Nozzles

1. NOZZLES J3008/7/1
NOZZLES
OBJECTIVES
General Objective : To understand the mechanism of flow in nozzles
Specific Objectives : At the end of the unit you should be able to :
 sketch and differentiate the types and shapes of nozzles
 define Critical Pressure Ratio
 calculate cross-sectional area, A and the temperature of a
throat at entrance and exit
 calculate maximum mass flow
 define and differentiate the use of nozzles in :
- steam turbine
- gas turbine
- jet engine
- flow measurement
- rocket propulsion
- steam injector
- injector
UNIT 7

2. NOZZLES J3008/7/2
7.0 INTRODUCTION
Nozzle
A nozzle is a device that increases the velocity of a fluid at the expense of pressure.
It is a duct of smoothly varying cross-sectional area in which a steadily flowing fluid
can be made to accelerate by a pressure drop along the duct.
There are many applications in practice which require a high-velocity stream of fluid,
and the nozzle is the best means of obtaining high-velocity, thus nozzles are used in
steam and gas turbines, in jet engines, in rocket motors, in flow measurement, and in
many other applications.
When a fluid is decelerated in a duct, causing a rise in pressure along the stream, then
the duct is called a diffuser; two applications in practice in which a diffuser is used are
the centrifugal compressor and the ram jet.
Nozzles and Diffusers
A nozzle is a device that increases the velocity of a fluid at the expense of pressure. A
common example would be a nozzle used at the end of a garden hose !
A diffuser is a device that increases the pressure of a fluid by slowing it down.
Several types of pumps operate by using shaft work to turn an impeller which will
increase the kinetic energy of the fluid, followed by a diffuser that converts some of
the kinetic energy to an increased pressure.
Nozzle Diffuser
Figure 7.1 Nozzle & Diffuser
INPUTINPUT

3. NOZZLES J3008/7/3
7.1 Types and shapes of nozzles
Typical nozzle cross-sectional areas of particular interest are shown in
Figure 7.2
Figure 7.2
a) Convergent Nozzle
Figure 7.3
• The convergent nozzle in which the cross-section converges from the entry
area to a minimum area which is the exit.
b) Convergent – divergent nozzle
Figure 7.4
• Figure 7.4 shows a convergent-divergent nozzle.
• It can be seen from the inlet area the nozzle converges to a minimum area
called the throat and then to the outlet area.
inlet throat outlet
Inlet Outlet

4. NOZZLES J3008/7/4
7.2 Critical Pressure Ratio
- It has been stated before, that the velocity at the throat of a correctly
designed nozzle is the velocity of sound.
- The flow-up to the throat is sub-sonic while the flow after the throat is
supersonic. It should be noted that a sonic or supersonic flow requires a
diverging duct to accelerate it.
- In the same way, for a nozzle that is convergent, the fluid will attain sonic
velocity at the exit if the pressure drop across the nozzle is large enough.
- The ratio of the pressure at the section where sonic velocity is attained to
the inlet pressure of a nozzle is called the critical pressure ratio.
- Critical temperature ratio, 1
2
11
1
−
=





=
−
γ
γ
γ
p
pc
T
Tc
- Critical pressure ratio,
( )1/
1
2
1
−






−
=
γγ
γp
pc
7.3 Maximum Mass Flow

5. NOZZLES J3008/7/5
- Consider a convergent nozzle expanding into space, the pressure of which
can be varied, while the inlet pressure remains fixed. The nozzle is shown
diagrammatically in the Figure 7.5.
- When the back pressure, pb is equal to p1,
then no fluid can flow through the nozzle.
As pb is reduced the mass flow through the
nozzles increases, since the enthalpy drop,
and hence the velocity increases.
- However, when the back pressure reaches
the critical value, it is found that no further
reduction in back pressure can affect the
mass flow.
- When the back pressure is exactly equal to the critical pressure, pc then the
velocity at exit is sonic and the mass flow through the nozzle is at a
maximum value. The exit pressure remains at pc, and the fluid expands
violently outside the nozzle down to the back pressure.
- It can be seen that the maximum mass flow through a convergent nozzle is
obtained when the pressure ratio across the nozzle is the critical pressure
ratio. Also, for a convergent-divergent nozzle, with sonic velocity at the
throat, the cross-sectional area of the throat fixes the mass flow through the
nozzle for fixed conditions.
- When a nozzle operates with the maximum mass flow, it is said to be
choked. A correctly designed convergent-divergent nozzle is always
choked.
7. 4 Cross-sectional area, A and temperature of a throat at entrance and exit
valve
 p1  Back
press, pb
Figure 7.5

