MODELING & SIMULATION OF ROCKET NOZZLE

MODELING & SIMULATION OF ROCKET NOZZLE Nirmith Kumar Mishra, Dr S Srinivas Prasad, Mr Ayub Padania Department of Aerospace Engineering MLR Institute of Technology Hyderabad, T.S Abstract This project develops a computer code which uses the Method of Characteristics and the Stream Function to define high efficiency nozzle contours for isentropic, inviscid, irrotational supersonic flows of any working fluid for any user-defined exit Mach number. The contours are compared to theoretical isentropic area ratios for the selected fluid and desired exit Mach number. The accuracy of the nozzle to produce the desired exit Mach number is also checked. The flow field of the nozzles created by the code are independently checked with the commercial Computational Fluid Dynamics (CFD) code ANSYS-FLUENT. ANSYSFLUENT predictions are used to verify the isentropic flow assumption and that the working fluid reached the user-defined desired exit Mach number. Key Words: Method of characteristics, Supersonic nozzle, Area ratio relation, Prandtl- Meyer expansion wave 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 (equivalently, the stagnation pressure of the jet must be above ambient). 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. In practice ambient pressure must be no higher than roughly 2-3 times the pressure in the supersonic gas at the exit for supersonic flow to leave the nozzle. 1. Introduction A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di 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 turbine, it is an essential part of the modern rocket engine, and it also sees use in supersonic jet engines. The nozzle was developed by Swedish inventor Gustaf de Laval in 1888 for use on a steam turbine. This principle was first used in a rocket engine by Robert Goddard. Very nearly all modern rocket engines that employ hot gas combustion use de Laval nozzles. A de Laval nozzle will only choke at the throat if the pressure and mass flow through the nozzle is Fig 1.1 Flow through C-D Nozzle The analysis of gas flow through de Laval nozzles involves a number of concepts and assumptions: 988 www.ijaegt.com

For simplicity, the gas is assumed to be an ideal gas. The gas flow is isentropic (i.e., at constant entropy). As a result the flow is reversible (frictionless and no dissipative losses), and adiabatic (i.e., there is no heat gained or lost). The gas flow is constant (i.e., steady) during the period of the propellant burn. The gas flow is along a straight line from gas inlet to exhaust gas exit (i.e., along the nozzle's axis of symmetry) The gas flow behavior is compressible since the flow is at very high velocities. 1.1 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. 2. Methodology 2.1 AREA MACH RELATION: For a subsonic flow (0 <= M < 1), as the area increases the velocity decreases and as the area decreases the velocity increases. For a supersonic flow (M > 1), an increase in velocity is associated with an increase in area and a decrease in velocity is associated with a decrease in area. For a sonic flow (M = 1) da=0. This corresponds to a local maxima or minima in the area distribution. Physically it corresponds to minimum area. da A = (1 M2 ) du U -------- 2.1 There it becomes clear that if one wants to accelerate a gas at rest to supersonic speeds, it must first be accelerated sub sonically in a convergent duct. The minimum area of the duct is called the throat. As soon as the sonic conditions are achieved at the throat, it must further be expanded to supersonic speeds in a divergent duct. The vice versa is also actually true. If one wants to decelerate a supersonic flow to subsonic speeds, it must first be decelerated to a sonic speed at the throat by means of a convergent duct and then be further decelerated in a divergent duct. Let us consider a duct as shown in the figure below. The Mach number, area and velocity at different stations in the duct are shown in the figure below 1.3 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 kilometre 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. Fig2.1 Geometry for Derivation of Area Mach Relation Here we assume that sonic flow exists at the throat of the nozzle. Hence all the parameters at the throat have an asterisk along with the notation to denote the sonic conditions. Applying the continuity equation to the above case 989 www.ijaegt.com

