Thickness, mass transfer boundary layer

Apart from the nature of the bulk flow, the hydrodynamic scenario close to the surfaces of drug particles has to be considered. The nature of the hydrodynamic boundary layer generated at a particle s surface may be laminar or turbulent regardless of the bulk flow characteristics. The turbulent boundary layer is considered to be thicker than the laminar layer. Nevertheless, mass transfer rates are usually increased with turbulence due to the presence of the viscous turbulent sub-layer. This is the part of the (total) turbulent boundary layer that constitutes the main resistance to the overall mass transfer in the case of turbulence. The development of a viscous turbulent sub-layer reduces the overall resistance to mass transfer since this viscous sub-layer is much narrower than the (total) laminar boundary layer. Thus, mass transfer from turbulent boundary layers is greater than would be calculated according to the total boundary layer thickness. 

Figure 6.4.16 shows the dimensionless profiles of temperature and NHg concentration as calculated by the finite element method for the angle / of 90 °, as indicated by the dashed line in the Figures 6.4.11 and 6.4.12. The mean value of the thickness of the boundary layer for mass transfer (5mass) is 140 xm and that for heat transfer 5heat is 100 p,m. 

Is there a chemical reaction of the solid with the liquid Sohd-liquid mixing operations involving chemical reactions often require a high relative velocity between the solid particle and the liquid—high local shear rate or agitation intensity—to minimize the thickness of the boundary layer for mass transfer. This is also due for the dissolution of a sparingly soluble solid, as discussed further in Chapter 13. 

The comparison of the magnitude of the two resistances clearly indicates whether tire metal or the slag mass transfer is rate-determining. A value for the ratio of the boundary layer thicknesses can be obtained from the Sherwood number, which is related to the Reynolds number and the Schmidt number, defined by... 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

Thus the driving force for fuming is approximately equal to that for free evaporation. Using dre experimental data, and the normal expression for mass transfer across a boundary layer, it is concluded that the boundary layer thickness which would account for this rate should be about 2 x 10 cm (Turkdogan et al., 1963). 

This can be further integrated from the wall to the boundary layer thickness y = 8, where the component is at the bulk concentration Cj,. Substituting / = - o and k = D/o, the mass-transfer coefficient yields the stagnant film model [Brian, Desalination by Reverse Osmosis, Merten (ed.), M.I.T. Press, Cambridge, Mass., 1966, pp. 161-292] ... 

The application of RHSE is primarily in the laminar boundary layer flow regime of Re < 15000, where the edge effect is negligible and the mass transfer theory has been confirmed by experimental investigations. An important consideration in the design of a practical RHSE system is to conform to the theoretical requirement that the boundary layer thickness be thin in comparison to the radius of the RHSE (<5 a). 

Early investigators assumed that this so-called diffusion layer was stagnant (Nernst-Whitman model), and that the concentration profile of the reacting ion was linear, with the film thickness <5N chosen to give the actual concentration gradient at the electrode. In reality, however, the thin diffusion layer is not stagnant, and the fictitious t5N is always smaller than the real mass-transfer boundary-layer thickness (Fig. 2). However, since the actual concentration profile tapers off gradually to the bulk value of the concentration, the well-defined Nernst diffusion layer thickness has retained a certain convenience in practical calculations. 

Unsuitable position of the reference electrode resulting in inclusion of a high ohmic potential drop between reference and working electrode. Moreover, when extended surfaces are used over which the mass transfer boundary layer thickness depends on position, a suitable number of independent reference electrodes should be used to measure local overpotentials on electrically isolated segments of the working electrode. 

The mass transfer boundary layer thickness, d, on a rotating disk electrode can be estimated by d = 1.6/J V a) where D is the substrate diffusion coefficient, v is the solution viscosity, and CO is the disk rotation speed. 

Higher tangential veloceties (or recirculation rates) should decrease the boundary layer thickness (6) and increase the mass transfer coefficient (k) in Equation 3 resulting in higher slopes of the flux vs concentration curve (Figure 11) without changing the gel-concentration. 

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 mass transfer rates for the case when d > d can easily be obtained from Eqs. 9 or 12 (see [48]). Using the surface renewal theory this case is not relevant because the boundary layer thickness is here considered to be infinite. 

The two mass transfer coefficients kG and kL give the ratio of the respective diffusion coefficients, D to the respective boundary layer thickness, xp... 

Provided the interphase mass transfer resistance (1 /k() is sufficiently large, the reactant concentration at the external pellet surface will drop almost to zero. Thus, we may neglect the surface concentration cs compared to the bulk concentration q>. With cs — 0 in eq 115, it is obvious that in this case the reaction will effectively follow a first-order rate law. Moreover, it is also clear that the temperature dependence of the effective reaction rate is controlled by the mass transfer coefficient k(. This exhibits basically the same temperature dependence as the bulk diffusivity Dm, since the boundary layer thickness 5 is virtually not affected by temperature (kf = Dm/<5). Thus, we have the rule of thumb that the effective activation energy of an isothermal, simple, nth order, irreversible reaction will be less than 5-lOkJmor1 when the overall reaction rate is controlled by interphase diffusion. 

