Dallavalle equation

In a series of papers, Chhabra (1995), Tripathi et al. (1994), and Tripathi and Chhabra (1995) presented the results of numerical calculations for the drag on spheroidal particles in a power law fluid in terms of CD = fn(tVRe, ). Darby (1996) analyzed these results and showed that this function can be expressed in a form equivalent to the Dallavalle equation, which applies over the entire range of n and tVRe as given by Chhabra. This equation is... 

Unfortunately, there are insufficient experimental data reported in the literature to verify or confirm any of these expressions. Thus, for lack of any other information, Eq. (11-49) is recommended, because it is based on the most detailed analysis. This can be extended beyond the Stokes flow region by incorporating Eq. (11-49) into the equivalent Dallavalle equation,... 

If Stokes flow is not applicable (or even if it is), the Dallavalle equation in the form of Eq. (11-16) can be used to determine the Reynolds number, and hence the diameter, of the smallest setting particle ... 

Alternatively, it may be necessary to determine the maximum capacity (e.g., flow rate, Q) at which particles of a given size, d, will (or will not) settle out. This can also be obtained directly from the Dallavalle equation in the form of Eq. (11-13), by solving for the unknown flow rate ... 

This analysis is based on the assumption that Stokes law applies, i.e., /VRe < 1. This is frequently a bad assumption, because many industrial centrifuges operate under conditions where iVRe > 1. If such is the case, an analytical solution to the problem is still possible by using the Dallavalle equation for CD, rearranged to solve for N t as follows ... 

The foregoing expressions give the suspension velocity (Fs) relative to the single particle free settling velocity, V0, i.e., the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 11, the Dallavalle equation can be used to calculate the single particle terminal velocity V0... 

This result can also be applied directly to coarse particle swarms. For fine particle systems, the suspending fluid properties are assumed to be modified by the fines in suspension, which necessitates modifying the fluid properties in the definitions of the Reynolds and Archimedes numbers accordingly. Furthermore, because the particle drag is a direct function of the local relative velocity between the fluid and the solid (the interstitial relative velocity, Fr), it is this velocity that must be used in the drag equations (e.g., the modified Dallavalle equation). Since Vr = Vs/(1 — Reynolds number and drag coefficient for the suspension (e.g., the particle swarm ) are (after Barnea and Mizrahi, 1973) ... 

If Ar< 15, the terminal velocity can be obtained from the Stokes equation If Ar > 15, the particle Reynolds number is obtained after rearranging Dallavalle equation ... 

For yVRe > 0.1 (or > 1, within 5%), a variety of expressions for Cq vs. NRe (mostly empirical) have been proposed in the literature. However, a simple and very useful equation, which represents the entire range of CD vs. lVRe reasonably well (within experimental error) up to about lVRe = 2 x 105 is given by Dallavalle (1948) ... 

Even more interesting and important than the foregoing unique method of measuring thermal conductivities of suspensions is the procedure used to calculate thermal conductivities theoretically. Orr and DallaValle noted that electrical and thermal fields are similar hence the usual equation for calculation of electrical conductivity of a suspension should also be applicable to thermal conductivities. Their extensive tabulated results support this contention to within 3 %. This equation is... 

Orr and DallaValle (05), on the basis of their extensive experimental results, recommend use of the equation... 

Dallavale defined a new shape factor that is modified from the correction factor. The Dallavalle shape factor can be useful for a log normal distribution because the shape of the size-hfcquency (density distribution) curve can be taken into account when combining Martin s correction factor with the Hatch-Choate equation (1,34,37). Applying the Hatch-Choate equation of dy, and from Table 6 to in Equation (79) and Sw in Equation (81) yields. 

It is not possible to assess the effect of the substrate surface independently however, the powder can be characterized with respect to effective size by means of a simple sieve analysis that can be used to get a cumulative size distribution. If the total fraction that passes each size sieve is plotted against the sieve opening on normal probability paper, the mean weight-diameter from the 50% point is obtained and the standard deviation from the sizes corresponding to the 13% and/or 83% diameter. The effective (surface mean) particle diameter can then be calculated by means of the equation given by Orr and Dallavalle [66] ... 

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