1. EXTROVERTSpace Propulsion 02 1 Thrust, Rocket Equation, Specific Impulse, Mass Ratio

2. EXTROVERTSpace Propulsion 02 2 Thrust comes from: a) Increase in momentum of the propellant fluid (momentum thrust) b) Pressure at the exit plane being higher than the outside pressure (pressure thrust). Where does the thrust act? In the rocket engine, the force is felt on the nozzle and the combustor walls, and is transmitted through the engine mountings to the rest of the vehicle. Effective Exhaust Velocity Thrust

3. EXTROVERTSpace Propulsion 02 3 Consider a rocket with effective exhaust velocity c e. As propellant is blasted out the exhaust nozzle, the mass of the vehicle decreases. This is substantial in the case of the rocket as compared to air-breathing engines, because all the propellant comes from inside the vehicle. From Newton's Second Law,. where M 1 is the initial mass, which includes the propellant, and M 2 is the mass after the propellant has been used up to achieve the velocity increment  V. Delta V and Mass Ratio

4. EXTROVERTSpace Propulsion 02 4 Specific Impulse of the system is where g is the standard value of acceleration due to gravity at sea-level (9.8m/s 2 ). Note that the unit of Specific Impulse is seconds. Using this definition, Mass Ratio of a rocket is Specific Impulse and Mass Ratio Note: Some organizations express Specific Impulse without dividing by g

5. EXTROVERTSpace Propulsion 02 5 For missions from Earth's surface to escape from earth's gravitational field, Mass Ratio is large. For specific impulse of 390 s, g = 9.8 m/s 2, and  V = 11,186 m/s (36700 fps), the mass ratio is 18.67. This means that the rocket at launch time must be at least 18.67 times as big as the spacecraft which is left after all the fuel is burned. To get a high specific impulse like 390 s, we have to use a costly system like liquid hydrogen - liquid oxygen. For earth orbit, the velocity increment  V needed is 25,000 fps, while 36,700fps will enable escape from Earth's gravitational field. Example:

6. EXTROVERTSpace Propulsion 02 6 To find the velocity increment required for various missions, we must calculate trajectories and orbits. This is done using Newton's Law of Gravitation: Here the lhs is the "radial force" of attraction due to gravitation, between two bodies; the big one of mass M, and the little one of mass m. The universal gravitational constant G is 6.670 * 10 -11 Nm 2 /kg 2. Newton's Law of Gravitation

7. EXTROVERTSpace Propulsion 02 7 Rocket Equation Including Drag and Gravity: Ref: Hill & Peterson, Chapter 10.. The rate of acceleration of the vehicle is Neglecting the air drag and gravity terms, we get the Ideal Rocket Equation

8. EXTROVERTSpace Propulsion 02 8 Drag Term in the Rocket Equation With density in kg/m 3 and h in meters, a = 1.2 and b = 2.9 x 10 -5 Roughly, density at 30,000 meters is about 1% of its sea-level value. where atmospheric density above Earth varies roughly as Drag Coefficients (typical) Note: drag coefficient peak is reached at around Mach 1.2. Inclination, deg. to flt. direction,CD low-speedCD peak @1.2CD @ MachCD @ Mach 2 00.060.150.13 4 0.1850.16 8 0.230.2

9. EXTROVERTSpace Propulsion 02 9 Gravity Term At 100 miles above the surface the change from the surface is still only about 5%.

10. EXTROVERTSpace Propulsion 02 10 Specific impulse of 390 s, g 0 = 9.8 m/s 2, and  V = 11186 m/s (36700 fps), Mass ratio is 18.67. This means that the rocket at launch time must be at least 18.67 times as big as the spacecraft which is left after all the fuel is burned. To get a high specific impulse like 390 s, we have to use a costly system like liquid hydrogen - liquid oxygen. Velocity increment  V for Low Earth Orbit: ~ 25,000 fps, Escape from Earth's gravitational field ~ 36,700fps Example

11. EXTROVERTSpace Propulsion 02 11 Single Stage Sounding Rocket Altitude at burnout, assuming it goes straight up: Neglecting drag If is constant, (Sounding rocket: not quite single-stage) NASA Goddard Space Flight center http://www.gsfc.nasa.gov

12. EXTROVERTSpace Propulsion 02 12 Single-Stage Sounding Rocket Going Straight Up (cont’d) Define Mass Ratio Substituting,

13. EXTROVERTSpace Propulsion 02 13 Equating kinetic energy at burnout with change in potential energy of the final mass Expression for Maximum Altitude Reached Note that at burnout, the sounding rocket is still moving fast upward.

