EME 1026 Fluid Mechanics Tri 1, 2014/20151 Tutorial 1 Density of an ideal gas 1. The density of oxygen contained in a tank is 2.0 kg/m 3 when the temperature is 25 C. Determine the gage pressure of the gas if the atmospheric pressure is 97 kPa. (58 kPa) Shear Stress 2. Crude oil having a viscosity of 9.52 x 10 -4 lb.s/ft 2 is contained between parallel plates. The bottom plate is fixed and the upper plate moves when a force P is applied. If the distance between the two plates is 0.1 in, what value of P is required to translate the plate with a velocity of 3 ft/s? The effective area of the upper plate is 200 in 2 . (0.476 lb) 3. Two layers of fluid is dragged along by the motion of an upper plate as shown in the figure below . The bottom plate is stationary. The top fluid puts a shear stress on upper plate, and the lower fluid puts a shear stress on the bottom plate. Determine the ratio of these two stresses. (1) 4. A 25-mm diameter shaft is pulled through a cylindrical bearing. The lubricant that fills the 0.3-mm gap between the shaft and the bearing is an oil having a kinematic viscosity 8.0 x 10 -4 m 2 /s and a specific gravity of 0.91. Determine the force P required to pull the shaft at a velocity of 3m/s. Assume the velocity distribution in the gap is linear. EME 1026 Fluid Mechanics Tri 1, 2014/2015 2 5. The viscosity of liquids can be measured through the use of a rotating cylinder viscometer of type illustrated in the figure below. In this device the outer cylinder is fixed and the inner cylinder is rotated with an angular velocity, . The torque required to develop is measured and the viscosity is calculated from these two measurements. Develop an equation relating , , , l, R o and R i . Neglect end effects and assume the velocity distribution in the gap is linear. (Diagram on page 7) ( i o R R w l Ri T 3 2 ) Cavitation 6. When a fluid flows through a sharp bend, low pressures may develop in localised regions of the bend. Estimate the minimum absolute pressure that can develop without causing cavitation if the fluid is water at 160 F. (4.74 psia) Surface Tension 7. Estimate the excess pressure inside a raindrop having a diameter of 3 mm. (97. 9 Pa) 8. An open, clean glass tube, having a diameter of 3 mm, is inserted vertically into a dish of mercury at 20 C. How far will the column of mercury in the tube be depressed? (3.00 mm) Manometry and Hydrostatic Force 9. For the inclined-tube manometer of the figure shown on the next page, the pressure in pipe A is 0.6 psi. The fluid in both pipes A and B is water and the gage fluid in the manometer has a specific gravity of 2.6. What is the pressure in pipe B corresponding to the differential reading shown? ( 0.224 psi) 10. A closed tank contains compressed air and oil ( SG oil = 0.90) as shown in the figure. A U-tube manometer using mercury ( SG HG =13.6) is connected to the tank. For column heights h 1 = 0.900 m , h 2 = 0.150 m and h 3 = 0.225m, determine the pressure reading of the gage. (Diagram on page 7) EME 1026 Fluid Mechanics Tri 1, 2014/2015 3 11. An inverted 0.1m diameter circular cylinder is partially filled with water and held in place as shown in the figure below. A force of 20 N is needed to pull the flat plate from the cylinder. Determine the air pressure within the cylinder. The plate is not fastened to the cylinder and has a negligible mass. (-4.51 kPa) 12. A homogeneous 4-ft wide, 8-ft long rectangular gate weighing 800 lb is held in place by a horizontal flexible cable as shown in the figure below. Water acts at the gate which is hinged at point A. Friction in the hinge is negligible. Determine the tension in the cable. (1350 lb) 13. An open tank has a vertical partition and on one side contains gasoline with a density = 700 kg/m 3 . At a depth of 4 m, as shown in the figure below. A rectangular gate that is 4 m high and 2 m wide and hinged at one end is located in the partition. Water is slowly added to the empty side of the tank. At what depth, h will the gate start to open? (3.55 m) EME 1026 Fluid Mechanics Tri 1, 2014/2015 4 14. A gate having the shape shown in the figure below is located in the vertical side of an open tank containing water. The gate is mounted on a horizontal shaft. (a) When the water level is at the top of the gate, determine the magnitude of the fluid force on the rectangular portion of the gate above the shaft and the magnitude of the fluid force on the semicircular portion of the gate below the shaft. (1060 kN (rectangle), 1010 kN ( semicircle)) (b) For this same fluid depth determine the moment of the force acting on the semicircular portion of the gate with respect to an axis which coincides with the shaft. ( 1.37 x 10 6 N.m) 15. The inclined face AD of the tank is a plane surface containing a gate ABC, which is hinged along line BC. The shape of the gate is shown in the plan view. If the tank contains water, determine the magnitude of the force that the water exerts on the gate. (599 lb) 16. A 4-m long curved gate is located in the side of a reservoir containing water as shown in the figure below. Determine the magnitude of the horizontal and vertical components of the force of the water on the gate. Will this force pass through point A? Explain. (882, 983 kN, yes) EME 1026 Fluid Mechanics Tri 1, 2014/2015 5 17. 18. A plug in the bottom of a pressurized tank is conical in shape as shown in the figure below. The air pressure is 50 kPa and the liquid in the tank has a specific weight of 27 kN/m 3 . Determine the magnitude, direction, and line of action of the force exerted on the curved surface of the cone within the tank due to the 50- kPa pressure and the liquid. (128 kN) Bouyancy and Flotation 19. The homogeneous wooden block A is 0.7m by 0.7m by 1.3 m and weighs 2.4 kN. The concrete block B (specific weight = 23.6 kN/m 3 ) is suspended from A by means of the slender cable causing A to float in the position indicated. Determine the volume of B. (0.0522 m 3 ) 20. The thin-walled, 1-m diameter tank of the figure shown is closed at one end and has a mass of 90 kg. The open end of the tank is lowered into the water and held in position shown by a steel block having a density of 7840 kg/m 3 . Assume that the air that is trapped in the tank is compressed at a constant temperature. Determine: (a) the reading on the pressure gage at the top of the tank, (18.9 kPa) A B 1.3 m 0.7 m water the figure. EME 1026 Fluid Mechanics Tri 1, 2014/2015 6 (b) the volume of the steel block. (0.208 m 3 ) Pressure variation with rigid body motion 21. A closed rectangular tank 1m wide and 2 m long contains gasoline to a depth of 1m. If the height of the tank is 1.5 m, what is the maximum horizontal acceleration ( along the long axis of the tank) that can develop before the gasoline begin to spill? (4.91m/s 2 ) 22. The open U-tube of the figure is partially filled with a liquid. When the device is accelerated with a horizontal acceleration a, a differential reading h develops between the manometer legs which are spaced at a distance l apart. Determine the relationship between a, l and h. ( h= al /g) 23. A closed, 0.4-m diameter cylindrical tank is completely filled with oil (SG=0.9) and rotates about its vertical longitudinal axis with an angular velocity of 40 rad/s. Determine the difference in pressure just under the vessel cover between a point on the circumference and a point on the axis. (28.8 kPa) EME 1026 Fluid Mechanics Tri 1, 2014/2015 7 Diagrams for Q 5 and Q 10 Q5 Q10