This chapter deals with the definitions and basic relations for the propulsive force, exhaust velocity, and efficiencies related to creating and converting energy; comparisons of various propulsion systems and the simultaneous performance of multiple propulsion systems are also presented. The basic principles of rocket propulsion are essentially those of mechanics, thermodynamics, and chemistry. Propulsion is achieved by applying a force to a vehicle, that is, accelerating it or, alternatively, maintaining a given velocity against a resisting force. The propulsive force derives from momentum changes that originate from ejecting propellant at high velocities, and the equations in this chapter apply to all such systems. Symbols used in all equations are defined at the end of the chapter. Wherever possible, the American Standard letter symbols for rocket propulsion (as given in Ref. 2–1) are used.

2.1 DEFINITIONS

The total impulse I_t is found from the thrust force F (which may vary with time) integrated over the time of its application t:

2–1

For constant thrust with negligibly short start and stop transients, this reduces to

2–2

Total impulse I_t is essentially proportional to the total energy released by or into all the propellant utilized by the propulsion system.

The specific impulse I_s represents the thrust per unit propellant “weight” flow rate. It is an important figure of merit of the performance of any rocket propulsion system, a concept similar to miles per gallon parameter as applied to automobiles. A higher number often indicates better performance. Values of I_s are given in many chapters of this book and the concept of optimum specific impulse for a particular mission is introduced later. If the total propellant mass flow rate is and the standard acceleration of gravity g₀ (with an average value at the Earth's sea level of 9.8066 m/sec² or 32.174 ft/sec²), then

2–3

This equation will give a time‐averaged specific impulse value in units of “seconds” for any rocket propulsion system and is particularly useful when thrust varies with time. During transient conditions (during start or thrust buildup or shutdown periods, or during a change of flow or thrust levels) values of I_s may be obtained by either the integral above or by using average values for F and for short time intervals. As written below, m_p represents the total effective propellant mass expelled.

2–4

In Chapters 3, 12, and 17, we present further discussions of the specific impulse concept. For constant propellant mass flow , constant thrust F, and negligible start or stop transients Eq. 2–3 simplifies,

2–5

At or near the Earth's surface, the product m_pg₀ is the effective propellant weight w, and its corresponding weight flow rate given by . But for space or in satellite outer orbits, mass that has been multiplied by an “arbitrary constant,” namely, g₀ does not represent the weight. In the Système International (SI) or metric system of units, I_s is in “seconds.” In the United States today, we still use the English Engineering (EE) system of units (foot, pound, second) for many chemical propulsion engineering, manufacturing, and test descriptions. In many past and some current U.S. publications, data, and contracts, the specific impulse has units of thrust (lbf) divided by weight flow rate of the propellants (lbf/sec), also yielding the unit of seconds; thus, the numerical values for I_s are the same in the EE and the SI systems. Note, however, that this unit for I_s does not indicate a measure of elapsed time but the thrust force per unit “weight flow rate.” In this book, we use the symbol I_s exclusively for the specific impulse, as listed in Ref. 2–1. For solid propellants and other rocket propulsion systems, the symbol I_sp is more commonly used to represent specific impulse, as listed in Ref. 2–2.

In actual rocket nozzles, the exhaust velocity is not really uniform over the entire exit cross section and such velocity profiles are difficult to measure accurately. A uniform axial velocity c is assumed for all calculations which employ one‐dimensional problem descriptions. This effective exhaust velocity c represents an average or mass‐equivalent velocity at which propellant is being ejected from the rocket vehicle. It is defined as

2–6

It is given either in meters per second or feet per second. Since c and I_s only differ by a constant (g₀), either one can be used as a measure of rocket performance. In the Russian literature c is used in lieu of I_s.

In solid propellant rockets, it is difficult to measure propellant flow rate accurately. Therefore, in ground tests, the specific impulse is often calculated from total impulse and the propellant weight (using the difference between initial and final rocket motor weights and Eq. 2–5). In turn, the total impulse is obtained from the integral of the measured thrust with time, using Eq. 2–1. In liquid propellant units, it is possible to measure thrust and instantaneous propellant flow rate and thus Eq. 2–3 is used for the calculation of specific impulse. Equation 2–4 allows yet another interpretation for specific impulse, namely, the amount of total impulse imparted to a vehicle per total sea‐level weight of propellant expended.

