Diffusion boundary layer local

The transport coefQcients are functions of the macroscopic and the local microscopic fluid movement, which is caused by the stirring of the gas hubbies. Additionally, the increase of the diffusion boundary layer during the reaction due to the formation of a pure Fe phase has to be taken into consideration for all three reaction routes (2)-(4). For the reaction rate, the following holds ... 

By coming to this understanding we have identified a requirement and a restriction of TS-PFR operation. The TS-PFR must be in local steady state with respect to adsorp-tion/desorption, pore diffusion, boundary layer diffusion, etc. There is no equivalent requirement for local steady state with respect to temperature as long as the gas phase... 

Of the main interest is the velocity distribution near the sphere in the diffusion boundary layer. Introduce local system of coordinates y,9), where the y-axis is perpendicular and 6 is tangential to the corresponding area element of the surface. Then r/a = 1 + y/a. Considering the case y/a 1, expand (10.69) as a power series in y/a. In a result, we obtain... 

MacLeod and Radke (1994) presented a solution to the above eqnations applicable whm both phases have uniform compositions initially, local eqndib-rium is maintained at the intraface, drop volume increases at a constant rate, and the thickness 6j of each diffusion boundary layer, which is on the order (Djl), is small in comparison with R. An interesting featnre of this analysis is that it predicts that a steady-state interfacial tension will be reached. Since R is large for these conditions, the last term of Equation 6.47 may be neglected and one finds... 

Some classification of parameters in their connection with physical or mechanical processes is to be done. The main parameter connecting hydrodynamic and diffusion parts of the film flow problem with surfactant is Marangoni number Ma. The both variants of positive (Ma > 0) and negative (Ma < 0) solutal systems are considered. The main hydrodynamic parameters are Re, 7 or equivalently S. 7. This two values determine the mean film thickness i/, mean velocity and flow rate as well as parameter k. The diffusion parameters Pe,co determine the local thickness of diffusion boundary layer h and smallness parameter e. Two values T, Di characterize the masstransfer of surfactant by the adsorption-desorption and the intensity of dissipation by the surface diffusion. Besides the limiting case of fast desorption (T = 0) the more general case (T 1) are considered. Intensity of the surfactant evaporation by parameter Bi is determined. The remaining parameter G gives an indication to the typical value of surface excess concentration A in comparison with c. ... 

Holtslag, A. A. M. and Boville, B. A. (1993). Local versus nonlocal boundary-layer diffusion in a global climate model, /. Clim. 6,1825-1842. 

Figure 8. Variation of the hydrodynamic boundary layer thickness (So, equation (26), continuous line), the diffusion layer thickness (<5,-, equation (34), dotted line) and the ensuing local flux (/, equation (32), dashed line) with respect to the distance from the leading edge (y) in the case of laminar flow parallel to an active plane (the surface is a sink for species i). Parameters /), = 10-9nrs, v= 10 3ms, c = lmolm-3, and v — 10-6 m2 s 1. Notice that c5, o (as required for the derivation of the flux equation (32)), and that the flux decreases when <5, increases...

At very low pH, the rate of dissolution is so fast that the rate is limited by the transport of the reacting species between the bulk of the solution and the surface of the mineral (Berner and Morse, 1974). The rate can then be described in terms of transport (molecular or turbulent diffusion) of the reactants and products through a stagnant boundary layer, 8. The thickness of this layer depends on the stirring and the local turbulence. (See Chapter 5 for a discussion of transport vs surface controlled processes.)... 

Ka can be defined as a gas-phase transfer coefficient, independent of the liquid layer, when the boundary concentration of the gas is fixed and independent of the average gas-phase concentration. In this case, the average and local gas-phase mass-transfer coefficients for such gases as sulfur dioxide, nitrogen dioxide, and ozone can be estimated from theoretical and experimental data for deposition of diffusion-range particles. This is done by extending the theory of particle diffusion in a boundary layer to the case in which the dimensionless Schmidt number, v/D, approaches 1 v is the kinematic viscosity of the gas, and D is the molecular diffusivity of the pollutant). Bell s results in a tubular bifurcation model predict that the transfer coefficient depends directly on the... 

Figure 5.18 shows the only reliable Nui c data available near the critical Reynolds number (XI). Since the data were taken with a side support, there is some effect on the separation and transition angles. Thus the values of Nuj are probably subject to error (R2, R3) although the trend with Re should be correct. At Re = 0.87 x 10 the Shi variation is similar to that shown at lower Re in Fig. 5.17. At Re = 1.76 x 10 the critical transition has already occurred, with the separation bubble accounting for the minimum in Nuj c at 0 — 110°. The maximum in Nuj at 0 = 125° reflects the increased transfer rate in the attached turbulent boundary layer. The local minimum at 0 = 160° is due to final separation. These angles do not agree exactly with those in Fig. 5.11 because of the crossflow support and the fact that angular diffusion shifts the...

To effect a solution, the boundary layer was considered to be divided into a large number of increments and for the element m, the local rate of diffusion of ammonia can be expressed ... 

D h) is the local diffusion coefficient. Modifying Levich s (II) boundary-layer solution by imposing the apparent first-order reaction at the surface of the rotating disk, one obtains Nu, the rate of deposition per unit area of disk ... 

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]. 

The membrane reactor shown in Fig. 6.5 consists of a tubular shell containing a tubular porous membrane. It defines two compartments, the inner and the outer (shell) compartments. The reactants are fed into the inner compartment where the reaction takes place. We can observe that when the reactants flow along the reactor, one or more of the reaction participants can diffuse through the porous membrane to the outer side. In this case, we assume that only one participant presents a radial diffusion. This process affects the local concentration state and the reaction rate that determine the state of the main reactant conversion. The rate of reaction of the wall diffusing species is influenced by the transfer resistance of the boundary layer (1/lq.) and by the wall thickness resistance (S/Dp). 

For the case of high Peclet numbers Pe 1), the concentration boundary layer is very thin compared to the local radius of curvature of the particle, and, thus, the curvature term as well as the tangential diffusion terms can be neglected as it has already been shown [1,8,11]. In that case, eq. (2) becomes parabolic on 0 and can be solved analytically in a manner quite similar to that of Levich, providing concentration profiles in the fluid phase of the form... 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

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