EXTROVERTSpace Propulsion 05 AE6450 Fall 2008 Lecture #5 Nozzles Types of nozzles Expansion Ratio criteria Nozzle heat transfer and materials issues In this lecture we will cover:

EXTROVERTSpace Propulsion 05 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: 1.Nozzle size (needed to get better performance) against nozzle weight penalty. 2.Complexity of the shape for shock-free performance vs. cost of fabrication

EXTROVERTSpace Propulsion 05 T-s Diagram (or h-s Diagram) for a nozzle: Ideal i: reactants at chamber pressure p 0 f: final mixture at combustion chamber stagnation conditions p 0,T 0 t: throat conditions. Mach 1, p *, T * e: exit conditions p e,T e (assuming fully expanded) i f t e T Entropy s p0p0 pepe p*p* T0T0 Q/c p

EXTROVERTSpace Propulsion 05 T-s Diagram (or h-s Diagram) for a nozzle: with losses i: reactants at chamber pressure p 0i f: final mixture at combustion chamber stagnation conditions p 0,T 0 t: throat conditions. Mach 1, p *, T * e: exit conditions p e,T e (assuming fully expanded). i f t e T Entropy s p0p0 pepe p*p* T0T0 Q/c p 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.

EXTROVERTSpace Propulsion 05 Recall – Expressions for Thrust Coefficient - 1 where A t is nozzle throat area and p 0 is chamber pressure (N/m 2 ) Thus, For sonic conditions at the throat, and

EXTROVERTSpace Propulsion 05 Using isentropic flow relations, and Thrust Coefficient Depends entirely on nozzle characteristics. The thrust coefficient is used to evaluate nozzle performance. Thrust Coefficient - 2

EXTROVERTSpace Propulsion 05 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, Characteristic Velocity c*

EXTROVERTSpace Propulsion 05 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.

EXTROVERTSpace Propulsion 05 Conical Nozzle · Easier to manufacture – for small thrusters · Divergence losses: Exit velocity is not all in the desired direction. a Here is the “thrust efficiency”, defined as ratio of actual to ideal thrust, accounting for flow divergence.  is the nozzle Area Ratio (ratio of exit area to throat area). Note: a can be as large as 12 to 18 degrees.

EXTROVERTSpace Propulsion 05 Geometric Representations of Nozzles :  x y R1R1 N 2R t RtRt Throat internal radius N Transition point from circular contour to conical contour. Located along at angle a downstream of throat. LNLN ReRe Exit radius Radius of curvature of the nozzle contraction R1R1 Cone nozzle Nozzle Length: LNLN For a conical nozzle, is a typical choice.

EXTROVERTSpace Propulsion 05 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.

EXTROVERTSpace Propulsion 05 Bell Nozzle An approximate shape can be formed from a parabola (after G.V.R. Rao)  x y’ R1R1 N RtRt L x’ Upstream of the throat  Downstream of the throat

EXTROVERTSpace Propulsion 05 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)

EXTROVERTSpace Propulsion 05 So, 1. and hence …………………..(A) 2. and ……………… (B) 3. ………………………….. (C) (from (A)) 4. …………… (D)

EXTROVERTSpace Propulsion 05 Equating (B) = (D) leads after some manipulation, to …………… (E) Also, squaring (B) …………… (F) Substituting for Q from (C), Eventually gives Down to one unknown. …… (G)

EXTROVERTSpace Propulsion 05 Use eqn. (G) to find Then either (F) or (E) to get S Then use (C) to get Q Then use (A) to get Tin 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 R 1 (i.e., 0.382R t ) f = 100%; 90% or 80% of a 15-degree cone.

EXTROVERTSpace Propulsion 05 Comparison of bell and cone nozzles For the same , 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

EXTROVERTSpace Propulsion 05 Types of Nozzles Types of nozzles Expansion Ratio criteria Nozzle heat transfer and materials issues In this lecture we will continue the discussion on nozzles:

EXTROVERTSpace Propulsion 05 E-D: Expansion-DeflectionSpike: “Plug”R-F: Reverse Flow H-F: Horizontal Flow Comparison of Optimal Nozzles (source: Huzel & Huang, 1967)

EXTROVERTSpace Propulsion 05 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.

EXTROVERTSpace Propulsion 05 Spike Nozzle Flow Geometries [from Berman and Crimp, 1961 ] External expansionInternal-external expansion Internal expansion

EXTROVERTSpace Propulsion 05 [from Berman and Crimp, 1961] Effect of Replacing Spike with Cone

EXTROVERTSpace Propulsion 05 Aerospike Truncates full spike and adds secondary base flow to help contour the inner flow. Source: Hill & Peterson, p

EXTROVERTSpace Propulsion 05 Aerospike Flow Features Rocketdyne RS2200. Flynn, 1996 Ruf and McConnaughey, 1997 Thrust = F thruster +F centerbody + Fbase

EXTROVERTSpace Propulsion 05 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.

EXTROVERTSpace Propulsion 05 Linear Aerospike Combustion chamber modules placed along the inside.

EXTROVERTSpace Propulsion 05 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) Rocketdyne, ’99

EXTROVERTSpace Propulsion 05 Combustor Geometry Spherical (usually for solid rockets Cylindrical Chamber AcAc AtAt LcLc q = contraction ratioFor a cylindrical chamber Chamber Volume Convergent part approximated as conical

EXTROVERTSpace Propulsion 05 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 D t is the throat diameter in centimeters. Relation plotted for 1/5 < A c /A t < 13.5 and 0.7cm < D t < 200cm

EXTROVERTSpace Propulsion 05 Ranges of Combustor Characteristic Length PropellantsCharacteristic Length L* Low(m)High(m) Liquid fluorine / hydrazine Liquid fluorine / gaseous H Liquid fluorine / liquid H Nitric acid/hydrazine N 2 O 4 / hydrazine Liquid O 2 / ammonia Liquid O 2 / gaseous H Liquid O 2 / liquid H Liquid O 2 / RP H 2 O 2 / RP-1 (incl. catalyst)

EXTROVERTSpace Propulsion 05 Contraction Ratio vs. Throat diameter Also, the contraction rate can be empirically determined as a function of D t. Chang gives a range from 2 to 5 for the e c for low thrust engines (smaller D t ), and from 1.3 to 2.5 for higher thrust engines (Larger D t ). Knowing L* and e c for a given A t and propellant, we can estimate the required combustor length, Lc, for the cylinder section.

EXTROVERTSpace Propulsion 05 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 . 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.

EXTROVERTSpace Propulsion 05 Losses (continued) 4. Nozzle chemistry (recombination) Recombination can actually increase C* above the frozen flow nozzle assumption as species recombine if equilibrium flow is assumed. if frozen flow is assumed and 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.