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How the will vary from subsonic to supersonic flow.
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5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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Dept. of ANE, MRCET Page 1
CHAPTER-1
INTRODUCTION TO ROCKET NOZZLES
1.1 Definition of nozzle: A Rocket Nozzle is a mechanical device which produces thrust
and is used to control the characteristics of fluid as it enters/exits an enclosed chamber or pipe.
Nozzle is a relatively simple device, just a specially shaped tube through which hot gases flow.
The types of nozzles can be explained as fallows[1].
1.2 Types of nozzles: There are 3 primary groups of nozzle
1.
Cone shaped nozzle
2. Bell nozzle
3. Annular nozzle
1.2.1 Cone:The cone shaped nozzles are used in Used in early rocket applications because of
simplicity and ease of construction. Cone gets its name from the fact that the walls diverge at aconstant angle. A small angle produces greater thrust, because it maximizes the axial component of exit
velocity and produces a high specific impulse. A small nozzle divergence angle causes most of the
momentum to be axial and thus give a high specific impulse, but the long nozzle has a penalty in rocket
propulsion system mass ,vehicle mass and also design complexity. A large divergence angle give short,
light weight designs, but the performance is low[1].
Figure 1.1- Conical nozzle
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1.2.2 Bell: There are two types in bell shaped nozzle. The primary type is Contour nozzle is
probably the most commonly used shaped rocket engine nozzle. It has a high angle expansion
section right behind the nozzle throat; this is followed by a gradual reversal of nozzle contour
slope so that the nozzle exit the divergence angle is small, usually less than a 10 degree half
angle[1].
Figure 1.2-Contoured nozzle
The second one is Convergent-Divergent nozzle. It is also called as De-Laval nozzle. It
is used to accelerate a hot, pressurizedgaspassing through it to asupersonic speed, and upon
expansion, to shape the exhaust flow so that the heat energy propelling the flow is maximally
converted into directedkinetic energy.Because of this, thenozzle is widely used in some types
ofsteam turbines,and is used as arocket engine nozzle.It also sees use in supersonicjet engines.
Figure 1.3- Convergent-Divergent nozzle
http://en.wikipedia.org/wiki/Gashttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/Nozzlehttp://en.wikipedia.org/wiki/Steam_turbineshttp://en.wikipedia.org/wiki/Rocket_engine_nozzlehttp://en.wikipedia.org/wiki/Jet_engineshttp://en.wikipedia.org/wiki/Jet_engineshttp://en.wikipedia.org/wiki/Rocket_engine_nozzlehttp://en.wikipedia.org/wiki/Steam_turbineshttp://en.wikipedia.org/wiki/Nozzlehttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/Supersonichttp://en.wikipedia.org/wiki/Gas5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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1.2.3 Annular: Annular nozzles are classified into three different types. They are Aero-spike,
Plug, Expansion-deflection nozzles. Considering an Aero-spike nozzle,Aero-spike the spike is
bowl-shaped with the exhaust exiting in a ring around the outer rim. In theory this requires an
infinitely long spike for best efficiency, but by blowing a small amount of gas out the center of a
shorter truncated spike, something similar can be achieved. In the linear Aero-spike the spike
consists of a tapered wedge-shaped plate, with exhaust exiting on either side at the "thick" end.
This design has the advantage of being stackable, allowing several smaller engines to be placed
in a row to make one larger engine while augmenting steering performance with the use of
individual engine throttle control[1].
Figure 1.4 - Aero-Spike nozzle
The plug nozzle is a type ofnozzlewhich includes a center body or plug around which
the working fluid flows. Plug nozzles have applications in aircraft, rockets, and numerous other
fluid flows. In rockets Plug nozzles belong to a class ofaltitude compensating nozzlesmuch like
the aero spikewhich, unlike traditional designs, maintains its efficiency at a wide range of
altitudes. The ideal contour of a plug nozzle is a long tapering 'spike' with a doughnut-shaped
combustion chamber situated at the base, hence sometimes this nozzle is also called a "spike
http://www.wikipedia.org/wiki/Nozzlehttp://www.wikipedia.org/wiki/Nozzlehttp://www.wikipedia.org/wiki/Nozzlehttp://www.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://www.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://www.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://www.wikipedia.org/wiki/Aerospike_enginehttp://www.wikipedia.org/wiki/Aerospike_enginehttp://www.wikipedia.org/wiki/Aerospike_enginehttp://www.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://www.wikipedia.org/wiki/Nozzle5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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nozzle". To save weight, this design is shortened without a large drop in efficiency. The exhaust
is confined by atmospheric pressure so that at different altitudes the varying pressures will allow
the exit area to change. This allows perfect atmospheric compensation. With the shortened
nozzle, the recirculation of trapped gases at the base of the plug causes a small thrust which
offsets the loss due to the non-ideal shape. Plug nozzles are used in aircraft typically withjet
enginesboth because of the annular shape of the turbine exhaust and for their altitude
compensating characteristics. For high speed aircraft, translating the plug or external cowl
provides a means of area control with relatively simple actuation. Plug nozzles have been shown
to provide noise reduction compared to traditionalconvergent-divergent nozzles.Weight and
cooling are typical concerns with aircraft plug nozzles[1].
Figure 1.5- Plug nozzle
The expansion-deflection nozzle is an advancedrocket nozzle which achievesaltitude
compensation through interaction of the exhaust gas with the atmosphere, much like theplug and
aero-spike nozzles.It appears much like a standard bell nozzle, but at the throat is a 'centrebody'
or 'pintle' which deflects the flow towards the walls. The exhaust gas flows past this in a more
outward direction than in standard bell nozzles while expanding before being turned towards the
exit. This allows for shorter nozzles than the standard design whilst maintaining nozzle
expansion ratios. Because of the atmospheric boundary, the atmospheric pressure affects the exit
area ratio so that atmospheric compensation can be obtained up to the geometric maximum
allowed by the specific nozzle. The nozzle operates in two distinct modes: open and closed. In
http://www.wikipedia.org/wiki/Jet_enginehttp://www.wikipedia.org/wiki/Jet_enginehttp://www.wikipedia.org/wiki/Jet_enginehttp://www.wikipedia.org/wiki/Jet_enginehttp://www.wikipedia.org/wiki/Convergent-divergent_nozzlehttp://www.wikipedia.org/wiki/Convergent-divergent_nozzlehttp://www.wikipedia.org/wiki/Convergent-divergent_nozzlehttp://en.wikipedia.org/wiki/Rocket_engine_nozzlehttp://en.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://en.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://en.wikipedia.org/wiki/Plug_nozzlehttp://en.wikipedia.org/wiki/Aerospike_enginehttp://en.wikipedia.org/wiki/Aerospike_enginehttp://en.wikipedia.org/wiki/Plug_nozzlehttp://en.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://en.wikipedia.org/wiki/Altitude_compensating_nozzlehttp://en.wikipedia.org/wiki/Rocket_engine_nozzlehttp://www.wikipedia.org/wiki/Convergent-divergent_nozzlehttp://www.wikipedia.org/wiki/Jet_enginehttp://www.wikipedia.org/wiki/Jet_engine5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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closed wake mode, the exhaust gas fills the entire nozzle exit area. The ambient pressure at
which the wake changes from open to closed modes is called the design pressure. If the ambient
pressure reduces any further, additional expansion will occur outside the nozzle much like a
standard bell nozzle and no altitude compensation effect will be gained. In open wake mode, the
exit area is dependant on the ambient pressure and the exhaust gas exits the nozzle as an annulus
as it does not fill the entire nozzle. Because the ambient pressure controls the exit area, the area
ratio should be perfectly compensating to the altitude up to the design pressure[1].
