CFD Rocket Nozzle

Report

On

“CFD Analysis of Rocket Nozzle”

Submitted in partial fulfillment of the requirements for the degree of

Bachelor of Technology

(Aerospace Engineering)

From

University of Petroleum and Energy Studies

Dehradun

By

Vipul Sharma (R180207060)

Ravi Shankar (R180207043)

Gaurav Sharma (R180207020)

Under the Guidance of

Dr. Ugur Guven

Department of Aerospace Engineering

University of Petroleum & Energy Studies

Dehradun

April 11, 2011

Calculation of velocity & Temperature profile of Rocket Nozzle using CFD in zero atmospheric

condition

A thesis submitted in partial fulfillment of the requirements for the Degree of

Bachelor of Technology

(Aerospace Engineering)

By

Gaurav Sharma

(R180207020)

Ravi Shankar

(R180207043)

Vipul Sharma

(R180207060)

Under the Guidance of

Dr. Ugur Guven

Professor of Aerospace Engineering (PhD)

Nuclear Science and Technology Engineer (M.Sc)

Approved

…………………………

Dean

College of Engineering

University of Petroleum & Energy Studies

Dehradun

April, 2011

CERTIFICATE

This is to certify that the work contained in this thesis titled “Calculation of velocity &

Temperature profile of Rocket Nozzle using CFD in zero atmospheric condition” has been

carried out by Gaurav Sharma, Ravi Shankar & Vipul Sharma under my supervision and has not

been submitted elsewhere for a degree.

Dr. UGUR GUVEN

Professor of Aerospace Engineering

(April 11, 2011)

TABLE OF CONTENTS

Page

TABLE OF CONTENTS i

LIST OF FIGURES iii

LIST OF TABLES vii

SUMMARY viii

1. ROCKET ENGINE…………………………………………………………………….….…1

1.1Terminologies used in rocket engines…………………………………………….…..3

1.2Principle of operation……………….……………………………….………………..3

1.3 Propellents used in rocket engine……………………………………………….…...4

1.3.1 Solid Propellant Rocket……………………………………………….…...4

1.3.2 Liquid Propellant Rocket…………………………………………………..5

1.4 Thermal Rocket Engine………………………………………………………………6

2. ROCKET ENGINE NOZZLE………………………………………………………………7

2.1 Types of nozzles……………………………………………………………………...8

2.1.1Jets…………………………………………………………………….……..8

2.1.2 High velocity nozzles……………………………………………………….9

2.1.3 Propelling nozzles……………………………………………………...….10

2.1.4 Magnetic nozzles…………………………………………………………..11

2.1.5 Spray nozzles……………………………………………………………...11

2.1.6 Vacuum nozzles…………………………………………………………...11

2.1.7 Shaping nozzles……………………………………………………………12

3. BASIC ROCKET EQUATIONS……………………………………………………………12

3.1 Newton‟s Third law of motion……………………………………………………….12

3.2 De laval nozzle equations……………...…………………………………………….12

3.3 Tsiolkovsky Equation…………….…………………………………………………13

3.4 Specific Impulse……………………………………………………………………...14

3.5 Delta V……………………………………………………………………………….15

4. THERMODYNAMICS OF ROCKET NOZZLE……………………………………...….16

5. THE EFFECT OF ATMOSPHERE ON THRUST AND OTHER PARAMTERS……..19

5.1 Development of Thrust Effect……………………………………………………….19

5.2 Exhaust Velocity……………………………………………………………………..23

5.3 Mass Flow Rate………………………………………………………………………25

6. SPACECRAFT NOZZLE AND GEOMETRY ANALYSIS……………………………..30

6.1 Calculated Values……………………………………………………………………33

6.2 Nozzle Geometry…………………………………………………………………….34

6.3 Parameters and initial boundary conditions………………………………………..35

7. ANALYSIS OF CONTOURS AND GRAPHS……………………………………………..36

7.1 Standard Nozzle……………………………………………………………………..36

7.2 Configuration1……………………………………………………………………….40

7.3 Configuration 2………………………………………………………………………44

7.4 Configuration 3………………………………………………………………………50

7.5 Configuration 4………………………………………………………………………56

7.6 Configuration 5………………………………………………………………………60

7.7 Configuration 6………………………………………………………………………66

8. CONCLUSION………………………………………………………………………………72

REFERENCES 73

LIST OF FIGURES

Page

Figure 1.1: Viking 5C rocket engine manufactured by France, Spain, Italy & Germany.........1

Figure 1.2: Action of exit pressure and atmospheric pressure on nozzle surface.....................3

Figure 1.3: Solid Propellant Rocket and liquid propellant rocket respectively........................5

Figure 1.4: A liquid-fuelled rocket engine................................................................................6

Figure 1.5: A solid-fuelled rocket engine..................................................................................7

Figure 2.1: Parts of rocket nozzle……………………………………………………………..8

Figure 2.2: Jet nozzle................................................................................................................9

Figure 2.3: Propelling Nozzle.................................................................................................10

Figure 2.4: Magnetic Nozzle...................................................................................................11

Figure 3.1: Graph of Tsiolkovsky‟s rocket equation with initial and final mass ratio............13

Figure 3.2: Mass-Ratio dependency on Delta V.....................................................................15

Figure 4.1: p-V diagram for a heat engine..............................................................................19

Figure 5.2: Gas flow through the nozzle.................................................................................20

Figure 5.3: Gas flow through nozzle.......................................................................................22

Figure 5.4: Gas velocity as a function of pressure ratio..........................................................25

Figure 5.5: Mass flow in the nozzle........................................................................................26

Figure 5.6: Variation of flow density through the nozzle.......................................................28

Figure 5.7: Area, velocity and flow density relative to the throat values as a function of the

pressure ratio............................................................................................................................29

Figure 6.1: Nozzle Expansion Geometry Effects for Thrust in Spacecraft.............................30

Figure 6.2: Convergent-Divergent Flow in a Spacecraft Nozzle……………………………31

Figure 6.3: Representation of a Spacecraft Nozzle Dimensions and Geometry.....................33

Figure 6.4: Nozzle Geometry (Edges).....................................................................................34

Figure 6.5: Nozzle Geometry (Mesh).....................................................................................34

Figure7.1 (a): Velocity contour of standard nozzle................................................................36

Figure7.1 (b): Velocity vs position graph of standard nozzle.................................................36

Figure 7.2 (a): Total Enthalpy contour of standard nozzle.....................................................37

Figure 7.2 (b): Total Enthalpy vs position graph of standard nozzle......................................37

Figure 7.3 (a): Total Pressure contour of standard nozzle......................................................38

Figure 7.3 (b): Total Pressure vs position graph of standard nozzle.......................................38

Figure 7.4 (a): Total temperature contour of standard nozzle.................................................39

Figure 7.4 (b): Total temperature vs position graph of standard nozzle.................................39

Figure7.5 (a): Velocity contour of changed configuration 1..................................................40

