Momentum eddy diffusivity

When representing rates of transfer of heat, mass, and momentum by eddy activity, the concepts of eddy thermal conductivity, eddy diffusivity, and eddy viscosity are sometimes useful. Extending the concepts of heat conduction, molecular diffusion, and molecular viscosity to include the transfer mechanisms by eddy activity, one can use Equations 2.13-2.15, which correspond to Equations 2.2,2.3, and 2.5, respectively. 

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. 

In the limit of c 1, equation 31 becomes equation 27. Karabelas (30) integrated equation 31 assuming that es = ef, where e/was the fluid momentum eddy diffusivity, and that the solids concentration was a function of the vertical coordinate only. He predicted the vertical solids concentration profile across a horizontal pipe to be... 

According to Prandtl (95), the particle diffusion coefficient is related to the liquid momentum eddy diffusivity e/ defined in terms of the turbulent shear stress (r0) and the time-average strain rate (7 )... 

Two things must occur before a particle in suspension in a fluid deposits on a surface to become part of the foulant layer. First the particle has to be transported to the surface by one or a combination of mechanisms including Brownian motion, turbulent diffusion, as described in Chapter 5, or by virtue of the momentum possessed by the particle. The size of the particle will have a large influence on the dominant mechanism. For instance very small particles would be expected to be subject to Brownian diffusion and eddy diffusion whereas the larger particles because of their mass would move under momentum forces. Having arrived at the surface the particle must stick if it is to be regarded as part of the foulant layer residing on the surface. 

The molecular momentum diffusivity ffp in mVs is a function only of the fluid molecular properties. However, the turbulent momentum eddy diffusivity e, depends on the fluid motion. In Eq. (3.10-29) we related e, to the Prandtl mixing length L as follows ... 

The term L dvfldy by Eq. (3.10-29) is the momentum eddy diffusivity e,. When this term is in the turbulent heat-transfer equation (5.7-24), it is called a, eddy thermal diffusivity. Then Eq. (5.7-24) becomes... 

Momentum eddy diffusivity For turbulent flow in circular pipes. Re = 50 000 to... 

Heat and mass eddy diffusivity The evidence is that with Prandtl and Schmidt numbers close to unity, as for most gases, the eddy diffusivities of heat and mass are equal to the momentum eddy diffusivity for all regions of turbulence [15], For turbulent fluids where Prandtl and Schmidt numbers exceed unity, the ratios E jand E /E will vary with location relative to the wall and in the turbulent core will lie generally in the range 1,2 to 1.3, with E and essentially equal [44, 62], For = 0 to 45, with Pr and Sc > 1, a critical analysis of the theoretical and experimental evidence [44] led to... 

A liquid mobile phase is far denser than a gas and, therefore, carries more momentum. Thus, in its progress through the interstices of the packing, violent eddies are formed in the inter-particular spaces which provides rapid solute transfer and, in effect, greatly increases the effective diffusivity. Thus, the resistance to mass transfer in that mobile phase which is situated in the interstices of the column is virtually zero. However, assuming the particles of packing are porous (i.e., silica based) the particles of packing will be filled with the mobile phase and so there will... 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

If there is a temperature gradient within the fluid, the eddies will be responsible for heat transfer and an eddy thermal diffusivity Ep may be defined in a similar way. It is suggested that, since the mechanism of transfer of heat by eddies is essentially the same as that for transfer of momentum, Eh is related to mixing length and velocity gradient in a similar manner. 

It has been assumed that the density is constant in writing these equations, which are therefore strictly valid only for incompressible flow. ed is called the eddy diffusivity and eh the eddy thermal diffusivity. Although s can be interpreted as the eddy diffusivity of momentum, it is usually called the eddy viscosity and sometimes by the better name eddy kinematic viscosity. 

If the fluid in the pipe is in turbulent flow, the effects of molecular diffusion will be supplemented by the action of the turbulent eddies, and a much higher rate of transfer of material will occur within the fluid. Because the turbulent eddies also give rise to momentum transfer, the velocity profile is much flatter and the dispersion due to the effects of the different velocities of the fluid elements will be correspondingly less. 

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

The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

Turbulent diffusion occurs because turbulent eddies are transporting mass, momentum, and energy over the eddy scale at the rotational velocity. This transport rate is generally orders of magnitude greater than the transport rate due to molecular motion. Thus, when a flow is turbulent, diffusion is normally ignored because e Z). The exception is very near the flow boundaries, where the eddy size (and turbulent diffusion coefficient) decreases to zero. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

There are shown in Fig. 19 values of the eddy diffusivity calculated from the measurements by Sherwood (SI6). These data show the same trends as were found in thermal transport, indicating that the values of eddy diffusivity are determined primarily from the transport of momentum for situations where the molecular Schmidt numbers of the components do not differ markedly from each other. 

At present analytical solutions of the equations describing the microscopic aspects of material transport in turbulent flow are not available. Nearly all the equations representing component balances are nonlinear in character even after many simplifications as to the form of the equation of state and the effect of the momentum transport upon the eddy diffusivity are made. For this reason it is not to be expected that, except by assumption of the Reynolds analogy or some simple consequence of this relationship, it will be possible to obtain analytical expressions to describe the spatial variation in concentration of a component under conditions of nonuniform material transport. 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

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... 

In turbulent pipe flow it is again also often convenient to write the turbulence quantities in terms of the eddy viscosity and diffusivity and when this is done the momentum and energy equations become ... 

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

Because an air packet and the molecules within it move as a unit, the eddy diffusion coefficients for different gaseous species are equal. In fact, Kj is often assumed to be the same for the transfer of gases, heat, and momentum (expressed in the same units), a relation that is referred to as the similarity principle. Therefore Kj is generally measured for the most... 

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