Fluid Mechanics
Module 5
ENERGY CONSERVATION
Energy Equation for Steady Flow of Incompressible Fluids
Energy and head. Energy is defined as the ability to do work. Work is the result of the application of a force through
distance and is generally defined mathematically as the product of a force and the distance traversed in the direction of
application.
Potential energy refers to the energy possessed by the element of fluid due to its elevation above a reference datum. In
terms of head it is called elevation head z.
Kinetic energy refers to the energy possessed by the element of fluid due to its velocity. In terms of head it is called
𝑽𝟐
velocity head , 𝟐𝒈.
Pressure energy or flow energy is the amount of work required to force the element of fluid across a certain distance
𝒑
against the pressure. In terms of head it is called pressure head, 𝜸.
Total energy. It is the sum of PE, KE AND FE. In terms of head it is the sum of elevation, velocity and pressure head.
𝑽𝟐
𝒑
𝑯 = 𝒛 + 𝟐𝒈 + 𝜸
Stagnation Pressure is the sum of the static and dynamic pressures.
Energy equation. The energy equation results from application of the principle of conservation of energy to fluid flow. In
the direction of flow, the energy principle is summarized by the general equation energy of entering fluid plus energy
added minus energy lost minus energy extracted equals energy at leaving fluid.
1. EULER’S EQUATION
Shearing stresses develop in a moving fluid because of the viscosity of the fluid. We know that for some common
fluids, such as air and water, the viscosity is small, and therefore it seems reasonable to assume that under some
circumstances we may be able to simply neglect the effect of viscosity, (and thus shearing stresses). Flow fields in
which the shearing stresses are assumed to be negligible are said to be inviscid, nonviscous, or frictionless.
Euler’s equations of motion apply to an inviscid flow in vector form.
𝝏𝑽
𝜌𝒈 − 𝛁𝒑 = 𝜌 [ 𝝏𝒕 + (𝑽 ∘ 𝛁)𝑽]
We will restrict our attention to steady flow so Euler’s equation in vector form becomes
𝜌𝒈 − 𝛁𝒑 = 𝜌[(𝑽 ∘ 𝛁)𝑽]
We wish to integrate this differential equation along some arbitrary streamline, and select the coordinate system
with the z axis vertical, with “up” being positive, so that, the acceleration of gravity vector can be expressed as
𝒈 = −𝑔𝛁𝒛
Thus;
−𝜌𝑔𝛁𝒛 − 𝛁𝒑 = 𝜌[(𝑽 ∘ 𝛁)𝑽]
𝟏
Also, it will be convenient to use the vector identity; (𝑽 ∘ 𝛁)𝑽 = 𝟐 𝛁(𝑽 ∘ 𝑽) − 𝑽 × (𝛁 × 𝐕) then
𝜌
−𝜌𝑔𝛁𝒛 − 𝛁𝒑 = 𝟐 𝛁(𝑽 ∘ 𝑽) − 𝜌𝑽 × (𝛁 × 𝐕)
Rearranging the equation in the form:
𝜌
𝛁(𝑽
𝟐
∘ 𝑽) + 𝜌𝑔𝛁𝒛 + 𝛁𝒑 = 𝜌𝑽 × (𝛁 × 𝐕) multiplying both sides by
and simplifying vector operations yield the following result:
𝑑𝑝
1
+ 2 𝑑(𝑉 2 ) + 𝑔𝑑𝑧 = 0
𝜌
where the change in p, V, and z is along the streamline. It can now be integrated to give
𝑉2
2
∫
𝑑𝑝
𝜌
+
∫
𝑑𝑝
𝛾
+ 2𝑔 + 𝑧 =constant
𝑉2
+ 𝑔𝑧 =constant
and if g is constant,
then
𝑑𝒔
𝜌
2.
Bernoulli’s Theorem: For inviscid, incompressible fluids, commonly called ideal fluids. The sum of the kinetic,
potential, and flow energies of a fluid particle is constant along a streamline during steady flow when the
compressibility and frictional effects are negligible.
