Rocket Performance MARYLAND U N I V E R S I T Y O F. Rocket Performance. ENAE 483/788D - Principles of Space Systems Design

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1 Lecture #06 September 17, 2015 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 David L. Akin - All rights reserved

2 Derivation of the Rocket Equation Momentum at time t: Momentum at time t+δt: Some algebraic manipulation gives: M = mv Take to limits and integrate: Δm V e m v-v e m-δm v M =(m m)(v + v)+ m(v V e ) mfinal m initial m v = mv e dm m = Vfinal V initial dv V e v+δv 2

3 The Rocket Equation Alternate forms Basic definitions/concepts Mass ratio r m final or R m initial m initial m final r m final = e V Ve m ( initial ) mfinal v = V e ln = V e ln r Nondimensional velocity change Velocity ratio 3 m initial V V e

4 Rocket Equation (First Look) Mass Ratio, (M final /M initial ) Typical Range for Launch to Low Earth Orbit Velocity Ratio, (ΔV/Ve)

5 Sources and Categories of Vehicle Mass Payload Propellants Structure Propulsion Avionics Power Mechanisms Thermal Etc. 5

6 Sources and Categories of Vehicle Mass Payload Propellants Inert Mass Structure Propulsion Avionics Power Mechanisms Thermal Etc. 6

7 Basic Vehicle Parameters Basic mass summary Inert mass fraction Payload fraction m o = m pl + m pr + m in δ m in m o = λ m pl m o = Parametric mass ratio r = λ + δ m in m pl + m pr + m in m pl m pl + m pr + m in m o =initial mass m pl =payload mass m pr =propellant mass m in =inert mass 7

8 A Word About Specific Impulse Defined as thrust/propellant used English units: lbs thrust/(lbs prop/sec)=sec Metric units: N thrust/(kg prop/sec)=m/sec Two ways to regard discrepancy - lbs is not mass in English units - should be slugs Isp = thrust/weight flow rate of propellant - if Isp is in seconds, V e = g o I sp If the real intent of specific impulse is I sp = Ṫ m and T = ṁv e then I sp = V e 8

9 Rocket Equation (including Inert Mass) Payload Fraction, (M payload /M initial ) Typical Range for Launch to Low Earth Orbit Inert Mass Fraction δ Velocity Ratio, (ΔV/Ve)

10 Limiting Performance Based on Inert Mass Asymptotic Velocity Ratio, (ΔV/Ve) Typical Feasible Design Range for Inert Mass Fraction Inert Mass Fraction, (M inert /M initial ) 10

11 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 StorabN2O4-A Delta 7925 Stage 2 13,367 15,394 StorabN2O4-A Titan II Stage 2 59,000 65,000 StorabN2O4-A Titan III Stage 2 77,200 83,600 StorabN2O4-A Titan IV Stage 2 77,200 87,000 StorabN2O4-A Proton Stage 3 110, ,000 StorabN2O4-A Titan II Stage 1 260, ,000 StorabN2O4-A Titan III Stage 1 294, ,000 StorabN2O4-A Titan IV Stage 1 340, ,000 StorabN2O4-A Proton Stage 2 330, ,000 StorabN2O4-A Proton Stage 1 904,000 1,004,000 StorabN2O4-A average standard deviation

12 Inert Mass Fractions for Existing LVs Inert Mass Fraction ,000 10,000 Gross Mass (MT) LOX/LH2 LOX/RP-1 Solid Storable 12

13 Regression Analysis Given a set of N data points (x i,y i ) Linear curve fit: y=ax+b find A and B to minimize sum squared error N error = (Ax i + B y i ) 2 Analytical solutions exist, or use Solver in Excel Power law fit: y=bx A error = i=1 Polynomial, exponential, many other fits possible i=1 N [A log(x i )+B log(y i )] 2 13

14 Solution of Least-Squares Linear Regression (error) A A N x 2 i + B i=1 N =2 (Ax i + B y i )x i =0 i=1 N x i i=1 (error) B N x i y i =0 i=1 N =2 (Ax i + B y i )=0 A i=1 N x i + NB i=1 N y i =0 i=1 A = N x i y i x i yi N x 2 i ( x i ) 2 B = yi x 2 i x i xi y i N x 2 i ( x i ) 2 14

15 Regression Analysis - Storables Inert Mass Fraction R 2 = ,000 Gross Mass (MT) 15

16 Regression Values for Design Parameters Vacuum Ve (m/sec) Inert Mass Fraction. Max ΔV (m/sec) LOX/LH ,070 LOX/RP Storables Solids

17 Revised Analysis With ε Instead of δ m in m in + m pr r = + =) = r r = m pl + m in m pl + m pr + m in r = m pl m pr +m in + m in m pr +m in m pl m pr +m in + m pr+m in m pr +m in r = + +1 where m pl m in + m pr 17 = r 1 r

