Evaluating the Molar Flux

We now consider four typical conditions that arise in mass transfer problems and show how the molar flux is evaluated in each instance. 

An expression for in terms of the concentration of A, C, for the case of EMCD can be found by first substituting Equation (11-14) into Equation (11-7)  

Dilute Concentrations. When the mole fraction of the diffusing solute and the bulk motion in the direction of the diffusion are small, the second term on the right-hand side of Equation (11-13) [i.e., y/s,( A + Wg)] can usually be 

External Diffusion Effects on Heterogeneous Reactions Chap. 11 

This approximation is almost always used for molecules diffusing within aqueous systems when the convective motion is small. For example, the mole fraction of a 1M solution of a solute diffusing in water whose molar concentration, Cw, is 

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We now need to evaluate the molar flux of A, Ja. that is superimposed on th molar average velocity V,... 

Evaluating the Molar Flux Boundar Conditions 765 Modeling Dififusion Without Reaction Temperature and Pressure Dependence /Dab 770... 

Experiments with different pressures in the chambers provide data for simultaneous determination of the diffusivities in and permeability B0 of a porous particle (Chapter 3). Allawi and Gunn [8] have described such experiments and the evaluation procedure, accounting for the variation of viscosity with composition. The permeability of the porous particle can also be found in separate experiments on single gas flow through the particle under a pressure difference. In this case the molar flux is given by the equation... 

The first step is to estimate the molar fluxes. This can be done as described above and elsewhere (Section 8.5). The mass transfer coefficients are calculated and the values of the discrepancy functions evaluated. To reestimate the molar fluxes we must evaluate the Jacobian matrix [J]. The elements of this matrix are obtained by differentiating the above equations with respect to the independent variables. These derivatives may be approximated by... 

The molar fluxes A, are then given by Eq. 8.5.3 with [jSlB] evaluated at some average composition (the arithmetic average of yg and is that used in practice) (Krishna, 1979d, 1981b)... 

If the total flux Af is not specified, an iterative approach is required for evaluation of the molar fluxes A -. A procedure based on repeated substitution is provided in Algorithm 8.7. A still more efficient procedure can be devised using Newton s method (cf. Algorithm 8.5) (Krishna and Taylor, 1986). 

The molar flux at the interface Ajq can be calculated by multiplying the diffusion flux by the appropriate bootstrap coefficient p evaluated at the interface composition... 

When taking these partial derivatives it must be remembered that, in general, the molar densities, the mass transfer coefficients, and thermodynamic properties are functions of temperature, pressure, and composition. In addition, H is a function of the molar fluxes. We have ignored most of these dependencies in deriving the expressions given above. The important exception is the dependence of the K values on temperature and composition that cannot be ignored. The derivatives of the K values with respect to the vapor mole fractions are zero in this case since the model used to evaluate the K values is independent of the vapor composition. 

The mass transfer rates can be evaluated from a model of mass and energy transfer in distillation such as those developed in Chapters 11 and 12. We review the necessary material here for convenience. The molar fluxes in each phase are given by... 

We will now evaluate the flux terms W. We have taken the time to derive the molar flux equations in this form because they are now in a form that is consistent with the pania differential equation (PDE) solver FEMLAB. which is included on the CD with this textbook,... 

Local Interfacial Molar Flux. Results for P(0 and Sc(t) via (11-199) and (11-202), respectively, represent basic information from which interphase mass transfer correlations can be developed. Gas-liquid mass transfer of mobile component A occurs because it is soluble in the liquid phase, and there is a nonzero radial component of the total molar flux of A, evaluated at r = R(t). Even though motion of the interface induces convective mass transfer in the radial direction, there is no relative velocity of the fluid with respect to the interface at r = R t). It should be emphasized that a convective contribution to interphase mass transfer in the radial direction occurs only when motion of the interface differs from Vr of the liquid at r = R. Hence, Pick s first law of diffusion is sufficient to calculate the molar flux of species A normal to the interface at r = R t) when f > 0 ... 

Estimate the external resistance to mass transfer by invoking continuity of the normal component of intrapellet fluxes at the gas/porous-solid interface. Then use interphase mass transfer coefficients within the gas-phase boundary layer surrounding the pellets to evaluate interfacial molar fluxes. 

The effective rate of reaction corresponds to the molar flux at the external surface 7i . Using the concentration profile evaluated for Z= 1 from Equation 2.171... 

If the heat release exceeds an optimal value, the adiabatic temperature (ATi) can be excessive, and an optimal temperature profile through the cycle should be kept to reach the highest efficiency. The efficiency of the system is a key parameter when evaluating a technology. The typical cold gas efficiency ( jcg) is given in (15.16) where QH2 and QCH4 are the molar fluxes of the hydrogen produced and fuel transformed. 

AG,r=d/2 represents the molar flux density of the adsorbed component in the gas phase as given by eq. (4) and evaluated at r = d/2. The transfer coefficient for external mriss transfer d is calculated from the well known dimensionless relations employing the Reynolds and Schmidt number (ref. 12). 

To evaluate the flux of CO we need the molar density and the mass transfer coefficient k = D/(. For a spherical film is obtained form the formula in Figure 8.3 the result is = 0.625 mm. The mass transfer coefficient therefore is... 

We require the density of the vapor mixture in order to calculate the low flux mass transfer coefficients. The molar density of the vapor may be estimated using the ideal gas law and, since the system is almost isothermal, may safely be assumed to be nearly constant. The mass density, however, is likely to vary considerably between the bulk and interface, since the molar masses of the three components in the vapor phase cover such a wide range. The mass density should, therefore, be evaluated with the average molar mass... 

