Mechanism of Turbulent Diffusion

In a turbulent flow there are two chief mechanisms of drop coagulation [2], that of turbulent diffusion and that of inertia. The inertial mechanism is based on the assumption that turbulent pulsations do not completely entrain the drop. As a result, relative velocities attained by drops due to turbulent pulsations depend on their masses. The difference in the pulsation velocities of drops of various radii causes their approach and leads to an increase of collision probability. The mechanism of turbulent diffusion is based on an assumption of full entrainment of drops by turbulent pulsations with scales, playing the chief role in the mechanism of approach of drops. Since drops move chaotically under the action of turbulent pulsations, their motion is similar to the phenomenon of diffusion and can be characterized by a coefficient of turbulent diffusion. 

We now estimate the growth rate of drops due to the mechanism of turbulent diffusion. For simplicity, drops are taken to be of identical size. The collision frequency is then equal to ... 

Mechanism of Turbulent Diffusion 487 The characteristic time of drop enlargement is thus equal to ... 

As earlier, consider ti to be the characteristic coagulation time of a polydisperse ensemble of drops, caused by the mechanism of turbulent diffusion due to the forces of hydrodynamic and molecular interactions. This time should be estimated. For typical values of the flow, Pq = 40 kg m , 2o = 5 x 10 m, Pq = 1.2 X 10 Pa-s, W = 5 X 10 m /m and distribution parameters of = 10 m, k = 3, one obtains 1/ti = 0.257 s. Thus, a twofold increase in drop radius occurs in a time t of 7 s. This time is almost two orders of magnitude higher than for a monodisperse distribution without regard to hydrodynamic and molecular forces. Such a big difference in characteristic times is undoubtedly caused not by taking into account the polydispersivity of the distribution, but as a result of considering the interaction forces. 

The estimation of Rav for characteristic parameter values shows that Rav where Aq = d/Re /" is the internal scale of turbulence. In a turbulent flow, both heat and mass exchange of drops with the gas are intensified, as compared to a quiescent medium. The delivery of substance and heat to or from the drop surface occurs via the mechanisms of turbulent diffusion and heat conductivity. The estimation of characteristic times of both processes, with the use of expressions for transport factors in a turbulent flow, has shown that in our case of small liquid phase volume concentrations, the heat equilibrium is established faster then the concentration equilibrium. In this context, it is possible to neglect the difference of gas and liquid temperatures, and to consider the temperatures of the drops and the gas to be equal. Let us keep all previously made assumptions, and in addition to these, assume that initially all drops have the same radius (21.24). Then the mass-exchange process for the considered drop is described by the same equations as before, in which the molar fluxes of components at the drop surface will be given by the appropriate expressions for diffusion fluxes as applied to particles suspended in a turbulent flow (see Section 16.2). In dimensionless variables (the bottom index 0" denotes a paramenter value at the initial conditions). 

Chapters Turbulent Diffusion. Turbulent diffusion is an important transport mechanism in the atmosphere, oceans, lakes, estuaries, and rivers. In fact, most of the atmosphere and surface waters of the Earth are turbulent. If you are going to work in any of these systems, it will be important to have at least a working knowledge of turbulent diffusion. 

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... 

Experimental investigations of turbulence diffusion. A factor in transportation of sediment in open channel flow. J Applied Mechanics, 12 A91-A100. 

As in the case of turbulent diffusion, the chemical flux is often expressed by Fick s first law, as shown in Eqs. [1-3] and [1-4], but in this case D is called a mechanical dispersion coefficient. Dispersion also occurs at much larger scales than that of soil particles for example, groundwater may detour around regions of relatively less permeable soil that are many cubic meters in volume. At this scale, the process is called macrodispersion. 

TURBULENT DIFFUSION The contribution of turbulent diffusion is a measure of heat transfer as a result of turbulent mixing of portions of the gas stream at different temperatures. As Singer and Wilhelm have pointed out, its value can be estimated from measurements of mass transfer radially by the same mechanism. On this basis... 

An aerosol issuing from a point source is dispersed in a steady turbulent plume in the atmosphere. Derive an expression for the variation of the extinction coeflicieni, h (Chapter 5), with position in the plume assuming that (a) the only mechanism affecting the light-scattering portion of the size distribution is turbulent diffusion and (b) the only mechanisms are turbulent diffusion and growth. 

The rate of dry deposition of particles depends on the particle size. Particles are transported down to the quasi-laminar layer near the surface by the same mechanisms of turbulent transport as are gases. The rate of transfer across the laminar layer is determined by the particle diffusivity, which depends on particle size. Particles with sizes in the range of 0.1 to 1.0 /rni diameter have atmospheric lifetimes with respect to dry deposition of about 10 days. 

Tha transport mechanisms of molecular diffusion and mass carried by eddy motion are again assumed edditive although the contribution of the molecular diffusivity term is quite small except in the region nenr a wall where eddy motion is limited. The eddy diffusivity is directly applicable to problems snch as the dispersion of particles or species (pollutants) from a souree into a homogeneously turbulent air stream in which there is little shear stress. The theories developed by Taylor.36 which have been confirmed by a number of experimental investigations, can describe these phenomena. Of more interest in chemical engineering applications is mass transfer from a turbolent fluid to a surface or an interface. In this instance, turbulent motion may he damped oni as the interface is approached aed the contributions of both molecolar and eddy diffusion processes must he considered. To accomplish this. 9ome description of the velocity profile as the interface is approached must be available. 