𝑝
( 𝛾1 +
𝑉1 2
2𝑔
+ 𝑧1 ) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
or
𝑝1
𝛾
+
𝑉1 2
2𝑔
+ 𝑧1 =
𝑝2
𝛾
+
𝑉2 2
2𝑔
+ 𝑧2
this is known as the Bernoulli’s equation.
For steady flow of incompressible fluids in which the change in internal energy is negligible with either energy
is added or extracted from the system, the equation becomes:
𝑝1
𝛾
+
𝑉1 2
2𝑔
+ 𝑧1 + 𝐻𝐴 − 𝐻𝐿 − 𝐻𝐸 =
𝑝2
𝛾
+
𝑉2 2
2𝑔
+ 𝑧2
Where: HA is the head added to the system, HE is the head extracted from the system,
HL is the head loss in the system.
Energy line. The energy line is a graphical representation of the energy at each section. With respect to a certain
datum, the total energy can be plotted at each representative section, and the line so obtained is a valuable tool in
many flow problems. The energy line will slope in the direction of flow except where energy is added by
mechanical devices.
Hydraulic grade line. The hydraulic grade line lies below the energy line by an amount equal to the velocity head at
the section. The two lines are parallel for all sections of equal cross-sectional area. The ordinate between the
center of the stream and the hydraulic grade line is the pressure head at the section.
Power Considerations in Fluid
Power. Fluid power can be calculated as follows;
P = QH
The mechanical energy can be defined as the form of energy that can be converted to mechanical work completely
and directly by an ideal mechanical device such as an ideal turbine. Kinetic and potential energies are the familiar forms
of mechanical energy.
The transfer of mechanical energy is usually accomplished by a rotating shaft, and thus mechanical work is often
referred to as shaft work. A pump or a fan receives shaft work (usually from an electric motor) and transfers it to the fluid
as mechanical energy (less frictional losses). A turbine, on the other hand, converts the mechanical energy of a fluid to
shaft work. In the absence of any irreversibility such as friction, mechanical energy can be converted entirely from one
mechanical form to another, and the mechanical efficiency of a device or process can be defined a
𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡
𝜂𝑚𝑒𝑐ℎ = 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡 𝑥100%
The mechanical efficiency should not be confused with the motor efficiency and the generator efficiency, which are
defined as
𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡
𝜂𝑚𝑜𝑡𝑜𝑟 =
𝑥100%
𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡
𝜂𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =
𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑢𝑡𝑝𝑢𝑡
𝑥100%
𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡
Problem Set:
1. Oil of specific gravity 0.75 is flowing through a 150 mm pipe under a pressure of 103 kpa. If the total energy relative to
a datum plane 2.4 m below the center of the pipe is 17.9m, determine the flow of oil.
a. 0.095m3/s
b. 0.765m3/s
c. 1.234m3/s
d. 0.543m3/s
2. A turbine is rated at 450 kW when the flow of water through it is 0.609 m3/s. Assuming an efficiency of 87%, what
head is acting on the turbine?
a. 86.8 m
b. 65.3m c. 76.5m d. 12.3m
3. A pipe carrying oil of specific gravity 0.877 changes in size from 150mm at section E to 450mm at section R. Section E
is 3.66m lower than R, and the pressure are 91 kpa and 60.3 kpa, respectively. If the discharge is 0.146m 3/s, determine
the lost head.
a. 3.4m
b. 4.5m
c. 5.4m
d. 2.4m
4. A 150-mm diameter jet of water is discharged from a nozzle into the air. The velocity of the jet is 36m/s. Find the power
in the jet.
a. 411 kW
b. 643kW
c. 342kW d. 765kW
5. Water flows upward in a vertical 300-mm pipe at the rate of 0.222m3/s. At point A in the pipe the pressure is 210 kpa.
At point B, 4.57m above A, the diameter is 600 mm, and the lost head A to B equals 1.83m. Determine the pressure at
B.
a. 152 kpa b. 176kpa c. 543kpa d. 342kpa
6. Water is to be siphon from a tank at the rate of 0.0892 m3/s. the flowing end of the siphon pipe must be 4.27m below the
water surface. The lost head terms are 1.5V2/2g from tank to summit of siphon and 1.0V2/2g from summit to end of
siphon. The summit is 1.52m above the water surface. Determine the size of siphon pipe in mm.
a. 150mm b. 165mm
c. 234mm
d. 125mm
7.