6. NOZZLES J3008/7/6
Consider a stream of fluid at pressure p1, enthalpy h1, and with a low velocity
C1. It is required to find the shape of duct which will cause the fluid to
accelerate to a high velocity as the pressure falls along the duct. It can be
assumed that the heat loss from the duct is negligibly small
(adiabatic flow, Q = 0), and it is clear that no work is done on or by the fluid
(W = 0). Applying the steady-flow energy equation :
W
C
hQ
C
h ++=++
22
2
2
2
1
2
1
Figure 7.6
• Applying the steady-flow energy equation, between section 1 and any other
section X-X where pressure 1p , enthalpy 1h , and with low velocity C1. It is
required to find the shape of duct which will cause the fluid to accelerate to
high velocity as the pressure falls along the duct. Figure 7.6
• It can be assumed that the heat loss from the duct is negligibly small, and it is
clear no work is done on or by the fluid. Applying the steady-flow energy
equation which is :
22
22
1
1
C
h
C
h +=+ ------------(1)
or can be written like these,
( ) 2
11
2
2 ChhC +−= ------------(2)
( ) 2
112 ChhC +−= ------------(3)
(where fluid velocity is C and h is an enthalpy)
X 2
A1
A2
h1
h2
C1
C2
X
1 X

7. NOZZLES J3008/7/7
• In most practical applications the velocity at the inlet to a nozzle is negligibly
small in comparison with the exit velocity. It can be seen from equation (5),
that a negligibly small velocity implies a very large area, and most nozzles are
in fact shaped at inlet in such a way that the nozzle converges rapidly over the
first fraction of its length :
( ) 2
112 ChhC +−=
• And neglecting C1 this gives,
( )hhC −= 12
• Since enthalpy is usually expressed in KJ/kg, then an additional constant of 103
will appear within the root sign if C is to be expressed in m/s,
( ) 72.44102 3
=×
(where 1 kJ=103
Nm)
• Hence, ( ) smhhC /72.44 1 −=
• If the area at the section X-X is A, and the specific volume is v :
v
CA
mrateflowMass =
•
, ------------(4)
or
C
v
m
A
velocitymassunitperArea =•
, ------------(5)
• Then substituting for the velocity C, from equation (3),
2
11 )(2 Chh
v
flowmassunitperArea
+−
=
Example 7.1

8. NOZZLES J3008/7/8
Air at 8.6 bar and 190°C expands at the rate of 4.5 kg/s through a convergent-
divergent nozzle into a space at 1.03 bar. Assuming that the inlet velocity is
negligible, calculate the throat and the exit cross-sectional areas of the nozzle.
The nozzle is shown diagrammatically in figure below. The critical pressure ratio is
given by,
1 C 2
8.6 bar 1.03 bar
C1=0 C2
( )
528.0
4.2
2
1
2 4.0
4.1
1
1
=





=





+
=
−γ
γ
γp
pc
barpc 54.46.8528.0 =×=
Also,
2.1
1
1
2
1
=
+
=
γT
Tc
KTc 8.385
2.1
273190
=
+
=
kgm
pc
RTc
vc /244.0
54.410
8.385287 3
5
=
×
×
==
Then,
( ) ( ) smRTC cc /3948.3852874.1 =××==
And,
( ) ( ){ }cpcc TTChhC −=−= 11 72.4477.44
( ){ } smCc /3948.385463005.175.44 =−=
To find the area of the throat,

9. NOZZLES J3008/7/9
2
00279.0
394
244.05.4
m
C
Vm
A
c
c
c =
×
==
•
26
27901000279.0 mmthroatofArea =×=
Using equation for a perfect gas,
( )
835.1
03.1
6.8
4.1/4.0/1
2
1
2
1
=





