Problem 11.3. Sedimentation Equilibrium

For various particle radii between 1 /u,m and 0.001 /u,m, determine the ratio of equilibrium concentrations, C2/C1, at a distance, x, 1 cm apart. The particles have a density of 2.0 gm/cm and the liquid is water with a viscosity of 0.01 poise (gm/cm/sec). 

Solution Using equation 11.29, we have the following results  

This shows that particles less than —0.01 /itm are subject to sedimentation equilibrimn over a distance of 1 cm. Bi er particles sediment and have a clear sedimentation interface. 

This same equilibrium can be determined for centrilugal sedimentation. The flux due to gravitational sedimentation at this plane is C dRJdt), and the flux due to diflusion is D dCldR. Because these two fluxes are equal at equilibrium, we have the following equation  

Cl and C2 are the concentrations at two points a distance i2,.i toi c2 apart, where sedimentation is at equilibrium. Again prolonged sedimentation will not change these two concentrations due to this equihbrium. 

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The problem at hand with Equation 74 is how to deal with the quantity , and this complication is what has stymied sedimentation equilibrium work with nonideal solutions containing polymeric solutes. This problem has been discussed by Goldberg (29), Wales (6), and Fujita (17). One way to overcome the problem is to assume that... 

The sedimentation equilibrium experiment requires much smaller volumes of solution, about 0.15 ml. With six-hole rotors and multichannel centerpieces (41) it is potentially possible to do fifteen experiments at the same time. For situations where the photoelectric scanner can be used one might (depending on the extinct coefficients) be able to go to much lower concentrations. Dust is no problem since the centrifugal field causes it to go to the cell bottom. For conventional sedimentation equilibrium experiments, the analysis of mixed associations under nonideal conditions may be virtually impossible. Also, sedimentation equilibrium experiments take time, although methods are available to reduce this somewhat (42, 43). For certain situations the combination of optical systems available to the ultracentrifuge may allow for the most precise analysis of a mixed association. The Archibald experiment may suffer some loss in precision since one must extrapolate the data to the cell extremes (rm and r6) to obtain MW(M, which must then be extrapolated to zero time. Nevertheless, all three methods indicate that it is quite possible to study mixed associations. We have indicated some approaches that could be used to overcome problems of nonideality, unequal refractive index increments, and unequal partial specific volumes. 

We are baffled as to the cause of the discrepancy between the calculated and measured intensity curves. The mathematics of the calculated curve needs to be refined to include the width of the slit above the rotor, which we had assumed to be negligible. Perhaps the effect of changing the slit width and the focus plane in the cell should be examined. Based on the good results obtained from sedimentation equilibrium experiments, we believe that the SIT vidicon, MIA electronics, and our software controlling the gathering of data are all behaving properly, but perhaps there are still problems to be solved. 

Buoyancy plays a role during the derivation of the sedimentation equilibrium and also in the broader sense with the ultracentrifuge. Someone said once with respect to the buoyancy that the most difficult problems in physical chemistry are the hydro-dynamic problems. Now, first buoyancy has to do with hydrostatics rather than with hydrodynamics. 

The source term in equation (2.22) requires a separate differential equation for Sp, which would incorporate the concentration of the compound in solution. We would thus have two equations that need to be solved simultaneously. However, most sorption rates are high, relative to the transport rates in sediments and soil. Thus, local equilibrium in adsorption and desorption is often a good assumption. It also simplifies the solution to a transport problem considerably. If we make that assumption, Sp changes in proportion to C alone, or... 

We have already noted that sedimentation and diffusion are opposing processes, the first tending to collect and the second to scatter. Let us now consider the circumstances under which these two tendencies equal each other. Once this condition is reached, of course, there will be no further macroscopic changes the system is at equilibrium. In order to formulate this problem, consider the unit cross section shown in Figure 2.15, in which the x direction is in the direction of either a gravitational or a centrifugal field. Suppose this field tends to pull... 

In these chapters, we focus on equilibrium situations and the associated problem of calculating the distribution of a compound between the different phases, when no net exchange occurs anymore. There are many situations in which it is correct to assume that phase transfer processes are fast compared to the other processes (e.g., transformations) determining a compound s fate. In such cases, it is appropriate to describe phase interchanges with an equilibrium approach. One example would be partitioning of compounds between a parcel of air and the aerosols suspended in it. Another case might be partitioning between the pore water and solids in sediment beds. 

In Section 19.2 we treated the phase problem by choosing a reference system (for instance, water) to which the concentrations of the chemicals in other phases are related by equilibrium distribution coefficients such as the Henry s law constant. Here we employ the same approach. The following derivation is valid for an arbitrary wall boundary with phase change. The mixed system B is selected as the reference system. In order to exemplify the situation, Fig. 19.9 shows the case in which system A represents a sediment column and system B is the water overlying the sediments. This case will be explicitly discussed in Box 19.1. 

Now the technique provides the basis for simulating concentrated suspensions at conditions extending from the diffusion-dominated equilibrium state to highly nonequilibrium states produced by shear or external forces. The results to date, e.g., for structure and viscosity, are promising but limited to a relatively small number of particles in two dimensions by the demands of the hydrodynamic calculation. Nonetheless, at least one simplified analytical approximation has emerged [44], As supercomputers increase in power and availability, many important problems—addressing non-Newtonian rheology, consolidation via sedimentation and filtration, phase transitions, and flocculation—should yield to the approach. 

Calculate the equilibrium between sedimentation and diffusion for the particle given in problem 9 assuming hindered settling at = 50%. 

The material considered in the preceding sections shows that the FB and EB methods can be successfully applied to the solution of these problems providing information not only on the equilibrium but also on the kinetic properties of polymer chains. This is particularly valuable in those cases in which the application of other more common methods is difficult or even impossible (e.g. sedimentation measurements of polymer solutions in sulfuric acid). 

The device used in this technique is a small stainless steel ball-mill containing two stainless steel or ceramic balls which crush and disaggregate the sample when the ball-mill is shaken (Fig. 5-25). This approach concentrates the loosely-bound adsorbed gases into the headspace of the ball-mill. Because of the equilibrium problem mentioned above under headspace techniques, this sampler was adapted by Whelan (1979) and Whelan et al. (1980) to ensure that lithified sediments and cuttings are completely broken up during analysis. 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

In summary, equilibrium calculations based on pore-water composition indicate that Fe carbonate or phosphate are unlikely to form below the top few centimeters of sediment and that siderite formation is unlikely generally. Undersaturations by a factor of A, p/IAP —100 are found. The presence of solid-phase sulfide is evidence for the formation of Fe sulfides. However, pore waters are not always saturated with respect to the common Fe-sulfide minerals and under-or supersaturations by a factor of —10 are calculated. These deviations may be due to such problems as organic complexing or cumulative analytical and sampling errors, but the possibility that other phases are influencing Fe concentrations cannot be excluded on the basis of these data. 

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