Axial dispersion coefficients solids

Fig. 10.16. Axial dispersion coefficients (solid line correlation, points CFD and experimental data).

TABLE 22-5 Comparison of Axial-Dispersion Coefficients for Several Liquid-Solid Contactors... 

The development of the equations for the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of v and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the element, as shown by the solid and dashed arrows respectively, in Fig. 4.12. 

Yerushalmi and Avidan (1985) suggest that the axial dispersion coefficient of solids in slugging and turbulent flow varies approximately linearly with the bed diameter, similar to Thiel and Potter (1978). The data are shown in Fig. 17 although May s results are probably in the bubbling fluidization regime rather than turbulent flow. 

Axial mixing in the liquid, induced by the upflow of the gas bubbles, can be substantial in commercial-scale bubble columns, especially in the chum turbulent regime. Due to typically small particle size, the axial dispersion of the solid catalyst in slurry bubble columns is expected to follow closely that of the liquid exceptions are high-density particles. The liquid axial mixing can be represented by an axial dispersion coefficient, which typically has the form... 

In gas/liquid/solid sparged columns the situation is somewhat more complicated. Just as in gas/liquid systems, different regimes have to be distinguished. Here the most important variables affecting the axial dispersion coefficient Dc l of the liquid, are dc and uG ... 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

This equation indicates that axial gas dispersion coefficient increases with gas velocity and solids circulation rate. This dependence of axial dispersion coefficient on gas velocity is at variance in the available literature. 

Computation of axial dispersion coefficients for solids was provided by Patience et al. (1991), in the following simple one-dimensional model using the data obtained from RTD measurements of a radioactive solid tracer in a 82.8 mm i.d. by 5 m high circulating fluidized bed, that is,... 

The axial dispersion coefficients Dag and )as can thus be obtained from their respective RTD, typical values of which are shown in Fig. 38. Figure 39 shows further that Dag increases with solids circulation rate but decreases with increasing gas velocity. Figure 40 shows that solids mixing is greater than gas, and Fig. 41 shows that fine particles mix better than coarse. 

Methods for evaluating the axial dispersion coefficient from RTD data As mentioned earlier, the one-parameter axial-dispersion model is widely used to correlate RTD data. The nature of the RTD depends upon the nature of the tracer input and the nature of the. flow, characteristics. For the RTD shown in Fig. 3-4 o), the axial dispersion coefficients for the liquid and solid phases can be obtained by fitting the equation... 

In the above equations, AG, AL, and As are the gas-phase, liquid-phase and calalyst-surface concentrations of the reacting species, ACi is the average gas-phase concentration at the reactor inlet, Z is the axial distance from the reactor inlet, L is the total length of the reactor, m = H/RgT, where H is the Henry s law constant (cm3 atm g-mol" ), Rg is the universal gas constant, and T is the temperature of the reactor. UG is the mean gas velocity, Us is the mean settling velocity of the particles, t is the time, k is the first-order rate constant, W is the catalyst loading, zc and ZP are the axial dispersion coefficients for the gas and solid phases, respectively. Following the studies of Imafuku et al.19 and Kato et al.,21 the axial dispersion coefficient for the liquid phase was assumed to be the same as that for the solid phase, w is the concentration of the particles and hG the fractional gas holdup. Other parameters have the same meaning as described earlier. 

The effects of suspended solid particles on liquid-phase axial dispersion in a cocurrent-upflow system have been studied by Schiigerl123 and Michelsen and Ostergaard.82 They showed that, in a three-phase column, the axial dispersion increases with gas rate. Unlike in a gas-liquid bubble-column, the liquid-phase axial dispersion coefficient in a three-phase column depends upon the liquid velocity. The nature of the effect is, however, dependent upon the gas rate and solids particle size. Similarly, the nature of the effect of solid size on the axial dispersion depends on the gas and liquid flow rates. 

Farkas and Leblond30 used the axial distribution of solids to calculate the values of the axial dispersion coefficient. The data were, however, not correlated to gas and liquid velocities. A method for characterizing the age distribution of suspended particles in a continuous bubble-column is given by Yamanaka et al.146... 

Recommendations For estimating the liquid- and solid-phase axial dispersion coefficients in a three-phase fluidized-bed column, use of Eqs. (9-37) and (9-39) are recommended. Future work on this subject should include the measurement of axial dispersion coefficients in the gas phase, particularly in large-diameter columns. 

We compare in Figure 6.20 two profiles that were calculated as numerical solutions of the equilibrium-dispersive model, using a linear isotherm. The first profile (solid line) is calculated with a single-site isotherm q = 26.4C) and an infinitely fast A/D kinetics (but a finite axial dispersion coefficient). The second profile (dashed line) uses a two-site isotherm model q — 24C - - 2.4C), which is identical to the single-site isotherm, and assumes infinitely fast A/D kinetics on the ordinary sites but slow A/D kinetics on the active sites. In both cases, the inverse Laplace transform of the general rate model given by Lenhoff [38] (Eqs. 6.65a to h) is used for the simulation. In the case of a surface with two t5q>es of adsorption sites, Eq. 6.65a is modified to take into accoimt the kinetics of adsorption-desorption on these two site types. 

Using the hodograph transform, Rhee and Amundson [3] have also shown that a plot of Q versus Q+i is a straight line (solid line in Figure 16.3), provided that these two components have the same axial dispersion coefficient (D ) and mass transfer rate constant (fcy), in addition to the competitive Langmuir isotherm behavior. The equation of this straight line is... 

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as Madonna. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of V and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the liquid phase volume of any element, n, as indicated in Fig. 4.24 below. Here j represents the diffusive flux, L the liquid flow rate and and Cla the concentration of any species A in both the solid and liquid phases, respectively. 

Since the particles retain their identity in the reactor and the rate equations are nonlinear, a residence time distribution model will be used. The particle slip velocity was found to be negligible as compared with the liquid-phase circulation velocities, which govern the dispersion coefQcient. It will be assumed that the axial dispersion coefficients for the solid and liquid phases are practically the same. Thus, the exit age distribution for the solid particles can be found by following the procedure for the liquid phase. 

Experimental extraction curves can be represented by this type of model, by fitting the kinetic coefficients (mass transfer coefficient to the fluid, effective transport coefficient in the solid, effective axial dispersion coefficient representing deviations from plug flow) to the experimental curves obtained fi om laboratory experiments. With optimized parameters, it is possible to model the whole extraction curve with reasonable accuracy. These parameters can be used to model the extraction curve for extractions in larger vessels, such as from a pilot plant. Therefore, the model can be used to determine the kinetic parameters from a laboratory experiment and they can be used for scaling up the extraction. 

---