=





=
− γγ
p
p
T
T
KT 252
835.1
463
2 ==
kgm
p
RT
v /702.0
03.110
252287 3
5
2
2
2 =
×
×
==
Then,
( ) ( ){ }21212 72.4472.44 TTchhC p −=−=
( ){ } smC /651252463005.172.442 =−=
Then to find the exit area,
2
2
2
2 00485.0
651
702.05.4
m
C
mv
A =
×
==
•
26
48501000485.0 mmareaExit =×=
ACTIVITY 7A

10. NOZZLES J3008/7/10
TEST YOUR UNDERSTANDING BEFORE YOU CONTINUE WITH THE NEXT
INPUT…!
7.1 Sketch two types of nozzles
7.2 Define :
(a) critical presssure ratio
(b) maximum mass flow
7.3 A fluid at 6.9 bar and 93°C enters a convergent nozzle with negligible
velocity, and expands isentropically into a space at 3.6 bar. Calculate the outlet
temperature and mass flow per m2
of exit area.
(a) when the fluids is helium (Cp=5.23 kJ/kgK)
(b) when the fluid is ethane (Cp=1.66 kJ/kgK)
Assume that both helium and ethane are perfect gases, and the respective
molecular weights as 4 and 30.
FEED BACK ON ACTIVITY 7A

11. NOZZLES J3008/7/11
7.1
b) Convergent Nozzle
b) Convergent – divergent nozzle
7.2
a) critical presssure ratio
- The ratio of the pressure at the section where sonic velocity is
attained to the inlet pressure of a nozzle.
b) maximum mass flow
- The flow through a convergent nozzle that can be obtained
when the pressure ratio across the nozzle is the critical pressure
ratio.
7.3 Solution :
a)
It is necessary first to calculate the critical pressure in order to discover
whether the nozzle is choked or not.
inlet throat outlet
Inlet Outlet

12. NOZZLES J3008/7/12
We know that,
M
R
R o
=
Therefore for helium,
KkgNmR /2079
4
8314
==
Then,
( )1−
=
γ
γR
Cp
So,
66.1
24.510
20791
5
=
×
==
−
pC
R
γ
γ
66.1
397.01
1
=
−
=∴γ
Then using equation for critical pressure ratio,
( )1/
1 1
2
−






−
=
γγ
γp
pc
( )
488.0
66.2
2
66.0/66.1
=





=
barpc 9.6488.0 ×=
barppressureCritical c 37.3=
The actual back pressure is 3.6 bar, hence in this case the fluid does not reach
the critical conditions and the nozzle is not choked. The nozzle is shown
diagrammatically in the figure below :
1 2

13. NOZZLES J3008/7/13
6.9 bar 3.6 bar
Then,
( )
295.1
6.3
9.6
397.0/1
2
1
2
1
=





=





=
− γγ
p
p
T
T
KT 5.282
295.1
27393
2 =
+
=
So,
( ) ( ){ }21212 72.4472.44 TTchhC p −=−=
( ) smC /9355.28236623.572.442 =−=
Also,
kgm
p
RT
v /63.1
6.310
5.2822079 3
5
2
2
2 =
×
×
==
So,
skg
V
CA
m /573
63.1
9351
2
22
=
×
==
•
skgareaexitmperflowMass /5732
=
b) Using the same prosedure for ethane, we have,

14. NOZZLES J3008/7/14
M
R
R o
=
Therefore for ethane ,
KkgNmR /1.277
30
8314
==
Then,
( )1−
=
γ
γR
Cp
So,
167.0
66.110
1.2771
3
=
×
==
−
pC
R
γ
γ
2.1
167.01
1
=
−
=∴γ
Then using equation for critical pressure ratio,
( )1/
1 1
2
−






−
=
γγ
γp
pc
( )
566.0
1.1
1
2.0/2.1
=





=
barpc 9.6566.0 ×=
barppressureCritical c 91.3=
The actual back pressure is 3.6 bar, hence in this case the fluid reaches critical
conditions at exit and the nozzle is choked. The expansion from the exit
pressure of 3.91 bar down to the back pressure of 3.6 bar must take place
outside the nozzle. The nozzle is shown diagrammatically in the figure below :
1 2
3.6 bar

