Diffusion with convection

The statement by Maxwell quoted earlier suggests that diffusion and convection always occur together, that one cannot occur without the other. This fact sets diffusion apart from many other phenomena. For example, thermal conduction can certainly occur without convection. In contrast, diffusion generates its own convection, so that understanding the process can be much more complicated, especially in concentrated solutions. 

At 6°C, the benzene vapor is dilute, and evaporation is limited by diffusion 

the benzene boils and flows evaporation is controlled by convection 

At 60 C, an intermediate case occurs in which both diffusion and convection are important 

To illustrate how diffusion and convection are interrelated, we consider the example shown in Fig. 3.1 -1. The physical system consists of a large reservoir of benzene connected to a large volume of air by means of a capillary tube. Benzene evaporates and moves through the capillary into the surrounding air. 

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Once the descriptive model has been realized, we need to make the mathematical model of the process, which can be used to identify the mean pore radius of the membrane pores and the associated tortuosity. Before starting with the establishment of the model, we consider that the elementary processes allowdng the gas flow through the membrane are a combination of Knudsen diffusion with convective flow. If we only take into account the linear part of the curve of the pressure increase with time then we can write ... 

Consider diffusion with convection in a coated wall reactor, where the reaction takes place at the wall. [9] The governing equation and boundary conditions for concentration are ... 

Example 15-2. Steady-state diffusion with convection High-ten erature evaporation... 

Choose an appropriate reference velocity Vj- f and solve Fick s model for steady-state binary diffusion with convection... 

Cl. For binary diffusion with convection, use Eqs. tlS-lSa. b, c) and the equivalent equations for conponent B to show that = Dg. ... 

The desorptive process may be analyzed before boiling. The key assumption is that the vapor and adsorbed phases are ia equiUbrium ia the bulk of the bed. This assumption eliminates iatraparticle resistances from further consideration and is reasonable for rotary kiln appHcations. The two remaining resistances are associated with hydrocarbon diffusion out of the bed and with convection from the bed surface to the bulk gases. The flux of species Fi from the desorbiag bed becomes... 

These three terms represent contributions to the flux from migration, diffusion, and convection, respectively. The bulk fluid velocity is determined from the equations of motion. Equation 25, with the convection term neglected, is frequently referred to as the Nemst-Planck equation. In systems containing charged species, ions experience a force from the electric field. This effect is called migration. The charge number of the ion is Eis Faraday s constant, is the ionic mobiUty, and O is the electric potential. The ionic mobiUty and the diffusion coefficient are related ... 

Hyperbolic Equations The most common situation yielding hyperbohc equations involves unsteady phenomena with convection. Two typical equations are the convective diffusive equation... 

The materials leaving containment are source terms for offsite convective-diffusion transport calculations. Codes. such as CRAC-2 calculate atmospheric diffusion with different probabilities of meteorological conditions to estimate the radiological health effects and costs. 

As their name suggests, these models are based on the physical principles of diffusion and convection, which govern the mixing process. According to the flow pattern, the reactor is divided into different zones with different flow characteristics. 

The cooling tower cools hot water tvith cool air by countercurrent (or cross-current) fiow of the tw o fluids past each other in a tower filled with packing. This involves both mass and heat transfer. The water surface that exists on the tower packing is covered with an air film assumed to be saturated at the water temperature. The heat is transferred between this film and the main body of air by diffusion and convection. Detailed presentations of the development of cooling tower theory are given in References 39 and 46. 

H. Steady-State Convective Diffusion with Simultaneous First-Order Irreversible... 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

Polyethylene-based membranes are manufactured for use in hazardous waste landfills, lagoons, and similar applications. Two of these products have been tested to determine their effectiveness as barriers against radon diffusion. (In most cases, diffusive flow is considered of little or no significance as a mechanism of radon entry compared with convective flow). A 20-mil high-density polyethylene tested 99.9% effective in blocking radon diffusion under neutral pressure conditions. A 30-mil low-density polyethylene tested 98% effective in blocking radon diffusion under neutral pressure conditions. 

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... 

Membrane transport represents a major application of mass transport theory in the pharmaceutical sciences [4], Since convection is not generally involved, we will use Fick s first and second laws to find flux and concentration across membranes in this section. We begin with the discussion of steady diffusion across a thin film and a membrane with or without aqueous diffusion resistance, followed by steady diffusion across the skin, and conclude this section with unsteady membrane diffusion and membrane diffusion with reaction. 

We deal with membrane diffusion no convection is involved (vz = 0). We also assume that the system is at steady state (dcjdt = 0). The generalized mass... 

Other researchers used flow between two parallel plates as the experimental and theoretical system to incorporate diffusion plus convection into their dissolution modeling and avoid film model approximations [10]. Though they did not consider adding reactions to their model, these workers did show that convection was an important phenomenon to consider in the mass transfer process associated with solid dissolution. In fact, the dissolution rate was found to correlate with flow as... 

Farmer (6) reviewed the various diffusion models for soil and developed solutions for several of these models. An appropriate model for field studies is a nonsteady state model that assumes that material is mixed into the soil to a depth L and then allowed to diffuse both to the surface and more deeply into the soil. Material diffusing to the surface is immediately removed by diffusion and convection in the air above the soil. The effect of this assumption is to make the concentration of a diffusing compound zero at the soil surface. With these boundary conditions the solution to Equation 8 can be converted to the useful form ... 

Evaporated film catalysts are virtually always used with a static gas phase, and with reactant gas pressures less than about 100 Torr. One thus relies upon gaseous diffusion and convection for transport to the catalyst surface. However, provided one is dealing with reaction times of the order of minutes to tens of minutes, gas phase transport has but a negligible effect on the reaction, provided none of the reaction volume is separated from the film by small bore tubulation. Beeck et al. (77) in fact originally used an all-glass magnetically coupled turbine for gas circulation, but this is only... 

When the diffusion coefficient is very small (or diffusion is slow compared with convection), the Peclet number will be large. In that case, extraneous diffusion will be included in the solution unless the mesh size (denoted by Ax) is small compared with the characteristic length of the problem. To avoid this problem (by keeping the factor small), very fine meshes must be used, and the smaller the diffusion coefficient, the smaller the required mesh size. 

The deposition of sub-micron aerosols in a hollow cast of human bronchi has recently been measured under realistic conditions (Cohen et al., in press). Typical data are shown in Figure 4. These are inconsistent with convective enhancement of deposition but support the classical treatment of deposition by diffusion (Chamberlain and Dyson, 1956). 

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