 Open Access
 Authors : S. Asha , G. Dhathri Naga Mohana , K. Sai. Priyanka , D. Govardhan
 Paper ID : IJERTV10IS050268
 Volume & Issue : Volume 10, Issue 05 (May 2021)
 Published (First Online): 28052021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Design of Minimum Length Nozzle Using Method of Characteristics
S. Asha1, G. Dhathri Naga Mohana2, K. Sai Priyanka3, D. Govardhan4
Research student1, Research student2, Assistant professor3, Head of the department4, Department of Aeronautical Engineering, Institute of Aeronautical Engineering, Telangana, India.
Abstract: The nozzle efficiency is largely affected by the nozzle contour. As a result, an optimum geometrical design of a solid rocket motor nozzle is designed in order to achieve maximum thrust and velocity. Two types of nozzle exit configurations are wellthoughtout in the design process, conical and contoured. A conical nozzle is used for its simplicity of production, but a contoured nozzle is preferred because it increases performance while reducing weight, a contour nozzle also adjusts the flow such that the exhaust product is in a less or more axial direction, reducing divergence losses. In this case, we'll look at a contourshaped nozzle that can optimise thrust as needed. Process of Designing a contour nozzle (minimum length nozzle) utilising method of characteristics (MOC) is carried out MOC is preferred owing to its widespread usage in the industry.
Keyword: Solid rocket motor, Nozzle, Trust, contour nozzle, minimum length nozzle, Method of characteristics.

INTRODUCTION TO ROCKET PROPULSION
Introduction
Rocket propulsion systems are classified in a variety of ways, including chemical, nuclear, and solar energy sources, as well as fundamental functions such as booster stages, sustained or upper stages, attitude control, orbit station maintaining, and the sort of vehicle they drive (aircraft, missile, assisted take off, space vehicle, etc.) or according to their size, propellant type, structure, or the number of rocket propulsion units utilised in a specific vehicle, as well as the thrust generation mechanism. The kind of propellant utilised in a rocket can be used to classify the propulsion system. Liquid and solid propellants are the two most common kinds.
Solid propellant rocket motor
Solid propellants (fuel/oxidizer) have been utilised in military applications such as missiles because they can be stored for a long period without considerable propellant deterioration and because they can be launched reliably most of the time. The grain is a solid propellant (or charge) that includes all of the chemical components required for full combustion. It is designed to burn smoothly and at a predefined
pace on all exposed internal grain surfaces once ignited. As the propellant is burnt and consumed, the inside cavity increases. The heated gases that arise travel through the supersonic nozzle, providing thrust. There are no feed systems or valves, and there are no or very few moving components throughout the system.

NOZZLE
The exhaust gas is accelerated out of the nozzle to create thrust in a divergent diverging nozzle, also known as a DE LAVAL nozzle. The heat generated in the combustion chamber of a solid rocket engine is generally converted into kinetic energy in the exhaust using a convergent divergent nozzle. Newton's 3rd law of motion describes how a rocket engine employs a nozzle to accelerate heated exhaust. The mass flow rate through the engine, the exit velocity of the flow, and the pressure at the engine's exit determine the amount of thrust generated. The rocket nozzle design determines the value of these three flow variables. The fundamental thermodynamic principles may be used to estimate throat area, nozzle half angle, and expansion ratio for simple nozzles with noncontoured conical outputs.
To decrease divergence loss, enhance particular impulse somewhat, and minimise nozzle length and mass, a more sophisticated contoured (or bellshaped) nozzle is utilised.
Minimum Length Supersonic Nozzle
Supersonic nozzles are used to increase a flow to desired supersonic levels in a number of technical applications. The use of supersonic nozzles is possible. There are two categories of nozzles: progressiveexpansion nozzles and nozzles with a minimum length. Gradualexpansion nozzles are widely employed in situations where maintaining a high quality flow at the appropriate exit conditions is critical. For other applications, the huge weight and length penalties associated with gradual expansion nozzles make them impractical, hence minimumlength nozzles with a sharp corner to give the initial expansion are usually utilised. The flow may be separated into simple and nonsimple areas for both gradualexpansion and minimumlength nozzles. Mach wave reflections and intersects describe a nonsimple area. It is preferable to limit the nonsimple zone as much as feasible in order to achieve the criterion of consistent conditions at the nozzle outlet. This can be accomplished by designing the nozzle surface so that no Mach waves (e.g., characteristics) are generated or reflected while the flow is straightened. As a result, the Method of Characteristics is used to build a supersonic nozzle that fits these parameters. The design of both a gradualexpansion and a minimum length nozzle is illustrated in this paper.

