UNITS AND DIMENSIONS 13
What will be the terminal falling velocity of a particle of diameter 10μm and of density
1600 kg/m^3 settling in a liquid of density 1000 kg/m^3 and of viscosity 0.001 Ns/m^2?
If Stokes’ Law applies for particle Reynolds numbers up to 0.2, what is the diameter
of the largest particle whose behaviour is governed by Stokes’ Law for this solid and
liquid?
Solution
The accelerating force due to gravityDmass of particlemass of liquid displacedg.
For a particle of radius r, volumeD 4 r^3 /3, or, in terms of diameter,d, volumeD
4 d^3 / 23 / 3 Dd^3 /6. Mass of particleDd^3 s/6, where (^) sis the density of the solid.
Mass of liquid displacedDd^3
/6, where is the density of the liquid, and accelerating
force due to gravityDd^3 s/ 6 d^3
/ 6 gDd^3 / 6
s
g.
At steady state, that is when the terminal velocity is attained, the accelerating force due
to gravity must equal the drag force on the particleF,or:d^3 / 6
s
gD 3 du 0
whereu 0 is the terminal velocity of the particle.
Thus: u 0 Dd^2 g/ 18
s
(i)
It is assumed that the resistance per unit projected area of the particle,R^0 , is a function
of particle diameter,d; liquid density, ; liquid viscosity, , and particle velocity,uor
R^0 Dfd,
, ,u. The dimensions of each variable areR^0 DM/LT^2 ,dDL, DM/L^3 ,
DM/LTanduDL/T. With 5 variables and 3 fundamental dimensions, there will be
5 3 D2 dimensionless groups. Takingd,
anduas the recurring set, then:
dL, LDd
M/L^3 , MD L^3 D
d^3
uL/T, TDL/uDd/u
Thus: dimensionless group 1:R^0 LT^2 /MDR^0 dd^2 /u^2 /
d^3 DR^0 /
u^2
dimensionless group 2: LT/MD dd/u/
d^3 D /du
and: R^0 /
u^2 Df /du
or: R^0 /
u^2 DKdu
/ nDKRen (ii)
In this way the experimental data should be plotted as the group (R/
u^2 )againstRe.
For this particular example,dD 10 μmD 10 ð 10 ^6 D 10 ^5 m; (^) sD1600 kg/m^3 ;
D1000 kg/m^3 and D 0 .001 Ns/m^2.
Thus, in equation (i): u 0 D 10 ^5 ^2 ð 9. 81 / 18 ð 0. 001 1600 1000
D 3. 27 ð 10 ^5 m/s or 0.033 mm/s
WhenReD 0 .2,du
/ D 0 .2 or when the terminal velocity is reached:
du 0 D 0. 2 /
D 0. 2 ð 0. 001 / 1000 D 2 ð 10 ^7
or: u 0 D 2 ð 10 ^7 /d