Equation for the mass transport

Mass transport models for multicomponent systems have been developed where the equilibrium interaction chemistry is solved independently of the mass transport equations which leads to a set of algebraic equations for the chemistry coupled to a set of differential equations for the mass transport. (Cederberg et al., 1985). 

The use of immobilized biocatalyst in large-scale industrial processes is presently a widespread technique. Thus, the description of the mass transport in spherical particles as support material is well known. Ferreira et al. [32] summarizes the most important variables and equations for the mass transport, the effect of the reaction rate on the concentration distribution in the spherical particles. 

Application Albery [99] andMa-tsuda [100] have derived an approximate equation for the mass transport-limited current for an -electron transfer reaction at a wall-jet electrode,... 

The adsorber model comprises a system of (i) three parabolic partial differential equations for the mass transport of each single component coupled by both sorption isotherm equations and an expression for the temperature dependence of rate coefficients (ii) two differential equations for chemical reaction and (iii) two parabolic partial differential equations for heat transfer. Beside time, the model contains three spatial coordinates that refer to the interstitial column volume, the macropore volume and the micropore volume and that may be of different geometry. The solution of the problem for which a module-wise algorithm was developed, is described in detail in refs. [103,104]. 

If we consider the Peclet number, which can be defined asPe = v LUD + D), where v is the true fluid velocity, L is an arbitrarily selected characteristic length, Deff is the effective diffusion coefficient, and D is the dispersion coefficient, it is generally considered that mass transport is affected by diffusion and not by advection. The equation for the mass transport in bentonite is expressed as... 

Because permeability in the standard units is represented in terms of gas flow rate per unit area per unit time, the basic equation for the mass transport of gas can be rearranged to give Eqn (8.16). 

The expression for the mass-transport-limiting current density may be employed together with the Nemst equation to deduce the complete current-potential response in a solution containing only oxidized or reduced species... 

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively. 

The ratio of the observed reaction rate to the rate in the absence of intraparticle mass and heat transfer resistance is defined as the elFectiveness factor. When the effectiveness factor is ignored, simulation results for catalytic reactors can be inaccurate. Since it is used extensively for simulation of large reaction systems, its fast computation is required to accelerate the simulation time and enhance the simulation accuracy. This problem is to solve the dimensionless equation describing the mass transport of the key component in a porous catalyst[l,2]... 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

Analogous equations for the unidirectional transport of heat and mass are the Fourier law and Fick s law, that are written, respectively, as ... 

The collective diffusion coefficient is thus relevant for the mass-transport at surfaces in systems, which are not in thermodynamic equilibrium. It generally depends on coverage. The above diffusion equation is widely employed to determine D, since the adsorbate concentration is a measurable quantity. In practice, frequently the decay of an adjusted coverage gradient is analyzed and diffusion equation is solved numerically or analytically for a given geometry. This task is considerably simplified when diffusion coefficients independent of coverage exist or may be assumed and ... 

The equations governing the mass transport of the electroactive species inside and outside the porous actuator with the corresponding initial and boundary condition are described and solved elsewhere [ 18] they result in a general solution for C (, /), representing the concentration profile of the species inside the porous actuator, Since the detection is carried out be the gate of the 1SFET, which is very closely located at the edge of the actuator (.r = 0). only the solution of C (0./) is of interest. The steady-state response for C (0./) is... 

In the model with two types of particles, (11.28) is solved for both of them. The catalyst particles are solved for all components, with the reforming kinetics. Like the equations for the buik transport, (11.28) is soived for four components and H2O is caicuiated from the sum of the mass fractions (11.50). The source term is then defined by ... 

The conservation equations for flow, mass transport, and reaction in porous media are formulated using a continuum approach in which a representative elemental volume (REV) is chosen which is assumed to be smaller than the length scale of the phenomena being monitored and within which the various properties are assumed to be constant. 

Transport Equations. In a system with fluid-filled porosity f/, A p reactive solid phases, and aqueous-phase species consisting of N, basis species (the smallest set of species needed to define the chemical system in the aqueous phase) and Nc complexes, the conservation equation for the mass of a component i is ... 

