School for Mechanical Engineering
& Applied Mathematics
SELF STUDY QUESTIONS
IMNDNG
NDip: Engineering: Mechanical
FLUID MECHANICS III
(MFM31BI / MFM32BI)
COMPILED BY:
Mr. JJ Du Preez
REVISED BY:
Mr. L Masheane
DATE REVISED: JANUARY 2017
-2-
SELF STUDY 1
1.
Fresh water flows through a pipe with a diameter of 0,3 m at a rate of 0,283 m3/s. The pipe
diameter enlarges suddenly to 0,6 m in diameter. The axis of the pipe is horizontal and the
water in a vertical tube connected to the larger pipe stands 0,36 m higher than the level in a
tube connected to the smaller pipe. Calculate the coefficient K if the shock loss is expressed
as K v2/2g, where v is the velocity in the smaller pipe.
(0,4986)
2.
A pipe, 0,093 m2 in area, carries a discharge of 283 kg/s of water. If it enlarges suddenly to
0,372 m2 and the pressure in the smaller section is 4,8 kPa, find: (a) the head lost, (b) the
pressure in the larger part, (c) the power needed to force the water through the enlargement.
(0,265 m; 6,54 kPa; 737 W)
3.
A pipeline carrying 0,236 m3/s is reduced suddenly from 450 mm to 300 mm in diameter.
Calculate the change in: (a) the total energy head, (b) the pressure energy head. Take
CC = 0,67.
(0,1379 m; 0,5939 m)
4.
Determine the loss of head due to friction in a cast-iron pipe 360 m long and 150 mm in
diameter which carries 0,042 m3/s. Take f = 0,005.
(13,8 m)
5.
Using the Chezy formula, find the loss of head in a pipe 120 m long and 75 mm in diameter
when the velocity of flow is 4,8 m/s. Take C = 54,6 SI units.
(49,5 m)
6.
Water is discharged from a reservoir through a pipe 1200 m long which is 400 mm in
diameter for the first 600 m of pipe length, and 250 mm in diameter for the rest. The far end
of the pipe is 30 m below the water level in the reservoir and f for the 400 mm pipe is 0,004
and for the 250 mm pipe 0,006. Calculate the flow, taking only friction into account.(0.151 m3/s)
7.
Two reservoirs whose difference of level is 13,5 m are connected by a pipe ABC, whose
highest point B is 1,5 m below the level in the upper reservoir A. The portion AB has a
diameter of 200 mm and the portion BC a diameter of 150 mm, the coefficient of friction for
each being 0,005. The total length of the pipe is 3 km. Find the maximum allowable length of
the portion AB if the pressure at B is not to be more than 3 m below atmospheric pressure.
Neglect the velocity head in the pipe and the shock losses.
(2034 m)
8.
A pipe of 50 mm in diameter and 45 m long is connected to a large tank, the entrance to the
pipe being 3 m below the surface. The lower end of the pipe which is 6 m below the upper
end is joined to a horizontal pipe of 100 mm in diameter and 75 m long, which discharges to
the atmosphere. Calculate the discharge taking into account all the losses. Take f = 0,008.
(4,65 x 10-3 m3/s)
9.
Two reservoirs, whose surface levels differ by 30 m, are connected by a pipe 0,6 m in
diameter and 3 km long. The pipeline crosses a ridge whose summit is 9 m above the level of
and 300 m distant from the higher reservoir. Find the minimum depth below the ridge at
which the pipe must be laid if the absolute pressure in the pipe is not to fall below 3 m of
water and calculate the discharge. Take f = 0,0075 and Hatm = 10,32 m of water.
(0,559 m3/s; 4,87 m)
-3-
10.
A pump supplies water to a nozzle of 25 mm in diameter through a pipe 180 m long and
75 mm in diameter. The nozzle is at a level 9 m above the pump and f for the pipe is 0,012.
Calculate the pressure required at the pump outlet to give a discharge of 8 dm3/s in the
pipeline.
(408 kPa)
11.
Calculate the maximum power available at the outlet end of a hydraulic pipeline which is
4,8 km long and 200 mm in diameter. The pressure at the inlet of the pipe is 6,9 MPa. Take
f = 0,007.
(378 kW)
12.
A pipeline, 1800 m long and 375 mm in diameter, is fitted with a nozzle, with an effective
diameter of 50 mm, at the pipe outlet. The coefficient of velocity for the nozzle is 0,972. If f
for the pipe is 0,005, calculate: (a) the velocity of the jet, (b) the discharge, (c) the power of
the jet. Assume that the pressure at the pipe inlet is 240 m.
(66 m/s; 0,1296 m3/s; 279,9 kW)
13.
Water is to be conveyed to a Pelton Wheel through a pipe 1200 m long with a fall between
open level and the nozzle of 126 m. If the output power is to be 300 kW with a turbine
efficiency of 70 %, calculate the smallest size pipe which could be employed. Take f = 0,008.
(459 mm)
14.
What is the maximum rate at which energy can be transmitted through a 150 mm diameter
pipeline, 3 km long, supplied with water at a pressure of 8,3 MPa? Take f = 0,01.
(257 kW)
15.
