Analogy Between Transport Diffusivities

The analogy between the fluctuating flux and diffusivities is obvious. The similarity of k—s, T — sr and — Bc models demonstrates that the fluctuation variance-dissipation pattern is the common methodology for closing the transport equation. Starting from this viewpoint, a unified model of computational transport has been suggested by Liu [5] as shown in subsequent section. Notice should be made that in spite of some newer CFD model is 

The coefficients C, Cto, and Cco are given differently by different authors the commonly accepted values are = 0.09, Cto = 0.11, Cco = 0.11 or 0.14, although slight change on these values does not give substantial difference in final simulated result. 

Moreover, the turbulent diffusivities, V(, oq, and D, obtained by the two-equation model as given above are applicable to aU directions, and therefore, they are isotropic. 

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The radial dispersion coefficient for this case is, of course, the average eddy diffusivity as discussed in works on turbulence (H9). If the various analogies between momentum, heat, and mass transport are used. 

Chapter 4 Mass, Heat, and Momentum Transport Analogies. The transport of mass, heat, and momentum is modeled with analogous transport equations, except for the source and sink terms. Another difference between these equations is the magnitude of the diffusive transport coefficients. The similarities and differences between the transport of mass, heat, and momentum and the solution of the transport equations will be investigated in this chapter. 

The immediate result of the above discussion is that the diffusion equation can be transformed into the differential equation for heat conduction by substitution of c by T and D by k. This analogy has the consequence that practically all mathematical solutions of the heat conductance equation are applicable to the diffusion equation. The analogy between diffusion and conductance should be kept in mind in the following discussion although the topic here will be mainly the treatment of the diffusion equation, which represents the most important process of mass transport. 

Equation 7.146 for the utilization factor corresponds to 7.107 for the case of heterogeneous catalysis with diffusional limitations. The analogy between 7.146 and 7.107 is complete when Shm = 1, i.e. when the reaction occurs simultaneously with diffusion throughout the complete liquid volume. The presence of a Sherwood number, besides the Hatta number, is needed to describe situations where a significant part of the reaction occurs in the bulk of the liquid, i.e. in series with the transport through the film. Such a situation is often encountered. Typical values for the Sherwood number are ... 

There is an important analogy between the Fokker-Planck-Kolmogorov equation and the property transport equation. Indeed, the term which contains A(t,x) describes the particle displacement by individual processes and the term which contains D(t,x) describes the left and right movement in each individual displacement or diffusion. We can notice the very good similarity between the transport and the Kolmogorov equation. In addition, many scientific works show that both... 

In conjunction with heat conduction we will also investigate mass diffusion. As a result of the analogy between these two molecular transport processes many results from heat conduction can be applied to mass diffusion. In particular the mathematical methods for the evaluation of concentration fields agree to a large extent with the solution methods for heat conduction problems. 

It is the fluctuating element of the velocity in a turbulent flow that drives the dispersion process. The foundation for determining the rate of dispersion was set out in papers by G. 1. Taylor, who first noted the ability of eddy motion in the atmosphere to diffuse matter in a manner analogous to molecular diffusion (though over much larger length scales) (Taylor 1915), and later identified the existence of a direct relation between the standard deviation in the displacement of a parcel of fluid (and thus any affinely transported particles) and the standard deviation of the velocity (which represents the root-mean-square value of the velocity fluctuations) (Taylor 1923). Roberts (1924) used the molecular diffusion analogy to derive concentration profiles... 

It is also possible (at least, in a formal way) to apply a fractional version of Richards equation to simulate one-dimensional water transport in horizontal columns. We were able to fit the FADE to data on horizontal water infiltration (data not shown). However, the parameter a had to be set to values greater than two to fit the experimental data. This range of a is theoretically unjustified (Benson et al., 1999 Meerschaert et al., 1999). This example serves as a reminder about the danger of drawing analogies between water and solute transport in soils, since the underlying physical processes are different, Particles of soil water moving faster than others are affected by the structure of pore surfaces and move in films rather than in bulk volume by convection. One possible way to model the water transport is to use the diffusivity model proposed by Jumarie (1992) ... 

Once again, 19a, mix and tc represent molecular transport properties on the right sides of these equations, but the analogy between Sc and Pr, as well as between the mass and heat transfer Peclet numbers (i.e.. Re Sc vs. Re Pr) is based on diffusivities. 

Once again, mass diffusivity J0a, mix and thermal conductivity tc in these expressions represent molecular transport properties via Pick s and Fourier s law, respectively. However, the fluid properties that appear in Sc and St should be interpreted as diffusivities, not molecular transport properties. In terms of the analogies between heat and mass transfer, sometimes 30A,mix represents a diffusivity, and other times it represents a molecular transport property. This ambiguity does not exist in the corresponding expressions for heat transfer. In general, 30a, mix represents a diffusivity in the mass transfer equation and in expressions for the boundary layer thickness Sc. 