The term specific propellant consumption corresponds to the reciprocal of the specific impulse and is not commonly used in rocket propulsion. It is used in automotive and air‐breathing duct propulsion systems. Typical values are listed in Table 1–2.

The mass ratio of the total vehicle or of a particular vehicle stage or of the propulsion system itself is defined to be the final mass m_f divided by the mass before rocket operation, m₀. Here, m_f consists of the mass of the vehicle or stage after the rocket has ceased to operate when all the useful propellant mass m_p has been consumed and ejected. The various terms are depicted in Fig. 4–1.

2–7

This equation applies to either a single or to a multistage vehicle; for the latter, the overall mass ratio is the product of the individual vehicle stage mass ratios. The final vehicle mass m_f has to include components such as guidance devices, navigation gear, the payload (e.g., scientific instruments or military warheads), flight control systems, communication devices, power supplies, tank structures, residual propellants, along with all the propulsion hardware. In some vehicles it may also include wings, fins, a crew, life support systems, reentry shields, landing gears, and the like. Typical values of can range from 60% for some tactical missiles down to 10% for some unmanned launch vehicle stages. This mass ratio is an important parameter for analyzing flight performance, as explained in Chapter 4. When the applies only to a single lower stage, then all its upper stages become part of its “payload.” It is important to specify when the applies either to a multiple‐stage vehicle, to a single stage, or to a particular propulsion system.

The propellant mass fraction ζ indicates the ratio of the useful propellant mass m_p to the initial mass m₀. It may apply to a vehicle, or a single stage, or to an entire rocket propulsion system.

2–8

2–9

2–10

Like the mass ratio , the propellant fraction ζ is used to describe a rocket propulsion system; its values will differ when applied to an entire vehicle, single or multistage. For a rocket propulsion system, the initial or loaded mass m₀ consists of the inert propulsion mass (the hardware necessary to burn and store the propellant) and the effective propellant mass. It would exclude masses of nonpropulsive components, such as payload or guidance devices (see Fig. 4–1). For example, in liquid propellant rocket engines the final or inert propulsion mass m_f would include propellant tanks, their feed and empty pressurization system (a turbopump and/or gas pressure system), one or more thrust chambers, various piping, fittings and valves, engine mounts or engine structures, filters, and some sensors. Any residual or unusable remaining propellant is normally considered to be part of the final inert mass m_f, as in this book. However, some rocket propulsion manufacturers and some literature assign residuals to be part of the propellant mass m_p. When applied to an entire rocket propulsion system, the value of the propellant mass fraction ζ indicates the quality of the design; a value of, say, 0.91 means that only 9% of the mass is inert rocket hardware, and this small fraction is needed to contain, feed, and burn the substantially larger mass of propellant; high values of ζ are desirable.

The impulse‐to‐weight ratio of a complete propulsion system is defined as the total impulse I_t divided by the initial (propellant‐loaded) vehicle sea‐level weight w₀. A high value indicates an efficient design. Under our assumptions of constant thrust and negligible start and stop transients, it can be expressed as

2–11

The thrust‐to‐weight ratio F/w₀ expresses the acceleration (in multiples of the Earth's surface acceleration of gravity, g₀) that an engine is capable of giving to its own loaded propulsion system mass. Values of F/w₀ are given in Table 2–1. The thrust‐to‐weight ratio is useful in comparing different types of rocket propulsion systems and/or in identifying launch capabilities. For constant thrust the maximum value of the thrust‐to‐weight ratio, or maximum acceleration, invariably occurs just before thrust termination (i.e., burnout) because the vehicle's mass has been diminished by the mass of useful propellant.

Engine Type	Specific Impulsea (sec)	Maximum Temperature (°C)	Thrust‐to‐Weight Ratiob	Propulsion Duration	Specific Powerc (kW/kg)	Typical Working Fluid	Status of Technology
Chemical—solid or liquid bipropellant, or hybrid	200–468	2500–4100	10−2–100	Seconds to a few minutes	10−1–103	Liquid or solid propellants	Flight proven
Liquid monopropellant	194–223	600–800	10−1–10−2	Seconds to minutes	0.02–200	N2H4	Flight proven
Resistojet	150–300	2900	10−2–10−4	Days	10−3–10−1	H2, N2H4	Flight proven
Arc heating—electrothermal	280–800	20,000	10−4–10−2	Days	10−3–1	N2H4, H2, NH3	Flight proven
Electromagnetic including pulsed plasma (PP)	700–2500	—	10−6–10−4	Weeks	10−3–1	H2 Solid for PP	Flight proven
Hall effect	1220–2150	—	10−4	Weeks		Xenon	Flight proven
Ion—electrostatic	1310–7650	—	10−6–10−4	Months, years	10−3–1	Xenon	Flight proven
Solar heating	400–700	1300	10−3–10−2	Days	10−2–1	H2	In development

^aAt psia and optimum gas expansion at sea level ().