Determine the diameter of pipe required to supply a turbine developing 1500 kW under a
gross head of 150 m. Assume pipe transmission efficiency of 95 % and a turbine efficiency of
86 %. The pipe is 3000 m long and Chezy's constant is 66 SI units.
(0,985 m)
16.
A pipeline transmits 260 kW over a distance of 2,4 km. If the supply pressure is 3,3 MPa,
calculate the number of 150 mm diameter pipes required to transmit this power with an
efficiency of 92 %. Take f = 0,01.
(6)
17.
Two reservoirs 4,8 km apart are connected by a pipeline which consists of a 150 mm diameter
pipe for the first 1,6 km, sloping 5,7 m/km, and a 225 mm diameter pipe for the remaining
distance, having a slope of 1,9 m/km. The water levels above the pipe openings in the
respective reservoirs are 6 m in the upper reservoir and 3,6 m in the lower reservoir. Taking
f = 0,0075, calculate the discharge through the pipeline. Take all the losses into account and
draw the hydraulic gradient and total energy gradient.
(16,3 dm3/s)
18.
Two tanks are joined by a 100 mm diameter pipe 30 m in length. The difference in level in
the tanks is 6 m and the ends of the pipe are 3 m under water. Both pipe ends are sharp.
Sketch the hydraulic gradient and calculate: (a) the velocity in the pipe, (b) the pressure head
in the pipe midway along its length. Take f = 0,01.
(2,95 m/s; 2,68 m)
19.
Two reservoirs having a difference of level of 6 m are connected by a single pipeline, with a
diameter of 600 mm, for the first 3000 m of pipe length. This pipe branches into two parallel
pipes, each with a diameter of 300 mm and a length of 3 km, which discharges the water into
the lower reservoir. Take f = 0,01. What will be total discharge?
(72,5 dm3/s)
-4-
20.
Two reservoirs, whose surface levels differ by a constant 66 m, are connected by a pipe
225 mm in diameter and 4 km long. The pipe branches at a distance of 1,6 km from the upper
reservoir and discharges some of the water directly to the atmosphere at a rate of 42,5 dm3/s.
Calculate the discharge into the lower reservoir, neglecting the shock losses. Take f = 0,009.
Sketch the hydraulic gradient for the pipeline.
(35,5 dm3/s)
21.
A reservoir, 60 m above the datum, supply water to a junction box through a 300 mm
diameter pipeline which is 1500 m in length. From the junction box discharges two other
pipes, each with a diameter of 300 mm and 1500 m long, water to two reservoirs whose
surface levels are 15 m and 30 m above the datum respectively. Determine the flow rate
entering each reservoir. Take f = 0,01.
(27,8 dm3/s; 90,2 dm3/s)
22.
Water flows from a reservoir through a pipe, 150 mm in diameter and 180 m long, to a point
13,5 m below the open surface of the reservoir. Here it branches into two pipes, each of
100 mm in diameter, one of which is 48 m long, discharging to the atmosphere at a point
18 m below the reservoir level, and the other 60 m long, discharging to atmosphere 24 m
below the reservoir level. Assuming a coefficient of friction of 0,008 and neglect the loss in
the junction, calculate the discharge from each pipe.
(19,5 dm3/s; 25,8 dm3/s)
23.
Water is discharged from a reservoir through a pipe 150 mm in diameter and 120 m long.
This pipe divides into two pipes each of 75 mm in diameter. One is 30 m long and discharges
into a second reservoir with water level 12 m below the first. The other is 60 m long and
discharges into a third reservoir with water level 24 m below the first. Taking f = 0,01 for
each pipe, find the discharge into each reservoir. Neglect all losses other than pipe friction.
(13,55 dm3/s; 15,35 dm3/s)
SELFSTUDY 2
1.
Fresh water with a viscosity of 0,012 Poise flow through a pipe 50 mm in diameter at a
discharge of 2,8 dm3/s. Calculate the pressure drop over a length of 6 m of pipe. Given that
f = 0,064 Re-0,23.
(2,488 kPa)
2.
Oil with a density of 880 kg/m3 flows under a head of 30 m through a pipe 300 mm in
diameter and 3 km in length. Due to cooling the viscosity changes along the length and may
be taken as 0,57 kg/ms over the first 1,5 km and 1,14 kg/ms over the second 1,5 km. Verify
that laminar flow conditions exist and determine the discharge. Hint: For laminar flow assume
that f = 16/Re.
(20,07 dm3/s)
3.
Oil of viscosity of 0,048 Pas flows through an 18 mm diameter pipe with a mean velocity of
0,3 m/s. Calculate the loss in pressure over as length of 45 m of pipe, as well as the velocity at
a distance of 3 mm from the wall of the pipe.
(64 kPa; 0,332 m/s)
4.
Oil of viscosity of 0,048 Pas flows through a 25 mm diameter pipe with a mean velocity of
0,3 m/s. Calculate the loss in pressure over a length of 30 m of pipe, as well as the velocity at
a distance of 6 mm from the wall of the pipe.
(22,1 kPa; 0,438 m/s)
5.
A pipe 75 mm in diameter and 900 mm long conveys oil with a specific gravity of 0,85 and a
kinematic viscosity of 3,3 stokes at a rate of 40 Mg per hour. Confirm that the flow is laminar
and find the power absorbed in overcoming friction in the pipe. (Hint: For laminar flow is
f = 16/Re).
