C. Unknown Viscosity

The viscosity of a Newtonian fluid can be determined by measuring the terminal velocity of a sphere of known diameter and density if the fluid density is known. If the Reynolds number is low enough for Stokes flow to apply (NRe < 0.1), then the viscosity can be determined directly by rearrangement of Eq. (11-10):

The Stokes flow criterion is rather stringent. (For example, a 1 mm diameter sphere would have to fall at a rate of 1 mm/s or slower in a fluid with a viscosity of 10 cP and SG = 1 to be in the Stokes range, which means that the density of the solid would have to be within 2% of the density of the fluid!) However, with only a slight loss in accuracy, the Dallavalle equation can be used to extend the useful range of this measurement to a much higher Reynolds number, as follows. From the known quantities, CD can be calculated from Eq. (11-11). The Dallavalle equation [Eq. (11-5)] can be rearranged to give NRe:

The viscosity can then be determined from the known value of NRe:

dVtP m

Note that when NRe > 1000, CD « 0.45 (constant). From Eq. (11-19), this gives m = 0! Although this may seem strange, it is consistent because in this range the drag is dominated by form (wake) drag and viscous forces are negligible. It should be evident that one cannot determine the viscosity from measurements made under conditions that are insensitive to viscosity, which means that the utility of Eq. (11-19) is limited in practice to approximately


All expressions so far have assumed that the particles are surrounded by an infinite sea of fluid, i.e. that the boundaries of the fluid container are far enough from the particle that their influence is negligible. For a falling particle, this might seem to be a reasonable assumption if d/D < 0.01, say, where D is the container diameter. However, the presence of the wall is felt by the particle over a much greater distance than one might expect. This is because as the particle falls it must displace an equal volume of fluid, which must flow back around the particle to fill the space just vacated by the particle. Thus the relative velocity between the particle and the adjacent fluid is much greater than it would be in an infinite fluid; i.e., the effective "free stream'' (relative) velocity is no longer zero, as it would be for an infinite stagnant fluid. A variety of analyses of this problem have been performed, as reviewed by Chhabra (1992). These represent the wall effect by a wall correction factor (Kw) which is a multiplier for the "infinite fluid'' terminal velocity. (this is also equivalent to correcting the Stokes' law drag force by a factor of Kw). The following equation due to Francis (see, e.g., Chhabra, 1992) is claimed to be valid for d/D < 0.97 and NRe < 1:

For larger Reynolds numbers, the following expression is claimed to be valid for d/D < 0.8 and NRe > 1000:

Although these wall correction factors appear to be independent of Reynolds number for small (Stokes) and large (> 1000) values of NRe, the value of Kw is a function of both NRe and d/D for intermediate Reynolds numbers (Chhabra, 1992).

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