Problems for Practice - Unit III: Dimensional Analysis and Model Studies

 1) Show that the resistance R to the motion of a sphere of diameter (D) moving with the uniform velocity (V) through a real fluid of density (ρ) dynamic viscosity (μ) is given by

2) Determine the dimensions of

(i) Force

(ii) Viscosity

(iii) Power

(iv) kinetic viscosity

Answer: 

MLT-2, ML-1T-1, ML2 T-3, L2T-1

3) The variables controlling the motion of a floating vessel through water are the drag force F, speed V, length L, density ρ and dynamic viscosity μ of water and acceleration due to gravity g. Derive expression for F by dimensional analysis

4) Frictional Torque T of a propeller is a turbulent flow depends on density of liquid ρ, viscosity μ, speed N, diameter of propeller shaft D. Using Buckingham's π theorem. Show that T =

5) An oil of specific gravity 0.9 and viscosity 0.03 poise is to be transported at the rate of 3000 lit/sec through a 1.5m diameter pipe. Test were conducted on a 15cm diameter pipe using water at 20°c. If the viscosity of water at 20°c is 0.01 poise find.

(i) velocity of flow in the model

(ii) Rate of flow in the model

Answer: 

(i) 5.09 m/s (ii) 80.9 lit/sec

6) In 1:30 model of a spillway, the velocity and discharge are 1.5 m/s, and 2.0 m3/sec respectively. Find the corresponding velocity and discharge in the prototype.

Answer: 

8.2 m/s, 9859 m3/s

7) An air duct is to be modelled to a scale of 1:20 and tested with water which is 50 times viscous and 800 times more dense than air. When tested under dynamically similar conditions, the pressure drop between two sections, in the model is 235 kpa. What is the corresponding pressure drop in the prototype.? 

Answer: 

183.3 pas.

8) The flow rate over a spill way is 125 m3/s. What is the maximum length scale factor of dynamically similar model if flow rate of 0.75 m3/s is available in the laboratory? On a part of such a model a, force of 3N is measured. What is the corresponding force on the prototype spill way

Answer: 

7.74 m/s, 1396. 45 N.

9) A model aeroplane has dimensions that are 1th/20 those of its prototype. If is desired to test ir in a pressure wind tunnel at a speed equal to that of the prototype. If the air temperature is the same wheat must be the corresponding pressure in the wind tunnel relative to atmospheric pressure?

Answer: 

Pm = 20 atm