(55,4 kW)
-5-
6.
The velocity along the centre-line of a 150 mm diameter pipe conveying oil is 3 m/s. The
viscosity of the oil is 1,2 Poises and its density is 900 kg/m3. Assuming that the velocity
distribution across the pipe is parabolic, calculate flow rate, as well as the shear stress in the
oil at the pipe wall. Also verify that the flow is laminar.
(26,51 dm3/s; 9,6 Pa)
7.
An oil cooler consists of tubes of 12 mm in diameter and 3,5 m long. Oil with a specific
gravity of 0,9 is pumped through the tubes at a velocity of 1,8 m/s. The viscosity of the oil
changes from 0,28 Poises to 1 Poises across the inlet and outlet of the tubes. It may be taken
to vary as a linear function of the length. Calculate the power required to pump the oil
through a group of 200 tubes.
(3,65 kW)
8.
Oil of viscosity of 0,048 Pas flows through a 50 mm diameter pipe with a mean velocity of
0,12 m/s. Calculate the loss in pressure over a length of 65 m of pipe, as well as the velocity
at a distance of 10 mm from the wall of the pipe.
(4,8 kPa; 0,153 m/s)
9.
Oil with a specific gravity of 0,9 and a kinematic viscosity of 0,00033 m2/s is pumped, at a
rate of 25 Mg/hour, through a 75 mm diameter pipe which is 1,5 km in length. Confirm that
the flow is laminar and find the power required by the pump. Assume a mechanical efficiency
of 70 % for the pump.
(Re = 396,9 SI units; 48,8 kW)
10.
Calculate the shear stress in the oil at the pipe wall for the data given in problem No 9.
(55,3 N/m2)
11.
Oil having a viscosity of 0,083 kg/ms flows between two very large parallel flat plates 24 mm
apart. The mean velocity of the oil is 0,15 m/s. What is the shearing stress at 6 mm and
12 mm from the lower plate?
(1,56 N/m2; 0)
12.
Oil of viscosity 0,8 Poises leaks from a container through a joint which is 0,6 m wide, 50 mm
long in the direction of flow. The gap between the parallel surfaces is 0,25 mm. Calculate the
volume of oil escaping per hour if the pressure difference between the inside and outside is
35 kPa.
(24,6 dm3/hour)
13.
A fluid with a density of 1260 kg/m3 and viscosity of 0,9 kg/ms flows through two infinite
large parallel flat plates 2 cm apart. If the flow rate is 0,5 dm3/s per unit width, calculate the
pressure drop per unit width if both plates are stationary.
(675 Pa/m)
14.
A smooth cylinder 50,1 mm in diameter and 100 mm long is placed with its axis vertical. If
the clearance space is entirely filled with oil of viscosity of 2,5 Poises, calculate the force
required to push a shaft of 50 mm in diameter through the cylinder with a velocity of 0,6 m/s.
(47,1 N)
15.
A piston with a diameter of 100 mm and 150 mm long slides concentrically in a stationary
cylinder of 100,1 mm in diameter. The clearance space is filled with oil of viscosity
0,175 kg/ms. Find the force required to slide the piston along the cylinder at a speed of 3 m/s
against the viscous resistance of the oil.
(494,8 N)
16.
The radial clearance between a plunger and the walls of a cylinder is 0,075 mm. The length of
the cylinder is 250 mm and its diameter is 100 mm. There is a difference in pressure of the
water on the two ends of the plunger of 207 kN/m2 and the dynamic viscosity of the water is
1,31 x 10-3 kg/ms. Treating the flow as if it occurred between parallel flat plates, calculate the
rate of leakage in dm3/hour.
(25,1 dm3/h)
-6-
17.
A storage tank containing oil of viscosity 0,7 Poises is cylindrical with its axis vertical and is
6 m in diameter. When the oil is under pressure at 345 kPa, leakage occurs at a
circumferential seam which consists of a riveted lap joint. The effective gap between the
plates is 0,025 mm and the plates overlap by 100 mm. The rivets reduce the effective
circumferential length of the opening by 40 %. Calculate the rate of leakage in dm3/h.
(2,61 dm3/h)
18.
A dashpot consists of a piston 143,5 mm in diameter working concentrically in a cylinder of
143,6 mm bore. The cylinder contains oil with a viscosity of 0,8 Poises. The length of the
piston is 250 mm. Calculate the force which must be applied to the piston to give it a velocity
of 0,003 m/s.
(3342 kN)
19.
The radial clearance between an hydraulic plunger and the cylinder wall is 0,1 mm. The
length of the plunger is 0,3 m and its diameter 100 mm. Find the velocity and the rate of
leakage past the plunger at an instant when the difference of pressure between the two ends of
the plunger is 9 m of water. Use µ = 1,31 x 10-3 kg/ms.
(0,187 m/s; 0,353 dm3/min)
20.
A shaft of 75 mm in diameter revolves concentrically in a bearing 150 mm in length. The
radial clearance is 1 mm and the speed is 3000 r/min. The viscosity of the oil is 3 Poises. Find
the resisting torque due to viscosity and the power absorbed in overcoming it.