NOZZLE DESIGN
For designing a profile of a nozzle Different methods have been proposed, Method of characteristics and G.V.R. Rao approximation method. Method of characteristics is the
method thats discussed below for designing a minimum length nozzle.
General Theory
The nonlinear differential equation of the velocity potential can be used to explain the physical circumstances of a two dimensional, steady, isentropic, irrotational flow. The technique of characteristics is a mathematical framework that may be used to discover solutions to the aforementioned velocity potential while meeting certain boundary conditions, resulting in the governing partial differential
Equation 3.2 shows the velocity potential by definition, and then equations 3.3 and 3.4 are derived.
(3.2)
(3.3)
(3.4)
equations (PDEs) becoming ordinary differential equations (ODEs).
Method of Characteristics
In a supersonic flow, characteristics are lines' oriented in specified directions along which disturbances (pressure waves) propagate. The Method of Characteristics (MOC) is a numerical approach for addressing twodimensional compressible flow problems, among other things. Flow parameters such as direction and velocity may be estimated at different sites across a flow field using this method.
The following are the three qualities of characteristics:
Property 1: A curve or line in a twodimensional supersonic flow along which physical disturbances propagate at the local speed of sound relative to the gas is a characteristic.
Property 2: A characteristic is a curve along which the flow qualities are continuous, even though the initial derivatives are discontinuous, and the derivatives are indefinite.
Property 3: A characteristic is a curve that may be used to convert the controlling partial differential equations into an ordinary differential equation (s).
Minimum Length Nozzle (MLN) using Method of Characteristics (MOC)
Characteristic Lines
TwoDimensional Irrotational Flow For steady, two dimensional, irrotational flow, Equation 3.1, which is the whole velocity potential
(3.1)
equation, is used to determine the characteristic lines. Note that is the velocity potential.
Equations 3.6, 3.7, and 3.8 are produced by substituting the equation 3.2 into equations 3.1, 3.3, and 3.4.
(3.6)
(3.7)
(3.8)
Equations 3.6 over 3.8 are a system of snchronized, linear, algebraic equations in the variables xx, yy, and xy. By applying Cramers rule, the explanation for xy is to be the following equation 3.9.
(3.9)
A point A and its surrounding neighbourhood in an arbitrary flow field are represented in Figure 3.1. The velocity potential's derivative,xy, has a specific value at point A. The solution for xy at point A for an arbitrary dx and dy option, and dy for an arbitrary direction away from point A specified by the dx and dy option. There are equivalent values of the change in velocity du and dv for the given dx and dy.
Figure 3.1 streamline geometry
The slope of the characteristic lines is shown below in equation 3.10 and equation 3.11.
(3.10)
(3.11)
There are three key points to remember:

When M > 1, each point of the flow field has two actual characteristics. Furthermore, equation 3.1 is a hyperbolic partial differential equation in this case.

If M = 1, each point of the flow has just one actual feature. A parabolic partial differential equation is Equation 3.1.

