STEAM NOZZLES

LECTURE NOTES – POWER PLANT
ENGINEERING
STEAM NOZZLES
VANITA THAKKAR
ASSOCIATE PROFESSOR
MECHANICAL ENGINEERING DEPARTMENT,
BABARIA INSTITUTE OF TECHNOLOGY,
VARNAMA, VADODARA

VANITA THAKKAR - BIT
2
INTRODUCTION
 A Power Plant / Power Station is an
industrial facility for generation of
Electric Power.
 It is a set-up consisting of systems and
sub-systems, equipments and auxiliaries
required for the generation of Electricity,
which involves conversion of energy
forms like chemical energy, heat energy
or gravitational potential energy into
Electrical Energy.

3
ENERGY CONVERSION PROCESS
The energy content in a primary source of energy, like
 Chemical Energy of a Fossil Fuel,
 Potential Energy of water stored at a height,
 Renewable / Non-conventional sources, like Solar
Thermal Energy, Wind energy, Geothermal Energy,
Tidal Energy, Wave Energy, etc.
is converted stage-wise to Mechanical Energy
(Rotational Energy) to obtain Electricity by creating
relative motion between a magnetic field and a
conductor.

4
THERMAL POWER PLANTS
 In Thermal Power Plants, mechanical power is produced by a
Heat Engine that transforms Thermal Energy, often from
Combustion of a Fuel, into Rotational Energy.
 Most Thermal Power Stations produce steam, and these are
sometimes called Steam Power Plants / Stations.
 Not all thermal energy can be transformed into mechanical
power, according to the Second Law of Thermodynamics.
Therefore, there is always heat lost to the environment.
 If this loss is employed as useful heat, for industrial
processes or distinct heating, the power plant is referred to as
a Cogeneration Power plant or CHP (combined heat-and-
power) plant.

5
RANKINE CYCLE
 A Thermal Power Plant is a
power plant in which the prime
mover is steam driven.
 Water is heated, turns into steam
in Boiler and spins a Steam
Turbine which either drives an
Electrical Generator or does
some other work, like Ship
Propulsion.
 After it passes through the
turbine, the steam is condensed
in a Condenser and recycled to
where it was heated.
 This is known as a Rankine cycle
– as shown in the figure.

6
MORE ABOUT RANKINE CYCLE
 The Rankine cycle is a
thermodynamic cycle which
converts heat into work.
 The heat is supplied externally
to a closed loop, which usually
uses water as the working
fluid.
 This cycle generates about
80% of all electric power used
throughout the world, including
virtually all solar thermal,
biomass, coal and nuclear
power plants.
 It is named after William John
Macquorn Rankine, a Scottish
polymath.

7
WILLIAM RANKINE
 The Rankine Cycle is named after William Rankine. Trained
as a civil engineer, William Rankine was appointed to the
chair of civil engineering and mechanics at Glasgow in
1855. He developed methods to solve the force distribution
in frame structures.
 He worked on heat, and attempted to derive Sadi Carnot's
law from his own hypothesis. His work was extended by
Maxwell.
 Rankine also wrote on fatigue in the metal of railway
axles, on Earth pressures in soil mechanics and the
stability of walls. He was elected a Fellow of the Royal
Society in 1853.
 Among his most important works are Manual of Applied
Mechanics (1858), Manual of the Steam Engine and Other
Prime Movers (1859) and On the Thermodynamic Theory of
Waves of Finite Longitudinal Disturbance.

8
PROCESSES IN RANKINE CYCLE
 There are Four processes in the Rankine cycle, each
changing the state of the working fluid.
 Process 1-2: Working fluid is PUMPED from low
to high pressure, as the fluid is a liquid at this
stage the pump requires little input energy.
 Process 2-3: The high pressure liquid enters a
BOILER where it is heated at constant pressure
by an external heat source to become a dry
saturated vapour (or wet vapour).
 Process 3-4: The dry saturated vapour expands
through a TURBINE, generating power. Due to
decrease in temperature and pressure of the vapour,
and some condensation may occur.
 Process 4-1: The wet vapour then enters a
CONDENSER where it is condensed at a
constant pressure and temperature to become
a saturated liquid. The pressure and temperature
of the condenser is fixed by the temperature of the
cooling coils as the fluid is undergoing a phase-
change.