(4,68 Nm; 1,47 kW)
21.
A shaft of 75 mm diameter revolves concentrically in a fixed tube of diameter 75,5 mm and
300 mm in length. The annular space is full of oil and it is found that a torque of 1 Nm is
required to drive the shaft at 2400 r/min. What is coefficient of viscosity of the oil? What
would be the critical velocity of this oil when flowing in a pipe of 75 mm in diameter if the
critical Reynolds number is taken as 2300 and the density of the oil is 960 kg/m3?
(0,01 Pas; 0,319 m/s)
22.
A shaft of 150 mm in diameter turns concentrically in a sleeve 150,15 mm in diameter and
225 mm long. The clearance space is filled with oil. The power required to turn the shaft at
1000 r/min is 2,2 kW. Calculate the viscosity of the oil.
(0,252 Poises)
23.
A shaft, 100 mm in diameter, revolves concentrically in a bearing 150 mm long. The radial
clearance is 1,25 mm and the speed is 3000 r/min. The viscosity of the oil is 2,8 Poises. Find
the resisting torque due to viscosity and the power absorbed in overcoming it.
(8,29 Nm; 2,6 kW)
24.
A shaft, 150 mm in diameter, runs in a 300 mm long bearing at a speed of 300 r/min. If an oil
of viscosity 1,54 Poises fills the space between the shaft and the bearing, find the power
absorbed due to viscous drag. Cr = 0,625 mm.
(1,934 kW)
25.
The thrust of a shaft is taken by a collar bearing. A thin film of oil of uniform thickness is
maintained between the surfaces of the collars and the bearing pads. The bearing surfaces are
three annular pads of inside diameter 100 mm and outside diameter 150 mm. The thickness of
the oil film is 0,25 mm and its viscosity is 0,85 Poises. Find the power lost in overcoming the
viscous torque in the bearing when the shaft rotates at 1400 r/min.
(874 W)
26.
The thrust of a shaft is taken by a collar bearing fitted with a forced lubrication system which
maintains a film of oil of constant thickness of 0,3 mm between the surface of the collar and
the surface of the bearings. The outer and inner diameters of the collar are 160 mm and
120 mm respectively. The viscosity of the oil is 0,12 kg/ms. Calculate the power lost in the
bearing due to viscous drag when the shaft rotates at 500 r/min.
(48 W)
-7-
SELF STUDY 3
1.
A vertical cylindrical tank, 0,6 m in diameter and 1,5 m high, has an orifice of 25 mm
diameter in the bottom. The discharge coefficient is 0,61. If the tank is originally full of
water, what time is required to lower the level by 0,9 m?
(192 sec)
2.
Water is discharging from a bell-mouthed orifice (Cd = 1) of 50 mm diameter in the base of a
tank having a surface area of 9 m2. How long will it take to reduce the depth in the tank from
1,2 m to 0,3 m above the orifice?
(1135 sec)
3.
A cylindrical vessel with its axis vertical is filled with water and discharges through an orifice
25 mm in diameter at the bottom with a coefficient of 0,623. If the diameter of the vessel is
0,6 m, find the time required for the water level to drop from 1,8 m to 0,6 m above the orifice
when the supply is cut off.
(237 sec)
4.
Discharge takes place from a 1 m diameter cylindrical tank whose axis is vertical, through a
25 mm diameter pipe and 3 m in length. The pipe is connected to the base of the tank and
discharges to atmosphere 2 m below the base. Initially the level in the tank is steady, water
entering and leaving at a constant rate of 2 litres per second. If the supply of water is suddenly
stopped, calculate the time required to empty it completely. Assume that the friction
coefficient for the pipe is constant at 0,01, and that the tank outlet is sharp.
(27 min 2 sec)
5.
A vertical cylindrical tank is 4,8 m in diameter and discharges through a pipe 90 m long and
225 mm in diameter. How long will it take for the water level in the tank to fall from 2,7 m
above the pipe exit to 1,2 m above that level? Assume that f = 0,01, and that the tank outlet is
sharp.
(470,8 sec)
6.
Two water tanks A and B, whose constant cross-sectional areas are 7,4 m2 and 3,7 m2
respectively, are connected by a 50 mm diameter pipe, 120 m long, for which f = 0,01. The
initial difference in the water levels are 1,5 m. Find the time taken for 2250 litres of water to
pass from tank A into tank B.
(42 min 25 sec)
7.
Two cylindrical tanks, with its axis vertical, stand on a horizontal floor. One is 1,8 m in
diameter, the other is 1,2 m in diameter, and they are joined by a pipe, 75 mm in diameter and
1,8 m long, with sharp entrance and exit. The tanks are partly filled with water and at a given
instant the level in the smaller tank is 1,2 m higher than that in the larger. Assuming f = 0,009,
calculate the time taken for the difference of levels to become 0,3 m.
(67,4 sec)
8.
Two vertical-sided reservoirs each have a surface area of 186 m2 and are connected by a
submerged opening of area 0,186 m2, which can be considered as an orifice with a coefficient
of discharge of 0,8. If the initial difference of surface levels is 2,7 m, how long will it be
before this difference is 1,2 m?
(2 min 34,5 sec)
SELF STUDY 4
1.
