International Journal of Mechanical Engineering and Technology (IJMET) Volume 7, Issue 5, September–October 2016, pp.221–240, Article ID: IJMET_07_05_024 Available online at http://iaeme.com/Home/issue/IJMET?Volume=7&Issue=5 Journal Impact Factor (2016): 9.2286 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6340 and ISSN Online: 0976-6359 © IAEME Publication
CONCEPTS AND CFD ANALYSIS OF DE-LAVAL NOZZLE
Malay S Patel, Sulochan D Mane and Manikant Raman Mechanical Engineering Department, Dr. D.Y Patil Institute of Engineering and Technology, Pune, India.
ABSTRACT Anozzle is a device designed to control the characteristics of fluid. It is mostly used to increase the velocity of fluid. A typical De-Laval nozzle is a nozzle which has a converging part, throat and diverging part. This paper aims at explaining most of the concepts related to De Laval nozzle. In this paper the principle of working of nozzle is discussed. Theoretical analysis of flow is also done at different points of nozzle. The variation of flow parameters like Pressure, Temperature, Velocity and Density is visualized using Computational Fluid Dynamics. The simulation of shockwave through CFD is also done. Key words: Rocket engine nozzle, De-Laval nozzle, Ideal gas, Computational Fluid dynamics, Ansys-Fluent. Cite this Article: Malay S Patel, Sulochan D Mane and Manikant Raman, Concepts and CFD Analysis of De-Laval Nozzle. International Journal of Mechanical Engineering and Technology, 7(5), 2016, pp. 221–240. http://iaeme.com/Home/issue/IJMET?Volume=7&Issue=5 1. INTRODUCTION Nomenclature: F= Force (N) P=Pressure (Pa) T=Temperature (K) V=Velocity (m/s) C= Velocity of sound (m/s) g=Gravitational acceleration (m/s2) ṁ = Mass flow rate (kg/s) ρ= Density (kg/m3) A=Cross sectional area (m2)
CP=Specific heat at constant pressure (J/kgK) h- Enthalpy (J) = Isentropic index
http://iaeme.com/Home/journal/IJMET 221 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman
R= Specific gas constant (J/kgK) Ma=Mach number Nozzle is a device which is designed to control properties of fluid like pressure, density, temperature and velocity. The major application of nozzle is to increase the velocity of flow by converting pressure and heat into kinetic energy. Mostly in rockets or air breathing engines, it is used to produce thrust to gain lift. Fluid of subsonic velocities can be accelerated to supersonic velocities using a rocket engine nozzle. To design a nozzle the major requirement is the magnitude of thrust produced by the nozzle. The altitude at which it operates and the properties of working fluid which govern the flow are its molecular weight, specific heat at constant pressure or volume and specific heat ratio. 2. LITERATURE REVIEW • “Theoretical and CFD Analysis of De Laval Nozzle” by Nikhil Deshpande and Suyash Vidwans. In this paper total CFD analysis of a model of De Laval nozzle has been done. In this paper comparison is done based on theoretical and CFD values. In our paper we have explained all the basic concepts of De Laval nozzle along with providing the CFD of each and every case(1). • “Modeling and simulation of Supersonic nozzle using computational fluid dynamics, by Venkatesh V and C Jayapal Reddy. In this paper theoretical and CFD analysis of various types of nozzles is given. Also thrust conditions and design parameters are discussed in this nozzle. We have taken diagrams and certain theory relevant to our topic(2).
2.1. Rocket or Air Breathing Nozzle
Figure 1 Thrust Operation The ultimate purpose of nozzle is to expand a gas as efficiently as possible to maximize the exit velocity. This will increase the thrust produced. The thrust produced is given by the equation.
