AE6450 Fall 2008 Lecture #5 Nozzles

1. AE6450 Fall 2008Lecture #5Nozzles In this lecture we will cover: • Types of nozzles • Expansion Ratio criteria • Nozzle heat transfer and materials issues

2. NOZZLES • The function of the rocket nozzle is to convert the thermal energy in the propellant into kinetic energy as efficiently as possible, in order to obtain high exhaust velocity along the desired direction. • The mass of a rocket nozzle is a large part of the engine mass. Many of the failures encountered in rocket engines are also traceable to failures of the nozzle – historical data suggest that 50% of solid rocket failures stemmed from nozzle problems. • The design of the nozzle must trade off: • Nozzle size (needed to get better performance) against nozzle weight penalty. • Complexity of the shape for shock-free performance vs. cost of fabrication

3. p0 T0 f p* t T i pe e Entropy s T-s Diagram (or h-s Diagram) for a nozzle: Ideal Q/cp i: reactants at chamber pressure p0 f: final mixture at combustion chamber stagnation conditions p0 ,T0 t: throat conditions. Mach 1, p*, T* e: exit conditions pe ,Te (assuming fully expanded)

4. T0 p0 f Q/cp p* T i t pe e Entropy s T-s Diagram (or h-s Diagram) for a nozzle: with losses Losses show up as increases in entropy for each step – usually accompanied by a loss of pressure. At the exhaust, note that exhaust temperature is higher than ideal when the final pressure is reached. i: reactants at chamber pressure p0i f: final mixture at combustion chamber stagnation conditions p0 ,T0 t: throat conditions. Mach 1, p*, T* e: exit conditions pe ,Te (assuming fully expanded).

5. Recall – Expressions for Thrust Coefficient - 1 where At is nozzle throat area and p0 is chamber pressure (N/m2) Thus, For sonic conditions at the throat, and

6. Thrust Coefficient - 2 Using isentropic flow relations, and Thrust Coefficient Depends entirely on nozzle characteristics. The thrust coefficient is used to evaluate nozzle performance.

7. Characteristic Velocity c* Used to characterize the performance of propellants and combustion chambers independent of the nozzle characteristics. where is the quantity in brackets. Note: So Characteristic velocity Assuming steady, quasi-1-dimensional, perfect gas. The condition for maximum thrust is ideal expansion: nozzle exit static pressure being equal to the outside pressure. In other words,

8. Nozzle Types The subsonic portion of the nozzle is quite insensitive to shape – the subsonic portion of the acceleration remains isentropic. The divergent nozzle is where the decisions come into play. • Conical Nozzle •  Easier to manufacture – for small thrusters •  Divergence losses: Exit velocity is not all in the desired direction. Bell Nozzle • Complex shape • Highest efficiency (near axial flow at exhaust) • Large base drag during atmospheric flight after burnout • Plug Nozzle or Aerospike Nozzle (linear or annular) (X-33, VentureStar, 1960s concept ) • Altitude compensating (see Chang 3-30c) • Expansion- Deflection Nozzle (E-D) • Shortest nozzle of the “enclosed” types.

9. Conical Nozzle ·        Easier to manufacture – for small thrusters ·        Divergence losses: Exit velocity is not all in the desired direction. a Note: a can be as large as 12 to 18 degrees. Here l is the “thrust efficiency”, defined as ratio of actual to ideal thrust , accounting for flow divergence. e is the nozzle Area Ratio (ratio of exit area to throat area).

10. Geometric Representations of Nozzles y Cone nozzle a 2Rt x N R1 LN Throat internal radius Rt Radius of curvature of the nozzle contraction R1 Transition point from circular contour to conical contour. Located along at angle a downstream of throat. N : LN Nozzle Length: Exit radius Re For a conical nozzle, is a typical choice.

11. Bell Nozzle • Complex shape • Highest efficiency (near axial flow at exhaust) • Large Base Drag during atmospheric flight after burnout A true bell nozzle is contoured to minimize the turning (compression) shock losses at the wall as the flow expands, but still turning the flow towards an axial exhaust. Note that the flow may still be under-, over- or fully-expanded at the exit, and hence shocks / expansions may exist downstream.

12. Bell Nozzle An approximate shape can be formed from a parabola (after G.V.R. Rao) qe y’ R1 q1 x’ N Rt x L Upstream of the throat Downstream of the throat

13. Parabolic Bell Nozzle Contour, cont’d (Rao, 1958) There are 4 unknowns in the rotated parabolic segment equation (P,Q,S,T) and 4 boundary conditions 1.  At N: 2. At e: Or, 3. At N: is given (Rao, 1958, plots) 2.     At e: is given (Rao, 1958, plots) (see Humble p. 225)

14. So, 1. and hence …………………..(A) 2. and ……………… (B) 3. (from (A)) ………………………….. (C) 4. …………… (D)

15. Equating (B) = (D) leads after some manipulation, to …………… (E) Also, squaring (B) Substituting for Q from (C), …………… (F) Eventually gives …… (G) Down to one unknown.