Consider particles of radius a Ao and assume that in the course of their motion in the liquid, they are completely entrained by turbulent pulsations that play the basic role in the mechanism of mutual approach of suspended particles. Then it can be assumed that particle transport is performed via isotropic turbulence. Since particles move chaotically in the liquid volume, their motion is similar to Brownian one and can be considered as diffusion with some effective factor of turbulent diffusion Dr. In the same manner as in the case of Brownian coagulation, it is possible to consider the diffusion flux of particles of radius U2 toward the test particle of radius Uj. The distribution of particles U2 is characterized by the stationary diffusion equation... 

The assessment of hydrodynamic and molecular interactions of drops can be made in the same manner as previously described for emulsions in Part V. Upon approach of the drops to each other under the action of turbulent pulsations up to distances smaller than Ao, they are subject to significant resistance from the environment, and the force of moleetilar attraction leads to collision and coalescence of the drops. If the basic mechanism of drop coagulation is that of turbulent diffusion, the coefficient of turbulent diffusion depends on the coefficient of hydrodynamic resistance [see Eqs. (11.70), (11.72), and (11.74)] and hence on the relative distance between the approaching drops ... 

Introduction. For many years mass-transfer coefficients, which were based primarily on empirical correlations, have been used in the design of process equipment. A better understanding of the mechanisms of turbulence is needed before we can give a theoretical explanation of convective-mass-transfer coefficients. Some theories of convective mass transfer, such as the eddy diffusivity theory, have been presented in this chapter. In the following sections we present briefly some of these theories and also discuss how they can be used to extend empirical correlations. 

Dissolved insecticides are transferred from soil solution to surface runoff through the concurrent mechanisms of molecular diffusion, raindrop impact induced turbulent diffusion, and shear stress induced mass transfer (63, 64). In addition, shallow interflow may contribute dissolved chemicals to surface runoff as it returns to the surface downslope or seeps into rills and furrows 65). Most studies of dissolved chemical transport into overland flow have employed inorganic tracers such as bromide, gypsum (CaS04 2H20) and 66, 67). The behavior of organophosphorus insecticides, however, is considerably more complex due to association with particulate and colloidal natural organic matter. 

FIGURE 1 9 Fickian transport by mechanical dispersion as water flows through a porous medium such as a soil. Seemingly random variations in the velocity of different parcels of water are caused by the tortuous and variable routes water must follow. This situation contrasts with that of Fig. 1.8, in which turbulence is responsible for the variability of fluid paths. Nevertheless, as in the case of turbulent diffusion, mass transport by mechanical dispersion is proportional to the concentration gradient and can be described by Fick s first law. 

Turbulent Diffusion FDmes. Laminar diffusion flames become turbulent with increasing Reynolds number (1,2). Some of the parameters that are affected by turbulence include flame speed, minimum ignition energy, flame stabilization, and rates of pollutant formation. Changes in flame stmcture are beHeved to be controlled entirely by fluid mechanics and physical transport processes (1,2,9). 

Eddy diffusion as a transport mechanism dominates turbulent flow at a planar electrode ia a duct. Close to the electrode, however, transport is by diffusion across a laminar sublayer. Because this sublayer is much thinner than the layer under laminar flow, higher mass-transfer rates under turbulent conditions result. Assuming an essentially constant reactant concentration, the limiting current under turbulent flow is expected to be iadependent of distance ia the direction of electrolyte flow. 

With good diy scrubbing sorbents, the controlling resistance for gas cleaning is external turbulent diffusion, which also depends on energy dissipated by viscous and by inertial mechanisms. It turns out to Be possible to correlate mass-transfer rate as a fimctiou of the fric tiou Factor. 

Concentration and temperature differences are reduced by bulk flow or circulation in a vessel. Fluid regions of different composition or temperature are reduced in thickness by bulk motion in which velocity gradients exist. This process is called bulk diffusion or Taylor diffusion (Brodkey, in Uhl and Gray, op. cit., vol. 1, p. 48). The turbulent and molecular diffusion reduces the difference between these regions. In laminar flow, Taylor diffusion and molecular diffusion are the mechanisms of concentration- and temperature-difference reduction. 

Lapse Rate and Atmospheric Stability Apart from mechanical interference with the steady flow of air caused by buildings and other obstacles, the most important fac tor that influences the degree of turbulence and hence the speed of diffusion in the lower air is the varia-... 

P. Clavin and F.A. Williams. Effects of molecular diffusion and of thermal expansion on the structure and dynamics of premixed flames in turbulent flows of large scale and low intensity. Journal of Fluid Mechanics, 116 251-282,1982. 

Micromixing comprises the mechanisms of stretching and shrinking of slabs discussed above, accompanied by molecular diffusion, which finally lead to homogenization at the molecular level. Contrary to turbulent macromixing it depends on viscosity. This has been proven experimentally by Bourne et al. (1989). 

There are only scant data on nasal deposition. The available studies reported utilized micron sized particles and the dominant mode of deposition is impaction. This is not the case for the particles considered here and diffusion and turbulent diffusion are the mechanisms of interest. George and... 

Turbulent diffusion flames. Annual Reviews of Fluid Mechanics 21, 101-135. 

Linear-eddy modelling of turbulent transport. Part 3. Mixing and differential diffusion in round jets. Journal of Fluid Mechanics 216, 411 —4-35. 

The vanishing effect of molecular diffusivity on turbulent dispersion Implications for turbulent mixing and the scalar flux. Journal of Fluid Mechanics 359, 299-312. 

---