18 Economy of Scale for Stage Size Stage#Inert#Mass#Frac6on#ε# 0.20# 0.18# 0.16# 0.14# 0.12# 0.10# 0.08# 0.06# 0.04# 0.02# 0.00# 1# 10# 100# 1,000# 10,000# Stage#Gross#Mass,#MT# Storable#MER# LOX/RPD1# Storables# LOX/LH2# Cryo#MER# 18

19 Stage Inert Mass Fraction Estimation LOX/LH2 =0.987 (M stage hkgi) storables = (M stage hkgi) Note: storables also pertains to LOX/ hydrocarbon

20 The Rocket Equation for Multiple Stages Assume two stages V 1 = V e1 ln V 2 = V e2 ln Assume V e1 =V e2 =V e ( mfinal1 m initial1 ( mfinal2 m initial2 ) ) = V e1 ln(r 1 ) = V e2 ln(r 2 ) V 1 + V 2 = V e ln(r 1 ) V e ln(r 2 )= V e ln(r 1 r 2 ) 20

21 Continued Look at Multistaging There s a historical tendency to define r 0 =r 1 r 2 V 1 + V 2 = V e ln (r 1 r 2 ) = V e ln (r 0 ) But it s important to remember that it s really V 1 + V 2 = V e ln (r 1 r 2 ) = V e ln m final1 m initial1 m final2 m initial2 And that r 0 has no physical significance, since m final1 = m initial2 r 0 = m final2 m initial1 21

22 Multistage Inert Mass Fraction Total inert mass fraction Convert to dimensionless parameters δ 0 = m in,1 + m in,2 + m in,3 m 0 δ 0 = m in,1 m 0 0 = General form of the equation n stages j 1 0 = + m in,2 m 0,2 m 0,2 m 0 22 j=1 = m in,1 m 0 + m in,2 m 0 + m in,3 m 0,3 m 0,3 m 0,2 m 0,2 m 0 j =1 + m in,3 m 0

23 Multistage Payload Fraction Total payload fraction (3 stage example) λ 0 = m pl m 0 = m pl m 0,3 m 0,3 m 0,2 m 0,2 m 0 Convert to dimensionless parameters λ 0 = λ 3 λ 2 λ 1 Generic form of the equation λ 0 = n stages j=1 λ j 23

24 Effect of Staging 0.25 Inert Mass Fraction δ=0.15 Payload Fraction stage 2 stage 3 stage 4 stage Velocity Ratio (ΔV/Ve) 24

25 Effect of ΔV Distribution 70 1st Stage: Solids 2nd Stage: LOX/LH2 Normalized Mass (kg/kg of payload) Stage 2 ΔV (m/sec) Total mass Stage 2 mass Stage 1 mass 25

26 ΔV Distribution and Design Parameters st Stage: Solids 2nd Stage: LOX/LH delta_0 lambda_0 lambda/delta Stage 2 ΔV (m/sec) 26

27 Lagrange Multipliers Given an objective function y = f(x) subject to constraint function z = g(x) Create a new objective function y = f(x)+ [g(x) z] Solve simultaneous equations y x = 0 y = 0 27

28 Optimum ΔV Distribution Between Stages Maximize payload fraction (2 stage case) subject to constraint function Create a new objective function λ o = λ 0 = λ 1 λ 2 =(r 1 δ 1 )(r 2 δ 2 ) V total = V 1 + V 2 ( e V 1 V e,1 δ 1 )( e V 2 V e,2 δ 2 ) + K [ V 1 + V 2 V total ] Very messy for partial derivatives 28

29 Optimum ΔV Distribution (continued) Use substitute objective function max (λ o ) max [ln (λ o )] Create a new constrained objective function ln (λ o )=ln (r 1 δ 1 )+ln (r 2 δ 2 )+K [ V 1 + V 2 V total ] Take partials and set equal to zero [ln (λ o )] r 1 =0 [ln (λ o )] r 2 =0 [ln (λ o )] K =0 29

30 Optimum ΔV Special Cases Generic partial of objective function [ln (λ o )] r i = 1 r i δ i + K V e,i r i =0 Case 1: δ 1 =δ 2 V e,1 =V e,2 r 1 = r 2 = V 1 = V 2 = V total 2 Same principle holds for n stages r 1 = r 2 = = r n = V 1 = V 2 = = V n = V total n 30

31 Sensitivity to Inert Mass ΔV for multistaged rocket where V tot = n stages k=1 V k = n k=1 m o,k = m pl + m pr,k + m in,k + m f,k = m pl + m in,k + nx j=k+1 V e,k ln nx j=k+1 m o,k m f,k (m pr,j + m in,j ) (m pr,j + m in,j ) 31

32 Finding Payload Sensitivity to Inert Mass Given the equation linking mass to ΔV, take and solve to find m pl m in,k ( V tot ) m pl = ( Vtot )=0 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) dm pl + ( V tot) m in,j dm in,j =0 32 k j=1 V e,j ( N l=1 V e,l ( ) 1 m o,j 1 m f,j ) 1 m o,l 1 m f,l