Figure7.5 (b): Velocity vs position graph of changed configuration 1...................................41

Figure 7.6 (a): Total Pressure contour of changed configuration 1.......................................41

Figure 7.6 (b): Total Pressure vs position graph of changed configuration 1.........................42

Figure 7.7 (a): Total Enthalpy contour of changed configuration 1.......................................42

Figure 7.7 (b): Total Enthalpy vs position graph of changed configuration 1........................43

Figure 7.8 (a): Total temperature contour of changed configuration 1...................................43

Figure 7.8 (b): Total temperature vs position graph of changed configuration 1...................44

Figure7.9 (a): Velocity contour of changed configuration 2..................................................45

Figure7.9 (b): Velocity vs position graph of changed configuration 2...................................45

Figure 7.10 (a): Total temperature contour of changed configuration 2.................................46

Figure 7.10 (b): Total temperature vs position graph of changed configuration 2.................47

Figure 7.11 (a): Total Pressure contour of changed configuration 2.....................................48

Figure 7.11 (b): Total Pressure vs position graph of changed configuration 2.......................48

Figure 7.12 (a): Total Enthalpy contour of changed configuration 2.....................................49

Figure 7.12 (b): Total Enthalpy vs position graph of changed configuration 2......................49

Figure7.13 (a): Velocity contour of changed configuration 3................................................50

Figure7.13 (b): Velocity vs position graph of changed configuration 3.................................51































LIST OF TABLE

Page

Table-3.1: Formulas used to define the nozzle geometry……………………………………..32

SUMMARY

The project is “Prediction of temperature profile and velocity at the exit of nozzle of rocket in

outer space using computational fluid dynamics.” This project involves the prediction of

temperature profile inside the rocket nozzle and using temperature profile prediction the velocity

at the exit of nozzle. The prediction code generated is used in outer-atmospheric conditions with

variation of pressure in vacuum consideration in outer-atmospheric condition. Fluent and Gambit

software are used for CFD analysis in the nozzle. By taking a standard configuration of the

nozzle the contours and graphs are obtained. Then by changing the configurations comparison is

done with standard configuration and best configuration is obtained.

1. ROCKET ENGINE

A rocket engine is a jet engine that uses specific propellant mass for forming high speed

propulsive exhaust jet. Rocket engines are reaction engines and obtain thrust in accordance with

Newton's third law. Since they need no external material to form their jet, rocket engines can be

used for spacecraft propulsion as well as terrestrial uses, such as missiles. Most rocket engines

are internal combustion engines, although non combusting forms also exist. Rocket engines are

in a group have maximum exhaust velocities, are the lightest, and are the least energy efficient of

all types of jet engines. The rockets are powered by exothermic chemical reactions of the rocket

propellant used.

Gas velocities from 2 to 4.5 kilometers per second can be achieved in rocket nozzles. The

nozzles which perform this feat are called DE Laval nozzles and consist of a convergent and

divergent section. The minimum flow area between the convergent and divergent section is

called the nozzle throat. The flow area at the end of the divergent section is called the nozzle exit

area from where the gases at their maximum possible releases from the engine.

Figure 1.1: Viking 5C rocket engine manufactured by France, Spain, Italy & Germany, adapted

from Url-1.

http://en.wikipedia.org/wiki/Jet_engine

http://en.wikipedia.org/wiki/Propellant

http://en.wikipedia.org/wiki/Jet_(fluid)

http://en.wikipedia.org/wiki/Reaction_engine

http://en.wikipedia.org/wiki/Newton%27s_third_law

http://en.wikipedia.org/wiki/Rocket

http://en.wikipedia.org/wiki/Spacecraft_propulsion

http://en.wikipedia.org/wiki/Missile

http://en.wikipedia.org/wiki/Internal_combustion_engine

Hot exhaust gases expand in the diverging section of the nozzle as the mach no. increases from

1.The pressure of these gases will decrease as energy is used to accelerate the gas to high

velocity. The nozzle is usually designed in such a way that the exit area is large enough as well

as the divergent part is long enough such that the pressure created in the combustion chamber is

reduced at the nozzle exit. It is under this condition that thrust is maximum and the nozzle is said

to be adapted, also called optimum or correct expansion. The basic equation of thrust used is as

follows:

(1.1)

Where

F = Thrust

q = Propellant mass flow rate

Ve = Velocity of exhaust gases

Pe = Pressure at nozzle exit

Pa = Ambient pressure

Ae = Area of nozzle exit

The product qVe is called the momentum, or velocity, thrust and the product (Pe-Pa)Ae is called

the pressure thrust. Here it can be seen that Ve and Pe are inversely proportional, that is, as one

increase the other decreases. If a nozzle is under-extended we have Pe & gt Pa and Ve is small.

For an over-extended nozzle we have Pe & lt Pa and Ve is large. Thus, momentum thrust and

pressure thrust are inversely proportional and, as we shall see, maximum thrust occurs when

Pe=Pa.

Figure 1.2: Action of exit pressure and atmospheric pressure on nozzle surface.

1.1Terminologies used in rocket engines

1. Solid-propellant rocket motor is basically the rockets which use solid propellant.

2. Liquid rockets (or liquid-propellant rocket engine) which uses one or more types of liquid

propellants that are held in tanks prior to burning.

3. Hybrid rockets are the rockets that contain solid propellant in combustion chamber and a

second liquid or gas propellant is added to permit it to burn.

4. Thermal rockets are rockets where the propellant is inert, but is heated by some power source

that can be as solar or nuclear power or beamed energy.

1.2 Principle of operation

Rocket engines produce thrust by creating a high-speed fluid exhaust. This fluid is generally

always a gas which is created by high pressure (10-200 bar) combustion of solid or liquid

propellants, consisting of fuel and oxidizer components, within a combustion chamber. The fluid

exhaust is then passed through a supersonic propelling nozzle which uses heat energy of the gas

to accelerate the exhaust gases to a very high speed, and from the Newton‟s third law the

reaction to this pushes the engine in the opposite direction.

In rocket engines, high temperatures and pressures are highly desirable for good performance as

this permits a longer nozzle to be fitted to the engine, which gives higher exhaust speeds, as well

as giving better thermodynamic efficiency.

http://en.wikipedia.org/wiki/Liquid_rocket

http://en.wikipedia.org/wiki/Hybrid_rocket

http://en.wikipedia.org/wiki/Thermal_rocket

http://en.wikipedia.org/wiki/Solar_thermal_rocket

http://en.wikipedia.org/wiki/Nuclear_thermal_rocket

http://en.wikipedia.org/wiki/Beamed_propulsion

http://en.wikipedia.org/wiki/Fluid

http://en.wikipedia.org/wiki/Rocket_propellant

http://en.wikipedia.org/wiki/Fuel

http://en.wikipedia.org/wiki/Oxidiser

http://en.wikipedia.org/wiki/Combustion_chamber

http://en.wikipedia.org/wiki/Propelling_nozzle

1.3 Propellents used in rocket engine

Generally rockets use either solid or liquid propellants for operation. In case of rockets propellant

does„t mean only fuel, it means both fuel and oxidizer. Oxidizer must be present for burning as

there is no air intake and no atmosphere in outer space. Jet engines have inlet wents from where it

draws oxygen into the engine from the atmosphere.

