Rocket Performance • • • • • • • • • The 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 © 2009 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu UNIVERSITY OF MARYLAND 1 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Derivation of the Rocket Equation Ve • Momentum at time t: Δm M = mv m-Δm v m v+Δv v-Ve • Momentum at time t+Δt: M = (m − ∆m)(V + ∆v) + ∆m(v − Ve ) • Some algebraic manipulation gives: m∆v = −∆mVe • Take to limits and integrate: ! mf inal dm =− m minitial UNIVERSITY OF MARYLAND 2 ! Vf inal Vinitial dv Ve Rocket Performance ENAE 483/788D - Principles of Space Systems Design The Rocket Equation • Alternate forms mf inal − ∆V r≡ = e Ve minitial ! " mf inal ∆v = −Ve ln = −Ve ln r minitial • Basic definitions/concepts – Mass ratio mf inal minitial r≡ or " ≡ minitial mf inal – Nondimensional velocity change ∆V “Velocity ratio” Ve UNIVERSITY OF MARYLAND 3 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Rocket Equation (First Look) Mass Ratio, (Mfinal/Minitial) 1 Typical Range for Launch to Low Earth Orbit 0.1 0.01 0.001 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) UNIVERSITY OF MARYLAND 4 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Sources and Categories of Vehicle Mass Payload Propellants Inert Mass Structure Propulsion Structure Avionics Propulsion Power Avionics Mechanisms Power Thermal Mechanisms Etc.Thermal Etc. UNIVERSITY OF MARYLAND 5 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Basic Vehicle Parameters • Basic mass summary mo = mpl + mpr + min • Inert mass fraction mo =initial mass mpl =payload mass mpr =propellant mass min =inert mass min min δ≡ = mo mpl + mpr + min • Payload fraction mpl mpl λ≡ = mo mpl + mpr + min • Parametric mass ratio r =λ+δ UNIVERSITY OF MARYLAND 6 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Rocket Equation (including Inert Mass) Payload Fraction, (Mpayload/Minitial) 1 Typical Range for Launch to Low Earth Orbit Inert Mass Fraction δ 0 0.1 0.05 0.1 0.15 0.01 0.2 0.001 0 2 4 6 8 Velocity Ratio, (ΔV/Ve) UNIVERSITY OF MARYLAND 7 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Limiting Performance Based on Inert Asymptotic Velocity Ratio, (ΔV/Ve) 5 4.5 Typical Feasible Design Range for Inert Mass Fraction 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Inert Mass Fraction, (Minert/Minitial) UNIVERSITY OF MARYLAND 8 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Regression Analysis of Existing Vehicles Veh/Stage Delta 6925 Stage 2 Delta 7925 Stage 2 Titan II Stage 2 Titan III Stage 2 Titan IV Stage 2 Proton Stage 3 Titan II Stage 1 Titan III Stage 1 Titan IV Stage 1 Proton Stage 2 Proton Stage 1 average standard deviation prop mass gross mass (lbs) (lbs) 13,367 15,394 13,367 15,394 59,000 65,000 77,200 83,600 77,200 87,000 110,000 123,000 260,000 269,000 294,000 310,000 340,000 359,000 330,000 365,000 904,000 1,004,000 UNIVERSITY OF MARYLAND 9 T y p e P r o p e l l a n t s Isp vac isp sl s i g m a eps delta (sec) (sec) Storable N2O4-A50 319.4 0.152 0.132 0.070 Storable N2O4-A50 319.4 0.152 0.132 0.065 Storable N2O4-A50 316.0 0.102 0.092 0.087 Storable N2O4-A50 316.0 0.083 0.077 0.055 Storable N2O4-A50 316.0 0.127 0.113 0.078 Storable N2O4-A50 315.0 