ρ u A = ρua--------- 2.2 A = ρ A ρ a = ρ ρ 0 u ρ 0 ρ a -------2.3 u Where ρ 0 is the stagnation density and is constant throughout the flow. We have ρ ρ 0 = 2 γ+1 1/(γ 1) ------2.4 ρ 0 ρ = 1 + γ 1 2 M2 1/(γ 1) ------ 2.5 u a 2 = M 2 = [(γ+1)/2]m2 -----2.6 M2 1+ γ 1 2 By performing certain mathematical manipulations we get A A 2 = ρ ρ 0 2 ρ 0 ρ 2 a u 2 ------ 2.7 A A 2 = 1 2 γ 1 1 + M 2 γ+1 2 M2 (γ+1) (γ 1) ----2.8 The above equation is called the area-mach number relation the equation tells us that the Mach number at any location in the duct is a function of the local throat area to the sonic throat area. Also the equation yields two values for M for a given area ratio, a subsonic value and a supersonic value. 2.2 Method Of Characteristics The physical conditions of a two-dimensional, steady, isentropic, irrotational flow can be expressed mathematically by the nonlinear differential equation of the velocity potential. The method of characteristics is a mathematical formulation that can be used to find solutions to the aforementioned velocity potential, satisfying given boundary conditions for which the governing partial differential equations (PDEs) become ordinary differential equations (ODEs). The latter only holds true along a special set of curves known as characteristic curves, which will be discussed in the next section. As a consequence of the special properties of the characteristic curves, the original problem of finding a solution to the velocity potential is replaced by the problem of constructing these characteristic curves in the physical plane. The method is founded on the fact that changes in fluid properties in supersonic flows occur across these characteristics, and are brought about by pressure waves propagating along the Mach lines of the flow, which are inclined at the Mach angle to the local velocity vector. The method of characteristics was first applied to supersonic flows by prandtl and Busemann in 1929 and has been much used since. This method supersonic nozzle design made the technique more accessible to engineers. In supersonic nozzle design the conventional twodimensional nozzle is usually considered to consist of several regions as shown in the figure, these are:- Contraction part, where the flow is entirely subsonic the throat region, where the flow accelerates from high subsonic to low subsonic speeds. The initial expansion region, where the slope of the counter increases up to its maximum value the straightening, or busemann region in which the processor area increases but the wall slope decreases to zero. The test section where the flow is uniform and parallel to the axis. 2.3 Characteristics Characteristics are unique in that the derivatives of the flow properties become unbounded along them. On all other curves, the derivatives are finite. Characteristics are defined by three properties as detailed by John and Keith A characteristic in a two-dimensional supersonic flow is a line along which physical disturbances are propagated at the local speed of sound relative to the gas. A characteristics is a cut across which flow properties are continuous, although they may have discontinuous first derivatives, and along which the derivatives are indeterminate. A characteristic is a cut along which the governing partial differential equations may be manipulated into an ordinary differential equations. Fluid particles travel along our clients propagating information regarding the condition of the flow. In supersonic flow, the cost equates travel along Mach lines propagating information regarding flow disturbances. this is described in the first property. The second property says that Macklin can be considered as an infinitesimally thin interface between two smooth and uniform, but different regions. The line is a boundary between continuous flows along the streamline passing through a field of these Mach waves, the derivative of the velocity and other properties may be discontinuous. The third property speaks for itself. ordinary differential equations are often easier to solve than partial differential equations. That is why this property is considered very important. Fig 2.2 Characteristics 990 www.ijaegt.com