When the concentration boundary layer is sufficiently thin the mass transport problem can be solved under the approximation that the solution velocity within the concentration boundary layer varies linearly with distance away from the surface. This is called the L6v que approximation (8, 9] and is satisfactory under conditions where convection is efficient compared with diffusion. More accurate treatments of mass transfer taking account of the full velocity profile can be obtained numerically [10, 11] but the Ldveque approximation has been shown to be valid for most practical electrodes and solution velocities. Using the L vSque approximation, the local value of the concentration boundary layer thickness, 8k, (determined by equating the calculated flux to the flux that would be obtained according to a Nernstian diffusion layer approximation that is with a linear variation of concentration across the boundary layer) is given by equation (10.6) [12]. 

For sufficiently large electrodes with a small vibration amplitude, aid < 1, a solution of the hydrodynamic problem is possible [58, 59]. As well as the periodic flow pattern, a steady secondary flow is induced as a consequence of the interaction of viscous and inertial effects in the boundary layer [13] as shown in Fig. 10.10. It is this flow which causes the enhancement of mass-transfer. The theory developed by Schlichting [13] and Jameson [58] applies when the time of oscillation, w l is small in comparison with the time taken for a species to diffuse across the hydrodynamic boundary layer (thickness SH= (v/a>)ln diffusion timescale 8h/D), i.e., when v/D t> 1. Re needs to be sufficiently high for the calculation to converge but sufficiently low such that the flow does not become turbulent. Experiment shows that, for large diameter wires (radius, r, — 1 cm), the condition is Re 2000. The solution Sh = 0.746Re1/2 Sc1/3(a/r)1/6, where Sh (the Sherwood number) = kmr/D and km is the mass-transfer coefficient,... 

Equation 8.2 shows how the net flux density of substance depends on its diffusion coefficient, Dj, and on the difference in its concentration, Ac] 1, across a distance Sbl of the air. The net flux density Jj is toward regions of lower Cj, which requires the negative sign associated with the concentration gradient and otherwise is incorporated into the definition of Acyin Equation 8.2. We will specifically consider the diffusion of water vapor and C02 toward lower concentrations in this chapter. Also, we will assume that the same boundary layer thickness (Sbl) derived for heat transfer (Eqs. 7.10-7.16) applies for mass transfer, an example of the similarity principle. Outside Sbl is a region of air turbulence, where we will assume that the concentrations of gases are the same as in the bulk atmosphere (an assumption that we will remove in Chapter 9, Section 9.IB). Equation 8.2 indicates that Jj equals Acbl multiplied by a conductance, gbl, or divided by a resistance, rbl. 

This overlapping will in fact reduce the available area for heat and mass transfer. During the present work, some boundary layer thicknesses were estimated for the experimental conditions of this work. As a result, the boundary layers only overlap for Reynolds numbers below 0.826. For the case of Reynolds numbers of 1.74 and 3.05 using the particle diameter of 0.035 cm., the boundary layers do not overlap.Table III shows some of the values obtained.Clearly, this effect cannot explain completely the low heat and mass transfer coefficients at low Reynolds numbers. 

Here /iq is the convective mass transfer coefficient for an unspecified geometry. For a given geometry, ho would contain the appropriate boundary layer thickness, or it would have to be determined by independent measurements giving correlations that permit /jq to be found from other parameters of the system. More interestingly, Eq. (22) should be compared to Eq. (179) in Chapter 6, which can be written as... 

Although diffusion of reacting species can be written in terms of the diffusivity and boundary layer thickness, the magnitude of 8 is unknown. Therefore, the mass-transfer coefficient is normally used. That is, the average molar flux from the bulk fluid to the solid surface is —x direction in Figure 6.2.1)... 

Fig. 19. Isoconcentration contours for a mass transfer boundary layer thickness equivalent to the notch depth d. (Figure and caption reprinted from Jordan and Tobias [57] by permission of the publisher, The Electrochemical Society, Inc.).

Fig. 20. Increase in flux as a function of distance along the perimeter for a mass transfer boundary layer thickness equivalent to the notch depth d. Note that the perimeter is longer than the distance in the X direction, hence the locations on the perimeter of the left corner, central trench, and right corner are also shown. (Figure and caption reprinted from Jordan and Tobias [57] by permission of the pubUsher, The Electrochemical Society, Inc.).

Tlie conversion and product selectivity is optimized when the gas mixing is improved, which reduces the boundary layer thickness at the catalyst surface and increases the mass transfer coefficient for a given channel dimension. The laminar flow through the honeycomb channels in the extrudate monolith results in a relatively thick boundary layer... 

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