Analysis of Multistage Rockets P M V Subbarao Professor Mechanical Engineering Department Maximization of reduction in the vehicle’s mass on the way to orbit ….

Multistage Rockets : Definitions Total mass of rocket, mt, may be written as sum of 3 primary components: Payload mass, mL Propellant mass, mp Structural mass, ms Includes everything but payload and propellant Engines, tanks, controls, etc. If rocket consumes all its propellant during firing, burnout mass consists of structure and payload:

Mass Reductions by A Rocket on the Way to Orbit

Details of Mass Vs Distance in an ith Stage

Structural coefficient of ith stage Mass Fractions Structural coefficient of ith stage Payload ratio in ith stage Payload ratio in final stage Propellant ratio of ith stage

Dynamic Mass of the Rocket : Mass fractions Mass ratio for ith stage

Rocket Equation for ith Stage

Velocity Increment in ith Stage Firing

Final Velocity of an n Stage Launch System The final velocity of an n stage launch system is the sum of the velocity gains from each stage. Total mass of Propellant consumed in n stage:

The Art of Multi-staging in Rockets Main idea is to minimize total propellant consumption required to launch a pay load at a given final velocity. Discard empty tanks and extra structure as rocket travels, so that this mass is not subjected to gravity losses Large engines used for initial high thrust phase, may produce excessive accelerations when propellant is nearly consumed. Multistage rocket is a series of individual vehicles or stages, each with its own structure, tanks and engines Each stage accelerates payload before being detached. Two points: Stages are ordered in number of firing. Analysis of multistage rockets is similar to that for single stage Payload for an particular stage is the mass of all subsequent stages

Design Evaluation of Launch vehicle Space (Ideal) velocity increment Payload fraction

The Indian : POLAR SATELLITE LAUNCH VEHICLE Polar Satellite Launch Vehicle (PSLV) is the third generation launch vehicle of India. It is the first Indian launch vehicle to be equipped with liquid stages. PSLV emerged as the reliable and versatile workhorse launch vehicle of India. It executed 39 consecutively successful missions by June 2017. During 1994-2017 period, the vehicle has launched 48 Indian satellites and 209 satellites for customers from abroad. Besides, the vehicle successfully launched two spacecraft – Chandrayaan-1 in 2008 and Mars Orbiter Spacecraft in 2013 – that later traveled to Moon and Mars respectively

TECHNICAL SPECIFICATIONS : PSLV for Sun-synchronous Polar Orbits Payload to SSPO: 1,750 kg PSLV can take up to 1,750 kg of payload to Sun-Synchronous Polar Orbits of 600 km altitude. Due to its unmatched reliability, PSLV has also been used to launch various satellites into Geosynchronous and Geostationary orbits, like satellites from the IRNSS constellation. Payload to Sub GTO: 1,425 kg

PSLV - Stage 1 First Stage: PS1 PSLV uses the S139 solid rocket motor that is augmented by 6 solid strap-on boosters. Engine: S139Fuel: HTPBMax. Thrust: 4800kN Strap-on Motors PSLV uses 6 solid rocket strap-on motors to augment the thrust provided by the first stage in its PSLV-G and PSLV-XL variants. However, strap-ons are not used in the core alone version (PSLV-CA). Fuel: HTPBMax. Thrust: 719 kN

PSLV - Stage 2 SLV uses an Earth storable liquid rocket engine for its second stage, know as the Vikas engine, developed by Liquid Propulsion Systems Centre. Engine: VikasFuel: UDMH + N2O4Max. Thrust: 799 kN

PSLV - Stage 3 The third stage of PSLV is a solid rocket motor that provides the upper stages high thrust after the atmospheric phase of the launch. Fuel: HTPBMax. Thrust: 240 kN

PSLV - Stage 4 The PS4 is the uppermost stage of PSLV, comprising of two Earth storable liquid engines. Engine: 2 x PS-4Fuel: MMH + MONMax. Thrust: 7.6 x 2 kN. Height: 44 m Diameter: 2.8 m Number of Stages: 4 Lift Off Mass: 320 tonnes (XL) Variants: 3 (PSLV-G, PSLV - CA, PSLV - XL) First Flight: September 20, 1993

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

Thermodynamic Design of ith Stage For an ith stage rocket engine: Where: ejects = propellant mass flow rate pe = exit pressure, paamb = ambient pressure Vejects = exit plane velocity, Ae = exit area 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.

Total Impulse (Itot) Defined as:: Thus the same total impulse may be obtained by high T, short tb (usually preferable), or low T, long tb

Effective Exhaust Velocity Convenient to define an effective exhaust velocity (Veff), 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

Thermodynamic Performance - Specific Impulse

Thermodynamic Performance - Specific Impulse Variable Parameters - Observations Strong pressure ratio effect - but rapidly diminishing returns after about 30:1. High Tc value desirable for high I - but gives problems with heat transfer into case walls and dissociation of combustion products – practical limit between about 2750 and 3500 K, depending on propellant. Low value of molecular weight desirable – favouring use of hydrogen-based fuels. Low values of  desirable.

Thrust Coefficient (CF) Maximum thrust when exhausting into a vacuum (e.g. in space), when: (11a)

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.