Eddy diffusion defined

I0-38Z ) is solved to give the temperature distribution from which the heat-transfer coefficient may be determined. The major difficulties in solving Eq. (5-38Z ) are in accurately defining the thickness of the various flow layers (laminar sublayer and buffer layer) and in obtaining a suitable relationship for prediction of the eddy diffusivities. For assistance in predicting eddy diffusivities, see Reichardt (NACA Tech. Memo 1408, 1957) and Strunk and Chao [Am. ln.st. Chem. Eng. J., 10, 269(1964)]. 

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... 

On a similar basis an eddy diffusivity for mass transfer Er> can be defined for systems in which concentration gradients exist as ... 

An additional equation is required to describe the turbulent flux. The usual approach is to define an eddy diffusivity Kj (with units of area/time) such that... 

If we define AT as the eddy diffusivity for momentum, the vertictil eddy diffusion coefficient under unstable conditions can be expressed as... 

The thermal motion of molecules of a given substance in a solvent medium causes dispersion and migration. If dispersion takes place by intermolecular forces acting within a gas, fluid, or solid, molecular diffusion takes place. In a turbulent medium, the migration of matter within it is defined as turbulent diffusion or eddy diffusion. Diffusional flux J is the product of linear concentration gradient dCldX multiphed by a proportionality factor generally defined as diffusion coefficient (D) (see section 4.11) ... 

In the case of turbulent flow it is possible to employ the same eddy concepts as were used in connection with the consideration of thermal transport. In the case of steady, uniform flow between parallel plates the eddy diffusivity may be defined by... 

Bomb products, rocket reentry bumup products, meteoritic dust, and other trace constituents will be redistributed rapidly within the mesosphere by large scale circulation, by eddy diffusion and, for particles of significant size, by gravitational sedimentation as well (1). Murgatroyd and Singleton (32) discuss the circulation of the mesosphere and indicate a meridional pattern, with ascent over the summer pole, descent over the winter pole, and a well defined flow from the summer hemisphere to the winter hemisphere above 50 km., with speeds of the order of meters sec."1 horizontally and cm. sec."1 vertically. 

Equations 2.61 and 2.65 account for the effect of ordinary and eddy diffusion in the zone broadening process. Now we need to express nonequilibrium effects which are concerned with the time the solute molecules spend in the two phases. Let us define a few more terms in order to set up some mathematical relation ships ... 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

The shearing stress, r, exerted by the wind on the ground entails a downwards flux of momentum. In the aerodynamic boundary layer above the surface, the momentum is transferred by the action of eddy diffusion on the velocity gradient. The friction velocity is defined by w = t/pa and is a measure of the intensity of the turbulent transfer. Near to a rough surface, the production of turbulance by mechanical forces... 

The mass transfer coefficient, K, is defined as the ratio of the mass transport controlled reaction rate to the concentration driving force. The concentration driving force will depend on both turbulent and bulk convection. Bulk convection depends on molecular diffusivity, while the turbulent component depends on eddy diffusivity (4). The mass transfer coefficient considers the combination of the two transport mechanisms, empirically. 

Let us define an eddy viscosity or eddy diffusivity for momentum eM such that... 

Horvath and Lin [10,11] showed how the different contributions to peak dispersion in a packed bed could be defined in an expression for H, which was applied [12] to CEC by Dittman et al. it was demonstrated that the major contribution of eddy diffusion to H in HPLC was considerably reduced in CEC. Minimal reduced plate heights h = (H/dp), generally found to be near 2 in HPLC, should be reduced to near 1 or less, giving rise to plate numbers over 10s for a 40-cm-long CEC column with 3-pm particles. Horvath and Lin showed [11] how their expression for H could be simplified to... 

In addition to this a turbulent diffusion coefficient or eddy diffusivity for mass transfer l)t (SI units m2/s) is defined by... 

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. 

The fluctuating coagulation tenns can be neglected because the concentration is approximately uniform away from the wails. Moreover, because the sy.siem is well-stitred. the concentration through the bulk Is approximately uniform up to a small distance. S. from the bottom of the chamber corresponding to the region where the eddy diffusion goes from its value in the bulk of the fluid to zero (at the wall). To a certain extent, this di.slance is arbitrary and need not be defined exactly for this analysis. 

Computed as explained in the text using initial stationary concentration (Ct 0) given by profile 1 in Figure 12 and new steady-state given by profile 2. Time to steady-state shown for different values of eddy diffusion coefficient (K) and advective velocity (U). Time to steady-state defined as the time when the oxygen concentration has attained 95% of the concentration difference between the new and initial steady-state profiles C — Ct 0 = 0.95(Ct x — Ct= 0). 

The eddy diffusion coefficient for mass transfer is defined in a similar fashion to the eddy viscosity ... 

Finally, it is necessary to assess the importance of transport by eddy diffusion relative to that by mean meridional motions. For this purpose, we define time constants for the transport over a characteristic distance dc as follows... 

Quantity is analogous to fi, the absolute viscosity. Also, in analogy with the kinematic viscosity v the quantity called the eddy diffusivity of momentum, is defined as e,j = EJp. 

The viscous sublayer occupies only a very small fraction of the total cross section. It has no sharp upper boundary, and its thickness is difficult to define. A transition layer exists immediately adjacent to the viscous sublayer in which both viscous shear and shear due to eddy diffusion exist. The transition layer, which is sometimes called a buffer layer, also is relatively thin. The bulk of the cross section of the flowing stream is occupied by entirely turbulent flow called the turbulent core. In the turbulent core viscous shear is negligible in comparison with that from eddy viscosity. 

On page 55 an eddy diffusivity for momentum transfer was defined. A corresponding eddy diffusivity for heat transfer can be defined by... 

A simple conceptual model for turbulent flow deals with eddies, small portions of fluid in the boundary layer that move about for a short time before losing their identity [8], The transport coefficient, which is defined as eddy diffusivity for momentum transfer eM, has the form... 

Similarly, eddy diffusivities for heat and mass transfer, eH and em, respectively, may be defined by the relations... 

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. 

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