Turbulent Eddy Diffusivity Model

For definiteness, we consider the transfer processes between a cylindrical wall and a turbulently flowing n-component fluid mixture. For condensation of vapor mixtures flowing inside a vertical tube, for example, the wall can be considered to be the surface of the liquid condensate film. We examine the phenomena occurring at any axial position in the tube, assuming that fully developed flow conditions are attained. For steady-state conditions, the equations of continuity of mass of component i (assuming no chemical reactions), Eqs. 1.3.7 take the form 

1 shows that is r invariant. It will prove to be more convenient to work in a coordinate system measuring the distance from the wall y 

Before proceeding further, it is convenient to define the following parameters and variables that incorporate information on the flow 

In terms of the reduced distance from the wall the boundary conditions are 

The first difficulty we encounter is that the position y, at which the bulk fluid-phase composition (o, is reached, is not known precisely. To overcome this shortcoming in our 

---

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. 

For computation of the fluxes themselves. Algorithms 8.1, 8.4, and 8.5 may be used more or less as written simply replace all quantities in molar units by the corresponding quantities in the turbulent eddy diffusivity model. 

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. 

In considering very many condenser simulations (not just those reviewed here) we have yet to find an application where the differences between any of the multicomponent film models that account for interaction effects (Krishna-Standart, 1976 Toor-Stewart-Prober, 1964 Krishna, 1979a-d Taylor-Smith, 1982) are significant. There is also very little difference between the turbulent eddy diffusivity model and the film models that use the Chilton-Colburn analogy (Taylor et al., 1986). This result is important because it indicates that the Chilton-Colburn analogy, widely used in design calculations, is unlikely to lead to large... 

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). 

Puttock, J.S., and Hunt, J.C.R. (1979) Turbulent diffusion from sources near obstacles with separated wakes. Part I. An eddy diffusivity model, Atmospheric Environment 13, 1-13. 

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. 

For Sc > 1000 c = 1/2 and for Sc < 500 c = 1/3 applied. This finding could be described in terms of the multi-parameter King model (King 1976), which contained a combination of molecular diffusion and turbulent eddy diffusion. 

There are a number of criticisms to this approach. First, the model is incomplete, since once growth begins it continues without limit. Nonlinear saturation and interactions with predators would be needed to stop this. The diffusion coefficient D certainly does not originate from the Brownian motion of the organisms, since this would be irrelevant to these processes above, say, on the millimeter scale. It is rather a turbulent eddy-diffusion coefficient aimed to... 

Eddy Diffusivity Models. The mean velocity data described in the previous section provide the bases for evaluating the eddy diffusivity for momentum (eddy viscosity) in heat transfer analyses of turbulent boundary layers. These analyses also require values of the turbulent Prandtl number for use with the eddy viscosity to define the eddy diffusivity of heat. The turbulent Prandtl number is usually treated as a constant that is determined from comparisons of predicted results with experimental heat transfer data. 

The horizontal liquid flow pattern is very complicated due to the mixing by vapor, dispersion, and the round cross section of the column. On single-pass trays, the latter results in the flow path, which first expands and then contracts. A rigorous modeling of this flow pattern is very difficult, and usually the situation is simplified by assuming that the liquid flow is unidirectional and the major deviation from the plug flow is the turbulent mixing or eddy diffusion. In [80], two different models, the eddy-diffusion model and the mixed pool model were developed and tested in the context of the rate-based approach for RD trays. The details of these models can be found in [81]. 

Turbulent (Eddy) dififusivity model in which a turbulent (eddy) diffusivity p, is introduced according to Eq. (1.7) to replace unknown pu Uj, called Bous-sinesq postulate. The p, is obtained by Eq. (1.14) where two unknowns k and a are involved. They are represented, respectively, by Eqs. (1.11a) and (1.13a). These equations should be further modeled to suit numerical computation. This model is commonly called fe e model. Several modifications have been made to extent its application. The weakness of this model is that the p, is isotropic and results in discrepancy when applying to the case of anisotropic flow. 

