D. P. Mishra
5.7 MULTISTAGE ROCKET ENGINES
The payload fraction LF and structural fraction SF are determined as LF m= l/m0=200 30 000 0 0066/ , = .
SF m m= s/ 0 =800 30 000 0 0266/ , = .
The velocity increment for an ideal case can be determined by using Equation 5.42:
DV =9 81 250. ´ ln
( )
30 8 34= . km s/The velocity can be attained ideally without taking into account drag and gravitational pull. In actual sense, it will be quite a low value.
In order to escape the earth’s gravity, velocity increment should be 11.2 km/s. Hence, it would not be possible to escape the earth’s grav- ity with this rocket engine. One solution would be to use a multistage engine, which is discussed in the following section.
5.7.1 Multistaging
Instead of a single large rocket, a series of rocket motors each with its own structure, tanks, and engines are used to enhance the velocity increment for the entire vehicle for the same Isp. As the propellant is consumed in each stage, its tank is dropped from the vehicle, at intervals. Thus, the propellant is not wasted in accelerating the unnecessary structural masses to attain a higher velocity. As a result, a launch vehicle can achieve higher velocity by using staging. Several kinds of staging have evolved over the years as shown in Figure 5.11 which will be discussed later. Now let us consider a tandem type shown in Figure 5.11a in which all the stages are placed in series on one another in the order of size. Generally, the first stage needs to impart higher thrust and total impulse as it is the largest mass to carry during launching operation. Hence, it is also known as the booster stage.
Lower thrusts are to be provided in subsequent stages. Note that the pay- load for the first stage would be equal to the total mass of all upper stages along with the actual payload to be placed in the orbit. The second stage
(b)
Two droppable strap-on booster
First stage Second
stage Payload
Droppable rocket with booster engines
and controls First stage Second stage Third
stage Payload
(a)
Booster Sustainer
(c )
(Booster)
FIGURE 5.11 Three types of multistaging: (a) tandem, (b) parallel, and (c) piggyback.
gets started when its velocity is almost equal to the velocity increment pro- vided by the first stage and releases the structural mass of the first stage, thus enhancing the mass ratio for getting higher velocity increment. If the velocity increment ΔV1 is contributed by the first stage and ΔV2 is contrib- uted by the second stage, then the total velocity increment at the end of the second stage of the operation would be equal to ΔV1 + ΔV2. We can gener- alize the total velocity increment for n stages of a rocket engine as follows:
DVn DV DV DVi DV
i n
= + + + = i
å
=1 2
1
(5.44)
Let us consider n rocket stages without considering the drag and grav- ity effects. The net ideal velocity increment for n rocket stages from Equation 5.44 becomes
DVn DV V MR V MR V MR
i n
i e e en n
= =
( )
+( )
+ +( )
å
= 11ln 1 2ln 2 ln (5.45)
By assuming the exit velocity of all stages to be the same, Equation 5.45 becomes
DVn DV V MR MR MR
i n
i e n
= =
(
×)
å
= 11ln 1 2 (5.46)
By noting that the initial mass of the successive stage is equal to the burnt- out mass of the previous stage, as discussed earlier, Equation 5.46 becomes
DV DV V m
m m m
m
m V m
n i
n
i e o
b
o o n
b n e o
= = æ ×
èç ö
ø÷ =
å
= 11 1
1 2
2 1
ln , ln
, , ,
, , b
,,
, 1
mb n
æ èç ö
ø÷ (5.47) Note that mo ,1/mb ,n is the ratio of the initial mass to final mass of the nth stage which represents the overall mass ratio of a multistage rocket engine.
If the mass ratio of each stage is the same, Equation 5.46 becomes
DVn DV V MR nV MR
i n
i e n
= =
( )
= e( )
å
= 11ln 1ln (5.48)
This equation can be considered in terms of payload fraction and struc- tural fraction by using Equation 5.37:
DV DV nV MR nV
LF SF
n i
n
i e i e
i i
= =
( )
= æ +èç ö
ø÷
å
= 11ln 1ln 1 (5.49)
where SFi and LFi are the structure and payload fractions of the ith stage, respectively. The overall payload fraction LFn is defined as the ratio of the final payload of the nth stage to the initial total mass as follows:
LF m m
m m
m m
m
m LF LF LF
n Ln Ln
n on n
L in
i n n
= = = =
( )
01 0 0 1-
2 01
... ; 1 (5.50)
Note that for all stages, payload fraction and structural fraction remain almost the same.