9
RANKINE CYCLE : PRACTICAL
CARNOT CYCLE
 The Rankine cycle is sometimes referred to as a
Practical Carnot cycle as, when an efficient
turbine is used, the TS diagram will begin to
resemble the Carnot cycle.
 The main difference is that a pump is used to
pressurize liquid instead of gas. This requires
about 1/100th (1%) as much energy than that in
compressing a gas in a compressor (as in the
Carnot cycle).

10
Thus, BASIC COMPONENTS OF
THERMAL POWER PLANT
 BOILER
 STEAM TURBINE : The Prime Mover
 CONDENSER
 FEED PUMP
Supported by various sub-systems / accessories /
equipments required for their proper, efficient
working and to ensure their proper working in
co-ordination with each other.

11
SCHEMATIC OF A STEAM POWER
PLANT

12
STEAM TURBINES
 A steam turbine is a mechanical device – a PRIME
MOVER - that extracts thermal energy from high pressure,
high temperature steam, and converts it into rotary motion.
 PRIME MOVER : A machine that transforms energy from
thermal or pressure form to mechanical form; typically
 ENGINE : A mechanical device used to produce
rotation to move vehicle or otherwise provide the force
needed to generate kinetic energy
OR
 TURBINE : Any of various rotary machines that use the
kinetic energy of a continuous stream of fluid - a
liquid or a gas - to turn a shaft.

13
HISTORICAL BACKGROUND OF
STEAM TURBINES
 The AEOLIPILE is considered to be the first
recorded steam engine or reaction steam
turbine. The name – derived from the Greek
words "aeolos" and "pila" – translates to "the
ball of Aeolus“ (Aeolus : Greek god of the
wind).
 It is a rocket style jet engine described in the
first century BC by Vitruvius in his treatise De
architectura. Later, in the first century AD, Hero
of Alexandria also described the instrument,
and many sources mistakenly give him the credit
for its invention.
 It used steam power directed through two jet
nozzles so as to cause a sphere to spin rapidly
on its axis.
 It was a stand-alone device, and was
presumably intended as a temple 'wonder', like
many of the other devices described in Hero’s
Pneumatica.

14
STEAM TURBINES (contd.)
 More than a thousand years later,
the first impact steam turbine with
practical applications was invented
in 1551 by Taqi al-Din in
Ottoman, Egypt, who described it
as a prime mover for rotating a
SPIT (a cooking aid – a long solid
rod used to hold food while it is
being cooked over a fire in a
fireplace or over a campfire, or
roasted in an oven ).
 Similar smoke jacks were later
described by John Wilkins in
1648 and Samuel Pepys in 1660.
 Another steam turbine device was
created by Italian Giovanni
Branca in 1629.
Spitted fowl are rotated by a
hand-crank and basted with a
long-handled spoon in this
illustration from the Romance of
Alexander, Bruges, 1338-44
(Bodleian Library).

15
STEAM TURBINES (contd.)
 The modern steam turbine was invented in 1884 by
the Englishman Sir Charles Parson, whose first
model was connected to a dynamo that generated
7.5 kW of electricity.
 The invention of Parson's steam turbine made cheap
and plentiful electricity possible and
revolutionized marine transport and naval
warfare.
 His patent was licensed and the turbine scaled-up
shortly after by an American, George
Westinghouse. A number of other variations of
turbines have been developed.
 The de Laval turbine (invented by Gustaf de Laval –
Swedish Engineer) accelerated the steam to full
speed before running it against a turbine blade
(IMPULSE TURBINE). This was good, as the turbine is
simpler, less expensive and does not need to be
pressure-proof. It can operate with any pressure of
steam. It is, however, considerably less efficient.
The Parson's turbine also turned
out to be relatively easy to scale-
up. His invention adopted for all
major world power stations. The
size of his generators had
increased from his first 7.5 kW set
up to units of 50,000 kW capacity.
Within Parson's lifetime the
generating capacity of a unit was
scaled-up by about 10,000 times.

16
ENERGY CONVERSION IN STEAM
TURBINE
Energy Conversion in Steam Turbine takes place in TWO STEPS :
 High Pressure, High Temperature Steam expands in Nozzles
and comes out at a high velocity.
 High velocity steam jets from Nozzles impinge on the blades
mounted on a wheel – Rotor – get deflected by an angle and
suffer a loss of momentum which is absorbed by the Rotor in
producing Torque.
Thus,
STEAM TURBINE = Assemblage of Nozzles and Blades
(mounted on Rotor).

