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  The International Journal Of Engineering And Science (Ijes) ||Volume||2 ||Issue|| 2 ||Pages|| 196-207 ||2013|| Issn: 2319  –   1813   Isbn: 2319  –   1805 www.theijes.com The IJES Page 196 Optimization of Divergent Angle of a Rocket Engine Nozzle Using Computational Fluid Dynamics   1 , Biju Kuttan P, 2, M Sajesh   1,2,  Department of Mechanical Engineering, NSS College of Engineering -----------------------------------------------------Abstract-------------------------------------------------------------   The CFD analysis of a rocket engine nozzle has been conducted to understand the phenomena of supersonic  flow through it at various divergent angles. A two-dimensional axi-symmetric model is used for the analysis and the governing equations were solved using the finite-volume method in ANSYS FLUENT  ®  software. The inlet boundary conditions were specified according to the available experimental information. The variations in the parameters like the Mach number, static pressure, turbulent intensity are being analyzed. The  phenomena of oblique shock are visualized and the travel of shock with divergence angle is visualized and it was found that at 15° the shock is completely eliminated from the nozzle. Also the Mach number is found have an increasing trend with increase in divergent angle thereby obtaining an optimal divergent angle which would eliminate the instabilities due to the shock and satisfy the thrust requirements for the rocket. Keywords   -  Mach number, oblique shock, finite-volume, turbulent kinetic energy, turbulent dissipation   ---------------------------------------------------------------------------------------------------------------------------------------- Date Of Submission: 16, February, 2013 Date Of Publication: 28, February 2013   --------------------------------------------------------------------------------------------------------------------------------------   I.   Introduction Computational Fluid Dynamics (CFD) is an engineering tool that assists experimentation. Its scope is not limited to fluid dynamics; CFD could be applied to any process which involves transport phenomena with it. To solve an engineering problem we can make use of various methods like the analytical method, experimental methods using prototypes. The analytical method is very complicated and difficult. The experimental methods are very costly. If any errors in the design were detected during the prototype testing, another prototype is to be made clarifying all the errors and again tested. This is a time-consuming as well as a cost-consuming process. The introduction of Computational Fluid Dynamics has overcome this difficulty as well as revolutionised the field of engineering. In CFD a problem is simulated in software and the transport equations associated with the problem is mathematically solved with computer assistance. Thus we would be able to predict the results of a problem before experimentation. The current work aims at determining an optimum divergent angle for the nozzle which would give the maximum outlet velocity and meet the thrust requirements. Flow instabilities might be created inside the nozzle due to the formation if shocks which reduce the exit mach number as well as thrust of the engine. This could be eliminated by varying the divergent angle. Here analysis has been conducted on nozzles with divergent angles 4°,7°, 10°, 13°, 15° . Experimentation using the prototypes of each divergent angle is a costly as well as a time consuming process. CFD proves to be an efficient tool to overcome these limitations. Here in this work the trend of various flow parameters are also analysed. II.   The Mathematical Model The mathematical model selected for this work is the standard K- ε model which is one of the Reynolds Averaged Navier-Stoke(RANS) model available in fluent. The standard K- ε model is the most widely used transport model. The standard K- ε model is a two -equation model and the two model equations are as follows:   The model equation for the turbulent kinetic energy K is: (1) = Rate of increase of K+ Convective transport= diffusive transport + Rate of production-Rate of destruction  Optimization Of Divergent Angle Of A Rocket  …   www.theijes.com The IJES Page 197 The model equation for the turbulent dissipation ε is : (2) = Rate of increase of ε+ Convective transport= diffusive transport + Rate of production -Rate of destruction The standard values of all the model constants as fitted with benchmark experiments are (Launder and Sharma, Letters in Heat and mass transport, 1974, 131-138): C μ = 0.09 ; σ k  =1.00 ; σ ε =1.30 ; C ε1 =1.44 ; C ε2 =1.92  Now the Reynolds stresses are found out using: (3) And the eddy-viscosity is evaluated as: (4) The major