Particle Eddy Diffusion Coefficient

Assume ihat ihe gas makes a well-defined sei of (urns in the annular space before exiting. Without sped lying the nature of the flow field, the number of turns necessary for complete removal of particles of size dp can be seen from Fig. 4.10 and (4,42) to be 

Smaller particles are removed to an extent that depends on their distance from the wall at the entry. 

The aerodynamic pattern is too complex for an exact analysis of the particle motion, and semi empiric at expressions are used for vaf (Fuchs, 19(S4), An approximate result of the integration of (4.44) often used in design applications has the form 

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

The analysis is similar to that used in Chapter 2 to derive the Stokes-Einstein relation for the diffusion coefficient. Again we consider only the one-dimensional problem. Particles originally present in the differential thickness around, v = 0 (Chapter 2) spread through the fluid a a result of the turbulent eddies. If the particles are much smaller than the size of the eddies, the equation of particle motion for Stokesian particles, based on (4.24) (see associated discussion), is 

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The nature of dispersion. The effect which the solid packing has on the flow pattern within a tubular reactor can sometimes be of sufficient magnitude to cause significant departures from plug flow conditions. The presence of solid particles in a tube causes elements of flowing gas to become displaced randomly and therefore produces a mixing effect. An eddy diffusion coefficient can be ascribed to this mixing effect and becomes superimposed on the transport processes which normally occur in unpacked tubes—either a molecular diffusion process at fairly low Reynolds... 

For turbulence it is convenient to describe particle flux in terms of an eddy diffusion coefficient, similar to a molecular diffusion coefficient. Unlike a molecular diffusion coefficient, however, the eddy diffusion coefficient is not constant for a given temperature and particle mobility, but decreases as the eddy approaches a surface. As particles are moved closer and closer to a surface by turbulence, the magnitude of their fluctuations to and from that surface diminishes, finally reaching a point where molecular diffusion predominates. As a result, in turbulent deposition, turbulence establishes a uniform aerosol concentration that extends to somewhere within the viscous sublayer. Then molecular diffusion or particle inertia transports the particles the rest of the way to the surface. 

Turbulent agglomeration. Far turbulent agglomeration two cases should be considered. First, if the inertia of the aerosol particles is approximately the same as that of the medium, the particles will move about with the same velocities as associated air parcels and can be characterized by a turbulence or eddy diffusion coefficient DT. This coefficient can have a value 104 to 106 times greater than aerosol diffusion coefficients. Turbulent agglomeration processes can be treated in a manner similar to conventional coagulation except that the larger diffusion coefficients are used. 

The conditions will only be maintained uniformly across the section of the bed if there is good lateral dispersion. This is often represented by a radial eddy diffusion coefficient In a packed bed division and recombination of streams around the particles promote this radial dispersion, and this has been analyzed as a random walk problem to predict a radial Peclet number Pe = of about 8. For Reynolds numbers above 10, a value... 

Numerical simulation of the eddy diffusion of particles in the turbulent core of a pipe flow indicates that for particles smaller than about 170 rm, particle and gas eddy diffusion coefficients are about the same (Uijltewaal, 1995). The studies were made for three Reynolds numbers 5500,18,3(X), and 42,000 with particles of about unit density and a pipe diameter of 5 cm. Hence for the usual ranges of interest in aero.sol dynamics, particle and gas eddy diffusion coefficients can be a.ssumed equal in the turbulent core. However, the viscou.s sublayer near the wall of a turbulent pipe flow alters the situation as discussed in the next section. 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

An important mixing operation involves bringing different molecular species together to obtain a chemical reaction. The components may be miscible liquids, immiscible liquids, solid particles and a liquid, a gas and a liquid, a gas and solid particles, or two gases. In some cases, temperature differences exist between an equipment surface and the bulk fluid, or between the suspended particles and the continuous phase fluid. The same mechanisms that enhance mass transfer by reducing the film thickness are used to promote heat transfer by increasing the temperature gradient in the film. These mechanisms are bulk flow, eddy diffusion, and molecular diffusion. The performance of equipment in which heat transfer occurs is expressed in terms of forced convective heat transfer coefficients. 