0.118 0.106 0.078 Storable N2O4-A50 296.0 0.035 0.033 0.027 Storable N2O4-A50 302.0 0.054 0.052 0.038 Storable N2O4-A50 302.0 0.056 0.053 0.039 Storable N2O4-A50 316.0 0.106 0.096 0.066 Storable N2O4-A50 316.0 285.0 0.111 0.100 0.065 312.2 285.0 0.100 0.089 0.061 8.1 0.039 0.033 0.019 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Inert Mass Fractions for Existing LVs 0.18 Inert Mass Fraction 0.16 0.14 0.12 LOX/LH2 0.10 LOX/RP-1 0.08 Solid 0.06 Storable 0.04 0.02 0.00 0 1 10 100 1,000 10,000 Gross Mass (MT) UNIVERSITY OF MARYLAND 10 Rocket Performance ENAE 483/788D - Principles of Space Systems 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 ! error = (Axi + B − yi )2 i=1 – Analytical solutions exist, or use Solver in Excel • Power law fit: y=BxA error = N ! 2 [A log(xi ) + B − log(yi )] i=1 • Polynomial, exponential, many other fits possible UNIVERSITY OF MARYLAND 11 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Solution of Least-Squares Linear N ! ∂(error) =2 (Axi + B − yi )xi = 0 ∂A i=1 A N ! x2i + B i=1 N ! xi − i=1 N ! N ! ∂(error) =2 (Axi + B − yi ) = 0 ∂B i=1 xi yi = 0 i=1 A N ! xi + N B − N ! i=1 A= N ! ! ! xi yi − xi yi ! 2 ! 2 N xi − ( xi ) UNIVERSITY OF MARYLAND 12 B= ! yi = 0 i=1 ! ! yi x2i − ! 2 N xi − ! xi xi yi ! 2 ( xi ) Rocket Performance ENAE 483/788D - Principles of Space Systems Design Inert Mass Fraction Regression Analysis - Storables 0.10 0.08 0.06 0.04 2 0.02 R = 0.0541 0.00 1 10 100 1,000 Gross Mass (MT) UNIVERSITY OF MARYLAND 13 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Regression Values for Design Parameters Vacuum Ve (m/sec) Inert Mass Fraction Max ΔV (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 UNIVERSITY OF MARYLAND 14 Rocket Performance ENAE 483/788D - Principles of Space Systems Design The Rocket Equation for Multiple Stages • Assume two stages ∆V1 = −Ve1 ln ! ∆V2 = −Ve2 ln ! mf inal1 minitial1 " = −Ve1 ln(r1 ) mf inal2 minitial2 " = −Ve2 ln(r2 ) • Assume Ve1=Ve2=Ve ∆V1 + ∆V2 = −Ve ln(r1 ) − Ve ln(r2 ) = −Ve ln(r1 r2 ) UNIVERSITY OF MARYLAND 15 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Continued Look at Multistaging • There’s a historical tendency to define r0=r1r2 ∆V1 + ∆V2 = −Ve ln (r1 r2 ) = −Ve ln (r0 ) • But it’s important to remember that it’s really ∆V1 + ∆V2 = −Ve ln (r1 r2 ) = −Ve ln ! mf inal1 mf inal2 minitial1 minitial2 • And that r0 has no physical significance, since mf inal1 != minitial2 UNIVERSITY OF MARYLAND 16 " mf inal2 ⇒ r0 = ! minitial1 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Multistage Inert Mass Fraction • Total inert mass fraction min,1 + min,2 + min,3 min,1 min,2 min,3 δ0 = = + + m0 m0 m0 m0 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 δ 0 = δ1 + δ 2 λ1 + δ 3 λ 2 λ 1 • General form of the equation n δ0 = " j−1 $ stages # ! δj λ! j=1 UNIVERSITY OF MARYLAND 17 !