Diffusion bulk flow

In addition to diffusion, bulk flow can occur within a reacting pellet (Beveridge and Goldie, 1968 Gower, 1971 Sohn and Sohn, 1980). This effect is considerably magnified for reactions with volume change such as... 

Mass transport in an adsorbent particle can be viewed as a combination of several mechanisms, as shown in Figure 14.8. Macropore and micropore diffnsion are shown and conld be considered as examples of intraparticle diffusion. Bulk flow or conveyance through the particle (e.g., via connected pores) is shown in that figure but, to be significant, requires high porosity or a large... 

In technological applications as well as in scientific experiments specific boundary conditions are often given, such as definite changes of the interfacial area. A schematic representation is given in Fig. 4.2. which shows various bulk and interfacial transport processes of surface active molecules diffusion in the bulk, interfacial diffusion, bulk flow of different origin, interfacial compression and dilation. 

Further development of this model incorporating diffusion, bulk flow, transmembrane flux, and matrix shrinkage [42-44] showed that the cell membrane is the main barrier to mass transfer only for single cells or thin slices of tissue. When the thickness of the sample increases, the extracellular space may become the limiting factor [45]. 

Multicomponent Diffusion. In multicomponent systems, the binary diffusion coefficient has to be replaced by an effective or mean diffusivity Although its rigorous computation from the binary coefficients is difficult, it may be estimated by one of several methods (27—29). Any degree of counterdiffusion, including the two special cases "equimolar counterdiffusion" and "no counterdiffusion" treated above, may arise in multicomponent gas absorption. The influence of bulk flow of material through the films is corrected for by the film factor concept (28). It is based on a slightly different form of equation 13 ... 

The generalized transport equation, equation 17, can be dissected into terms describing bulk flow (term 2), turbulent diffusion (term 3) and other processes, eg, sources or chemical reactions (term 4), each having an impact on the time evolution of the transported property. In many systems, such as urban smog, the processes have very different time scales and can be viewed as being relatively independent over a short time period, allowing the equation to be "spht" into separate operators. This greatly shortens solution times (74). The solution sequence 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 ... 

Equations (13-111) to (13-114), (13-118) and (13-119), contain terms, Njj, for rates of mass transfer of components from the vapor phase to the liquid phase (rates are negative if transfer is from the liquid phase to the vapor phase). These rates are estimated from diffusive and bulk-flow contributions, where the former are based on interfacial area, average mole-fraction driving forces, and mass-... 

Intraparticle convection can also occur in packed beds when the adsorbent particles have very large and well-connected pores. Although, in general, bulk flow through the pores of the adsorbent particles is only a small frac tion of the total flow, intraparticle convection can affec t the transport of veiy slowly diffusing species such as macromolecules. The driving force for convec tion, in this case, is the... 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

The material balance with bulk flow in the axial direction z and diffusion in the radial direction rwith diffusivity D gives rise to the equation... 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

Closed system input and output by bulk flow only (i.e., no diffusion takes plaee aeross the system boundaries). 

Geometrieally defined boundaries within one phase aeross whieh exehanges oeeur eidier by bulk flow or by moleeular diffusion (e.g., nutrient inlet and outlet pipes). 

GASFLOW models geometrically complex containments, buildings, and ventilation systems with multiple compartments and internal structures. It calculates gas and aerosol behavior of low-speed buoyancy driven flows, diffusion-dominated flows, and turbulent flows dunng deflagrations. It models condensation in the bulk fluid regions heat transfer to wall and internal stmetures by convection, radiation, and condensation chemical kinetics of combustion of hydrogen or hydrocarbon.s fluid turbulence and the transport, deposition, and entrainment of discrete particles. 

The mechanism by which analytes are transported in a non-discriminate manner (i.e. via bulk flow) in an electrophoresis capillary is termed electroosmosis. Eigure 9.1 depicts the inside of a fused silica capillary and illustrates the source that supports electroosmotic flow. Adjacent to the negatively charged capillary wall are specifically adsorbed counterions, which make up the fairly immobile Stern layer. The excess ions just outside the Stern layer form the diffuse layer, which is mobile under the influence of an electric field. The substantial frictional forces between molecules in solution allow for the movement of the diffuse layer to pull the bulk... 

As noted previously, mixing in highly viscous liquids is slow both at the molecular scale, on account of the low values of diffusivity, as well as at the macroscopic scale, due to poor bulk flow. Whereas in low viscosity liquids momentum can be transferred from a rotating impeller through a relatively large body of fluid, in highly viscous liquids only... 

Total transfer = Transfer by diffusion + Transfer by bulk flow. 

In a packed absorption column, the flow pattern is similar to that in a packed distillation column but the vapour stream is replaced by a mixture of carrier gas and solute gas. The solute diffuses through the gas phase to the liquid surface where it dissolves and is then transferred to the bulk of the liquid. In this case there is no mass transfer of the carrier fluid and the transfer rate of solute is supplemented by bulk flow. 

As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. 

In many applications of mass transfer the solute reacts with the medium as in the case, for example, of the absorption of carbon dioxide in an alkaline solution. The mass transfer rate then decreases in the direction of diffusion as a result of the reaction. Considering the unidirectional molecular diffusion of a component A through a distance Sy over area A. then, neglecting the effects of bulk flow, a material balance for an irreversible reaction of order n gives ... 

In a gas absorption process, the solute gas A diffuses into a solvent liquid with which it reacts. The mass transfer is one of steady state unidirectional molecular diffusion and the concentration of A is always sufficiently small for bulk flow to be negligible. Under these conditions the reaction is first order with respect to the solute A. 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

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 ... 

The diffusivity of water vapour in air is 2.4 x It) 5 nr/s and the mass transfer resistance is equivalent to that of a stagnant gas film of thickness 0.25 mm. Neglect the effects of bulk flow. 

The void fraction should be the total void fraction including the pore volume. We now distinguish Stotai from the superficial void fraction used in the Ergun equation and in the packed-bed correlations of Chapter 9. The pore volume is accessible to gas molecules and can constitute a substantial fraction of the gas-phase volume. It is included in reaction rate calculations through the use of the total void fraction. The superficial void fraction ignores the pore volume. It is the appropriate parameter for the hydrodynamic calculations because fluid velocities go to zero at the external surface of the catalyst particles. The pore volume is accessible by diffusion, not bulk flow. 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

Both pH and the availability of nutrient ions in soil play important roles in rhizo-sphere dynamics and are often dependent on one another. Nutrient ions move in soil toward plant roots either by mass flow with the soil water or by diffusion. Mass flow is the result of bulk convective movements of the soil solution toward roots, whereas diffusion occurs in response to a concentration gradient for a particular ion, which results from its absorption by the root and depletion from the... 

Upon formation of a metal chelate or complex, the next rate-limiting step in delivering iron to the cell is the diffusion of iron complexes through the. soil in response to diffusion gradients. In the vicinity of plant roots, metal chelates and complexes may also move by bulk flow in the transpiration stream as water moves from the soil into the plant. However, depending on their charge characteristics and hydrophobicity, metal chelators and complexes can become adsorbed to clay and organic matter, which may then decrease their mobility and bioavail-... 

---