F = ṁ Ve + (Pe − P0)Ae (1) For equation 1and figure 1, [Ref (2)]
2.2. Expansion Ratio The expansion ratio of the nozzle is the ratio of exit area to throat area.
A ε = e (2) A∗ A rocket generally does not travel through a single altitude, thus while designing a path over which a rocket is to travel is considered so that the expansion ratio that maximizes the performance is selected. Other factors like nozzle weight, length, manufacturability, heat transfer and aerodynamic characteristic are also considered, while designing thus altering the shape of nozzle.
http://iaeme.com/Home/journal/IJMET 222 [email protected] Concepts and CFD Analysis of De-Laval Nozzle
3. DE-LAVAL NOZZLE De-Laval nozzle is mainly used to achieve supersonic velocities. De-Laval nozzle or Converging Diverging Nozzle is the most widely used type of nozzle. It is in the shape of a tube pinched at the middle, making a perfectly balanced asymmetric hourglass shape. It is widely used in supersonic jet engines and steam turbines. They also have certain application in jet stream within astrophysics. Converging nozzle has a cross section reducing up to the throat at which the fluid gains maximum velocity. However in a converging nozzle the fluid can be accelerated up to sonic speed [Ma=1]. Thus to increase the velocity of fluid up to supersonic speed [Ma>1] a diverging part is attached to the nozzle. This type of nozzle is known as Converging Diverging Nozzle or De-Laval Nozzle.
3.1. Operation The speed of a gas (subsonic flow) will increase if the pipe carrying it narrows, because the mass flow rate is constant. The gas flowing is assumed to be isentropic. In a subsonic flow the gas is compressible and sound will propagate. At the throat, where cross section area is at its minimum, the gas velocity locally becomes sonic [Ma=1], a condition called chocked flow. As the nozzle cross sectional area increases the gas begins to expand, and the gas flow increase to supersonic velocities, where a sound wave will not propagate backwards through the gas as viewed in the frame of reference of the nozzle [Ma>1] It is not sufficient to force a fluid through nozzle to gain the supersonic velocity. Sometimes, the fluid may decelerate in the diverging section instead of accelerating if the back pressure is not in the range. Thus the state of nozzle flow is determined by the Overall Pressure Ratio
P O. P. R = b (3) P0 Consider the following conditions, (Ref Figure. 2)
1. When P0=Pb
When P0=Pb, There is no pressure difference and thus no flow through nozzle.
2. When P0>Pb>Pc The flow remains subsonic through the nozzle. The fluid velocity increases in converging section and reaches to maximum at the throat, but the velocity is still subsonic at the throat. Thus in the diverging part, fluid loses its energy and diverging part acts as diffuser. The pressure reduces up to throat and again increases in the diverging section.
3. When Pb=Pc The throat pressure becomes P* and fluid gains the sonic velocity at the throat. P* is the minimum pressure at the throat and the velocity obtained is maximum velocity achieved by converging nozzle. Further reduction does not influence flow through the converging section, but influences flow through diverging section.
4. When PC>Pb>Pe The fluid that achieves the sonic velocity continues to accelerate to supersonic velocities in the diverging section. As pressure decreases, acceleration comes to sudden stop. Normal shocks are developed at the section between throat and exit plane and thus there is sudden drop in the velocity and flow gets decelerated. Flow through shock is highly irreversible and cannot be approximated as isentropic flow.
When Pb=Pe shockwave forms at exit plane. Thus the supersonic flow through the nozzle becomes subsonic before leaving the nozzle as it crosses the normal shock.
5. When Pe>Pb>0
The flow in diverging section is supersonic and fluid expands to PF at the nozzle exit with no normal shock. Thus analysis of flow can be done as isentropic flow. When Pb=PF however, the pressure inside the nozzle increases from PF to Pb irreversibly in the wake of nozzle creating oblique shocks. The behavior of
http://iaeme.com/Home/journal/IJMET 223 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman fluid expansion process is governed by exhaust pressure as well as pressure of external environment into which it exhausts. For minimum conversion of thermal energy into thrust, the exit pressure will be equal to the ambient pressure.
Where Pb is back pressure, Pe is exit pressure. PA , PB, PC are pressures at points A, B and C.
Figure 2 Pressure conditions
3.2. Expansion Analysis of Nozzle There are four cases to be taken into consideration (Ref Figure 3):
Pe is exit pressure and Pamb is atmospheric pressure.
