Diffusive Boundary Layer and Turbulence

The Hayduk-Laudie relationship, described in Chapter 3, allows us to compute a value of Dta = 5 X 10 m /s. We can determine various values of Pt/Pf, as given in Table E8.5.1. 

All that is left is to determine the equilibrium partial pressure for toluene, Pf  

As long as the water concentration is 55g/m, the equilibrium partial pressure of toxaphene will be 7.97 x 10 atm. 

The edge of the concentration boundary layer is typically determined to be where Pt/Pf = 0.01. That would be where erf [z/(4 = 0-99 or where z = 

66(Dtaty. The thickness of the concentration boundary layer, 8c, is also given in Table E8.5.1 for each time. At 15 min and 28 hrs, of course, the value of 8c is not realistic because the atmosphere is not quiescent (stagnant), and turbulence would mix the toxaphene into the surrounding air. 

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The eddy diffusion coefficients that we introduced in Chapter 5 were steady quantities, using mean turbulence quantities (e.g., the temporal mean of u C). This temporal mean character of eddy diffusion coefficients can be misleading in determining the thickness of a diffusive boundary layer because of the importance of unsteady characteristics. We will review some conceptual theories of mass transfer that have been put forward to describe the interaction of the diffusive boundary layer and turbulence. 

Liu Z, Dreybrodt W. Dissolution kinetics of calcium carbonate minerals in H20-C02 solutions in turbulent flow—the role of the diffusion boundary layer and the slow reaction H20 + CO2 <-> H+ + HC03. Geochim Cosmochim Acta 1997 61(14) 2879-2889. 

From this equation, the dependency of the mass transfer coefficient yff,- on the diffusion coefficient D, and the boundary layer thickness d of the fluid flow, may be seen. The laminar boundary layer and turbulent bulk cannot be distinguished exactly, due to the continous transition the boundary layer thickness 3 is, therefore, a formal complementary variable. 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

The slope of the lines presented in Figure 5 is defined as k(q/v). The q/v term defines the turnover of the tank contents or what is commonly referred to as the retention time. When q is increased, the liquid contacts the carbon more often and the removal of pesticides should increase, however, the efficiency term, k, can be a function of q. As the waste flow rate is increased, the fluid velocity around each carbon particle increases, thereby increasing system turbulence and compressing the liquid boundary layer. The residence time within the carbon bed is also decreased at higher liquid flow rates, which will reduce the time available for the pesticides to diffuse from the bulk liquid into the liquid boundary layer and into the carbon pores. From inspection of Table II, the pesticide concentration also effects the efficiency factor, k can only be determined experimentally and is valid only for the equipment and conditions tested. 

The Reynolds number is the ratio of inertial to viscous forces and depends on the fluid properties, bulk velocity, and boundary layer thickness. Turbulence characteristics vary with Reynolds number in boundary layers [40], Thus, variation in the contributing factors for the Reynolds number ultimately influences the turbulent mixing and plume structure. Further, the fluid environment, air or water, affects both the Reynolds number and the molecular diffusivity of the chemical compounds. 

In Example 8.5, we saw how the diffusive boundary layer could grow. The boundary never achieves 1 m in thickness or even 5 cm in thickness, because of the interaction of turbulence and the boundary layer thickness. The diffusive boundary layer is continually trying to grow, just as the boundary layer of Example 8.5. However, turbulent eddies periodically sweep down and mix a portion of the diffusive boundary layer with the remainder of the fluid. It is this unsteady character of the turbulence... 

Figure 3.3. Various features of diffusion and convection associated with crystal growth in solution (a) in a beaker and (b) around a crystal. The crystal is denoted by the shaded area. Shown are the diffusion boundary layer (db) the bulk diffusion (D) the convection due to thermal or gravity difference (T) Marangoni convection (M) buoyancy-driven convection (B) laminar flow, turbulent flow (F) Berg effect (be) smooth interface (S) rough interface (R) growth unit (g). The attachment and detachment of the solute (solid line) and the solvent (open line) are illustrated in (b).

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

The leaf resistance is in series with that of the boundary layer, Thus the total resistance for the flow of water vapor from the site of evaporation to the turbulent air surrounding a leaf (r al) is 4- (see Fig. 8-5). We now must face the complication created by the two leaf surfaces representing parallel pathways for the diffusion of water vapor from the interior of a leaf to the surrounding turbulent air. We will represent the leaf resistance for the pathway through the upper (adaxial) surface by rj afu and that for the lower (abaxial) surface by r af. Each of these resistances is in series with that of an air boundary layer—and for the upper and the lower leaf surfaces, respectively. The parallel arrangement of the pathways through the upper and the lower surfaces of a leaf leads to the following total resistance for the diffusion of water vapor from a leaf ... 

In a laminar flow, the fluid moves in smooth layers or films. There is relatively little mixing and, consequently, the velocity gradients are small and the shear stresses are low. The thickness of the laminar boundary layer increases with the distance from the start of the boundary layer and decreases with Re. For the laminar boundary layer, the average width of the diffusion layer, 8, is approximately Re 1/2, but it is Re 1 / 5 for a turbulent flow. 

Reverse osmosis is a cross-flow process and, as in any dynamic hydraulic process, the fluid adjacent to the membrane moves slower than the main stream. While the main stream flow may be turbulent, the layer next to the membrane surface is laminar. This thin, laminar flow film is called the boundary layer. When water permeates through the membrane, nearly all of the salt remains behind in the boundary layer next to the membrane. The salt must then diffuse across the boundary layer and back into the bulk stream. This results in a boundary layer with a salt concentration which is more concentrated than the bulk stream. The effect has been termed concentration polarization, and it is defined by the following equation ... 

A fundamental problem in performing a turbulent flow analysis involves determining the eddy diffusivities as a function of the mean properties of the flow. Unlike the molecular diffusivities, which are strictly fluid properties, the eddy diffusivities depend strongly on the nature of the flow they can vary from point to point in a boundary layer, and the specific variation can be determined only from experimental data. 

In considering the volatilization of contaminants, the factors that must be considered are (a) escape from the interface, (b) diffusion through the surface boundary layer, and (c) turbulent diffusion in the atmosphere. - The escape from the surface depends mainly on the vapor pressure of the contaminant at a given temperature, the molecular weight, and Henry s coefficient. After the contaminant has escaped from the surface, it must diffuse outward in the stagnant boundary layer that is normally present. Then, the contaminant will be transported away from the stagnant layer by advection and turbulent diffusion... 

In the vegetated area, the turbulence diffusivity exists only in the bed roughness boundary layer, and the kinematic eddy viscosity will be formulated as follows ... 

In the regions where a turbulent boundary layer and a viscous sublayer exist, the calculations for the transfer coefficients depend upon the expression chosen for the variation of the eddy diffusivities with distance from the wall. Alternatively, a great reliance is put upon experimental measurements. Results will be summarized later. o... 

At 700°C and 1 atm this leads to a diffusion constant of 0.81 cm /s The flow field around a superheater tube is very complex involving both laminar and turbulent boundary layers and the estimation of the local boundary layer thicknesses (velocity, diffusion and thermal boundary layers) around the tube requires computer simulations with computational fluid dynamic (CFD) software packages. However, for this rough analysis an average value of the thermal boundary layer thickness around the tube is enough and can be estimated if the average Nusselt number around the tube is known... 

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