^bRatio of thrust force to full propulsion system sea level weight (with propellants, but without payload).

^cKinetic power per unit exhaust mass flow.

2.2 THRUST

Thrust is the force produced by the rocket propulsion system acting at the vehicle's center of mass. It is a reaction force, experienced by vehicle's structure from the ejection of propellant at high velocities (the same phenomenon that pushes a garden hose backward or makes a gun recoil). Momentum is a vector quantity defined as the product of mass times its vector velocity. In rocket propulsion, relatively small amounts of propellant mass carried within the vehicle are ejected at high velocities.

The thrust, due to a change in momentum, is shown below (a derivation can be found in earlier editions of this book). Here, the exit gas velocity is assumed constant, uniform, and purely axial, and when the mass flow rate is constant, thrust is itself constant. Often, this idealized thrust can actually be close to the actual thrust.

2–12

But this force represents the total propulsive force only when the nozzle exit pressure equals the ambient pressure.

The pressure of the surrounding fluid (i.e., the local atmosphere) gives rise to the second component of thrust. Figure 2–1 shows a schematic of an external uniform pressure acting on the outer surfaces of a rocket chamber as well as the changing gas pressures inside of a typical thermal rocket propulsion system. The direction and length of the arrows indicates the relative magnitude of the pressure forces. Axial thrust can be determined by integrating all the pressures acting on areas that have a projection on the plane normal to the nozzle axis. Any forces acting radially outward may be appreciable but do not contribute to the thrust when rocket chambers are axially symmetric.

For a fixed nozzle geometry, changes in ambient pressure due to variations in altitude during flight result in imbalances between the atmospheric pressure p₃ and the propellant gas pressure p₂ at the exit plane of the nozzle. For steady operation in a homogeneous atmosphere, the total thrust can be shown to equal (this equation is derived using the control volume approach in gas dynamics, see Refs. 2–3 and 2–4):

2–13

The first term is the momentum thrust given by the product of the propellant mass flow rate and its exhaust velocity relative to the vehicle. The second term represents the pressure thrust, consisting of the product of the cross‐sectional area at the nozzle exit A₂ (where the exhaust jet leaves the vehicle) and the difference between the gas pressure at the exit and the ambient fluid pressure (p₂ may differ from p₃ only in supersonic nozzle exhausts). When the exit gas pressure is less than the surrounding fluid pressure, the pressure thrust is negative. Because this condition gives a lower thrust and is undesirable for other reasons (discussed in Chapter 3), rocket nozzles are usually designed so that their exhaust pressures equal or are slightly higher than the ambient pressure.

When the ambient or atmospheric pressure equals the exhaust pressure, the pressure term is zero and the thrust is the same as in Eq. 2–12. In the vacuum of space p₃ = 0 and the pressure thrust becomes a maximum,

2–14

Most nozzles are have their area ratio designed so that their exhaust pressure will equal the surrounding air pressure (i.e., ) somewhere at or above sea level. For any fixed nozzle configuration this can occur only at one altitude, and this location is referred to as nozzle operation at its optimum expansion ratio; this case is treated in Chapter 3.

Equation 2–13 shows that thrust is independent of a rocket unit's flight velocity. Because changes in ambient pressure affect the pressure thrust, rocket thrust varies noticeably with altitude. Since atmospheric pressure decreases with increasing altitude, thrust and specific impulse will increase as the vehicle reaches higher altitudes. On Earth, this increase in pressure thrust due to altitude changes can amount to between 10 and 30% of the sea‐level thrust; the fact that as flights gain altitude, atmospheric pressure continuously diminishes causing the thrust to increase is a unique feature of rocket propulsion systems. Table 8–1 shows the sea level and high‐altitude thrust for several liquid rocket engines. Appendix 2 gives the properties of the standard atmosphere (the ambient pressure, p₃) as a function of altitude.