16. Use eqn. (G) to find Then either (F) or (E) to get S Then use (C) to get Q Then use (A) to get T in terms of the original X,Y axes. Recall: and where L is generally a fraction of that for a 15-degree half-angle cone with the same R1 (i.e., 0.382Rt) f = 100%; 90% or 80% of a 15-degree cone.

17. Comparison of bell and cone nozzles For the same e, we would expect A bell nozzle, while more complex to build, will generally yield a more efficient exhaust than a cone in a shorter nozzle length. Same nozzle efficiency factor can be reached with about 70% of the length of a cone nozzle. Alternatively, efficiency factor can be increased from about 98% for a cone to about 99.2% for a bell of the same length

18. Types of Nozzles In this lecture we will continue the discussion on nozzles: • Types of nozzles • Expansion Ratio criteria • Nozzle heat transfer and materials issues

19. Comparison of Optimal Nozzles (source: Huzel & Huang, 1967) H-F: Horizontal Flow E-D: Expansion-Deflection Spike: “Plug” R-F: Reverse Flow http://www.aerospaceweb.org/design/aerospike/shapes.shtml

20. Annular Nozzles Expansion-Deflection Nozzle and Plug Nozzle • Substantially shorter than conical or bell nozzles for the same thrust and area ratio. • Design-point performance is nearly as good. • Off-design performance is better under conditions where conventional nozzles would be over-expanded. • Common feature: Free shear layer bounding nozzle flow.

21. Spike Nozzle Flow Geometries [from Berman and Crimp, 1961] External expansion Internal-external expansion Internal expansion http://www.aerospaceweb.org/design/aerospike/inflow.shtml

22. Effect of Replacing Spike with Cone [from Berman and Crimp, 1961] http://www.aerospaceweb.org/design/aerospike/inflow.shtml

23. Aerospike Source: Hill & Peterson, p. 540 Truncates full spike and adds secondary base flow to help contour the inner flow. http://www.aerospaceweb.org/design/aerospike/aerospike.shtml

24. Aerospike Flow Features Rocketdyne RS2200. Flynn, 1996 Ruf and McConnaughey, 1997 Thrust = Fthruster+Fcenterbody+Fbase http://www.aerospaceweb.org/design/aerospike/aerodynamics.shtml

25. Theoretical Performance Comparison [from Huzel and Huang, 1967] Note: The aerospike is susbtantially better at low altitudes where the bell and cone are likely to be overexpanded. http://www.aerospaceweb.org/design/aerospike/compensation.shtml

26. Linear Aerospike Combustion chamber modules placed along the inside.

27. Advantages and Disadvantages Advantages • Much smaller size • Altitude compensation • Lower chamber pressure / higher expansion ratio: safer • Lower vehicle drag • Lower vehicle weight • Modular: lower manufacture and repair costs; lower downtime • Thrust Vectoring: mass flow through individual elements can be controlled. Disadvantages • Heating. • More complex geometry • ??? (not much experience to-date) http://www.aerospaceweb.org/design/aerospike/x33.shtml Rocketdyne, ’99

28. Ac At Combustor Geometry q Spherical (usually for solid rockets Lc Cylindrical Chamber For a cylindrical chamber = contraction ratio Chamber Volume Convergent part approximated as conical

29. Scaling relation between the combustion-chamber length and Contraction Ratio Assuming we get theta from the throat geometry, how do we find ? Characteristic Length L* L* is empirically determined for various propellant combinations – includes mixing, burn rates, etc. (table 5.6 in Humble). where Dt is the throat diameter in centimeters. Relation plotted for 1/5 < Ac/At < 13.5 and 0.7cm < Dt < 200cm

30. Ranges of Combustor Characteristic Length

31. Contraction Ratio vs. Throat diameter Also, the contraction rate can be empirically determined as a function of Dt. Chang gives a range from 2 to 5 for the ec for low thrust engines (smaller Dt), and from 1.3 to 2.5 for higher thrust engines (Larger Dt). Knowing L* and ec for a given At and propellant, we can estimate the required combustor length, Lc, for the cylinder section.

32. Sources of losses 1. We have already discussed the effect of non-axial flow at the nozzle exit. Other sources of losses 2. Friction: Small in large engines, but can be significant for large e. 3. Heat loss - Exit velocity is lower than expected due to enthalpy losses (fairly small). Non-uniform flow at exit - can be a big effect if the nozzle flow is separated inside. - Open cycle mass flow losses Note that the mas flow rate through the throat, which is what goes through the nozzle, may be less than the propellant consumption rate, because some of the propellant is diverted to run the pumps or not recaptured after use as coolant.

33. Losses (continued) 4. Nozzle chemistry (recombination) Recombination can actually increase C* above the frozen flow nozzle assumption as species recombine if frozen flow is assumed and if equilibrium flow is assumed. Taken together, all of these corrections will yield a “real” lambda of 0.9 to 0.98 (Humble pp.206) depending on the cycle, nozzle, throat losses, etc.