The boundary layer for mass transfer

To apply this relation, Gz must be smaller than 0.03. Under the present conditions, Gz = 10 , indicating that the boundary layer for mass transfer is in development. As a result, the Sherwood number is orders of magnitude higher than the lower limit of 2.96. At a stirrer speed of 150 rpm, liquid velocity (vl) in the monolith channels is around 0.5 m/s [53]. With an estimated Sc of 1350... 

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. 

These rate determining steps are shown in Figure 5.6. As the reaction is written in equation (5.26), mass transfer in the boundary layer and mass transfer by diffusion in the product layer can be limiting for the reactant gas. A, making its way in from the bulk gas to the unreacted core, or for the product gas, R, making its way out. 

What is important to recognize from the discussion is that the boundary condition for mass transfer through a semipermeable membrane is directly analogous to that for a mixed heterogeneous reaction. A consequence of this is that what is said about the one problem can be translated to the other, despite the somewhat different physics and chemistry. The example of reverse osmosis is therefore used as an illustration of a mixed heterogeneous reaction. The major part of the discussion will, however, be confined to the developing layer, where... 

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). 

Next, consider the gradients of temperature. If the reaction is exothermic, the center of the particle tends to be hotter, and conversely for an endothermic reaction. Two sets of gradients are thus indicated in Figure 8.9. Heat transfer through the particle is primarily by conduction, and between exterior particle surface (Ts) and bulk gas (Tg) by combined convection-conduction across a thermal boundary layer, shown for convenience in Figure 8.9 to coincide with the gas film for mass transfer. (The quantities T0, ATp, A7y, and AT, are used in Section 8.5.5.)... 

Equation 9.1-15 equates the rate of heat transfer by conduction at the surface to the rate of heat transfer by conduction/convection across a thermal boundary layer exterior to the particle (corresponding to the gas film for mass transfer), expressed in terms of a film coefficient, h, and the difference in temperature between bulk gas at Tg and particle surface at Ts ... 

Although multiplicities of the effectiveness factor have also been detected experimentally, these are of minor importance practically, since for industrial processes and catalysts, Prater numbers above 0.1 are less common. On the contrary, effectiveness factors above unity in real systems are frequently encountered, although the dominating part of the overall heat transfer resistance normally lies in the external boundary layer rather than inside the catalyst pellet. For mass transfer the opposite holds the dominating diffusional resistance is normally located within the pellet, whereas the interphase mass transfer most frequently plays a minor role (high space velocity). 

The Schmidt number for the mass transfer is analogous to the Prandtl number for heat transfer. Its physical implication means the relative thickness of the hydrodynamic layer and mass-transfer boundary layer. The ratio of the velocity boundary layer (S) to concentration boundary layer (Sc) is governed by the Schmidt number. The relationship is given by... 

Feed-side and strip-side concentration polarization result in a reduction in the driving force for mass transfer. There is a decrease in water activity at the feed-membrane interface and an increase at the strip-membrane interface. This results in a reduction in the water vapor pressure gradient across the membrane. The feed side and strip side mass transfer co-efficients, Kf and K, respectively, can be expressed in terms of the solute diffusion co-efficient in the boundary layer, D, ... 

In the preceding section, it was assumed that the concentration at the surface is known. In other words, the surface concentration can be taken as equal to the concentration in the bulk of the fluid phase. This is true only if there is essentially no mass transfer resistance across the film (or boundary layer) surrounding the pellet. If such a resistance is present, the actual surface concentration will be lower for the reactant and higher for the product than in the fluid bulk. A Sherwood number for mass transfer would then be needed in the analysis. Several empirical correlations have been proposed that relate the Sherwood number (for mass transfer) to the Reynolds number (for flow) and Schmidt... 

The current model is a step closer toward a reliable working description of the sulfite oxidation rate in scrubber slurries. By incorporating a boundary layer description of the film around each particle, this model predicts the conditions at the particle surface which drive the mass transfer. The interfacial area for mass transfer was discovered to be more closely represented by a sphere than by a plate-like shape. From the model, using highly catalyzed experiments, a mass transfer coefficient of 0.015 cm sec-1 was found - quite close to literature correlation predictions. 

