MAE 5380: ROCKETS AND MISSION ANALYSIS Basics of Chemical Rocket Performance Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11 Schematic Diagram of a Conventional Liquid Rocket Motor (Figure 11.1 in Sforza)

Northrop Grumman: LOX/LH2 Engine

Northrop Grumman: LOX/LH2 Engine

SUMMARY OF KEY EXIT VELOCITY, Ue, EQUATIONS For high Ue (high Isp), desire Propellants with low molecular weight, M Propellant mixtures with large heat release, QR High combustion chamber pressure, P02 NOTE: Sometimes subscript 2 is dropped, but still conditions in combustion chamber

SUMMARY OF KEY THRUST, T, EQUATIONS Comparison to best theoretical Measure from actual rocket (parameters that can be easily measured on a thrust stand)

OVERVIEW Next page shows a plot of thrust ratio vs. area ratio Figure compares two non-dimensional numbers Abscissa is ratio of nozzle exit area to minimum area, or nozzle exit area to throat area (minimum area always occurs at throat), Ae/A* Ordinate is ratio of thrust with diverging to converging nozzle, T/Tconv Curve is plotted for constant ratio of specific heats, g = cp/cv = 1.2 Curve would shift for g = 1.4 or any other value Curves correspond to various ratios of Pa/P0 Pa/P0 = ambient (atmospheric) to combustion chamber pressure P0 is approximately constant for most rockets Compare with Figure 11.7 (page 448) in Sforza

T/Tconv versus Ae/A*

Figure 11.7 (page 448) in Sforza

COMPARISON OF CONVERGING vs. DIVERGING NOZZLES Examine ratio of thrusts, with and without a diverging section Examine performance benefit of having diverging portion Metric of comparison: Excellent Web Site: http://www.engapplets.vt.edu/fluids/CDnozzle/cdinfo.html Chamber, Pa Chamber P0 Chamber P0 Converging Nozzle Converging-Diverging Nozzle

COMMENTS: CONVERGING NOZZLE (CTconv) For nozzle with only a converging section → analysis is straightforward Pa/P0 is varied in equation Evaluate at Me = 1 Sonic exit condition For converging nozzle Ae/A* = 1

THRUST COEFFICIENT, CTconv, FOR CONVERGING NOZZLES Maximum Thrust Coefficient when Pa = 0 (expansion into a vacuum) Ae/A*=1

COMMENTS: DIVERGING NOZZLE (CT) Requires more analysis than simple converging nozzle IMPORTANT POINT: We can not vary Pe/P0 and Ae/A* independently Connected through Mach Number, Me Expression for Pe/P0 Vary Pa/P0 and Ae/A* Given A/A* → 2 Me Solutions Subsonic and Supersonic

MACH NUMBER vs. A/A* Differences in Cp/Cv Amplified as M ↑ For Given A/A* → 2 Solutions Subsonic and Supersonic Mach Highly Sensitive Region: Small Changes in A/A* → Large Changes in M

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE WHAT DID WE DO HERE? Set Pa/P0 = 0.05, g = 1.2 For any Ae/A* determine supersonic Me Using this Me calculate P0/Pe Calculate CT Plot CT/CTconv (or T/Tconv) as function of Ae/A* (which is equivalent to plotting CT as a function of Me (supersonic)) Function is Maximized when Pe = Pa

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Maximum Thrust (Pe = Pa) Diverging Portion Increases Thrust In terms of calculation, we could allow T/Tconv to become negative, but as we will soon see, we can deal with this part of the curve more realistically Diverging Portion Reduces Thrust

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Nozzle is Ideally Expanded Pe = Pa Curve can also tell us where Pe > or < Pa IF: Pe > Pa Nozzle is Under-Expanded IF: Pe < Pa Nozzle is Over-Expanded

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Nozzle is Ideally Expanded (Pe = Pa) Nozzle is Under-Expanded (Pe > Pa) Nozzle is Over-Expanded (Pe < Pa)