In general, the overall balance for the mass transport streams (Equations 6.21 and 6.22) at the column inlet and outlet has to be fulfilled. In Equation 6.91 the closed boundary condition is obtained by setting the dispersion coefficient outside the column equal to zero. In open systems, the column stretches to infinity and in these limits concentration changes are zero. 

Finally, the effects of retardation and sorption/desorption were included in the model. Kaolinite has a low CEC (about 0.01 mg/g) and low buffering capacity (Mitchell, 1993), hence retardation for ff+ transport may not be significant. However, due to the change in pH environment, the tortuosity and the presence of other species, there may be a significant influence of retardation to the transport of H+ (Acar and Alshawabkeh, 1993). An experimentally obtained retardation factor of Jid was applied to the rate of concentration change term for the mass transport equation of H+. 

Recently, Kandwal et al. [104] developed a mathematical model for the mass transport prediction of metal ion permeation through the hollow-fiber contactors. They have written the mass flux equations by using Pick s first law of diffusion, and taking mass balance in the single hollow fiber and feed tank, the following equation was derived ... 

For theoretical analysis of sintering, it is necessary to establish the equations for diffusional mass transport. These equations can be solved when subject to... 

For the mass transport processes considered here the law of mass conservation must be applied the mass of molecules diffusing from a volimie element (or the mass difihising into the element) must be equal to the reduction (or increase) of molecule mass in the volume element. Mathematically, a continuity equation results from such a conservation law ... 

The same calculation as made in Chapter 1 for the mass transport by diffusion through a thin sheet of thickness dx leads to the main equation of heat conduction under transient conditions ... 

The theoretical analysis of sintering requires the formulation of equations for diffusional mass transport and their solution subject to the appropriate boundary conditions. There are two equivalent formulations sintering can be viewed in terms of the diffusion of atoms or the diffusion of vacancies. 

It becomes clear when Equations 11.41 and 11.42 are compared that the empirical Equation 11.41 is basically the Tafel Equation 11.42 plus terms to account for the mass transport resistance loss and the iR loss. It is then fair to say that the Tafel equation is valid only in the absence of mass transport resistance and when the Ohmic resistance is corrected. 

To predict significant transport effects, respective criteria are needed. This is particularly important for kinetic studies in laboratory reactors to minimize the intrusion of diffusion and heat conduction. Consequently, these criteria are deduced. The equations for the mass and heat transfer and the respective parameters such as a, fi, and Deffhave already addressed in Sections 3.1 and 3.2, but some equations are subsequently repeated. 

The two equations for the mass and heat balance, Eqs. (4.10.125) and (4.10.126) or the dimensionless forms represented by Eqs. (4.10.127), (4.10.128) and (4.10.130), consider that the flow in a packed bed deviates from the ideal pattern because of radial variations in velocity and mixing effects due to the presence of the packing. To avoid the difficulties involved in a rigorous and complicated hydrodynamic treatment, these mixing effects as well as the (in most cases negligible contributions of) molecular diffusion and heat conduction in the solid and fluid phase are combined by effective dispersion coefficients for mass and heat transport in the radial and axial direction (D x, Drad. rad. and X x)- Thus, the fluxes are expressed by formulas analogous to Pick s law for mass transfer by diffusion and Fourier s law for heat transfer by conduction, and Eqs. (4.10.125) and (4.10.126) superimpose these fluxes upon those resulting from convection. These different dispersion processes can be described as follows (see also the Sections 4.10.6.4 and 4.10.7.3) ... 

Conditions for constant surface concentration can be formulated on the basis of equations describing the mass transport. For instance, it follows from Eq. (3.12) that iS should be kept constant under steady-state conditions. This condition... 

At first we tried to solve the problem in study by a simultaneous numerical analysis of the Navier-Stokes equations and the mass transport equation. Since by this method serious numerical instabilities were encountered we choose an alternative procedure. This consists of calculating first, at each grid point in the tube, the axial and radial velocities, after which these values are used to solve the mass transport equation. In solving this equation care is taken that the grid used is identical to that for solving the velocity field. 

This equation clearly states that the diffusion coefficient results from the product of a purely thermodynamic factor, a, and a purely kinetic factor, the mobility, L, as long as the chemical potential is the true driving force for the mass transport process. 

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