Solid rocket propellants are dry and contain the combination of fuel & oxidizer. Generally the

fuel is hydrocarbons and the oxidizer contains oxygen. Liquid propellants are often gases but

have been compressed until they change into liquids. In these propellants the fuel & oxidizer are

kept in different chambers or containers. Fuel and oxidizer are mixed together just before

ingnition.

1.3.1 Solid Propellant Rocket

A rockets with solid propellants have simple engines. It contains nozzle, a case, insulation,

propellant and an igniter. The case of the engine is generally made of relatively thin metal. There

is lining of insulation present to prevent the propellant from burning through rapidly. The

propellant itself is also packed inside the insulation layer. Many solid-propellant rocket engines

contain a hollow core that runs through the block of propellant. Rockets that do not contain the

hollow core are to be ignited at the lower end and burning of the propellants proceeds gradually

from one end to the other. mostly burning takes place on the surface of the propellant. However,

to get higher thrust, the hollow core is used which increases the surface of the propellants

available for burning.

The propellants burn from the inside out at a much higher rate producing higher speed as mass

flows out of the nozzle at a higher rate resulting in greater thrust. Propellant cores can be of

different shapes to increase the area for burning one of the most commonly used is star shaped

core. There are different kinds of igniters used for ignition of solid propellants.

The solid-propellant engines have nozzle as the opening at the back of the rocket for the exit of

gases. The nozzle is used to increase the the acceleration of the gases and hence increase the

thrust.

The insulation is used to prevent it from heat produced from the gases. Small pieces of the

insulation get very hot and break away from the nozzle. As they are blown away, heat is carried

away with them.

1.3.2 Liquid Propellant Rocket

Liquid propellent is the other kind of propellant used in rocket engines. Liquid propellant can be

either pumped or fed into the engine by pressure. Liquid propellant engines are more complicated

as compared to solid propellant engines. Liquid propellants have separate chambers for the

storage of fuel and oxidizer.

The fuel used in liquid propellant rockets are generally kerosene or liquid hydrogen and the

oxidizer used is liquid oxygen these are combined before ignition inside a cavity known as the

combustion chamber. In this the propellants burn and build up high temperatures and pressures.

The gases expand and escape through the nozzle. To get the most power combustion must be

proper. Fuel is injected through the small nozzles present in the combustion chamber.

Combustion chamber operates under high pressures therefore the propellants are forced inside.

Weight is one of the most important factor in any rocket. If the rocket is heavy then it will require

more thrust for the lift off. Due to pumps and hozes these engines are more heavy and complex

then solid propellant rocket engines. Weight can be decreased by making the nozzle walls with

lighter metal but it should not be thin as the high temperature gases will melt the walls to prevent

this cooling system is used.

Figure 1.3: Solid Propellant Rocket and liquid propellant rocket respectively.

1.4 Thermal Rocket Engine

Thermal rocket is basically when rocket engine is treated as heat engine. Thermal rocket is the

basis of all the launchers and almost all the space propulsions. Rocket engine doesn‟t have any

external medium for the propulsion to act upon, no external oxidant for fuel so the thermal rocket

helps to understand propulsion better.

The thermal rocket converts heat energy generated by burning of fuel and propellant in the

combustion chamber into kinetic energy of the exhaust. The exhaust gases have momentum

which provides thrust. Rocket engine is just like all other heat engines since the conversion of

heat into work is same whether work is done on a piston, or on a steam of exhaust gases.

A liquid-fuelled rocket engine consists of combustion chamber in which fuel and oxidant are

pumped, and an expansion nozzle which converts the high-pressure hot gas, produced by the

combustion chamber, into high velocity exhaust stream.

Figure 1.4: A liquid-fuelled rocket engine

A solid-fuelled rocket engine operates in the same way as liquid propelled rocket engine, except

that the fuel and oxidants are pre-mixed in solid form, and are contained within the combustion

chamber.

Figure 1.5: A solid-fuelled rocket engine

The nozzles of both are identical in form that both have the same principle of operation. The

combustion which takes place in combustion chamber can be any chemical reaction which

produces heat. It can be simple oxidation of fuel by liquid oxygen.

2. ROCKET ENGINE NOZZLE

A nozzle is used to give the direction to the gases coming out of the combustion chamber.

Nozzle is a tube with variable cross-sectional area. Nozzles are generally used to control the rate

of flow, speed, direction, mass, shape, and/or the pressure of the exhaust stream that emerges

from them.

The nozzle is used to convert the chemical-thermal energy generated in the combustion chamber

into kinetic energy. The nozzle converts the low velocity, high pressure, high temperature gas in

the combustion chamber into high velocity gas of lower pressure and temperature.

The general range of exhaust velocity is 2 to 4.5 kilometer per second. The convergent and

divergent (also known as convergent-divergent nozzle) type of nozzle is known as DE-LAVAL

nozzle. Throat is the portion with minimum area is a convergent-divergent nozzle. The divergent

part of the nozzle is known as nozzle exit area or nozzle exit.

Figure 2.1: Parts of rocket nozzle, adapted from Url-2

In the convergent section the pressure of the exhaust gases will increase and as the hot gases

expand through the diverging section attaining high velocities (from continuity equation). In

nozzle the combustion chamber pressure is decreased as the flow propagates towards the exit as

compared to the ambient pressure (i.e. pressure outside the nozzle), this result in maximum

expansion (known as optimum expansion) and nozzle is known as adapted.

2.1 Types of nozzles

2.1.1Jets

A gas jet is a nozzle made for the ejection of gas or fluid in the flow stream into the surrounding

environment. It is also known as fluid jet or hydro jet. These types of jets are generally present in

household equipments like gas stoves, ovens or barbecues. In early days when there was no

electricity then the gas jets were used for light. Other fluid jets are used where flow regulation is

required like in carburetors smooth orifices are used for the regulation of the fuel flow into an

engine.

Another type of jet is the laminar jet. This is basically a water jet with a streamlined flow. These

types of nozzles are often used in fountains.

http://en.wikipedia.org/wiki/Gas_stove

http://en.wikipedia.org/wiki/Oven

http://en.wikipedia.org/wiki/Barbecue

http://en.wikipedia.org/wiki/Gas_light

http://en.wikipedia.org/wiki/Carburetor

http://en.wikipedia.org/wiki/Calibrated_orifice

http://en.wikipedia.org/wiki/Gasoline

http://en.wikipedia.org/wiki/Laminar

http://en.wikipedia.org/w/index.php?title=Laminar_jet&action=edit&redlink=1

Nozzles which are used for feeding hot blast into a blast furnace are called tuyeres.