1. Pe 2. Pe=Pamb In this situation, the flow is perfectly expanded and thus it provides maximum efficiency. 3. Pamb 4. Pe< http://iaeme.com/Home/journal/IJMET 224 [email protected] Concepts and CFD Analysis of De-Laval Nozzle This ambient pressure is too large compared to the exit pressure. Hence it is given the name grossly expanded nozzle. Figure 3 Expansion analysis Some typical values of exhaust gas velocity Ve for rocket engines burning various propellants using De Laval nozzle are: • 1700 to 2900 m/s (3800 to 6500 mph) for liquid monopropellants. • 2900 to 4500 m/s (6500 to 10,100 mph) for liquid bipropellants. • 2100 to 3200 m/s (4700 to 7200 mph) for solid propellants. 3.3. Equations and Formulae for De Laval Nozzle Basic equations: The continuity equation is given by, 휌1 퐴1푉1 = 휌2퐴2푉2(3) The ideal gas equation is given by, P=ρRT (4) For an ideal gas speed of sound is given by, 푐 = √푘푅푇 (5) Mach number is defined as ratio of speed of flow to the speed of sound, 푉 푀푎 = (6) 푐 Stagnation Properties: During a stagnation process, the kinetic energy of fluid is converted to enthalpy which results in an increase in fluid temperature and pressure. The properties of fluid at stagnation state are called stagnation properties. T0 is stagnation temperature and P0 is stagnation pressure.T0 is temperature an ideal gas attains when it is brought to rest adiabatically.ρ0 is stagnation density. 푉2 푇0 = 푇 + (7) 2퐶푝 P0 is pressure it attains when fluid is brought to rest adiabatically. http://iaeme.com/Home/journal/IJMET 225 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman 푘 푃0 푇0 = ( )푘−1 (8) 푃 푇 1 휌0 푇0 = ( )퐾−1 (9) 휌 푇 Nozzle equations: The Mach number variation is given by, 푘+1 퐴 1 2 푘−1 = (( ) (1 + ( ) 푀푎2))2(푘−1) (10) 퐴∗ 푀푎 푘+1 2 퐴∗is area of nozzle at the throat. The pressure variation is given by, 푘 푃0 퐾−1 = (1 + ( ) 푀푎2)푘−1 (11) 푃 2 The temperature variation is given by, 푇 푘−1 0 = (1 + ( ) 푀푎2) (12) 푇 2 The density variation is given by, 1 휌0 푘−1 = (1 + ( )푀푎2)푘−1 (13) 휌 2 The mass flow rate is given by, 푘 푚̇ = 푃퐴푀푎√ (14) 푅푇 3.4. Shockwave For some back pressure values, abrupt changes in fluid properties occur in a very thin section of a De Laval nozzle under supersonic flow conditions, creating a shockwave. 3.4.1. Normal shocks Shockwaves that occur in a plane normal to the direction of flow are normal shocks. The flow is supersonic before the shock and subsonic afterwards. Therefore the flow must change from supersonic to subsonic if a shockwave is to occur. The larger the Mach number before the shock, the stronger the shock will be. In the limiting case of Ma=1, the shock wave simply becomes a sound wave. Figure 4 Shockwave http://iaeme.com/Home/journal/IJMET 226 [email protected] Concepts and CFD Analysis of De-Laval Nozzle According to conservation of energy principle, the stagnation enthalpy remains constant across the shock; h01=h02. The stagnation temperature is also constant T01=T02. The stagnation pressure decreases across the shock, because of irreversibility, while the ordinary temperature rises drastically because of the conversion of kinetic energy into enthalpy due to a large drop in fluid velocity. Figure 5 Variation of parameters in shockwave The equations between various properties before and after the shock for an ideal gas are as follows: The temperature ratio is given by 푘−1 1+( )푀푎2 푇2 2 1 = ( 푘−1 ) (15) 푇1 1+( )푀푎2 2 2 The pressure ratio for Fanno line is given by, √ 2 푘−1 푃 푀푎1 1+푀푎1( ) 2 = 2 (16) 푃 푘−1 1 푀푎 √1+푀푎2( ) 2 2 2 The pressure ratio for Rayleigh line is given by, 2 푃2 1+푘푀푎1 = 2 (17) 푃1 1+푘푀푎2 The Mach number relation for before and after shockwave is given by, 2 푀푎2+( ) 2 1 푘−1 푀푎2 = 푘 (18) 2푀푎2( )−1 1 푘−1 Equations 5 to 18 citation ‘Fluid Mechanics’ by Cengel and Cimbala. 