Figure 1.6-Expansion-Deflection nozzle
1.3Nozzle Functions:
The nozzle serves as a back pressure control for the engine and an acceleration device gas
thermal energy to kinetic energy. A secondary function of the nozzle is to provide required thrust
reversing and thrust vectoring[2].
1.3.1 Engine Backpressure Control
The throat area of the nozzle is one of the main means available to control the thrust and
fuel consumption characteristics. The throat area of the nozzle is fixed by selection of specific
values for the engine design parameters and design mass flow rate. Changing the nozzle throat
area from its original value will change the engine design and operating characteristics of the
engine at both on- and off-design conditions.
Large changes in the exhaust nozzle throat area are required for afterburning engines to
compensate for the large changes in total temperature leaving the afterburner. The variable-area
nozzle required for an afterburning engine can also be used for backpressure control at its non-
afterburning settings. One advantage of the variable-area exhaust nozzle is that it improves the
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starting of the engine. Opening the nozzle throat area to its maximum value reduces the
backpressure on the turbine and increases its expansion ratio[2].
1.3.2 Thrust Reversing and Thrust Vectoring
The need for thrust reversing and thrust vectoring is normally determined by the required
aircraft and engine performance. Thrust reversers are used on commercial transports to
supplement the brakes. In-flight thrust reversal has been shown to enhance combat effectiveness
of fighter aircraft.
Two basic types of thrust reversers are used: the cascade-blocker type and the clamshell
type as shown in the figure below:
Figure 1.7- Thrust reversing
In cascade blocker type the primary nozzle exit is blocked off, and cascades are opened in
the upstream portion of the nozzle duct to reverse the flow. In the clamshell type, the exhaust jet
is split and reversed by the clamshell. Since both types usually provide a change in effective
throat area during deployment most reversers are designed such that the effective nozzle throat
area increases.
The exhaust system of the Concorde, the supersonic passenger aircraft, has two nozzles, a
primary nozzle and a secondary nozzle. The secondary nozzle is positioned as a convergent
nozzle for take-off and as a divergent nozzle for supersonic cruise.
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Thrust vectoring nozzles for combat aircraft has increased in the past. Vectoring nozzles
have been used on vertical take-off and landing aircraft, and are proposed for future fighters to
improve maneuvering and augment lift in combat.
Figure 1.8- Thrust Vectoring
Thrust vector control is effective only while the propulsion system is creating thrust. At
other stages of flight, separate mechanisms are required for attitude andflight pathcontrol.
Nominally, theline of actionof the thrust vector of arocket nozzlepasses through thevehicle'scenter of mass,generating zero netmomentabout the mass center. It is possible to
generatepitch and yawmoments by deflecting the main rocket thrust vector so that it does not
pass through the mass center. Because the line of action is generally oriented nearly parallel to
therollaxis, roll control usually requires the use of two or more separately hinged nozzles or a
separate system altogether, such asfins,or vanes in the exhaust plume of the rocket engine,
deflecting the main thrust.
Thrust vectoring for manyliquid rocketsis achieved bygimballingtherocket engine.
This often involves moving the entirecombustion chamberand outer engine bell as on theTitan
II's twin first stage motors, or even the entire engine assembly including the
relatedfuel andoxidizerpumps. Such a system was used on theSaturn Vand theSpace
Shuttle[2].
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CHAPTER-2
LITERATURE SURVEY
2.1Introduction:
The purpose of this applet is to simulate the operation of a converging-diverging nozzle, perhaps
the most important and basic piece of engineering hardware associated with propulsion and the
high speed flow of gases. This device was invented by Carl de Laval toward the end of the l9th
century and is thus often referred to as the 'de Laval' nozzle. This applet is intended to help
students of compressible aerodynamics visualize the flow through this type of nozzle at a range
of conditions.
2.2Technical-Background:
The usual configuration for a converging diverging nozzle is shown in the figure. Gas flows
through the nozzle from a region of high pressure to one of low pressure. The chamber is usually
big enough so that any flow velocities here are negligible. The pressure here is denoted by the
symbol Pc. Gas flows from the chamber into the converging portion of the nozzle, past the throat,
through the diverging portion and then exhausts into the ambient as a jet. The pressure of the
ambient is referred to as the 'back pressure' and given the symbol Pb[5].
Fig 2.1- Convergent-Divergent Nozzle Configuration
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2.2.1-SimpleExample:
To get a basic feel for the behavior of the nozzle imagine performing the simple
experiment shown in figure 2. Here we use a converging diverging nozzle to connect two air
cylinders. Cylinder A contains air at high pressure, and takes the place of the chamber. The CD
nozzle exhausts this air into cylinder B, which takes the place of the tank.
Imagine you are controlling the pressure in cylinder B, and measuring the resulting mass flow
rate through the nozzle. You may expect that the lower you make the pressure in B the more
mass flow you'll get through the nozzle. This is true, but only up to a point. If you lower the back
pressure enough you come to a place where the flow rate suddenly stops increasing all together
and it doesn't matter how much lower you make the back pressure (even if you make it a
vacuum) you can't get any more mass flow out of the nozzle. We say that the nozzle has become
'choked'. You could delay this behavior by making the nozzle throat bigger (e.g. grey line) but
eventually the same thing would happen. The nozzle will become choked even if you eliminated
the throat altogether and just had a converging nozzle.
