Turbulent Schmidt Number Model

3 Conventional Turbulent Mass Diffusivity Model 3.3.1 Turbulent Schmidt Number Model 

By considering the analogy between mass transfer and fluid flow, the turbulent mass diffusivity Dt may be analogous to the turbulent diffusivity (eddy diffusivity) 

In the literature, Sct is usually assumed to be a constant ranging from 0.5 to 1.2 for different processes and operating conditions. Although this is the simplest way to obtain D, yet the correct value of Sct is hard to guess. Moreover, the relationship between and fit is complicated as seen from Eq. (3.4) for Dt and Eq. (1.7) for fit, the assumption of constant Sct throughout the process and equipment cannot be proved and remains questionable. 

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This equation contains three new terms, namely flux of scalar variance, production of variance and dissipation of scalar variance, which require further modeling to close the equation. The flux terms are usually closed by invoking the gradient diffusion model (with turbulent Schmidt number, aj, of about 0.7). This modeled form is already incorporated in Eq. (5.21). The variance production term is modeled by invoking an analogy with turbulence energy production (Spalding, 1971) ... 

Homogeneous, isotropic turbulence cannot be assumed in the free jets. The authors in [541], therefore, utilized the Phoenics program (CHAM Ltd.) in connection with the slower diazotization reaction. The constants of the /c/e-turbulence model are adapted to well-known pictures of flow patterns and the turbulent Schmidt number determined to be Scturb = 1. It thereby succeeds in achieving the best description of the decoloration length. 

The turbulent Schmidt numbers (a ) were included for all variables that are modeled via the gradient and Boussinesq hypotheses. These Schmidt numbers were set to 1.0. The only exception was [Pg.1149]

The turbulent diffusivity, Dt, is assumed to be proportional to the turbulent viscosity. 5q is the turbulent Schmidt number with a typical value of 0.7. Equation (12.5.2-1) assumes that the scalar flux and the mean species concentration gradient are aligned, in contrast with the scalar flux model (Annex 12.5.2.A). This is strictly valid only for isotropic turbulence. Nevertheless, (12.5.2-1) is frequently applied in CFD codes. 

Concentration distribution for the calculation of tray efficiency As stated in Chap. 3, the conventional way of using turbulent Schmidt number Sc, model for predicting the concentration distribution is not dependable for the reason that the correct Sc, is not only hard to guess but also it is varying throughout the process. Hence the recently developed c — Bc two-equation model and the Reynolds mass flux model are recommended to use as described in the subsequent sections. 

Over the last decades, the application of computational fluid dynamics (CFD) to study the velocity and temperature profiles in packed column has been frequently reported [1-5]. However, for the prediction of concentration profile, the method commonly employed is by guessing an empirical turbulent Schmidt number Sc, or by using experimentally determined turbulent mass diffusivity D, obtained by using the inert tracer technique under the condition of no mass transfer [6, 7]. Nevertheless, the use of such empirical methods of computation, as pointed out in Chap. 3, is unreliable and not always possible. To overcome these drawbacks, the development of rigorous mathematical model is the best choice. 

In general, the scalar Taylor microscale will be a function of the Schmidt number. However, for fully developed turbulent flows,18 l.,p L and /, Sc 1/2Xg. Thus, a model for non-equilibrium scalar mixing could be formulated in terms of a dynamic model for Xassociated with working in terms of the scalar spatial correlation function, a simpler approach is to work with the scalar energy spectrum defined next. 

As in Section 2.1 for the turbulent energy spectrum, a model scalar energy spectrum can be developed to describe lop(n). However, one must account for the effect of the Schmidt number. For Sc < 1, the scalar-dissipation wavenumbers, defined by19... 

The first term on the right-hand side of Eq. (257) is provided by the quasisteady model, whereas the second represents the contribution of the transient process. Measurements of the mass transfer coefficients from the dissolution of the wall of a tube into a turbulent liquid having Schmidt numbers as large as 10s could be correlated with the expression [56]... 

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 ... 

The rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

The turbulent viscosity i/j is determined using the WALE model [330], similar to the Smagorinski model, but with an improved behavior near solid boundaries. Similarly, a subgrid-scale diffusive flux vector Jfor species Jk = p (uYfc — uYfc) and a subgrid-scale heat flux vector if = p(uE — uE) appear and are modeled following the same expressions as in section 10.1, using filtered quantities and introducing a turbulent diffusivity = Pt/Sc], and a thermal diffusivity Aj = ptCp/Pr. The turbulent Schmidt and Prandtl numbers are fixed to 1 and 0.9 respectively. 

The first model suggested for these dimensionless groups is named the Reynolds analogy. Reyuolds suggested that in fully developed turbulent flow heat, mass and momentum are transported as a result of the same eddy motion mechanisms, thus both the turbulent Prandtl and Schmidt numbers are assumed equal to unity ... 

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