advantages of this model are that it is relatively simple to implement, it leads to stable calculations, and it is a widely validated turbulence model. The known limitation of this model is that its performance is very poor for flows with strong separation, large streamline curvature and high swirling components. Despite of all these limitations, the model is widely accepted model for initial level screening of alternate designs in compressible flows, combustion engineering etc. 2.1 Analysis Procedure The analysis was run on ANSYS FLUENT ®  software. The geometry of the nozzle was created using the Geometry workbench of ANSYS. A two-dimensional geometry of the nozzle was created. The dimensions and the inlet boundary conditions of the nozzle were obtained from [1] and are as shown in the Table 2.1  below: Table I: Nozzle dimensions and boundary conditions Inlet width(m) 1.000 Throat width (m) 0.304 Exit width(m) 0.861 Throat radius of curvature(m) 0.228 Convergent length(m) 0.640 Convergent angle(°) 30 Divergent angle(°) 15 Total pressure(bar) 44.10 Total temperature(K) 3400 Mass flow rate(Kg/s) 826.0 The next task was to mesh the geometry created. The created geometry was imported to the meshing workbench. The mesh used was tetrahedral mesh. The face-mapped meshing option was employed to the geometry in order to avoid the resolution errors. The mesh was refined to the third degree using the refinement option of the workbench. After meshing, the inlet, the axis and the outlet boundaries were named. This meshed geometry is now imported to the FLUENT workbench. In the FLUENT workbench the settings made are as tabulated:  Optimization Of Divergent Angle Of A Rocket  …   www.theijes.com The IJES Page 198 Table II: Problem setup Table III: Solution The results are obtained when the solution converges. Now we can find the variations in the parameters as follows: Table IV: Results This procedure is continued for various configurations of the nozzle and is compared. III. RESULTS AND DISCUSSION   3.1 Case1: divergent angle = 4° 3.1Mach number Figure.3.1 General Solver type : Density-based 2D Space: Axi-symmetric Models Energy equation : On Viscous model : standard k- ε model  Materials Air-Ideal gas Boundary Conditions Inlet-mass-flow boundary Enter mass flow rate =826kg/s Temperature =3400k Axis-axis boundary Outlet-pressure outlet boundary condition Solution controls Courant number=0.8 Solution initialization Compute from : inlet Run calculation Check case, Enter number of iterations Click Calculation Graphics and animation Use contour option to get the mach number contour, static pressure contour, total temperature contour, turbulent intensity contour Plots Use XY plots to plot the mach number Vs position, static pressure Vs position plots  Optimization Of Divergent Angle Of A Rocket  …   www.theijes.com The IJES Page 199 The mach contour of the nozzle with the divergent angle of 4° shows the formation of oblique shock in the divergent section. Across the first shock, the Mach number suddenly drops from 2.00 Mach to 1.10 Mach. After this the velocity of flow again increases before the next shock occurs. This shock wave produced is reflected from the walls of the nozzle and it forms another shock in the divergent section itself. At this point, the velocity drops from 2.10Mach to 1.45Mach. It can be seen that the velocity again increases and it reaches 2.20Mach at the exit of the nozzle. The positions where the shock occurs can be determined from the Mach Vs  position plot as shown in fig3.2. Figure.3.2 It is found that shock occurs at the position 1m from the inlet section and the shock formed due to the reflection of the wave is formed at 2m from the inlet section. The velocity magnitude is found to increase as we move from inlet to exit. The velocity at the inlet is 0.0853Mach (sub-sonic). At the throat section the velocity varies from 0.931Mach to 1.04Mach. The velocity at the exit is found to be 2.2Mach (super-sonic). 3.2 Static pressure Figure.3.3 Static pressure is the pressure that is exerted by a fluid. Specifically, it is the pressure measured when the fluid is still, or at rest. The above figure reveals the fact that the gas gets expanded in the nozzle exit. The static  pressure in the inlet is observed to be 2.39 e+06 Pa and as we move towards the throat there is a decrease and the value at the throat is found out to be 1.67e+06 Pa. After the throat, there is a sudden increase in the static  pressure at the axis which indicates the occurrence of the shock. After the shock there is a slight decrease in the pressure but it again rises at the second shock. Then it reduces to a value of 1.15e+05Pa at the exit section due to the expansion of the fluid towards the exit of the nozzle.
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