3.4.2. Oblique Shocks When a shockwave is inclined to the flow direction, it is called as an oblique shock. Oblique shock is generally generated at the nose of the nozzle during downstream flow. Oblique shock generally occurs when supersonic flow is encountered at a corner that effectively turns the flow into itself and compresses it. It can be converted to a normal shock by a Galilean transformation. An ideal gas (k=1.4) initially at a pressure of 2MPa and at a temperature of 300K enters a converging diverging nozzle at negligible velocity. The variation of pressure, temperature, velocity and density of flow across the nozzle is calculated theoretically and also through computer simulation (CFD).The model is chosen as follows http://iaeme.com/Home/journal/IJMET 227 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman Using equations (2) to (11) the variation of the above parameters is obtained and are summarized in the tables given below, Table 1 Mach number Variation Section 푨 X coordinate Mach number ∗ 푨 (m) converging 1.333 0 0.51 converging 1.167 0.21 0.60 throat 1 0.42 1 diverging 1.301 0.63 1.66 diverging 1.6 0.84 1.94 Table 2 Temperature Variation Section 푨 X coordinate Temperature 푨∗ (m) (K) converging 1.333 0 285.567 converging 1.167 0.21 278.502 throat 1 0.42 256.326 diverging 1.301 0.63 193.425 diverging 1.6 0.84 171.519 Table 3 Pressure Variation Section 푨 X coordinate Pressure(Pa) ∗ 푨 (m) converging 1.333 0 1.68× 106 converging 1.167 0.21 1.54× 106 Throat 1 0.42 1.05× 106 diverging 1.301 0.63 4.30× 105 diverging 1.6 0.84 2.82× 105 http://iaeme.com/Home/journal/IJMET 228 [email protected] Concepts and CFD Analysis of De-Laval Nozzle Table 4 Velocity Variation Section 푨 X coordinate Velocity(m/s) 푨∗ (m) converging 1.333 0 172.754 converging 1.167 0.21 207.399 throat 1 0.42 320.717 diverging 1.301 0.63 462.774 diverging 1.6 0.84 509.287 Table 5 Density Variation Section 푨 X coordinate Density(kg/m3) ∗ 푨 (m) converging 1.333 0 20.465 converging 1.167 0.21 19.522 throat 1 0.42 14.725 diverging 1.301 0.63 7.75172 diverging 1.6 0.84 5.7112 5. COMPUTER SIMULATION OF DE LAVAL NOZZLE Figure 5 CAD model of nozzle http://iaeme.com/Home/journal/IJMET 229 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman Computer simulation of nozzle was done using computational fluid dynamics (CFD). CFD is method to solve complex problems involving fluid flow. The above conditions were taken as mentioned in the previous section and the computer simulation that is the analysis was done using Ansys-Fluent. When we performed this analysis, we found out the variation of parameters as we did in the case of theoretical treatment. The CFD Analysis was done in the following steps: • Modelling • Meshing • Pre processing • Solver/Processing • Post processing 5.1. Modeling The modeling of the De Laval nozzle was done in Ansys design modeler. The points were plotted first using construction point command and then they were joined using 3D curves. Lines from points option was also used. A 2D surface was finally created using surface from edges command. The nozzle dimensions are as follows Parameter Dimension Total length of nozzle (mm) 840 Inlet diameter (mm) 280 Outlet diameter (mm) 336 Throat diameter (mm) 210 Convergence angle (deg) 30.55 Divergence angle (deg) 54.532 http://iaeme.com/Home/journal/IJMET 230 [email protected] Concepts and CFD Analysis of De-Laval Nozzle 5.2. Meshing Figure 6 Meshed model of nozzle The model created