Fig 2.2-A Simple experiment
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The reason for this behavior has to do with the way the flows behave at Mach 1, that is
when the flow speed reaches the speed of sound. In a steady internal flow the Mach number can
only reach 1 at a minimum in the cross-sectional area. When the nozzle is not choked, the flow
through it is entirely subsonic and, if you lower the back pressure a little, the flow goes faster and
the flow rate increases. As you lower the back pressure further the flow speed at the throat
eventually reaches the speed of sound . Any further lowering of the back pressure cannot
accelerate the flow through the nozzle anymore, because that would entail moving the point
where M=1 away from the throat where the area is a minimum, and so the flow gets stuck. The
flow pattern downstream of the nozzle can still change if you lower the back pressure further, but
the mass flow rate is now fixed because the flow in the throat is now fixed too.
The changes in the flow pattern after the nozzle has become choked are not veryimportant in our thought experiment because they do not change the mass flow rate. They are,
however, very important however if you were using this nozzle to accelerate the flow out of a jet
engine or rocket and create propulsion, or if you just want to understand how high-speed flows
work[5].
2.3 The flow pattern:
Figure 2.3a shows the flow through the nozzle when it is completely subsonic. The flow
accelerates out of the chamber through the converging section, reaching its maximum speed at
the throat. The flow then decelerates through the diverging section and exhausts into the ambient
as a subsonic jet. Lowering the back pressure in this state increases the flow speed everywhere in
the nozzle.
Lower it far enough and we eventually get to the situation shown in figure 2.3b. The flow
pattern is exactly the same as in subsonic flow, except that the flow speed at the throat has just
reached Mach 1. Flow through the nozzle is now choked since further reductions in the backpressure can't move the point of M=1 away from the throat. However, the flow pattern in the
diverging section does change as you lower the back pressure further.
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Fig 2.3-Flow pattern
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As Pb is lowered below that needed to just choke the flow a region of supersonic flow
forms just downstream of the throat. Unlike a subsonic flow, the supersonic flow accelerates as
the area gets bigger. This region of supersonic acceleration is terminated by a normal shock
wave. The shock wave produces a near-instantaneous deceleration of the flow to subsonic speed.
This subsonic flow then decelerates through the remainder of the diverging section and exhausts
as a subsonic jet. In this regime if you lower or raise the back pressure you increase or decrease
the length of supersonic flow in the diverging section before the shock wave[5].
If you lower Pbenough you can extend the supersonic region all the way down the nozzle
until the shock is sitting at the nozzle exit . Because you have a very long region of acceleration
in this case the flow speed just before the shock will be very large in this case. However, after
the shock the flow in the jet will still be subsonic.
Lowering the back pressure further causes the shock to bend out into the jet and a
complex pattern of shocks and reflections is set up in the jet which will now involve a mixture of
subsonic and supersonic flow, or just supersonic flow. Because the shock is no longer
perpendicular to the flow near the nozzle walls, it deflects it inward as it leaves the exit
producing an initially contracting jet. We refer to this as over expanded flow because in this case
the pressure at the nozzle exit is lower than that in the ambient- that is the flow has been
expanded by the nozzle to much.
A further lowering of the back pressure changes and weakens the wave pattern in the jet.
Eventually we will have lowered the back pressure enough so that it is now equal to the pressure
at the nozzle exit. In this case, the waves in the jet disappear altogether , and the jet will be
uniformly supersonic. This situation, since it is often desirable, is referred to as the 'design
condition'.
Finally, if we lower the back pressure even further we will create a new imbalance
between the exit and back pressures ,figure 2.3g. In this situation what we call expansion waves
form at the nozzle exit, initially turning the flow at the jet edges outward in a plume and setting
up a different type of complex wave pattern.
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2.3.1 The pressure distribution in the nozzle:
A plot of the pressure distribution along the nozzle provides a good way of summarizing
its behavior. To understand how the pressure behaves you have to remember only a few basic
rules
When the flow accelerates the pressure drops
The pressure rises instantaneously across a shock
The pressure throughout the jet is always the same as the ambient (i.e. the back pressure)
unless the jet is supersonic and there are shocks or expansion waves in the jet to produce
pressure differences.
The pressure falls across an expansion wave.
Fig 2.4-Pressure distribution along the nozzle labels refer to flow regime 2.3
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The labels on figure 2.4 indicate the back pressure and pressure distribution for each of
the flow regimes illustrated in figure 2.3. Notice how, once the flow is choked, the pressure
distribution in the converging section doesn't change with the back pressure at all.
2.4 Operating Instructions for the applet:
All of the above description is quite a lot to understand and remember without actually
having a converging diverging nozzle to look at. This is the ideal of theapplet - to give you a
model of a nozzle that you can play around with and get experience of.
To start the program, go to theappletpage and press the button labeled 'Start!' a window
like that shown below will appear[5].
http://www.engapplets.vt.edu/fluids/CDnozzle/index.htmlhttp://www.engapplets.vt.edu/fluids/CDnozzle/index.htmlhttp://www.engapplets.vt.edu/fluids/CDnozzle/index.htmlhttp://www.engapplets.vt.edu/fluids/CDnozzle/index.html5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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On the left hand side of the window there are three panels used for plotting the flow
conditions in the nozzle. The top panel, shaded gray, is used to show the shape of the nozzle and
a color contour map of the temperature distribution within it. Initially this region will be blank,
note that the temperature distribution behaves qualitatively like the pressure distribution. The
middle panel is used to display the pressure as a function of distance down the nozzle, and the
lower panel displays the Mach number as a function of distance. When results are displayed, the
horizontal axes of these three panels all line up so the association between features on the
different plots can easily be observed. On the top right of the applet window a graphic is
displayed showing an actual rocket nozzle in a test stand. Below this is a yellow information
panel, and then text areas where you can enter k the ratio of specific heats for the gas in the
nozzle, and Pb/Pcthe pressure ratio that is driving the flow through the nozzle. Below are a series
of six buttons used to control the actions of the applet[5].
To begin press the 'Design Nozzle' button, which should bring up a window like that
shown in the figure. On the right of the window there is a text area that allows you to enter the
ratio of the exit area Aeto the throat area At. This must be greater than 1. The larger the ratio, the
higher the Mach number of the flow that your nozzle will produce it may be difficult to see all
the results clearly on the plots. Type in '4' and press the 'Set' button. The graph on the left shows
the shape of the nozzle, chamber on the left, exit on the right. The program assumes you are
dealing with an axisymmetric nozzle so, for example, your nozzle will appear as having an exit
with a diameter of twice that at the throat. You can change the shape of the diverging section by
clicking the area shaded with '+' signs close to the line representing the diverging section. Note
that you can't move the throat, or create a diverging section with a maximum in area - the
program will warn you if either of these occurs. When you are satisfied with the shape, press the
'Done' button.
You can compute and display the flow through the nozzle in one of two ways. The mostdirect way is to enter a value for the back pressure in the text area labeled 'Pb/Pc'. Enter '0.5' and
press the 'Compute' button. Almost instantaneously the results should be plotted as shown below.
