Rocket Performance MARYLAND

Transcription

1 Rocket Perforance The rocket equation Mass ratio and perforance Structural and payload ass fractions Multistaging Optial V distribution between stages Trade-off ratios Parallel staging Modular staging 22 David. Akin - All rights reserved Rocket Perforance Principles of Space Systes Design

2 Derivation of the Rocket Equation Moentu at tie t: M v Moentu at tie t+ t: M ( )( v+ v) + ( v V e ) Soe algebraic anipulation gives: v Take to liits and integrate: V e final d initial V V final initial dv V e Rocket Perforance Principles of Space Systes Design

3 The Rocket Equation r Alternate fors final initial e Basic definitions/concepts Mass ratio V V e r V V final or R initial Nondiensional velocity change Velocity ratio e final ln Ve ln( r) initial V V e initial final Rocket Perforance Principles of Space Systes Design

4 Rocket Equation (First ook) Mass Ratio, (M final /M initial ).. Typical Range for aunch to ow Earth Orbit Velocity Ratio, ( V/Ve) Rocket Perforance Principles of Space Systes Design

5 Sources and Categories of Vehicle Mass Payload Propellants Inert Mass Structure Propulsion Avionics Mechaniss Theral Etc. Rocket Perforance Principles of Space Systes Design

6 Basic Vehicle Paraeters Basic ass suary + p+ i Inert ass fraction δ i Payload fraction λ Paraetric ass ratio i + + p i + + p i r λ + p i δ initial ass payload ass propellant ass inert ass Rocket Perforance Principles of Space Systes Design

7 Rocket Equation (including Inert Mass) Payload Fraction, (M payload /M initial )... Typical Range for aunch to ow Earth Orbit Velocity Ratio, ( V/Ve) Inert Mass Fraction δ Rocket Perforance Principles of Space Systes Design

8 The Rocket Equation for Multiple Stages Assue two stages V V V e, V 2 e, 2 Assue V e, V e,2 V e final, ln Ve, ln( r) initial, final, 2 ln Ve, 2ln( r2) initial, 2 V + V V ln( r) V ln( r ) V ln( rr ) 2 e e 2 e 2 Rocket Perforance Principles of Space Systes Design

9 Continued ook at Multistaging Converting to asses V+ V2 Veln( rr 2) Veln Keep in ind that final, ~ initial,2 final, initial, final, 2 initial, 2 final, 2 V+ V2 Veln( rr 2) Veln Ve ln( r ), r has no physical significance! initial Rocket Perforance Principles of Space Systes Design

10 Multistage Vehicle Paraeters Inert ass fraction n stages j δ ij, δ λ j l Payload fraction λ j l n stages i λ Payload ass/inert ass ratio i λ δ Rocket Perforance Principles of Space Systes Design

11 Effect of Staging Inert Mass Fraction δ.2 Payload Fraction, (M payload /M initial ) stage 2 stage 3 stage 4 stage Velocity Ratio, ( V/Ve) Rocket Perforance Principles of Space Systes Design

12 Effect of V Distribution. st Stage: OX/H2 2nd Stage: OX/H labda delta lad/del st Stage Delta-V (/sec) Rocket Perforance Principles of Space Systes Design

13 agrange Multipliers Given an objective function y f( x) subject to constraint function z g( x) Create a new objective function y f( x) + λ g( x) z Solve siultaneous equations y x y λ [ ] Rocket Perforance Principles of Space Systes Design

14 Optiu V Distribution Between Stages Maxiize payload fraction (2 stage case) λ λ λ ( r δ )( r δ ) subject to constraint function Vtotal V + V 2 Create a new objective function V V 2 λ Ve δ Ve2 ( e )( e δ ) + K V + V V 2 2 Very essy for partial derivatives! [ ] Total Rocket Perforance Principles of Space Systes Design

