1.0 TITLE

The measurement of drag coefficient of spheres.

2.0 OBJECTIVE

To measure the drag coefficients of spheres over several decades of particle Reynolds

number.

3.0 INTRODUCTION

3.1 Definition of Drag Coefficient for Flow Past Immersed Objects.

The flow of fluids outside bodies appears in many engineering applications

and other processing applications. This occurs, for example, in flow past

spheres in settling, flow through packed beds in drying and filtration, flow past

tubes in heat exchangers and others. It is useful to be able to predict the

frictional losses and/or the force on the submerged objects in these various

applications.

In the examples of fluid friction inside conduits that we considered, the

transfer of momentum perpendicular to the surface resulted in a tangential

shear stress or drag on the smooth surface parallel to the direction flow. This

force exerted by the fluid on the solid in the direction of flow is called skin or

wall drag. For any surface in contact with a flowing fluid, skin friction will

exist. In addition to skin friction, if the fluid is not flowing parallel to the

surface but must change directions to pass around a solid body such as a

sphere, significant additional frictional losses will occur and this is called form

drag.

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3.1.1 Drag Coefficient

In treating fluid flow through pipes and channels, a friction factor, defined as

the ratio of the shear stress to the product of the velocity head and density, was

shown to be used. An analogous factor, called the drag coefficient, is used for

immersed solids. Consider a smooth sphere immersed in a flowing fluid and at

a distance from the solid boundary of the stream sufficient for the approaching

stream to be at a uniform velocity. Define the projected area of the solid body

as the area obtained by projecting the body on a plane perpendicular to the

direction of flow. Denote the projected area by AP. For a sphere, the projected

area is that of a great circle, or ( π4 )D2

p, where DP is the diameter. If FDis the

total drag, the average drag per unit projected area is FD / A p. Just as the

friction factor f is defined as the ratio of τ w to the product of the density of the

fluid and the velocity head, so the drag coefficient CD is defined as the ratio of

FD / A pto this same product or

CD ≡

F D

Ap

ρuO2

2

Where u0 is the velocity of the approaching stream. For particles having

shapes other than spherical, it is necessary to specify the size and geometric

form of the body and its orientation with respect to the direction of flow of the

fluid. One major dimension is chosen as the characteristic length, and the other

important dimensions are given as ratios to the chosen one. For cylinders the

diameter D is taken as the defining dimensions, and the length expressed as

the L/D ratio. The orientation between the particle and the stream is also

specified. For a cylinder, the angle formed by the axis of the cylinder and the

2

direction of flow is sufficient. Then the projected area is can be determined as

LDP, where L is the length of the cylinder. For a cylinder with its axis parallel

to the direction f flow, Ap is( π4 )D 2

p, the same as for sphere of the same

diameter.

From dimension analysis, the drag coefficient of a smooth solid in an

incompressible fluid depends upon a Reynolds number and the necessary

shape ratios. For given shape

CD=Φ(ℜP)

The Reynold’s number for a particle in a fluid is defined as

ℜP ≡G0G p

u

Where D p=characteristic length

G0=uo ρ

A different CD versus ℜP relation exists for each shape and orientation. The

relation must in general be determined experimentally, although a well-

substantiated theoretical equation exists for smooth spheres at low Reynolds

numbers. Drag coefficients for compressible fluids increase with an increase

in the Mach number when the latter becomes greater than about 0.6.

Coefficients in supersonic flow are generally greater than in subsonic flow.

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4.0 APPARATUS

1. Glass tubes.

2. Drain valves, V1 & V2

3. Knob

4. Ball valves, V3 & V4

4

Glass tubes

Drain valves

Ball valves

Knob

5.0 METHOD

The Instructions Methods

1. Fill the two glass tubes with clear liquids

of different viscosities (cooking oil and

detergent).

The two glass tubes is filled with cooking oil

and another one is filled up with detergent.

2. Mild steel sphere with diameter 3.175mm

are dropped one at a time from the top of the

tubes and allowed to fall to the bottom.

Start the experiment with diameter of

3.175mm by dropping it at one time from top

of the tubes and allowed to fall to bottom.

