Turbulent diffusivity boundary condition

In summary, for a non-stationary turbulent reacting flow wherein all scalars can be assumed to have identical molecular diffusivities and the initial and boundary conditions are uniform/constant, the -component molar concentration vector... 

Table 5.1 shows that, with the boundary conditions present in most environmental flows (i.e., the Earth s surface, ocean top and bottom, river or lake bottom), turbulent flow would be the predominant condition. One exception that is important for interfacial mass transfer would be very close to an interface, such as air-solid, solid-liquid, or air-water interfaces, where the distance from the interface is too small for turbulence to occur. Because turbulence is an important source of mass transfer, the lack of turbulence very near the interface is also significant for mass transfer, where diffusion once again becomes the predominant transport mechanism. This will be discussed further in Chapter 8. 

It should be recognized that the boundary conditions of the problem will establish the value of the hydrodynamic velocity, u. In the case of most turbulent flows the indirect influence of molecular diffusion on the hydro-dynamic velocity can be neglected. It should be emphasized that the hydrodynamic velocity is the time-average point velocity in Reynolds sense (R2). Under unsteady, nonuniform conditions of flow between parallel plates the material balance may be expressed for turbulent flow in the following form ... 

An alternative approach, developed by chemical engineers as well, is the surface renewal model by Higbie (1935) and Danckwerts (1951). It applies to highly turbulent conditions in which new surfaces are continuously formed by breaking waves, by air bubbles entrapped in the water, and by water droplets ejected into the air. Here the interface is described as a diffusive boundary. 

To understand the principal idea of Deacon s model we have to remember the key assumption of the film model according to which a bottleneck boundary is described by an abrupt drop of diffusivity, for instance, from turbulent to molecular conditions (see Fig. 19.3a). Yet, theories on turbulence at a boundary derived from fluid dynamics show that this drop is gradual and that the thickness of the transition zone from fully turbulent to molecular conditions depends on the viscosity of the fluid. In Whitman s film model this effect is incorporated in the film thicknesses, 8a and 8W (Eq. 20-17). In addition, the film thickness depends on the intensity of turbulent kinetic energy production at the interface as, for instance, demonstrated by the relationship between wind velocity and exchange velocity (Figs. 20.2 and 20.3). 

Transport (advection and diffusion) of tracers (both passive and reactive) is performed on-line at each meteorological time-step using WAF scheme for advection and a true (second order) diffusion, with diffusion coefficient carefully estimated from experiments (Tampieri and Maurizi 2007). Vertical diffusion is performed using ID diffusion equation with a diffusion coefficient estimated by means of an k-l turbulence closure scheme. Dry deposition is computed through the resistance-analogy scheme and is provided as a boundary condition to the vertical diffusion equation. Furthermore, vertical redistribution of tracers due to moist convection is parameterized consistently with the Kain-Frisch scheme used in the meteorological part for moist convection. Transport of chemical species is performed in mass units while gas chemistry is computed in ppm. 

At low Re, the viscous effects dominate inertial effects and a completely laminar flow occurs. In the laminar flow system, fluid streams flow parallel to each other and the velocity at any location within the fluid stream is invariant with time when boundary conditions are constant. This implies that convective mass transfer occurs only in the direction of the fluid flow, and mixing can be achieved only by molecular diffusion [37]. By contrast, at high Re the opposite is true. The flow is dominated by inertial forces and characterized by a turbulent flow. In a turbulent flow, the fluid exhibits motion that is random in both space and time, and there are convective mass transports in all directions [38]. 

The work of Higbte21 in 1935 provided the foundation for the penemoton theory, which supposes that turbulence transports eddies from the bulk of the phase to the interface, where they remain fora short but constant time before being displaced back into the interior of the phase to be mixed with the bulk fluid. Solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion, in accordance with Pick s Second Law and appropriate boundary conditions ... 

Consider the coalescence of drops with fiilly retarded (delayed) surfaces (which means they behave as rigid particles) in a developed turbulent flow of a lowconcentrated emulsion. We make the assumption that the size of drops is much smaller than the inner scale of turbulence R Ao), and that drops are non-deformed, and thus incapable of breakage. Under these conditions, and taking into account the hydrodynamic interaction of drops, the factor of mutual diffusion of drops is given by the expression (11.70). To determine the collision frequency of drops with radii Ri and Ri (Ri < Ri), it is necessary to solve the diffusion equation (11.36) with boundary conditions (11.39). Place the origin of a spherical system of coordinates (r, 0,0) into the center of the larger particle of radius i i. If interaction forces between drops are spherically symmetrical, Eq. (11.36) with boundary conditions (11.39) assumes the form... 

It is well known that multiphase flow in tubular canals is accompanied by fibering. Phase disengagement leads to reduction of specific surface of reacted flows and consequently to additional diffusion limitations under fast chemical and mass-exchange physical process realization under polymers synthesis. This determines the expediency of determination of boundary condition of homogeneous and fibered regions of reaction mixture flow formation in tubular turbulent apparatus. 

In search of a solution including turbulence production by heat convection, we need additional boundary conditions. The first assumption is that energy diffusion and pressure transport cancel each other (Cj = 0). The second assumption is that the ratio between energy... 

This simply assumes that axial dispersion (D m. s ) is superimposed onto plug flow. Axial dispersion may be caused by a velocity profile in the radial direction or statistical dispersion in a packing or turbulent diffusion or by any physicochemical process which delayes some particles with respect to others. The model parameter is the axial PECLET number, Pe = uL/D, or its reciprocal, the dispersion number, D /uL. Depending on the boundary conditions assumed at the reactor inlet and outlet (which are different from those of the simple assumptions above), a lot of mathematical formulae can be found in the literature for the RTD [3]. This is often academic as in the range of usefulness of the model (small deviation from plug flow, say Pe > 20) all conditions lead to res-... 

As shown in Fig. 6.17a, Dt is high at x = 0.11 and very low at x = 0.17 indicates the turbulent diffusion rate is low at the vicinity of column bottom and soon rapidly increases toward column top. Such tendency is kept in radial direction from column center to about rlR — 0.8 and then diminishes to almost zero toward the column wall. The around column center HR < 0.1) fluctuates obviously may be due to 0C/0y there is almost zero so as to affect the inlet boundary condition. After HR <0.1, Dtj, is almost zero means the turbulent effect in radial direction is negligible. 

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