33 Trade-off Ratio Example: Gemini-Titan II Initial Mass (kg) Stage 1 Stage 2 150,500 32,630 Final Mass (kg) 39, Ve (m/sec) dm

34 Payload Sensitivity to Propellant Mass In a similar manner, solve to find m pl m pr,k = ( Vtot )=0 k j=1 V e,j N l=1 V e,l ( This equation gives the change in payload mass as a function of additional propellant mass (all other parameters held constant) ( ) 1 m o,j ) 1 m o,l 1 m f,l 34

35 Trade-off Ratio Example: Gemini-Titan II Initial Mass (kg) Stage 1 Stage 2 150,500 32,630 Final Mass (kg) 39, Ve (m/sec) dm dm

36 Payload Sensitivity to Exhaust Velocity Use the same technique to find the change in payload resulting from additional exhaust velocity for stage k m pl V e,k = ( Vtot )=0 ( ) k j=1 ln mo,j m f,j N l=1 V e,l ( ) 1 m o,l 1 m f,l This trade-off ratio (unlike the ones for inert and propellant masses) has units - kg/(m/sec) 36

37 Trade-off Ratio Example: Gemini-Titan II Initial Mass (kg) Stage 1 Stage 2 150,500 32,630 Final Mass (kg) 39, Ve (m/sec) dm dm dm (kg/m/sec)

38 Parallel Staging Multiple dissimilar engines burning simultaneously Frequently a result of upgrades to operational systems General case requires brute force numerical performance analysis 38

39 Parallel-Staging Rocket Equation Momentum at time t: M = mv Momentum at time t+δt: (subscript b =boosters; c =core vehicle) M = (m m b m c )(v + v) + m b (v V e,b ) + m c (v V e,c ) Assume thrust (and mass flow rates) constant 39

40 Parallel-Staging Rocket Equation Rocket equation during booster burn V = V e ln m final m initial = V e ln m in,b +m in,c + m pr,c +m 0,2 m in,b +m pr,b +m in,c +m pr,c +m 0,2 χ where = fraction of core propellant remaining after booster burnout, and where V e = V e,bṁ b +V e,c ṁ c ṁ b +ṁ c = V e,bm pr,b +V e,c (1 χ)m pr,c m pr,b +(1 χ)m pr,c 40

41 Analyzing Parallel-Staging Performance Parallel stages break down into pseudo-serial stages: Stage 0 (boosters and core) V 0 = V e ln Stage 1 (core alone) V 1 = V e,c ln Subsequent stages are as before ( ) min,b +m in,c +χm pr,c +m 0,2 m in,b +m pr,b +m in,c +m pr,c +m 0,2 m in,c +m 0,2 m in,c + m pr,c +m 0,2 41

42 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 42

43 Shuttle Parallel Staging Example V e,b = gi sp,e =(9.8)(269) = 2636 m sec V e = χ = = (1, 007, 000) (721, 000)(1.7417) 1, 007, , 000(1.7417) V 0 = 2921 ln 862, 000 3, 062, 000 =3702m sec 154, 900 V 1 = 4459 ln 689, 700 =6659m sec V tot =10, 360 m sec 43 V e,c =4459 m sec =2921 m sec

44 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 44

45 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 Conversions m in m in + m pr = ε = δ 1 λ 45 m in m mod σ = σ m in m pr δ 1 δ λ

46 Rocket Equation for Modular Boosters Assuming n modules in stage 1, If all 3 stages use same modules, n j for stage j, r 1 = where ρ pl n(m in )+m o2 n(m in + m pr )+m o2 = r 1 = n 1ε + n 2 + n 3 + ρ pl n 1 + n 2 + n 3 + ρ pl m pl m mod ; m mod = m in + m pr 46 nε + mo2 m mod n + m o2 m mod

47 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, V e 2842 m/sec) 47

48 Conestoga 1620 Performance 4th stage Δ V V 4 = V e4 ln m f4 = 2842 ln m o4 Treat like three-stage modular vehicle; M pl =3100 kg = m in m mod = = pl = = 3104 m sec m pl = 3100 m mod = n 1 = 4; n 2 = 2; n 3 =1 48

49 Constellation 1620 Performance (cont.) r 1 = n 1 + n 2 + n 3 + pl = n 1 + n 2 + n 3 + pl r 2 = n 2 + n 3 + pl = n 2 + n 3 + pl r 3 = n 3 + pl = n 3 + pl V 1 = 1814 m sec ; V 2 = 2116 m sec V 3 = 3223 m sec ; V 4 = 3104 m V total = 10, 257 m sec sec = = =0.3103

50 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 50

51 OTRAG

52 Today s Tools Mass ratios Estimation of vehicle masses from v and inert mass fractions (both and ) Regression analysis Staging calculations Optimization of v distribution between stages Trade-off ratios Parallel staging calculations Modular vehicle calculations 52