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NOZZLE NOZZLE NOZZLE NOZZLE THEORY THEORY THEORY THEORY Alessandro Corsini Department of Mechanical & Aerospace Engineering Sapienza University of Rome, Italy

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NOZZLE NOZZLE NOZZLE NOZZLE THEORYTHEORYTHEORYTHEORY

Alessandro Corsini

Department of Mechanical & Aerospace EngineeringSapienza University of Rome, Italy

2

GOVERNING EQUATIONS

3

• WHAT IS A NOZZLE?

A nozzle is a mechanical device that control the characteristics of a flow, as it exits from a chamber into a receiving medium.

A nozzle is generally a duct or a pipe (rectilinear or curvilinear) with varying sections

• WHERE DO I FIND IT?

Nozzles are used to accelerate fluid being part of stator cascades of power-producing turbomachinery

Nozzles find application in every field of engineering, from engines to atomiser to plastic extruder.

4

THE IDEAL NOZZLE

The ideal nozzle is a quasi-one-dimensional nozzle flow under the following

assumptions:

1. Working fluid is homogeneous

2. Working fluid is a gas and free of condensed phases

3. The working substance obeys the perfect gas law

4. The flow is adiabatic, i.e. there is no heat transfer with the walls

5. There is no friction on the wall, i.e. boundary layers are neglected

6. Nozzle flow is free of shock waves or discontinuities

5

THE IDEAL NOZZLE

The ideal nozzle is a quasi-one-dimensional nozzle flow under the following

assumptions:

7. Flow is stationary, i.e. time-dependent phenomena are neglected

8. Outlet velocity is perfectly axial

9. Flow quantities are circumferentially uniform, i.e. quantities vary only

along the axial direction

10. No chemical reactions

Under this assumptions, the ideal nozzle shows departure from real nozzles

of less than 6%.

6

First Law of Thermodynamics

In an adiabatic, no-shaft work process, the flow entropy does not change.

∆s = 0

Furthermore, the total stagnation enthalpy h0 is given by:

2

0constant

2

vh h= + =

Where v is velocity of the gas

7

CONSERVATION OF ENERGY

The conservation of energy for isentropic flow between two sections, hereafter denoted with the subscripts x and y, can be written as:

CONSERVATION OF MASS

Also known as continuity equation, states that along every section the mass flow rate remains the same

where A and V are the area of passage and the specific volume respectively.

( ) ( )2 21

2x y y x p x yh h v v c T T− = − = −

where Cp is the specific heat at constant pressure.

An increase/decrease of enthalpy or thermal content leads to a

decrease/increase of kinetic energy

/x y

m m m Av V= = =& & &

8

It is written as:

with p the pressure, R’ is the universal gas constant (8314.3 J/Kg-mole-K)

and n the number of moles of the gas mixture. Further relationships to be

considered:

PERFECT GAS LAW

x x x x

Rp V T RT

n

′= =

( )

/

/ 1

p v

p v

P

k c c

c c R

c kR k

=

− =

= −

with k the heat capacity ratio

9

For isentropic flow process, the relations between two points x and y reads as:

Looking at the previous expression, during an isentropic expansion Vy>Vx :

• the pressure drops substantially

• the absolute temperature also drops

• specific volume increases

PERFECT GAS LAW

( )( )

( )1 / 1

/ / /k k k

x y x y y xT T p p V V− −

= =

10

When a flow is stopped isentropically, the starting conditions are known as

stagnation conditions, denoted with the 0 subscripts.

