Diffusive flow

Diffusion depends on a concentration gradient being present (the gas flows from an area of high concentration to an area with a lower concentration). Molecular diffusion can be modelled using a number of laws but most of these are complex. If it is assumed that isobaric conditions are prevalent and there is no pressure driven flow, combined with relatively low concentrations of the gas being considered and high soil permeability, then Pick s law can be used. 

Pick s law is also a component of the model used by Johnson and Ettinger to model gas or vapour migration into buildings. According to Pick s law the rate of mass transfer of a gas or vapour by diffusion can be estimated as follows (USEPA, 2003)  

E = rate of mass transfer due to diffusion (g/d) A = area through which migration occurs (m ) 

Diffusion can be estimated through both the capillary zone and the unsaturated zone using different values for the effective diffusion coefficient. 

This is the equation used in the Johnson and Ettinger (1991) model for vapour diffusion in the ground. Values of the diffusion coefficient of methane for various soils and rocks have been given in CIRIA Report 152 (O Riordan and Milloy, 1995). 

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Dawes, W., 1995, A Simulation of the Unsteady Interaction of a Centrifugal Impeller with its Vaned Diffuser Flows Analysis, ASME Journal of Turbomachinery, Vol. 117, pp. 213-222. 

We now consider the resistance force caused by the diffusion. This force resists the diffusion flow in a porous material together with Writing the linear momentum equation for component A in accordance with Eq. (4.302),... 

The term p/(p-p ) derives from assumption (4.313) representing the effects of Stefan flow. If e = 1, Eq. (4.318) gives the diffusion flow in a free space. 

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

The concentration of the remaining oxidation centered on the relaxed film at any oxidation time is defined by the difference between the density of charge stored in the point at which the film attains an oxidation steady state at the working potential and large polarization times and the charge density stored after a given polarization time [< j(0]-So the diffusion flow of ions is given by... 

Neale, GH Nader, WK, Prediction of Transport Processes Within Porous Media Diffusive Flow Processes Within a Homogeneous Swarm of Spherical Particles, AIChE Journal 19, 112, 1973. 

Figure 1.10. Balance region showing convective and diffusive flows in and out.

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

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. 

Another available product has two faces of aluminum foil over a core of glass scrim webbing it is coated with asphalt. The membrane is 0.012 in. thick. This material has not been tested as a barrier against diffusive flow of radon, but its performance should be similar to that of other foil-faced products. Seams are sealed with aluminum tape. 

An intact polythene membrane within the concrete base of a building will prevent pressure driven flow of radon into the building from the soil, even if the concrete is cracked. Diffusive flow of radon into the building will also be reduced because of the comparatively low diffusion coefficient of radon in polythene ( v 10 7 cm2 s"1). No significant improvement was achieved by substituting a 50 ym sheet of mylar for polythene (mylar diffusion coefficient x 10"11 cm2 s"1). In this case additional difficulties were experienced in sealing the less flexible material to the walls. 

In PF, the transport of material through a vessel is by convective or bulk flow. All elements of fluid, at a particular axial position in the direction of flow, have the same concentration and axial velocity (no radial variation). We can imagine this ideal flow being blurred or dispersed by backmixing of material as a result of local disturbances (eddies, vortices, etc.). This can be treated as a diffusive flow superimposed on the convective flow. If the disturbances are essentially axial in direction and not radial, we refer to this as axial dispersion, and the flow as dispersed plug flow (DPF). (Radial dispersion may also be significant, but we consider only axial dispersion here.)... 

In considering axial dispersion as a diffusive flow, we assume that Fick s first law applies, with the diffusion or effective diffusion coefficient (equation 8.5-4) replaced by an axial dispersion coefficient, D,. Thus, for unsteady-state behavior with respect to a species A (e.g., a tracer), the molar flux (NA) of A at an axial position x is... 

This diffusive flow must be taken into account in the derivation of the material-balance or continuity equation in terms of A. The result is the axial dispersion or dispersed plug flow (DPF) model for nonideal flow. It is a single-parameter model, the parameter being DL or its equivalent as a dimensionless parameter. It was originally developed to describe relatively small departures from PF in pipes and packed beds, that is, for relatively small amounts of backmixing, but, in principle, can be used for any degree of backmixing. 

In a fixed-bed catalytic reactor for a fluid-solid reaction, the solid catalyst is present as a bed of relatively small individual particles, randomly oriented and fixed in position. The fluid moves by convective flow through the spaces between the particles. There may also be diffusive flow or transport within the particles, as described in Chapter 8. The relevant kinetics of such reactions are treated in Section 8.5. The fluid may be either a gas or liquid, but we concentrate primarily on catalyzed gas-phase reactions, more common in this situation. We also focus on steady-state operation, thus ignoring any implications of catalyst deactivation with time (Section 8.6). The importance of fixed-bed catalytic reactors can be appreciated from their use in the manufacture of such large-tonnage products as sulfuric acid, ammonia, and methanol (see Figures 1.4,11.5, and 11.6, respectively). 

The two flux equations of importance to subsurface transport are Darcy s law for the advective flow of water and other liquids and Fick s law for the diffusive flow of molecules and gases. These laws are independently discussed below. 

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