DV DV nV MR nV
LF SF
n i
n
i e i e
n n i
= =
( )
=( )
+æ è çç ç
ö ø
÷÷
= ÷
å
11ln 1ln 11 (5.51)
The variation of change in velocity with exit velocity is plotted in Figure 5.12 for structural fraction for each stage and payload fraction for all stages.
0 1 2 3 4
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
ΔV/Ve
Number of stages, n SFi= 0.15
LFn= 0.015
FIGURE 5.12 Variation of velocity increment with number staging.
It may be noted that for two stages, velocity increment increases by 50%
compared to a single stage. As a result, the initial takeoff mass for the same payload decreases considerably. But with subsequent increase in stages, the velocity increment increases asymptotically. In other words, the gain in initial mass diminishes gradually with an increase in the number of stages. Hence, it is not prudent to go beyond four stages as the gain in ini- tial mass for lifting the same payload would be substantial to compensate the enhanced complexities in the operation with an increase in number of stages. That is the reason why three stages are popular across the globe. For example, the space shuttle, GSLV, Titan V, Delta, and so on are some of the space vehicles which have adopted three stages of rocket engine.
Apart from the tandem type of multistaging, several other types have been designed and developed for space launch vehicles. Three of the most popular multistaging types are (1) tandem, (2) parallel, and (3) piggyback, which are depicted schematically in Figure 5.11. The first one is the tandem or series multistaging (see Figure 5.11a) in which stages are placed verti- cally on top of each other from larger mass to smaller mass as described earlier. This is the most commonly used method of multistaging as it is quite simple and very effective in achieving higher velocity. However, the ejection timing of a used stage and ignition timing of the next stage are critical for its successful operation. In a partial multistaging system, three or four strap-on booster rockets are used along with vertical rocket config- urations which are dropped during the flight. Nowadays, both tandem and parallel stages are adopted together in the first stage of a space vehicle as this provides extra initial thrust to overcome the initial gravity losses and atmospheric drag. Of course, strap-on boosters are jettisoned to reduce the weight. The piggyback configuration has been used in the U.S. space shuttle in which two large solid boosters and expendable external propel- lant tanks are used to feed the shuttle’s onboard main. The orbiter (aircraft) is carried as piggyback to make this vehicle reusable.
REVIEW QUESTIONS
1. What are the major forces acting on a rocket engine during its flight in the earth’s atmosphere?
2. Derive a rocket equation mentioning assumptions made by you.
What are the terms which play an important role during its flight in the earth’s atmosphere?
3. What is the mass ratio for a class-one truck with a gross vehicle weight of 2225 kg which can carry 8 tons of materials?
4. What is meant by burnout distance? Derive an expression for it.
5. What is a sounding rocket? Derive a relationship for coasting height when a sounding rocket is fired.
6. How is the flight trajectory of a rocket estimated during its flight in the earth’s atmosphere? Devise a procedure for it.
7. What is meant by escape velocity? What will be the escape velocity required to leave the moon?
8. Derive an expression for the time period required to revolve around a circular orbit.
9. Derive an expression for the total energy required to place a satellite on an orbit around the earth.
10. What is meant by a geosynchronous orbit? Derive an expression for the radius of a geosynchronous orbit.
11. How is an elliptical orbit different from a circular orbit? Is the ellipti- cal orbit preferred over a circular orbit? Explain why.
12. What is meant by Hohmann’s transfer orbit? How is it different from Obert’s transfer orbit?
13. Derive an expression for the orbit velocity for an elliptical orbit.
14. What are the kinds of orbits one can use for launching a satellite around the earth?
15. Define payload fraction. How is it different from payload coefficient?
16. What is meant by structural coefficient? Derive an expression to show how the structural and payload coefficients can be related to mass fraction MR.
17. What is meant by multistaging? What are the advantages of multi- staging over a single-stage rocket engine?
18. What is the optimum number of stages that can be used for a space vehicle? Justify your answer by deriving a mathematical expres- sion for it.
19. What are the common types of multistaging used for launching space vehicles?
20. Discuss them by mentioning their pros and cons.
PROBLEMS
5.1 Determine the velocity and period of revolution of an artificial