THE TSIOLKOVSKY ROCKET EQUATION

Force Balance on A Rocket

Let

For a typical launch vehicle headed to an orbit, aerodynamic drag losses are typically quite small, on the order of 100 to 500 m/sec. Gravitational losses are larger, generally ranging from 700 to 1200 m/sec depending on the shape of the trajectory to orbit. By far the largest term is the equation for the space velocity increment.

REACHING ORBIT The lowest altitude where a stable orbit can be maintained, is at an altitude of 185 km. This requires an Orbital velocity approximately 7777 m/sec. To reach this velocity from a Space Center, a rocket requires an ideal velocity increment of 9050 m/sec. The velocity due to the rotation of the Earth is approximately 427 m/sec, assuming gravitational plus drag losses of 1700 m/sec. A Hydrogen-Oxygen system with an effective average exhaust velocity (from sealevel to vacuum) of 4000 m/sec would require Mi/ Mf = 9.7.

Geostationary orbit A circular geosynchronous orbit in the plane of the Earth's equator has a radius of approximately 42,164 km (26,199 mi) from the center of the Earth. A satellite in such an orbit is at an altitude of approximately 35,786 km (22,236 mi) above mean sea level. It maintains the same position relative to the Earth's surface. If one could see a satellite in geostationary orbit, it would appear to hover at the same point in the sky. Orbital velocity is 11,066 km/hr= 3.07 km/sec (6,876 miles/hr).

Travel Cycle of Modern Spacecrafts

MULTISTAGE ROCKETS With current technology and fuels, a single stage rocket to orbit is still not possible. It is necessary to reach orbit using a multistage system where a certain fraction of the vehicle mass is dropped off 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.

ANALYSIS OF MULTISTAGE ROCKETS M0i : The total initial mass of the ith stage prior to firing including the payload mass,ie, the mass of i, i+1, i+2, i+3,...., n stages. Mpi : The mass of propellant in the ith stage.

MOMENTUM BALANCE FOR A ROCKET Rocket mass X Acceleration = Thrust – Drag -gravity effect

Rocket Principles High pressure/temperature/velocity exhaust gases provided through combustion and expansion through nozzle of suitable fuel and oxidiser mixture. A rocket carries both the fuel and oxidiser onboard the vehicle whereas an air-breather engine takes in its oxygen supply from the atmosphere.

Criteria of Performance Specific to rockets only. thrust specific impulse total impulse effective exhaust velocity thrust coefficient characteristic velocity

Thrust (F) For a rocket engine: Where: = propellant mass flow rate pe = exit pressure, paamb = ambient pressure Uejects = exit plane velocity, Ae = exit area

Specific Impulse (I or Isp) The ratio of thrust / ejects mass flow rate is used to define a rocket’s specific impulse-best measure of overall performance of rocket motor. In SI terms, the units of I are m/s or Ns/kg. In the US: with units of seconds - multiply by g (i.e. 9.80665 m/s2) in order to obtain SI units of m/s or Ns/kg. Losses mean typical values are 92% to 98% of ideal values.

Total Impulse (Itot) Defined as: where tb = time of burning If FT is constant during burn:

Thus the same total impulse may be obtained by either : high FT, short tb (usually preferable), or low FT, long tb Also, for constant propellant consumption (ejects) rate:

Effective Exhaust Velocity (c) Convenient to define an effective exhaust velocity (c), where:

Thrust Coefficient (CF) Defined as: where pc = combustion chamber pressure, At = nozzle throat area Depends primarily on (pc/pa) so a good indicator of nozzle performance – dominated by pressure ratio.

Characteristic Velocity (c*) Defined as: (6) Calculated from standard test data. It is independent of nozzle performance and is therefore used as a measure of combustion efficiency – dominated by Tc (combustion chamber temperature).

Thermodynamic Performance - Thrust Parameters affecting thrust are primarily: mass flow rate exhaust velocity exhaust pressure nozzle exit area

Thrust Coefficient (CF) - Observations More desirable to run a rocket under-expanded (to left of optimum line) rather than over-expanded. Uses shorter nozzle with reduced weight and size. Increasing pressure ratio improves performance but improvements diminish above about 30/1. Large nozzle exit area required at high pressure ratios – implications for space applications.

Actual Rocket Performance Performance may be affected by any of the following deviations to simplifying assumptions: Properties of products of combustion vary with static temperature and thus position in nozzle. Specific heats of combustion products vary with temperature. Non-isentropic flow in nozzle. Heat loss to case and nozzle walls. Pressure drop in combustion chamber due to heat release. Power required for pumping liquid propellants. Suspended particles present in exhaust gas.

Internal Ballistics Liquid propellant engines store fuel and oxidiser separately - then introduced into combustion chamber. Solid propellant motors use propellant mixture containing all material required for combustion. Majority of modern GW use solid propellant rocket motors, mainly due to simplicity and storage advantages. Internal ballistics is study of combustion process of solid propellant.

Solid Propellant Combustion Combustion chamber is high pressure tank containing propellant charge at whose surface burning occurs. No arrangement made for its control – charge ignited and left to itself so must self-regulate to avoid explosion. Certain measure of control provided by charge and combustion chamber design and with inhibitor coatings.