3. The passage between the 1m marks on the

wall of the tubes being timed with a stop

watch and recorded.

Time is recorded by watching at the passage

between 1m marks on the wall.

4. When each sphere arrives at the recess in

the base of the tubes, it is removed by turning

the valve through 180° by rotating the knob

at the bottom of the glass tube, then open the

ball valve, V3 or V4.

The sphere is removed from the base of the

tubes by rotating the knob at the bottom of

the glass tube, and then only open the ball

valve.

5. After ejecting the sphere, the ball valve

and knobs should be returned to the operating

position.

The sphere is ejected and returned the knob

to the operating position.

6. Repeat the experiment for step 2 until 5

with different dimension and material of the

sphere.

The experiment is repeated by changing the

diameter size of the sphere and the material

of the sphere.

7. Calculate viscosity µ of liquid used. The viscosity of the liquid used is calculated.

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6.0 RESULTS AND DISCUSSION

RESULTS

Fluid : Cooking Oil

Object shape

MaterialLength/diameter

(mm)Mass (kg)

Time taken (s) Drag coefficient

CD oil

Reynolds No Re oil

Viscosity (kg/sm)1 2 3 Average

Sphere

Mild steel

3.175 0.000142 3.50 3.50 3.60 3.53 4.2559 5.59 0.14605.000 0.000526 1.90 2.00 1.80 1.90 3.8893 2.97 0.80546.350 0.001050 1.40 1.50 1.40 1.43 1.2709 18.74 0.21528.000 0.002040 1.10 1.10 1.10 1.10 0.9338 25.70 0.25709.525 0.003530 0.95 0.95 0.98 0.96 0.8546 27.87 0.3233

Stainless steel

3.175 0.000142 3.81 3.84 3.59 3.75 4.8022 4.96 0.15515.000 0.000525 1.91 1.99 2.00 1.97 4.1811 2.76 0.83516.350 0.001040 1.62 1.58 1.53 1.58 1.5345 15.52 0.23528.000 0.002048 1.12 1.11 1.26 1.16 1.0443 22.98 0.27259.525 0.003527 1.08 0.99 0.99 1.02 0.9648 24.68 0.3435

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Fluid : Detergent

Object shape

MaterialLength/diameter

(mm)Mass (kg)

Time taken (s) Drag coefficient

CD det

Reynolds No Re det

Viscosity (kg/sm)1 2 3 Average

Sphere

Mild steel

3.175 0.000142 1.00 1.10 1.20 1.10 0.4292 55.45 0.04575.000 0.000526 0.68 0.75 0.72 0.72 0.5800 19.89 0.30656.350 0.001050 0.62 0.60 0.61 0.61 0.2402 99.13 0.09228.000 0.002040 0.50 0.53 0.52 0.52 0.2168 110.72 0.12209.525 0.003530 0.48 0.47 0.49 0.48 0.2220 107.28 0.1624

Stainless steel

3.175 0.000142 0.98 1.10 1.02 1.03 0.3763 63.24 0.04285.000 0.000525 0.83 0.80 0.86 0.83 0.7707 14.97 0.35336.350 0.001040 0.73 0.68 0.69 0.70 0.3129 76.15 0.10468.000 0.002048 0.60 0.59 0.57 0.59 0.2807 85.52 0.13929.525 0.003527 0.53 0.55 0.54 0.54 0.2809 84.77 0.1827

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Additional info : Oil

Material Weight (N) γ s (N

m3) γ f (N

m3) v (m/s) µ (Ns/m2) V (m3 ¿ r (m)

Mild steel

0.0014 81599.35 8907.48 0.2833 0.1460 1.7157 ×10−8 0.00160.0052 79449.96 8907.48 0.5263 0.8054 6.5450 ×10−8 0.00520.0103 75040.07 8907.48 0.6993 0.2152 1.3726 ×10−7 0.00320.0200 74604.60 8907.48 0.9091 0.2570 2.6808 ×10−7 0.00400.0346 74689.69 8907.48 1.0417 0.3233 4.6325 × 10−7 0.0048

