Dispersion coefficients measurement

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

The eddy dispersion coefficient has been measured and correlated empirically as... 

A related measure of efficiency is the equivalent number of stages erkngof CSTR battery with the same variance as the measured RTD. Practically, in some cases 5 or 6 stages may be taken to approximate plug flow. The dispersion coefficient also is a measure of deviation... 

The distribution of tracer molecule residence times in the reactor is the result of molecular diffusion and turbulent mixing if tlie Reynolds number exceeds a critical value. Additionally, a non-uniform velocity profile causes different portions of the tracer to move at different rates, and this results in a spreading of the measured response at the reactor outlet. The dispersion coefficient D (m /sec) represents this result in the tracer cloud. Therefore, a large D indicates a rapid spreading of the tracer curve, a small D indicates slow spreading, and D = 0 means no spreading (hence, plug flow). 

Stemerding (S16) has reported dispersion measurements in a column filled with 13-mm Raschig rings with water and air in countercurrent flow. The dispersion coefficient was observed to be essentially independent of the water flow rate and dependent on the air flow rate only. For increasing air flow rates, the dispersion coefficient passed through a maximum. 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Chapter 15 provides additional discussion of the axial dispersion model and of methods for measuring dispersion coefficients. A more advanced account is given in... 

When analysing the standard deviation value, which measures the dispersion of measurements, the effect of heteroscedasticity, already discussed in connection with the measurement of vapour pressure, is noted ie the dependency between standard deviation and average (the higher the average, the greater the dispersion of measurements). One way to make this unfortunate property obvious when it comes to analysing data is to calculate the coefficient of variation for each distribution (C 0- If it is more or less constant, there is heteroscedasticity. 

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

Saunders, A.E. and Korgel, B.A. (2004) Second virial coefficient measurements of dilute gold nanocrystal dispersions using small-angle x-ray scattering. Journal of Physical Chemistry B, 108 (43), 16732-16738. 

The open ends boundary conditions apply when the measuring points are some distances from the ends. Such an arrangement is used in making accurate measurements of dispersion coefficients. 

Longitudinal dispersion coefficients can be readily obtained by injecting a pulse of tracer into the bed in such a way that radial concentration gradients are eliminated, and measuring the change in shape of the pulse as it passes through the bed. Since dC/dr is then zero, equation 4.34 becomes ... 

Kang, Fan and Kim(96) measured coefficients for heat transfer from a cone-shaped heater to beds of glass particles fluidised by water. They also found that the heat transfer coefficient passed through a maximum as the liquid velocity was increased. The heat transfer rate was strongly influenced by the axial dispersion coefficient for the particles, indicating the importance of convective heat transfer by the particles. The region adjacent to the surface of the heater was found to contribute the greater part of the resistance to heat transfer. 

In determining how the dispersion coefficients depend on travel time one may employ atmospheric diffusion theory or the results of experiments. Because of the difficulty of performing puff experiments, however, the coefficients are usually inferred not from instantaneous releases but from continuous releases. Thus, the dispersion coefficients derived from such experiments are essentially a measure of the size of the plume envelope formed by sampling a real meandering plume emitted from a... 

The Gaussian plume model estimates the average pheromone flux by multiplying the measured odor concentration by mean wind speed, using the following formula (Elkinton etal, 1984). Everything is the same as in the Sutton model, except that ay and az, respectively, replace the terms Cy and Cz of the Sutton model. Dispersion coefficients are determined for each experiment separately. 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

This work was extended by De Michelis and Calvelo (1994) who measured dispersion coefficients for 1cm cubes and for 1cm x 1cm x 1.5 cm cuboids. The data were again correlated by an expression taking the form of equation 3.30, with similar exponents on bed height (2.46 and 2.58 for cubes and cuboids respectively) and gas velocity (3.13 and 3.34 respectively) but coefficients of 0.110 and 0.256 respecfively fhe coefficient in equation 3.30 increased with decreasing particle sphericity. 

The remainder of this section will be devoted to describing the methods for measuring dispersion coefficients and the resulting correlations of the data. 

As has been discussed, the usual method of finding the dispersion coefficients is to inject a tracer of some sort into the system. The tracer concentration is then measured downstream, and the dispersion coefficients may be found from an analysis of the concentration data. For these tracer experiments there are no chemical reactions, and so r = 0. Also the source term is given by... 

If a pulse of tracer is injected into a flowing stream, this discontinuity spreads out as it moves with the fluid past a downstream measurement point. For a fixed distance between the injection point and measurement point, the amount of spreading depends on the intensity of dispersion in the system, and this spread can be used to characterize quantitatively the dispersion phenomenon. Levenspiel and Smith (L16) first showed that the variance, or second moment, of the tracer curve conveniently relates this spread to the dispersion coefficient. 

Inspection of Fig. 8 shows that there is considerable scatter in the data. Part of this may be due to the fact that we are attempting to represent a complex phenomenon with a single parameter, the dispersion coefficient. Errors would also be caused by the common practice of taking measurements at or beyond the exit of the packed section. This neglect of end conditions could lead to large errors in the calculated dispersion coefficients, as pointed out by Bischoff and Levenspiel (B14). Also, all the analyses were based on the assumption of having a perfect pulse, step. 

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