=1 Rocket Performance ENAE 483/788D - Principles of Space Systems 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 = λ3 λ2 λ1 • Generic form of the equation n λ0 = stages ! λj j=1 UNIVERSITY OF MARYLAND 18 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Effect of Staging Inert Mass Fraction δ=0.15 0.25 Payload Fraction 0.20 1 stage 0.15 2 stage 3 stage 0.10 4 stage 0.05 0.00 0 1 2 3 4 5 Velocity Ratio (ΔV/Ve) UNIVERSITY OF MARYLAND 19 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Effect of ΔV Distribution 1st Stage: Solids Normalized Mass (kg/kg of payload) 70 2nd Stage: LOX/LH2 60 50 40 Total mass Stage 2 mass 30 Stage 1 mass 20 10 0 2000 3000 4000 5000 6000 7000 8000 9000 10000 Stage 2 ΔV (m/sec) UNIVERSITY OF MARYLAND 20 Rocket Performance ENAE 483/788D - Principles of Space Systems Design ΔV Distribution and Design Parameters 1st Stage: Solids 0.45 2nd Stage: LOX/LH2 0.4 0.35 0.3 delta_0 0.25 lambda_0 0.2 lambda/delta 0.15 0.1 0.05 0 2000 3000 4000 5000 6000 7000 8000 9000 10000 Stage 2 ΔV (m/sec) UNIVERSITY OF MARYLAND 21 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Lagrange Multipliers • Given an objective function y = f (x) subject to constraint function z = g(x) • Create a new objective function z = f (x) + λ[g(x) − z] • Solve simultaneous equations ∂y =0 ∂x UNIVERSITY OF MARYLAND 22 ∂y =0 ∂λ Rocket Performance ENAE 483/788D - Principles of Space Systems Design Optimum ΔV Distribution Between Stages • Maximize payload fraction (2 stage case) λ0 = λ1 λ2 = (r1 − δ1 )(r2 − δ2 ) subject to constraint function ∆Vtotal = ∆V1 + ∆V2 • Create a new objective function λo = ! e −∆V1 Ve,1 " ! −∆V " 2 Ve,2 e − δ1 − δ2 + K [∆V1 + ∆V2 − ∆Vtotal ] ➥Very messy for partial derivatives! UNIVERSITY OF MARYLAND 23 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Optimum ΔV Distribution (continued) • Use substitute objective function max (λo ) ⇐⇒ max [ln (λo )] • Create a new constrained objective function ln (λo ) = ln (r1 − δ1 ) + ln (r2 − δ2 ) + K [∆V1 + ∆V2 − ∆Vtotal ] • Take partials and set equal to zero ∂ [ln (λo )] =0 ∂r1 ∂ [ln (λo )] =0 ∂r2 UNIVERSITY OF MARYLAND 24 ∂ [ln (λo )] =0 ∂K Rocket Performance ENAE 483/788D - Principles of Space Systems Design Optimum ΔV Special Cases • “Generic” partial of objective function ∂ [ln (λo )] 1 Ve,i = +K =0 ∂ri ri − δ i ri • Case 1: δ1=δ2 Ve,1=Ve,2 ∆Vtotal r1 = r2 =⇒ ∆V1 = ∆V2 = 2 • Same principle holds for n stages r1 = r2 = · · · = rn =⇒ ∆Vtotal ∆V1 = ∆V2 = · · · = ∆Vn = n UNIVERSITY OF MARYLAND 25 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Sensitivity to Inert Mass ΔV for multistaged rocket ∆Vtot = where n stages ! ∆Vk = n ! Ve,k ln k=1 k=1 mo,k = mpl + mpr,k + min.k + n ! " mo,k mf,k # (mpr,j + min,j ) j=k+1 mf,k = mpl + min.k + n ! (mpr,j + min,j ) j=k+1 UNIVERSITY OF MARYLAND 26 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Finding Payload Sensitivity to Inert Mass • Given the equation linking mass to ΔV, take ∂(∆Vtot ) ∂(∆Vtot ) dmpl + dmin,j = 0 ∂mpl ∂min,j and solve to find ! ∂mpl !! ∂min,k ! "k ∂(∆Vtot )=0 # $ − j=1 Ve,j m1o,j − m1f,j $ # = " N 1 1 − V "=1 e," mo,! mf,! • This equation shows the “trade-off ratio” Δpayload resulting from a change in inert mass for stage k (for a vehicle with N total stages) UNIVERSITY OF MARYLAND 27 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Trade-off Ratio Example: Gemini-Titan II Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097 dmpl/dmin,k -0.1164 -1 UNIVERSITY OF MARYLAND 28 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Payload Sensitivity to Propellant Mass • In a similar