An undershot waterwheel consists of a series of flat vanes, mounted radially on a wheel of
large diameter, which are struck normally by a jet of water 0,3 m in diameter. If the velocity
of the water leaving the nozzle is 7,5 m/s and the velocity of the vanes is 4,8 m/s, what is the
force exerted by the jet on the vanes, the work done per second and the hydraulic efficiency?
(1,431 kN; 6,87 kW; 46 %)
-8-
2.
A flat plate is struck normally by a jet of water 50 mm in diameter with a velocity of 18 m/s.
Calculate (a) the force on the plate when it is stationary, (b) the force on the plate when it
moves in the same direction as the jet at 6 m/s, (c) the work done per second and the
hydraulic efficiency in case (b).
(636 N; 283 N; 1,696 kW; 29,6 %)
3.
A square plate, mass 12,7 kg, of uniform thickness and 300 mm edges, is hung so that it can
swing freely about its upper horizontal edge. A horizontal jet of water, 20 mm in diameter,
strikes the plate with a velocity of 15 m/s normally at its centre when the plate hangs vertical.
Find: (a) what force must be applied at the lower edge of the plate to keep it vertical, (b) what
inclination to the vertical the plate will assume under the action of the jet if allowed to swing
freely.
(35,34 N ; 34,57° )
4.
A rectangular plate of mass 5,45 kg is suspended vertically by a hinge on the top horizontal
edge. The centre of gravity of the plate is 10 cm from the hinge. A horizontal jet of water of
25 mm in diameter, whose axis is 15 cm below the hinge, impinges normally on the plate with
a velocity of 5,65 m/s. Find the horizontal force applied at the centre gravity to maintain the
plate in its vertical position. Find the alteration of the velocity of the jet if the plate is
deflected through an angle of 30 degrees, and the same horizontal force continues to act at the
centre of gravity of the plate.
(23,5 N; 2,35 m/s)
5.
A jet of water, 50 mm in diameter, with a velocity of 18 m/s, strikes a flat plate inclined at an
angle of 25° to the axis of the jet. Determine the normal force exerted on the plate when: (a)
the plate is stationary, (b) the plate is moving at 4,5 m/s in the direction of the jet, and (c)
determine the power transferred and the hydraulic efficiency for case (b).
(268,9 N; 151,2 N; 287,6 W; 5 %)
6.
A jet of water, 75 mm in diameter, strikes a flat plate with a velocity of 24 m/s. The normal to
the plate is inclined at 30° to the axis of the jet. Calculate the normal force on the plate when:
(a) the plate is stationary, (b) the plate has a velocity of 12 m/s in the same direction as the jet.
(2,2 kN; 0,551 kN)
7.
A tanker discharges a jet of water horizontally backwards with a velocity of 4,8 m/s. If the
rate of discharge is 85 dm3/s, what force is required to keep the tanker at rest?
(407,99 N)
8.
Calculate the reaction of a jet on a nozzle connected to a pipe. The diameter of the pipe is
0,1 m and the mean velocity of flow in the pipe is 15 m/s. The pressure in the pipe is
2070 kPa. Take Cv = 0,96.
(5,5 kN)
9.
A horizontal pipe gradually reduces in diameter from 300 mm to 150 mm. Determine the
thrust exerted on the reducer if at the larger end the pressure is 275 kPa and the velocity of the
water is 3 m/s.
(13,863 kN)
10.
Water flows through a 910 mm diameter pipe at the end of which there is a reducer
connecting to a 590 mm diameter pipe. If the pressure at the entrance to the reducer is
400 kPa and the velocity is 2 m/s, determine the resultant thrust on the reducer, assuming that
the frictional loss of head in the reducer is 1,2 m.
(153 kN)
11.
A jet-propelled vessel takes in water through ducts amidships and discharges it through ducts
astern. The discharge is 34 m3/min and the velocity of flow through the ducts is 9 m/s. The
speed of the vessel is 4,5 m/s. Calculate the magnitude of the propulsive force.
(2,55 kN)
-9-
12.
A jet of water, 6,5 cm2 in cross-sectional area, moving at 12 m/s, is turned through an angle of
135° by a curved plate. The plate is moving at 4,5 m/s in the same direction as the jet.
Neglecting any loss of velocity by shock or friction, find the power transferred to the plate.
(281 W)
13.
A jet of water, 75 mm in diameter, with velocity 21 m/s flows tangentially onto a stationary
vane which deflects it through 120°. What is the magnitude and direction, referred to the
direction of the jet, of the resultant force on the vane? If this jet flows onto a series of vanes
similarly oriented with regard to it but moving in the direction of the jet with a velocity of
10,5 m/s, determine: (a) the force on the system of vanes in the direction of motion, (b) the
work done per second, (c) the efficiency.
(3375 N; 30°; 1461 N; 15,342 kW; 75 %)
14.
A jet of water of 50 mm in diameter, having a velocity of 24 m/s impinges tangentially on a
series of vanes which, when stationary, deflect the jet through an angle of 120°. Calculate the
magnitude of the force on the vanes in the direction of motion when they are: (a) stationary,
(b) moving with a velocity of 9 m/s in the same direction as the jet. (c) Calculate the work
done per second and the hydraulic efficiency for case (b).