by the above dimensions was meshed in mesh mode of Ansys component systems. Meshing is nothing but converting an infinite number of particles model of finite number of particles. The details of mesh were Physics preference: CFD. The relevance of meshing was set to 100. The proximity and curvature option was selected, since nozzle is a curved object. The smoothing was high and num cells across gap were set to 50. The meshing was fine. Therefore number modes were calculated as 12760 and number of elements was 12427. Mapped face meshing was used for the two faces. The relevance center was set to fine and the elements used were of quadrilateral shape. 5.3. Pre Processing The next step of the CFD after meshing is pre processing. In pre processing appropriate boundry conditions are applied to the meshed model. The pre processing was done in Ansys Fluent. Table 6 Preprocessing Details General Solver Type: Density Based 2D space: Planer Time: Steady Materials Density: Ideal Gas 퐶푝: 1006.43 퐽/푘푔퐾 k: 1.4 Viscosity: 1.81*10-5 Pas Thermal Conductivity: 0.02619W/mK Mean molecular mass: 28.966 g/mole http://iaeme.com/Home/journal/IJMET 231 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman Boundary Conditions Inlet Pressure = 20×105 Pa Inlet temperature = 300K Outlet temperature = 171.519K Outlet Pressure = 2.5×105 Pa Table 7 Preprocessing for Shockwave General Solver type: density based 2D space: Planar Time: Steady Materials Density: Ideal Gas Cp: 10006.43 J/kgK k: 1.4 Viscosity: 1.81×10-5PaS Thermal Conductivity: 0.02619 W/mk Mean Molecular mass:28.966 g/mole Boundary Condition Inlet pressure = 20×105Pa Inlet temperature = 300K Outlet pressure = 16×105 5.4. Solver The next step is solver. In solver the solution is initialized and calculation is proceeded with the desired number of iterations. it is the most important port of CFD analysis .Using Ansys -fluent, it is possible to solve the governing equation related to flow physical properties Table 8 Solver Details Solver Details Solution Controls Courant number = 5 Solution Initialization Compute from = inlet Run Calculation Number of iterations = 1000 The solution was terminated after 648 iterations http://iaeme.com/Home/journal/IJMET 232 [email protected] Concepts and CFD Analysis of De-Laval Nozzle Figure 6 Solution initialization Table 9 Solver Details for Shockwave Solver Details Solution Controls Courant number = 5 Solution Initialization Compute from = inlet Run Calculation Number of iterations = 2000 The solution was terminated after 1639 iterations Figure 7 Solution initialization for shockwave http://iaeme.com/Home/journal/IJMET 233 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman 5.5. Post Processing/Result 5.5.1. Velocity Variation Velocity goes on increasing from left to right. It is minimum at the inlet section and maximum at the outlet section. Velocity is equal to sonic velocity at the throat. Minimum velocity is 161.571m/s. and maximum velocity is 525.648 m/s. Figure 8 Velocity contour and graph (m/s) 5.5.2. Pressure Variation Pressure is maximum at the inlet section and minimum at the outlet section Maximum value of pressure is 1.713×106 Pa and minimum value is 2.32×105. http://iaeme.com/Home/journal/IJMET 234 [email protected] Concepts and CFD Analysis of De-Laval Nozzle Figure 9 Pressure contour and graph (Pa) 5.5.3. Temperature Variation Temperature variation is similar to pressure variation. It is maximum at the inlet and minimum at the outlet. Maximum value of temperature is 287.082K and the minimum value of temperature 163.15K. http://iaeme.com/Home/journal/IJMET 235 