Rocket Performance • Te rocket equation • Mass ratio and performance • Structural and payload mass fractions • Regression analysis • Multistaging • Optimal ΔV distribution between stages • Trade-off ratios • Parallel staging • Modular staging © 2016 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu U N I V E R S I T Y O F Rocket Performance MARYLAND 1 ENAE 791 - Launch and Entry Vehicle Design Derivation of the Rocket Equation V • Momentum at time t: e m-Δm v m M = mv Δm v+Δv v-V • Momentum at time t+Δt: e M =(m − ∆m)(V +∆v)+∆m(v − Ve) • Some algebraic manipulation gives: ! m∆v = −∆mVe • Take to limits and integrate: m V final dm final dv = − ! m ! Ve minitial Vinitial U N I V E R S I T Y O F Rocket Performance MARYLAND 2 ENAE 791 - Launch and Entry Vehicle Design The Rocket Equation mfinal − ∆V • Alternate forms r ≡ = e Ve m initial mfinal ∆v = −Ve ln = −Ve ln r ! !minitial " • Basic definitions/concepts m m – Mass ratio r ≡ final or ℜ≡ initial ! minitial mfinal – Nondimensional velocity change “Velocity ratio” ∆V ⌫ ⌘ Ve U N I V E R S I T Y O F Rocket Performance MARYLAND 3 ENAE 791 - Launch and Entry Vehicle Design Rocket Equation (First Look) 1 Typical Range for Launch to ) Low Earth Orbit initial 0.1 /M nal fi 0.01 Mass Ratio, (M Mass 0.001 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) U N I V E R S I T Y O F Rocket Performance MARYLAND 4 ENAE 791 - Launch and Entry Vehicle Design Sources and Categories of Vehicle Mass Payload Propellants Structure Propulsion Avionics Power Mechanisms Thermal Etc. U N I V E R S I T Y O F Rocket Performance MARYLAND 5 ENAE 791 - Launch and Entry Vehicle Design Sources and Categories of Vehicle Mass Payload Propellants Inert Mass Structure Propulsion Avionics Power Mechanisms Thermal Etc. U N I V E R S I T Y O F Rocket Performance MARYLAND 6 ENAE 791 - Launch and Entry Vehicle Design Cost and Weight Distribution - Atlas V U N I V E R S I T Y O F Rocket Performance MARYLAND 7 ENAE 791 - Launch and Entry Vehicle Design Cost and Weight - Atlas V First Stage propellant mass Propellant mass fraction PMF = total stage mass U N I V E R S I T Y O F Rocket Performance MARYLAND 8 ENAE 791 - Launch and Entry Vehicle Design Cost and Weight - Atlas V Second Stage U N I V E R S I T Y O F Rocket Performance MARYLAND 9 ENAE 791 - Launch and Entry Vehicle Design Basic Vehicle Parameters • Basic mass summary mo =initial mass mo = mpl + mpr + min mpl =payload mass • Inert mass fraction mpr =propellant mass ! m m min =inert mass δ ≡ in = in ! mo mpl + mpr + min • Payload fraction ! m m λ ≡ pl = pl ! mo mpl + mpr + min • Parametric mass ratio r = λ + δ U N I V E R S I T Y O F Rocket Performance MARYLAND 10 ENAE 791 - Launch and Entry Vehicle Design Rocket Equation (including Inert Mass) 1 ) Typical Range for Launch to Low initial Earth Orbit Inert Mass /M Fraction δ 0.1 0 0.05 payload 0.1 0.15 0.01 0.2 0.001 Payload Fraction, (M Fraction, Payload 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) U N I V E R S I T Y O F Rocket Performance MARYLAND 11 ENAE 791 - Launch and Entry Vehicle Design Limiting Performance Based on Inert Mass 5 4.5 Typical Feasible Design Range for V/Ve) 4 Inert Mass Fraction Δ 3.5 3 2.5 2 1.5 1 0.5 0 