Turbulent Mass Diffusivity Model

Turbulent mass diffusivity models This category of models is conventional, which features to evaluate the unknown mJc in terms of a new variable the turbulent mass diffusivity D,. The following models belong to this category  

Reynolds mass flux models This category of models features to solve the unknown —u[d directly instead of in terms of D,. This category of models includes 

From Eq. (3.4), the fluctuating mass flux wjc can be also expressed as proportional to the negative gradient of C as follows  

In chemical engineering literature, Eq. (3.4a) is substantially the well-known Pick s law, which states that the mass flux flow is proportional to the negative concentration gradient due to the fact that the flow of mass flux is from high to the low concentration, or the flow any mass flux should be under negative concentration gradient. 

If Ui can be found from CFD, there are only two unknown variables in foregoing equation C and D,. The closure of Eq. (3.5) relies on the evaluation of Df 

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Conventional Turbulent Mass Diffusivity Model 3.3.1 Turbulent Schmidt Number Model... 

This chapter describes models of mass transfer in turbulent conditions. Beginning with a brief survey of turbulent eddy diffusivity models we develop solutions to the binary mass transport equations at length before presenting the corresponding multicomponent results. 

ANALYSIS NIq are asked to determine the rate of production of methane. We shall use the turbulent eddy diffusivity model to represent the transport processes in the gas phase. The mass fluxes are given by Eq. 10.4.24... 

We shall, as before, use Newton s method to solve all the independent equations simultaneously. The independent variables that are to be determined by iteration are the fluxes and the interface compositions and temperature. However, the use of the turbulent eddy diffusion model for the vapor-phase mass transport means that the mass fluxes and n 2, and the molar fluxes Aj and A2, appear in the set of model equations. These fluxes are related by... 

A comparison of the interactive film models that use the Chilton-Colburn analogy to obtain the heat and mass transfer coefficients with the turbulent eddy diffusivity models. 

A number of investigators used the wetted-wall column data of Modine to test multicomponent mass transfer models (Krishna, 1979, 1981 Furno et al., 1986 Bandrowski and Kubaczka, 1991). Krishna (1979b, 1981a) tested the Krishna-Standart (1976) multicomponent film model and also the linearized theory of Toor (1964) and Stewart and Prober (1964). Furno et al. (1986) used the same data to evaluate the turbulent eddy diffusion model of Chapter 10 (see Example 11.5.3) as well as the explicit methods of Section 8.5. Bandrowski and Kubaczka (1991) evaluated a more complicated method based on the development in Section 8.3.5. The results shown here are from Furno et al. (1986). 

Besides applying the postulation similar to the Boussinesq s (or Pick s law) to solve the Reynolds mass flux — mJc in terms of isotropic turbulent mass diffusivity Dt as described in preceding Sect. 3.2 by c — Sc two-equation model, another model has been recently developed to solve the anisotropic Reynolds mass flux —M-c directly instead of using D, to close the turbulent species mass conservation equation. The Reynolds mass flux model discussed in this section could be known as a result following the turbulence closure postulations for the second-order closure turbulence model in the book of Chen and Jaw [23]. 

Using the Reynolds mass flux model, the directional —wjc can be calculated separately as —u c, —u c and — anisotropic turbulent mass diffusivity can be obtained. 

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. 

The turbulent mass diffusivity of the reactive species D, can be obtained according to c — Sc model as follows ... 

The turbulent thermal diffusivity at can also be calculated by using two-equation model as shown in Fig. 7.5, in which, similar to the turbulent mass diffusivity Dj, the t reaches almost steady condition after traveling a distance about 50-fold of the effective catalyst diameter from the entrance and decreases sharply afterward. 

Skaret presents a general air and contaminant mass flow model for a space where the air volume, ventilation, filtration, and contaminant emission have been divided for both the zones and the turbulent mixing (diffusion) between the zones is included. A time-dependent behavior of the concentration in the zones with constant pollutant flow rate is presented. 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

In fight of this analogy, we anticipate that the effect of turbulence may be dealt with in a similar manner like the random motion of molecules for which die gradient-flux law of diffusion (Eq. 18-6) has been developed. In addition, the mass transfer model (Eq. 18-4) may provide an alternative tool for describing the effect of turbulence on transport... 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation of rates of mass and energy transport in multicomponent systems. Multicomponent mass transfer coefficients are defined in Chapter 1, Chapter 8 develops the multicomponent film model, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considers models based on turbulent eddy diffusion. Chapter 11 shows how the additional complication of simultaneous mass and energy transfer may be handled. 

Matrix of mass transfer rate factors in linearized film model (Eq. 8.4.4) [ - ] Matrix of mass transfer rate factors in turbulent diffusion model (Eq. 10.3.9) [-]... 

In the mesoscale model, setting Tf = 0 forces the fluid velocity seen by the particles to be equal to the mass-average fluid velocity. This would be appropriate, for example, for one-way coupling wherein the particles do not disturb the fluid. In general, fluctuations in the fluid generated by the presence of other particles or microscale turbulence could be modeled by adding a phase-space diffusion term for Vf, similar to those used for macroscale turbulence (Minier Peirano, 2001). The time scale Tf would then correspond to the dissipation time scale of the microscale turbulence. 

This model has several limitations. The film model assumes that mass transfer is controlled by the liquid-phase film, which is often not the case because the interface characteristics can be the limiting factor (Linek et al., 2005a). The liquid film thickness and diffusivity may not be constant over the bubble surface or swarm of bubbles. Experiments also indicate that mass transfer does not have a linear dependence on diffusivity. Azbel (1981) indicates that others have shown that turbulence can have such a significant effect on mass transfer such that eddy turbulence becomes the controlling mechanism in which diffusivity does not play a role. In most instances, however, eddy turbulence and diffusivity combine to play a significant role in mass transfer (Azbel, 1981). 

As fast liquid-phase chemical processes obey the laws of mass and heat transfer, a macrokinetic approach is required for their calculation. The calculation of a fast liquid-phase chemical reaction, on the basis of hydrodynamic and diffusion models, revealed fundamental aspects of carrying out fast processes in turbulent flows ... 

As M (the initial mass transfer rate which determines the initial slope of the dissolution curve) and E (the equilibrium concentration) reduce, the water is less plumbosolvent (less lead dissolves curves A to C) and these factors can be determined by stagnation sampling at appropriate reference houses or by laboratory plumbosolvency testing. Curves A1 and A2 differ in shape as a consequence of the relationship between the 30 minutes stagnation and equilibrium concentrations, which vary for individual waters (Hayes, 2008). The exponential curve and the assumption of plug flow are both approximations, but they enable the computational demands of the model to be greatly reduced. Extensive research (Hayes, 2002 and Van der Leer et al, 2002) has demonstrated that these approximations are adequate when compared to the more scientifically exact diffusion model and the three dimensional simulation of turbulent flow. 

Turbulent diffusivity based closure models for the scalar fluxes describing turbulent transport of species relate the scalar flux to the mean species concentration gradient according to Reynolds analogy between turbulent momentum and mass transport. The standard gradient-diffusion model can be written ... 

A dynamic mathematical model of the three-phase reactor system with catalyst particles in static elements was derived, which consists of the following ingredients simultaneous reaction and diffusion in porous catalyst particles plug flow and axial dispersion in the bulk gas and liquid phases effective mass transport and turbulence at the boundary domain of the metal network and a mass transfer model for the gas-liquid interface. 

Keywords Computational mass transfer Reynolds averaging Closure of time-averaged mass transfer equation Two-equation model Turbulent mass transfer diffusivity Reynolds mass flux model... 

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