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman Figure 10 Temperature contour and graph (K) 5.5.4. Density Variation Density of the flow is maximum at the inlet and minimum at outlet. Maximum value of density is 20.8171kg/m3 and the minimum value is 4.99 kg/m3. Figure 11 Density contour and graph (kg/m3) http://iaeme.com/Home/journal/IJMET 236 [email protected] Concepts and CFD Analysis of De-Laval Nozzle 5.5.5. Shockwave Shockwave results across the flow are as follows. a) Pressure variation due to shockwave Figure 12 Pressure variation in shockwave b) Velocity variation due to shockwave Figure 13 Velocity variation during shockwave c) Temperature variation due to shockwave Figure 14 Temperature variation in shockwave http://iaeme.com/Home/journal/IJMET 237 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman d) Density variation due to shockwave Figure 15 Density variation in shockwave 6. CONCLUSION Table 10 Velocity Comparison Section 푨 Velocity(m/s) Velocity(m/s) ∗ 푨 Theoretical CFD converging 1.333 172.754 161.57 converging 1.167 207.399 205.22 throat 1 320.717 302.25 diverging 1.301 462.774 433.46 diverging 1.6 509.287 525.64 Table 11 Pressure Comparison Section 푨 Pressure(Pa) Pressure 푨∗ Theoretical CFD converging 1.333 1.68× 106 1.71×106 converging 1.167 1.54× 106 1.50× 106 throat 1 1.05× 106 1.15×106 diverging 1.301 4.30× 105 5.70×105 diverging 1.6 2.82× 105 2.32×105 http://iaeme.com/Home/journal/IJMET 238 [email protected] Concepts and CFD Analysis of De-Laval Nozzle Table 12 Temperature Comparison Section 푨 Temperature(K) Temperature CFD 푨∗ Theoretical converging 1.333 285.567 287.082 converging 1.167 278.502 279.569 throat 1 256.326 255.232 diverging 1.301 193.425 208.223 diverging 1.6 171.519 163.152 Table 13 Density Comparison Section 푨 Density(kg/m3) Density(kg/m3) 푨∗ Theoretical CFD converging 1.333 20.46 20.81 converging 1.167 19.52 19.38 throat 1 14.72 15.56 diverging 1.301 7.751 9.28 diverging 1.6 5.711 4.99 By this paper we have explained the basic concepts connected with converging diverging nozzle. Through this study and analysis, we have found out the values of variation of nozzle parameters by theoretical method that is by using formulae and also using CFD. Thus we conclude that the values by both these methods are similar to each other. The comparison of these values is summarized in the following tables. ACKNOWLEDGEMENTS We thank Mrs. V. Vandana and Mrs. Rupali Patil without whose help this project would not have been possible. REFERENCE [1] Nikhil Deshpande and Suyash Vidwans, “Theoretical and CFD Analysis of De Laval Nozzle” , vol. 2,No 4,April 2014. [2] Venkatesh V and C Jayapal Reddy, “Modelling and simulation of Supersonic nozzle using computational fluid dynamics,”vol. 2,2015, No 6, pp. 16-27. [3] Mr. Lijo P Varghese, Mr. Rajiv Saxena And Dr. R.R. Lal, Analysis of the Effect of Nozzle Hole Diameter on CI Engine Performance Using Karanja Oil-Diesel Blends. International Journal of Mechanical Engineering and Technology (IJMET), 4(4), 2013,pp. 79–88. [4] Yunus Cengel and John Cimbala,”Fluid Mechanics” 3rd edition, pp. 659-723. http://iaeme.com/Home/journal/IJMET 239 [email protected] Malay S Patel, Sulochan D Mane and Manikant Raman [5] Yunus Cengel and Michael Boles, “Thermodyanmics an engineering approach,”5th edition,pp.823-907 [6] Rathakrishnan,”Gas Dynamics”,pp.(62-73) [7] Nagesha S, Vimala Narayanan, S. Ganesan and K. S.Shashishekar, CFD Analysis in an Ejector of Gas Turbine Engine Test BED. International Journal of Mechanical Engineering and Technology (IJMET), 5(9), 2014,pp. 101–108. http://iaeme.com/Home/journal/IJMET 240 [email protected]