Stainless steel

0.0014 81599.35 8907.48 0.2667 0.1551 1.7157 ×10−8 0.00160.0052 79449.96 8907.48 0.5076 0.8351 6.5450 ×10−8 0.00520.0102 74311.53 8907.48 0.6329 0.2352 1.3726 ×10−7 0.00320.0201 74977.62 8907.48 0.8621 0.2725 2.6808 ×10−7 0.00400.0346 74689.69 8907.48 0.9804 0.3435 4.6325 × 10−7 0.0048

γ s=

ρg/WV

/ Mg

V

v=1mT

V= 43

π r3

8

W =mgAdditional info : Detergent

Material Weight (N) γ s (N

m3) γ f (N

m3) v (m/s) µ (Ns/m2) V (m3 ¿ r (m)

Mild steel

0.0014 81599.35 8613.18 0.9091 0.0457 1.7157 ×10−8 0.00160.0052 79449.96 8613.18 1.3889 0.3065 6.5450 ×10−8 0.00520.0103 75040.07 8613.18 1.6393 0.0922 1.3726 ×10−7 0.00320.0200 74604.60 8613.18 1.9231 0.1220 2.6808 ×10−7 0.00400.0346 74689.69 8613.18 2.0833 0.1624 4.6325 × 10−7 0.0048

Stainless steel

0.0014 81599.35 8613.18 0.9709 0.0428 1.7157 ×10−8 0.00160.0052 79449.96 8613.18 1.2048 0.3533 6.5450 ×10−8 0.00520.0102 74311.53 8613.18 1.4286 0.1046 1.3726 ×10−7 0.00320.0201 74977.62 8613.18 1.6949 0.1392 2.6808 ×10−7 0.00400.0346 74689.69 8613.18 1.8519 0.1827 4.6325 × 10−7 0.0048

γ s=

ρg/WV

/ Mg

V

v=1mT

V= 43

π r3

W =mg

9

5.59 2.97 18.74 25.7 27.870

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Mild Steel Sphere in Oil

Mild Steel Sphere in Oil

Reynold's Number

Drag Coefficient

10

55.45 19.89 99.13 110.72 107.280

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Mild Steel Sphere in Detergent

Mild Steel Sphere in Detergent

Reynold'sNumber

Drag Coefficient

11

4.96 2.76 15.52 22.98 24.680

1

2

3

4

5

6

Stainless Steel Sphere in Oil

Stainless Steel Sphere in Oil

Reynold's Number

Drag Coefficient

12

63.24 14.97 76.15 85.52 84.770

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Stainless Steel Sphere in Detergent

Stainless Steel Sphere in Detergent

Reynold'sNumber

DragCoefficient

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DISCUSSION

1. Drag force exerted on the solid which move in a fluid is due to viscosity of the fluid. It is made up of two components, surface drag and form drag.

2. Relationship of the drag force with the coefficient is

F=CD ×AρV 2

23. The resultant force on the sphere must equal to the specific weight difference times

the volume of the displaced liquid, strokes expression,

F=4 π r3

3¿

4. The measurement of the drag coefficients of spheres are as follows:

CD=83

r ¿¿

5. Reynolds number

ℜ= ρVDμ

6. The fall velocity of a sphere V is calculated as 1/T, where T is the take time taken by the sphere to fall between the 1m marks. So, in order to calculate the Re, we need to find and determine the viscosity first. Stokes law can be used to find µ as long as Re < 1. The procedure is to select the smallest sphere of the lightest material provided, measure T, calculate V and use this value to find viscosity from following equation.

μ=29

r2¿¿

7. From the above equation we can find the Reynolds number

ℜ= ρVDμ

8. It is possible to measure the value of T for the remaining spheres, and calculate the drag coefficient and the Reynolds number. To get the accurate reading, it is advisable to average the number of T obtained over repeated drops of the sphere.

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7.0 CONCLUSION

A measurement technique was developed enabling time measurement of spheres falling in

fluids with great accuracy. A proposed mathematical model that includes a new drag

coefficient correction factor enables more precise evaluation of drag coefficients.

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