Page 101 - Modelling in Transport Phenomena A Conceptual Approach

4.3. FLOW PAST A SINGLE SPHERE 81

The value of X is calculated from Eq. (4.3-21) as

4
x=- gDP(PP - P)
3 Put2
- - (9.8)(5 x 10-~)(1000 - 910)
4
- = 7536
3 (910)(9.26 x 10-4)2
Substitution of the value of X into Eq. (4.3-23) gives the Reynolds number as

X
- 24 [1+ 120 (7536)-20/11] 4/11 = 3.2 x lov3
-
7536
Hence, the viscosity of the fluid is

p=- DPVtP
Rep
- (5 x 10-3)(9.26 x 10-4)(910) = 1.32 kg/ m. s
-
3.2 x 10-3

4.3.1.2 Deviations from ideal behavior
It should be noted that Eqs. (4.3-4) and (4.3-10) are only valid for a single spherical
particle falling in an unbounded fluid. The presence of container walls and other
particles as well as any deviations from spherical shape affect the terminal velocity
of particles. For example, as a result of the upflow of displaced fluid in a suspension
of uniform particles, the settling velocity of particles in suspension is slower than the
terminal velocity of a single particle of the same size. The most general empirical
equation relating the settling velocity to the volume fraction of particles, w, is given
bv
ut (suspension)
= (1 -W)n (4.3-25)
vt(sing1e sphere)
where the exponent n depends on the Reynolds number based on the terminal
velocity of a particle in an unbounded fluid. In the literature, values of n are
reported as
4.65 - 5.00 Rep < 2
n={ (4.3-26)
2.30 - 2.65 500 5 Rep 5 2 x lo5
The particle shape is another factor affecting terminal velocity. The terminal
velocity of a non-spherical particle is less than that of a spherical one by a factor
of sphericity, 4, i.e.,
ut( non-spherical) (4.3-27)
ut (spherical) =4<1
Sphericity is defined as the ratio of the surface area of a sphere having the same
volume as the non-spherical particle to the actual surface area of the particle.