2.3 EXHAUST VELOCITY

The effective exhaust velocity as defined by Eq. 2–6 applies to all mass‐expulsion thrusters. From Eq. 2–13 and for constant propellant mass flow it can be modified to give the equation below. As before, g₀ is a constant whose numerical value equals the average acceleration of gravity at sea level and does not vary with altitude.

2–15

In Eq. 2–6 the value of c may be determined from measurements of thrust and propellant flow. When , the value of the effective exhaust velocity c equals , the average actual nozzle exhaust velocity of the propellant gases. But even when and , the second term of the right‐hand side of Eq. 2–15 usually remains small in relation to ; thus, the effective exhaust velocity always stays relatively close in value to the actual exhaust velocity. At the Earth's surface and when , the thrust, from Eq. 2–13, can be simply written as (the second version below is less restricted)

2–16

The characteristic velocity , pronounced “cee‐star,” is a term frequently used in the rocket propulsion literature. It is defined as

2–17

The characteristic velocity , though not a physical velocity, is used for comparing the relative performance of different chemical rocket propulsion system designs and propellants; it may be readily determined from measurements of , p₁, and A_t. Being essentially independent of nozzle characteristics, may also be related to the efficiency of the combustion process. However, the specific impulse I_s and the effective exhaust velocity c remain functions of nozzle geometry (such as the nozzle area ratio A₂/A_t, as shown in Chapter 3). Some typical values of I_s and are given in Tables 5–5 and 5–6.

2.4 ENERGY AND EFFICIENCIES

Although efficiencies are not commonly used directly in designing rocket propulsion systems, they permit an understanding of the energy balance in these systems. Their definitions arbitrarily depend on the losses considered and any consistent set of efficiencies, such as the set presented in this section, is satisfactory in evaluating energy losses. As stated previously, two types of energy conversion processes occur in all propulsion systems, namely, the production of energy, which is most often the conversion of stored energy into available energy and, subsequently, its conversion to the form with which a reaction thrust can be obtained. The kinetic energy of ejected matter is the main useful form of energy for propulsion. The power of the jet P_jet is the time derivative of this energy, and for a constant gas exit velocity ( and at sea level), this is a function of I_s and F:

2–18

The term specific power is sometimes used as a measure of the utility of the mass of the propulsion system, including its power source; it equals the jet power divided by the loaded propulsion system mass, P_jet/m₀. For electrical propulsion systems that need to include a massive, relatively inefficient energy source, specific power can be much lower than that for chemical rockets. The source of energy input to a rocket propulsion system is different in different thruster types; for chemical rockets this energy is created solely by combustion. The maximum energy available in chemical propellants is their heat of combustion per unit of propellant mass Q_R; the power input to a chemical engine is given by

2–19

where J is a units‐conversion constant (see Appendix 1). A significant portion of the energy may leave the nozzle as residual enthalpy in the exhaust gases and is unavailable for conversion into kinetic energy. This is analogous to the energy lost in the hot exhaust gases of internal combustion engines.

The combustion efficiency for chemical rockets is the ratio of the actual energy released to the ideal heat of reaction per unit of propellant mass and represents a measure of the source efficiency. Its value can be high (nearly 94 to 99%). When the power input P_chem is multiplied by the combustion efficiency, it becomes the power available to the propulsive device, where it is then converted into the kinetic power of the exhaust jet. In electric propulsion the analogous efficiency is the power conversion efficiency; with solar cells this efficiency has a relatively low value because it depends on the efficiency of converting solar radiation energy into electric power (presently between 10 and 30%).

The power transmitted to the vehicle at any one instant of time is defined in terms of the product of the thrust of the propulsion system F and the vehicle velocity u:

2–20

The internal efficiency of a rocket propulsion system reflects the effectiveness of converting the system's input energy to the propulsion device into kinetic energy of ejected matter; for example, for a chemical unit it is the ratio of the kinetic power of the ejected gases expressed by Eq. 2–18 divided by the power input of the chemical reaction, Eq. 2–19. The energy balance diagram for a chemical rocket (Fig. 2–2) shows typical losses. The internal efficiency η_int may be expressed as

2–21

Any object moving through a fluid medium affects the fluid (i.e., stirs it) in ways that may hinder its motion and/or require extra energy expenditures. This is one consequence of skin friction, which can be substantial. The propulsive efficiency η_p (Fig. 2–3) reflects this energy cost for rocket vehicles. The equation that determines how much exhaust kinetic energy is useful for propelling a rocket vehicle is defined as

2–22

where F is the thrust, u the absolute vehicle velocity, c the effective rocket exhaust velocity with respect to the vehicle, the propellant mass flow rate, and η_p the desired propulsive efficiency. This propulsive efficiency becomes one when the forward vehicle velocity is exactly equal to the effective exhaust velocity; here any residual kinetic energy becomes zero and the exhaust gases effectively stand still in space.