15 Optiu V Distribution (continued) Use substitute objective function ax( λ ) ax ln( λ ) [ ] Create a new constrained objective function ln( λ ) ln( r δ ) + ln( r δ ) 2 2 [ ] + K V + V ln( r) + V ln( r ) Total e e2 2 Take partials and set equal to zero ln( λ) r ln( λ) r 2 ln( λ ) K Rocket Perforance Principles of Space Systes Design

16 Optiu V Special Cases Generic partial of objective function ln( λ ) r Case : δ δ 2 V e V e2 Case 2: δ δ 2 V e V e2 More coplex cases have to be done nuerically K V ei + r δ r i i i r r V V 2 2 r r 2 δ δ 2 i Rocket Perforance Principles of Space Systes Design

17 Sensitivity to Inert Mass V for ultistaged rocket n stages n Vtot Vk V ek, ln k k ok, fk, Vtot d Vtot d + ij n ok, pk, ik, p, j i, j j k+ n fk, ik, pj, ij, j k ij, ( ), ( ) Rocket Perforance Principles of Space Systes Design

18 Trade-off Ratio: Payload< >Inert Mass ik, V Total k j N l V V e, j e, l oj o, l f,, f, j l Rocket Perforance Principles of Space Systes Design

19 Rocket Perforance Principles of Space Systes Design Trade-off Ratio : Payload< >Propellant V V pk V e j oj j k e o f N Total,,,,,, l l l l

20 Trade-off Ratio: Payload< >Exhaust Velocity V ek, V Total N l V k j e, l ok, ln ok, o l f,, l Rocket Perforance Principles of Space Systes Design

21 Trade-off Ratio Exaples: Saturn IB Stage (Ve, M/sec) Stage Initial Mass (kg) Stage Final Mass (kg) ik, pk, V ek, (2568) 588, 8, (426) 44, 38, V Note that has units of ek, kg sec Rocket Perforance Principles of Space Systes Design

22 Parallel Staging Multiple dissiilar engines burning siultaneously Frequently a result of upgrades to operational systes General case requires brute force nuerical perforance analysis Rocket Perforance Principles of Space Systes Design

23 Parallel-Staging Rocket Equation Moentu at tie t: M v Moentu at tie t+ t: (subscript b boosters; c core vehicle) M ( b c)( v+ v) + ( v V ) + ( v V ) b eb, c ec, Assue thrust (and ass flow rates) constant Rocket Perforance Principles of Space Systes Design

24 Parallel-Staging Rocket Equation Rocket equation during booster burn final ib, + ic, + χ pc, + o, 2 V Ve ln Ve ln initial where V e V + V + V + V χ eb, b ec, c eb, pb, ec, pc, b c ib, pb, ic, pc, o, 2 ( ) ( χ) + pb, pc, χ and fraction of core propellant left when booster propellant is depleted Rocket Perforance Principles of Space Systes Design

25 Analyzing Parallel-Staging Perforance Parallel stages break down into pseudo-serial stages: Stage (boosters and core) ib, + ic, + χ pc, + o, 2 V Ve ln ib, pb, ic, pc, o, 2 Stage (core alone) ic+, o, 2 V V ec, ln ic+ pc+, χ, o, 2 Subsequent stages are as before Rocket Perforance Principles of Space Systes Design

26 Modular Staging Use identical odules to for ultiple stages Have to cluster odules on lower stages to ake up for nonideal V distributions Advantageous fro production and developent cost standpoints Rocket Perforance Principles of Space Systes Design

27 Module Analysis All odules have the sae inert ass and propellant ass Because δ varies with payload ass, not all odules have the sae δ! Introduce two new paraeters i ε σ i + i p p Conversions δ δ ε σ λ δ λ Rocket Perforance Principles of Space Systes Design

28 Rocket Equation for Modular Boosters Assuing n odules in stage, nε + n ( i)+ o, 2 o r n ( i + p)+ o, 2 n + If all 3 stages use sae odules, n j for stage j, where ρ r nε+ n2 + n3 + ρ n + n + n + ρ 2 3 ; od i + od o, 2,od o, 2 o,od p Rocket Perforance Principles of Space Systes Design