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The flow you have computed corresponds to case c in figure 2.3 above, i.e. flow with a
shock in the nozzle . On the top left of the frame a contour map of the flow temperature is
plotted, normalized on the temperature in the chamber. Notice how the temperature falls as the
flow accelerates up to and past the throat, and then suddenly rises in the shock wave. The center
left plot shows the pressure distribution and beneath that is plotted the Mach number distribution.
Notice how the Mach number is 1 at the throat in this case, and how the Mach number drops
from super to subsonic across the shock wave. If you want you can access the numerical results
of this calculation by pressing the 'Export data' button, copying out the numbers and pasting
them into another application, like Excel, or Notepad.
The second way to compute the flow is the most useful if you want to see the whole
range of phenomena present in the flow at different back pressures. To do this press the 'AutoRun' button. The program begins slowly lowers and raises the back pressure computing in small
increments the entire flow and displaying the results. The net effect is an animation of what
occurs in the nozzle as you raise and lower the back pressure. You can stop the animation at any
time by pressing 'Stop'. To leave the applet you should press the 'Quit' button.
2.4.1 How the applet works:
The applet works by computing the flow using the one dimensional equations for the
isentropic flow of a perfect gas, and the Rankine Hugoniot relations for normal shock waves in
perfect gases. You can learn about these relations by reading, form example, Modern
Compressible Flow, 2nd Edition, 1990, by John D. Anderson Jr. You can use the Compressible
Aerodynamics Calculator to help you use these relations in your own calculations.
This applet is help us to check the results and it is also capable of designing the nozzle
when we give area ratio to it[5].
http://www.aoe.vt.edu/aoe3114/calc.htmlhttp://www.aoe.vt.edu/aoe3114/calc.htmlhttp://www.aoe.vt.edu/aoe3114/calc.htmlhttp://www.aoe.vt.edu/aoe3114/calc.html5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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CHAPTER-3
OPERATION OF CONVERGENT-DIVERGENT NOZZLE
AND ISENTROPIC RELATIONS
3.1 Operation of Convergent-Divergent nozzles :
A De Laval nozzle convergent-divergent nozzle, CD nozzle is a tube that is pinched in
the middle, making a carefully balanced, asymmetric hourglass-shape. It is used to accelerate a
hot, pressurized gas passing through it to a supersonic speed, and upon expansion, to shape the
exhaust flow so that the heat energy propelling the flow is maximally converted into directed
kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines, and is
used as a rocket engine nozzle. These are also used in supersonic jet engines.
Its operation relies on the different properties of gases flowing at subsonic and
supersonic speeds. The speed of a subsonic flow of gas will increase if the pipe carrying it
narrows because the mass flow rate is constant. The gas flow through a de Laval nozzle is
isentropic At subsonic flow the gas is compressible, a small pressure wave, will propagate
through it. At the throat, where the cross sectional areais a minimum, the gas velocity locally
becomes sonic , a condition called choked flow. As the nozzle cross sectional area increases
the gas begins to expand and the gas flow increases to supersonic velocities where a sound wave
will not propagate backwards through the gas.
A de Laval nozzle will only choke at the throat if the pressure and mass flow through the
nozzle is sufficient to reach sonic speeds, otherwise no supersonic flow is achieved and it will act
as a venturi tube, this requires the entry pressure to the nozzle to be significantly above ambient
at all times. In addition, the pressure of the gas at the exit of the expansion portion of the exhaust
of a nozzle must not be too low. Because pressure cannot travel upstream through the supersonic
flow, the exit pressure can be significantly below ambient pressure it exhausts into, but if it is too
far below ambient, then the flow will cease to be supersonic, or the flow will separate within the
expansion portion of the nozzle, forming an unstable jet that may flop around within the nozzle,
possibly damaging it.
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Figure 3.1- Operation of De-Laval nozzle
3.2 Analysis of gas flow in De Laval nozzles:
It involves a number of concepts and assumptions.
1.The gas is assumed to be an ideal gas.
2.The gas flow is isentropic. As a result the flow is reversible and adiabatic.
3.The gas flow is constant during the period of the propellant burn.
4.The gas flow is along a straight line from gas inlet to exhaust gas exit[8].
3.3 Isentropic Process: Isentropic process is one in which, that the process takes placefrom initiation to completion without an increase or decrease in the entropy of the system, thatmeans the entropy of the system remains constant. It can be proven that anyreversibleadiabatic
process is an isentropic process. A simple more common definition of isentropic would be one
that produces "No change in entropy".
An isentropic flow is aflow that is both adiabatic and reversible. That is, no heat is added
to the flow, and no energy transformations occur due tofriction ordissipative effects. For an
isentropic flow of a perfect gas, several relations can be derived to define the pressure, density
and temperature along a streamline.
3.3.1Entropy: Entropy is a measure of the number of specific ways in whichathermodynamic system may be arranged, often taken to be a measure ofdisorder,or a measure
of progressing towardsthermodynamic equilibrium. The entropy of an isolated system never
http://en.wikipedia.org/wiki/Reversible_process_(thermodynamics)http://en.wikipedia.org/wiki/Adiabatic_processhttp://en.wikipedia.org/wiki/Adiabatic_processhttp://en.wikipedia.org/wiki/Fluid_dynamicshttp://en.wikipedia.org/wiki/Frictionhttp://en.wikipedia.org/wiki/Dissipationhttp://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Entropy_(order_and_disorder)http://en.wikipedia.org/wiki/Thermodynamic_equilibriumhttp://en.wikipedia.org/wiki/Thermodynamic_equilibriumhttp://en.wikipedia.org/wiki/Entropy_(order_and_disorder)http://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Dissipationhttp://en.wikipedia.org/wiki/Frictionhttp://en.wikipedia.org/wiki/Fluid_dynamicshttp://en.wikipedia.org/wiki/Adiabatic_processhttp://en.wikipedia.org/wiki/Adiabatic_processhttp://en.wikipedia.org/wiki/Reversible_process_(thermodynamics)5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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decreases, because isolated systems spontaneously evolve towards thermodynamic equilibrium,
which is the state ofmaximum entropy.
3.3.2Enthalpy: Enthalpy is a measure of the totalenergyof athermodynamic system. Itincludes theinternal energy,which is the energy required to create a system, and the amount of
energy required to make room for it by displacing itsenvironmentand establishing its volume
and pressure.
Enthalpy is athermodynamic potential. It is astate functionand anextensive quantity.
The unit of measurement in theInternational System of Units(SI) for enthalpy is thejoule,but
other historical, conventional units are still in use, such as the small and the largecalorie.
3.3.3Internal Energy: Internal energy is theenergy contained by athermodynamic system.
