Chapter 8 Multistage Rockets 8.1 Notation With current technology and fuels, and without greatly increasing the e ective I sp by air-breathing, a single stage rocket to Earth orbit is still not possible. So it is necessary to reach orbit using a multistage system where a certain fraction of the vehicle mass is dropped o after use, thus allowing the non-payload mass carried to orbit to be as small as possible. The final velocity of an n stage launch system is the sum of the velocity gains from each stage. V n v 1 + v 2 + v 3 +...+ v n (8.1) The performance of an n -stage system can be optimized by proper selection of the structural mass, propellant mass and specific impulse of each of the n stages. Let the index i refer to the ith stage of an n stage launch system.the structural and propellant parameters of the system are as follows. M 0i - The total initial mass of the ith vehicle prior to firing including the payload mass, ie, the mass of i, i + 1, i + 2, i + 3,..., n stages. M pi - The mass of propellant in the ith stage. M si - Structural mass of the ith stage alone including the mass of its engine, controllers and instrumentation as well as any residual propellant which is not expended by the end of the stage burn. M L - The payload mass 8-1
CHAPTER 8. MULTISTAGE ROCKETS 8-2 Figure 8.1 schematically shows a three stage rocket at each regime of flight. Define the following variables. Payload ratio Figure 8.1: Three stage rocket notation. M 0(i+1) i M 0i M 0(i+1) n M 0(n+1) M 0n M 0(n+1) M L M 0n M L (8.2) Structural coefficient " i M Si M 0i M 0(i+1) M Si M Si + M Pi (8.3) Mass ratio
CHAPTER 8. MULTISTAGE ROCKETS 8-3 Ideal velocity increment R i M 0i M 0i M Pi 1+ i " i + i. (8.4) V n C i ln (R i ) 1+ C i ln i " i + i. (8.5) Payload fraction M L M 01 M02 M03 M04 M 01 M 02 M 03 1 2 3... 1+ 1 1+ 2 1+ 3 ML... M 0n n 1+ n. (8.6) Take the logarithm of (8.6) to express the payload fraction as a sum in terms of the payload ratios ln ( ) ln i 1+ i. (8.7) 8.2 The variational problem The structural coe cients, " i and e ective exhaust velocities, C i, are known constants based on some prior choice of propellants and structural design for each stage. The question is: how should we distribute the total mass of the vehicle among the various stages? In other words, given V n, choose the distribution of stage masses so as to maximize the payload fraction,. It turns out that the alternative statement; given maximize the final velocity V n, leads to the same distribution of stage masses. The mathematical problem is to maximize for fixed ln ( ) G ( 1, 2, 3,..., n) (8.8) V n F ( 1, 2, 3,..., n) (8.9)
CHAPTER 8. MULTISTAGE ROCKETS 8-4 or, equivalently, maximize (8.9) for fixed (8.8). The approach is to vary the payload ratios, ( 1, 2, 3,..., n), so as to maximize. Near a maximum, a small change in the i will not change G. @G G i 0 (8.10) The basic idea is shown in Figure 8.2. Figure 8.2: Variation of G near a maximum. The i are not independent, they must be chosen so that V n is kept constant. @F F i 0 (8.11) Thus only n 1 of the i can be treated as independent. Without loss of generality let s choose n to be determined in terms of the other payload ratios. The sums (8.10) and (8.11) are 1 1 @G @F @G i + @F i + n 0 n 0 9 >. (8.12) >; Use the second sum in (8.12) to replace n in the first
CHAPTER 8. MULTISTAGE ROCKETS 8-5 1 @G + 1 @F i 0 (8.13) where @F @G / (8.14) plays the role of a Lagrange multiplier. Since the equality (8.13) must hold for arbitrary i, the coe cients in brackets must be individually zero. @G From the definition of given by (8.14) + 1 @F 0; i 1, 2, 3,...,n 1 (8.15) @G + 1 @F 0. (8.16) We now have n + 1 equations in the n +1unknowns( 1, 2, 3,..., n, ). @G V n + 1 @F 0; i 1, 2, 3,...,n 1+ i C i ln " i + i 9 > >; (8.17) If we supply the expressions for F and G in (8.17) the result for the optimal set of payload ratios is (no sum on the index i) i " i (C i C i " i ). (8.18) The Lagrange multiplier is determined from the expression for V n. V n X Ci C i ln " i C i (8.19) Note that has units of velocity. Finally, the optimum overall payload fraction is
CHAPTER 8. MULTISTAGE ROCKETS 8-6 ln ( ) ln " i (C i C i " i + " i ). (8.20) 8.3 Example - exhaust velocity and structural coe cient the same for all stages Let C i C and " i " be the same for all stages. In this case The payload ratio is C 1 "e ( Vn nc ). (8.21) 1 "e( Vn nc ) e ( Vn nc ) 1. (8.22) The payload fraction is 1 "e ( Vn nc ) (1 ") e ( Vn nc )! n (8.23) and the mass ratio is R e ( Vn nc ). (8.24) Consider a liquid oxygen, kerosene system. Take the specific impulse to be 360 sec implying C 3528 m/ sec; a very high performance system. Let V n 9077 m/ sec needed to reach orbital speed. The structural coe cient is " 0.1 and let the number of stages be n 3. The stage design results are 2696 m/ sec, 0.563, R 2.3575 and the payload fraction is 0.047. (8.25) Less than 5% of the overall mass of the vehicle is payload. It is of interest to see how much better we can do by increasing the number of stages in this problem. Equation (8.23) is plotted in Figure 8.3 using the parameters of the problem.
CHAPTER 8. MULTISTAGE ROCKETS 8-7 Figure 8.3: Payload fraction as a function of number of stages for a constant parameter high performance launch vehicle. It is clear that beyond three stages, there is very little increase in payload. Note also that one stage cannot make orbit even with zero payload for the assumed value of ". 8.4 Problems Problem 1 - A two stage rocket is to be used to put a payload of 1000 kg into low earth orbit. The vehicle will be launched from Kennedy Space Center where the speed of rotation of the Earth is 427 m/ sec. Assume gravitational velocity losses of about 1200 m/ sec and aerodynamic velocity losses of 500 m/ sec. The first stage burns kerosene and oxygen producing a mean specific impulse of 320 sec averaged over the flight, while the upper stage burns hydrogen and oxygen with an average specific impulse of 450 sec. The structural coe cient of the first stage is 0.05 and that of the second is 0.07. Determine the payload ratios and the total mass of the vehicle. Suppose the same vehicle is to be used to launch a satellite into a north-south orbit from a launch complex on Kodiak island in Alaska. How does the mass of the payload change? Problem 2 - A group of universities join together to launch a four stage rocket with a small payload to the Moon. The fourth stage needs to reach the earth escape velocity of 11, 176 m/ sec. The vehicle will be launched from Kennedy Space Center where the speed of rotation of the Earth is 427 m/ sec. Assume gravitational velocity losses of about 1500 m/ sec and aerodynamic velocity losses of 600 m/ sec. To keep cost down, four stages with the same e ective exhaust velocity C and structural coe cient " are used. Each stage burns kerosene and oxygen producing a mean specific impulse of 330 sec averaged over each segment of the flight. The structural coe cient of each stage is " 0.1. Is the payload
CHAPTER 8. MULTISTAGE ROCKETS 8-8 fraction greater than zero? Problem 3 - A low-cost four stage rocket is to be used to launch small payloads to orbit. The concept proposed for the system utilizes propellants that are safe and cheap but provide a specific impulse of only 200 sec. All four stages are identical. What structural e ciency is required to reach orbit with a finite payload?