17
NOZZLES
 A NOZZLE is a device, a DUCT of
smoothly varying c/s area, that
increases the velocity of a fluid at the
expense of pressure.
 The chief use of nozzle is to produce a
jet of steam (or gas) of high velocity
to produce thrust for the propulsion of
rocket motors and jet engines and to
drive steam or gas turbines.
 A DIFFUSER is a device that increases
the pressure of a fluid by slowing it
down.
 Diffusers are used in compressors,
combustion chambers etc.

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TYPES OF NOZZLES
TWO types of nozzles :
 Convergent : The cross section of the
nozzle tapers to a smaller section to
allow for changes which occur due to
changes in velocity, specific volume,
dryness fraction – as the fluid
expands. It has lower Expansion
Ratio and hence lower outlet
velocities.
 THE SMALLEST SECTION OF THE
NOZZLE IS CALLED THROAT.
 Convergent – Divergent (c-d nozzle) :
The nozzle which converges to throat
and diverges afterwards. It has
higher Expansion Ratio – as addition
of divergent portion produces steam
at higher velocities.
CONVERGENT NOZZLE
CONVERGENT-DIVERGENT NOZZLE

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FLOW THRO’ NOZZLES : VELOCITY AND HEAT
DROP
ASSUMPTIONS :
 Steady state
 Adiabatic boundaries (Fluid Velocity
is very high, so there is no time
available for heat exchange with
surroundings.)
 Equilibrium states at inlet and outlet
 Mass average velocities adequate for
calculations
 No shaft work
 Change in potential energy is
negligible.
1 2
Consider fluid flow through a nozzle,
initially at :
pressure = p1; Enthalpy = h1, inlet velocity
= V1
Outlet conditions are respectively p2, h2
and V2 (denoted by subscript 2).
According to Steady Flow Energy Equation
(SFEE) :
(h1 + V1
2/2) = (h2 + V2
2/2)
=> (h1 - h2 ) = (V2
2/2 - V1
2/2 )

20
FLOW THRO’ NOZZLES : VELOCITY AND
HEAT DROP (contd.)
=> V2 = √{V1
2 + 2 (h1 – h2)}
Since, V1 << V2,
V2 = √{2 (h1 – h2)} OR
V2 = √{2 ∆h}
Since, enthalpy is usually expressed in
kJ/kg, and velocity in m/s,
V2 = √1000{2 ∆h}
=> V2 = [44.72√∆h] m/s ….. (1)
When the expanding fluid is a vapour, ∆h is
called Rankine Heat Drop, which can be
found from Mollier Chart or Steam Tables
for steam.
For Gases : pvγ = Constant
∆h = Cp.∆T

21
CROSS SECTION AREA AT OUTLET FOR
GIVEN INLET AREA
From Continuity Equation :
(A1V1/v1) = (A2V2/v2) = m = (AV/v) (in general);
Where,
A = Area (m2);
v = specific volume (m3/kg);
m = mass flow rate (kg/s)
Area per unit mass flow, (A/m) = (v/V)
i.e., (A/m) = (v/ 44.72√∆h) [From (1)]
Thus, it can be seen that : for calculating variations in area of the
nozzle, it is essential to know how expansion takes place, i.e.
how specific volume and enthalpy vary in the duct.

22
2. GENERAL RELATIONSHIP BETWEEN AREA, VELOCITY AND
PRESSURE (Effect of Area on Flow Properties in Isentropic Flow)
 To Study : the effect of Area variation on the velocity and pressure.
 Here, we determine the effect of change in area, A, on the velocity V, and the
pressure p.
From Steady Flow Energy Equation (SFEE) between two planes at an
infinitesimal distance apart : dQ = dh + d(V2/2) +dW
For adiabatic flow in a Nozzle / Diffuser : dQ = 0 and dW = 0
Hence, dh + d(V2/2) = 0 ……………. (1)
From Second Law of Thermodynamics, for reversible flow between planes
at infinitesimal distance : dQ = T.ds
From First Law of Thermodynamics : (dQ)rev = du + p.dv
Thus, Tds = du + p.dv = [dh – d(pv)] + p.dv = dh – v.dp
For an isentropic process, dQ = T.ds = 0, hence, dh = v.dp ………….. (2)
From (1) and (2) : v.dp + d(V2/2) = 0
i.e., dp/ρ + d(V2/2) = 0 ……………… (3) [as v = 1/ρ]

23
GENERAL RELATIONSHIP BETWEEN AREA, VELOCITY AND
PRESSURE (Effect of Area on Flow Properties in Isentropic Flow) –
contd.
Dividing by ρ/V2, ………….. (4)
Logarithmic Differentiation of Continuity Equation, (AV/v) = constant :
[ln A+ lnV – lnv = contant];
Differentiating :
(dA/A) + (dV/V) – (dv/v) = 0
i.e. …………. (5)
From (4) and (5) : OR

24
contd.
For Isentropic Process : [a2 = dp/dρ]
Hence :
……….. (5)
Thus, we see that
For Ma<1 an area change causes a pressure change
of the same sign, i.e. positive dA means positive dp
for Ma<1.
For Ma>1, an area change causes a pressure change
of opposite sign, i.e. positive dA means negative
dp for Ma>1.

