1. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 1 Coupling Between Fuel Mass Transfer and Free-Stream Mass Flow in Hybrid Combustion Brian Cantwell Stanford University

2. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 2 Empirical regression rate equation Burning rate law: where G is the port mass flux Typically m is small and 0.4 < n < 0.7. Marxman suggested m=0.2, n=0.8

3. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 3 Substitute and separate variables. A simplification that is often used is to let the regression rate depend on oxidizer mass flux Integrate In this approximation the radius is constant along the port and the leading constant a in the regression rate equation depends on the O/F ratio and n. Simplified approach

4. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 4 The regression rate equation is The mass flow rate increase along the port is determined by the rate at which mass is swept up from the fuel surface. These first order PDEs need to be solved simultaneously for the local mass flow rate and port radius. Solve the nonlinear coupled mass-flow-regression-rate problem

5. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 5 The coupled problem can be solved exactly for the case. The increase in port surface area exactly balances the decrease in mass flux and so the mass flux at any axial position is constant for constant oxidizer mass flow rate. The O/F ratio at the end of the port is constant. The case n=1/2

6. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 6 Karabeyoglu, M.A., Cantwell, B.J. and Zilliac, G., Development of Scalable Space-Time Averaged Regression Rate Expressions For Hybrid Rockets, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & ExhibitAIAA 2005-3544 10 - 13 July 2005, Tucson, Arizona Numerical solution of the coupled problem Numerical solution captures the “coning” effect due to the mass flow increase in the port as well as the port minimum due to a nonzero m exponent on x. Initial and final port geometry data from four tests were used to estimate a, n and m in the full space-time coupled problem.

7. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 7 Let the fuel be a semi-infinite block filling the right half plane in 3-D. At a finite oxidizer mass flow rate is initiated along the x-axis. At first the mass flux is infinite but as the port opens up the mass flux drops to reasonable values in the range of interest to a designer. For constant oxidizer mass flow rate the coupled equations admit a similarity solution. This solution is relevant to the design of a working system. Similarity solution Cantwell, B. J., Similarity solution of fuel mass transfer, port mass flux coupling in hybrid propulsion, J. Engr. Math. Vol 84 issue 1, Feb 2014. Special issue in rememberance of Milton Van Dyke.

8. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 8 Nondimensionalize variables. Scales can be defined in terms of the basic parameters of the problem. Solve for characteristic length, time and mass. Governing parameters

9. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 9 Dimensionless variables and equations Dimensionless governing equations Note The boundary conditions of the semi-infinite problem defined earlier are Define

10. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 10 These equations admit a three dimensional Lie algebra Group operators The group holds invariant the constant oxidizer mass flow rate in the port

11. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 11 Finite group

12. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 12 The group X c is used to construct similarity variables. Similarity variables The governing equations reduce to a pair of coupled ODEs

13. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 13 Finally the problem reduces to an autonomous system Further reduction to an autonomous pair of ODEs This system of ODEs admits a dilation group. Construct new similarity variables.

14. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 14 Critical points The system has two critical points at and Normalize by the coordinates of the nonzero critical point Finally Notice the absence of m

15. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 15 Connection back to R and J The preservation of is accomplished through the asymptotic behavior of

16. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 16 The problem boils down to the solution of a single first order ODE

17. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 17 Phase portrait One trajectory in the phase portrait preserves the boundary condition. Along this trajectory.

18. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 18 Solution V versus U for n=0.62 Recall

19. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 19 Numerical approximation to and Solution in terms of

20. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 20 Dimensionless radius and mass flow solutions.

21. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 21 Dimensionless radius and mass flow solutions.

22. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 22 Parameters of some typical oxidizer/fuel combinations

23. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 23 Comparison with NASA Ames test 4L-05 O/F at the end of the port looks very good over the entire burn O/F at the end of the port looks very good over the entire burn

24. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 24 Comparison with NASA Ames test 4L-05 Excellent agreement with port geometry and O/F

25. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 25 O/F comparison between coupled and uncoupled regression rate formulations

26. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 26 Photo - Kevin Lohner

27. Presentation to the Hybrid Rocket Technical Committee, 50 th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 29, 2014 27 O/F at the end of the port much too high. need a longer port O/F at the end of the port much too high. need a longer port