staging and clustering of rockets

        The achievable values of specific impulse and structural mass fraction , encountered in practice are such that it is very difficult for a single rocket to provide the velocities required for orbiting the Earth and for escaping from it. An increase in propellant mass, if provided to give larger value of impulse , makes the rocket unwieldy and heavy and leads to smaller values of the thrust to mass ratio.

                                                                                                                               

                If a series of rockets were put one on top of the other and operated sequentially In stages one after the other , the total velocity would correspond to the sum of the ideal velocities provided  by the operation of each stage of the rocket. The above fig. gives a schematic diagram of a two stage rocket with the first stage rocket taking off from the ground and followed by the operation of 2^nd stage. The 2^nd stage gets started when its velocity is equal to the velocity provided by the 1ststageV_1.i.e. the velocity at the end of the operation of the 1^st stage . If V₂ is the velocity contributed by the 2^ndstage , the velocity at the end of operation of the 2^nd stage is V₁ + V₂.

                 In general, for a rocket having n stages , this stage continues with each successive stage provided an incremental velocity . Denoting the ideal velocity provided by each stage as V₁,V₂,…..V_n respectively , the total ideal velocity of a n stage rocket is: 

                 

                                            V=V₁+V₂+….+V_n

              The used rocket stages are discarded once their propellant is consumed in order to reduce the inert mass and hence improve the mass fraction of the upper stages . This procedure of operating a series of rockets sequentially  is known as staging of rockets .

    

                    Most of the rockets used for putting space crafts into orbit have 2 to 4 rocket stages , and are referred to as satellite launch vehicles. The first stage rocket of a satellite launch vehicle , which takes off from ground , is the heaviest and it carries the upper rocket stages on it . It is known as the Booster stage . The payload of the initial rocket stage comprises the upper stage rockets  in addition to the actual payload . If the mass ratio of the booster stage is R_m1 and the mass ratios of the upper stages are R_m2,R_m3,…..R_mnfor a rocket with n stages in series, the net ideal velocity of the rocket is the sum of the individual velocity increment  of each stage .

      

                        V =V_j1ln(1/R_m1)+ V_j2ln(1/R_m2) + ….+ V_jnln(R_mn)

We get             V=V_jln(M_i,1/M_f,n)

Where             V_j=V₁=V₂……..=V_n

       

                       V=V_jln (1/R_m)ⁿ =nV_jln (1/R_m)

           Multiple rocket stages are therefore , used to obtain higher values of V. As the No.of stages goes up , the operational complexity and cost of the multi-stage rocket increases. The No.of stages cannot  therefore , be arbitrarily increased. The main aim has been to enhance the values of V_j and decrease the structural mass fraction , keeping the No.of stage as small as possible . The present trend is to work towards a Two stage to orbit (TSTO) launch vehicle. A single stage to orbit(SSTO) vehicle would perhaps be possible if very advanced materials of construction are devised to reduce th structural and inert mass and have energetic  propellants which provide larger values of V_j than obtainable at present.

            The initial mass of a multi stage rocket is quiet large considering that the mass of all the rocket stages above booster stage have to be carried by it . The thrust of the booster rocket should be sufficient for the vehicle to accelerate and take off from ground . If the thrust of a single booster rocket is is insufficient a No.of rockets are clustered together and operated simultaneously to generate the large values of thrust required. Clustering is like parallel staging . Two booster rockets could be strapped on either side of the core rocket to get sufficient acceleration for take off. The strapped rockets are known as strap on rockets .