Transport Equation with Turbulent Diffusion Coefficients

Mass Transport Equation with Turbuient Diffusion Coefficients 

In this section, we will derive the most common equations for dealing with mass transport in a turbulent flow. Beginning with equation (2.14), we will take the temporal mean of the entire equation and eventually end up with an equation that incorporates turbulent diffusion coefficients. 

One of the conclusions of our consideration of the flux through a control volume was equation (2.14)  

In a turbulent flow field, equation (2.14) is difficult to apply because C,u,v, and w are all highly variable functions of time and space. Osborne Reynolds (1895) reduced the complexities of applying equation (2.14) to a turbulent flow by taking the temporal 

Equation (5.5), the change of a temporal mean over time, may seem like a misnomer, but it will be left in to identify changes in C over a longer time period than At. Continuing, 

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B. MASS TRANSPORT EQUATION WITH TURBULENT DIFFUSION COEFFICIENTS... 

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. 

We will apply equation (5.20) to solve for the concentration profile of suspended sediment in a river, with some simplifying assumptions. Suspended sediment is generally considered similar to a solute, in that it is a scalar quantity in equation (5.20), except that it has a settling velocity. We will also change our notation, in that the bars over the temporal mean values will be dropped. This is a common protocol in turbulent transport and will be followed here for conformity. Thus, if an eddy diffusion coefficient, e, is in the transport equation,... 

To represent the partially premixed turbulent combustion of a refinery gas in the heater, a combination of the flamelet formulations for premixed and nonpremixed combustion was used [16]. The standard k-e model was used for turbulent flow calculations. The effect of turbulence on the mixture fraction was accounted for by integrating a beta-PDF derived from the local mixture fraction and mixture fraction variance, which were in turn obtained by solving their respective transport equations. A relatively simple approach was used to compute radiant heat transfer—a diffusion model with a constant absorption coefficient (0.1 m i). 

Many chemicals escape quite rapidly from the aqueous phase, with half-lives on the order of minutes to hours, whereas others may remain for such long periods that other chemical and physical mechanisms govern their ultimate fates. The factors that affect the rate of volatilization of a chemical from aqueous solution (or its uptake from the gas phase by water) are complex, including the concentration of the compound and its profile with depth, Henry s law constant and diffusion coefficient for the compound, mass transport coefficients for the chemical both in air and water, wind speed, turbulence of the water body, the presence of modifying substrates such as adsorbents in the solution, and the temperature of the water. Many of these data can be estimated by laboratory measurements (Thomas, 1990), but extrapolation to a natural situation is often less than fully successful. Equations for computing rate constants for volatilization have been developed by Liss and Slater (1974) and Mackay and Leinonen (1975), whereas the effects of natural and forced aeration on the volatilization of chemicals from ponds, lakes, and streams have been discussed by Thibodeaux (1979). 

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