Equation (7) showed that the term for mobile phase mass transfer was an indefinite combination of a several factors, including particle size dP, column diameter dc, and the diffusion coefficient DM in the mobile phase. It includes the old concept of eddy diffusion, as noted earlier, and it is significantly important only in LC. It is indeed complex, and only the most important parameters will be discussed here the paper by Hawkes6 should be consulted for further details. 

It will be useful to develop an understanding of the relationship between turbulent flow field and dispersion. Measurement techniques need to be developed for the measurement of turbulent flows in multiphase systems. The relationship between eddy diffusion and the dispersion coefficient needs to be brought out over a wide range of particle sizes, setthng velocities, column diameters, column heights, phase velocities, and physical properties. 

The velocity-independent term A characterises the contribution of eddy (radial) diffusion to band broadening and is a function of the size and the distribution of interparticle channels and of possible non-uniformiiies in the packed bed (coefficient A.) it is directly proportional to the mean diameter of the column packing particles, dp. The term B describes the effect of the molecular (longitudinal) diffusion in the axial direction and is directly proportional to the solute diffusion coefficient in the mobile phase, D, . The obstruction factor y takes into account the hindrance to the rate of diffusion by the particle skeleton. 

Resolution and separations in SEC are described by the van Deemter equation (Eq. 14.7). The main contributions to zone broadening, in order of importance, are as follows. The first factor involves the kinetics of partitioning between the mobile phase and the gel there is a limited rate at which equilibrium can be established, and this rate depends on the solute s diffusion coefficient. Second, differences in the lengths of different stream paths in the packed bed of irregularly shaped particles result in eddy diffusion with an ideally packed column, this is not a significant problem, but in practice, eddy diffusion may be significant. Finally, longitudinal... 

The bulk diffusion coefficient of lipase was estimated [55]. The dispersion coefficient is used to characterize the axial dispersion in a packed bed. This parameter accounts for the dispersion due to molecular diffusion as well as eddy diffusion due to velocity differences around the particles. A correlation used to estimate the dispersion coefficient Dm in fixed beds was developed by Chung and Wen [56]. 

Taylor eddy diffusion coefiicieni for the turbiijeni iliiid (Goldstein. 1938, p. 217). However, the correlation coefficient in (4.54) applies to the gas velocities over the path of the particle. Heavy particles move slowly and cannot follow the fluid eddies that surge around them. Thus the time scale that should be employed in (4,54) ranges between the Lagrangian scale for small particles that follow the gas and the Eulerian lime scale for heavy particles that remain almost fixed (Fricdlander, 1957),... 

The A term corresponds to the eddy diffusion which describes the irregular flow through the packed particles in a column causing different pathways and different exit times for the solute molecules. The B term is the longitudinal molecular diffusion or random diffusion along the column. The last term C, corresponds to the mass transfer in the stationary phase. This mass transfer occurs between the mobile and stationary phase of the chromatographic system and is dependant on several factors such as particle size, column diameter and diffusion coefficient. 

According to Prandtl (95), the particle diffusion coefficient is related to the liquid momentum eddy diffusivity e/ defined in terms of the turbulent shear stress (r0) and the time-average strain rate (7 )... 

The A term depends on eddy dispersion and is proportional to the diameter of the particles, axial diffusion constant B is proportional to the molecular diffiisivity D of the solute. The mass transfer effect, the C terms, includes mass transfer outside the particles—Cm proportional to dpDu and mass transfer in the solid or the coated liquid phase—C5 proportional to dJ/Dg, where is the ffim ffikkness and is the diffusion coefficient in the stationary phase. 

The first term in Eq. 3, ApP, includes the contribution of eddy diffusion to band broadening as well as that of mass transfer of the solute through the mobile phase. This contribution of this mobile-phase mass transfer to this term, //, increases with the square of the stationary-phase particle diameter d. It is also inversely proportional to the diffusion coefficient of the solute in the mobile phase, D, according to... 

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