2.4 Discretation of Equations, Boundary Condition and Stream Function Analysis. Discretizing the Characteristic and Compatibility Equations To implement the characteristic and compatibility equations into a computer code for designing supersonic nozzle contours, the equations for axisymmetric, irrotational, inviscid flow developed in Appendix A must be discretized with boundary conditions defined and applied. The first step in designing a computer code is to discretize the characteristic and compatibility equations. They are rewritten below dr = tan(θ α) ------------ 2.9 dx char 1 dr d(θ + α) = ------------- 2.10(a) M 2 1 cot θ r calculate the nozzle contours. The list of variables required is described in Table 2.1 with description below. The program then passes the necessary input variables to the subroutines that need them. All input variables are passed to subroutines. The subroutines calculate the contour of an nozzle, respectively, as well as their truncated versions if applicable. Axisymmetric, the subroutine that calculates the annular nozzle contour only requires input variables Beta, DeltaVAeroD, Gamma and M exit. A fourth subroutine, PMtoMA, is used in calculating the Mach numbers of the points in the flow field and will be discussed last. Once all subroutines return their solutions, subroutine Supersonic Nozzle plots their nozzle contours. d(θ α) = (along C Characteristic) 1 dr ------------- 2.10 (b) M 2 1+cot θ r (along C + Characteristic) dr = tan(θ α) ------------ 2.10 (a) dx C dr = tan(θ + α) ------------ 2.10 (b) dx C + Using the Forward Difference Technique and rearranging equations 2.10a and b yields r i+1 tan(θ i α i ). x i+1 = r j tan(θ i α i ). x i -- -------- 2.11 (a) (along C Characteristic) r i+1 tan(θ i + α i ). x i+1 = r j+1 tan(θ i + α i ). x i+1 -------2.11 (b) (along C + Characteristic) Note that all variables with subscript i are known quantities and variables with subscript i+1 are unknown quantities, the discretized characteristic equations that will define the location in the x-r space where the C- and C+ characteristics curves intersect. This collection of points is called the Characteristic Net. 2.5 Computer Program Calculation Details The supersonic nozzle discussed above is combined into one program that will calculate the nozzles' contour using the Method of Characteristics and the Stream Function. A brief description of the subroutines developed for this paper and their associated flow charts are included below. The complete set Matlab source code and program flowcharts are available in Appendix B. Since nozzle type is based on isentropic relations, the codes' error can be directly quantified using isentropic area ratios for given desired exit Mach number and ratio of specific heats. The program begins by asking the user for all necessary design variables the program will need to Program Variable Beta DeltaVAeroD Gamma M exit Percent Table 2.1 Description The throat multiplier that will be used to calculate the radius of the arc used in the expansion region for annular and internal-external aerospike nozzles The desired incremental step size of the Prandtl-Meyer expansion angle used in the calculation. It is also used as the x-space step direction for determining the x- component of the starting point for the "backward" C- characteristic Ratio of Specific Heats of the working fluid Desired Exit Mach Number The % of the ideal length the user would like in the event they choose to calculate truncated versions of the aerospike nozzles 991 www.ijaegt.com

3. CFD setup Fig: 2.3The divergent curve obtained by running code in Matlab x_cord y_cord 0.049979 1.00125 0.099833 1.004996 0.149438 1.011229 0.198669 1.019933 0.247404 1.031088 0.29552 1.044664 0.342898 1.060627 2.987181 1.905588 4.797082 2.379298 6.954327 2.830017 9.700147 3.26939 13.44639 3.695064 18.73308 4.056261 26.82458 42104 Table 2.2 Coordinates obtained by running the code 6 4 2 0 y_cord 0 10 20 30 Fig 2.4 The graph obtained from table 2.2 y_cord 3.1 Complex Chemical Equilibrium Composition and Application Program The Complex Chemical Equilibrium Composition and Application (CEA) Program developed by NASA uses the minimization of Gibb's Free Energy to predict the composition of the exhaust products of a combustion system. In doing so, the properties of the exhaust fluid are predicted using mass averaging of the species produced by the combustion system. The CEA program has multiple subroutines to choose from for different combustion systems. Since we are analyzing rocket