Figure 2.2: Jet nozzle, adapted from Url-3

2.1.2 High velocity nozzles

The main goal is to increase the kinetic energy of the fluid at the expense of its pressure and

energy. Nozzles can be defined as convergent (narrowing down from a wide diameter to a smaller

diameter in the direction of the flow) or divergent (expanding from a smaller diameter to a larger

one).Convergent part of the nozzles accelerates subsonic fluids. If the pressure ratio of the nozzle

is high enough the flow will reach sonic velocity at the narrowest point (i.e. the nozzle throat).

This condition of the nozzle choked condition.

On increasing the nozzle pressure ratio further will not increase the throat Mach number beyond

unity. Downstream flow is free to expand to supersonic velocities. Divergent nozzles slow fluids,

if the flow is subsonic, but accelerate sonic or supersonic fluids.

Convergent-divergent nozzles can therefore accelerate fluids that have choked in the convergent

section to supersonic speeds. This process is more efficient than allowing a convergent nozzle to

expand supersonically externally. The shape of the divergent section also ensures that the

http://en.wikipedia.org/wiki/Hot_blast

http://en.wikipedia.org/wiki/Blast_furnace

http://en.wikipedia.org/wiki/Tuyere

http://en.wikipedia.org/wiki/Kinetic_energy

http://en.wikipedia.org/wiki/Pressure

http://en.wikipedia.org/wiki/Mach_number

http://en.wikipedia.org/wiki/De_laval_nozzle

direction of the escaping gases is directly backwards, as any sideways component would not

contribute to thrust.

2.1.3 Propelling nozzles

A jet exhaust produces a net thrust from the energy obtained from combusting fuel which is

added to the inducted air. This hot air is passed through a high speed nozzle, a propelling nozzle

which drastically increases its kinetic energy.

For a particular mass flow, greater thrust is obtained with a higher exhaust velocity, but the best

energy efficiency is obtained when the exhaust speed is well matched with the airspeed.

However, no jet aircraft can maintain velocity while exceeding its exhaust jet speed, due to

momentum considerations. Supersonic jet engines, like those employed in fighters & commercial

aircraft (e.g. Concorde), need high exhaust speeds. Therefore supersonic aircraft use a

convergent divergent nozzle despite weight and cost penalties. Subsonic jet engines employ

relatively low, subsonic, exhaust velocities. They thus employ simple convergent nozzles. In

addition, bypass nozzles are employed giving even lower speeds.

Rocket motors use convergent-divergent nozzles with very large area ratios so as to maximize

thrust and exhaust velocity and thus extremely high nozzle pressure ratios are employed. Mass

flow is at a premium since all the propulsive mass is carried with vehicle, and very high exhaust

speeds are desirable.

http://en.wikipedia.org/wiki/Fighter_aircraft

http://en.wikipedia.org/wiki/Concorde

http://en.wikipedia.org/wiki/Bypass_ratio

http://en.wikipedia.org/wiki/Rocket_motor

Figure 2.3: Propelling Nozzle, adapted from Url-4

2.1.4 Magnetic nozzles

Magnetic nozzles have also been proposed for some types of propulsion in which the flow of

plasma is directed by magnetic fields instead of walls made of solid matter.

Figure 2.4: Magnetic Nozzle, adapted from Url-5

2.1.5 Spray nozzles

Many nozzles produce a very fine spray of liquids. Atomizer nozzles are used for spray painting,

perfumes, carburetors for internal combustion engines, spray on deodorants, antiperspirants and

many other uses. Air-Aspirating Nozzle-uses an opening in the cone shaped nozzle to inject air

into a stream of water based foam (CAFS/AFFF/FFFP) to make the concentrate "foam up". Most

commonly found on foam extinguishers and foam hand lines. Swirl nozzles inject the liquid in

tangentially, and it spirals into the center and then exits through the central hole. Due to the

vortexing this causes the spray to come out in a cone shape.(Ref…Wikipedia.org)

2.1.6 Vacuum nozzles

Vacuum cleaner nozzles come in several different shapes.

http://en.wikipedia.org/wiki/Plasma_%28physics%29

http://en.wikipedia.org/wiki/Magnetic_field

http://en.wikipedia.org/wiki/Atomizer_nozzle

http://en.wikipedia.org/wiki/Carburettor

http://en.wikipedia.org/wiki/Internal_combustion_engine

http://en.wikipedia.org/wiki/Deodorant

http://en.wikipedia.org/wiki/Antiperspirant

http://en.wikipedia.org/wiki/Vacuum_cleaner

2.1.7 Shaping nozzles

Some nozzles are shaped to produce a stream that is of a particular shape. For example extrusion

molding is a way of producing lengths of metals or plastics or other materials with a particular

cross-section. This nozzle is typically referred to as a die.

3. BASIC ROCKET EQUATIONS

3.1 Newton’s Third law of motion: To every action there is always opposed an equal reaction:

or the mutual actions of two bodies upon each other are always equal.

3.2 De Laval nozzle equations:

I Conservation of Mass: ṁ= ρ V A= constant.

II Conservation of Momentum:

III Isentropic Flow:

IV Combine with Momentum:

V Combine with Mass:

http://en.wikipedia.org/wiki/Extrusion_molding

http://en.wikipedia.org/wiki/Extrusion_molding

http://en.wikipedia.org/wiki/Die_%28manufacturing%29

For subsonic flow (M<1), increase in area (dA>0) causes flow velocity to decrease (dV<0)

For supersonic flow (M>1), increase in area (dA>0) causes flow velocity to increase (dV>0)

3.3 Tsiolkovosky equation

The accelerating force is represented by Newton‟s law as:

In the above equation, the thrust of the rocket is expressed in terms of the mass flow rate, m, and

the effective exhaust velocity, .

So, the resulting formula which tsiolkovsky obtained for the vehicle velocity v is

Here M0is the mass of the rocket at ignition (initial), and M is the current mass of the rocket

s the exhaust velocity.

This is the simple formula is the basis of all rocket propulsion. The velocity increases with time

as the fuel burned.

Figure 3.1: Graph of Tsiolkovsky‟s rocket equation with initial and final mass ratio.

The rocket equation shows that the final speed depends upon only on two factors which are the

final mass ratio and the exhaust velocity. It does not depend on the thrust, size of the rocket

engine and also on the time of rocket burns or any other parameter.

From the above graph, it is inferred that the rocket can travel faster than the speed of its exhaust.

The fact is that the exhaust is not pushing against anything at all, and once it has left the rocket

engine‟s nozzle has no effect on rocket. All the actions take place inside the rocket where a

constant accelerating force is being exerted on the inner walls of the combustion chamber and

the inside of the nozzle. This implies that the speed of the rocket depends on magnitude of the

exhaust velocity but it can itself be much greater.

3.4 Specific Impulse: Specific impulse (abbreviated Isp) is a way to describe the efficiency of

rocket and jet engines. It represents the impulse per unit amount of propellant used.

The equation for specific impulse is:

g

VI e

sp

(3.8)

Where,

Ve = the exhaust velocity of the spacecraft (m/s).

g = Acceleration due to gravity.

