Mass transfer boundary-layer models

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. 

The influence of a wall on the turbulent transport of scalar (species or enthalpy) at the wall can also be modeled using the wall function approach, similar to that described earlier for modeling momentum transport at the wall. It must be noted that the thermal or mass transfer boundary layer will, in general, be of different thickness than the momentum boundary layer and may change from fluid to fluid. For example, the thermal boundary layer of a high Prandtl number fluid (e.g. oil) is much less than its momentum boundary layer. The wall functions for the enthalpy equations in the form of temperature T can be written as ... 

Hence, surface-averaged reaction velocity constants, inferred from plug-flow simulations, are underestimated because convective diffusion in the mass transfer boundary layer adjacent to the catalytic surface is not modeled correctly. 

Many additional studies have been conducted with the boundary layer model by taking into account the variation of physical properties with composition (or temperature) and by relaxing the assumption that Vy = 0 at y = 0 when mass transfer is occurring. Under conditions of high mass transfer rates one finds that mass transfer to the plate decreases the thickness of the mass transfer boundary layer while a mass flux away from the wall increases the boundary layer thickness The analogous problem of uniform flux at the plate has also been solved. Skelland describes a number of additional mass transfer boundary layer problems such as developing hydrodynamic and mass transfin- profiles in the entrance region of parallel flat plates and round tubes. 

Another concept sometimes used as a basis for comparison and correlation of mass transfer data in columns is the Clulton-Colbum analogy (35). This semi-empirical relationship was developed for correlating mass- and heat-transfer data in pipes and is based on the turbulent boundary layer model... 

From the viewpoint of the chemical engineer, the application of the fundamentals just discussed takes the form of mass-transfer theory, and the constants Kj are called mass-transfer coefficients. However, some care must be exercised in determining the potential difference between two phases. One cannot simply take the difference between the thermodynamic activities in two phases without ensuring that they are adjusted to the same reference state. The point is illustrated with a brief consideration of the boundary-layer model. 

Rehder et al. (2004) measured the dissociation rates of methane and carbon dioxide hydrates in seawater during a seafloor experiment. The seafloor conditions provided constant temperature and pressure conditions, and enabled heat transfer limitations to be largely eliminated. Hydrate dissociation was caused by differences in concentration of the guest molecule in the hydrate surface and in the bulk solution. In this case, a solubility-controlled boundary layer model (mass transfer limited) was able to predict the dissociation data. The results showed that carbon dioxide hydrate dissociated much more rapidly than methane hydrate due to the higher solubility in water of carbon dioxide compared to methane. 

II is a function of hydrodynamic parameters of the model. Unfortunately, these parameters which describe the effect of hydrodynamics do not correspond to any physical quantity nor can they be Independently evaluated. For some models, the value of w is a constant. For example, the penetration and surface renewal models (Danckwerts, 31) predict w 0.5, while for the boundary layer model w 2/3. The film-penetration model, on the other hand, predicts that w varies between 0.5 and 1 (Toor and Marchello, 32). Knowledge of the effect of dlffuslvlty on k Is needed in evaluating the various mass transfer models. Calderbank (13) reported a value of 0.5 Linek et al. (22) used oxygen, Helium and argon. The reported diffusion coefficients for helium and similar gases vary widely. Since in the present work three different temperatures have been used, the value of w can be determined much more accurately. Figure 4... 

There is a great deal of theoretical and experimental information from micrometeorological research on the transfer of momentum, heat, and mass at solid and liquid surfaces and across their associated air boundary layers (hence the term boundary layer models for relationships arising from this approach). Based on the analogy between transfer of momentum and mass, it has been shown that k is proportional to the friction velocity in air (u ) and that k is also proportional to Sc. Apart from an assumption that the surface was smooth and rigid, it was also necessary to assume continuity of stress across the interface in order to convert the velocity profile in air to the equivalent profile in the water (Deacon, 1977). The relationship developed by Deacon is as follows ... 

J. J. Carberry, A Boundary Layer Model of Fluid Particle Mass Transfer in Fixed Beds, A. I. Ch. E. Journal, 6,460 (1960). 

A number of nonequilibrium models fall into the general framework described above. The differences between models are due primarily to the models of flow and mass transfer on a tray (or within a section of packed column). Young and Stewart (1990), for example, use collocation techniques to solve a boundary layer model of cross-flow on a tray. An alternative approach that builds on the models of mass and energy transfer described in Chapters 11 and 12 has been developed in a series of papers by Taylor and co-workers (Krishnamurthy and Taylor, 1985a-c, 1986 Taylor et al., 1992). The latter model and some illustrations of its use are presented in this chapter. 

A significant criticism of the above model is thei the high mass transfer flux should influence the effective Aim thickness, yet no allowance has been utede for this in tbe film thsory model. A boundary layer model to he described shortly allows for the effect of the high flux On hydrodynamics. 

Many numerical and series solutions for the laminar boundary layer model of mass transfer are available for situations such as flow in coeduits under conditions of fully developed or developing concentration or velocity profiles. Skellaed31 provides a particularly good summary of these results. The laminar boundary layer model has been extended to predict tha effects of high mass transfer flux on the mass transfer coefficient from a flat plate. The results of this work ate shown in Fig. 2.4-2 and. in com rest to the other theories, iedicate a Schmith number dependence of Ihe correction factor. 

Four of the simplest and best known of the theories of mass transfer from flowing streams are (1) the stagnant-film model, (2) the penetration model, (3) the surface-renewal model, and (4) the turbulent boundary-layer model... 

Among the first ones to present a complete theoretical approach are Burton, Prim, and Slitcher (1953). The authors developed a mathematical formulation of the problem for metallic systems with partial solid solubility crystallized from the melt in a suspension process. Their theory is based on a boundary layer model and does not account for the mutual dependence of heat and mass transfer but regards the influence of heat transfer as negligible. 

Hence, the Schmidt nnmber is 3600. If spherical pellets of solid sucrose with a diameter of 1 mm fall through this 20 wt% aqueous solution at a settling velocity of 6.5 cm/min, then the Reynolds number is 0.60, which corresponds to the upper limit of creeping flow. The mass transfer Peclet number is about 2000. The assumptions embedded in the boundary layer model are justified, as sucrose dissolves in solution. At the equatorial position of the pellet where 9 = njl radians. 

ABSTRACT Voltammetric and thermoelectrochemical (TEC) transfer function measurements have been carried out to study the eleetrodeposition of silver from nitric and tartaric solutions. For an isothermal cell, the observed increase of the limiting current is due to the diffusion coefficient increase and to the mass transport boundary layer decrease when bath temperature increases. In a non-isothermal cell, through the use of sine wave temperature modulation, the TEC transfer function measurements show a typical mass transport responses and typical adsorption relaxation in middle frequency domain. The experimental data are in good accordance with previously developed model and permit to determine the diffusion activation energy and the densification coefficients of silver ions in this media. 

Carberry [72] compiled several experimental data on fixed-bed mass transfer and derived a boundary layer model with A = 1.15, B = 1, and Be, = p Uidp/p. Wakao et al. [73] proposed the following correlation ... 

Carberry JJ (1960) A boundary-layer model of fluid-particle mass transfer in mixed beds. AIChE J 4 460... 

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. 

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

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] ... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

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