manner, solve to find ! ∂mpl !! ∂mpr,k ! ∂(∆Vtot )=0 "k # $ 1 V j=1 e,j mo,j $ # 1 1 − V "=1 e," mo,! mf,! − =" N • This equation gives the change in payload mass as a function of additional propellant mass (all other parameters held constant) UNIVERSITY OF MARYLAND 29 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Trade-off Ratio Example: Gemini-Titan II Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097 dmpl/dmin,k -0.1164 -1 dmpl/dmpr,k 0.04124 0.2443 UNIVERSITY OF MARYLAND 30 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Payload Sensitivity to Exhaust Velocity • Use the same technique to find the change in payload resulting from additional exhaust velocity for stage k ! ∂mpl !! ∂Ve,k ! ∂(∆Vtot )=0 =" N "k j=1 "=1 Ve," ln # # mo,j mf,j 1 mo,! − $ 1 mf,! $ • This trade-off ratio (unlike the ones for inert and propellant masses) has units - kg/(m/sec) UNIVERSITY OF MARYLAND 31 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Trade-off Ratio Example: Gemini-Titan II Stage 1 Stage 2 Initial Mass (kg) 150,500 32,630 Final Mass (kg) 39,370 6099 Ve (m/sec) 2900 3097 dmpl/dmin,k -0.1164 -1 dmpl/dmpr,k 0.04124 0.2443 dmpl/dVe,k 2.870 6.459 (kg/m/sec) UNIVERSITY OF MARYLAND 32 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Parallel Staging • Multiple dissimilar engines burning simultaneously • Frequently a result of upgrades to operational systems • General case requires “brute force” numerical performance analysis UNIVERSITY OF MARYLAND 33 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Parallel-Staging Rocket Equation • Momentum at time t: M = mv • Momentum at time t+Δt: (subscript “b”=boosters; “c”=core vehicle) M = (m − ∆mb − ∆mc )(v + ∆v) +∆mb (v − Ve,b ) + ∆mc (v − Ve,c ) • Assume thrust (and mass flow rates) constant UNIVERSITY OF MARYLAND 34 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Parallel-Staging Rocket Equation • Rocket equation during booster burn ∆V = −V̄e ln ! mf inal minitial " = −V¯e ln ! min,b +min,c +χmpr,c +m0,2 min,b +mpr,b +min,c +mpr,c +m0,2 where χ= fraction of core propellant remaining after booster burnout, and where V̄e = Ve,b ṁb +Ve,c ṁc ṁb +ṁc UNIVERSITY OF MARYLAND 35 = " Ve,b mpr,b +Ve,c (1−χ)mpr,c mpr,b +(1−χ)mpr,c Rocket Performance ENAE 483/788D - Principles of Space Systems Design Analyzing Parallel-Staging Performance Parallel stages break down into pseudo-serial stages: • Stage “0” (boosters and core) ∆V0 = −V̄e ln ! min,b +min,c +χmpr,c +m0,2 min,b +mpr,b +min,c +mpr,c +m0,2 • Stage “1” (core alone) ∆V1 = −Ve,c ln ! min,c +m0,2 min,c +χmpr,c +m0,2 • Subsequent stages are as before UNIVERSITY OF MARYLAND 36 " " Rocket Performance ENAE 483/788D - Principles of Space Systems Design Parallel Staging Example: Space Shuttle • 2 x solid rocket boosters (data below for single SRB) – – – – Gross mass 589,670 kg Empty mass 86,183 kg Isp 269 sec Burn time 124 sec • External tank (space shuttle main engines) – – – – Gross mass 750,975 kg Empty mass 29,930 kg Isp 455 sec Burn time 480 sec • “Payload” (orbiter + P/L) 125,000 kg UNIVERSITY OF MARYLAND 37 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Shuttle Parallel Staging Example Ve,b = gIsp,e m m = (9.8)(269) = 2636 Ve,c = 4459 sec sec 