Where T is the absolute fluid static temperature. In adiabatic flows, the

stagnation temperature remains constant. By combining the previous two

expressions:

PERFECT GAS LAW

( )2

0/ 2 pT T v c= +

( )( )

( )/ 1

2

0 0/ 1 / 2 /

k k k

pp p v c T V V

− = + =

11

The velocity of sound a can be written as:

equal to 343.8 m/s in air at 293K. Thus, the Mach number can be defined as

the ratio between the current flow velocity and the local velocity of sound:

MACH NUMBER

a kRT=

/ /M v a v kRT= =

Flow

condition

M < 1

M = 1

M > 1

subsonic flow

supersonic flow

sonic flow

12

IDEAL NOZZLE

13

IDEAL NOZZLE

Inlet

At,Vt,pt

Ax,Vx,px

Ay,Vy,py

Outlet

Throat

Example of a converging – diverging supersonic nozzle

14

Stagnation temperature can therefore be written as a function of the Mach

number:

TEMPERATURE RATIO

( )( )

2

0/ 2

/ 1

p

p

T T v c

c kR k

= +

= −

( ) ( )

( ) ( ) ( )

( ) ( )

( )

2

0

2

0

2

0

2

0

1 / 2

1 / 2

1 / 2

11 1

2

T T v k kR

T T M kRT k kR

T T M kRT k kR

T T k M

= + −

= + −

= + −

= + −

15

Thus, Mach number is also computed as:

PRESSURE RATIO

02

11

TM

k T

= −

By looking back at the stagnation pressure equation we get:

( )( )

( ) ( )( )

( )( )

/ 12

0

/ 12

0

/ 1

2

0

1 / 2

1 / 2

11 1

2

k k

p

k k

p

k k

p p v c T

p p M kRT c T

p p k M

= +

= +

= + −

16

The area ratio for a nozzle tiwht isentropic flow can be written in term of Mach

AREA RATIO

( )

( )

( 1)/( 1)2

2

1 1 / 2

1 1 / 2

k k

yy x

x y x

k MA M

A M k M

+ − + − =

+ −

When the ratio between inlet and throath area A1/At is small, between 3 and 6,

the passage is convergent. In supersonic flow, the nozzle section diverges and

the ratio A2/At varies between 15 and 30 at M=4.

17Relationship of area ratio, pressure ratio and temperature ratio as a

function of Mach number, with Mx = 1.0

18

From the conservation of energy we can derive the exit velocity:

ISENTROPIC FLOW TROUGH NOZZLES

By substituting the formula for enthalpy hx = cp(Tx-Ty) in the former expression:

( ) ( )

( )

2 2

1 2 2 1 1 2

2

2 1 2 1

1

2

2

ph h v v c T T

v h h v

− = − = −

= − +

( )

( )

2

2 1 2 1

2 21

2 1 2 1 1 1

2

2

2 2 11 1

pv c T T v

Tk kv R T T v RT v

k k T

= − +

= − + = − +

− −

19

ISENTROPIC FLOW TROUGH NOZZLES

21

2 1 1

2

2 11

Tkv RT v

k T

= − +

and considering the temperature ratio:

( )( )

( )1 / 1

/ / /k k k

x y x y y xT T p p V V− −

= =

( 1)/ ( 1)/

22 2

2 1 1 1

1 1

2 1 2 11 1

k k k k

p pk kv RT v RT

k p k p

− − = − + ≈ −

− −

This expression is still viable for any two points inside the nozzle. When the

inlet section is large compared to the nozzle throat section, the term v1 can

be neglected.

If A1 >> At

20

ISENTROPIC FLOW TROUGH NOZZLES

The maximum theorical outlet velocity can be therefore calculated by

considering an expansion into the vacuum, i.e. p2 = 0.

( 1)/

2

2 1

1

2 11

k k

pkv RT

k p

− = −

2,max 02

1

kv RT

k=

21

NOZZLE FLOW

Nozzles that consist of a convergent

section followed by a divergent

section are also called De Laval

nozzles.

• From the continuity eq. area is

inversely proportional to the ratio

v/V.

• A maximum in the curve v/V

because in the convergent

section velocity increases more

than the volume, whereas in the

divergent section trend is

opposite.

22

NOZZLE FLOW

The minimum nozzle area is called throat area, hereafter denoted with the

subscript t. We define:

2

t

A

Aε =

nozzle area

expansion rate

The maximum gas flow per unit area occurs at the throat. Here the gas

ratio is only a function of the ratio of specific heats k, obtained by setting

M=1.

