Dispersion in turbulent flow

5(10cm)(500cm/sec) = 2, 500cm /sec The concentration change is significant when z = /AEt 

About one percent of the pipeline will contain mixed gases. 

We now recognize that dispersion can be described by the mathematics of diffusion but that it requires flow. When such flow exists, dispersion is much faster than diffusion. It has a different physical origin than the small-scale, Brownian motion of molecules. Interestingly, its physical origin is completely different for dispersion in turbulent flow than in laminar flow. 

In this section we discuss the origins of dispersion in turbulent flow. This discussion is especially relevant to problems common in environmental engineering, problems like pollutant dilution in rivers or the spreading of plumes. Not surprisingly, the origin of the effect turns out to be a consequence of turbulent fluctuations in velocity and concentration. The coupling between these fluctuations is the cause of dispersion. In more informal terms, gusts and eddies cause dispersion. 

To show how turbulence affects dispersion, we return to the mass balances developed in general terms in Section 3.4. For example, for flow described in Cartesian coordinates, we have from Table 3.4-2  

---

Sommerfeld, M. (1990), Numerical simulation of the particle dispersion in turbulent flow the importance of particle lift forces and particle/wall collision models, in Numerical Methods for Multiphase Flows, Vol. 91, ASME, New York. 

In this regime the typical distance from the origin of motion increases as the square root of time. Thus, the dispersion in turbulent flows at long times is analogous to molecular diffusion or random walks with independent increments and comparison of Eq. (2.24) with (2.16) relates the turbulent diffusion coefficient, Dt, to the integral of the Lagrangian correlation function, Tl, as... 

A.Berlemont, P.Desjonqueres. A Lagrangian approach for the prediction of particle dispersion in turbulent flows. Third workshop on two phase flow predictions. Belgrade. June 1986. 

Eddy diffusion or turbulent diffusion is dispersion in turbulent flows caused by the motions of large groups of molecules called eddies, this motion is measured as the turbulent velocity fluctuations. The turbulent diffusivity, Dt, is a conceptual analogy to Dab but is a property of the local flow rather than of the fluid. 

In Section 4.1, we give a simple example of dispersion to illustrate the similarities to and differences from diffusion. We discuss dispersion coefficients for environmental and industrial situations in Section 4.2. In Section 4.3, we discuss how diffusion and flow interact to produce dispersion in turbulent flow. In Section 4.4, we make similar calculations for laminant flow. Overall, the material is presented at an elementary level, partly because it is unevenly understood at any other level and partly because more detail seems outside the scope of this book. 

At what velocity will dispersion in turbulent flow be smallest ... 

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

The mechanism of suspension is related to the type of flow pattern obtained. Suspended types of flow are usually attributable to dispersion of the particles by the action of the turbulent eddies in the fluid. In turbulent flow, the vertical component of the eddy velocity will lie between one-seventh and one-fifth of the forward velocity of the fluid and, if this is more than the terminal falling velocity of the particles, they will tend to be supported in the fluid. In practice it is found that this mechanism is not as effective as might be thought because there is a tendency for the particles to damp out the eddy currents. 

In oil and gas well cementing operations, polyethyleneimine phosphonate-derivative dispersants enhance the flow behavior of the cement slurry [422]. The slurry can be pumped in turbulent flow, thereby forming a bond between the well casing and the rock formation. 

If the fluid in the pipe is in turbulent flow, the effects of molecular diffusion will be supplemented by the action of the turbulent eddies, and a much higher rate of transfer of material will occur within the fluid. Because the turbulent eddies also give rise to momentum transfer, the velocity profile is much flatter and the dispersion due to the effects of the different velocities of the fluid elements will be correspondingly less. 

Huang, C. H. (1979). Theory of dispersion in turbulent shear flow. Atmos. Environ. 13,453-463. 

Figures 13.15 and 13.16 show the findings for flow in pipes. This model represents turbulent flow, but only represents streamline flow in pipes when the pipe is long enough to achieve radial uniformity of a pulse of tracer. For liquids this may require a rather long pipe, and Fig. 13.16 shows these results. Note that molecular diffusion strongly affects the rate of dispersion in laminar flow. At low flow rate it promotes dispersion at higher flow rate it has the opposite effect.

Another interesting application of the data in Fig. 2.20 for dispersion coefficients in turbulent flow is in calculating the mixing that occurs in long pipelines. Many refined petroleum products are distributed by pipelines which may extend over hundreds of kilometres. The same pipeline is used to convey several different products, each... 

Leonard, R. A., G. J. Bernstein, R. H. Pelto, and A. A. Ziegler. 1981. Liquid-liquid Dispersion in Turbulent Couette Flow. AIChE J. 27, 495-503. 

More recent work (17) has covered viscous drops in turbulent flow, over a wide range of mixer types (SMV, SMX, SMXL, SMR, SMF), diameters (up to 80 mm), lengths, and dispersed-phase viscosities (up to 70 mPa) ... 

Thus in turbulent flow, the dispersion coefficient is independent of the diffusion coefficient, but in laminar flow, the dispersion coefficient depends inversely on the diffusion coefficient. This counterintuitive inverse dependence, the result of axial convection coupled with radial diffusion, is the foundation of the Goulay equation describing peak spreading in chromatography. We now return from this dispersion tangent back to diffusion and in particular, to mass transfer. 

For the case of horizontal gas mixing, Werther and co-workers [70, 71] have shown that, for the bed solids they used (quartz sand, dp = 0.13 mm, Geldart group B), horizontal gas mixing in the top part of the circulating fluidized bed in the core zone can be described by the model for gas dispersion in turbulent singlephase flow [72]. The Peclet number... 

Hyperbolic Model for Describing Dispersion Effects in Turbulent flow in a Tube... 

The extension of the two-mode axial dispersion model to the case of fully developed turbulent flow in a pipe could be achieved by starting with the time-smoothed (Reynolds-averaged) CDR equation, given by Eq. (117), where the reaction rate term R(C) in Eq. (117) is replaced by the Reynolds-averaged reaction rate term Rav(C), and the molecular diffusivity Dm / is replaced by the effective diffusivity Dej- in turbulent flows given by... 

---