14. EXTROVERTSpace Propulsion 02 14 Example Values given: Ce = 3000 m/s R = 10 tb = 30s Hmax = ??

15. EXTROVERTSpace Propulsion 02 15 Chemical Rockets Burn time of existing rockets is ~ 30 to 200 seconds. Payload ratio Structure coefficient Thus, Definitions Payload Mass Structure (incl. engine) Mass Propellant Mass

16. EXTROVERTSpace Propulsion 02 16 Multistaging -1 Total initial mass of i-th stage prior to firing, include its effective payload. Total mass of i-th stage after burnout, include its effective payload. Payload of last stage. Structural mass of i-th stage; include engine controls, instruments.

17. EXTROVERTSpace Propulsion 02 17 Payload of stage is mass of all subsequent stages. Structural coefficient of stage: If stage contains no propellant at burnout: Multistaging - 2

18. EXTROVERTSpace Propulsion 02 18 Mass ratio of stage i.e.,. Similar stages: same and Multistaging - 3

19. EXTROVERTSpace Propulsion 02 19 If are equal, Multistaging - 4

20. EXTROVERTSpace Propulsion 02 20 Structural coefficient Multistaging - 5

21. EXTROVERTSpace Propulsion 02 21 EngineJ2H1 Thrust, kN1023 FuelHydrogenHydrocarbon Engine Mass, Millions of grams1.6220.921 Engine Mass Fraction0.0240.014 Tank Mass/ Propellant Mass0.0460.016 Eq. Exhaust Vel. m/s41752891 Specific Impulse, seconds426295 Apollo engines (Source: Hill Peterson, page 479)

22. EXTROVERTSpace Propulsion 02 22 Stage123 EngineF-1J-2 FuelRP1 HydrocarbonLH2 Number of engines551 Total thrust, Newton33 Million4.45 Million0.89 Million Total Initial Mass, Kg 2.780 Million0.677 Million0.215 Million Propellant mass, kg1.997 Million0.429 Million0.109 Million Structure & engine, kg 0.106 Million0.0326 Million0.0257 Million Structure mass fraction 0.050.0710.191 Payload fraction0.3210.4660.603 Saturn V Apollo 11 Flight Configuration

23. EXTROVERTSpace Propulsion 02 23 The idea of “Thrust Coefficient” where F is the thrust, At is the nozzle throat area and P0 is the combustion chamber stagnation pressure. Basically a higher thrust coefficient means a better usage of the available stagnation pressure in converting to thrust. The thrust coefficient has values ranging from 0.8 to 1.9. Note also that a plot of thrust coefficient vs. altitude for a given nozzle will give the variation of thrust with altitude for a given chamber pressure and nozzle throat area. The thrust coefficient is also used to compare different nozzle designs for given constraints. In the following we will use gas dynamics to derive expressions for the thrust coefficient in terms of gas properties.

24. EXTROVERTSpace Propulsion 02 24 Thrust Coefficient - 1 where A t is nozzle throat area and p 0 is chamber pressure (N/m 2 ) Thus, For sonic conditions at the throat, and

25. EXTROVERTSpace Propulsion 02 25 Using isentropic flow relations, and Thrust Coefficient Depends entirely on nozzle characteristics. The thrust coefficient is used to evaluate nozzle performance. Thrust Coefficient - 2

26. EXTROVERTSpace Propulsion 02 26 Characteristic Exhaust Velocity c* Used to characterize the performance of propellants and combustion chambers independent of the nozzle characteristics. where is the quantity in brackets. Note: So Characteristic exhaust velocity Assuming steady, quasi-1-dimensional, perfect gas. The condition for maximum thrust is ideal expansion: nozzle exit static pressure being equal to the outside pressure. In other words,

27. EXTROVERTSpace Propulsion 02 27 We’ll end Lecture 2 here, and go on to discuss orbits before getting back to compressible Flow and chemistry considerations. The purposes are: 1.To enable mission calculations. 2.To give everyone a chance to look at the content so far and see how much they need review of compressible flow material. Please browse the web links in the first lecture. End of Section 2