The higher the specific impulse, the less propellant is needed to gain a given amount of

momentum.

3.5 Delta V: From the Tsiolkovsky equation we get,

final

initialexhaust

M

MVV ln

(3.9)

where

Minitial : The initial mass of the space vehicle

Mfinal : The final mass of the space vehicle.

As specific impulse goes down, so does Delta V.

Figure 3.2: Mass-Ratio dependency on Delta V

The implication of Tsiolkvosky equation is:

a) The spacecraft needs to have either a very large Ve (exhaust gas velocity) or

b) The spacecraft will need to have a very high proportion of m /mo.

4. THERMODYNAMICS OF ROCKET NOZZLE

a) b)

Figure 4.1: p-V diagram for a heat engine,adapted from Gas dynamics by Rathakrishnan

Figure: (a) From 1 to 2 the fuel-air mixture is being compressed; then after ignition, it expands at

constant pressure from 2 to 3as the piston moves downward. This is followed by adiabetic

expansion, 3 to 4, as the gas does further work and cools. The final stroke, 4 to 1, shows the gas

being exhausted at constant pressure.

Figure: (b) Rocket engine analogue to the internal combustion engine. There is no inlet stroke, as

the rocket operates in a continuous manner.

The thermodynamics of the exhaust nozzle play a very important role in the performance

characterstics of the spacecraft.

From Thermodynamic relations:

JQve 2

(4.1)

Where ev =exhaust velocity of the gas, J = mechanical equivalent of heat, Q = amount of heat

liberated in kJ / g of reaction product.

It is possible to derive a better equation for the isentropic expansion of gases in order to form a

better equation for the exhaust velocity of the gas.

The ideal gas equation of state is defined as:

TM

RJpv (4.2)

Where, R= Universal gas Constant, M = Molecular weight of the gas

In this equation, it is assumed in the spacecraft that there is zero velocity in the combustion

chamber and that there is an ideal isentropic expansion of the gases from the exhaust chamber.

From the thermodynamic equation (1.12) with the above ideal gas conditions:

)(22

HHJv ce (4.3)

Hence, by using the above equations with the ideal gas equation, the stream velocity of the

exhaust gas is found to be:

11

2 11

2

c

ccP

Pvpv (4.4)

Where

Pc is pressure of the main chamber

P is the gas pressure at that instant

Vc is the volume of the main chamber and,

is the latent heat of vaporization

Through the values presented above, it is possible to examine the stream velocity of the

spacecraft and thus the performance of the spacecraft. This provides the means to alternate the

design of the spacecraft where necessary.

If the velocity of the exhaust gas equation is written in the best possible way so that the final

velocity of the spacecraft can be seen, then give:

11

11

2c

ece

P

PT

M

RJv

(4.5)

Where,

ve = speed of the gas at the exhaust point,

Pe= gas pressure at the exhaust,

Tc =Temperature of the combustion chamber.

From the Tsiolkovsky‟s equation, the maximum velocity of the spacecraft as the function of the

velocity of the exhaust gas can be defined as:

o

em

mvV lnmax (4.6)

Thus, by combining equations, it is possible to get an equation for the maximum velocity of the

spacecraft:

o

c

ec mm

P

PT

M

RJV /ln1

12

11

max (4.7)

Above equation gives the velocity of the spacecraft as a function of temperature. As Tc is

increased, the final Vmax will also increase accordingly. This is very useful, since the temperature

is the main operational parameter of spacecraft propulsion. This way, direct link up between

obtaining high temperature for faster speeds is seen with the equation above. This will become

handy for understanding the role of nuclear heating of a propellant in order to get a higher

exhaust speed.

5. THE EFFECT OF ATMOSPHERE ON THRUST AND OTHER PARAMTERS

5.1 Development of Thrust Effect

The effective exhaust velocity is that velocity which, when combined with the actual mass flow

in the exhaust stream, produces the measured thrust, F=mve, where m is mass flow rate, ve is the

effective exhaust velocity; ve combines the true exhaust velocity with the effects of atmospheric

pressure and the pressure in the exhaust stream, into one parameter. The true exhaust velocity is

also function of temperature and the pressure in the combustion chamber besides these.

The energy of the gases in the combustion chamber is represented by temperature and pressure

has to be converted into velocity, this occurs when the gas expands and cools while it passes

through the nozzle. The velocity rises very rapidly, passing the speed of sound as it crosses throat

there after it continues to accelerate until it exits. The reaction of nozzle wall to the gas pressure

provides accelerating force, as the gas expands. That‟s why most of the thrust is generated due to

nozzle itself.

Figure 5.1: Forces in the combustion chamber and exhaust nozzle

The above figure represents the action of the gas pressure on the combustion chamber and the

exhaust nozzle; this force which accelerates the rocket. It also shows the reaction of the walls of

the combustion chamber and the exhaust nozzle acting on the gas contained by them, which is the

force that accelerates the exhaust gas.

The force accelerating the exhaust gas is equal to the surface integral of the pressure, taken over

the whole inner surface of the chamberand nozzle.

Figure 5.2: Gas flow through the nozzle

The gas flowing through the nozzle is forced by the pressure gradient from the combustion

chamber to the exit. At any point in the nozzle, the pressure upstream is greater than the pressure

downstream. The net accelerating force acting on the shaded portion is

( . )

Where A is the cross-sectional area at any given point, and the pressure gradient is dp/dx.

Thrust is the combination of effective exhaust velocity and the actual mass flow in the exhaust

i.e. F=mve, m is the mass flow rate and ve is the effective exhaust velocity. This effective

velocity is the combination of true velocity of the exhaust gases, effect of atmospheric pressure

and pressure of the exhaust stream, besides this true velocity also includes the conditions like the

temperature and pressure in the combustion chamber.

The energy released by the chemical reactions in the combustion chamber is represented by the

conditions of the chamber i.e. its pressure and temperature. This energy is converted into

velocity while expanding through the nozzle. The expanding gases produce the accelerating

force to the exhaust gases as reaction from nozzle wall. As a result the thrust developed is mostly

due to the nozzle. Thrust equation relates all the aspects like true exhaust velocity, combustion

chamber pressure and atmospheric pressure.

The accelerating force from eqn. 5.1

F =

There is another force called retarding force acts other than this, it can be seen in Figure 5.2

Figure 5.3: Gas flow through nozzle

Net accelerating force acting on shaded portion is

dF=pA-(p-dp)A (5.3)

Where A is the cross-sectional area at any point, pA is the outward force.

Considering the Newton‟s Law‟s application to exhaust gases

(5.4)

Retarding Force acting on the rocket due to atmosphere

(5.5)

The atmospheric pressure is the main reason for the retardation, so net force is

(5.6)

Substituting the value of from equation 2.3

(5.7)

This is the thrust equation

This equation can be used to examine performance characteristics.