8.
9.
10.
11.
The theoretical velocity of flow through an orifice 3m below the surface of water in a tall tank is;
a. 7.67m/s
b. 9.87m/s
c. 3.24m/s d. 1.23m/s
The average velocity in a pipe flowing full of incompressible liquid is 3m/s. After passing through a conical section
that reduces streams cross-sectional area to ¼ of its previous value, the velocity after the conical section is:
a. 12 b. 6
c. 3/2
d. 3/4
A liquid of specific gravity 1.75 flows in a 6cm horizontal pipe. The total energy at a certain point in the flow is 80m.
The elevation of the pipe above a fixed datum is 2.6m. If the pressure at the specified point is 75kpa, determine the
velocity of flow and the power available at that point.
a. 37.85m/s; 146,979 W b. 44.29, 84372 c. 35.43, 98262 d. 33.38, 78933
If the velocity of water is 8m/s and the pressure is 140kPa on the discharge side of a pump. What is the head of the
pump if the velocity is 4m/s and pressure is 90kpa before the pump?
a. 7.54m b. 7.98m
c. 8.21m d. 6.82m
A 300mmx75mm venture meter is inserted in a 300mm diameter pipeline where water flows at 55L/s. Neglecting
losses, compute the drop in pressure head from the inlet to the throat.
a. 7.9m b. 4.3m c. 6.4m
d. 5.4m
12. Water flows at 0.2m3/s through a 300mm diameter, 120m long pipe under a pressure difference of 280mm Hg.
Compute the friction factor.
a. 0.0233
b. 0.0435
c. 0.0154
d. 0.0876
13. A pipeline leads from one reservoir to another which has its water surface 10m lower. For a discharge of 1m 3/s,
determine the losses in meters and in kilowatts.
a. 97.9
b. 98.3 c. 96.9 d. 99.8
14. Oil with specific gravity of 0.75 is flowing through a 6 in. pipe under a pressure of 15psi. If the total energy relative to
a datum plane 8 ft below the center of the pipe is 58.6ft-lb/lb. determine the flowrate of oil.
a. 3.32cfs b. 3.69cfs c. 4.2cfs d. 4.13cfs
15. If the total available head of a stream flowing at a rate of 300cfs is 25 ft, what is the theoretical horsepower available?
a. 851
b. 890
c. 922 d. 873
16. Horizontal orifice under a constant head of 1.3m issues a jet which hits a point 5m below the centerline of the orifice
and 5m horizontally from the vena contracta. What is the coefficient of velocity?
a. 0.98 b. 0.95 c. 0.9 d. 0.89
17. What is the discharge capacity if a concrete pipe culvert 4ft in diameter and 10m long if the difference in water level
at the outlet is 1.52m. Assume coefficient of discharge of 0.74.
a. 4.73
b. 3.63
c. 6.83
d. 5.89
18. A vertical jet of water thru a nozzle supports a load of 150N. The velocity and diameter of the jet at the nozzle tip are
17.46m/s and 3cm. find the distance of load from the nozzle tip in meters.
a. 8m b. 5m
c. 10m
d. 3m
19. An orifice at the side of a tank is located 1m above the bottom of the tank which is resting on the ground. The jet of
water strikes a distance of 2.75m horizontally away from the orifice with C v=0.98. The height of the tank is 4m and it
is filled with water 2m depth and on top of it is another liquid having a depth of 1m. determine the specific gravity of
the liquid.