25
contd.
From (4) and (5) :
….(6)
Thus, we see that
For Ma<1, an area change
causes a velocity change of
opposite sign, i.e. positive dA
means negative dV for Ma<1.
For Ma>1, an area change
causes a velocity change of
same sign, i.e. i.e. positive dA
means positive dV for Ma>1

26
contd.
 If a nozzle is used to obtain a supersonic
stream staring from low speeds at the inlet,
then the Mach number should increase from
Ma=0 near the inlet to Ma>1 at the exit.
 It is clear that the nozzle must converge in
the subsonic portion and diverge in the
supersonic portion.
 Such a nozzle is called a convergent-
divergent nozzle (De Laval Nozzle).
 It is also clear that the Mach number must
be unity at the throat, where the area is
neither increasing nor decreasing. This is
consistent with Equation (6), which shows
that dV can be non-zero at the throat only
if Ma=1. It also follows that the sonic
velocity can be achieved only at the
throat of a nozzle or a diffuser.

27
More about Flow through c-d nozzle
Ma may not necessarily be unity at the throat.
According to Eq. (6),
such situations are possible when Ma≠1 at the
throat and if dV=0.
 The flow in a convergent-divergent duct may be
subsonic everywhere with Ma increasing in the
convergent portion and decreasing in the
divergent portion with Ma≠1 at the throat (see
upper Fig.). The first part of the duct is acting
as a nozzle, whereas the second part is acting
as a diffuser.
 Alternatively, in a convergent-divergent duct in
which the flow is supersonic everywhere with
Ma decreasing in the convergent part and
increasing in the divergent part and again at
the throat Ma≠1 (see lower Fig.). Here, First
part : Diffuser; Second Part : Nozzle.

28
4. CONDITION FOR MAXIMUM
FLOW THROUGH A NOZZLE
Consider fluid flow through a nozzle, initially at :
pressure = p1; Enthalpy = h1, inlet velocity = V1
Outlet conditions are respectively p2, h2 and V2
(denoted by subscript 2).
According to Steady Flow Energy Equation (SFEE) :
(h1 + V1
2/2) = (h2 + V2
2/2)
=> (h1 - h2 ) = (V2
2/2 - V1
2/2 )
=> V2 = √{V1
2 + 2 (h1 – h2)}
Since, V1 << V2,
V2 = √{2 (h1 – h2)} OR V2 = √{2 ∆h}
From Second Law of
Thermodynamics, for reversible flow
between planes at infinitesimal
distance : dQ = T.ds
From First Law of Thermodynamics:
(dQ)rev = du + p.dv
Thus, Tds = du + p.dv
= [dh – d(pv)] + p.dv
= dh – v.dp
For an isentropic process,
dQ = T.ds = 0,
hence, dh = v.dp ………….. (1)

29
CONDITION FOR MAXIMUM FLOW
THROUGH A NOZZLE (contd.)
=>
Assuming that the pressure and volume of steam during
expansion obey the law pvn = constant, where n is the
isentropic index,
( )
1 1 1
2 2 2 1
1
n n
n n n
n
p v p p
n
− −
  −  
= −  
−    

30
i.e., ∆h
Now, mass flow rate : =>
Therefore, the mass flow rate at the exit of the nozzle :
….. (2)
OR …....(2)
(As V = √{2 ∆h})
( )
1 1
2 2 1 1 1
1
n
n n
n
p v p v p
n
−
 −
− 
−  

31
The exit pressure, p2 determines the for a given inlet condition.
There will be only one value of pressure ratio called critical
pressure ratio which gives maximum discharge (mass flow
rate).
The mass flow rate is maximum when,
i.e.
For maximum , ……(3)
i.e., the discharge through the nozzle will be the maximum at the
critical pressure ratio (p*/p1), shown above.