While it is desirable to use energy economically and thus have high efficiencies, there is also the need for minimizing the expenditure of ejected mass, which in many cases is more important than minimizing the energy. With nuclear‐reactor energy and some solar‐energy sources, for example, there is an almost unlimited amount of heat energy available; yet the vehicle can carry only a limited amount of propellant. For a given thrust, economy of mass expenditures in the working fluid can be obtained when the exhaust velocity is high. Because the specific impulse is proportional to the exhaust velocity, it therefore measures this propellant mass economy.

2.5 MULTIPLE PROPULSION SYSTEMS

The relationships below are used for determining the total or overall (subscript oa) thrust and the overall mass flow of propellants for a group of propulsion systems (two or more) firing in parallel (i.e., in the same direction at the same time). These relationships apply to liquid propellant rocket engines, solid propellant rocket motors, electrical propulsion systems, hybrid propulsion systems, and any combinations of these. Many space launch vehicles and larger missiles use multiple propulsion systems. The Space Shuttle, for example, had three large liquid engines and two large solid motors firing jointly at liftoff.

Overall thrust, F_oa, is needed for determining the vehicle's flight path and overall mass flow rate, , is needed for determining the vehicle's mass; together they determine the overall specific impulse, (I_s)_oa:

2–23

2–24

2–25

For liquid propellant rocket engines with a turbopump and a gas generator, there is a separate turbine outlet flow that is usually dumped overboard through a pipe and a nozzle (see Fig. 1–4), which needs to be included in the equations above and this is treated in Examples 2–3 and 11–1.

2.6 TYPICAL PERFORMANCE VALUES

Typical values of representative performance parameters for different types of rocket propulsion are given in Table 2–1 and in Fig. 2–4.

Chemical rocket propulsion systems have relatively low values of specific impulse, relatively light machinery (i.e., low engine total mass), with very high thrust capabilities, and therefore can provide high acceleration and high specific power. At the other extreme, ion‐propulsion thrusters have a very high specific impulse, but they carry massive electrical power systems to deliver the power necessary for such high ejection velocities. The very low acceleration potential for electrical propulsion units (and for others using solar energy) usually requires long accelerating periods and thus these systems are best suited for missions where powered flight times can be long. The low thrust magnitudes of electrical systems also imply that they are not useful in fields of strong gravitational gradients (e.g., for Earth takeoff or landing) but are best suited for flight missions in space.

Performance of rocket propulsion systems also depends on their application. Typical applications are shown in several chapters of this book (see Index).

Chemical systems (solid and liquid propellant rockets) are presently the most developed and are widely used for many different vehicle applications. They are described in Chapters 5 to 16. Electrical propulsion has also been in operation in many space flight applications (see Chapter 17). Other types are still in their exploratory or development phase.

The accelerations shown in Figure 2–4 are directly related to the magnitudes of the thrust that is applied by the various rocket propulsion systems depicted. The effective exhaust velocities are related the required jet power per unit thrust (see Eq. 2–18) and the resulting input power (which is proportional to the square of the velocity and depends on the internal efficiency, Eq. 2–21) would turn out to be very high for electric propulsion systems unless the thrust itself is reduced to comparatively low values. As of 2015, electrical power levels of about 100 kW are expected to be available in space.

In Figure 2–4, hybrid rocket propulsion systems (see Chapter 16) are not shown separately because they are included as part of liquid and solid chemical propellants. Compressed (cold) gases stored at ambient temperatures have been used for many years for roll control in larger vehicles, for complete attitude control of smaller flight vehicles, and in rocket toys. Nuclear fission rocket propulsion systems as shown in Figure 2–4 represent analytical estimates; their development has been stopped. Hall and ion thrusters are part of the electrostatic and electromagnetic entry in this figure; they have flown successfully in many space applications (see Chapter 17).

