solved problems in hydrostatic

There is a simplest explaination of fluid mechanics

1. The diagram shows a pump delivering water through as pipe 30 mm bore to a tank. Find the pressure at point (1) when the flow rate is 0.0014 m 3 /s of water. The loss of pressure due to friction is 50 Kpa. Solution

2. Since Mean velocity in pipe = Applying Bernoulli’s Eq. between point “1” and the surface of the tank …………………(1)   Taking the datum passes through point “1” The pressure on the surface is zero (gage pressure) & P 2 = 50 KPa. The velocity at “1” is 1.98 m/s and at the surface is zero. Substituting into eq. (1) gives: =  

3. The diagram shows a tank that is drained by a horizontal pipe. Calculate pressure head at point (2) when the valve is partly closed so that the flow rate is reduced to 0. 020 m 3 /s. The pressure loss is equal to 2 m head. Solution

4. Applying Bernoulli’s equation (in head form) between point “1” and “2” gives or   Taking a horizontal datum through point “2” Point “1” in the free surface, then Z 1 = 0, and also V 1 is assumed negligible. Since Q =0.02 m 3 /s and = =  

5. A pump “A” whose characteristics are given in the following table, is used to pump water from an open tank through 40 m of 70 mm diameter pipe of friction factor = 0.02 to another tank in which the surface level of water is 5.0 m above that in the supply tank. a . Determine the flow rat e when the pump is operated at 1450 rev/min. (00857 m 3 /s) Head – Flow Characteristics of pump (A) when operating at 1450 rev/ min, It is desired to increase the flow rate and 3 possibilities are under investigation. To install a second identical pump in series with pump A. To install a second identical pump in parallel with pump A. To increase the speed of the pump by 10%. Predict the flow rate that would occur in each of these situations. Head (m) 9.75 8.83 7.73 6.90 5.50 3.83 Flow rate (L/s) 6.22 7.57 8.36 9.55 10.73

6. Q (L/S) H (m)

7. Q (L/S) H (m)

8. Water of h q 18 m / 3 V = flow rate has to be transported by the equipment shown in the figure. a) How wide pipe do we need to fulfill this task? b) Determine the maximal dike height where the transport is possible? (theoretical answer)

9. Valve Branch B, L B Branch A, L A Large tank Pump Branch C, L C Large tank Large tank 1 2 3 V A V B V C J

10. A turbine discharges 2.0 m 3 /s of water into a vertical draft tube as shown in the figure. The diameter of the tube is 0.80 m at “A”. If the head loss in the draft tube can be assumed as 1.5 times the velocity head at “A”, Estimate the pressure at “A”. (Ans. -30.32 kPa )  T A 1.20 m Dia. Da = 0.80 m 3.5 m B

11. A 90N rectangular solid block slides down a 30 o plan. The plane is lubricated by a 3 mm thick film of oil of relative density 0.90 and viscosity 8.0 poise. If the contact area is 0.3 m 2 (see the shown figure), Estimate the terminal velocity the block. At the terminal velocity, the sum of the forces acting on the block in the direction of its motion is zero. Hence Where: = shear stress on the block, and Area of the block   Solution: “w”= 90 N V = terminal velocity  

12. At the terminal velocity, the sum of the forces acting on the block in the direction of its motion is zero. Hence …….(1) where: y = thickness of oil film = shear stress on the block, and Area of the block Substituting the various values in eq. (1).  

14. A thin plate is placed between two flat surfaces “y” cm apart such that the viscosity of the plate are respectively. Determine the position of the thin plate such that the viscous resistance to uniform motion of the thin plate is minimum (assume “y” to be very small).   Solution: Let “y” be the distance of the thin flat plate from the top flat surface and “V” = velocity of the thin plate (see the attach figure). Since “ ϑ ” The shear stress on the top portion  

15. The gap on the bottom = h - y And t The total shear stress on the top and bottom of the plate Where: A & V are the area and velocity of the thin plate For F t to be minimum, From which or  

16. Thin flat plate h (h-y) y 1   2   V

17. A cylindrical shaft of 90 mm diameter rotates about a vertical axis inside a fixed cylindrical tube of length 50 cm and 95 mm internal diameter. If the space between the tube and the shaft is filled by a lubricant of dynamic viscosity 2.0 poise, Determine the power required to overcome viscous resistance when the shaft is rotated at a speed of 240 rpm (see the shown figure). Solution:

18. Two vertical plates are placed with a gap of 15 mm between them and the gap is filled with glycerin ( density of glycerin = 1260 kg/m 3 and  = 1.5 kg. S/ m 2 ) . An 80 cm square, 3 mm thick steel plate weighting 110 N is placed exactly midway in the glycerin - filled gap of the vertical plates. If it required to pull the steel plate vertically upwards at a constant velocity of 15 cm/s, Estimate the force required, (neglect the resistance at the edges of the plate).

