Diffusion boundary layer equation

At high Peclet numbers, the diffusion flux to the surface of the first solid particle can be found by solving the conventional diffusion boundary layer equation the presence of the second solid particle influences only by changing the velocity... 

We treat first the capture by a collector of submicrometer-size particles undergoing Brownian motion in a low-speed flow of velocity U. The collector is taken to be a sphere of radius a and is assumed to be ideal in that all of the particles that impinge on its surface stick to it (Fig. 8.3.1). Because the Brownian particle diffusivities D - kTIGiruap, where is the particle radius, are typically about a thousand times smaller than the molecular diffusivities, the diffusion Peclet number (Ua/D) is generally very large compared with unity. The diffusive flux of the particles to the surface is therefore governed by the steady, convective, diffusion boundary layer equation, with the particles treated as diffusing points. ... 

We may write the convective diffusion boundary layer equation in spherical coordinates (r,8) in the form... 

There are various ways of using Equations 8 and 9 to obtain information about the solidification process. The simplest one is to do an order of magnitude analyses, OMA, of these equations. This yields immediately that on a relative basis the first, second and third terms are of order 1/t, U/6 and DL/62, where 6 is the approximate thickness of the diffusion boundary layer. Equating the first and last terms gives... 

In a boundary layer equation the mass center is considered with the help of the velocity (u, Uy, u ) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity v, giving a formula for Vax /x, but not for /ax - x useful approach is offered by Eq. (4.268c), using the artificial multiplication factor (v - ax /... 

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 ... 

Again, the form of the concentration profile in the diffusion boundary layer depends on the conditions which are assumed to exist at the surface and in the fluid stream. For the conditions corresponding to those used in consideration of the thermal boundary layer, that is constant concentrations both in the stream outside the boundary layer and at the surface, the concentration profile is of similar form to that given by equation 11.70 ... 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

Four strategies are generally employed to demonstrate mass transfer limitation in aquatic systems. Most commonly, measured uptake rates are simply compared with calculated maximal mass transfer rates (equation (17)) (e.g. [48,49]). Uptake rates can also be compared under different flow conditions (e.g. [52,55,56,84]), or by varying the biomass under identical flow conditions (e.g. [85]). Finally, several recent, innovative experiments have demonstrated diffusion boundary layers using microsensors [50,51]. Of the documented examples of diffusion limitation, three major cases have been identified ... 

It is perhaps wise to begin by questioning the conceptual simplicity of the uptake process as described by equation (35) and the assumptions given in Section 6.1.2. As discussed above, the Michaelis constant, Km, is determined by steady-state methods and represents a complex function of many rate constants [114,186,281]. For example, in the presence of a diffusion boundary layer, the apparent Michaelis-Menten constant will be too large, due to the depletion of metal near the reactive surface [9,282,283], In this case, a modified flux equation, taking into account a diffusion boundary layer and a first-order carrier-mediated uptake can be taken into account by the Best equation [9] (see Chapter 4 for a discussion of the limitations) or by other similar derivations [282] ... 

The basic assumption is that the rotating filter creates a laminar flow field that can be completely described mathematically. The thickness of the diffusion boundary layer (5) is calculated as a function of the rotational speed (to), viscosity, density, and diffusion coefficient (D). The thickness is expressed by the Levich equation, originally derived for electrochemical reactions occurring at a rotating disk electrode ... 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

Equations for the thickness 8 of the diffusion boundary layer can be obtained by solving Eq. (372). Equations (369a) and (369b) are compatible with Eq. (373) only if... 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

Wilson (79, 80) pointed out that A is not the dimensionless thickness of the diffusion boundary layer scaled with D/Vg, as originally suggested by Burton et al. (74), except in the limit at which the velocity field in the layer is dominated by the bulk flow, that is, X >> 1. In this case, the analysis reduces to the one first presented by Levich (81), and the integral in equation 25 is approximated as follows ... 