Asymptotic Velocity Ratio, ( Velocity Asymptotic 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Inert Mass Fraction, (Minert/Minitial) U N I V E R S I T Y O F Rocket Performance MARYLAND 12 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis of Existing Vehicles Veh/Stage prop mass gross mass Type Propellants Isp vac isp sl sigma eps delta (lbs) (lbs) (sec) (sec) Delta 6925 Stage 2 13,367 15,394 StorableN2O4-A50 319.4 0.152 0.132 0.070 Delta 7925 Stage 2 13,367 15,394 StorableN2O4-A50 319.4 0.152 0.132 0.065 Titan II Stage 2 59,000 65,000 StorableN2O4-A50 316.0 0.102 0.092 0.087 Titan III Stage 2 77,200 83,600 StorableN2O4-A50 316.0 0.083 0.077 0.055 Titan IV Stage 2 77,200 87,000 StorableN2O4-A50 316.0 0.127 0.113 0.078 Proton Stage 3 110,000 123,000 StorableN2O4-A50 315.0 0.118 0.106 0.078 Titan II Stage 1 260,000 269,000 StorableN2O4-A50 296.0 0.035 0.033 0.027 Titan III Stage 1 294,000 310,000 StorableN2O4-A50 302.0 0.054 0.052 0.038 Titan IV Stage 1 340,000 359,000 StorableN2O4-A50 302.0 0.056 0.053 0.039 Proton Stage 2 330,000 365,000 StorableN2O4-A50 316.0 0.106 0.096 0.066 Proton Stage 1 904,000 1,004,000 StorableN2O4-A50 316.0 285.0 0.111 0.100 0.065 average 312.2 285.0 0.100 0.089 0.061 standard deviation 8.1 0.039 0.033 0.019 U N I V E R S I T Y O F Rocket Performance MARYLAND 13 ENAE 791 - Launch and Entry Vehicle Design Inert Mass Fraction Data for Existing LVs 0.18 0.16 0.14 0.12 LOX/LH2 0.10 LOX/RP-1 0.08 Solid 0.06 Storable Inert Mass Fraction Mass Inert 0.04 0.02 0.00 0 1 10 100 1,000 10,000 Gross Mass (MT) U N I V E R S I T Y O F Rocket Performance MARYLAND 14 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis • Given a set of N data points (xi,yi) • Linear curve fit: y=Ax+B – find A and B to minimize sum squared error N ! 2 error = !(Axi + B − yi) ! i=1 – Analytical solutions exist, or use Solver in Excel A • Power law fit: y=BxN 2 ! error = ! [A log(xi)+B − log(yi)] i=1 • Polynomial, exponential, many other fits possible U N I V E R S I T Y O F Rocket Performance MARYLAND 15 ENAE 791 - Launch and Entry Vehicle Design Solution of Least-Squares Linear Regression N ∂(error) =2!(Axi + B − yi)xi =0 ∂A i=1 N ∂(error) =2 (Axi + B − yi)=0 ∂B ! N N N i=1 2 A xi + B xi − xiyi =0 ! ! ! N N i=1 i=1 i=1 A ! xi + NB − ! yi =0 i=1 i=1 2 N ! xiyi − ! xi ! yi ! yi ! xi − ! xi ! xiyi B A = 2 2 = 2 2 N ! xi − (! xi) N ! xi − (! xi) U N I V E R S I T Y O F Rocket Performance MARYLAND 16 ENAE 791 - Launch and Entry Vehicle Design Regression Analysis - Storables 0.10 0.08 0.06 0.04 0.02 R2 = 0.0541 0.00 Inert Mass Fraction Mass Inert 1 10 100 1,000 Gross Mass (MT) U N I V E R S I T Y O F Rocket Performance MARYLAND 17 ENAE 791 - Launch and Entry Vehicle Design Regression Values for Design Parameters Vacuum Ve Inert Mass Max ΔV (m/sec) Fraction (m/sec) LOX/LH2 4273 0.075 11,070 LOX/RP-1 3136 0.063 8664 Storables 3058 0.061 8575 Solids 2773 0.087 6783 U N I V E R S I T Y O F Rocket Performance MARYLAND 18 ENAE 791 - Launch and Entry Vehicle Design Economy of Scale for Stage Size 0.20# M 0.18# ✏ = inert 0.16# Minert + Mprop 0.14# 0.12# 0.10# 0.08# 0.06# 0.04# Stage#Inert#Mass#Frac6on#ε# 0.02# PMF =1 ✏ 0.00# − 1# 10# 100# 1,000# 