If M is less than one, the properties are imaginary, and equation 3.1 is an elliptic partial differential equation.
Because each point in a flow has two real characteristics, the method of characteristics (MOC) becomes a suitable strategy for solving supersonic flow. The equation 3.10 for a steady, twodimensional supersonic flow is investigated. Study the streamline as shown in Figure 3.1. At point A, u = Vcos and v = Vsin Equation 3.10 develops equation 3.12 shown below.
(3.11)
Since, sin (1/ M), equation 3.12 is obtained as follows.
(3.12)
Therefore, the following equation 3.13 is obtained.
(3.13)
The slope of the characteristic lines is given by equation 3.14, which is derived from trigonometry and algebra.
(3.14)
Figure 3.2 depicts a graphical representation of equation
3.14. The streamline establishes an angle with the x axis at point A. There are two characteristics that passthrough point A, one at an angle above the streamline and the other at an angle below it.
Mach lines are the distinguishing lines. The angle + characteristic is referred to as a C+ characteristic, and it is a leftrunning characteristic. A C characteristic is a right running characteristic that is given by the angle – Because the flow parameters change from point to point in the flow, the characteristics are curved.
Figure 3.2 left and right characteristic lines
Compatibility Equations
The compatibility equation is the following equation 3.15
(3.15)
To maintain the flow field derivatives small, N is zero only when D is zero, according to equation 3.9. When D = 0, only directions along the characteristic lines are examined, and when N = 0, only directions along the characteristic lines are examined. As a result, equation 3.15 is only valid along characteristic lines. As a result, equation 3.16 is as follows:
(3.16)
The compatibility equation therefore is shown in equation
3.18 below
(3.18)
The equation's negative version is applied to the C characteristic, whereas the positive form is applied to the C+ characteristic. For PrandtlMeyer flow, equation 3.18 is equal to equation 3.19 given below.
(3.19)
Equation 3.19 is substituted by algebraic compatibility equations 3.20 and 3.21 since the equations are equivalent
(3.20)
(3.21)
Compatibility Equations Point by Point Along the Characteristics
Internal Flow
If the flow field conditions are known at two points in the flow, the conditions at a third point can be found. The third point is located by the intersection of the C characteristic through the first point and the C+ characteristic through the second point, as shown in Figure 3.3 below.
Figure 3.3unit process for MOC
The and for the third point are found in terms of the known values of K+ and K as shown in equation 3.22 and equation 3.23 below
(3.22)
(3.23)
As a result of the known values at the first and second points, the flow conditions at the third point may now be calculated. Equation 3.24 is used to get the Mach number.
(3.24)
After calculating the Mach number, the pressure, temperature, and density may be determined using isentropic flow relations, as illustrated in the equations below.
Assume that the characteristics are straightline segments connecting the grid points, with average slopes. C characteristic across the first point is drawn as a straight line with an average slope angle. As indicated in the equation below, the C+ characteristic across the second point is drawn as a straight line with an average slope angle.
Supersonic Nozzle Design
In order to increase the speed of an internal steady flow via a duct from subsonic to supersonic, the duct must be convergentdivergent in form, as shown in Figure 3.4.
Assume that the sonic line is perfectly straight. In the throat, the flow increases to sonic speed. The duct splits downstream of the sonic line. The expansion section in Figure 3.5 is shortened to a point in minimal length nozzles, and the expansion occurs by a centred PrandtlMeyer wave emerging from a sharpcentre throat with an angle wmax,
ML, as shown in Figure 3.5. The length of the supersonic nozzle, L, is the smallest value that can be achieved while maintaining shockfree, isentropic flow.
Assume that the exit Mach numbers of the nozzles in Figures
3.4 and 3.5 are the same. The expansion contour of the minimumlength nozzle depicted in Figure 3.5 exhibits a sharp corner at point a. Only two wave systems meet the fluid: rightrunning waves from point a and leftrunning waves from point d. In equation 3.25, the wall's expansion angle downstream of the neck is illustrated.
Figure 3.4 schematic of supersonic nozzle design by MOC.
Figure 3.5 schematic of minimum length nozzle
(3.25)
It is point e when the centreline Mach number matches the design exit Mach number. The expansion section comes to an end at point c, which fixes its length as well as the value of wmax. The number of nodes is calculated using the formula below.
Figure 3.6 Schematic of characteristic lines for minimum length nozzle


RESULTS AND DISCUSSION
By compressing the expansion section, the design of a minimal length supersonic nozzle is capable of constructing a minimal length nozzle.
The entire length of the nozzle is reduced when the expansion portion is contracted. The length of the supersonic nozzle is kept to a minimum in the preceding design because the expansion section is kept to a minimum.
In actuality, the expansion part has been compressed to point at the throat's end. using the PrandtlMeyer equation the characteristic lines are solved in MATLAB all the points are plotted against nozzle axis in which different characteristic lines are seen
Figure. 4.1 plot of the contour curve with the characteristic lines in MATLAB
Figure 4.2 plot of wall points
Figure 4.3 points reflected on the wall
These points are later used for plotting a 2D domain FOR CFD analysis You must also choose an optimal number of features, which should be large enough to form a curved (bellshaped) contour towards the end of the process. Make sure your lines are straight. MOC's overall efficacy may be measured.

CONCLUSION
There are a variety of uses for supersonic nozzles. Normally, they are exposed to a complicated flow pattern. To achieve the high precision and huge calculations required by current highspeed applications, a computer is required. As a result, a computerised approximation method could be a preferable way to address such an issue. The most acceptable approach to utilise with the supersonic nozzle design is the characteristic method.
By doing several iterations, we can improve our results. Slight modfications in the input parameters algorithm We were able to effectively use the method of characteristics to produce a nozzle shape for a nozzle with a minimum length. The nozzle points acquired may be imported into any CAD programme for further refining and production. To achieve the greatest outcomes in line with the input specification, it is strongly suggested to combine this strategy with an iterative design approach.

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