2.7 VARIABLE THRUST

Most operational rocket propulsion systems have essentially constant propellant mass flow producing constant thrust or slightly increasing thrust with altitude. Only some flight missions require large thrust changes during flight; Table 2–2 shows several applications; the ones that require randomly variable thrust use predominantly liquid propellant rocket engines. Some applications require high thrust during a short initial period followed by a pre‐programmed low thrust for the main flight portion (typically 20 to 35% of full thrust); these use predominantly solid propellant rocket motors. Section 8.8 describes how liquid propellant rocket engines can be designed and controlled for randomly variable thrust. Section 12.3 explains how the grain of solid rocket motors can be designed to give predetermined thrust changes.

Some solid and liquid propellant experimental propulsion systems have used variable nozzle throat areas (achieved with a variable position “tapered pintle” at the nozzle throat) and one experimental version has flown. To date, there has been no published information on production and implementation of such systems.

SYMBOLS

A	area, m2 (ft2)
At	nozzle throat area, m2 (ft2)
A2	exit area of nozzle, m2 (ft2)
c	effective exhaust velocity, m/sec (ft/sec)
	characteristic velocity, m/sec (ft/sec)
E	energy, J (ft‐lbf)
F	thrust force, N (lbf)
Foa	overall force, N (lbf)
g0	average sea‐level acceleration of gravity, 9.81 m/sec2 (32.2 ft/sec2), [at equator 9.781, at poles 9.833 m/sec2]
Is	specific impulse, sec
(Is)oa	overall specific impulse, sec
It	impulse or total impulse, N‐sec (lbf‐sec)
J	conversion factor or mechanical equivalent of heat, 4.184 J/cal or 1055 J/Btu or 778 ft‐lbf/Btu
m	mass, kg (slugs, ‐weight at sea level)
moa	overall mass, kg
	mass flow rate, kg/sec (lbm/sec)
mf	final mass (after rocket propellant is ejected), kg (lbm or slugs)
mp	propellant mass, kg (lbm or slugs)
m0	initial mass (before rocket propellant is ejected), kg (lbm or slugs)
	mass ratio (mf/m0)
p	pressure, pascal [Pa] or N/m2 (lbf/ft2)
p3	ambient or atmospheric pressure, Pa (lbf/ft2)
p2	rocket gas pressure at nozzle exit, Pa (lbf/ft2)
p1	chamber pressure, Pa (lbf/ft2)
P	power, J/sec (ft‐lbf/sec)
Ps	specific power, J/sec‐kg (ft‐lbf/sec‐lbm)
QR	heat of reaction per unit propellant, J/kg (Btu/lbm)
t	time, sec
u	vehicle velocity, m/sec (ft/sec)
	gas velocity leaving the nozzle, m/sec (ft/sec)
w	weight, N or kg‐m/sec2 (lbf)
	weight flow rate, N/sec (lbf/sec)
w0	initial weight, N or kg‐m/sec2 (lbf)

PROBLEMS

When solving problems, three appendixes (see end of book) may be helpful:

• Appendix 1. Conversion Factors and Constants
• Appendix 2. Properties of the Earth's Standard Atmosphere
• Appendix 3. Summary of Key Equations

1. A jet of water hits a stationary flat plate in the manner shown below.

   

   1. a If 50 kg per minute flows at an absolute velocity of 200 m/sec, what will be the force on the plate?
   2. b What will this force be when the plate moves in the flow direction at u = 50 km/h? Explain your methodology.Answers: 167 N; 144 N.
2. The following data are given for a certain rocket unit: thrust, 8896 N; propellant consumption, 3.867 kg/sec; velocity of vehicle, 400 m/sec; energy content of propellant, 6.911 MJ/kg. Assume 100% combustion efficiency.Determine (a) the effective velocity; (b) the kinetic jet energy rate per unit flow of propellant; (c) the internal efficiency; (d) the propulsive efficiency; (e) the overall efficiency; (f) the specific impulse; (g) the specific propellant consumption.Answers: (a) 2300 m/sec; (b) 2.645 MJ/kg; (c) 38.3%; (d) 33.7%; (e) 13.3%; (f) 234.7 sec; (g) 0.00426 sec⁻¹.
3. A certain rocket engine (flying horizontally) has an effective exhaust velocity of 7000 ft/sec; it consumes 280 lbm/sec of propellant mass, and liberates 2400 Btu/lbm. The unit operates for 65 sec. Construct a set of curves plotting the propulsive, internal, and overall efficiencies versus the velocity ratio . The rated flight velocity equals 5000 ft/sec. Calculate (a) the specific impulse; (b) the total impulse; (c) the mass of propellants required; (d) the volume that the propellants occupy if their average specific gravity is 0.925. Neglect gravity and drag.Answer: (a) 217.4 sec; (b) 3,960,000 lbf‐sec; (c) 18,200 lbm; (d) 315 ft³.
4. For the rocket in Problem 2, calculate the specific power, assuming a propulsion system dry mass of 80 kg and a duration of 3 min.
5. A Russian rocket engine (RD‐110 with LOX‐kerosene) consists of four thrust chambers supplied by a single turbopump. The exhaust from the turbine of the turbopump then is ducted to four vernier nozzles (which can be rotated to provide some control of the flight path). Using the information below, determine the thrust and mass flow rate of the four vernier gas nozzles. For individual thrust chambers (vacuum):
   