15. NOZZLES J3008/7/15
6.9 bar 3.91 bar
Then,
1.1
1
2
2
1
=





+
=
γT
Tc
KTT c 7.332
1.1
366
2 ===
So,
( ) ( ) smRTCC cc /3337.3321.2772.12 =××== γ
Also,
kgm
p
RT
v /236.0
91.310
7.3321.277 3
5
2
2
2 =
×
×
==
So,
skg
V
CA
m /1412
236.0
3331
2
22
=
×
==
•
skgareaexitmperflowMass /14122
=
7.4 The nozzle can be used in the following application :
INPUTINPUT

16. NOZZLES J3008/7/16
Steam turbine, gas turbine, jet engine, flow measurement, rocket propulsion,
steam injector and an injector itself.
But do you know that :
All jet engines have a nozzle at the back of the engine. It is
the exhaust duct of the engine. The air from the turbine
blades and the engine mixes together in the nozzle and makes
a big force that blasts out at the back of the engine. It is this
power that pushes, or thrust, the airplane forward.
a) Steam Turbine
• Of all the heat engines and prime movers the steam turbine is the nearest to
the ideal and it is widely used in power plants and in all industries where
power and/or heat is needed for processes; such as pulp mills, refineries,
petro-chemical plants, food processing plants, desalination plants, refuse
incinerating and district heating plants.
• Operation principle : In principle, the impulse steam turbine consists of a
casing containing stationary steam nozzles and a rotor with moving or
rotating buckets. The steam passes through the stationary nozzles and is
directed at high velocity against the rotor buckets causing the rotor to rotate
at high speed.
• The following events take place in the nozzles:
− The steam pressure decreases.
− The enthalpy of the steam decreases.
− The steam velocity increases
− The volume of the steam increases.
b) Gas Turbine
• A gas turbine has a compressor, combustion chamber, and turbine. The
turbine and the compressor are on the same shaft. The compressor raises
the pressure of atmospheric air and sends this air to the combustion
chamber. Here, a fuel (oil, gas, or pulverized coal) burns, raising the
temperature and increasing the heat energy. The hot gas in the turbine

17. NOZZLES J3008/7/17
expands to develop mechanical energy, as expanding steam does in a
steam turbine.
• The basic parts of a turbine are the rotor, which has blades projecting
radially from its periphery; and nozzles, through which the gas is
expanded and directed. The conversion of kinetic energy to mechanical
energy occurs at the blades. The basic distinction between the types of
turbines is the manner in which the gas causes the turbine rotor to move.
• The main use for the gas turbine in the present day is in the air-craft field,
and the large unit of a gas turbine is used for electric power generation and
for marine propulsion.
c) Jet Engine
• Jet engines move the airplane forward with a great force that is produced
by a tremendous thrust and causes the plane to fly very fast.
• All jet engines, which are also called gas turbines, work on the same
principle. The engine sucks air in at the front with a fan. A compressor
Translated from a Korean text :
The development of gas turbine
can make us fly in the sky, explore
the seven seas and generate
electric power that we use
everyday to make life better !

18. NOZZLES J3008/7/18
raises the pressure of the air. The compressor is made up of fans with many
blades and attached to a shaft. The blades compress the air. The
compressed air is then sprayed with fuel and an electric spark lights the
mixture. The burning gases expand and blast out through the nozzle, at the
back of the engine. As the jets of gas shoot backward, the engine and the
aircraft are thrust forward as shown in Figure 7.7.
• In a jet engine airplane, thrust is a result of hot gases (exhaust) rushing out
of the engine's nozzle. The action of the gases rapidly moving backward
causes a reaction in the air. The air puts out a force equal to the thrust, but
in the opposite direction, moving the airplane forward.
Figure 7.7
d) Flow Measurement
• A nozzle is used frequently as a flow meter by inserting it into a
pipeline and measuring the pressure drop or the differential between the
inlet and the throat. This pressure must be kept small, and is measured by a
water or mercury manometer.

