Fundamentals: Relevant Terms
….. pushing change of momentum
Momentum (kgm/s) mass (m) x velocity (v)
Impulse (kgm/s or Ns) change of momentum
Force (N) rate of change of momentum
Specific Impulse (Ns/kg) Impulse delivered per unit mass of propellant
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Rocket Principle and Rocket Equation
Principle: Pushing itself forward by constantly ejecting out material stored
within it.
Ziolkovsky Equation (Rocket Equation) [early 1900’s]
V = Vj ln (Mi/Mf)
where, V, velocity increment
Vj, efflux velocity
Mi, initial mass
Mf, final mass
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Mass Ratio Of a Rocket
Mass Ratio Rm = Mf/Mi
where, Mf = final mass after all the propellant has been consumed
and Mi = initial mass before the rocket operation
ie. V = Vj ln (1/ Rm)
We can show that, V = Vj ln [( + + )/ ( + )]
where, = payload mass fraction, Mu/Mi
= structural mass fraction, Ms/Mi and
= propellant mass fraction, Mp/Mi
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Desirable parameters of rocket
V = Vj ln [( + + )/ ( + )]
or, = e-(V / Vj) -
i.e., to achieve higher values of payload mass fraction ,
(i) Vj needs to be large
(ii) needs to be small
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Payload mass fraction, Vs (V/Vj) and
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Rockets having small propellant mass fraction
i.e., Mp < Mu , Ms
V ~ Vj [ (/ ( + )]
= Vj (Mp/Mf)
~ Vj (Mp/Mi)
Mp = pVp
where, p the density of the propellant,
Vp the volume of the propellant in the rocket
V pVj as Vp and Mf are fairly fixed
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Performance parameters of a rocket
Impulse (I)
change in momentum = mpVj [Ns]
Specific Impulse (Isp)
impulse delivered per unit mass of propellant = I/mp = Vj [m/s, Ns/kg, s]
Density Impulse = Isp * propellant bulk density
Thrust (F)
rate of change of impulse = dI/dt = pVj [N]
Impulse to mass ratio
ratio of total impulse to the initial mass of rocket = I/Mi
Thrust to mass ratio
ratio of the thrust to the initial mass = F/Mi
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Propulsive Efficiency
Energy balance
diagram for a
chemical rocket
Propulsive efficiency determines how much of the k.e. of the exhaust jet is
useful for propelling a vehicle
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Propulsive Efficiency
Propulsive efficiency (p): ratio of the rate of work done by the rocket to the
rate of energy supplied
Rate of work done = Thrust power = Thrust (F) * Flight velocity (V)
Rate of energy supplied = Thrust power + residual kinetic jet power
i.e.
The propulsive efficiency is maximum when the forward vehicle velocity is
exactly equal to the exhaust velocity
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V/Vj
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Staging and Clustering of Rockets
liquid prop.
solid prop.
liquid prop.
solid prop (booster rocket).
strap-on rockets
(parallel staging)
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Multistaging
Mu Mu
Mu
1/2Mp 1/2Mp
Mp
For eg. for a rocket of 1000 tonnes 1/2Mp
Vj = 2700 m/s, Mu = 1 ton, Ms = 10 tonnes
V single stage = 5959 m/s
For a 2 stage rocket each with half the Mp
and Ms we get, 1/2Ms
V1 = 1590 m/s and
V2 = 5752 m/s
ie. V2-stage = 7342 m/s
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Staging and Clustering of Rockets (continued)
the total velocity at the end of the operation is equal to the sum of the ideal
velocities provided by the operation of each stage
i.e., the total ideal velocity at the end of a n stage rocket
V = V1 + V2 + V3 + V4 …….+ Vn where, V1, V2…… Vn
are the velocity provided by each stage
or V = Vj1 ln (1/ Rm1) + Vj2 ln (1/ Rm2) +……….+ Vjn ln (1/ Rmn)
= Vj ln [(1/ Rm1) (1/ Rm2)… (1/ Rmn)] for equal Vj at the end
of each stages
= Vj ln [(Mi/Mf)1(Mi/Mf)2.... (Mi/Mf)n] in terms of intial and
final masses
= Vj ln [(Mi,1/Mf,n)] with intial mass of the
successive stage equal to
the final mass of previous
stage i.e., Mf,1 = Mi,2, Mf,2
= Mi,3...
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Staging and Clustering of Rockets (continued)
In case of equal mass ratios at all stages,
i.e., V = Vj ln (1/ Rm)n for Rm1 = Rm2 =…Rmn = Rm
= n Vj ln (1/ Rm)
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Staging and Clustering of Rockets (continued)
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