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Nominal Range of Pa/P0 Decreasing Back Pressure or Increasing Altitude

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Line of Maximum Thrust: Connects Locus of Maxima For each value of Pa/P0 there is an optimum area ratio for nozzle

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE Small Ratios of Pa/P0 Require Very Large Area Ratios

EXAMPLE: ROCKET LAUNCH Ae/A* = 20 Burnout (Under-Expanded) ↑ Vertical Flight Max Thrust (Ideally Expanded) Launch (Over-Expanded) Notice we are closer to Optimum Thrust on Under-Expanded Side

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE What can physically happen to supersonic flow in this region? For this combination of pressure ratios and area ratios, a shock enters nozzle

MODEL OF SHOCK IN EXIT PLANE We can plot shock line by located a shock at exit plane of nozzle Requires 1 additional equation Flow across a normal shock to connect conditions For a given g only one Pa/P0 for which a normal shock will locate in plane of a nozzle of given area ratio Ae/A*

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE On this line a normal shock wave located at exit of nozzle

PERFORMANCE CHARACTERISTIC OF A 1-D ISENTROPIC NOZZLE If Pe reduced substantially below Pa flow can separate A rough approximation for this condition is: Pe/Pa < 0.4 NOTE: Axial thrust direction is not usually altered by separation and CT can actually be increased over non-separated case

THRUST COEFFICIENT PLOTS Taken from: Rocket Propulsion Elements, 6th Edition, by G. P. Sutton Notation p1 = p0 p2 = pe p3 = pa CF = CT A2/At = e = Ae/A* k = g = cp/cv Comments: Plots are only CF (CT), they are not normalized by CTconv as in Figure 11.3 Large region of separated flow Asymptotic behavior as p1/p3 → ∞ pa/p0 → ∞ in H&P

THRUST COEFFICIENT VS. NOZZLE AREA RATIO FOR g=1.2

OPTIMUM EXPANSION SUMMARY

KEY POINTS ON PERFORMANCE CURVE How does a rocket flying vertically move on Performance Curve? High Pa/P0 to Low Pa/P0 P0 usually remains ~ constant during flight Pa ↓ as altitude ↑ As Pa/P0 ↓ very large Ae/A* for maximum thrust How does optimal Ae/A* vary as rocket flies vertically? Required Ae/A* for maximum thrust increases as rocket altitude increases If T/Tconv < 1, diverging portion of rocket is hindrance Actual rockets never operate in this region Best nozzle gives best performance (Isp, range, etc.) over flight envelope If nozzle operation is still unclear, lecture on operation of C-D nozzles coming soon

COMMENTS ON ACTUAL NOZZLES Model of thermal rocket thrust chamber performance Model has many simplifications → measure of best theoretical performance Actual rockets benefit from diverging nozzle portion, operate above T/Tconv =1 Actual thrust chambers (non-idealities important to consider) Pressure losses associated with combustion process Actual flow in nozzle is not isentropic Friction Heat losses Shocks within nozzle Chemistry Frozen Flow: Propellant composition remains constant Shifting Equilibrium: Composition changes with propellant temperature Actual shape of nozzle affects performance

SUMMARY: WHAT HAVE WE DONE? Simplified model of thermal rocket thrust chamber Model resulted in connection between thermodynamics and exit velocity, Ue Propellants with low molecular weight to achieve high exit velocity (high Isp) Desirable to have propellant mixtures with large QR/M Desirable to have high combustion chamber pressure, P0 For a given thrust, higher P0 leads to lower A* (smaller rocket) Increasing P0 leads to difficulties (stress, heat transfer, chemical issues) Model resulted in connection between thermodynamics, geometry and exit velocity Developed Characteristic Velocity, c*, and Thrust Coefficient, CT Compare actual rockets to theoretical predictions Developed plot of Performance Characteristics of a 1-D isentropic rocket nozzle BASICS OF THERMAL (CHEMICAL) ROCKET PROPULSION AND PERFORMANCE