nozzles, the rocket subroutine was chosen to predict the exhaust properties. Within the rocket subroutine, the finite area combustion chamber was utilized because the test chamber of the test apparatus is small with an interior radius of 1.25 inches. To complete the simulation, the pressure at the injector, chamber to throat area ratio, oxidizer and fuel chemical formulas and amounts with respect to the desired oxidizer to fuel ratio must all be entered. Using conditions from a previous single firing of the test apparatus, the CEA program was used to predict the ratio of specific heats, chamber pressure and temperature for the exhaust fluid. The results from the CEA program give information for three planes in the apparatus, at the injector, at the end of the combustion chamber and at the throat of the nozzle. The ratio of specific heats predicted at the throat is used as the input for the supersonic nozzle program discussed in Section 3.0. The chamber pressure and temperature are used as boundary conditions in the CFD simulations discussed in the next few subsections. Table 3.1 gives the inputted data used for the CEA simulation. All nozzles designed were assumed to have the same combustion system and working fluid. Table 3.1 CEA Program Inputs Subroutine Rocket Combustion Finite Area Chamber: Chamber to 44.44 Throat Area Ratio: Initial Pressure: 360 pisa Combustion 3800 K Temperature (estimate) Reactants Found N2O (Nitrous in the Oxide) Thermodynamic Library: Reactants with C 224 H 155 O 27 N User-Provided (Papi 94) Amount: 320 kg Amount: 12 kg 992 www.ijaegt.com

Names and Properties: C 667 H 999 O 5 (HTPB) Amount: 88 kg 3.2 Annular Nozzle CFD Simulation In order to run a simulation of the flow in supersonic annular nozzles, the nozzles must be built virtually so that a mesh can be generated in the fluid region. The supersonic nozzle program described in section 2 produces a set of points which define the nozzle's contour. These points are imported into Ansys. A mesh generating program used to mesh the fluid domain of the simulation. It is important to note that the points generated by the supersonic nozzle program only yields points of the wall contour after the throat. Since the fluid experiences few losses in the convergent section of a supersonic nozzle, the user can design the convergent section of the nozzle given the known geometry of the combustion chamber. All points are connected to produce a 2D axisymmetric virtual geometry. Figure 3,1 below shows the typical geometry and boundary conditions used to simulate an annular nozzle. Since the nozzle contours were built on the assumption of inviscid, irrotational, isentropic flow, the CFD simulations need to reflect this. The inviscid assumption was satisfied by selecting the inviscid model for the simulations. The isentropic assumption, which implies irrotationality, was achieved by assigning the specific heat at constant pressure as a constant property of the working fluid. 4. Results & Discussion 4.1 Theoretical Accuracy of Computer Code The first check of accuracy for the program was comparing the desired exit Mach number with the exit Mach number calculated by the program. Table 4.1 below shows the percent difference between the desired and computer calculated exit Mach numbers. Table 4.1 also shows how the code becomes more accurate as a smaller change in Prandtl-Meyer expansion angle is used during calculations. Since the equations were based on isentropic flow theory, the accuracy of the code was also checked by calculating the exit to throat area ratio using equation 4.1 substituting in the user- defined ratio of specific heats and computer calculated exit Mach number. This yields the theoretical area ratio for the Mach number actually calculated by the program. A A 2 = 1 2 γ 1 1 + M 2 γ+1 2 M2 (γ+1) (γ 1) ---- 4.1 The theoretical and computer calculated isentropic area ratios for the desired exit Mach number were also compared for a user-defined ratio of specific heats in Table 4.1 Fig: 3.1 Typical Annular Nozzle CFD boundary conditions Once the geometry of the nozzle has been virtually created, the fluid region can be meshed. Fig 3.2 gives a typical the meshed geometry of an annular nozzle. Fig 4.1 Typical entropy contour for annular nozzle Fig :3.2 Typical Annular Mesh it can be seen in fig 4.1 that the large majority of the fluid domain demonstrates constant entropy signifying that the isentropic flow assumption is valid. The region near the wall contour where the entropy is changing is a result of the discontinuities in the wall contour. Since the wall contour was defined by a set of points that were connected by straight line segments, it is discontinuous at the points that connect them. The change in entropy in 993 www.ijaegt.com