   For overall engine with verniers (vacuum):
   
   Answers: 5.37 kN, 2.32 kg/sec.
6. A certain rocket engine has a specific impulse of 250 sec. What range of vehicle velocities (u, in units of ft/sec) would keep the propulsive efficiencies at or greater than 80%. Also, how could rocket–vehicle staging be used to maintain these high propulsive efficiencies for the range of vehicle velocities encountered during launch?Answers: 4021 to 16,085 ft/sec; design upper stages with increasing I_s.
7. For a solid propellant rocket motor with a sea‐level thrust of 207,000 lbf, determine: (a) the (constant) propellant mass flow rate and the specific impulse I_s at sea level, (b) the altitude for optimum nozzle expansion as well as the thrust and specific impulse at this optimum condition and (c) at vacuum conditions. The initial total mass of the rocket motor is 50,000 lbm and its propellant mass fraction is 0.90. The residual propellant (called slivers, combustion stops when the chamber pressure falls below a deflagration limit) amounts to 3 % of the burnt. The burn time is 50 seconds; the nozzle throat area (A_t) is 164.2 in.² and its area ratio (A₂/A_t) is 10. The chamber pressure (p₁) is 780 psia and the pressure ratio (p₁/p₂) across the nozzle may be taken as 90.0. Neglect any start/stop transients and use the information in Appendix 2. Answers: (a) = 873 lbm/sec, 237 sec., (b) ., (c) .
8. During the boost phase of the Atlas V, the RD‐180 engine operates together with three solid propellant rocket motors (SRBs) for the initial stage. For the remaining thrust time, the RD‐180 operates alone. Using the information given in Table 1–3, calculate the overall effective exhaust velocity for the vehicle during the initial combined thrust operation.Answer: 309 sec.
9. Using the values given in Table 2–1, choose three propulsion systems and calculate the total impulse for a fixed propellant mass of 20 kg.
10. Using the MA‐3 rocket engine information given in Example 2–3, calculate the overall specific impulse at sea level and at altitude, and compare these with I_s values for the individual booster engines, the sustainer engine, and the individual vernier engines.Answers: (SL) and 258 sec (altitude)
11. Determine the mass ratio and the mass of propellant used to produce thrust for a solid propellant rocket motor that has an inert mass of 82.0 kg. The motor mass becomes 824.5 kg after loading the propellant. For safety reasons, the igniter is not installed until shortly before motor operation; this igniter has a mass of 5.50 kg of which 3.50 kg is igniter propellant. Upon inspection after firing, the motor is found to have some unburned residual propellant and a motor mass of 106.0 kg.Answers: = 0.1277, propellant burned .

REFERENCES

1. 2–1. “American National Standard Letter Symbols for Rocket Propulsion,” ASME Publication Y 10.14, 1959.
2. 2–2. “Solid Propulsion Nomenclature Guide,” CPIA Publication 80, Chemical Propulsion Information Agency, Johns Hopkins University, Laurel, MD, May 1965, 18 pages.
3. 2–3. P. G. Hill and C. R. Peterson, Mechanics and Thermodynamics of Propulsion, Addison‐Wesley, Reading, MA, 1992. [Paperback edition, 2009]
4. 2–4. R. D. Zucker and O. Biblarz, Fundamentals of Gas Dynamics, 2nd ed., John Wiley & Sons, Hoboken, NJ, 2002.