( )( )/ 1

1/ 2 / 1

k k

tp p k−

= +

pt is also known as the critical pressure, with common values of the ratio

reported between 0.53 and 0.57.

23

NOZZLE FLOW

At the point of critical pressure (the throat), the Mach number is equal to

one, therefore the specific volume and temperature can be written as:

Because of M = 1, throat velocity can be immediately derived by the

definition of Mach number:

( )( )

( )

1/ 1

1

1

1 / 2

2 / 1

k

t

t

V V k

T T k

−= +

= +

1

2

1t t

kv a M kRT kRT RT

k= = = =

+

The throat velocity is clearly the sonic velocity and it can be directly derived

from the inlet temperature T1.

24

TYPE OF NOZZLES

Based on the flow condition at the exit section, we find three different types

of nozzles

25

For a supersonic nozzle the ratio between the throat and any downstream

area x, as follows:

1/ ( 1)/1/( 1)

1 1

1 11

2 1

k k kk

t t x x x

x x t

A V v p pk k

A V v p k p

−− + + = = −

The mass flow through a supersonic nozzle is proportional to

the throat area At and pressure p1. In fact, given:

[ ]( 1)/( 1)

1

1

2 / ( 1)k k

t t

t

t

kA vm A p k

V kRT

+ −+

= =&

26

UNDER- / OVER- EXPANDED NOZZLES

27

Under-expanded nozzles:

• discharge the fluid at an exit pressure greater than the external pressure;

• the exit area is too small for un optimum area ratio;

• expansion of the fluid is incomplete and must take place outside.

Over-expanded nozzles:

• discharge the fluid at lower pressure than the exterior;

• the exit area is too large for optimum;

• expansion is completed in the nozzle entirely.

28

Distribution of pressures in a converging-

diverging nozzles for different flow

conditions.

Over-expanded nozzles:

• Curve AB shows the optimum

expansion.

• AC and AD show the variation of

pressure for increasingly higher

external pressures.

• To compensate external pressure, we

find a sudden expansion (point I).

• Point I is accompanied by the

separation of the flow from the walls.

• Such phenomena are also called

compression waves.

29

Four possible flow conditions:

1. external pressure is below the nozzle exit pressure: the nozzle will flow fully with

external expansion waves at the exit.

2. external pressure is slightly higher than the nozzle exit pressure: nozzle is flowing

fully. This behaviour is found for ration between external and exit pressure

between 25% and 40%. Shock waves persist outside the exit section.

3. external pressure is dramatically higher than nozzle exit pressure: the diameter of

the supersonic jet is smaller than the nozzle exit diameter. Axial-symmetric

separation occurs on the walls. The point of separation travels downstream with

decreasing external pressure. Shock waves outside the nozzle.

4. exit pressure is just below the values of the inlet pressure: the pressure ratio is

below the critical pressure ratio, subsonic low prevails throughout the entire

nozzle.

30

Three flow conditions during operating conditions of a rocket

31

NOZZLES CONFIGURATIONS

32

Most of the proven nozzle configurations are directly derived from rockets and propulsions.

Simplified configurations of nozzles and flow effects

33

• Nozzles and chambers are usually of circular cross section and have a converging

section and a throat in the narrowest location, followed by a diverging section;

• The converging section has low impact on the overall performances, and any radius,

cone angle, wall contour curve or nozzle inlet shape is satisfactory;

• The throat contour is also not very critical to performance, any radius or other curve

is usually acceptable. The only relevant factor is the wall friction. Wall surface needs

to be smoother as possible to minimize radial absorption and heat transfer;

Length comparison of several type of nozzles

34

Cone- and Bell-Shaped Nozzles

The conical nozzles is the oldest and simplest configuration.

Bell vs conical nozzles: design parameters

35

Cone-Shaped Nozzles

A theoretical correction factor λ is usually applied to the nozzle exit

momentum of an ideal rocket with a conical nozzle exhaust.

( )1

1 cos2

λ α= +

λ is the ratio between the momentum of the gases in a nozzle with a finite

nozzle angle of 2α and the momentum of an ideal one-dimensional axial flow.

