Dispersion coefficients concentration fluctuations

Figure 5-3 shows the effect of the dispersion coefficient on the tracer distribution in a soil column. Of course, the figure shows a 10-fold increase in dispersion coefficient quickly dampens out fluctuations in the concentration distribution curve leading to a near steady-state concentration profile for x > 200 cm. By con-... 

The QELS method can be utilized to evaluate Z)(oo) of a single polymer in concentrated solutions. To this end, chemically labeled chains are dispersed at a very low concentration in a given concentrated solution of unlabeled chains of the same or different species and 5(fc, r) data for the labeled species are analyzed by the method established for dilute solutions. QELS measurements on concentrated solutions in which polymer chains entangle with one another usually reveal a decay of 5(fc, r) much faster than that observed in dilute solutions. This decay is associated with the relaxation of local concentration fluctuations, and defines a dynamic quantity called the cooperative diffusion coefficient and discussed in Chapter 7. 

The most commonly used technique for determining 5 is photon correlation spectroscopy (PCS) [also known as quasi-elastic light scattering (QELS)]. PCS has become one of the standard tools of the trade for the colloid chemist. In this technique concentration fluctuations arising from the diffusive motion of the dispersion particles give rise to fluctuations in the dielectric constant of the medium are monitored photometrically. These fluctuations decay exponentially with a time constant related to the diffusion coefficient, Ds, of the scatterer, which can in turn be related to its hydrodynamic radius through the Stokes-Einstein equation ... 

The model of Eqn. (10-36) is based on flie assumption that mixing of species A in the direction of flow (z-direction) is proportional to the concentration gradient (dCA/dz), i.e., mixing is Fickian in nature. The parameter D is known as flie axial dispersion coefficient. In general, D is not flie same as flie molecular diffusion coefficient Da. Except in rare cases, D is much greater than flie molecular diffusion coefficient because D includes flie effects of radial and temporal velocity fluctuations, in addition to molecular diffusion. 

In earlier sections of this chapter we saw how components in smoke plumes and pipelines sometimes spread much more rapidly than expected. The concentrations of these component pulses could be described by diffusion equations but by using new dispersion coefficients. In turbulent flow, these dispersion coefficients were the result of coupled fluctuations of concentration and velocity. 

Dispersion can also occur in laminar flow but for completely different reasons. This is not surprising because laminar flow has no sudden concentration or velocity fluctuations. In this section we discuss one example of dispersion in laminar flow. This leads to an accurate prediction of the dispersion coefficient. Tliis particular example is so instructive that it is worth including in detail. 

The reason for this inconsistency is that long-range fluctuations dominate behavior near any spinodal limit, including a consolute point. When fluctuations of concentration and of fluid velocity couple, diffusion occurs. Under ordinary conditions, the concentration fluctuations are dominated by motion of single molecules, but near the critical point, these fluctuations exist even when the average fluid velocity is zero. The result is like a turbulent dispersion coefficient but without flow. 

It turns out that turbulent diffusion can be described with Fick s laws of diffusion that were introduced in the previous section, except that the molecular diffusion coefficient is to be replaced by an eddy or turbulent diffusivity E. In contrast to molecular diffusivities, eddy dififusivities are dependent only on the phase motion and are thus identical for the transport of different chemicals and even for the transport of heat. What part of the movement of a turbulent fluid is considered to contribute to mean advective motion and what is random fluctuation (and therefore interpreted as turbulent diffusion) depends on the spatial and temporal scale of the system under investigation. This implies that eddy diffusion coefficients are scale dependent, increasing with system size. Eddy diffusivities in the ocean and atmosphere are typically anisotropic, having much large values in the horizontal than in the vertical dimension. One reason is that the horizontal extension of these spheres is much larger than their vertical extension, which is limited to approximately 10 km. The density stratification of large water bodies further limits turbulence in the vertical dimension, as does a temperature inversion in the atmosphere. Eddy diffusivities in water bodies and the atmosphere can be empirically determined with the help of tracer compounds. These are naturally occurring or deliberately released compounds with well-estabhshed sources and sinks. Their concentrations are easily measured so that their dispersion can be observed readily. 

Scalar isotropic pressure Pg in the continuous phase approximately equals the mean fluid pressure, and particulate stresses P, are expressible through derivatives of w and scalar isotropic pressure p, in the dispersed phase in accordance with Equation 4.4. Pressure p, is a function of suspension volume concentration and of particle fluctuation temperature defined by equation of state (4.6) for particulate pseudogas. Osmotic pressure function G(())) appearing in Equation 4.6 is given by either Equation 4.8, 4.9, or by some other equation that follows from some other statistical pseudo-gas theory. Dispersed phase dynamic viscosity coefficient p, and particle fluctuation energy transfer coefficient q, that appear in Equation 4.4 also can be represented as functions of fluctuation temperature T and concentration < > in conformity with the formulae in Equations 5.5 and 5.7. Force nf of interphase interaction per unit suspension volume approximately equals the force in Equation 3.2 multiplied by the particle number concentration. Finally, coefficients and a are determined in Equation 4.11 and 4.12, respectively. 

Here, to and k are the fluctuation frequency and wave-number, respectively, H(x) designates the Heaviside step function, D is the particle self-diffusion coefficient identified in Equation 4.5 and Equation 4.7, and is understood as the maximal wave number possible in a dispersion containing spherical particles of radius a, particle volume concentration being equal to ([). 

---