Turbulent-diffusivity-based models

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size 

The random displacement of fluid particles along the scalar gradient creates a diffusive flux proportional to the characteristic velocity and the characteristic concentration 

17 This is strictly true only at high Reynolds numbers, but most CFD codes do not account for molecular effects seen at low to moderate Reynolds numbers. 

18 Flows with shear do not exhibit simple gradient transport. See, for example, the homogeneous shear flow results reported by Rogers et al. (1986). 

The scalar flux is then proportional to a turbulent-diffusion coefficient  

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The turbulent diffusivity defined by (4.74) is proportional to the turbulent viscosity defined by (4.46). Turbulent-diffusivity-based models for the scalar flux extend this idea to arbitrary mean scalar gradients. The standard gradient-diffusion model has the form... 

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. 

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 ... 

In contrast to Section IV,M, where the turbulent diffusivity was employed to derive an expression for the mass transfer coefficient, in this section expression (401), which is based on a physical model, constitutes the starting point. Concerning the renewal frequency s, the following dimensional considerations can lead to useful expressions. The state of turbulence near the interface can be characterized by a characteristic velocity ua = (gSf)i/2, the dynamic viscosity rj, the surface tension a, and the density p. Therefore... 

Compared with the more traditional theories based on turbulent diffusivity, the use of physical models enables a clearer understanding of the mechanism of turbulent transfer. A comparison of both theories shows that they are not contradictory and provide the same final results. Both start with transport equations that are valid in laminar flow. The more traditional theory involves... 

In this equation, represents the effective lateral diffusion/dispersion constant for laminar flow a value on the order of is suitable, and for turbulent flow the molecular diffusion coefficient in this expression should be replaced by the turbulent diffusion coefficient. Based on these simple relations it can be calculated that under typical conditions, lateral reactant transport takes place only over distances of a few subchannels. In other words, if the gas velocity through the wall channels differs much from the velocity through the central subchannels, the nonuniform flow profile can have a significant effect on the overall reactant conversion. In these situations the CBS model can be expected to give a better estimate of the reactor performance than the CB model. 

The thin him (or stagnant layer) model is based on the assumption that a dissolved chemical has a uniform concentration throughout a surface water body, due to turbulent diffusion, except in a very thin layer at the water s surface. A similar assumption is made concerning the chemical concentration in overlying air. Within a few micrometers or millimeters of the water-air interface, it is assumed that the eddies responsible for turbulent diffusion are suppressed therefore, chemical transport in this thin layer (or him) can only occur by molecular diffusion, which is considered to be the rate-limiting step of air-water exchange (Fig. 2-14) (Liss and Slater, 1974). 

Numerical simulations of Sardesai s experiments are discussed by Webb and Sardesai (1981) and Webb (1982) (who used the Krishna-Standart (1976), Toor-Stewart-Prober (1964) and effective diffusivity methods to calculate the condensation rates), McNaught (1983a, b) (who used the equilibrium model of Silver, 1947), and Furno et al. (1986) (who used the turbulent diffusion models of Chapter 10 in addition to methods based on film theory). It is the results of the last named that are presented here. 

However, to solve the heat and mass transfer equations an additional modeling problem has to be overcome. While there are sufficient measurements of the turbulent velocity field available to validate the different i>t modeling concepts proposed in the literature, experimental difficulties have prevented the development of any direct modeling concepts for determining the turbulent conductivity at, and the turbulent diffusivity Dt parameters. Nevertheless, alternative semi-empirical modeling approaches emerged based on the hypothesis that it might be possible to calculate the turbulent conductivity and diffusivity coefficients from the turbulent viscosity provided that sufficient parameterizations were derived for Prj and Scj. 

Traditional turbulence-diffusion models (based on the boundary layer adjoining a solid wall) imply that n = 2/3, but a value n = 1/2 is appropriate for a boundary layer adjoining a free surface (Jahne and Haussecker 1998). The appropriate value of n depends on the wind stress and the surfactant loading of the surface. Soloviev and Schluessel (1994) have described a procedure for estimating gas transfer velocities from measurements of heat transport, assuming that transport is adequately described by the classical (Danckwerts) surface renewal model. The key relationship can be written in the form ... 

The theoretical description of the turbulent mixing of reactants in tubular devices is based on the following model assumptions the medium is a Newtonian incompressible medium, and the flow is axis-symmetrical and nontwisted turbulent flow can be described by the standard model [16], with such parameters as specific kinetic energy of turbulence K and the velocity of its dissipation e and the coefficient of turbulent diffusion is equal to the kinematic coefficient of turbulent viscosity D, = Vj- =... 

The results of the modelling of liquid flows, based on Navier-Stokes equations and the C- model of turbulence, demonstrate that the turbulent diffusion coefficient increases decreases) for reactors with a radial input of reactants (P2- and P3-type mixers), especially for reactors with conical widening at the input of the liquid flows (P4, P5). For P5-type reactors, the time of mixing decreases approximately tenfold compared with the Pl-type at given flow parameters (Figure 2.5). An increase of Dfuji, results in a faster equalisation of the reactant concentration profile. Analysis of the construction of the reaction devices confirms that drops of hydraulic pressure... 

The US DOE had a major effort to understand the many variables affecting the performance of a bubble column reactor. Dudukovic and Toseland [75] outlined the cooperative study by Air Products and Chemicals (APC), Ohio State University (OSU), Sandia National Laboratory (SNL), and Washington University in St. Louis (WU). The efforts of this group have developed valuable unique experimental techniques for the measurement of gas holdup, velocity, and eddy diffusivities in bubble columns. They have obtained data that allows improved insight in churn-turbulent flow and have assessed the impact of various effects (internals, solid concentration, high gas velocity, pressure, etc.). General ideal flow pattern-based models do not reflect bubble column reality to date the models are based on a combination where some parameters are evaluated from first principles and some from the database. 

Perhaps the most detained lagrandian model is the one of Durbin and coworkers [31], first devised in order to simulate the turbulent diffusion only,. and later extended to take into account a two species bimolecular reaction [32]. It is based on a stochastic simulation of the random walk of fluid particles, but is able also to provide the probability density function of the position (and then of the composition), within the entrance section of the reactor, of two fluid particles which would be at the same later time at a given point within the reactor. Owen to this new... 

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