the flow field is a propagation of these discontinuities. Table 4.1 Code Accuracy for γ = 1. 4 M a = 3.0 β = 1. 0 A exit Com r throat =1.0 (Dimensionless) v = 0.05 5.6 588 A exit theo A exit 33. %er 63 % M a Comp 3.2 489 M a %error 8.3 0% A exit 5.3 Com 635 A exit 5.5 Com 1% v = 0.025 5.5 827 31. 84 % 3.2 455 8.1 8% 5.3 463 4.4 2% v = 0.01 4.4 882 5,9 9% 3.0 289 0.9 6% 4.3 527 3.1 1% v = 0.005 4.4 938 6.1 2% 3.0 323 1.0 8% 4.3 668 2.9 1% v v = = 0.0025 0.001 4.3 4.3 940 3.7 6% 3.0 093 3.0 067 722 2.8 5% 815 3.4 7% 0.3 1% 0.2 2% 617 2.8 1% Fig 4.3 Contours of Dynamic Pressure Fig 4.4 Static Temperature contours The exit Mach number is checked by the Mach contours of the simulation as well as having ANSYS-FLUENT calculate the area-weighted Mach number at the exit plane of the nozzle. The area weighted Mach number calculated by ANSYS-FLUENT is compared to the Mach number calculated by the program in Section 2 and the desired exit Mach number. Figure shows the typical Mach contours of an annular nozzle designed for a Mach number of 3.0 Fig 4.5 Entropy Plot curve Fig 4.6 Mach Plot curve Fig: Contours of Mach Number 994 www.ijaegt.com

5. Conclusions The code developed in this project proves to be a useful tool in creating annular and supersonic nozzle contours for isentropic, irrotational, inviscid flow. The program exhibits increasing accuracy in the exit Mach number and exit area ratio as the incremental Prandtl- Meyer expansion angle decreases. This accuracy increase is independent of fluid or desired exit Mach number. The exit Mach number of the nozzles calculated with the program described in Section 3 shows good agreement with the ANSYS-FLUENT simulated density contour, exit Mach numbers, dynamic pressure, static temperature contour, entropy curve, mach plot curve. The code developed in this project will enable the researchers to investigate other types of rocket nozzles besides conical. It will further advance the researchers in achieving their ultimate goal, designing a supersonic convergent divergent nozzle 6. Scope for Future Work In pursuit of their ultimate goal of developing a low-cost alternative launch platform for satellites, the research continues. A comparison of the theoretical and measured thrusts produced by each nozzle has to be conducted. These tests will validate the accuracy of the program for real-world flows and also give an indication of the energy losses observed in the nozzles. The testing will also enable the researchers to compare the thrust/weight/cost ratios to develop the most cost effective rocket engine. 7. Design of supersonic wind tunnel using Method of characteristics by Mr.Y D Dwivedi, Mr. B.Parvathavadhani. K, Mr.Nirmith Kumar Mishra. International Journal of Futuristic Science Engineering and Technology Vol.1,Issue 04.ISSN 2320 4486 Authors Profile First Author: Nirmith Kumar Mishra received B.Tech Aeronautical Degree from MLR Institute of Technology in 2012.Currently pursuing M.Tech in Aerospace Engineering at MLR Institute of Technology, Dundigal Hyderabad. Research interests are Aerodynamics, Performance, Stability & Control of aircraft. Second Author: Dr. S.Srinivas Prasad working as Professor & Head of Department in Aerospace Engineering at MLR Institute of Technology, Dundigal Hyderabad. Third Author: Mr.Ayub Padania working as Assistant professor in Aerospace Engineering at MLR Institute of Technology, Dundigal Hyderabad References 1. Anderson, JD., 2001, Fundamentals of Aerodynamics, 3rd Edition, pp. 532-537, pp.555-585. 2. Anderson, JD., 1982, Modern Compressible Flow with Historical Perspective, pp. 268-270, pp. 282-286. 3. Shapiro, AH., 1953, The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. I, pp. 294-295. 4. Shapiro, AH., 1954, The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. II, pp. 694-695. 5. Sutton, GP, Rocket Propulsion Elements, 7th Edition 6. Design and Analysis of Rocket Nozzle Contours for Launching Pico-Satellites By Brandon Lee Denton 995 www.ijaegt.com