# DRAG COEFFICIENT FOR SPHERES

AIM:
To determine the drag coefficient for spheres.

APPARATUS AND MATERIALS

Fluid particle system and accessories, stop watch.

Theory

Drag force depends on number of variables and one of them is the shape of the moving body and the fluid involved.

The equilibrium of the ball under the steady state condition requires that the drag force and the buoyancy force must be added up to balance the weight of the ball.

For a ball of diameter D moving at a velocity V in an engine oil or glycerol of viscosity the drag is governed by stokes law.

Drag = 3 VD                                                                                   1

If the density of the falling body is s and the density of the engine oil or glycerol is f, then the buoyance force and the weight are given by;

Fg = f g                                                                                         2

W = s g                                                                                        3

For a spherical ball, the volume is given by;

V = 4

At equilibrium,

Drag + Buoyancy = weight

Therefore,

3 VD + g = g                                                               5

Hence,

V = 6

Equation 6 is called Stokes law

The density of the steel ball or PVC as the case may be is obtained by using = 7

And volume from equation 4 is

V =  s = 8

The density of the engine oil and glycerol may be determined using density bottle and the electronic balance in the laboratory. The velocity is Distance/Time (d/t) but in this case it is the height of fall between two fixed marks on the glass column and the time is recorded with your stop watch.

Therefore,

V = Where H = Height of fall

T = Time of fall

The net gravitational force F1 on the particle is given by

F1 = ( 9

The retardation force F2 on the sphere from the fluid is given by the equation;

F2 = CDSP 10

Where CD = dimensionless drag coefficient

SP = Projected area of the sphere

At equilibrium, when the sphere attains a constant velocity, F1 = F2 and so equations 9 and 10 yield = CDSp 11

For a spherical particle, SP the projected area of the sphere in a plane perpendicular to the direction of the engine oil/glycerol stream is given as

Area = SP = 12

Inserting equation 12 into equation 11 gives

V = 13

V is the terminal settling or falling velocity of the sphere (PVC or steel balls)

Equation 12 can be rearranged to give

CD = 4D 14

Where CD = is drag coefficient

Reynold’s number Re =  = density of the fluid

V = terminal velocity

D = diameter of the sphere = viscosity of the fluid

In case of surface drag around a sphere, it can be shown that

CD = = 15

Equipment and instrument

1.      Large vertical glass clear cylinders filled with glycerin and oil.

2.      Stainless steel spheres of different diameter

3.      Stop watch

4.      Specific gravity bottle

Experimental Procedure

1.      Determine the density of the fluid, steel balls and also the weight and diameter of the steel ball

2.      Gently drop the balls into the glycerin column or the oil column

3.      Determine the time taken for the sphere of different diameters to fall through the tube length (1m) marked along the vertical cylinder for the two columns.

4.      Using the column below repeat for balls with different diameters

Table of values

 S/N Sphere Weight (g) Diameter (mm) Density (g/ml) Time (s) Velocity (m/s)

Data treatment

1.      Evaluate the viscosity of the fluid using stokes law, using the smallest sphere diameter

2.      Calculate the terminal velocity for various sphere with equation 13

3.      Determine the drag coefficient CD with equation 14 for all the values of sphere

4.      Determine also the Reynolds number of each sphere

5.      Plot graph of settling velocity against ball diameter for each sphere and fluid on the same graph

6.      Plot graph of CD vs Re on log graph paper

7.      What can you conclude from the shape of the CD/Re curve?

8.      Why is the CD/Re plot drawn on a log-log paper?

Data

g = 9.18m/s2 f = 1.046g/cm3 (Glycerol) f = 0.95g/cm3 (Castor oil) f = 1.00g/cm3 (water) f = 0.92g/cm3 (engine oil)

Level of Study: 300 Level