Convection and Diffusion

The transfer of mass from one point to the other may take place by two different modes, namely, diffusion and convection. The basic mechanisms for these modes of mass transfer are similar to those for heat transfer discussed in 4.2 and 4.3. Specifically, the mechanism for mass convection is analogous to heat convection and that for mass diffusion is analogous to heat conduction. 

Mass convection may be due to the bulk motion of the carrier gas or may be associated with the drift of the solute through the carrier gas as a result of net forces applied directly to the solute (e.g., centrifugation due to centrifugal force). Here, we discuss only the convection due to bulk motion of the carrier gas. 

For a binary mixture, the convective molar flux, Jc, for species A can be expressed as 

For multicomponent mixtures, the overall molar flux for species i in the mixture can be expressed by 

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

The permeabUity of ceUular polymers to gases and vapors depends on the fraction of open ceUs as weU as the polymer-phase composition and state. The presence of open ceUs in a foam allows gases and vapors to permeate the ceU stmcture by diffusion and convection dow, yielding very large permeation rates. In closed-ceUed foams the permeation of gases or vapors is governed by composition of the polymer phase, gas composition, density, and ceUular stmcture of the foam (194,199,215,218,219). 

A fundamental difference exists between the assumptions of the homogeneous and porous membrane models. For the homogeneous models, it is assumed that the membrane is nonporous, that is, transport takes place between the interstitial spaces of the polymer chains or polymer nodules, usually by diffusion. For the porous models, it is assumed that transport takes place through pores that mn the length of the membrane barrier layer. As a result, transport can occur by both diffusion and convection through the pores. Whereas both conceptual models have had some success in predicting RO separations, the question of whether an RO membrane is truly homogeneous, ie, has no pores, or is porous, is still a point of debate. No available technique can definitively answer this question. Two models, one nonporous and diffusion-based, the other pore-based, are discussed herein. 

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 ... 

Influence of Chemical Reactions on Uq and When a chemical reaction occurs, the transfer rate may be influenced by the chemical reac tion as well as by the purely physical processes of diffusion and convection within the two phases. Since this situation is common in gas absorption, gas absorption will be the focus of this discussion. One must consider the impacts of chemical equilibrium and reac tion kinetics on the absorption rate in addition to accounting for the effec ts of gas solubility, diffusivity, and system hydrodynamics. 

In what follows, both macromixing and micromixing models will be introduced and a compartmental mixing model, the segregated feed model (SFM), will be discussed in detail. It will be used in Chapter 8 to model the influence of the hydrodynamics on a meso- and microscale on continuous and semibatch precipitation where using CFD, diffusive and convective mixing parameters in the reactor are determined. 

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. 

Transport of a species in solution to and from an electrode/solution interface may occur by migration, diffusion and convection although in any specific system they will not necessarily be of equal importance. However, at the steady state all steps involved in the electrode reaction must proceed at the same rate, irrespective of whether the rate is controlled by a slow step in the charge transfer process or by the rate of transport to or from the electrode surface. It follows that the rate of transport must equal the rate of charge transfer ... 

The gaseous components must be transferred from the bulk gaseous phase to the bulk liquid phase. The components are transferred to the gas-liquid interface by convection and diffusion in the gas and from the interface by diffusion and convection in the liquid. 

Equation (15) is derived under the assumption that the amount of adsorbed component transferred by flow or diffusion of the solid phase may be neglected. This assumption is clearly justified in cases of fixed-bed operation, and it is believed to be permissible in many cases of slurries or fluidized beds, since the absolute amount of adsorbed component will probably be quite low due to its low diffusivity in the interior of the catalyst pellet. The assumption can, however, be waived by including in Eq. (15) the appropriate diffusive and convective terms. 

Anodic shipping voltammetry (ASV) is the most widely used form of stripping analysis, hi this case, the metals are preconcenhated by elechodeposition into a small-volume mercury electrode (a tiiin mercury film or a hanging mercury drop). The preconcenhation is done by catiiodic deposition at a controlled tune and potential. The deposition potential is usually 0.3-0.5 V more negative than E° for the least easily reduced metal ion to be determined. The metal ions reach die mercury electrode by diffusion and convection, where diey are reduced and concentrated as amalgams ... 

The gas-phase mass flux of species k at the surface is a combination of diffusive and convective processes. 

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. 

Under realistic conditions a balance is secured during current flow because of additional mechanisms of mass transport in the electrolyte diffusion and convection. The initial inbalance between the rates of migration and reaction brings about a change in component concentrations next to the electrode surfaces, and thus gives rise to concentration gradients. As a result, a diffusion flux develops for each component. Moreover, in liquid electrolytes, hydrodynamic flows bringing about convective fluxes Ji j of the dissolved reaction components will almost always arise. 

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. 

The diffusive and convective terms in Eq. (20-10) are the same as in nonelectrolytic mass transfer. The ionic mobility Uj, (g mol cm )/(J-s), can be related to the ionic-diffusion coefficient D, cmVs, and the ionic conductance of the ith species X, cmV(f2-g equivalent) ... 

By locating the anode entirely upstream from the ionized gas volume, collection of long range beta particles is minimized in the displaced coaxial cylinder design, and the direction of gas flow minimizes diffusion and convection of electrons to the collector electrode. However, the free electrons are sufficiently mobile that modest pulse voltages (e.g., 50 V) are adequate to cause the electrons to move against the gas flow and be collected during. this time. 

In a pa(dced bad the aoblla diase flows through a tortuous channel systes and lateral eass transfer can take place by a coabination of diffusion and convection. The diffusion contribution can be approKiaated by equation (1.28)... 

If there is an external force acting in the same direction on solute molecules, the velocity of these molecules is vz and the resulting flux is cvz. Therefore, the total flux, nz, due to both diffusion and convection is... 

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... 

Land disposal sites result in soil contamination through leachate migration. The composition of the substances produced depends principally on the type of wastes present and the decomposition in the landfill (aerobic or anaerobic). The adjacent soil can be contaminated by direct horizontal leaching of surface runoff vertical leaching and transfer of gases from decomposition by diffusion and convection. The disposal of... 

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 ... 

Another group of surveys has focused on the direct modeling of some effective transport phenomena which are essential for predicting parameters that have an important role in underground gas sequestration process such as diffusivity and convection. Azin et al., in 2013, have conducted study regarding correct measurement of diffusivity coefficient [114]. 

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... 

A general transport equation describing the rate of change of the radon activity concentration in the pore space results from combining the effects of diffusion and convection ... 

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