Week # 2MR Chapter 2

• Tutorial #2• MR # 2.1, 2.4, 2.8.

• To be discussed on Jan. 25, 2017.

• By either volunteer or class list.

MARTIN RHODES (2008) Introduction to Particle Technology , 2nd Edition. Publisher John Wiley & Son, Chichester, West Sussex, England.

Motion of solid particles in a fluid

For a sphere

Stoke’s law

Standard drag curve for motion of a sphere in a fluid

Reynolds number ranges for single particle drag coefficient correlations

At higher relative velocity, the inertia of fluid begins to dominate.

Four regions are identified: Stoke’s law, intermediate, newton’s law, boundary layer separation.

Table 2.1 (Schiller and Naumann (1933) : Accuracy around 7%.

Single Particle Terminal Velocity

Special Cases

• Newton’s law region:12( )

1.74 p fT

f

x gU

Intermediate region:

0.71.1 0.29 0.43, , ,T p f fU x

To calculate UT and x

• (a) To calculate UT, for a given size x,

• (b) To calculate size x, for a given UT,

32

2

( )4Re3

f p fD

x gC

Independent of UT

3 2

( )4Re 3

P fD

P T f

gCU

Independent of size x

Particles falling under gravity through a fluid

Method for estimating terminal velocity for a given size of particle and vice versa

Non-spherical particles

Drag coefficient CD versus Reynolds number ReP for particles of sphericity ranging from 0.125 to 1.0

Effect of boundaries on terminal velocity

Sand particles falling from rest in air (particle density, 2600 kg/m3)

When a particle is falling through a fluid in the presence of a solid boundary the terminalVelocity reached by the particle is less than that for an infinite fluid.

Following Francis (1933), wall factor ( )/w Df U U

Limiting particle size for Stoke’s law in water

Limiting particle size for Stoke’s law in air

850

• Where the plotted line intersects the standard drag curve for a sphere ( = 1), Rep = 130.

• The diameter can be calculated from:

Re 130 f v TP

x U

Hence sphere diameter, xv = 619 m.

• For a cube having the same terminal velocity under the same conditions, the same CD vesus Rep relationship applies, only the standard drag curve is that for a cube( = 0.806)