The difficulty with Eq, (26-58) is that it is impossible to determine the velocity at every point, since an adequate turbulence model does not currently exist, The solution is to rewrite the concentration and velocity in terms of an average and stochastic quantity C = (C) -t- C Uj = (uj) + Uj, where the brackets denote the average value and the prime denotes the stochastic, or deviation variable. It is also helpful to define an eddy diffusivity Kj (with units of area/time) as... 

In the absence of diffusion, all hydrodynamic models show infinite variances. This is a consequence of the zero-slip condition of hydrodynamics that forces Vz = 0 at the walls of a vessel. In real systems, molecular diffusion will ultimately remove molecules from the stagnant regions near walls. For real systems, W t) will asymptotically approach an exponential distribution and will have finite moments of all orders. However, molecular diffusivities are low for liquids, and may be large indeed. This fact suggests the general inappropriateness of using to characterize the residence time distribution in a laminar flow system. Turbulent flow is less of a problem due to eddy diffusion that typically results in an exponentially decreasing tail at fairly low multiples of the mean residence time. 

The term numerical diffusion describes the effect of artificial diffusive fluxes which are induced by discretization errors. This effect becomes visible when the transport of quantities with small diffusivities [with the exact meaning of small yet to be specified in Eq. (42)] is considered. In macroscopic systems such small diffusivities are rarely found, at least when being looked at from a phenomenological point of view. The reason for the reduced importance of numerical diffusion in many macroscopic systems lies in the turbulent nature of most macro flows. The turbulent velocity fluctuations induce an effective diffusivity of comparatively large magnitude which includes transport effects due to turbulent eddies [1]. The effective diffusivity often dominates the numerical diffusivity. In contrast, micro flows are often laminar, and especially for liquid flows numerical diffusion can become the major effect limiting the accuracy of the model predictions. 

When electrically insulated strip or spot electrodes are embedded in a large electrode, and turbulent flow is fully developed, the steady mass-transfer rate gives information about the eddy diffusivity in the viscous sublayer very close to the electrode (see Section VI,C below). The fluctuating rate does not give information about velocity variations, and is markedly affected by the size of the electrode. The longitudinal, circumferential, and time scales of the mass-transfer fluctuations led Hanratty (H2) to postulate a surface renewal model with fixed time intervals based on the median energy frequency. 

Deviation from the ideal plug flow can be described by the dispersion model, which uses the axial eddy diffusivity (m s ) as an indicator of the degree of mixing in the flow direction. If the flow in a tube is plug flow, the axial dispersion is zero. On the other hand, if the fluid in a tube is perfectly mixed, the axial dispersion is infinity. For turbulent flow in a tube, the dimensionless Peclet number (Pe) deflned by the tube diameter (v dlE-Q is correlated as a function of the Reynolds number, as shown in Figure 10.3 [3] dz is the axial eddy diffusivity, d is the tube diameter, and v is the velocity of liquid averaged over the cross section of the flow channel. 

The coefficient Ex is called the turbulent (or eddy) diffusion coefficient it has the same dimension as the molecular diffusion coefficient [L2 1]. The index x indicates the coordinate axis along which the transport occurs. Note that the turbulentjliffusion coefficient can be interpreted as the product of a mean transport distance Lx times a mean velocity v = (Aa At) l Egex, as found in the random walk model, Eq. 18-7. 

For the case of turbulent flow, the basic transport laws have to be changed to account for turbulence, and many models have been developed since the time of Reynolds. He tried to keep the simple form of Eqns. (3.4-1), (3.4-2), and (3.4-3), but with transport coefficients modified with the so-called eddy diffusivities that are added to the molecular values. So, the net effect of turbulence would be to increase the transfer rates. 

Turbulent flow results because of the radial mixing of layers to equalize flowrates. The mixing of the layers is due to the increased eddies and mass transfer occurs by eddy diffusivity. Turbulent diffusivity increases in proportion to mean flow velocity as depicted in Figure 2.10B. Figure 2.IOC represents plug flow which is unattainable in practice but does suggest a model... 

---