Mixing by molecular diffusion

Two large reservoirs of gas are connected by a pipe of length 2L with a full-bore valve at its midpoint. Initially a gas A fills one reservoir and the pipe up to the valve and gas B fills the other reservoir and the remainder of the pipe. The valve is opened rapidly and the gases in the pipe mix by molecular diffusion,... 

Mixing by molecular diffusion. This is the ultimate and finally the only process really able to mix the components of a fluid to the molecular scale. The time constant for this process is the diffusion time t = yLy is a shape factor and L is the ratio of the volume to the external surface area of the particle. For instance let us consider various shapes slabs (thickness 2R, case of lamellar structure with striation thickness 6 = 2R), long cylinders (diameter 2R, case of filamentous structure), and spheres (diameter 2R, case of spherical aggregates). 

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

The characteristic time of micromixing (mixing by molecular diffusion) can be, in this case, found in the following equation ... 

Let us consider an aggregate which is subjected to a uniform flow with no velocity gradient (Figure 6.1a). All the points of the aggregate are converted with the same velocity and there is no deformation of the structure. In that case, the flow has no effect on mixing and the problem is brought back to the previous one of mixing by molecular diffusion. 

As molten polymers are high viscosity flnids, they cannot be readily subjected to turbulent flow, which otherwise would be a very efficient method of mixing from a practical standpoint. Neither will they readily mix by molecular diffusion like gases and low viscosity liqnids. 

In the case of high differences between two timescales r and to, either Dan S> 1 or Dan 1 is possible. In the case of Dan 3> 1 (to 3> ), the processes are referred to as diffusion limited, that is, diffusion rate determines the reaction rate. In the case of Dan -C 1 (Jr 3> to), the reaction partners can be mixed by molecular diffusion fast enough so that the reaction is not mixing influenced, but reaction limited. If both timescales are nearly the same (i.e., Dan 1). then diffusion and reaction compete with each other, that is, the reaction depends on the mixing intensity. These considerations can be transferred to micromixing in a microreactor. The mixing time (, ) is defined as... 

Micromixing Mixing among molecules of different ages (i.e., mixing between macrofluid clumps). Mixing on a scale smaller tlian tlie minimum eddy size or minimum striation diickness by molecular diffusion. 

For a chemical reaction such as combustion to proceed, mixing of the reactants on a molecular scale is necessary. However, molecular diffusion is a very slow process. Dilution of a 10-m diameter sphere of pure hydrocarbons, for instance, down to a flammable composition in its center by molecular diffusion alone takes more than a year. On the other hand, only a few seconds are required for a similar dilution by molecular diffusion of a 1-cm sphere. Thus, dilution by molecular diffusion is most effective on small-scale fluctuations in the composition. These fluctuations are continuously generated by turbulent convective motion. 

We first explain the setting of reactors for all CFD simulations. We used Fluent 6.2 as a CFD code. Each reactant fluid is split into laminated fluid segments at the reactor inlet. The flow in reactors was assumed to be laminar flow. Thus, the reactants mix only by molecular diffusion, and reactions take place fi om the interface between each reactant fluid. The reaction formulas and the rate equations of multiple reactions proceeding in reactors were as follows A + B R, ri = A iCaCb B + R S, t2 = CbCr, where R was the desired product and S was the by-product. The other assumptions were as follows the diffusion coefficient of every component was 10" m /s the reactants reacted isothermally, that is, k was fixed at... 

At high Reynolds numbers, we can again expect the small scales of the scalar field to be nearly isotropic. In the classical picture of turbulent mixing, one speaks of scalar eddies produced at large scales, with a distinct directional orientation, that lose their directional preference as they cascade down to small scales where they are dissipated by molecular diffusion. 

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

Bakd et al. theoretically analyzed simultaneous gas flow and diffusion in Weibel s symmetric model. Th applied a time-varying flow with simultaneous longitudinal diffusion and concluded that convective mixing is much less important than mixing induced by molecular diffusion. 

The concept of separate regions of dilution by molecular diffusion and turbulent mixing is of major importance in understanding the exchange of gases at land surface and for the identification of physical factors that impede the dispersion process under various microclimates. 

Dispersion models, as just stated, are useful mainly to represent flow in empty tubes and packed beds, which is much closer to the ideal case of plug flow than to the opposite extreme of backmix flow. In empty tubes, the mixing is caused by molecular diffusion and turbulent diffusion, superposed on the velocity-profile effect. In packed beds, mixing is caused both by splitting of the fluid streams as they flow around the particles and by the variations in velocity across the bed. 

Transport of gas to the surface. Assuming mixing occurs by molecular diffusion rather than by mechanical or convective processes, the characteristic times for gas-phase diffusion to the surface are in the range 10 l(l-10-4 s for droplets with radii from 10 5 to 10 2 cm, respectively. 

Transport of the dissolved species within the aqueous phase. Because diffusion in liquids is much slower than in gases, the characteristic times for diffusion within the droplet itself are much greater (by about four orders of magnitude) than for diffusion of the gas to the droplet surface (again assuming mixing only by molecular diffusion). Thus the times are —10 f>—1 s... 

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