Analysis of Flow Through Converget-divergent Nozzle

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
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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
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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
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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].
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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].
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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)


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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_thermodynamics


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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.html


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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 pres=pow(temp,(gamma/(gamma-1)));

return pres;

}

double calc_dens(double mac,double gamma)

{


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;


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;


return temp;

}

double calc_pres(double mac,double gamma)

{

//pres=(temp)^(gamma/gamma-1)


double pres=pow(temp,(gamma/(gamma-1)));

return pres;

}

double calc_dens(double mac,double gamma)

{


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;


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;


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


}

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


}

}

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

Analysis of Flow Through Converget-divergent Nozzle

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