It is amacroscopicproperty. It is the energy needed to create the system but excludes the energy
to displace the system's surroundings, the kinetic energy of motion of the system as a whole, and
the potential energy of the system as a whole due to external force fields.
Though it is a macroscopic quantity, internal energy can be explained in microscopic
terms by two components. One is the microscopic kinetic energy due to the microscopic motion
of the system's particles . The other is the potential energy associated with the microscopicforces, including thechemical bonds, between the particles, and with the staticrest mass
energy of the constituents of matter[9].
3.4 Isentropic Relations:Consider a gas is forced through a tube, the gas molecules are
deflected by the walls of the tube. If the speed of the gas is much less than the speed of sound of
the gas, thedensity of the gas remains constant and the velocity of the flow increases. However,
as the speed of the flow approaches thespeed of sound we must considercompressibility
effects on the gas. The density of the gas varies from one location to the next. Considering flow
through a tube, as shown in the figure, if the flow is very gradually compressed and then
gradually expanded , the flow conditions return to their original values. We say that such a
process is reversible. From a consideration of thesecond law of thermodynamics, a reversible
http://en.wikipedia.org/wiki/Maximum_entropy_thermodynamicshttp://www.wikipedia.org/wiki/Energyhttp://www.wikipedia.org/wiki/Energyhttp://www.wikipedia.org/wiki/Energyhttp://www.wikipedia.org/wiki/Thermodynamic_systemhttp://www.wikipedia.org/wiki/Thermodynamic_systemhttp://www.wikipedia.org/wiki/Thermodynamic_systemhttp://www.wikipedia.org/wiki/Internal_energyhttp://www.wikipedia.org/wiki/Internal_energyhttp://www.wikipedia.org/wiki/Internal_energyhttp://www.wikipedia.org/wiki/Environment_(systems)http://www.wikipedia.org/wiki/Environment_(systems)http://www.wikipedia.org/wiki/Environment_(systems)http://www.wikipedia.org/wiki/Thermodynamic_potentialhttp://www.wikipedia.org/wiki/Thermodynamic_potentialhttp://www.wikipedia.org/wiki/Thermodynamic_potentialhttp://www.wikipedia.org/wiki/State_functionhttp://www.wikipedia.org/wiki/State_functionhttp://www.wikipedia.org/wiki/State_functionhttp://www.wikipedia.org/wiki/Extensivehttp://www.wikipedia.org/wiki/Extensivehttp://www.wikipedia.org/wiki/Extensivehttp://www.wikipedia.org/wiki/International_System_of_Unitshttp://www.wikipedia.org/wiki/International_System_of_Unitshttp://www.wikipedia.org/wiki/International_System_of_Unitshttp://www.wikipedia.org/wiki/Joulehttp://www.wikipedia.org/wiki/Caloriehttp://www.wikipedia.org/wiki/Caloriehttp://www.wikipedia.org/wiki/Caloriehttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Macroscopichttp://en.wikipedia.org/wiki/Chemical_bondshttp://en.wikipedia.org/wiki/Mass-energy_equivalencehttp://en.wikipedia.org/wiki/Mass-energy_equivalencehttps://www.grc.nasa.gov/www/k-12/airplane/fluden.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/sound.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/airsim.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/airsim.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/thermo2.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/thermo2.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/airsim.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/airsim.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/sound.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/fluden.htmlhttp://en.wikipedia.org/wiki/Mass-energy_equivalencehttp://en.wikipedia.org/wiki/Mass-energy_equivalencehttp://en.wikipedia.org/wiki/Chemical_bondshttp://en.wikipedia.org/wiki/Macroscopichttp://en.wikipedia.org/wiki/Thermodynamic_systemhttp://en.wikipedia.org/wiki/Energyhttp://www.wikipedia.org/wiki/Caloriehttp://www.wikipedia.org/wiki/Joulehttp://www.wikipedia.org/wiki/International_System_of_Unitshttp://www.wikipedia.org/wiki/Extensivehttp://www.wikipedia.org/wiki/State_functionhttp://www.wikipedia.org/wiki/Thermodynamic_potentialhttp://www.wikipedia.org/wiki/Environment_(systems)http://www.wikipedia.org/wiki/Internal_energyhttp://www.wikipedia.org/wiki/Thermodynamic_systemhttp://www.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Maximum_entropy_thermodynamics5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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flow maintains a constant value ofentropy.Engineers call this type of flow an isentropic flow; a
combination of the Greek word "iso" and entropy[4].
Figure 3.2- Flow through nozzle
Isentropic flows occur when the change in flow variables is small and gradual, such as
the ideal flow through thenozzle shown above. The generation ofsound waves is an isentropic
process. Asupersonic flow that is turned while the flow area increases is also isentropic. We call
this an isentropicexpansionbecause of the area increase. If a supersonic flow is turned abruptly
and the flow area decreases,shock waves are generated and the flow is irreversible. The
isentropic relations are no longer valid and the flow is governed by theoblique ornormal shock
relations.
The isentropic relations can be written as fallows.
(
)
The above equation gives the ratio of total temperature to the static temperature at a point
in a flow as a function of Mach number M at that point.
(
)
https://www.grc.nasa.gov/www/k-12/airplane/entropy.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/nozzle.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/sndwave.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/mach.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/expans.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/shock.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/oblique.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/normal.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/normal.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/oblique.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/shock.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/expans.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/mach.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/sndwave.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/nozzle.htmlhttps://www.grc.nasa.gov/www/k-12/airplane/entropy.html5/27/2018 Analysis of Flow Through Converget-divergent Nozzle
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The above equation gives the ratio of total pressure to the static pressure at a point in a
flow as a function of Mach number M at that point for =1.4 air at standard condition.
(
)
The above equation gives the ratio of total density to the static density at a point in a flow
as a function of Mach number M at that point for =1.4 air at standard condition[1].
3.5 AREA-MACH NUMBER RELATION:
The equation for the area-mach number relation can be written as fallows.
() [
(
)]
The above equation is known as Area-Mach number relation. The above equation tell us
that
M=f(
That is the mach number at any location in the duct is a function of ratio of the local duct
area to the sonic throat area. If we have a tube with changing area, like thenozzle shown on the
above, the maximum mass flow rate through the system occurs when the flow is choked at the
smallest area. This location is called the throat of the nozzle. The conservation of mass specifies
that the mass flow rate through a nozzle is a constant. If no heat is added, and there are no
pressure losses in the nozzle, the total pressure and temperature are also constant. By substituting
the sonic conditions into the mass flow rate equation, and doing some algebra, we can relate the
Mach number M at any location in the nozzle to the ratio between the area A at that location and
the area of the throat A*.The above area-mach number helps to find the flow parameters like
pressure ratio, density ratio, temperature ratio .