19. 15 mm Gap 6 mm t s =3 mm Steel Plate V = 15 cm/s F F S F S w e

20. A hydraulic lift used for lifting automobiles has a 25 cm diameter ram which slides in a 25.018 cm diameter cylinder. The annular space being filled with oil having a kinematic viscosity of 3.7 cm 2 /s and relative density of 0.85. If the rate of travel of the ram is 15 cm/s, estimate: The frictional resistance when 3.30 m of ram is engaged in the cylinder. Solution: Since “ ϑ ” The shear stress Frictional resistance,  

21. Pressure has the dimension of: Is usually expressed in “Pascal” ( ), kilo Pascal Kpa ) If “h” is the height of a column of a fluid and “  g =  ” = in bars (= 10 5 Pa) or atmospheres =( number of standard atmospheric pressure values. The pressures are commonly indicated as gage pressures and unless a pressure is specifically marked absolute the pressure is treated as gage pressure . The atmosphere, however, is an exception and is an absolute pressure.   Absolute pressure = Local atmospheric pressure + Gage pressure

22. Gage pressure : is commonly measured by a Bourdon gage. Difference in pressure is measured by a manometers. Local atmospheric pressure : (i.e. the absolute pressure of the atmosphere at a place) is measured by a mercury barometer. The local atmospheric pressure varies with the elevation above mean see level and local meteorological conditions. For engineering application, a standard atmospheric pressure at mean sea level at 15 o C is often used. The value of this standard atmospheric pressure (called 1 atmosphere) is: 1 atm. = 760 mm of mercury =10.34 m of water = 101.33 KPa = 101.33 m bar

23. For the system shown in figure, calculate the height “H” of oil at which the rectangular hinged gate will just begin to rotate counterclockwise. Gate 1.5x 0.6 m Oil S.G.= 0.80 Oil S.G. = 0.80 Air H 1.5 m 30 KPa Hinge F 1 F 2 Oil Pressure Air Pressure 1.5 m (a) (b)

24. Solution Assume the force due to oil F 1 where: = = To calculate the center of pressure of F 1   And the force due to air F 2 It acts at a distance of 0.75 below the hinge (see the shown sketch)   Continue by your own

25. Answer A siphon consisting of a pipe of 15 cm diameter is used to empty kerosene oil (S. G. = 0.80) from tank “A”. The siphon discharges to the atmosphere at an elevation of 1.00 m. The oil surface in the tank is at elevation of 4.00 m. The centerline of the siphon pipe at its highest point “c” is at an elevation of 5.50 m. Estimate: The discharge in the pipe, The pressure at point “C”, The losses in the pipe can be assumed to be 0.50 m up to the summit and 1.20 m from the summit to the outlet.

26. Consider points “1” and “2” at the surface of the oil and at the end of the siphon (see the shown figure). By applying Bernoulli’s equation, we have: Continue by your own +   Assuming P 1 = P 2 =P atm = 0 (gage) and taking a reference datum through point “1”, Z 1 = 0 +   ………………     and =5.05 m/s    

27. For a hydraulic machine shown in the figure, the following data are available: Flow : From “A” to “B” Discharge : 200 L/s of water Diameters : at “A” 20 cm; at “B” 30 cm Elevation (m) : at “A” 105 m; at “B” 100 m Pressures : at “A” 100 kPa ; at “B” 200 kPa . Is this machine a pump or a turbine? Calculate the power input or output depending on whether it is pump or a turbine. ( Ans. The machine is a pump with P =6.965 kw)

28. A reaction turbine has a supply pipe of 0.8 m diameter and a draft tube with a diameter of 1.20 m at the turbine and expanding gradually downwards. The tail-water surface is 4.0 m below the centerline of the supply pipe at the turbine. For a discharge of 1.10 m 3 /s, the pressure head just upstream the turbine is 40 m. Estimate the power output of the turbine by assuming 85% efficiency, Also, determine the pressure at the commencement of the draft tube next to the turbine, which is 3.50 m above the tail-water, (see the attach figure). (Ans. P = 405 KW; P B = -34.79 kPa )

29. In a siphon the summit is 4 m above the water level in the reservoir from which the flow is being discharged out. If the head loss from the inlet of the siphon to the summit is 2 m and the velocity head at the summit is 0.5 m the pressure at the summit is: – 63.64 Kpa (b) - 9.0 m of water (c) 6.50 m of water (abs) (d) - 39.16 KPa. A liquid jet issues out from a nozzle inclined at an angle of 60 o to the horizontal and directed upwards. If the velocity of the jet at the nozzle is 18 m/s the maximum vertical distance attained by the jet, measured above the point of exit from the nozzle is: (a) 14.30 m (b) 16.51 m (c) 4.12 m (d) 12.39 m .

