Richardson-Zaki correlation

Another approximation, one of the most enduring empirical correlations in multiphase systems, is the Richardson-Zaki correlation for a single particle in a suspension (3) ... 

Asif M. Generalized Richardson-Zaki correlation for liquid fluidization of binary solids. Chem Eng Technol 21 77-82, 1998. 

Engineers often use the Lewis-Bowerman and Richardson-Zaki correlations to characterize fluidized beds by determining porosity as a function of the flow velocity. These correlations, empirical in nature, are recapitulated in Table 15.2. While the diameter of the column is taken into account by the Richardson-Zaki correlation, its effect remains very weak as long as the size of the particles is small compared to the diameter of the colunrn. Comparing the Lewis-Bowerman and Richardson-Zaki correlations with Ergun s relation, expressed in the form [15.39],... 

The Richardson-Zaki coefficient (ri) can be calculated from the following correlations ... 

For uniform gas-solid flows, the well-known Richardson and Zaki correlation [Richardson and Zaki, 1954 Growther and Whitehead, 1978] given next can be applied... 

The Reynolds number (based on the terminal settling velocity) was estimated from the Gahleo number using the correlations given by Lali et al. (1989). The terminal setthng velocity was then calculated from the Reynolds number and the physical properties. The Richardson-Zaki index m was estimated using the correlations given by Richardson (1971). 

Although both expressions are commonly used, they fail to predict some important macroscopic properties of solid-fluid suspensions, such as the expansion and sedimentation profiles. To overcome this limitation Mazzei Lettieri (2007) developed a relationship for the drag coefficient that is based on the empirical correlation by Richardson Zaki (1954) describing the expansion profiles of homogeneous fluid-solid suspensions. Its main feature resides in the fact that the expression is consistent with the Richardson and Zaki correlation over the whole range of fluid-dynamic regimes and for any value of the suspension void fraction. It has the following formulation ... 

Resin manufacturers often express the fluidization properties of thejr resins in terms of the percentage expension of the bed related to its pseked depth. This expansion cannot be calculatad directly from the above equation since a range of bead sizes is present. Typical data are shown in Fig. 13.3-1, taken from a Rohm and Haas pamphlet. These curves can be correlated by an equation giving the expension as a fouction of the 1.5 power of the flow rale. An equation of almost identical form can be derived from the Richardson-Zaki equation by expressing the expansion in cerma of the void fraction and using appropriate values for n. 

The expansion of a fluidized bed of resin is an important variable in determining how much resin will be held in a vessel. This property of a resin is measured very readily in a simple laboratory experiment. For uniform bead size, correlation of data can be done with the Richardson-Zaki equation expressed in the form... 

This relationship, known as the Richardson-Zaki equation, has been modified in order to provide a better correlation of the experimental data [14]. 

With these two equations, an expression of the Richardson--Zaki type applied to the particulate phase and a correlation function between Sq and Uj an iterative method of calculation of e was proposed. The (gq + e ) results predicted with this method and the experimental values differed significantly,... 

The relation between superficial velocity (U) and bed voidage (e) in a fluidised bed can be described by the classical correlation first postulated by Richardson and Zaki 49... 

Richardson and Zaki (1954) formulated sectionwise correlations of the exponent n to the terminal Reynolds number Re = dputpf/p. 

From the preceding manipulations on dimensionless numbers, it can be seen that, if the fixed and fluidized beds are to be considered comprehensively, functional plots of Ar—Re with c as the curve parameter are most desirable, as shown in Fig. 6. If only the fluidized state is of interest, then the Ar1/3-Ly1/3 plot will facilitate independent evaluation of dp and u, as shown in Fig. 7. It has already been observed that the family of curves shown in Figs. 6 and 7 is filled in by using the relation given by Eq. (2.2). Instead of the sectionwise correlations of Richardson and Zaki, n has been correlated to Re, empirically as a continuous function, as shown in Fig. 8. 

Equation (9-31) assumes no interaction between bubbles. Equation (9-32) is proposed by Turner.135 Equation (9-33) is the correlation of Richardson and Zaki,114 where the presence of other bubbles increases the effective viscosity of liquid. For gas-liquid systems, Wallis142-143 proposed m = 2. 

The motion of swarms of droplets is, generally, mnch slower than that of single droplets. The influence of drop concentration is accounted for by an empirical correlation (Richardson and Zaki 1954) developed for fluidized beds of rigid particles ... 

Kato et al. [43] used the pressure profile method and the electroconductivity probe in a bidimensional column. The liquid hold-ups obtained by the two methods agreed well except near the top of the fluidized bed. Because this study involved experimentation with air, aqueous solutions of carboxymethyl cellulose at different concentrations and glass particles of different sizes (0.42 mm, 0.66 mm, 1.2 mm, 2.2 mm) the effects of superficial liquid velocity, liquid viscosity and particle size on the liquid hold-up were considered. With this data a correlation similar to the one derived by Richardson and Zaki for liquid-solid fluidized beds was proposed [43]. 

The settling rate and settling time can be estimated using e.g. the Richardson and Zaki equation (2.42). For slurries of irregular particles, however, the assumptions in the correlation are exceeded and the settling rate then becomes more difficult to calculate. Consequently, the Jar Test (see Chapter 2) is frequently used to determine R and r in practice. 

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