Find the sedimentation velocities of (i) a pollen grain, diameter 10 µm, density 0.8 g cm^−3, and (ii) a hailstone, diameter 6 mm, density 0.5 g cm^−3. (Hint: You will have to estimate the Reynolds number and decide whether to use the trial-and-error method described in the text).

Question

Find the sedimentation velocities of (i) a pollen grain, diameter 10 µm, density 0.8 g c[latex]m^{−3}[/latex], and (ii) a hailstone, diameter 6 mm, density 0.5 g c[latex]m^{−3}[/latex]. (Hint: You will have to estimate the Reynolds number and decide whether to use the trial-and error method described in the text).

Accepted Answer

i. If the pollen grain obeys Stokes’ Law, then [latex] V_{s} = 2 ho gr^{2} / 9 ho_{a} v [/latex]. Hence [latex] V_{s} = 2 imes 0.8 imes 10^{6} imes 9.81 imes (5 imes 10^{-6} )^{2} / 9 imes 1.29 imes 10^{3} imes 15 imes 10^{-6} = 2.4 \; mm \; s^{-1} [/latex].

To determine whether using Stokes’ Law is appropriate, calculate the particle Reynolds number [latex]Re_{p}[/latex].

[latex] Re_{p} = 2.4 imes 10^{-3} imes 10 imes 10^{-6} /(15 imes 10^{-6} ) = 1.6 imes 10^{-3} [/latex].

So Stokes’ Law is valid.

ii. In this case, Stokes’ Law predicts that [latex] V_{s} = 545 \; m \; s^{-1} [/latex], but then [latex]Re_{p}[/latex] = 109 × 10³, so Stokes’ Law clearly is not adequate for calculating [latex] V_{s}[/latex]. Using the iterative method described in the text, with a first guess for drag coefficient of 0.44 eventually yields [latex] V_{s} = 13 \; m \; s^{-1} [/latex]. There is an interesting discussion of drag coefficients at

Question 12.1: Find the sedimentation velocities of (i) a pollen grain, dia...

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