Effective exhaust velocity expression is

(5.8)

5.2 Exhaust Velocity

The propellants are mixed and ignited in the combustion chamber; the mixture is then heated and

expands through the nozzle. The assumptions made before finding the change in internal energy

are for deriving exhaust velocity kinetic energy can be equated to enthalpy change of the exhaust

gas, all of this is carried out in isentropic conditions i.e. no heat exchange takes place between

gas and nozzle walls.

The change in internal energy

(5.9)

Where cP is specific heat at constant pressure, Tc is temperature of combustion chamber and Te is

temperature of exiting gas.

The change in energy equals the kinetic energy gain

(5.10)

Thus the square of exhaust velocity is

(5.11)

There is a sensor present in the chamber that helps in measuring the temperature of the chamber.

The temperature has nothing to do with the design of the chamber but it depends on propellant

mixture. As the temperature of the exhaust depends on degree of expansion (i.e. nozzle design) it

is difficult to measure it.

Equations for adiabatic or isentropic expansion:

(5.12)

(5.13)

Where γ is the ratio of specific heat of the exhaust gases at constant pressure to that at constant

volume, for air, its value is 1.3 at normal temperature and pressure (NTP).

Relation of specific heat, gas constant and molecular weight of exhaust gases:

(5.14)

Where R is universal gas constant and M is the molecular weight of the exhaust gases.

Substituting for Te and cP, velocity can be expressed by:

(5.15)

The ratio (pe/pc)(γ-1)/γ

is the expression for temperature difference in terms of pressure difference

between entrance (i.e. combustion chamber) and exit of nozzle.

For a perfect nozzle pe is zero in vacuum. The maximum value of exhaust velocity is

(5.16)

This equation demonstrates the most efficient rocket as in which delivers maximum thrust in

vacuum since thrust is proportional to exhaust velocity. Figure 2 shows this example.

Figure 5.4: Gas velocity as a function of pressure ratio, adapted from Rocket and Spacecraft

Propulsion Martin J L Turner Third edition

Velocity has strong dependency on γ for small values of γ but as the value of γ approaches 1 then

whole expression tends to infinity. The velocity is directly proportional to combustion

temperature. Velocity also depends on molecular weight of gases exiting nozzle but inversely

which is indirectly or in latter stages helps in selection of optimized propellant. If the operating

temperature can be kept high then low molecular weight propellants can be used as they are of

significant advantage. For a rocket engine the most needed significant performance indicator.

5.3 Mass Flow Rate

Now mass flow rate is the only left term in the thrust equation i.e. m. This depends on the

conditions in nozzle and combustion chamber. Mass flow rate will be determined by the pressure

difference between entry and exit of nozzle together with the cross sectional area.

As all the propellant that enters the system has to exit under steady flow so the mass flow ret will

remain constant throughout nozzle.

Figure 5.5: Mass flow in the nozzle, adapted from Rocket and Spacecraft Propulsion Martin J L

Turner Third edition

The mass flow rate can be expressed as:

(5.17)

Where m is the mass flow rate (constant), u is velocity and A is the cross sectional area.

The velocity at a given point is:

(5.18)

The mass flow rate can be expressed as:

(5.19)

In this equation using gas laws ρ can be found out

(5.20)

(5.21)

Density can be expressed in two ways:

(5.22)

(5.23)

Density in the combustion chamber can be expressed as:

(5.24)

The above value of ρ can be substituted in mass flow equation

(5.25)

(5.26)

Mass flow rate per unit area of cross section of the nozzle is not a constant value.

This is shown in Figure 5.6.

Figure 5.6: Variation of flow density through the nozzle, adapted from Rocket and Spacecraft

Propulsion Martin J L Turner Third edition

Figure 5.6 shows the working of rocket nozzle. The flow density first increases and then

decreases. The flow density starts decreasing as soon as the pressure reaches 60% the value of

pressure in the combustion chamber.

Ideal cross section area of the nozzle for any pressure is:

(5.27)

Pressure ratio of initial value and combustion chamber:

(5.28)

For γ=1.2 the pressure ratio will be equal to 0.57. Putting this in equation for exhaust velocity we

get:

(5.29)

Differentiating the equation of mass flow rate per unit cross sectional area (Throat area):

(5.30)

Figure 5.7 shows the velocity and density of the flow at throat.

Figure 5.7: Area, velocity and flow density relative to the throat values as a function of the

pressure ratio, adapted from Rocket and Spacecraft Propulsion Martin J L Turner Third edition

Since mass flow rate is constant everywhere so this expression can also be used

(5.31)

The thermodynamic thrust equation is the result of substitution of values of velocity and mass

flow rate in expression (2.6):

(5.32)

This is the full thermodynamic thrust equation of rocket containing Newtonian thrust (due to

mass ejection), accelerating force (due to static pressure) and retarding force (due to atmospheric

pressure).

6. SPACECRAFT NOZZLE GEOMETRY ANALYSIS

Analysis of nozzle geometry is a very important parameter in determining the performance

characteristics of a spacecraft. Proper geometrical nozzle design can regulate the exhaust in such

a way that maximum effective velocity can be reached in a spacecraft. Both under and over

expansion nozzles can be problematic for flight of a spacecraft.

Figure 6.1: Nozzle Expansion Geometry Effects for Thrust in Spacecraft

Convergent and divergent nozzle is the most commonly used nozzle since in using it the

propellant can be heated in combustion chamber. After getting heated the propellant first

converges at the throat of the nozzle and then expands under constant temperature in the

divergent part. The expansion of the gases takes place the exhaust velocity will increase to

supersonic, the area of the nozzle will also contribute in increasing the velocity to supersonic

speed. The exhaust will give the rocket forward momentum since there is an opposite reaction is

applied to propellant‟s momentum. Both proper design and analysis will affect the final exhaust

velocity of the spacecraft.

Role of geometry comes into play after the proper conditions are reached. As per the dimensions

(i.e. Length & Shape) of the nozzle appropriate exit profile can be achieved as seen in Figure 6.2.

Figure 6.2: Convergent-Divergent Flow in a Spacecraft Nozzle

There are many equations that are used to find out the nozzle geometry. These equations are as

follows:

Table-6.1: Formulas used to define the nozzle geometry.

e

thrust

V

Fm.

Mass Flow in Rockets (3.1)

t

e

A

A Expansion Ratio in Nozzles

(3.2)

cu

c

t

TR

MP

mA

1

1

.

1

2

Area of the Nozzle Throat

(3.3)

2

ee rA Exit Area of the Nozzle (3.4)

tc AA 3 Convergence area in Nozzle (3.5)

tt

Ar Radius of the Throat

(3.6)

cc

Ar Combustion radius

(3.7)

tan

1edn

AL Diverging Nozzle Length

(3.8)

tan

1ccn

AL Length of the Converging Nozzle

(3.9)

2

*

c

tc

r

LAL Length of the Combustion Chamber

(3.10)

6.1 Calculated Values

According to the above equations the specifications of the nozzle are as follows:-

1. Expansion Ratio in nozzle: - 72.44.

2. Area of Nozzle Throat:- 0.0127 m2.

3. Exit Area of the Nozzle:- 0.92 m2.

4. Convergence Area in Nozzle:- 0.038 m2.

5. Radius of the Throat: - 0.064 m.

6. Converging Nozzle Radius: - 0.11 m.

7. Diverging Nozzle Radius:- 0.56 m.

8. Diverging Nozzle Length: - 2.02 m.

9. Length of the converging Nozzle: - 0.64 m.

Figure 6.3: Representation of a Spacecraft Nozzle Dimensions and Geometry.