a. 0.98 b. 0.82 c. 0.91 d. 0.86
20. The water in a large lake is to be used to generate electricity by the installation of a hydraulic turbine–generator at a
location where the depth of the water is 50 m. Water is to be supplied at a rate of 5000 kg/s. If the electric power
generated is measured to be 1862 kW and the generator efficiency is 95 percent, determine (a) the overall efficiency
of the turbine–generator, (b) the mechanical efficiency of the turbine, and (c) the shaft power supplied by the turbine
to the generator. Ans. a. 0.76 b. 0.80 c. 1964kW
21. A piezometer and a Pitot tube are tapped into a horizontal water pipe, as shown in figure below, to measure static and
stagnation (static ' dynamic) pressures. For the indicated water column heights, determine the velocity at the center of
the pipe. Ans. 1.53m/s
22. The pump of a water distribution system is powered by a 15-kW electric motor whose efficiency is 90 percent. The
water flow rate through the pump is 50 L/s. The diameters of the inlet and outlet pipes are the same, and the elevation
difference across the pump is negligible. If the pressures at the inlet and outlet of the pump are measured to be 100
kPa and 300 kPa (absolute), respectively, determine (a) the mechanical efficiency of the pump and (b) the temperature
rise of water as it flows through the pump due to the mechanical inefficiency. Ans. a. 74.1% b. 0.017C
23. In a hydroelectric power plant, 100 m3/s of water flows from an elevation of 120 m to a turbine, where electric power
is generated. The total irreversible head loss in the piping system from point 1 to point 2 (excluding the turbine unit) is
determined to be 35 m. If the overall efficiency of the turbine–generator is 80 percent, estimate the electric power
output.
24. A fan is to be selected to cool a computer case whose dimensions are 12 cm x 40 cm x 40 cm. Half of the volume in
the case is expected to be filled with components and the other half to be air space. A 5-cm-diameter hole is available
at the back of the case for the installation of the fan that is to replace the air in the void spaces of the case once every
second. Small low-power fan–motor combined units are available in the market and their efficiency is estimated to be
30 percent. Determine (a) the wattage of the fan–motor unit to be purchased and (b) the pressure difference across the
fan. Take the air density to be 1.20 kg/m3.
25. Air flows through a pipe at a rate of 200 L/s. The pipe consists of two sections of diameters 20 cm and 10 cm with a
smooth reducing section that connects them. The pressure difference between the two pipe sections is measured by a
26.
27.
28.
29.
30.
31.
water manometer. Neglecting frictional effects, determine the differential height of water between the two pipe
sections. Take the air density to be 1.20 kg/m3. Answer: 3.7 cm
Water is flowing through a Venturi meter whose diameter is 7 cm at the entrance part and 4 cm at the throat. The
pressure is measured to be 430 kPa at the entrance and 120 kPa at the throat. Neglecting frictional effects, determine
the flow rate of water. Answer: 0.538 m3/s
Water flows at a rate of 0.035 m3 /s in a horizontal pipe whose diameter is reduced from 15 cm to 8 cm by a reducer.
If the pressure at the centerline is measured to be 470 kPa and 440 kPa before and after the reducer, respectively,
determine the irreversible head loss in the reducer. Take the kinetic energy correction factors to be 1.05. Answer: 0.68
m
In a hydroelectric power plant, water flows from an elevation of 240 ft to a turbine, where electric power is generated.
For an overall turbine–generator efficiency of 83 percent, determine the minimum flow rate required to generate 100
kW of electricity. Answer: 370 lbm/s
Water flows at a rate of 20 L/s through a horizontal pipe whose diameter is constant at 3 cm. The pressure drop across
a valve in the pipe is measured to be 2 kPa. Determine the irreversible head loss of the valve, and the useful pumping
power needed to overcome the resulting pressure drop. Answers: 0.204 m, 40 W
A fireboat is to fight fires at coastal areas by drawing seawater with a density of 1030 kg/m 3 through a 20-cmdiameter pipe at a rate of 0.1 m3/s and discharging it through a hose nozzle with an exit diameter of 5 cm. The total
irreversible head loss of the system is 3 m, and the position of the nozzle is 4 m above sea level. For a pump
efficiency of 70percent, determine the required shaft power input to the pump and the water discharge velocity.
Answers: 200 kW, 50.9 m/s
A 2-m-high large tank is initially filled with water. The tank water surface is open to the atmosphere, and a sharpedged 10-cm-diameter orifice at the bottom drains to the atmosphere through a horizontal 100-m-long pipe. If the total
irreversible head loss of the system is determined to be 1.5 m, determine the initial velocity of the water from the tank.
Disregard the effect of the kinetic energy correction factors. Answer: 3.13 m/s