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Figure 3.3- Area-Mach number relation
The above graph says that for an area ratio there are two mach numbers are existing.
Based on this relation the program coding is done. In the coding we gave two conditions. They
are subsonic and supersonic considering subsonic condition an supersonic condition. In the case
of subsonic the mach number is lies between 0 to 1 whereas in supersonic region the mach value
is lies between 1 and .The coding ask us to input the area ratio and it will print the respective
mach numbers as output. With the help of mach number the coding can able to print the flow
parameters like temperature ratio, pressure ratio, density ratio respectively[4].
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CHAPTER-4
PROGRAM CODING
4.1 Introduction to C-language:
C is animperative language. It was designed to be compiled using a relatively
straightforwardcompiler, to provide low-level access to memory, to provide language constructs
that map efficiently to machine instructions, and to require minimalrun-time support. C was
therefore useful for many applications that had formerly been coded in assembly language.
Despite its low-level capabilities, the language was designed to encouragecross-platform
programming. A standards-compliant andportably written C program can be compiled for a very
wide variety of computer platforms and operating systems with few changes to its source code.
The language has become available on a very wide range of platforms, from
embeddedmicrocontrollers tosupercomputers.
4.2 Characteristics of C-language:
There is a small, fixed number of keywords, including a full set of flow of control
primitives: if/else, for, while, do/while and switch.
There are a large number of arithmetical and logical operators, such as +,+=,++ etc.
More than one assignment may be performed in a single statement.
Function return values can be ignored when not needed.
Typing is static, but weakly enforced: all data has a type, but implicit conversions can be
performed; for instance, characters can be used as integers.
Declaration syntax mimics usage context. C has no "define" keyword; instead, a
statement beginning with the name of a type is taken as a declaration. There is no
"function" keyword; instead, a function is indicated by the parentheses of an argument
list.
Low-level access to computer memory is possible by converting machine addresses to
typed pointers.
Procedures are a special case of function, with an un typed return type void.
Functions may not be defined within the lexical scope of other functions[3].
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4.3 Program Coding: In coding we developed a c-program to get the pressure ratio,
Temperature ratio, Density ratio, Area ratio at the required co-ordinates of nozzles. The required
relations for the coding purpose are Area-Mach relation and Pressure, Temperature, Density
Ratios. The C- concepts used for the C-program are mainly Arrays, Functions, Files. The
purpose of coding is we can get accurate results. Once the coding is done we are able to get the
pressure, temperature, density and area ratios of any nozzle[3].
'C' PROGRAMMING CODE FOR CONVERGENT- DIVERGENT NOZZLE
4.4 ISENTROPIC TABLE GENERATION WITH THE HELP OF CODING
#include
#include
int main()
{
float gamma=1.4,M,T,P,d,A,a,b;
printf(" Mach Temp_ratio pres_ratio den_ratio area_mach relatn ");
for(M=0.1;M
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}
return 0;
}
RESULT:
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4.5 If the area ratio is given then the parameters like pressure ratio,
temperature ratio, density ratio, can be evolved by the following code.
#include
#includedouble calc_area(double mac,double gamma)
{
double area=0;
double a=(1+((gamma-1)/2)*mac*mac);
double b=((2/(gamma+1))*a);
double c=pow(b,((gamma+1)/(gamma-1)));
area=sqrt((1/(mac*mac))*c);
return area;
}
double calc_temp(double mac,double gamma)
{
//t=1+((gamma-1)/2)*mac*mac;
double temp=1+((gamma-1)/2)*mac*mac;
return temp;
}
double calc_pres(double mac,double gamma)
{
//pres=(temp)^(gamma/gamma-1)
double temp=1+((gamma-1)/2)*mac*mac;
double pres=pow(temp,(gamma/(gamma-1)));
return pres;
}
double calc_dens(double mac,double gamma)
{
double temp=1+((gamma-1)/2)*mac*mac;
double dens=pow(temp,(1/(gamma-1)));
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return dens;
}
double calc_subsonic_mac(double tarea,double gamma)
{
//0 to 1 region
double mac1=0;
double mac2=1;
double mid,carea,error;
while(1)
{
mid=(mac1+mac2)/2;
carea=calc_area(mid,gamma);
error=tarea-carea;
if((error=0))
break;
if((error>=-0.001)&&(error0)
mac2=mid;
else
mac1=mid;
}
return mid;
}
double calc_supersonic_mac(double tarea,double gamma)
{
//1- infinity region
double mac1=1;
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double mac2=2;
double mid,carea,error;
while(1)
{
carea=calc_area(mac2,gamma);
error=tarea-carea;
//add other condition also
if(error=-0.001)&&(error0)
mac1=mid;
else
mac2=mid;
}
return mid;
}
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int main()
{
double m1,m2,tarea,gamma=1.4,t1,t2,p1,p2,d1,d2;
int ch;
while(1)
{
printf("\nEnter the area value:");
scanf("%lf",&tarea);
//mac m1,m2
m1=calc_subsonic_mac(tarea,gamma);
printf("\n\nMachnumber in subsonic case: %lf",m1);
m2=calc_supersonic_mac(tarea,gamma);
printf("\n\nMach number in supersonic case: %lf",m2);
//temp t1,t2
t1=calc_temp(m1,gamma);
printf("\n\ntemparature_ratio for subsonic case: %lf ",t1);t2=calc_temp(m2,gamma);
printf("\n\ntemparature_ratio for supersonic case: %lf ",t2);
//pressure p1,p2
p1=calc_pres(m1,gamma);
printf("\n\npressure_ratio for subsonic case: %lf ",p1);
p2=calc_pres(m2,gamma);
printf("\n\npressure_ratio for supersonic case: %lf ",p2);
//density d1,d2
d1=calc_dens(m1,gamma);
printf("\n\ndensity_ratio for subsonic case: %lf ",d1);
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d2=calc_dens(m2,gamma);
printf("\n\ndensity_ratio for supersonic case: %lf ",d2);
printf("\n\nDo you want to exit(0 or 1):");
scanf("%d",&ch);
if(ch==1)
break;
}
}
RESULT:
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4.6 If the nozzle contours are given then the parameters like pressure ratio,
temperature ratio, density ratio, can be evolved by the following code.#include
#include
double calc_area(double mac,double gamma)
{
double area=0;
double a=(1+((gamma-1)/2)*mac*mac);
double b=((2/(gamma+1))*a);
double c=pow(b,((gamma+1)/(gamma-1)));
area=sqrt((1/(mac*mac))*c);
return area;
}
double calc_temp(double mac,double gamma)
{
//t=1+((gamma-1)/2)*mac*mac;
double temp=1+((gamma-1)/2)*mac*mac;
return temp;
}