6.2 Nozzle Geometry: -

By using the above parameters the geometry of the nozzle in Gambit is as follows:

Figure 6.4: Nozzle Geometry (Edges)

Now accordingly the meshing is done and the initial parameters are given.

Figure 6.5: Nozzle Geometry (Mesh)

6.3 Parameters and initial boundary conditions

The parameters according to the nozzle conditions are given which are as follows:

Accordingly the geometry of the nozzle is divided into zones and boundary conditions given to

these are: -

1. The inside nozzle surfaces are given as WALL.

2. The inlet is given as mass flow inlet.

3. The outlet is given as Pressure outlet.

Now, the fluent condition to the standard nozzle geometry is as follows: -

1. Density based solver

2. 2-D space

3. Absolute velocity formulation

4. Energy equation included.

5. Viscous model- K epsilon

6. Materials are Hydrogen and Titanium for fluid flow and wall respectively.

7. Operating Pressure = 0

8. Mass flow rate (Inlet) = 100 kg/s.

9. Inlet Pressure = 20MPa.

10. Supersonic/initial gauge pressure (less than/ slightly less than Inlet) = 6000000 N/m2.

11. Inlet temperature = 1000 K.

12. Outlet Gauge pressure and temperature are 0 and 2.73 k respectively.

13. Wall material is Titanium.

7. ANALYSIS OF CONTOURS AND GRAPHS: -

7.1 Standard Nozzle

Velocity Contour and Plot

Figure7.1 (a): Velocity contour of standard nozzle

Figure7.1 (b): Velocity vs position graph of standard nozzle

As seen from the above figure that there is an increase in the velocity value at the throat which is

shown in the graph at x = 0.5. After that by following the supersonic properties the value

gradually increases to 10,000 m/s at the exit which is approximately 60% increment.

Enthalpy contour and Plot

Figure 7.2 (a): Total Enthalpy contour of standard nozzle

Figure 7.2 (b): Total Enthalpy vs position graph of standard nozzle

Enthalpy is defined as the measure of the total energy of a thermodynamic system. It includes the

internal energy, which is the energy required to create a system, and the amount of energy

required to make room for it by displacing its environment and establishing its volume and

pressure. It is positive for endothermic reactions and negative for exothermic reactions

(Ref...wikipedia.org). Therefore in the rocket engine the heat is liberated therefore the above

graph and contour are verified.

Pressure contour and plot

Figure 7.3 (a): Total Pressure contour of standard nozzle

Figure 7.3 (b): Total Pressure vs position graph of standard nozzle

http://en.wikipedia.org/wiki/Energy

http://en.wikipedia.org/wiki/Thermodynamic_system

http://en.wikipedia.org/wiki/Internal_energy

http://en.wikipedia.org/wiki/Environment_%28systems%29

As seen from the above figure that the pressure value till x = 0.5m is gradually decreasing as the

fluid the fluid velocity is increasing and then there is sudden drop of pressure at x = 2.01m as

there is a sudden expansion and velocity is increased drastically.

Total temperature contour and plot

Figure 7.4 (a): Total temperature contour of standard nozzle

Figure 7.4 (b): Total temperature vs position graph of standard nozzle

As the wall temperature is 1000k the fluid near the wall has a higher temperature than the fluid

near the centre. This is due to the heat exchange between solid and fluid.

Now as the contours and plots of the standard nozzle are obtained, changing the configuration of

the nozzle changes the efficiency. This can be evaluated by comparing the graphs and contours

of the standard nozzle with the changed configurations. Some of the configurations are as

follows: -

7.2 Configuration1.

In this configuration the convergent radius is increased from 0.11m to 0.22 m. Apart from the

convergent area other configurations are same as above. The contours and plots are as follows: -

Velocity contour and plot

Figure7.5 (a): Velocity contour of changed configuration 1.

Figure7.5 (b): Velocity vs position graph of changed configuration 1.

In this configuration according to the graph the initial velocity is 3000m/s and the velocity at the

exit is 8000 m/s. The increment in the speed is approximately 183.3%, but at the same time the

output velocity is lesser than the standard value.

Total pressure contour and plot

Figure 7.6 (a): Total Pressure contour of changed configuration 1

Figure 7.6 (b): Total Pressure vs position graph of changed configuration 1

Initially at the inlet or the converging part of the nozzle the pressure is very high as the velocity

is starting to increase and then at the throat(x = 0.5m) there is a slight drop of pressure due to

increment of velocity at the throat. Now at the end part of divergent nozzle(x = 2.01m) there is a

sudden drop of pressure as the velocity drastically increase.

Enthalpy contour and plot

Figure 7.7 (a): Total Enthalpy contour of changed configuration 1

Figure 7.7 (b): Total Enthalpy vs position graph of changed configuration 1

Enthalpy is defined as how much energy is converted into work. Therefore in the above graph it

is seen that as the fluid is leaving the nozzle the enthalpy is gradually decreasing and at the end

of divergent section the enthalpy is decreased. The overall process is exothermic and has a

negative difference of enthalpy.

Temperature contour and plot

Figure 7.8 (a): Total temperature contour of changed configuration 1.

Figure 7.8 (b): Total temperature vs position graph of changed configuration 1.

In the above figure it is clearly seen that there is a formation of viscous sub layer at the wall and

due to friction there is an increment in temperature and at the end of divergent part due to sudden

expansion in velocity there is a rise in temperature.

7.3 Configuration 2.

In this configuration the throat radius is increased from 0.064m to 0.08m.All the other

dimensions are same. The graph and contours are as follows: -

Velocity contour and graph



In this configuration the velocity at the throat is increased (as seen in contour) and before the exit

there is a sudden increment of velocity, but at the exit the velocity is slightly decreased. The

increment in the velocity is approximately 50% but the exit velocity is lesser as compared to the

standard configuration.

Total Temperature contour and plot



In the above contour it is seen that there is an increment of viscous sub layer as due to friction

the temperature is transferred more to the fluid flowing in the nozzle. The initial and final

temperatures are approximately same.

Total Pressure contour and plot

Figure 7.11 (a): Total Pressure contour of changed configuration 2.


In this configuration the inlet pressure is lesser as compared to the standard nozzle that means the

inlet velocity is high. But at the throat the pressure drop is more which means the throat velocity

is higher.




On comparing the total enthalpy with the standard configuration due to increase in throat area the

enthalpy is increased at the throat as well as divergent part.