19. NOZZLES J3008/7/19
• A convergent nozzle can be used in a pipeline as shown in the Figure
7.8. The different levels in the manometer is ∆ wp / , where ∆ p is the
pressure difference between section 1 and 2, and w is the specific weight
of the manometer liquid.
• Eddies are set up as the fluid leaves the nozzle and the kinetic energy
of the jet is dissipated irreversibly. This means that some of the pressure
drop, ∆ p , is not recovered, and so the nozzle causes a loss of pressure in
the pipeline.
• The pressure loss can be reduced by using a convergent-divergent
nozzle in the pipeline. The pressure loss can be reduced by using a
convergent-divergent nozzle as shown in Figure 7.9. Since the nozzle in
Figure 7.9 is far from choked condition, it acts as a venturi meter. The flow
is expanded down to the throat at section 2, and diffused from 2 to 3.
• In this way, the pressure drops to the throat, ∆ p , is almost
completely recovered in the diffuser portion, and the pressure loss in the
pipeline due to the venturi meter is much smaller than that due to a
convergent nozzle.
Figure 7.8 Figure 7.9
Convergent Nozzle Convergent-Divergent Nozzle
e) Rocket Propulsion
• One very important use of the nozzle is
as a means of propolsion. Since the fluid
flowing through the nozzle is accelerated
relative to the nozzle, then by Nowton’s
In 1926, Robert
Goddard tested the
first liquid-propellant
rocket engine. His
engine used gasoline
and liquid oxygen.
The basic idea is
simple. In most
liquid-propellant
rocket engines, a
fuel and an oxidizer
(for example,
gasoline and liquid
oxygen) are
pumped into a
combustion
chamber. There
they burn to create
a high-pressure and
high-velocity stream
of hot gases. These
gases flow through
a nozzle that
accelerates them
further (5,000 to
10,000 mph exit
velocities being
typical), and then
they leave the
engine.

20. NOZZLES J3008/7/20
third law, it follows that the fluid exerts a
trust on the nozzle in the opposite direction
to the fluid flow.
• In the jet aeroplane and the ram-jet the
atmospheric air is drawn in, compressed,
heated, and allowed to expand through a
nozzle, leaving the aircraft at high velocity ;
the rate of change of momentum of the air
backwards relative to the aircraft gives a
reactive forward trust to the aircraft.
• In order to achieve jet-propelled flight in
space, where there is no atmosphere to be
drawn into the vehicle, it is necessary that
the fuel plus its oxidant should be carried in
the rocket. This is known as the rocket
propolsion.
• A rocket operating on a chemical fuel
consists of tanks containing the chemical
propollent, and a rocket motor (or rocket
engine) which consists of a combustion
chamber and a convergent-divergent nozzle.
Some way of introducing the propellant
from the tanks to the combustion chamber
is also necessary, and this can be done by
using a pump or by having an additional
tank of compressed nitrogen.
• When a pump is used it can be driven by
a small turbine using the propellant as fuel.
A simple line diagram of a rocket is shown
in Figure 7.10.
f) Steam Injector
• Steam injector is widely used in the steam locomotives and is one of
the components used in the nuclear power plants.
Figure 7.10
Adapted from
www.HowStuffWorks.com

21. NOZZLES J3008/7/21
• In the steam engine of the steam locomotive, the water supply to the
boiler is provided by two live steam injectors, or one live steam and one
exhaust injector on larger locomotives. Injectors work because steam
under the same pressure and conditions flows from a contracted nozzle at
a much greater velocity than water. The steam cone, or nozzle, regulates
the quantity of steam used by the injector. It is both convergent and
divergent in order to direct the flow of steam into the combining cone and
gives it the maximum possible velocity. (Figure 7.11)
• The condensation of the steam jet and the transfer of its energy to the
water takes place in the combining cone which receives the steam and
water. In condensing, the steam gives up its velocity to the water, which is
then further accelerated by the vacuum in the combining cone caused by
the reduction in the volume of the steam when condensed by the water.
• At the inlet end is a jet consisting of a mixture of steam and water,
while the outlet end has a jet of hot water flowing at high velocity but very
low in pressure. Steam injectors are very efficient and waste very little heat
as the steam used is returned to the boiler as hot water.
Figure 7.11
g) Injector
• One of an example of an injector is a fuel injector. It is an electronically
controlled valve. It is supplied with pressurized fuel by the fuel pump in
Water
Delivery pipe
Steam
Boiler

