Diffusional molar flux

The radial component of the diffusional molar flux of mobile component A, is evaluated at the solid-liquid interface (i.e., at r = R, y = 0, 0 > 0, = 0) to calculate the local mass transfer coefficient fee. locai ... 

The diffusional molar flux of component A is expressed via Pick s law in terms of the concentration gradient of A, only. Coupling between the diffusional mass flux of one species and all the independent mass fractions in the liquid phase is avoided by modeling this multicomponent diffusion problem as if it were a pseudobinary mixture. 

Step 10. Rewrite the mass transfer equation in terms of the diffusional molar flux of A away from the catalytic surface (i.e., in the s direction). In other words,... 

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) ... 

When mass transfer occurs by diffusion only, the diffusional molar flux can be predicted using Pick s law (1855)... 

However, when accounting for diffusional effects, the molar flow rate of species A, FA, in a specific direction z, is the product of molar flux in that direction, IFaz the cross-sectional area normal to the direction of flow,/ , ... 

The relationship between the molar flux according to (1.178) and the diffusional flux from (1.179) is appropriately defined as... 

In Chapter 11 we discuss the molar fluxes in some detail, but for now us just say they consist of a diffusional component. —D, dCJdz), and a ci vective flow component. U.Cj... 

Now, calculate the normal component of the total local molar flux of species A at the nondeformable zero-shear interface. Since the radial component of the flnid velocity vector vanishes at r = R, species A is transported across the interface exclusively via concentration diffnsion (i.e.. Pick s law). Then, the diffusional flux of species A in the radial direction, evalnated at the interface, is equated to the product of a local mass transfer coefficient and the overall concentration driving force for mass transfer (i.e., Ca. equilibrium — CA.buik)- The... 

These conditions on the diffusional mass flux and the molecular flux of thermal energy, the latter of which includes Fourier s law and the interdiffusional contribution, allow one to relate temperature and reactant molar density within the pellet. If n is the local coordinate measured in the direction of n, then equations (27-20) and (27-21) can be combined as follows ... 

Furthermore, if it is appropriate to consider the molar flux being governed by a diffusional process in the pellet macropores,... 

Here (v) is the observable transmembrane solvent velocity, and P is the membrane solute diffusional permeability. The permeability in turn is defined as the ratio of the molar flux of solute transport, moles/area-time, to the solute concentration difference causing this transport The most familiar examples of low-Pe devices are blood oxygenators and hemodialyzers. High-Pe systems include micro-, ultra-, and nano-filtration and reverse osmosis. The design and operation of membrane separators is discussed in some detail in standard references [Ho and Sirkar, 1992 Noble and Stern, 1995], and a summary of useful predictions is provided in Section 5.4. 

As a result of the diffusional process, there is no net overall molecular flux arising from diffusion in a binary mixture, the two components being transferred at equal and opposite rates. In the process of equimolecular counterdiffusion which occurs, for example, in a distillation column when the two components have equal molar latent heats, the diffusional velocities are the same as the velocities of the molecular species relative to the walls of the equipment or the phase boundary. 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

Equation 1.70 shows that the molar diffusional flux of component A in the y-direction is proportional to the concentration gradient of that component. The constant of proportionality is the molecular diffusivity 2. Similarly, equation 1.69 shows that the heat flux is proportional to the gradient of the quantity pCpT, which represents the. concentration of thermal energy. The constant of proportionality klpCp, which is often denoted by a, is the thermal diffusivity and this, like 2, has the units m2/s. 

It should be emphasized that the flux vectors for which expressions have been given in Eqs. (28) through (36) are all defined here as fluxes with respect to the mass average velocity. Not all authors use this convention, and considerable confusion has resulted in the definition of the energy flux and the mass flux. Mass fluxes with respect to molar average velocity, stationary coordinates, and the velocity of one component (such as the solvent, for example) are all to be found in the literature on diffusional processes. Research workers in the field of diffusion should be meticulous in specifying the frame of reference for fluxes used in writing up their research work. In the next section this important matter is considered in detail for two-component systems. 

A reference system with the average molar velocity is called the particle reference system. Other reference systems and velocities are available in the literature [1.21], The diffusional flux in one system can be transferred to any other system, as is shown in the example which follows. 

The minus sign appears because the normal vector n points outwards from the area dA, and the molar flow into the area should be positive. The diffusional flux is defined by (1.158) as... 

The first step in our CRE algorithm is the mole balance, which we now need to extend to include the molar fiux, and diffusional effects. The molar flow rate of A in a given direction, such as the z direction down the length of a tubular reactor, is just the product of the flux. (mol/m s), and the cross-sectional area. Ac (nt ), that is. 

The molar diffusional flux of reactant A toward the catalytic surface is governed by Pick s law with a concentration-independent binary molecular diffusion coefficient. Thermal (Soret), pressure, and forced diffusion are neglected relative to concentration diffusion. 

One more algebraic eqnation is required to solve for all unknown molar densities at Zk+i It is not advantageons to write the mass balance at the catalytic surface (i.e., at xatx-i-i) because the no-slip boundary condition at the wall stipnlates that convective transport is identically zero. Hence, one relies on the radiation boundary condition to generate eqnation (23-46). Diffusional flux of reactants toward the catalytic snrface, evalnated at the surface, is written in terms of a backward difference expression for a first-derivative that is second-order correct, via equation (23-40). This is illnstrated below at Xwaii = x x+i for equispaced data ... 

Step 11. Write all the boundary conditions that are required to solve this boundary layer problem. It is important to remember that the rate of reactant transport by concentration difhision toward the catalytic surface is balanced by the rate of disappearance of A via first-order irreversible chemical kinetics (i.e., ksCpJ, where is the reaction velocity constant for the heterogeneous surface-catalyzed reaction. At very small distances from the inlet, the concentration of A is not very different from Cao at z = 0. If the mass transfer equation were written in terms of Ca, then the solution is trivial if the boundary conditions state that the molar density of reactant A is Cao at the inlet, the wall, and far from the wall if z is not too large. However, when the mass transfer equation is written in terms of Jas, the boundary condition at the catalytic surface can be characterized by constant flux at = 0 instead of, simply, constant composition. Furthermore, the constant flux boundary condition at the catalytic surface for small z is different from the values of Jas at the reactor inlet, and far from the wall. Hence, it is advantageous to rewrite the mass transfer equation in terms of diffusional flux away from the catalytic surface, Jas. 

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