Mass transfer boundary layer thickness dimensionless

T Tobias number (ratio of mass transport to ohmic resistance), dimensionless Wa Wagner number, (ratio of activation to ohmic resistance), dimensionless aa,ac, transfer coefficients, anodic and cathodic, respectively, dimensionless 8C equivalent mass transfer boundary layer thickness (Nemst-type), cm r overpotential, V... 

In dimensionless notation, the generalized expression for the simplified mass transfer boundary layer thickness is... 

This second-order ordinary differential equation given by (16-4), which represents the mass balance for one-dimensional diffusion and chemical reaction, is very simple to integrate. The reactant molar density is a quadratic function of the spatial coordinate rj. Conceptual difficulty arises for zeroth-order kinetics because it is necessary to introduce a critical dimensionless spatial coordinate, ilcriticai. which has the following physically realistic definition. When jcriticai which is a function of the intrapellet Damkohler number, takes on values between 0 and 1, regions within the central core of the catalyst are inaccessible to reactants because the rate of chemical reaction is much faster than the rate of intrapellet diffusion. The thickness of the dimensionless mass transfer boundary layer for reactant A, measured inward from the external surface of the catalyst,... 

Dimensionless Molar Density. The final form of the mass transfer equation for Cp, y, t), which will be used to calculate the concentration profile and boundary layer thickness of species A in the liquid phase, is... 

This result can be written in terms of the important dimensionless numbers for mass and heat transfer. A completely dimensionless expression is obtained via division of the boundary layer thickness by the cylindrical radius / . If the Reynolds number is defined using R as the characteristic length, instead of the cylindrical diameter, then... 

The Nernst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation = aib (32)... 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

In Chap. 9, we considered the solution of this equation in the limit Re 1, where the velocity distribution could be approximated by means of solutions of the creeping-flow equations. When Pe 1, we found that the fluid was heated (or cooled) significantly in only a very thin thermal boundary layer of 0(Pe l/3) in thickness, immediately adjacent to the surface of a no-slip body, or () Pe l/2) in thickness if the surface were a slip surface with finite interfacial velocities. We may recall that the governing convection di ffusion equation for mass transfer of a single solute in a solvent takes the same form as (111) except that 6 now stands for a dimensionless solute concentration, and the Peclet number is now the product of Reynolds number and Schmidt number,... 

We have the following picture of mass transfer near the surface of a rotating disk. The dimensionless concentration exponentially decreases away from the disk. At a distance z 6, this variable is close to its nonperturbed value and practically does not vary any more. At large Schmidt numbers, the concentration mainly varies is a thin layer (of thickness Sc-1 3) adjacent to the disk surface. This region is called a diffusion boundary layer. 

For low wind speeds, one might ask, at what characteristic depth or distance from the interface is the rate of advective transfer of normal dissolved gases away from the interface equal to diffusive transfer To answer this question, we can use the dimensionless Peclet number, (dV/D), which expresses the relative importance of mass transfer by advection to transfer by diffusion. In the Peclet number, d can be taken as the thickness of the diffusive layer, V the velocity and D as the gas diffusivity in the water phase. If we take V as the piston velocity with an appropriately low value of about 1 cm h, and a typical diffusivity for gases in water of about 10" cm s the thickness of the boundary layer can be determined for a Peclet number, Pe= 1, i.e. at a distance from the interface where advective and diffusive transport are comparable. Under these conditions, d is... 

Sherwood number (Sh) A dimensionless measure of the ratio of convective mass transfer to molecular mass transfer. If the mass transfer coefficient k is defined in terms of the film theory, then Sh is a measure of the ratio of hydraulic diameter to the thickness of the boundary layer. See Section 6.5. 

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. 

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