Mass balance diffusion

Most of the known IE kinetic problems have been solved by the use of a single mass-balanced diffusion equation [1-3,7-11,14-24,34-43]. They are, on this basis, identified as one component systems and the diffusion rate for the invading B ion is controlled by the concentration gradient of this ion alone. In these cases the effective interdiffusion coefficient depends on the ion concentrations and the equilibrium constants of the chemical reaction between both ions in the ion exchanger [2-3,7-12,16-22,23,23,30,32,34,42,32-34]. 

Equimolar Counterdiffusion. Just as unidirectional diffusion through stagnant films represents the situation in an ideally simple gas absorption process, equimolar counterdiffusion prevails as another special case in ideal distillation columns. In this case, the total molar flows and are constant, and the mass balance is given by equation 35. As shown eadier, noj/g factors have to be included in the derivation and the height of the packing is... 

Material Balances Whenever mass-transfer applications involve equipment of specific dimensions, flux equations alone are inadequate to assess results. A material balance or continuity equation must also be used. When the geometiy is simple, macroscopic balances suffice. The following equation is an overall mass balance for such a unit having bulk-flow ports and ports or interfaces through which diffusive flux can occur ... 

An industrial chemical reacdor is a complex device in which heat transfer, mass transfer, diffusion, and friction may occur along with chemical reaction, and it must be safe and controllable. In large vessels, questions of mixing of reactants, flow distribution, residence time distribution, and efficient utilization of the surface of porous catalysts also arise. A particular process can be dominated by one of these factors or by several of them for example, a reactor may on occasion be predominantly a heat exchanger or a mass-transfer device. A successful commercial unit is an economic balance of all these factors. 

Its molecular diffusivity should be low and it should be conserved (i.e., a mass balance on it must be possible). 

MASS BALANCE unit volume transfer (diffusion) per unit volume reaction) Empirically deiennined flux specified (3) Concen tration specified (1. 2b) Mass flux specified (2a.4) ... 

For the mass balance of component A, diffusion velocity and the corresponding diffusion factor are defined with regard to the mean molar velocity V, defined by the equation... 

If a concentration gradient exists within a fluid flowing over a surface, mass transfer will take place, and the whole of the resistance to transfer can be regarded as lying within a diffusion boundary layer in the vicinity of the surface. If the concentration gradients, and hence the mass transfer rates, are small, variations in physical properties may be neglected and it can be shown that the velocity and thermal boundary layers are unaffected 55. For low concentrations of the diffusing component, the effects of bulk flow will be small and the mass balance equation for component A is ... 

Then the diffusion equation for the fluctuation of the metal ion concentration is given by Eq. (68), and the mass balance at the film/solution interface is expressed by Eq. (69). These fluctuation equations are also solved with the same boundary condition as shown in Eq. (70). 

Uncharged reaction components are transported by diffusion and convection, even though their migration fluxes are zero. The total flux density Jj of species j is the algebraic (vector) sum of densities of all flux types, and the overall equation for mass balance must be written not as Eq. (4.1) but as... 

It was shown in Section 1.8 that in addition to ion migration, diffusion and convection fluxes are a substantial part of mass transport during current flow through electrolyte solutions, securing a mass balance in the system. In the present chapter these processes are discussed in more detail. 

Fluid density and component brownian diffusivity D are also assumed constant. A steady-state component mass balance can be written for component concentration c ... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

The presence of a liquid layer on the surface of the filter cake will cause solute to diffuse from the top layer of cake into the liquid. Also if disturbed the layer of liquid will mix with the surface layer of filter cake. This effect can be incorporated into the digital simulation by assuming a given initial depth of liquid as an additional segment of the bed which mixes at time t=0 with the top cake segment. The initial concentrations in the liquid layer and top cake segment are then found by an initial mass balance. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

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. 

Fick s second law of diffusion can be derived from Fick s first law by using a mass balance approach. Consider the differential fluid element shown in Figure 4. This differential fluid element is simply a small cube of liquid or gas, with volume Ax Ay Az, and will be defined as the system for the mass balance. Assume now that component A enters the cube at position x by diffusion and exits the cube at x + Ax by the same mechanism. For the moment, assume that no diffusion occurs in the y or z directions and that the faces of the cube that are perpendicular to the y and z axes thus are impermeable to the diffusion of A. Under these conditions, the component mass balance for A in this system is... 

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

With this equation, we can now discuss a generalized mass balance equation. We still use Figure 1 to show the derivation. Based on Eq. (5), the net contribution by diffusion and convection now becomes... 

This is a generalized mass balance equation in one dimension. If diffusion and convection occur in other directions, the generalized mass balance equation becomes... 

In the previous section, we detailed diffusion equations and generalized mass balance equations. We now turn to their practical uses in the pharmaceutical sciences. Mass transport problems can be classified as steady or unsteady. In steady mass transport there is no change of concentration with time [3], characterized mathematically by... 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

The CAT model estimates not only the extent of drug absorption, but also the rate of drug absorption that makes it possible to couple the CAT model to pharmacokinetic models to estimate plasma concentration profiles. The CAT model has been used to estimate the rate of absorption for saturable and region-depen-dent drugs, such as cefatrizine [67], In this case, the model simultaneously considers passive diffusion, saturable absorption, GI degradation, and transit. The mass balance equation, Eq. (51), needs to be rewritten to include all these processes ... 

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). 

The environmental compartments are represented by boxes and the concentration of a chemical in these boxes is affected by processes that cause mass flows of the chemical to and from the boxes. The chemical can be input into a box from outside the system, output from a box to outside the system, or transported by means of advective or diffusive processes to and from other boxes. A mass balance equation can be written for each of the boxes representing the mass flow of the chemical. Generally, the magnitude of these mass flows depends on the concentration of the chemical in the boxes. If mathematical expressions which relate the mass flows to the concentrations are available, the set of mass balance equations (one for... 

The deposition rate of the attached fraction, plotted in Figure 3, is calculated from the aerosol size distribution assuming diffusion and electrophoresis to be the most important deposition mechanisms (Raes et al.,1985a). The accuracy of the absolute values was checked by forming the aerosol mass balance after the generation of a high aerosol concentration.In Table II is compared the decay of the... 

Figure 5. The same data as in figure 4b plotted together with the diffusion battery data. Note, the CN counter cannot see any particle below 4.2 nm. According to rough mass balance estimate 1( - 108 small clusters per as could be present.

Mass Balance of a Single Cylindrical Pore and Diffusive... 

To be useful in modeling electrolyte sorption, a theory needs to describe hydrolysis and the mineral surface, account for electrical charge there, and provide for mass balance on the sorbing sites. In addition, an internally consistent and sufficiently broad database of sorption reactions should accompany the theory. Of the approaches available, a class known as surface complexation models (e.g., Adamson, 1976 Stumm, 1992) reflect such an ideal most closely. This class includes the double layer model (also known as the diffuse layer model) and the triple layer model (e.g., Westall and Hohl, 1980 Sverjensky, 1993). 

The redox reactions taking place in a sewer biofilm require that diffusion of both electron donor and electron acceptor be considered. The steady state mass balance for these two components is [cf. Equation (2.20)] ... 

For an incompressible liquid (i.e. a liquid with an invariant density which implies that the mass balance at any point leads to div v = 0) the time dependency of the concentration is given by the divergence of the flux, as defined by equation (13). Mathematically, the divergence of the gradient is the Laplacian operator V2, also frequently denoted as A. Thus, for a case of diffusion and flow, equation (10) becomes ... 

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