32
Comparing this with the results of sonic properties – Refer : GENERAL
RELATIONSHIP BETWEEN AREA, VELOCITY AND PRESSURE (Effect of Area on
Flow Properties in Isentropic Flow) – the critical pressure occurs at the
throat for Ma = 1.
The critical pressure ratio is defined as the ratio of pressure at the throat to
the inlet pressure, for choked flow (MAX. MASS FLOW CONDITION), i.e.
when Ma = 1 at throat.
n For P2/p1 or p*/ p1
1.4 for diatomic gases 0.528
1.3 for super saturated steam 0.546
1.135 for dry saturated steam 0.58
1.035 + 0.1x wet steam with dryness fraction x
(Dr. Zenner’s Equation)

33
MAXIMUM DISCHARGE THROUGH
THE NOZZLE
( ) ( )
( ) ( )
2 1
1 12 2
1 1max
1 1
1
.
2
1
n
n n n n
n n
n n
A n
m p v
v n
+
− −
 
   
      
       + +          
 
=   − 
( ) ( )
( )
2 1
1 12 2
1 1max
1 1
1
.
2
1
n
n n
n n
A n
m p v
v n
+
− −
 
   
−     + +    
 
=   − 
( ) ( )
( )
2 1
1 11 2 2
max
1 1
1
.
2
1
n
n n
n n
pn
m A
n v
+
− −
 
   
−     + +    
 
=   − 

34
MAXIMUM DISCHARGE THROUGH THE
NOZZLE (contd.)
( ) ( )
( )
2 1
1 1
1
1
1 2
1max
1
1
. 2
2
1 1
n
n n
n
n
n
pn
m A
n v n
  + 
−    − −  
+
 −   
−   +  
 
   
=     − +  
( ) ( )
( )
( )
1
1
1
1
1 2
1max
1
1
. 2
2
1 1
n
n
n
n
n
pn
m A
n v n
−
−
+
 −
  
−   +  
 
   
=     − +  
( ) ( )
1
1
1
1 2
1max
1
1
. 2
2
1 1
n
n
n
pn
m A
n v n
−
+
−    −   +  
   
=     − +  
( )
1
1
1
max
1
. 2 1
2
1 1 2
n
npn n
m A
n v n
+
−  −   
=       − +    
1
1
1
max
1
. 2
. .,
1
n
np
i e m A n
v n
+
−  
=   
+  

35
CONCLUSIONS REGARDING MAXIMUM
DISCHARGE CONDITIONS
From the equation
it is clear that Maximum mass flow or CHOKED FLOW :
 depends only on the initial condition of the steam
(p1, v1) and the throat area.
 does not depend on the final pressure of the steam,
i.e. at the exit of the nozzle.
The addition of divergent part of the nozzle after the
throat does not affect the discharge of steam
passing through the nozzle, but it only accelerates the
steam leaving the nozzle.
1
1
1
max
1
. 2
1
n
np
m A n
v n
+
−   
=    
+  

36
CONCLUSIONS REGARDING MAXIMUM
DISCHARGE CONDITIONS (contd.)
Discharge through nozzle increases as the pressure at
the throat of nozzle (p2) decreases, when the supply
pressure p1 is constant.
Keeping the inlet pressure p1 constant, when the nozzle
outlet pressure, p2 reaches the critical value given by
:
the discharge becomes maximum and after that the
throat pressure and mass flow rate remain constant
irrespective of the pressure at the exit.

37
VELOCITY AT THROAT IS SONIC UNDER
MAXIMUM DISCHARGE CONDITIONS
Under maximum discharge conditions, when the
pressure ratio at the throat of the nozzle has critical
pressure value, the velocity at throat – critical velocity
= sonic velocity.
Refer : GENERAL RELATIONSHIP BETWEEN AREA,
VELOCITY AND PRESSURE (Effect of Area on Flow
Properties in Isentropic Flow)

38
MAXIMUM VELOCITY OF STEAM
(CRITICAL VELOCITY)
From Continuity Equation : =>
From equations (2) and (3), i.e.
&
Thus, the velocity is also dependent on the initial conditions of the
steam.
( )max 1 1
1
2
1 1
n n
V p v
n n
  − 
=     − +   ( )max 1 12
1
n
V p v
n
 
=   + 
( )
1
1
max 1 1
2
2 1
1 1
n
n n
nn
V p v
n n
−
−
 
      
−      − +        

39
CHOKED FLOW THROUGH A
CONVERGENT NOZZLE ….
WHEN A NOZZLE OPERATES UNDER MAXIMUM
FLOW CONDITIONS, IT IS SAID TO BE CHOKED.
Consider a convergent nozzle with constant inlet
pressure p1, whose back pressure pb can be varied by
a valve.
When pb= p1, there is no fluid flow through the nozzle.
As pb decreases, mass flow through the nozzle increases,
as enthalpy decreases and velocity increases.
When pb = p*, i.e. CRITICAL PRESSURE, further
reduction in pb cannot affect the mass flow.
Also, when pb = p*, velocity at the exit is sonic and
mass flow rate through the nozzle is maximum.

