with a corresponding value of Km (cf. equations (35) and (36)) that imposes an upper limit on the ambient concentration of the metal that can be present in order to avoid saturation of the surface ligands. More importantly, as pointed out by Hudson and Morel [7], this condition also imposes a lower limit on the carrier concentration. Since the complexation rate is proportional to the metal concentration and the total number of carriers, for very low ambient metal concentrations, a large number of carriers are required if cellular requirements are to be satisfied. 

This boundary-layer theory applies to mass-transfer controlled systems where the membrane permeation rate is independent of pressure, for there is no pressure term in the model. In such cases it has been proposed that, as the concentration at the membrane increases, the solute eventually precipitates on the membrane surface. This layer of precipitated solute is known as the gel-layer, and the theory has thus become known as the gel-polarisation model proposed by Micii i i.si 0). Under such conditions C, in equation 8.15 becomes replaced by a constant Cq the concentration of solute in the gel-layer, and ... 

The membrane and diffusion-media modeling equations apply to the same variables in the same phase in the catalyst layer. The rate of evaporation or condensation, eq 39, relates the water concentration in the gas and liquid phases. For the water content and chemical potential in the membrane, various approaches can be used, as discussed in section 4.2. If liquid water exists, a supersaturated isotherm can be used, or the liquid pressure can be assumed to be either continuous or related through a mass-transfer coefficient. If there is only water vapor, an isotherm is used. To relate the reactant and product concentrations, potentials, and currents in the phases within the catalyst layer, kinetic expressions (eqs 12 and 13) are used along with zero values for the divergence of the total current (eq 27). 

Application of Eq. (15.1) to the liquid membrane process highlights one of the main advantages of the process, i.e., the high solute distribution coefficient that can be obtained between phases 3 and 1. However, another factor that must be considered when evaluating a separation process performance is the kinetics of transfer, which is given in a general form by Eq. (15.4). This equation indicates that the transfer rate in the contactor increases with both the interfacial flux and the specific interfacial area. 

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