480 − 124 = 0.7417 χ= 480 m 2636(1, 007, 000) + 4459(721, 000)(1 − .7417) = 2921 V̄e = 1, 007, 000 + 721, 000(1 − .7417) sec m 862, 000 = 3702 ∆V0 = −2921 ln 3, 062, 000 sec m 154, 900 = 6659 ∆V1 = −4459 ln 689, 700 sec m ∆Vtot = 10, 360 sec UNIVERSITY OF MARYLAND 38 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Modular Staging • Use identical modules to form multiple stages • Have to cluster modules on lower stages to make up for nonideal ΔV distributions • Advantageous from production and development cost standpoints UNIVERSITY OF MARYLAND 39 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Module Analysis • All modules have the same inert mass and propellant mass • Because δ varies with payload mass, not all modules have the same δ! • Introduce two new parameters min min ε≡ = min + mpr mmod min σ≡ mpr • Conversions δ ε= 1−λ δ σ= 1−δ−λ UNIVERSITY OF MARYLAND 40 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Rocket Equation for Modular Boosters • Assuming n modules in stage 1, nε + n(min ) + mo2 r1 = = n(min + mpr ) + mo2 n+ mo2 mmod mo2 mmod • If all 3 stages use same modules, nj for stage j, where n1 ε + n2 + n3 + ρpl r1 = n1 + n2 + n3 + ρpl mpl ρpl ≡ ; mmod = min + mpr mmod UNIVERSITY OF MARYLAND 41 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Example: Conestoga 1620 (EER) • • • • • • Small launch vehicle (1 flight, 1 failure) Payload 900 kg Module gross mass 11,400 kg Module empty mass 1,400 kg Exhaust velocity 2754 m/sec Staging pattern – – – – 1st stage - 4 modules 2nd stage - 2 modules 3rd stage - 1 module 4th stage - Star 48V (gross mass 2200 kg, empty mass 140 kg, Ve 2842 m/sec) UNIVERSITY OF MARYLAND 42 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Conestoga 1620 Performance • 4th stage ∆V mf 4 900 + 140 m ∆V4 = −Ve4 ln = −2842 ln = 3104 mo4 900 + 2200 sec • Treat like three-stage modular vehicle; Mpl=3100 kg min 1400 != = = 0.1228 mmod 11400 mpl 3100 ρpl = = = 0.2719 mmod 11400 n1 = 4; n2 = 2; n3 = 1 UNIVERSITY OF MARYLAND 43 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Constellation 1620 Performance (cont.) n1 ! + n2 + n3 + ρpl 4 × 0.1228 + 2 + 1 + 0.2719 r1 = = = 0.5175 n1 + n2 + n3 + ρpl 4 + 2 + 1 + 0.2719 n2 ! + n3 + ρpl 2 × 0.1228 + 1 + 0.2719 r2 = = = 0.4638 n2 + n3 + ρpl 2 + 1 + 0.2719 n3 ! + ρpl 1 × 0.1228 + 0.2719 r3 = = = 0.3103 n3 + ρpl 1 + 0.2719 m m V1 = 1814 ; V2 = 2116 sec sec m m V3 = 3223 ; V4 = 3104 sec sec m Vtotal = 10, 257 sec UNIVERSITY OF MARYLAND 44 Rocket Performance ENAE 483/788D - Principles of Space Systems Design Discussion about Modular Vehicles • Modularity has several advantages – Saves money (smaller modules cost less to develop) – Saves money (larger production run = lower cost/ module) – Allows resizing launch vehicles to match payloads • Trick is to optimize number of stages, number of modules/stage to minimize total number of modules • Generally close to optimum by doubling number of modules at each lower stage • Have to worry about packing factors, complexity UNIVERSITY OF MARYLAND 45 Rocket Performance ENAE 483/788D - Principles of Space Systems Design OTRAG - 1977-1983 UNIVERSITY OF MARYLAND 46 Rocket Performance ENAE 483/788D - Principles of Space Systems Design