(1,696 kN; 1,061 kN; 9,54 kW; 70,3 %)
15.
A jet of water, flowing at a rate of 20 kg/s and at a velocity of 25 m/s, impinges on a series of
vanes moving at 12 m/s in a direction making an angle of 25° to the jet. Determine the inlet
angle of the blade for no shock entry. If the vane exit angle is 150° to the direction of motion
and the power developed if friction reduces the water velocity relative to the vanes by 20 %
during its passage over the vanes.
(45°; 421,1 N; 5,053 kW)
16.
A jet of water discharges 13,6 kg/s at 24 m/s in a direction making 30° to the direction of
motion of a series of curved vanes moving at 10,5 m/s. If the outlet angle of the vanes is 20°,
determine: (a) the inlet angle of the vanes such that there is no shock at entry, (b) the work
done per second on the wheel.
(49,4°; 3,589 kW)
17.
A nozzle, having a coefficient of velocity of 0,96, operates under a head of 90 m of water and
directs a 50 mm diameter jet of water on to a ring of axial flow impulse blades, which have an
inlet angle of 40°, measured relative to the direction of blade motion. The blades turn the
water through an angle of 105° and because of friction the velocity of the water relative to the
blades is reduced by 15 % during its passage over the blades. The blade speed is to be 18 m/s
and the water is to flow on to the blades without shock. Calculate: (a) the angle which the line
of the jet will make with the direction of motion of the blades, (b) the power developed by the
blade ring, (c) the hydraulic efficiency.
(23,3°; 51,8 kW; 80,4 %)
18.
A jet of fresh water, moving at a velocity of 25 m/s, strikes a series of vanes on the periphery
of a turbine wheel which moves at a pitch line velocity of 12 m/s in a direction making an
angle of 20° to the jet. The vane outlet angle is 20°. Neglect the frictional losses across the
vanes. Calculate: (a) the inlet vane angle for no shock, (b) the power/kg of water exerted by
the wheel, (c) the efficiency of the system.
(36,65°; 299 W; 95,7 %)
19.
A jet of water with a velocity of 30 m/s strikes a series of vanes moving at a speed of 15 m/s
in a direction making an angle of 30° to the jet. The jet leaves the vanes relative at 15°.
Neglect friction loss across the vanes. Calculate: (a) the vane inlet angle for no shock, (b) the
work done by the wheel/kg of water/sec, (c) the hydraulic efficiency. (53,8°; 434 W; 96,4 %)
- 10 -
20.
A jet of fresh water strikes a single vane type of turbine at 30° with the direction of motion,
and leaves it at 160° absolute with the direction of motion. The absolute velocity at inlet is
25 m/s and the vane moves at a velocity of 12 m/s. Ignore friction across the vane. Calculate:
(a) the vane inlet angle and outlet angle, (b) the power/kg of water, (c) the hydraulic
efficiency.
(52,33°; 4,94°; 304,6 W; 97,5 %)
21.
A technician employed at a company which manufacture turbines must determine the required
height of a reservoir above a certain turbine which has to develop 2,635 kW. The diameter of
the jet is 100 mm. The jet strikes the vanes tangentially. The vanes deflect the water through
an angle of 110°. The speed of the vanes is 5 m/s in the direction of the jet. Cv = 0,65. Ignore
the losses in the supply pipe and across the vanes. Find: (a) the jet velocity to generate the
required power, (b) the height of the water level in the reservoir above to the turbine.
(10 m/s; 12,06 m)
22.
A stream of fresh water flows through a rectangular duct and collides against the flat blades
of an undershot water wheel. The blades are 1,2 m in width, 0,5 m high and are fully in
contact with the water. The pressure head is 4 m and the pitch circle diameter of the blades is
5 m. The wheel turns at a speed of 20 r/min. Find: (a) the force exerted on each blade, (b) the
power transmitted to the wheel, assuming that one blade is continuously in contact with the
stream, (c) the hydraulic efficiency of the wheel.
(19,26 kN; 100,8 kW; 48,3 %)
SELF STUDY 5
1.
It is found that at a free vortex, which started in a large reservoir, is a point on the free surface
at a diameter of 300 mm, 75 mm below the level of the free surface of the water in the
reservoir. What will be the level of the surface at a diameter of 600 mm below the water level
in the reservoir?
(18,75 mm)
2.
A cylindrical tank has a height of 1,2 m and a diameter of 0,6 m and is three-quarters filled
with water when stationary. If the tank rotates about its vertical axis at 150 r/min some of the
water is spilled over the sides. Calculate: (a) the depth of the water below the depression of
the vortex while the cylinder rotate, (b) the total depth of the water in the cylinder when it is
stationary again, (c) the quantity of water spilled in litres.
(68 mm; 634 mm; 75,2 dm3)
3.
A cylindrical vessel is 1,5 m high and 0,8 m in diameter. It rotates about its vertical axis at
120 r/min. The vessel is open at the top, and was completely filled with water before the
rotation started. Calculate how many litres of water are left in the vessel after the rotation.
(430,3 dm3)
4.
A cylindrical tank has a diameter of 0,6 m, a height of 1,15 m and is open to the atmosphere
at the top. It rotates about its vertical axis at a speed of 105 r/min. Before the rotation started
the tank was full of water. Calculate how many water remain in the tank after the rotation.