10,000# Stage#Gross#Mass,#MT# Storable#MER# LOX/RPD1# Storables# LOX/LH2# Cryo#MER# U N I V E R S I T Y O F Rocket Performance MARYLAND 19 ENAE 791 - Launch and Entry Vehicle Design Stage Inert Mass Fraction Estimation 0.183 ✏ =0.987 (M kg )− LOX/LH2 stageh i 0.275 ✏ =1.6062 (M kg )− storables stageh i U N I V E R S I T Y O F Rocket Performance MARYLAND 20 ENAE 791 - Launch and Entry Vehicle Design The Rocket Equation for Multiple Stages • Assume two stages ! mfinal1 ! ∆V1 = −Ve1 ln = −Ve1 ln(r1) !minitial1 " ! m ! V −V final2 −V r ∆ 2 = e2 ln m = e2 ln( 2) ! ! initial2 " • Assume Ve1=Ve2=Ve ∆V1 +∆V2 = −Ve ln(r1) − Ve ln(r2)=−Ve ln(r1r2) U N I V E R S I T Y O F Rocket Performance MARYLAND 21 ENAE 791 - Launch and Entry Vehicle Design Continued Look at Multistaging • Tere’s a historical tendency to define r0=r1r2 ! ∆V + ∆V = V ln (r r ) = V ln (r ) 1 2 − e 1 2 − e 0 ! • But it’s important to remember that it’s really ! m m ∆V + ∆V = V ln (r r ) = V ln final1 final2 ! 1 2 − e 1 2 − e m m initial1 initial2 ⇥ • And that r0 has no physical significance, since mfinal2 mfinal1 = minitial2 r0 = ⇥ ⇒ ⇥ minitial1 U N I V E R S I T Y O F Rocket Performance MARYLAND 22 ENAE 791 - Launch and Entry Vehicle Design Multistage Inert Mass Fraction • Total inert mass fraction ! m + m + m m m m δ = in,1 in,2 in,3 = in,1 + in,2 + in,3 0 m m m m ! 0 0 0 0 min,1 min,2 m0,2 min,3 m0,3 m0,2 ! δ0 = + + m0 m0,2 m0 m0,3 m0,2 m0 • Convert to dimensionless parameters ! δ0 = δ1 + δ2λ1 + δ3λ2λ1 • General form of the equation n stages j−1 δ δ λ 0 = ! [ j " ℓ] j=1 ℓ=1 U N I V E R S I T Y O F Rocket Performance MARYLAND 23 ENAE 791 - Launch and Entry Vehicle Design Multistage Payload Fraction • Total payload fraction (3 stage example) ! mpl mpl m0,3 m0,2 λ0 = = ! m0 m0,3 m0,2 m0 • Convert to dimensionless parameters ! λ0 = λ3λ2λ1 • Generic form of the equation n stages λ λ 0 = ! j j=1 U N I V E R S I T Y O F Rocket Performance MARYLAND 24 ENAE 791 - Launch and Entry Vehicle Design Effect of δ and ΔV/Ve on Payload 0.9000.7000.4000.2500.0900.0600.020 0.018 ∆V 0.3500.8000.080 0.0500.600 = 12340.0255 0.7000.0700.2000.016 0.300 Ve 0.0140.500 0.6000.0600.040 0.250 0.1500.012 1 stage 0.400 0.5000.050 2 stages 0.2000.0300.010 3 stages 0.3000.4000.040 0.1000.008 0.150 4 stages 0.3000.0300.020 0.2000.006 0.100 Total Payload Fraction Payload Total 0.0500.0040.2000.020 0.010 0.0500.100 0.0020.1000.010 0.000 0 0.2 0.4 0.60.6 0.80.8 11 Inert Mass Fraction U N I V E R S I T Y O F Rocket Performance MARYLAND 25 ENAE 791 - Launch and Entry Vehicle Design Effect of Staging Inert Mass Fraction δ=0.15 0.25 Single Stage Three Stage Two Stage Four Stage 0.20 0.15 1 stage 2 stage 3 stage 0.10 4 stage Payload Fraction 0.05 0.00 0 1 2 3 4 5 Velocity Ratio (ΔV/Ve) U N I V E R S I T Y O F Rocket Performance MARYLAND 26 ENAE 791 - Launch and Entry Vehicle Design Trade Space for Number of Stages 4 3.5 Solids 3 LOX/RP-1 Four Stage Storables 2.5 Three Stage 2 LOX/LH2 1.5 Two Stage Velocity Ratio DV/Ve Ratio Velocity Velocity Ratio DV/Ve Ratio Velocity 1 Single Stage 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Inert Mass Fraction U N I V E R S I T Y O F Rocket Performance MARYLAND 27 ENAE 791 - Launch and Entry Vehicle Design Effect of ΔV Distribution