double calc_pres(double mac,double gamma)
{
//pres=(temp)^(gamma/gamma-1)
double temp=1+((gamma-1)/2)*mac*mac;
double pres=pow(temp,(gamma/(gamma-1)));
return pres;
}
double calc_dens(double mac,double gamma)
{
double temp=1+((gamma-1)/2)*mac*mac;
double dens=pow(temp,(1/(gamma-1)));
return dens;
}
double calc_subsonic_mac(double tarea,double gamma)
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{
//0 to 1 region
double mac1=0;
double mac2=1;
double mid,carea,error;
while(1)
{
mid=(mac1+mac2)/2;
carea=calc_area(mid,gamma);
error=tarea-carea;
if((error=0))
break;
if((error>=-0.001)&&(error0)
mac2=mid;
else
mac1=mid;
}
return mid;
}
double calc_supersonic_mac(double tarea,double gamma)
{
//1- infinity region
double mac1=1;
double mac2=2;
double mid,carea,error;
while(1)
{
carea=calc_area(mac2,gamma);
error=tarea-carea;
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//add other condition also
if(error=-0.001)&&(error0)
mac1=mid;
else
mac2=mid;
}
return mid;
}
int main()
{
double xth,yth,throat_area,m,tarea,carea,error,gamma=1.4,t,p,d,x[1000],y[1000];
int i,ch,ncords;
FILE *fptr=fopen("nozzle.txt","r");
FILE *fptr2=fopen("results.txt","w");
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fprintf(fptr2,"x\t\ty\t\tcase\tmac\t\ttemp_ratio\tpres_ratio\tdens_ratio\tarea_ratio\n");
fscanf(fptr,"%d",&ncords);
for(i=0;i
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fprintf(fptr2,"%lf %lf SUB %lf %lf %lf %lf
%lf\n",x[i],y[i],m,t,p,d,tarea);
}
else if(x[i]>xth)
{
//supersonic
tarea=3.14*y[i]*y[i];
tarea=tarea/throat_area;
m=calc_supersonic_mac(tarea,gamma);
t=calc_temp(m,gamma);
p=calc_pres(m,gamma);
d=calc_dens(m,gamma);
printf("\n\nSuperSonicMac: %lf \nTemp_ratio : %lf \n Pres_ratio : %lf\n Dens_ratio: %lf\n
area_ratio: %lf ",m,t,p,d,tarea);
fprintf(fptr2,"%lf %lf SUP %lf %lf %lf %lf
%lf\n",x[i],y[i],m,t,p,d,tarea);
}
else
{
tarea=3.14*y[i]*y[i];
tarea=tarea/throat_area;
m=calc_supersonic_mac(tarea,gamma);
t=calc_temp(m,gamma);
p=calc_pres(m,gamma);
d=calc_dens(m,gamma);
printf("\n\nThroat Mac: %lf \nTemp_ratio : %lf \n Pres_ratio : %lf\n Dens_ratio: %lf\n
area_ratio: %lf ",m,t,p,d,tarea);
fprintf(fptr2,"%lf %lf THR %lf %lf %lf %lf
%lf\n",x[i],y[i],m,t,p,d,tarea);
}
}
fclose(fptr);
fclose(fptr2);
}
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RESULTS:
Nozzle Contours
X(m) Y(m)
0 0.4
0.212 0.2
1.70494 0.6
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CHAPTER-5
DESIGN OF CONVERGRNT-DIVERGENT NOZZLE IN CFD
5.1 Overview of CFD:
Over the last twenty to thirty years considerable progress has been achieved, and the
field of Computational fluid Dynamics is reaching a mature stage, where most of the basic
methodology is, and will remain, well established.
Computational fluid dynamics is one of the branches of fluid mechanics that uses
numerical methods and algorithms to solve and analyze problem that involve fluid flows.
Computational fluid dynamics technology will enable you to study the dynamics of things that
flow. Using CFD, you can built a computational model that represents a system or device that
you want to study. Then you apply the fluid flow physics and chemistry to this virtual prototype,
and the software will output a prediction of the fluid dynamics and related physical phenomena.
Therefore, CFD is a sophisticated computationally-based design and analysis technique. CFD
software gives you the power to simulate flow of gases and liquids, heat and mass transfer,
moving bodies, multiphase physics, chemical reaction, fluid-structure interaction and acoustics
through computer modeling. Using CFD software, you can build a virtual prototype of the
system or device that you wish to analyze and then apply real-world physics and chemistry to the
model, and the software will provide you with images and data, which predict the performance
of that design. CFD is predicting what will happen, quantitatively, when fluids flow, often with
the complications of simultaneous flow of heat, mass transfer example: perspiration, dissolution,
phase changes chemical reaction, mechanical movement stresses and displacement of immersed
or surrounding solids[6].
Until recently, CFD has only been effectively utilized with in the aerospace and automotive
industries because of high software costs and powerful computational requirements. With the
development of computers that have high speed processing capability, it is now possible to run
the majority of CFD models. CFD can be used in almost all industrial and non- industrial
applications-starting with aerodynamics and gas turbine design automotive engineering, turbo-
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machinery, chemical processes, marine engineering, environmental and biomedical engineering,
metrology, hydrology and oceanography etc.
5.1.1 Advantages of Computational Fluid Dynamics:
With the rapid development of digital computers, CFD is poised to remain at the forefront of
cutting edge research in the sciences of fluid dynamics and heat transfer. Also the emergence of
CFD as a practical tool in modern engineering practices is steadily attracting much interest and
appeal. There are many advantages in considering CFD.
5.2 Use of CFD: Knowing how fluids will flow, and what will be their quantitative effects on
the solids with which they are in contact, assists:-
Building-services engineers and architects to provide comfortable and safe human
environments;
Power-plant designers to attain maximum efficiency, and reduce release of pollutants;
Chemical engineers to maximize the yields from their reactors and processing equipment;
Land-, air- and marine-vehicle designers to achieve maximum performance, at least cost;
Risk-and-hazard analysts, and safety engineers, to predict how much damage tostructures, equipment, human beings, animals and vegetation will be caused by fires,
explosions and blast waves.
5.2.1Applications of CFD: CFD is used by engineers and scientists in a wide range of field.
Typical applications include:
Process industry: Mixing vessels, chemical reactors
Building services: Ventilation of buildings, such as atriums
Health and safety: Investigation the effects of fire and smoke
Motor industry: Combustion modeling, car aerodynamics
Electronics: Heat transfer within and around circuit boards
Environmental:Dispersion of pollutants in air or water[6]
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5.3 Problem Definition:
Traditionally, the design of the convergent-divergent nozzle is done in order to predict
the flow parameters. With the recent developments in computational techniques and
computational fluid dynamics, CFD tools are used to optimize the nozzle design.