However, the two-sink model as well as other existing adsorption (sink) models do not seem to be able to describe the strong asymmetry between the adsorption/desorption of VOCs on/from indoor surface materials (the desorption process is much slower than the adsorption process). Diffusion combined with internal adsorption is assumed to be capable of explaining the observed asymmetry. Diffusion mechanisms have been considered to play a role in interactions of VOCs with indoor sinks. Dunn and Chen (1993) proposed and tested three unified, diffusion-limited mathematical models to account for such interactions. The phrase unified relates to the ability of the model to predict both the ad/absorption and desorption phases. This is a very important aspect of modeling test chamber kinetics because in actual applications of chamber studies to indoor air quality (lAQ), we will never be able to predict when we will be in an accumulation or decay phase, so that the same model must apply to both. Development of such models is underway by different research groups. An excellent reference, in which the theoretical bases of most of the recently developed sorption models are reviewed, is the paper by Axley and Lorenzetti (1993). The authors proposed four generic families of models formulated as mass transport modules that can be combined with existing lAQ models. These models include processes such as equilibrium adsorption, boundary layer diffusion, porous adsorbent diffusion transport, and conveetion-diffusion transport. In their paper, the authors present applications of these models and propose criteria for selection of models that are based on the boundary layer/conduction heat transfer problem. 

Two types of mass- transfer can be distinguished for catalysis with heterogeneous catalyst particles. External mass transfer refers to molecular transport between the bulk reaction mixture and the surface of the enzyme particle through a boundary layer. Internal mass transfer is the molecular transport inside the solid enzyme phase. Internal mass transfer occurs within the pores of the catalyst particle to and from the particle surface. Figure 4.9-4 illustrates the definitions of external and internal mass transfer. 

As mentioned above, the preceding equation is also useful to calculate fluid temperature profiles via boundary layer heat transfer if one replaces Ca by T and 50a, mix by thermal diffusivity a. The dimensionless profile for mass transfer is constructed as follows ... 

In the description of mass transfer processes another fluid layer is frequently postulated, viz. the stagnant film (see Figure 6.8) or, as it is sometimes called, the effective film for mass transfer , 8. This hypothetical film is not the same thing as the more fundamental diffusion boundary layer 6n, but it may be considered to be of the same order of magnitude. 

Figure 2.6 Mass transfer in a chemically mediated membrane process. The chemical potential gradient from the bulk feed to the bulk permeate streams is the driving force for mass transfer (shown as yellow line). Three main resistances to mass transfer are shown—fluid boundary layers on the feed and permeate sides of the membrane and diffusion through the membrane.

The viscosity /j and density p used are the actual flowing mixture of solute A and fluid B. If the mixture is dilute, properties of the pure fluid B can be used. The Prandtl number c pjk for heat transfer is analogous to the Schmidt number for mass transfer. The Schmidt number is the ratio of the shear component for diffusivity pip to the diffusivity for mass transfer and it physically relates the relative thickness of the hydrodynamic layer and mass-transfer boundary layer. 

Film diffusion may lead to concentration gradients in the boundary layer. For example, the concentration becomes practically zero at the outer surface, if the intrinsic rate constant (more precisely the effective internal rate constant reflecting the interplay of intrinsic kinetics and pore diffusion) exceeds by far the external mass transfer coefficient. 

Figure 4.5.26 shows the radial concentration distribution in a porous spherical particle with diameter 2tp according to Eq. (4.5.115) for two values of the Thiele modulus ( reversible fot the example of a gas phase free of B (cB,g = 0) and Defr/O tp) = 0.05 and K = l. Note that in the case of high values of ( reversible (S>5 in Figure 4.5.26), the external mass transfer determines the effective reaction rate, that is, the equilibrium concentrations are almost reached within the porous particle (for the example of Figure 4.5.26, K<, = 1 and Ca,equilibrium = Cb,equilibrium = 0.5c J, and the concentrations vary strongly in the boundary layer, for example,...

The driving potential for mass transfer in convective drying in air is commonly considered to be differences of vapor concentration across a boundary layer. An alternative view is that mass transfer in steam occurs by bulk flow due to a pressure difference. The vapor pressure at the surface where evaporation occurs is thought to be slightly higher than the free stream pressure, and causes bulk flow of vapor into the gas stream. Surface temperatures slightly higher than the saturation temperature have been measured. 

Figure 9-13 Profile of the concentration of reactant A (Ca) through the boundary layer for the case where rjkylc/kc 1. The reaction is controlled by transport of reactant A from the bulk fluid to the external surface of the catalyst particle, i.e., by external mass transfer.

In considering the effect of mass transfer on the boiling of a multicomponent mixture, both the boiling mechanism and the driving force for transport must be examined (17—20). Moreover, the process is strongly influenced by the effects of convective flow on the boundary layer. In Reference 20 both effects have been taken into consideration to obtain a general correlation based on mechanistic reasoning that fits all available data within 15%. 

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... 

Figures 4.34 and 4.35 represent two extreme cases. Drying processes represent the case shown in Fig. 4.34 and distillation processes represent Fig. 4.35. Neither case represents a convective mass transfer case while the gas flow is in the boundary layer, other flows are Stefan flow and turbulence. Thus Eqs. (4.243) and (4.244) can seldom be used in practice, but their forms are used in determining the mass transfer factor for different cases.

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