As shown in Chapter 2, transfer of heat by conduction is due to random molecular motions, and there is an obvious analogy between the two processes of heat conduction and of mass transport by diffusion. Pick in 1855 recognized this fact, and put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Pourier (1822). 

This contribution arises exclusively from the transport in the fluid and corresponds to the mixing that is described by the efEective diffusion in radial direction, D . When the analogy between heat and mass transfer is complete the following relation may be written ... 

In general, exchange reactions should be analyzed with respect to both transport between micelles (diffusion) and expulsion/insertion. In complete analogy with chemical reaction kinetics known in physical chemistry (see, e.g., Atkins [59]) this can be formulated as ... 

Turbulent diffusivity based closure models for the scalar fluxes describing turbulent transport of species relate the scalar flux to the mean species concentration gradient according to Reynolds analogy between turbulent momentum and mass transport. The standard gradient-diffusion model can be written ... 

Our treatment so far has made occasional reference to heat transfer, primarily to draw the reader s attention to tiie analogies that exist between the transport of heat and mass. For example, in Chapter 1 we highlighted the similarities between the rate laws governing convective and diffusive heat and mass transfer. The analogy between tiie two phenomena when dealing witii co-current or countercurrent operations has been brought out on several occasions, notably Illustration 8.7. 

Analogies between the transport phenomena can be developed from the velocity, concentration, and temperature distributions from which the eddy diffusivities of Eqs. (3.34) to (3.38) were obtained. These data are not extremely... 

Abstract In this chapter, the two CMT models, i.e., c — Eci model and Reynolds mass flux model (in standard, hybrid, and algebraic forms) are used for simulating the chemical absorption of CO2 in packed column by using MEA, AMP, and NaOH separately and their simulated results are closely checked with the experimental data. It is noted that the radial distribution of Di is similar to a, but quite different from fit. It means that the conventional assumption on the analogy between the momentum transfer and the mass transfer in turbulent fluids is unjustifled, and thus, the use of CMT method for simulation is necessary. In the analysis of the simulation results, some transport phenomena are interpreted in terms of the co-action or counteraction of the turbulent mass flux diffusion. 

It is seen that we are comparing kinematic viscosity, thermal diffusivity, and dif-fusivity of the medium for both air and water. In air, these numbers are all of the same order of magnirnde, meaning that air provides a similar resistance to the transport of momenmm, heat, and mass. In water, there is one order of magnitude or more difference between kinematic viscosity, thermal diffusion coefficient, and mass diffusion coefficient. Also provided in Table 9.1 are the Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat and mass transport are more precise for air than for water, and the techniques applied to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are on the liquid side of the interface. 

Mass Transfer A nalogy It is obvious by comparison that there is a direct analogy between heat and mass hansfer Each has a developing region, each can be transported by direct molecular collision (conduction or diffusion), and each can be transported by fluid motion (convection). Indeed, Fourier s law of heat conduction is mathematically identical to Pick s law of diffusion. 

In this section the analogy between heat and mass transfer is introduced and used to solve problems. The specific estimation relationships for permeants in polymers are discussed in Section 4.2 with the emphasis placed on gas-polymer systems. This section provides the necessary formulas for a first approximation of the diffusivity, solubility, and permeability, and their dependence on temperature. Non-Fickian transport, which is frequently present in high activity permeants in glassy polymers, is introduced in Section 4.3. Convective mass transfer coefficients are discussed in Section 4.4, and the analogies between mass and heat transfer are used to solve problems involving convective mass transfer. Finally, in Section 4.5 the solution to Design Problem III is presented. 

The rapid transport of the linear, flexible polymer was found to be markedly dependent on the concentration of the second polymer. While no systematic studies were performed on these ternary systems, it was argued that the rapid rates of transport could be understood in terms of the dominance of strong thermodynamic interactions between polymer components overcoming the effect of frictional interactions this would give rise to increasing apparent diffusion coefficients with concentration 28-45i. This is analogous to the resulting interplay of these parameters associated with binary diffusion of polymers. 

In addition to overall mass conservation, we are concerned with the conservation laws for individual chemical species. Beginning in a way analogous to the approach for the overall mass-conservation equation, we seek an equation for the rate of change of the mass of species k, mk. Here the extensive variable is N = mu and the intensive variable is the mass fraction, T = mk/m. Homogeneous chemical reaction can produce species within the system, and species can be transported into the system by molecular diffusion. There is convective transport as well, but it represented on the left-hand side through the substantial derivative. Thus, in the Eulerian framework, using the relationship between the system and the control volume yields... 

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