Richardson and Zaki equation

Interestingly, it turns out that despite the differing functions of the expansion index, , the correction factor, g(e), is approximately constant at the extremes of the flow regimes due to the particular values of the expansion index. The value of n is not known entirely theoretically but can be obtained from the Richardson and Zaki equation. Thus At low RCp ( = 4.6)... 

According to this, the simple Richardson and Zaki equation is an obvious choice in practice. Such relation can be expressed as... 

Calculate the minimum area and the diameter of a thickener with a circular basin to treat 0.2 m /s of a slurry which contains 20 pm particles of silica (density 2600 kgW) suspended in water (density 1000kg/m and viscosity 0.001 Ns/m ) at a concentration of 650kg/m. Assume that the slurry cannot be tested and do your calculations on the basis of the Richardson and Zaki equation (equations 18.12 and 18.15 combined), i.e. u u = (notation as in chapter 18). Take the underflow concentration as 1560kg/m and also calculate the underflow volumetric flow rate assuming total separation of all solids. Ans. 1326.5 m, 41.1 m, 0.083 m /s. 

Chabra et al have shown that the Richardson and Zaki equation (represented by equations 18.12 and 18.15) can be used for prediction of sedimentation velocities with power law fluids in the range of flow index 0.8 < n < 1. This was to some extent confirmed by an experimental investigation using a conductivity meter although much larger discrepancies were found than for Newtonian fluids, attributed to the observed development of local nonhomogeneities. 

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. 

The following semi-empirical equation relates the (hindered) settling velocity of a slurry of particles to the settling velocity of a single particle, known as the Richardson and Zaki (1954) (RZ) equation. The RZ equation is also used for liquid fluidization whereby particles are supported by an up-flow of fluid. 

This result follows from the Richardson-Zaki equation. In their original work, Richardson and Zaki (1954) studied batch sedimentation, in particular the settling of coarse solid particles through a liquid in a vertical cylinder with a closed bottom. Richardson and Zaki found that the settling speed uc of the equal-sized particles in the concentrated suspension was related to the terminal settling speed u, of a single particle in a large expanse of liquid by the equation... 

Richardson and Zaki (1954) showed that in the Reynolds number range Rep<0.2, the velocity uc of a suspension of coarse spherical particles in water relative to a fixed horizontal plane is given by the equation... 

Values of the exponent n were determined by Richardson and Zaki(56) by plotting n against d/dt with Re () as parameter, as shown in Figure 5.16. Equations for evaluation of n are given in Table 5.1, over the stated ranges of Ga and Re Q. 

Richardson and Zaki(11) found that m, corresponded closely to u0, the free settling velocity of a particle in an infinite medium, for work on sedimentation as discussed in Chapter 5, although u, was somewhat less than n0 in fluidisation. The following equation for fluidisation was presented ... 

For a fluidized standpipe, the drag force of particles balances the pressure head as a result of the weight of solids. If the Richardson and Zaki form of equation (Eq. (8.55)) is proposed for the drag force, derive an expression for the leakage flow of gas in this standpipe. Discuss the effect of particle size on the leakage flow assuming all other conditions are maintained constant. 

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 difference between the relationships in equations 20 and 21 over the range of solids concentration (0 < c < 40%) is small. Therefore, the much simpler equation of Richardson and Zaki is more suitable in practice. 

The measured free-fall velocities, wq, were corrected for the wall effect to give the free-fall velocities in an infinite medium, wo(oo) with the equation (Richardson and Zaki, 1954)... 

In a liquid fluidized bed the superficial liquor velocity, Ws, and the bed voidage, , are related through the Richardson and Zaki (1954) equation ... 

Although these equations have been shown to be reasonably accurate for spheres, there is little published information on the behaviour of non-spherical particles. Richardson and Zaki give a factor by which z must be multiplied when non-spherical particles are being handled, but this was only verified for particle Reynolds numbers above 500. A further difficulty is that when djD is... 

This is vahd for beds operated with a gas or a hquid. The solid curves (= const.) for expanding homogeneous (liquid) fluidized beds are based on equations presented by Anderson (Anderson 1961) compared witli Richardson and Zaki (Richardson and Zaki 1954). In the case of heterogeneous (low pressure gas) fluidized beds only the boundaries between the minimum fluidizing velocity ( 0.4) and the pneumatic transport (e 1) are well known. The curve for... 

For non-flocculated systems Richardson and Zaki [1954] were able to draw an analogy between sedimentation and fluidisation and diowed that the settling rate is linked to the terminal settling velocity of the particles by the voidage or volumetric concentration of the solids, raised to a power that is a function of the particle Reynolds number. These relationships are described by the equation ... 

For a given particle material, the exponent n is an empirical function of Rci (Richardson and Zaki, 1954). Then a set of specified gas and solids velocities are fed into above equations to obtain the voidage expression. Figure 24 gives such a contour with exponent n = 4.4. 

Perhaps the simplest and certainly the most widely used, if not always the most accurate, of the empirical equations proposed in the literature for predicting the expansion of a liquid fluidized bed is that of Richardson and Zaki (1954), which was anticipated by several prior investigators (Hancock, 1937/38 Lewis et al., 1949 Jottrand, 1952 Lewis and Bowerman, 1952). In its most primitive form this equation is simply... 

Eq. (12) assumes that the liquid divides itself between the solids-free wakes and the particulate region of the fluidized bed. On the other hand, Eq. (13) considers that the interstitial velocity of the liquid can be represented by an equation similar to the one proposed by Richardson and Zaki [14] ... 

Satisfactory agreement between the measured bed porosities and the predicted values was found [27]. An average deviation of 3.7% - 5.6% between eq. (27) and the measured e value was obtained. The constant 2.65 on eq. (27) for Rep < 500 was later on revised by Dakshinamurty et al. [28] to 2.85. Eq. (27) shows9 however, a significant limitation because it fails to predict the porosity for zero gas velocity conditions. In fact when uq- 0, the bed porosity is not zero and it should certainly approach the predictions of Richardson and Zaki s equation [14]. 

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