(247 dm3)
5.
A closed cylindrical tank has a diameter of 650 mm and rotates about its vertical axis at a
speed of 600 r/min. If the tank is fully filled with water, calculate the pressure generated
inside the cylinder.
(208,5 kPa)
6.
A cylindrical drum, with its axis vertical, has a inside- diameter of 600 mm and is completely
filled with water. An impeller, with a diameter of 200 mm, rotates at the bottom of the drum
at a speed of 120 r/min. Calculate: (a) the velocities of the water on radii of 75 mm and
225 mm respectively, (b) the corresponding pressure heads at these radii, measured from the
bottom of the forced vortex.
(0,942 m/s; 0,559 m/s; 45,2 mm; 145,1 mm)
- 11 -
7.
A compound vortex formed in a large reservoir having a free surface comprises a central
forced vortex surrounded by a free vortex. The assembly is completely developed. The
separation between the forced- and free vortex occurs at a radius of 200 mm. The total
depression is 900 mm. Find the rotational speed of the forced vortex.
(141,9 r/min)
8.
The impeller of a centrifugal pump discharges fresh water at a velocity of 20 m/s against a
pressure of 12 m of water. The pump must deliver the water against a pressure of 20 m of
water. Calculate the required diameter of the vortex chamber if the impeller has a diameter of
600 mm.
(769,7 mm)
9.
A centrifugal pump rotates at a speed of 800 r/min with its delivery valve closed. The
impeller has a diameter of 250 mm. The pressure at the inlet to the impeller is -25 mm of
mercury. Assume that a forced vortex is generated in the impeller. Find now the pressure at
the periphery of the impeller.
(51,49 kPa)
10.
Water is discharged from the impeller of a centrifugal pump against a pressure of 10,5 m and
a tangential velocity component of 14 m/s. The impeller has a diameter of 60 cm. Calculate
the required diameter of the vortex chamber to increase the delivery pressure to 15 m.
(809 mm)
11.
A centrifugal pump has a diameter of 260 mm and a vortex chamber with a diameter of
340 mm. Calculate the required speed for the pump if a pressure of 18 m of water is gained
inside the vortex chamber.
(2142,5 r/min)
SELF STUDY 6
1.
A single-acting reciprocating piston pump has a piston diameter of 200 mm and a stroke
length of 600 mm. The speed of the pump is 20 r/min. Discharge occurs through a pipe with a
length of 45 m and a diameter of 100 mm, with f = 0,008. Calculate the saving in power if an
air vessel is fitted near the delivery side of the pump.
(161,75W)
2.
The vertical suction pipe of a single-acting piston pump is 2 m in length and 50 mm in
diameter. The piston has a diameter of 150 mm and a stroke length of 300 mm. The pump is
1,9 m above the water level in the sump. Find the maximum allowable speed for the pump
just before separation occurs. Assume the atmospheric pressure to be 10,3 m of water, and
that separation will occur if the absolute pressure drop below 1,8 m of water.
(46,8 r/min)
3.
The piston diameter of a single-acting reciprocating piston pump is 115 mm and a stroke
length of 230 mm. The diameter of the suction pipe is 90 mm and 4,2 m long. If separation
occurs at an absolute pressure of 1,2 m, calculate the maximum allowable speed at which the
pump can be operated before separation will occur. Assume a barometer reading of 757,5 mm
of mercury. The water level in the sump is 3 m below the pump centre. Find also the power
needed to overcome friction in the suction pipe at this speed. ( f = 0,01)
(83,2 r/min; 5,5 W)
4.
A double-acting reciprocating piston pump which has a stroke length of 350 mm and a bore
of 175 mm, takes water in from a sump 3 m below and deliver it at a height of 46 m above the
pump level. Both the suction and delivery pipes have a diameter of 100 mm and lengths of
6 m and 75 m respectively. The pump piston follows the motion of simple harmonic at
40 double strokes per minute. Large air vessels are fitted on both sides of the pump. On the
suction side is the air vessel 1,5 m and on the delivery side 4,5 m away from the pump. The
coefficient of friction for both pipes is 0,008. Calculate the pressure difference across the
piston at the beginning of the stroke.
(57,5 m)
- 12 -
5.
A double-acting reciprocating pump has a piston diameter of 200 mm and a stroke of 0,6 m
and runs at 20 r/min. It discharges through a 150 mm main 75 m long ( f = 0,0075) with a
vertical lift of 45 m. Assuming the piston to have s.h.m. and that no air vessel is used, sketch
the part of the indicator card corresponding to discharge giving the heads in the cylinder at the
ends and middle of the stroke. Neglect friction at the discharge valve. (62,9 m, 45,96 m, 27,1 m)
6.
A single-acting reciprocating pump has a plunger diameter of 250 mm and a stroke of
450 mm. The delivery pipe is 110 mm diameter and 48 m long. If the plunger moves with
simple harmonic motion, find the power saved in overcoming friction in the delivery pipe by
the provision of a large air vessel on this pipe close to the cylinder when the pump is driven at
20 rev/min, taking ƒ = 0,01.
(215 W)
7.