40
CHOKED FLOW THROUGH A
CONVERGENT NOZZLE (contd.)
If pb is reduced below p*, mass flow
remains maximum, exit pressure p2
remains at p* and fluid expands
violently outside the nozzle down to pb.
Thus maximum flow through a convergent
nozzle is obtained when the pressure
ratio across the nozzle is critical pressure
ratio.

41
CHOKED FLOW THROUGH A C-D
NOZZLE
For a C-D Nozzle with sonic velocity
at the throat, the cross-section
area of the throat fixes the mass
flow through the nozzle for fixed
inlet conditions.
A correctly designed C-D Nozzle is
always choked.

42
EFFECT OF FRICTION AND NOZZLE
EFFECIENCY
For steam flowing through a nozzle, its final velocity for a given
pressure drop is reduced due to :
 Friction between nozzle surface and steam.
 Internal friction of steam itself.
 Shock losses.
Most of the frictional losses occur between the throat and exit in
c-d nozzle, producing following effects :
 Expansion is no more isentropic.
 Enthalpy drop is reduced.
 Final dryness fraction of steam increases
(kinetic energy  heat, due to friction and gets absorbed.)
 Specific Volume of steam increases.
(steam becomes more dry due to frictional reheating.)

43
EFFECIENCY (contd.)
Point 1 : nozzle entrance, DRY
SATURATED STEAM.
Line 1-2 : steam expansion from
entrance to throat (friction
neglected)
Line 2-3 : steam expansion from
throat to exit (friction
neglected).
Friction occurs mainly between
throat and exit, hence,
considering friction, heat drop is
represented by Line 2-3’.

44
EFFECIENCY (contd.)
The horizontal line from point 3’
cuts the pressure line on which
point 3 lies on 2’, which represents
the final condition of the steam
(nozzle exit).
Hence actual process of expansion
is given by : Line 1-2-2’.
Dryness Fraction of steam is more
at 2’ than at 3. Hence, the effect
of friction is to improve the
quality of steam.

45
EFFECIENCY (contd.)
Point 4 : nozzle entrance,
SUPERHEATED STEAM.
Line 4-5 : steam expansion
from entrance to throat
(friction neglected)
Line 5-6 : steam expansion
from throat to exit (friction
neglected)
Friction occurs mainly between
throat and exit, hence,
considering friction, heat drop
is represented by Line 4-6’.

46
EFFECIENCY (contd.)
The horizontal line from point 6’
cuts the pressure line on which
point 6 lies on 5’, which
represents the final condition of
the steam (nozzle exit).
Hence actual process of
expansion is given by : Line 4-5-
5’.
Dryness Fraction of steam is
more at 5’ than at 6. Hence, the
effect of friction is to superheat
the steam.

47
EFFECIENCY (contd.)
NOZZLE EFFICIENCY : The ratio of ACTUAL
ENTHALPY DROP to ISENTROPIC ENTHALPY
DROP between the same pressure.
Nozzle efficiency :
(FOR DRY, SATURATED STEAM)
(FOR SUPERHEATED STEAM)
1 3'
zzle
1 3
h h
h h
Νο
−
η =
−
4 6'
zzle
4 6
h h
h h
Νο
−
η =
−

48
Why friction is considered only between
throat and exit in c-d nozzle ?
Reasons why friction is assumed between throat and exit only
for c-d nozzle :
1.Small inlet velocity in the convergent portion, hence friction
losses are also small.
2.No losses due to turbulence in convergent part.
3.Divergent part of the nozzle is longer to avoid flow
separation.
4.Velocity is high in divergent portion.
5.Angle of divergence is less than 20o.
6.There are losses in the divergent portion due to turbulence.

