As we discussed above, thrust is the force generated by the rocket engine that propels a rocket along its trajectory through the air and space. Specifically, the engine houses a combustion process where a gas is heated and expanded and then forced out the rear of the rocket in the opposite direction as that of the motion of the rocket itself. As we will discuss in Chapter 4, this gas is forced out the back of the rocket through a converg- ing, diverging nozzle system that accelerates the gas flow. For now, just con- sider the fact that the gas flow is accelerated out of the rocket engine and that that is the main purpose of the rocket engine—to accelerate exhaust gas flow.
To better understand the thrust generated by this accelerated gas flow, a few physics concepts need to be discussed. In classical physics, there is a phenomenon where momentum is imparted to a baseball by a bat (assum- ing the batter keeps his eye on the ball) called impulse. Impulse, I (sometimes called total impulse), in its purist sense is defined as the total integrated force with respect to time and is written as
I=
∫
Fdt. (3.7) Here, F is force as a constant or a function of time and dt is the incremental time change variable. Recall from Newton’s Second Law that force can be written as the derivative of momentum with respect to time orI Fdt dp
dtdt dp p
=
∫
=∫
=∫
=∆ . (3.8)Equation 3.8 is referred to as the impulse-momentum theorem. And once again, the theorem basically tells us that a force applied to an object over a given amount of time produces an effect and that effect is an impulse. Another way to think of impulse is that it is the change in momentum of an object due to an applied force. Recall the baseball’s momentum being changed by hitting a bat. Also note from Equation 3.8 that impulse will have the same units as momentum of kg m/sec or a N∙s. Integrating Equation 3.8 yields
I Ft= = ∆ . p (3.9)
Considering a force generated by a changing mass with a constant velocity, the impulse can be written as
I v dm dtdt p m m v m m i f f i =
∫
=∆ =(
−)
. (3.10)As our discussion continues, it will become clear why Equation 3.10 might be useful. Clearly, if we have a rocket that is in need of a course change or even a lift-off, we must change its momentum and Equation 3.10 shows us how. If we start off with an initial mass for the rocket and apply an impulse by ejecting propellant mass out of the back of it at a constant exhaust velocity, we can rewrite the equation as
I=
(
m m vi− f)
= ∆mpropellant ev. (3.11) Solving for the exhaust velocity results inI
mpropellant v e
∆ = . (3.12)
Before we go farther with this line of reasoning, we need to identify some- thing. We said in Equation 3.6 that we defined thrust as something more
than simply the m-dot times the exhaust velocity. Reexamining that equa- tion and defining a new parameter called the equivalent velocity or sometimes the effective exhaust velocity, C, we rewrite Equation 3.6 as
FThrust=mv e+
(
P P Ae− o)
e=mC (3.13) and we see that Equation 3.11 and Equation 3.12 should really be the equiva- lent velocity rather than just the exhaust velocity, thus,I
mpropellant C
∆ = . (3.14)
This equation tells us that the total impulse imparted to a rocket divided by the propellant mass ejected is equal to the equivalent velocity. The rela- tion is useful in describing the total rocket thrust, but it doesn’t really tell us anything about the rocket itself. By defining a new parameter, we can make some very powerful assessments with Equation 3.14.
That parameter is known as the specific impulse, Isp. This is a more useful parameter (given in seconds only) and is written as
I I m g C g sp propellant = = ∆ . (3.15)
What Equation 3.15 tells us is that the Isp of a rocket engine is the total number of seconds that the rocket can deliver thrust equal to the weight of the total propellant mass under acceleration due to one standard Earth gravity, g. This is an efficiency number that we use to
describe rocket engines. The higher the Isp the more efficiently the engine can apply ∆-ν to the spacecraft. Table 3.1 shows some specific impulse values for vari- ous rocket systems. From that data it becomes clear that specific impulse is not the only important parameter when it comes to discussing rockets. For example, a launch vehicle engine will have Isp typically around 200 to 500 seconds. An ion engine for a deep space mission will have much higher Ispvalues, upwards of 3,000 sec- onds. Why are they so different and why would we use one over the other?
Launch vehicles typically are employed to lift very heavy payloads into orbit or on an interplanetary trajectory as quickly as possible. The key com- ponent needed for such missions is thrust—as much thrust as usually can be applied within engineering capabilities. Applying this much thrust in such
TABLE 3.1
Specific Impulse for Various Rockets Rocket Isp in sec SSME 363 RS-68 365 SRB 269 NSTAR 3,100 NERVA 800
a short time requires a large amount of propellant mass. This is why launch vehicle rockets are mostly propellant and oxidizer with small areas on top for payloads. An example of this is the Delta IV heavy launch vehicle that employs three common booster core liquid hydrogen fuel-, liquid oxygen oxi- dizer-driven Boeing/Rocketdyne RS-68 engines as discussed in Chapter 1. Each of the three engines can supply just under 3.4 mega-Newtons (MN) of thrust with an Isp of 362 sec at sea level.
On the other hand, interplanetary missions (after launch) typically need to apply thrust continuously for a long period of time. These are usually the small payloads that are atop the launch vehicles, thus they have very little mass budget for fuel. This means that they cannot apply large thrusts for long periods of time or they will run out of fuel. Hence, a more propel- lant efficient engine, such as an ion thruster, is needed. The ion thrusters use small amounts of propellant mass at a time, but accelerate that mass to very high equivalent velocities. Figure 3.2 illustrates images of NASA’s Deep Space Probe 1 that used an ion engine that only generated 0.09 N of thrust, but had an Isp of over 3,100 sec.
3.2.1 Example 3.1: Isp of the Space Shuttle Main Engines
The three Space Shuttle Main Engines (SSMEs) of the Orbiter each provide about 1.8 MN of thrust with an Isp of 363 sec at sea level. What is the mass flow rate of an SSME?
The first step is to determine the equivalent velocity, C, of the engine. From Equation 3.15, we see that C = g Isp = 3557.4 m/sec. Then from Equation 3.13, we can solve for the mass flow rate, m, which is
m F C thrust = =1 8 10× = 3557 4 505 99 6 . . . kg/sec.
In other words, one SSME uses about a half ton of propellant each second. Don’t forget that there are three of them.