In this configuration the radius of the throat is reduced to 0.04 m and other specifications are

same as standard configurations.




In this configuration the velocity is suddenly increased at the throat from 6000m/s to 10000 m/s

due to throat properties and then the velocity suddenly increases to 12000 m/s before the throat at

x = 2.01m and then reduces to 10500m/s (approx.) at the exit.

Temperature contour and plot



In this configuration the temperature at the throat suddenly increases to 1100k as the throat area

is very small, therefore there is a complete transfer of temperature due to friction to the air at the

centre at the throat area.




Here the enthalpy is suddenly dropped at the throat region and after the throat it increases

slightly which means that the phenomenon of total energy which has to be converted into work is

not been done. Also at x=2.01m again enthalpy is decreasing.


Figure 7.16 (a): Total Pressure contour of changed configuration 3.


The total pressure is drastically decreasing at the throat area and then slightly increasing till

x=2m which shows that there may be a sudden expansion. The decrement in pressure is

approximately 240%.


In this configuration the exit radius is increased to 0.76m. It is increased from 0.56m which is in

standard nozzle. Apart from this all the other dimensions are same.


Figure7.17 (a): Velocity contour of changed configuration 4

Figure7.17 (b): Velocity vs position graph of changed configuration 4

As the exit diameter is increased the velocity field is getting weaker near the exit area and

therefore the desired increment of velocity is not up to the desired value. The increment in

velocity is approximately 33%.


Figure 7.18 (a): Total temperature contour of changed configuration 4

Figure 7.18 (b): Total temperature vs position graph of changed configuration 4

When the divergent area is increased there is more transfer of heat from nozzle surface to the

centre. The viscous sub layer is thick near the exit area as seen in the figure.

Total pressure contour and plot



The total pressure is decreased drastically near the exit on the divergent side of the nozzle. It can

be seen by blue area in the contour. This shows that when the diameter is increased more than a

certain value the pressure becomes weak near the exit area.

Total Enthalpy contour and plot



In this configuration the enthalpy is maximum in the convergent area of the nozzle. As moving

forward towards the exit area the enthalpy decreases and due to larger divergent area there is a

sudden drop in enthalpy just before the exit.


In this configuration the exit radius is increased to 0.36m. It is increased from 0.56m which is in

standard nozzle. Apart from this all the other dimensions are same.




In this configuration as the divergent nozzle area is lesser as compared to standard nozzle the

increment of velocity is very large. The inlet velocity is 4770m/s and the exit velocity is

1100m/s. The increment is approximately 130% .




In this configuration the viscous sub layer is very thin. Also, the transfer of heat from the nozzle

surface to the fluid flowing is very less due to high velocity. The total temperature is increased

approximately 100k.




In this configuration, at the throat area there is a gradual decrease in pressure which shows that

the velocity is gradually increased in the converging zone. This defines that the fluid is following

the supersonic flow properties. There is also a sudden decrement in pressure which shows an

expansion .The total decrement in pressure is approximately 57%.

Total Enthalpy contour and plot



In this configuration there is a sudden decrement in enthalpy at x = 2.25m as compared to

gradual decrement of enthalpy at the throat area.


In this configuration the length of divergent part is decreased to 1.8m. It is decreased from 2.02m

which is in standard nozzle. Apart from this all the other dimensions are same.




In this configuration there is a sudden increment of velocity at the throat and approximately 33%

increment overall. Now the sudden increase point has been shifted from x = 2.01m to x = 1.85 m

as the divergent part length is decreased.




Here there is a sudden increment in temperature at the throat as the heat transfer is more at the

throat region. The overall increment in temperature is 50k.




In this configuration the pressure is gradually decreased at the throat region and there is sudden

drop in total pressure at x = 1.8m where there is a sudden increment in velocity graph. The

overall pressure decrement is approximately 62.5%.

Total enthalpy contour and plot



Here there is a gradual decrement in enthalpy similarly to the pressure. At the throat region the

enthalpy is gradually decreased and a sudden decrement at x = 1.8m. The decrement is due to the

drastic expansion fan formation which increases the velocity and decreases the pressure and

enthalpy.

8. CONCLUSION

Configuration 1

As the inlet area increases the increment in velocity at throat is less as compared to original

configuration. In the divergent part of the nozzle the abrupt increment is more as compared to

original configuration but the velocity at the exit is less as compared to the original. The viscous

heating at throat is less as compared to the original configuration. The decrement in pressure in

convergent part is gradual as compared to the original configuration. Overall change in enthalpy

is same but in divergent section more energy is converted into work as compared to original

nozzle.

Configuration 2

There is not much change in velocity but the velocity at the exit is slightly less than the original

configuration. The viscous heating is less as compared to the original nozzle. There is not much

change in the pressure and enthalpy curves.

Configuration 3

When the throat area is decreased the change in velocity at throat is more as compared to the

original configuration. Viscous heating after throat is more as compared to the original

configuration. The decrement in pressure in converging section is more abrupt near the throat as

compared to the original configuration. There is sudden decrement of enthalpy at throat i.e. more

work can be drawn from energy at throat.

Configuration 4

On increment in exit area the change in velocity at throat is more as compared to original

configuration. The viscous heating at the walls near the exit is more as compared to the original

configuration. There is more pressure decrement at throat as compared to the original

configuration and there is less efficiency in terms of pressure. Enthalpy is more or less same as

the pressure.

Configuration 5

On decreasing the divergent area the increment in velocity is comparatively less as compared to

the original nozzle. At throat there is less heat transfer & less viscous heating. Pressure is gradual

walls of divergent part and also there is less decrement in convergent part. Near the exit area

more energy is converted into work.

Configuration 6

All the other parameters are almost same as the original configuration but there is less viscous

heating and the velocity at the exit point is less as compared to the original nozzle.

By studying all the configurations on the basis of velocity and temperature contours & their

respective plots the original configuration is appropriate and better than other configurations.

This can be concluded on the basis that where there is less viscous heating the velocity at the exit

is lesser than the original configuration and where the exit velocity is equal or more as compared

to the original nozzle the viscous heating is also more. For optimum performance the viscous

heating should be less and the velocity change from inlet to outlet should be more. The original

configuration is having maximum difference in velocities from inlet to outlet with lesser viscous

heating effects.

REFERENCES

Lucy Rogers, 2008: Its Only Rocket Science

Martin J. L. Turner, 2009: Rocket and Spacecraft Propulsion Third Edition,

Dr. K. M. Pandey and S.K.Yadav, International journal of chemical engineering and

applications,Vol.1, No.4, December 2010: CFD Analysis of Rocket Nozzle with four inlets.

Url-1 <http://en.wikipedia.org/wiki/file:viking5C.jpg>, accessed on 10.04.2011.

Url-2 <http://nasa.gov/rocketnozzle.jpg>, accessed on 10.04.2011

Url-3 <http://google.com/images/jetnozzle.jpg>, accessed on 10.04.2011

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CFD Rocket Nozzle