Convective diffusion, of mass

A derivation of the convective diffusion equation for axisymmetric flow in a pipe is given in Appendix 8.1. A reasonably general version of the equation is 

The da/dt term in Equation 8.18 corresponds to unsteady operation. It will be used in Chapter 16 where the method of false transients is introduced as a solution technique for PDEs. It is also used to smdy unsteady reactors in Chapter 14. There are two 

Radial convection,yr(T z)da/dr, arises only when VAr, z) changes in the axial direction. A change in viscosity due to changes in concentration or temperature can excite 

The axial diffusion terms can be dropped for macroscale reactors for the reasons given in Section 8.2.4. Thus the equation to be solved in most of this chapter is 

Compare Equation 8.20 to Equation 8.2. The only new phenomenon is radial diffusion that now allows transfer of mass between streamlines. Equation 8.20 assumes that the diffusivity is constant. A more rigorous design equation is 

Molecules must come into contact for a reaction to occur, and the mechanism for the contact is molecular motion. This is also the mechanism for diffusion. Diffusion is inherently important whenever reactions occur, but there are some reactor design problems where diffusion need not be explicitly considered, e.g., tubular reactors that satisfy the Merrill and Hamrin criterion. Equation (8.3). For other reactors, a detailed accounting for molecular diffusion may be critical to the design. 

Diffusion is important in reactors with unmixed feed streams since the initial mixing of reactants must occur inside the reactor under reacting conditions. Diffusion can be a slow process, and the reaction rate will often be limited by diffusion rather than by the intrinsic reaction rate that would prevail if the reactants were premixed. Thus, diffusion can be expected to be important in tubular reactors with unmixed feed streams. Its effects are difficult to calculate, and normal design practice is to use premixed feeds whenever possible. 

Laminar flow reactors have concentration and temperature gradients in both the radial and axial directions. The radial gradient normally has a much greater effect on reactor performance. The diffusive flux is a vector that depends on concentration gradients. The flux in the axial direction is 

As a first approximation, the concentration gradient in the axial direction is 

The relatively short distance in the radial direction leads to much higher diffusion rates. In most of what follows, axial diffusion will be ignored. 

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The unsteady version of the convective diffusion equation is obtained just by adding a time derivative to the steady version. Equation (8.32) for the convective diffusion of mass becomes... 

The dimensionless equations for the convective diffusion of mass and heat are... 

This section applies the method of lines to the equations for convective diffusion of mass and heat. It is convenient to replace r hy r = i Ar, i = 0,..., I, for coding... 

Let us consider the transport of one component i in a liquid solution. Any disequilibration in the solution is assumed to be due to macroscopic motion of the liquid (i.e. flow) and to gradients in the concentration c,. Temperature gradients are assumed to be negligible. The transport of the solute i is then governed by two different modes of transport, namely, molecular diffusion through the solvent medium, and drag by the moving liquid. The combination of these two types of transport processes is usually denoted as the convective diffusion of the solute in the liquid [25] or diffusion-advection mass transport [48,49], The relative contribution of advection to total transport is characterised by the nondimensional Peclet number [32,48,49], while the relative increase in transport over pure diffusion due to advection is Sh - 1, where Sh is the nondimensional Sherwood number [28,32,33,49,50]. 

Diffusion is caused by a gradient in chemical potential (which in most cases is approximated by a concentration gradient). It is predominant when the migration and convection modes of mass transport are... 

In addition to the Navier-Stokes equations, the convective diffusion or mass balance equations need to be considered. Filtration is included in the simulation by preventing convection or diffusion of the retained species. The porosity of the membrane is assumed to decrease exponentially with time as a result of fouling. Wai and Fumeaux [1990] modeled the filtration of a 0.2 pm membrane with a central transverse filtrate outlet across the membrane support. They performed transient calculations to predict the flux reduction as a function of time due to fouling. Different membrane or membrane reactor designs can be evaluated by CFD with an ever decreasing amount of computational time. 

The mass transfer coefficient (3C with SI units of m/s or m3/(sm2) is defined using these equations. It is a measure of the volumetric flow transferred per area. The concentration difference Aca defines the mass transfer coefficient. A useful choice of the decisive concentration difference for mass transfer has to be made. A good example of this is for mass transfer in a liquid film, see Fig. 1.41 where the concentration difference cA0 — cM between the wall and the surface of the film would be a a suitable choice. The mass transfer coefficient is generally dependent on the type of flow, whether it is laminar or turbulent, the physical properties of the material, the geometry of the system and also fairly often the concentration difference Aca. When a fluid flows over a quiescent surface, with which a substance will be exchanged, a thin layer develops close to the surface. In this layer the flow velocity is small and drops to zero at the surface. Therefore close to the surface the convective part of mass transfer is very low and the diffusive part, which is often decisive in mass transfer, dominates. 

Basically, three mechanisms are responsible for mass transport inside an electrochemical cell diffusion, migration, and convection. Diffusion is mass transport because of concentration gradients, i.e., variations in the concentration of a species with position. Diffusion occurs mainly near the electrode surface because of gradients created by the consumption of species that undergo redox reactions and are incorporated into the deposit. This incorporation process depletes the deposition species near the electrode, generating the concentration gradient. 

This chapter uses FEMLAB to solve problems involving two- or three-dimensional flow (2D or 3D) and diffusion of mass and/or heat. Convection and diffusion of mass... 

Since mass transfer must be rate determining, first, we consider the problem of convective diffusion of the /-species moving toward the surface. The steady-state conditions must be fulfilled so we can say that in the presence of an effective supporting electrolyte... 

The treatment of many process liquors often results in an electrochemical reaction which is under mass transport control due to the restricted convective-diffusion of species to (or from) the electrode (figure 4). This is particularly true in dilute liquors, where the bulk reactant concentration, Cg is low. The situation may be characterised by a mass transport coefficient,... 

I. Convection at a boundary. For the case where convection occurs outside in the fluid and the concentration of the fluid outside is suddenly changed to c , we can make a mass balance on the outside slab in Fig. 7.7-1. Following the methods used for heat transfer to derive Eq. (5.4-7), we write rate of mass entering by convection — rate of mass leaving by diffusion = rate of mass accumulation in At hours. 

The convective fluxes of mass and heat substantially increase the purely diffusive mass transfer and the purely conductive heat transfer for large droplets. The expressions for the convective fluxes are not valid for very high droplet number concentrations. We therefore believe it prudent to make estimates using two model options including and excluding convective mass and heat transfer. 

If the entire electrode supports a reaction under mass transport control via convective diffusion of the reactant or product, then the local current density is given by ... 

Accumulation of mass = net convective flow of mass + net diffusion of mass + production of mass... 

Concentration gradients for the analyte in the absence of convection, showing the time-dependent change in diffusion as a method of mass transport. 

Concentration gradient for the analyte showing the effects of diffusion and convection as methods of mass transport. 

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