49
EFFECIENCY (contd.)
Exit velocity under isentropic conditions = V3
Let, Actual exit velocity = V2’
By steady flow energy equation (SFEE) :
or (ISENTROPIC CONDITIONS)
or (ACTUAL CONDITIONS)
2 2
1 2'
1 2'
2 2
V V
h h+ = +
2 2
2' 1
1 2'
2
V V
h h
−
− =
22
31
1 3
2 2
VV
h h+ = +
2 2
2' 1
Nozzle 2 2
3 1
Nozzle Efficiency,
V V
V V
−
η =
−
2 2
3 1
1 3
2
V V
h h
−
− =

50
EFFECIENCY (contd.)
As V1 <<< V2’, V3
Value of Nozzle Efficiency depends on :
1.Nozzle material.
2.Workmanship in manufacture of nozzle.
3.Shape and size of nozzle.
4.Angle of divergence.
5.Nature of fluid flowing and its state.
6.Fluid velocity.
7.Friction
8.Turbulence in nozzle flow passages.
2
2'
Nozzle 2
3
Nozzle Efficiency,
V
V
η =

51
FACTORS AFFECTING NOZZLE
EFFECIENCY
Nozzle Type Velocity Coefficient
Roughly cast nozzles 0.93 to 0.94
Machined nozzles 0.95 to 0.96
Smoothly milled nozzles 0.96 to 0.97
VELOCITY COEFFICIENT : Ratio of ACTUAL EXIT
VELOCITY to the EXIT VELOCITY WHEN THE FLOW IS
ISENTROPIC between the same pressure.
Thus, Velocity Coefficient is the square root of Nozzle
Efficiency when the inlet velocity is assumed to be
negligible.
2 '
Nozzle
3
Velocity Coefficient
V
V
= = η

52
SUPERSATURATED FLOW
When steam flows / expands through a nozzle, it
would be normally expected that the discharge
of steam through the nozzle would be slightly
less than the theoretical value.
But, experiments on flow of wet steam show that
Discharge is slightly greater than that calculated
by the formulae. This can be explained as
follows :
 The converging part of the nozzle is so short and
the steam velocity is so high that the molecules
of steam have insufficient time to collect and
form droplets.
 So, normal condensation does not take place.

53
SUPERSATURATED FLOW (contd.)
 There is a phase of rapid expansion,
which is said to be metastable and it
produces supersaturated state.
 In supersaturated state, steam is
undercooled to a temperature less than
that corresponding to its pressure.
 So, density of the steam increases and
hence the weight of discharge.

54
SUPERSATURATED FLOW –
WILSON’S LINE
Prof. Wilson showed it experimentally that :
“When dry saturated steam is suddenly
expanded in absence of dust, it does not
condense until its density is about EIGHT TIMES
that of the saturated vapour at the same
pressure.”
The limiting condition of under-cooling at
which condensation commences and is
assumed to restore conditions of normal
thermal equilibrium is called WILSON LINE.

55
SUPERSATURATED FLOW (contd.)
Consider the h-s diagram
shown in the figure.
The process 1-2 is the
isentropic expansion. The
change of phase should
begin to occur at point 2.
BUT, Vapour continues
to expand in a dry state.

56
SUPERSATURATED FLOW(contd.)
Steam remains in an
unnatural superheated
state until its density is
about eight times that of
the saturated vapour
density at the same
pressure (As per Prof.
Wilson’s Experiment).
When this limit is reached,
the steam will suddenly
condense.

57
Point 3 is achieved by extension of
the curvature of constant pressure
line p3 from the superheated
region which strikes the vertical
expansion line at 3 and through
which Wilson line also passes.
The point 3 corresponds to a
metastable equilibrium state of
the vapour.
The process 2-3 shows expansion
under super-saturation condition
which is not in thermal
equilibrium.
It is also called under cooling.

58
At any pressure between p2 and
p3 i.e., within the superheated
zone, the temperature of the
vapours is lower than the
saturation temperature
corresponding to that pressure.
Since at 3, the limit of
supersaturation is reached, the
steam now condenses
instantaneously to its normal
state at the constant pressure
and constant enthalpy which is
shown by the horizontal line 3-3’
where 3’ is on normal wet area
pressure line of the same
pressure p3.

59
3’-4’ is again isentropic
expansion in thermal
equilibrium.
To be noted that 4 and 4’ are on
the same pressure line.
Effect of supersaturation :
•Increase in entropy and specific
volume of the steam.
•Slight reduction in the enthalpy
drop during expansion and
corresponding reduction in final
velocity.

60
Effect of supersaturation (contd.) :
•Final dryness fraction increases.
•Density of supersaturated steam is
more than that for equilibrium
conditions [As no condensation
during supersaturated expansion
=> supersaturation temperature <
saturation temperature
corresponding to the pressure].
•Thus, Measured discharge (=>
mass) is greater than that
theoretically calculated.