36

Cone-Shaped Nozzles

Small divergence angle cause most of the momentum to be axial and thus

has high thrust, but has high penalties in term of weight and complexity. A

large divergence gives short and lightweight design, but the performance

are lower.

37

Bell-Shaped Nozzles

• The length of a bell nozzle is usually given a fraction of the length of a

reference conical nozzle with a 15 deg half angle.

• An 80% bell nozzle has a length that is 20% shorter than a 15 deg cone

with the same aria ratio.

38

Bell-Shaped Nozzles

• Design of the bells can be approximated with a parabola

• Parabola is tangent θi at point I and has an exit angle θe at point E

• Design is scarcely influenced by the throat approach radius ri and the

39

Two-Step Nozzles

Several correction to the Bell- and Cone-

shaped nozzles, to compensate the effect of

altitude. Some of them are:

• Extendable nozzles: actuators are used to

increase the length of the nozzle during

flights. Complex and requires a power

supply.

• Droppable insert concept: a mechanism is

used to enlarge the section area.

• Dual bell nozzles: two shortened bell

nozzles are combined into one with a bump

or inflection point between them

Top to bottom: extendable nozzles, droppable

inserts, dual-bell

40

REAL NOZZLES

41

REAL NOZZLES

• The flow is two-dimensional but axisymmetric;

• Temperatures and velocities are not uniform over any section;

• Exit velocity v2 can be seen as a function of the radius r:

The 11 assumptions at the begin of the chapter cease to be universally true.

It is possible to apply them through:

1. Empirical correction factors

2. More accurate computing of the phenomena involved

2

2, 20

2

2average

v v rdrA

π= ∫

42

REAL NOZZLES

Losses in real nozzles can be ascribed to:

1. Divergence of the flow at the exist section

2. Low nozzle contraction ratios A1/At cause pressure losses in the chamber

3. Wall friction reduce the performances up to 1.5%

4. Solid particles or liquid roplets reduce performance up to 5%

5. Unsteady combustion and oscillating phenomena are responsible of small

deflections

6. Transient conditions lead to lower performance

7. Erosion leads to non-optimal design of the throat section

8. Non-uniform gas composition could lead to incomplete combustion / non efficient

mixing

9. Undesired chemical reactions may change gas composition slightly

10. Real gas properties are not fixed but changed along the nozzle

11. Operations at out-of-optimum design points usually lead to lower performances

43

Boundary Layer

Real nozzles have a viscous boundary layer next to the nozzles walls.

Immediately next to the wall the flow velocity

is zero. Here the low-velocity flow is laminar

and subsonic, whereas in higher-velocity

regions the flow is supersonic and can

become turbulent.

In the boundary layer the local temperature

can be substantially higher than the core

flow, due to conversion of kinetic energy into

thermal energy. However, the layer

immediately at contact with the wall will be

cooler due to the heat transfer with the wall.

This deflection from the theoretical design

can lead to a thrust lost of 1%.

44

Multiphase Flow

• Real flows contain many small liquid droplets and solid particles that are accelerated

by the gas.

• Common examples are aluminium powders, ion oxides, propellants with beryllium or

boron.

• If the diameter of particles is small (<0.005 mm), the multiphase effect can neglected,

as particles are in thermal equilibrium with the main flow.

• If the diameter is larger, the inertia of the particles grows as the cube of the diameter,

while the drag force increases with the square.

• Larger particles do not move at the same speed as the core flow and reach higher

temperature than the smaller.

45

Multiphase Flow

• Theoretical model can be therefore be corrected to account multi-phasing.

• Governing equations can be rewritten as a function of the particle fraction β, defined

as the mass of particles divided by the total mass. Denoting with s and g subscripts

the two phases (solid and gas), we derive:

( )

( )

( )( )

,

,

,

1

1

/

(1 )

1

1

p g s

g

g g

g

p g s

v g s

h c T c T

V V

p R T V

R R

c ck

c c

β β

β

β

β β

β β

= − +

= −

=

= −

− +=

− +

46

Nozzle Guide Vanes in Turbomachinery

Gas turbines