The geometry of nozzle is as fallows
Figure 5.1- Nozzle geometry
5.4 ICEM CFD AND ANSYS CFX:
The basic steps involved in solving any CFD Analysis problem:
Pre-processing:
1. Creation of geometry
2. Mesh generation
3.
Selection of Physics and Fluid properties4. Specification of Boundary conditions.
Solution:
5. Initialization of Solver Control.
6. Monitoring Convergence.
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Result Report and visualization-Post process:
Obtaining:
X-Y Plots
Contour plots
5.4.1 Preprocess:
5.4.1.1 Creation of geometry:
The first step in any CFD analysis is the definition and creation of geometry. Several
sections are created and then lofted to obtain the required shape of the nozzle.
Figure 5.2- Modeling of nozzle
5.4.1.2 Mesh generation: The second step-mesh generation-constitutes one of the most
important steps during the pre-process stage after the definition of the domain geometry. CFD
requires the subdivision of domain into a number of smaller, non-overlapping sub domains in
order to solve the flow physics within the domain geometry that has been created. This results in
the generation of mesh of cells overlaying the whole domain geometry. The accuracy of a CFD
solution is governed by the number of cells in the mesh within the computational domain. In
general, the provision of a large number of cells leads to the attainment of an accurate solution.
The nozzle geometry is imported into the ICEM CFD software for meshing. An unstructured
mesh is created for the nozzle.
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Figure 5.3 - Mesh model of convergent-divergent nozzle
5.4.1.3 Selection of Physics and Fluid properties:
The generated mesh is imported to the ANSYS CFX. The flow material is selected as air.
Domain - Default Domain
Type Fluid
Location LIVE
Materials
Air Ideal Gas
Fluid Definition Material Library
Morphology Continuous Fluid
Settings
Buoyancy Model Non BuoyantDomain Motion Stationary
Reference Pressure 1.0000e+00 [bar]
Heat Transfer Model Total Energy
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5.4.1.4 Boundary Condition:
A CFD user needs to define appropriate boundary conditions that mimic the real physical
representation of the fluid flow into a solvable CFD problem.
Figure 5.4- Convergent-divergent nozzle boundary condition
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Domain Boundaries
Default Domain
Boundary - in1
Type INLET
Location IN1
Settings
Flow Direction Normal to Boundary Condition
Flow Regime Subsonic
Heat Transfer Total Temperature
Total Temperature 1.1692e+03 [K]
Mass And Momentum Total Pressure
Relative Pressure 1.1687e+02 [bar]
Turbulence Medium Intensity and Eddy Viscosity Ratio
Boundary - out1
Type OUTLET
Location OUT1
Settings
Flow Regime Subsonic
Mass And Momentum Static Pressure
Relative Pressure 0.0000e+00 [bar]
Boundary - Default Domain Default
Type WALL
Location ARC1, CONV1, DIV1
Settings
Heat Transfer AdiabaticMass And Momentum No Slip Wall
Wall Roughness Smooth Wall
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5.4.1.5 Solution:
After specifying the Boundary condition, the CFX solver control is to be initialized to
obtain the solution.
5.4.1.6 Results Report and Visualization-Post processer:
The next step after the solution is to post process the results obtained from the CFX solver
manager. The results are post processed and are presented in the report as: contour plots and x-y
plots.
Figure 5.5 -Mach number contour
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Figure 5.6 - Plot between the Nozzle length versus Mach number
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Figure 5.7 - Pressure contour
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Figure 5.8 - Plot between Nozzle length versus pressure
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Figure 5.9 - Density contour
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Figure 5.10- Plot between Nozzle length versus Density
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Figure 5.11- Temperature contour
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-
Figure 5.12- Plot between Nozzle length versus Pressure ratio
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Figure 5.13 -Plot between Nozzle ratio versus Temperature ratio
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The below mentioned values are taken from the CFD design analysis with the help probe tool in
order to check the results
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CHAPTER-6
FUTURE SCOPE
The present thesis is on the modeling and analysis of convergent-divergent nozzle in
CFD and programming for that nozzle with the help of c-programming. To validate the obtained
results from program code used CFD was used.
The work is divided into two parts
1.Program coding for the nozzle contours
2.Modeling and Analysis of convergent-divergent nozzle in CFD.
5.1 Program coding for the nozzle contours:
In this part we were focused on C-language. Program is developed for the nozzle
contours. The input for the coding is nozzle contours and the output will be the flow parameters
like pressure ratio, temperature ratio, density ratio. This program is helpful for any kind of nozzle
contours. Advantage of this program is this can be stored for a long time and output will get
easily and quickly.
5.2 Modeling and Analysis of convergent-divergent nozzle:
The modeling of convergent-divergent nozzle is done with the help of given geometry
and analysis of that nozzle is completed by applying boundary conditions and CFD tools. This
Analysis is done for the validation of program code.
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CHAPTER-7
CONCLUSION
Conclusion:
The coding is extensively being used now a days due to its simplicity and accurate
results. C is one of the most widely used programming languages of all time. Basically, a
program has been developed to find out all the parameters of the nozzle and the results obtained
here are compared with CFD simulation. Both the results are compared as fallows.
Parameters Coding CFD Error
Mach number 3.806152 3.8 0.16
Pressure ratio 116.8682 110.8 5.1923
Temperature ratio 3.897359 3.8346 1.61029
The error between the flow parameters like Mach number, Pressure ratio, Temperature
ratio are calculated.
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References:
[1] Nozzle Types, Nozzle Functions, http://en.wikipedia.org/wiki/Rocket_engine_nozzle
[2] George P.Suttton, Oscar Biblarz,"Rocket Propulsion Elements", ISBN-978-81-265-2577-
5,7th
Edition,Wiley India Private Limited, 2010.
[3] P.J. Deitel, H.M Deitel,"C - How To Program",ISBN-978-81-203-3495-3,5th
Edition PHI
Learning Private Limited, 2009.
[4] John D.Anderson, Jr,"Modern Compressible Flow With Historical Perspective",
ISBN:9780071241366, 3rd
Edition, Mc Graw Hill Education, 2004.
[5] Convergent-Divergent nozzle,http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html
[6] CFD,http://en.wikipedia.org/wiki/Computational_fluid_dynamics
[7] Flow through nozzles,https://www.grc.nasa.gov/www/k-12/airplane/nozzled.html
[8] Operation of de-laval nozzle,http://en.wikipedia.org/wiki/De_Laval_nozzle
[9] Isentropic process,http://en.wikipedia.org/wiki/Isentropic