22. NOZZLES J3008/7/22
your car, and it is capable of opening and closing many times per second.
Figure 7.12
• When the injector is energized, an electromagnet moves a plunger that
opens the valve, allowing the pressurized fuel to squirt out through a tiny
nozzle. The nozzle is designed to atomize the fuel -- to make the fuel as
fine a mist as possible so that it can burn easily.
• The injectors are mounted in the intake manifold so that they spray fuel
directly at the intake valves. A pipe called the fuel rail supplies pressurized
fuel to all of the injectors.
Figure 7.12
Adapted from www.HowStuffWork.com
ACTIVITY 7B
A fuel injector firing

23. NOZZLES J3008/7/23
TEST YOUR UNDERSTANDING BEFORE YOU CONTINUE WITH THE NEXT
INPUT…!
4
1 s t a u b i n
h
3 p o p l i o n
u 2
r 7
6 t o t p
9 t r
5 f w e t
11 r s
c e
10 b o e r u
n
12 a l e 8 j t
l
13 I j e o r
s
Horizontal :
1. The _____________ is widely used in power plants and all industries where power
or heat is needed for processes.
3. Marine ___________use the gas turbine to delevop mechanical energy.
5. A nozzle is used as a _____________ by inserting it into a pipeline.

24. NOZZLES J3008/7/24
6. A differential between the inlet and the _________ of a flow meter is called the
pressure drop.
8. In the______aeroplane the atmospheric air is drawn in, compressed, heated, and
allowed to expand through a nozzle.
10. In the steam engine of steam locomotive, the water supply to the _______ is
provided by two live steam injectors.
13. When an injector is energized, an electromagnet moves a pluger that opens the
______,allowing the pressurized fuel to squirt out through a tiny nozzle.
Vertical :
2. The basic parts of a turbine are the __________, which has blades projecting
radially from its periphery.
4. Jet engines move the airplane forward with a great force that is produced by a
tremendous __________ and causes the plane to fly very fast.
7. The ______________ can be reduced by using a convergent-divergent nozzle in
the pipeline.
9. A rocket operating on a chemical ________ consists of tanks containing the
chemical propellent.
11. The steam ______ or nozzle regulates the quantity of steam used by the injector.
FEEDBACK ON ACTIVITY 7B

25. NOZZLES J3008/7/25
4
1 s t e a m t u r b i n e
h
3 p r o p u l s i o n
u 2
s r 7
6 t h r o a t p
9 t r
5 f l o w m e t e r
11 u r s
c e s
10 b o i l e r u
n r
12 v a l v e 8 j e t
l
o
13 I n j e c t o r s
s
SELF-ASSESSMENT

26. NOZZLES J3008/7/26
You are approaching success. Try all the questions in this self-assessment section
and check your answers with those given in the Feedback on Self-Assessment. If you
face any problems, discuss it with your lecturer. Good luck.
7.1 Calculate the throat and exit areas of a nozzle to expand air at the rate of 4.5
kg/s from 8.3 bar, 327°C into a space at 1.38 bar. Neglect the inlet velocity
and assume isentropic flow.
7.2 It is required to produce a stream of helium at the rate of 0.1 kg/s travelling at
sonic velocity at a temperature of 15°C. Calculate the inlet pressure and
temperature required assuming a back pressure of 1.013 bar and negligible
inlet velocity. Calculate also the exit area of the nozzle. Assume isentropic
flow and helium is a perfect gas of molecular weight = 4 and γ =1.66.
7.3 Recalculate problem 1 assuming a coefficient of discharge is 0.97 and nozzle
efficiency is 0.92.
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27. NOZZLES J3008/7/27
Answers :
7.1 3290 mm2
, 4850 mm2
7.2 2.077 bar, 110°C , 592 mm2
7.3 The throat diameter = 20.5 mm & the exit diameter = 34 mm