A double-acting reciprocating pump is used to raise water to a height of 42 m through a
delivery pipe of diameter 75 mm and length 81 m. The pump speed is 180 r/min, the stroke is
250 mm and the piston diameter is 115 mm. A large air vessel is fitted in the delivery pipe at
6 m from the cylinder, measured along the pipe. Determine the absolute pressure at the end of
each delivery stroke, given that the friction coefficient for the pipe is 0,007. It should be
assumed that the piston moves with simple harmonic motion, that the effect of the piston rod
is negligible, and that the atmospheric pressure is 10,2 m of water.
(6,94 m of water)
8.
A double-acting single-cylinder reciprocating pump of 190 mm bore and 380 mm stroke runs
at 36 double strokes/min, suction head 3,6 m, and discharge head 30 m. The length of the
suction pipe is 9 m and of the discharge pipe 60 m, and the diameter of each pipe 100 mm.
Large air vessels are provided 3 m away from the pump on the suction side, and 6 m away on
the discharge side, both measured along the pipelines, ƒ = 0,008. Neglecting entrance and exit
losses for the pipes, estimate for the beginning of the stroke: (a) the head in the two ends of
the cylinder, (b) the load on the piston rod, neglecting the size of the piston rod and assuming
simple harmonic motion.
(34,04 m, -4,75 m, 10,8 kN)
9.
A single-acting reciprocating pump has a piston of 200 mm diameter and 600 mm stroke. It
run at 20 r/min with simple harmonic motion. Delivery is through a 100 mm diameter pipe of
length 45 m for which ƒ = 0,008. Find the power which would be saved by fitting an air
vessel to the delivery side assuming that there would then be no acceleration in the pipe. (66 W)
10.
Sketch theoretical indicator diagrams for a single-cylinder single-acting reciprocating pump
not fitted with air vessels. Use your diagram to explain clearly the effect of acceleration and
friction on both suction and delivery strokes. Assuming simple harmonic motion of the piston,
develop an expression for the acceleration head in the cylinder at the beginning of the suction
stroke in such a pump. The following data relate to a pump of the type described above:
length of suction pipe, 9 m; diameter of suction pipe, 75 mm; suction lift, 3 m; plunger
diameter, 125 mm; stroke, 300 mm; speed, 30 rev/min. Calculate the theoretical absolute
pressure head in meter of water at the beginning and end of the suction stroke. Barometric
pressure corresponds to 10,2 m of water.
(3,43 m, 10,97 m)
11.
A reciprocating pump has three single-acting cylinders the pistons of which are operated by
cranks at 120° apart. The pistons are 75 mm diameter and have a stroke of 150 mm. The
cylinders discharge water into a single pipe 50 mm diameter and 60 m long. The pump speed
is 60 r/min and there is no air vessel on the delivery side. Give diagrams on a crank angle base
showing how the water velocity and acceleration in the discharge pipe differ from those
obtained with a single-acting single-cylinder pump of the above dimensions and speed. If the
pipe discharges to air 30 m above the level of the cylinders calculate the range of pressure in
the pipe just beyond the pump. Take ƒ for the pipe as 0,01.
(52,46 to 11,66 m of water)
- 13 -
12.
Explain the object of fitting an air vessel on (a) the suction side and (b) the delivery side of a
reciprocating pump. A single-acting reciprocating pump with plunger diameter of 100 mm
and stroke of 150 mm has a speed of 75 r/min. The centre of the pump cylinder is 1,5 m above
the level of the water in the sump. The 75 mm diameter suction pipe is 7,2 m long. The level
of the delivery tank is 30 m above the centre of the pump cylinder, and the 63 mm diameter
delivery pipe is 75 m long. The friction coefficient for the suction and delivery pipes is 0,01.
There is no air vessel on the suction side, but one, which may be assumed to be perfectly
efficient, is installed on the delivery side. Assuming that the plunger moves horizontally with
simple harmonic motion, determine (a) the pressure on the plunger at the beginning, middle
and end of the suction stroke; (b) the water power of the pump. Also obtain the pressure on
the plunger at the beginning of the delivery stroke if no air vessel had been fitted on the
delivery side.
((a) –7,56 m, -l,71m, +4,56 m of water, (b) 465W, 119m of water)
13.
A reciprocating pump has a cylinder of 75 mm bore x 150 mm stroke and draws water from a
sump whose level is 1,5 m below the axis of the pump. If the suction pipe is 2,4 m long and
50 m in diameter, find the speed of the pump in rev/min at which separation occurs if this
takes place at a vacuum head of 7,9m of water. Assume simple harmonic motion of the piston.
If ƒ = 0,01 for the pipe, what is the friction head at mid-stroke when running at this speed?
(119 r/min, 0,435 m)
14.
A double-acting single-cylinder reciprocating pump has a cylinder diameter of 150 mm and
450 mm stroke. The suction and delivery pipes have diameters and lengths of 100 mm, 6 m
and 75 mm, 60 m respectively. The sump is 4,5 m below and the reservoir 45 m above the
centre-line of the pump. If the pump runs at 60 rev/min, determine the power of the driving
motor if its efficiency is 0,85. Take ƒ = 0,005 and assume simple harmonic motion for the
plunger.
(12,6 kW)