Schmidt number molecular

The problems that arise when experiments are carried out in a greatly reduced scale can be overcome if the Reynolds number is high and the flow pattern is governed mainly by fully developed turbulence. It is possible to ignore the Reynolds number, the Schmidt number, and the Prandtl number because the structure of the turbulence and the flow pattern at a sufficiently high level of velocity will be similar at different supply velocities and therefore independent of the Reynolds number. The transport of thermal energy and mass by turbulent eddies will likewise dominate the molecular diffusion and will therefore also be independent of the Prandtl number and the Schmidt number. 

The average Nusselt number is not very sensitive to changes in gas velocity and Reynolds number, certainly no more than (Re)I/3. The Sherwood number can be calculated with the same formula as the Nusselt number, with the substitution of the Schmidt number for the Prandtl number. While the Prandtl number of nearly all gases at all temperatures is 0.7 the Schmidt number for various molecules in air does depend on temperature and molecular type, having the value of 0.23 for H2, 0.81 for CO, and 1.60 for benzene. 

NPei and NRtt are based on the equivalent sphere diameters and on the nominal velocities ug and which in turn are based on the holdup of gas and liquid. The Schmidt number is included in the correlation partly because the range of variables covers part of the laminar-flow region (NRei < 1) and the transition region (1 < NRtl < 100) where molecular diffusion may contribute to axial mixing, and partly because the kinematic viscosity (changes of which were found to have no effect on axial mixing) is thereby eliminated from the correlation. 

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

When the concentration profile is fully developed, the mass-transfer rate becomes independent of the transfer length. Spalding (S20a) has given a theory of turbulent convective transfer based on the hypothesis that profiles of velocity, total (molecular plus eddy) viscosity, and total diffusivity possess a universal character. In that case the transfer rate k + can be written in terms of a single universal function of the transfer length L and fluid properties (expressed as a molecular and a turbulent Schmidt number) ... 

The Schmidt and Prandtl numbers must be evaluated in order to be able to determine concentration and temperature differences between the bulk fluid and the external surface of the catalyst. The Schmidt number for naphthalene in the mixture may be evaluated using the ordinary molecular diffusivity employed earlier, the viscosity of the mixture, and the fluid density. 

For a free-falling spherical particle of radius R moving with velocity u relative to a fluid of density p and viscosity p, and in which the molecular diffusion coefficient (for species A) is DA, the Ranz-Marshall correlation relates the Sherwood number (Sh), which incorporates kAg, to the Schmidt number (Sc) and the Reynolds number (Re) ... 

The sub-grid-scale turbulent Schmidt number has a value of Scsgs 0.4 (Pitsch and Steiner 2000), and controls the magnitude of the SGS turbulent diffusion. Note that due to the filtering process, the filtered scalar field will be considerably smoother than the original field. For high-Schmidt-number scalars, the molecular diffusion coefficient (T) will be much smaller than the SGS diffusivity, and can thus usually be neglected. 

The Schmidt number is the ratio of kinematic viscosity to molecular diffusivity. Considering liquids in general and dissolution media in particular, the values for the kinematic viscosity usually exceed those for diffusion coefficients by a factor of 103 to 104. Thus, Prandtl or Schmidt numbers of about 103 are usually obtained. Subsequently, and in contrast to the classical concept of the boundary layer, Re numbers of magnitude of about Re > 0.01 are sufficient to generate Peclet numbers greater than 1 and to justify the hydrodynamic boundary layer concept for particle-liquid dissolution systems (Re Pr = Pe). It can be shown that [(9), term 10.15, nomenclature adapted]... 

The transport of an adsorbable species from the bulk fluid flowing around an individual bead is a problem of molecular diffusion. With the fluid in motion the rate of transport to the surface of a bead or pellet of adsorbent material is generally treated as a linear driving force. Eor gas phase separations there are a variety of correlations available to describe the mass transport to the surface in terms of the particle Reynolds number, the Schmidt number, the size of the adsorbent particle and of course the binary diffusivity of the species of interest. 

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... 

Parameter for molecular diffusion model In a moving zone, equivalent to the reciprocal of Peclet number, dispersion number Reynolds number, Re Prandtle number, Pr Schmidt number Sc... 

Deissler (D3) recently extended the analysis of thermal and material transfer associated with turbulent flow in tubes to include the behavior of fluids with high molecular Prandtl and Schmidt numbers. If the variation in molecular properties of the fluid with position are neglected, the following expression for the temperature distribution was suggested (D3) ... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

The eddy Schmidt number on the basis of the Reynolds analogy would be unity throughout the flowing stream. It is probable that the Reynolds analogy (R2) is no more applicable to the transport of material than it is to thermal transport. If the speculative analysis of Jenkins (J5) can be extended to material transport, the effect of molecular Schmidt number upon the eddy Schmidt number may be estimated. However, in this instance, since the molecular Schmidt number varies with each component, it is to be expected that the eddy Schmidt number will also vary... 

As indicated in Eq. (49), the total Schmidt number is a function of the component involved as well as of the position in the stream. It approaches the local molecular Schmidt number as the boundary flow is reached and... 

Values of the concentration parameter material transport, similar to those established for thermal transport, have been developed by Deissler. They take into account changes in fluid properties with concentration. Since such detailed information usually is not readily available for a particular system, it does not appear worth while to discuss further ramifications of Deisgler s analysis. It should be emphasized that all the foregoing is based upon the premise that the eddy diffusivity is isotropic and at all points in the shear flow is numerically equal to the eddy viscosity. As has been indicated earlier, the eddy Schmidt number undoubtedly is influenced by the local value of the molecular Schmidt number, which in turn is a function of the component in transport and the state of the phase. 

Deacon s intention was to separate the viscosity effect from the wind effect, so that the new model would be able to describe the change of via due to a change of water or air temperature (i.e., of viscosity) at constant wind speed. Deacon concluded that mass transfer at the interface must be controlled by the simultaneous influence of two related processes, that is, by the transport of chemicals (described by molecular diffusivity Dia), and by the transport of turbulence (described by the coefficient of kinematic viscosity va). Note that v has the same dimension as Dia. Thus, the ratio between the two quantities is nondimensional. It is called the Schmidt Number, Scia ... 

Table 20.3 Molecular Diffusivities, Kinematic Viscosities, and Schmidt Numbers (ScJW) in Water for Selected Chemicals... 

In the film model of Whitman the water-phase exchange velocity, v,w, is a function of the molecular diffusion coefficient of the chemical, while in Deacon s boundary layer model v[W depends on the Schmidt Number Sc W. Explain the reason for this difference. 

In these relationships, Ya, is the mass fraction that enters the channel, Da is the diffusion coefficient of species A relative to the bulk fluid, and IV is the mean molecular weight. Based on the scaling and the nondimensional groups, discuss the circumstances under which the axial diffusion may be neglected. Show also how the nondimensional axial coordinate z may be written in terms of the Reynolds and Schmidt numbers. 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

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