61
Degree of super heat =
p3 = Limiting saturation pressure
p3s = Saturation pressure at
temperature T3 shown on T-s diagram
Degree of undercooling = T3s – T3
T3s is the saturation temperature at p3
T3 = Supersaturated steam temperature
at point 3 which is the limit of
supersaturation.
Supersaturated vapour behaves like
supersaturated steam (n = 1.3).
3
3s
p
p

62
Important Note : Problems on
supersaturated flow cannot be
solved on Mollier Chart UNLESS
Wilson Line is drawn on it.

63
EFFECT OF VARIATION IN BACK
PRESSURE
Nozzles are designed for maximum
discharge for given area.
When exit pressure(p2) differs from the value
for which the nozzle is designed (p*), the
flow conditions in the nozzle change.
When pb < p*, : UNDEREXPANDING
NOZZLE – Expansion of fluid to design
pressure inside nozzle;
At outlet fluid expands violently and
irreversibly down to pb.

64
EFFECT OF VARIATION IN BACK
PRESSURE (contd.)
When pb > p*, : OVEREXPANDING
NOZZLE.
In Convergent Nozzle : Reduction in
mass flow rate from the nozzle.
In C-D Nozzle : Expansion followed
by Recompression.

65
EFFECT OF VARIATION IN BACK PRESSURE
(contd.) : CONVERGENT NOZZLE
Consider a convergent nozzle
connected to a large reservoir of
fluid.
The reservoir is assumed to be so
large that the inlet flow
conditions to the nozzle remain
constant.
The back pressure (pb) is varied
by a valve – as shown in the
figure.
When inlet pressure, p1 = pb :
No fluid flow – curve (a).
On reducing pb : flow starts.

66
With reduction in pb : mass
flow rate and velocity
increase; pb > p* – curve (b)
 OVEREXPANSION.
When pb = p* : maximum
mass flow rate / discharge
through the nozzle – curve (c)
: CHOKED FLOW through the
nozzle.
Upto this point and
excluding this point :
OVEREXPANDING
CONDITIONS.

67
With further reduction in
pb i.e. pb < p* : mass flow
rate, velocity and specific
volume do not change.
The fluid expands
violently and irreversibly
outside the nozzle to pb :
UNDEREXPANSION,
during which the pressure
oscillates and shock wave
is formed.

68
(contd.) : C-D NOZZLE
Consider a C-D nozzle
connected to a large reservoir
of fluid.
The reservoir is assumed to be so
large that the inlet flow
conditions to the nozzle remain
constant.
The back pressure (pb) is varied
by a valve – as shown in the
figure.
When inlet pressure, p1 = pb :
No fluid flow – curve (a).
On reducing pb : flow starts.

69
With reduction in pb : mass
flow rate and velocity
increase; pb > p* – curve (a).
Here :
Throat pressure is > p*.
Velocity at throat is
subsonic.
Divergent portion acts as
Subsonic Diffuser.
Nozzle acts as Venturimeter.

70
With further
reduction in pb :
When throat
pressure = p* :
maximum mass
flow rate /
discharge through
the nozzle – curve
(b) : CHOKED
FLOW through the
nozzle.

71
On Further reduction in pb – Curve
(c) :
No change in the conditions at
throat.
Mass flow rate remains constant.
Fluid velocity increases in the
divergent portion, making velocity
supersonic.
At some point, due to divergence,
supersonic steam is decelerated =>
shock wave is generated.
Shock wave : Abrupt rise in
pressure, increase in entropy and
velocity reduces from supersonic to
subsonic.

72
On Further reduction in pb :
Shock wave travels down
the nozzle.
A stage comes when pb =
design exit pressure –
Curve (d).
In this condition, flow is
accelerated continuously
from sonic velocity at the
throat to supersonic
velocity at exit.

73
Finally, on reduction in pb
below designed value :
Expansion takes place
outside the nozzle, as in
convergent nozzle :
UNDEREXPANSION –
Curve (e) – a series of
irreversible compressions
through shock-waves,
alternated with
irreversible expansions,
until the pressure = pb.

74
THANKS !!!
VANITA THAKKAR
ASSISTANT PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING,
FACULTY OF TECHNOLOGY AND ENGINEERING,
CHAROTAR UNIVERSITY OF SCIENCE AND
ENGINEERING, CHANGA,
DIST. : ANAND (GUJARAT)

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