30. A Pump delivers 50 L/s of water and delivers 7.5 kw of power to the system. The head developed by the pump: 7.5 m (b) 5 m, (c) 1.53 m (d) 15.32 m. In a hydro-power project a turbine has a head of 50 m. The discharge in the feeding penstock is 3.60 m 3 /s. If a head loss of 5 m takes place due to losses, and a power of 1000 kw is extracted, the residual head downstream of the turbine is: 5.0 m (b) 10.95 m (c) 15.95 m (d) 20.95 m.

31. In a turbine having a flow of 1.20 m 3 /s the net head is 120 m. If the efficiency of the turbine is 90% the shaft power developed in kw, is: 1440 (b) 160 (c) 1566 (d) 1269

32. In a siphon the summit is 4 m above the water level in the reservoir from which the flow is being discharged out. If the head loss from the inlet of the siphon to the summit is 2 m and the velocity head at the summit is 0.50 m, the pressure at the summit is: -63.64 Kpa - 9.0 m of water, 6.5 m of water (absolute) - 39.16 Kpa. A liquid jet issues out from a nozzle inclined at an angle of 60 o to the horizontal and directed upwards. If the velocity of the jet at the nozzle is 18 m/s the maximum vertical distance attained by the jet, measured above the point of exit from the nozzle is: 14.30 m 16.51 m, 4.12 m, 12.39 m.

33. A pipeline delivering water from a reservoir is shown in the figure. A pump at “M” adds energy to the flow and 45 L/s of water is discharged to atmosphere at the outlet. Calculate the power delivered by the pump. Assume the head loss in the pipes as two times the velocity head at the suction side and 10 times the velocity head in the delivery pipe. Draw a neat sketch showing energy line and hydraulic grade lines. (Ans. P = 5.218 kw)

34. A centrifugal pump has an axial inlet of 12 cm diameter and an impeller of 30 cm diameter. The width of the impeller at the outlet is 25 mm (as shown in the figure). 5% of the outlet area can be assumed to be occupied by the blade. For a flow of 80 L/s, estimate: The radial component of velocity at the outlet of the impeller. What is the axial velocity in the inlet pipe? (Ans. V = 3.575 m/s & 7.074 m/s ) 30 cm dia. 12 cm dia. 5 mm V a V a

35. Solution Discharge and area at the outlet ) = Redial velocity at the outer edge of the impeller The continuity equation at the inlet gives,  

36. A conical valve of weight 2.5 KN deeps water from flowing out of a tank as shown in the figure. It is held in position by a counter weight of 10 KN connected by a string passing over frictionless pullies as in the figure. Find: the maximum height “H” of water in the tank, which will make the device to function without any leak. (Ans. H = 1.06 m) Weight of 1 0 KN 2.5 KN Cone 60 o Water 1.0 m H

37. For the container shown in the figure, Estimate the resultant force on the hemispherical bottom. (Ans. F y =52.46 KN, F x =0 “due to symmetry”) Air of 15 KPa Oil of S.G. = 0.75 3. 0 m 1. 50 m hemisphere

38. A cylinder of diameter 0.6 m is located in water as shown in the figure. The cylinder and the wall are smooth. If the length of the cylinder is 1.5 m, find: Its weight The resultant force exerted by the wall on the cylinder, and The resultant moment about the center of the cylinder due to water forces on the cylinder. (Ans. 4410 KN, 661.5 KN, zero) r = 3 m o

39. The counterweight pivot gate shown in the figure, controls the flow from a tank. The gate is rectangular and is 3m x 2 m. Determine the value of the counterweight “W” such that the upstream water can be 1.5 m. (Ans. W = 84.8 KN) 3.0 m 0.6 m 1.5 m w Pivot

40. Calculate the force “F” required to hold the hinged door shown in the figure in closed position. The door is a 0.5 m square. An air pressure of 30 Kpa acts over the water surface. (Ans. F = 7.217 KN) Air of 30 KPa Water Hinge Rectangular Door 2.50 m 0.50 m 0.50 m Door F