To complete this text we offer a selection of exercises. Some of the exercises are of a theoretical nature and we ask you to verify some of the equations presented in the text or to derive established results in other ways. In other exercises we invite you to extend the theoretical treatments in ways that we have not considered in detail (although in some cases the results are available in the literature). Still other exercises are of a computational nature and we invite you to compute mass transfer coefficients, molar fluxes, composition profiles, and other quantities. In order to gain some familiarity with the various methods described in the text we strongly recommend solving the computational problems. Hand calculation is extremely instructive but does get to be a bit tiresome after a while. There are also exercises that require you to evaluate papers in the literature. Finally, we have provided exercises that can be assigned as term projects for students, perhaps as a replacement for a final examination. 

The initial condition for the ODEs are Nb (0) - Ng/ and No (0) = Nh (0) = NjiO) = 0> Because the number of moles are conserved in both reactions, the total molar flux does not change with reactor volume. The volumetric flowrate can be evaluated from Equation 4.74, which in this case reduces to,... 

The total molar flux of mobile component A with respect to a stationary reference frame Na is evaluated at the solid-liquid interface when 0 > 0. It is necessary to consider the component of this flux in the normal coordinate (i.e., radial)... 

Answer The mass transfer calculation is based on the normal component of the total molar flux of species A, evaluated at the solid-liquid interface. Convection and diffusion contribute to the total molar flux of species A. For thermal energy transfer in a pure fluid, one must consider contributions from convection, conduction, a reversible pressure work term, and an irreversible viscous work term. Complete expressions for the total flux of speeies mass and energy are provided in Table 19.2-2 of Bird et al. (2002, p. 588). When the normal component of these fluxes is evaluated at the solid-liquid interface, where the normal component of the mass-averaged velocity vector vanishes, the mass and heat transfer problems require evaluations of Pick s law and Fourier s law, respectively. The coefficients of proportionality between flux and gradient in these molecular transport laws represent molecular transport properties (i.e., a, mix and kxc). In terms of the mass transfer problem, one focuses on the solid-liquid interface for x > 0 ... 

Step 17. If the local diffusional molar flux of reactant A (a) toward the catalytic surface, and (b) evaluated at the surface, is used to define the following local mass transfer coefficient, fcc,iocai(z) ... 

Zhu and Chen (1998) prepared cross-linked PVA composite catalytic membranes on porous ceramic plate for PV separation and PV-esterification coupling. The composite catalytic membrane was evaluated through the PV and a model system of n-butyl alcohol esterification coupling with the PV. The conversion of n-butyl alcohol reached 95% when a cross-linked PVA catalytic manbrane was used. The order of permeation fluxes was water > acid > alcohol > acetate and the total flux was greater than 0.5 kg/m h during the reaction time. The order of the separation selectivities of membranes was water-acetate > water-alcohol > water-acid. The parameters such as temperature, initial molar ratio of acid to alcohol, and catalyst concentration could be changed in order to attain the optimum of the PV-esterification coupling operation. 

The authors theoretically evaluated the performance of the as-prepared membrane, assuming catalyticaUy active nickel complexes dissolved in the IL film. In Figure 21.20, the ratio of absorbed propylene and propane molar fluxes versus... 

Once these control experiments were performed, we could evaluate the capacity of bexarotene to block the formation of amyloid pores induced by Ap peptides. In these experiments, SH-SY5Y cells were treated with Ap25-35 (or Api-42) in presence of bexarotene (molar ratio 1 1). Under these conditions, we did not detect any increase of Ca fluxes (Fig. 14.12A, right panel). This effect was observed not only when bexarotene and Ap25-35 were premixed for 1 h before addition to the cells but also when both compoxmds were simultaneous added into the cells. Finally, we demonstrated that bexarotene can inhibit the elevation of Ca entry induced by Api-42 (Fig. 14.12B-C). The photomicrographs taken during the experiments indicated that the inhibitory effect of bexarotene was xmiformly distributed over the cell culture (Fig. 14.12B). Overall, tiiese data show that bexarotene is a potent blocker of amyloid pore formation. 

Once the velocities are known, the fluxes, both mass and molar, can be evaluated. The flux is a vector quantity, and its magnitude denotes the mass (or moles) passing through a unit area per unit time. Depending on the velocity we choose, we can define mass and molar fluxes relative to stationary coordinates, relative to the mass average velocity, v, and relative to the molar average velocity,Thus, the mass and molar fluxes relative to fixed coordinates are defined as... 

The transport coefficients for the evaluation of the gas phase molar flux of the adsorptive as used in eq. (4) are defined as... 

Model Equations to Predict Deposition Rate. Appropriate constitutive expressions are needed to evaluate each of the rate terms in the component molar balances. The final model equations must predict the deposition rate, r(d, ZnS), as a function of independent control variables—component incident fluxes, r(i, Zn) and r(i, S), and the substrate temperature. 

The previous integrated equations are valid only for irradiation with monochromatic light or, approximately, with light transmitted by an interference filter sufficiently narrow to assume that the reactant has a constant molar absorption coefficient in the entire band. Otherwise, in the case of uneven absorbance of the reactant and constant emission of the lamp in the spectral range of the filter, the relation between the overall incident photon flux and the overall absorbed photon flux can be evaluated by considering a continuous sequence of very narrow (e.g., 1 nm) wavelength intervals (Fig. 4.5) in this way, the relation can be numerically calculated by means of the equation... 

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