Note that equation (5.1) follows immediately from Fick s laws on the assumption of a quasi-stationary distribution of the concentration of components within the diffusion boundary layer. Indeed, if in this layer cAl t 0, then the second Fick law yields cA x const. It means that the distribution of the concentration of component A within this layer is close to linear (Fig. 5.1). Anywhere outside of this layer, the concentration of A is assumed to be the same and equal to an instantaneous value, c. This implies sufficiently intensive agitation of the liquid. In such a case, the flow of A atoms across the diffusion boundary layer under the condition of constancy of the surface area of the dissolving solid is... 

Indeed, it seems obvious that a variation of the concentration of any dissolving solid substance in a liquid is directly proportional to both the area of its surface contacting with the liquid and the difference between the saturation concentration (solubility) at a given temperature and the instantaneous concentration of A in B, and is inversely proportional to the volume of the liquid phase. Therefore, the general form of equation (5.1) remains unchanged for either dissolution regime of any solid in any liquid. The difference lies in the character of the dependence of the dissolution-rate constant, k, upon the thickness, 5, of the diffusion boundary layer. 

According to V.G. Levich299 (see also Refs 300, 302, 304), the thickness of the diffusion boundary layer at the rotating disc surface is determined by the equation... 

This equation allows the determination of the diffusion coefficient, D, of the atoms of the dissolving substance across the diffusion boundary layer, knowing the value of the dissolution-rate constant, k, and vice versa. It is essential to remember, however, that equation (5.6) holds for Schmidt s numbers, Sc, exceeding 1000. 

Although the assumption of a quasistationary distribution of the concentration of component A within the diffusion boundary layer seems to be very rough, nevertheless under conditions of sufficiently intensive convection the dissolution kinetics of solids in liquids is well described by equations (5.1) and (5.8)-(5.10) (see Refs 301, 303, 304, 306-308). Clearly, these equations are generally applicable at a low solubility of the solid in the liquid phase (about 10-100 kg m or up to 5 mass %). Note that they may also describe fairly well the dissolution process in systems of much higher solubility. An example is the Al-Ni binary system in which the solubility of nickel in aluminium amounts to 10 mass % even at a relatively low temperature of 700°C (in comparison with the melting point of aluminium, 660°C).308... 

Knowing an experimental value of k, it is possible to evaluate the diffusion coefficient of the atoms of a dissolving solid substance across the diffusion boundary layer at the solid-liquid interface into the bulk of the liquid phase using equations (5.6) and (5.7). Its calculation includes two steps. First, an approximate value of D is calculated from equation (5.6). Then, the Schmidt number, Sc, and the correction factor, /, is found (see Table 5.1). The final, precise value is evaluated from equation (5.7). In most cases, the results of these calculations do not differ by more than 10 %. Values of the diffusion coefficient of some transition metals in liquid aluminium are presented in Table 5.9.303... 

Consider a sphere of radius a held fixed in a creeping flow field with approach velocity U. The fluid contains Brownian particles having a diffusion coefficient D. Should the Peclet number 2aUiDUj have a value much greater than unity, the diffusion boundary layer will be sufficiently thin so that curvature effects and tangential diffusion are negligible. Under these conditions the convective-diffusion equation assumes the following form ... 

When the Bom, double-layer, and van der Waals forces act over distances that are short compared to the diffusion boundary-layer thickness, and when the e forces form an energy hairier, the adsorption and desorption rates may be calculated by lumping the effect of the interactions into a boundary condition on the usual ccm-vective-diffusion equation. This condition takes the form of a first-order, reversible reaction on the collector s surface. The apparent rate constants and equilibrium collector capacity are explicitly related to the interaction profile and are shown to have the Arrhenius form. They do not depend on the collector geometry or flow pattern. 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

To determine the net rale of adsorption of particles suspended in a fluid that is flowing over the collector, one may then solve the usual convective-diffusion equation subject to a reversible first-order reaction as the boundary condition, provided the diffusion boundary layer is much thicker than the interaction boundary layer. 

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