Rocket Performance Parameters and Units
[281] American rocket research and development during 19451959 used the English system of units. As NASA has directed that metric units be used in all publications, including this one, the conventional units have been converted using E. A. Mechtly, "The International System of Units: Physical Constants and Conversion Factors," (NASA SP7012, 1973). The international system, designated SI in all languages, mandates newtons for force, newtons per square meter for pressure, and joules per second per square meter for heat transfer rates. In the interest of readability and as a concession to those brought up in the English system, a few compromises have been made. These are described in the following discussion of the major performance parameters used in rocketry.
Thrust. Force, expressed in newtons (N), kilonewtons (kN), and meganewtons (MN). A newton is that force that gives 1 kilogram (kg) of mass an acceleration of 1 meter per second per second (m/s^{2}). To convert from 1 pound force to newtons, multiply by 4.448. This has usually been rounded off to 4.45 for conversions in the text. To offset the unfamiliarity of the newton, thrust is normally expressed in pounds in parentheses.
[282] Propellant flow. Expressed in kilograms per second (kg/s). 1 lb = 0.4536 kg.
Mixture ratio. The proportions in which fuel and oxidizer are burned in the combustion chamber. Mixture ratio is expressed in several ways, one of the most Common is the ratio of mass flow of oxidizer to mass flow of fuel, abbreviated O/F; sometimes the inverse is used. Another way of expressing mixture ratio, popular in the 1950s, is the percentage of the total mass flow that is fuel; a third is the molar ratio of one to the other. A mole is the mass equivalent of the molecular weight of the fuel and oxidizer. Sometimes a stoichiometric mixture ratio is mentioned; this means the exact proportion of fuel and oxidizer for complete combustion. As an example,
is the stoichiometric mixture. In this example, one mole of hydrogen combines with a half mole of oxygen to produce a mole of water. Expressed in mass units, 2.016 grams of hydrogen plus 16 grams of oxygen produce 18.016 grams of water. The O/F is 7.9, percent fuel is 11, and molar oxidizer to fuel ratio is 1/2.
Specific impulse and exhaust velocity. In 1903 Tsiolkovskiy, and other Europeans after him, expressed rocket engine performance in terms of the velocity of the exhaust emerging from the nozzle in meters per second (m/s). This made sense because the rocket exhaust velocity was a term in the equation expressing the velocity of a rocketpropelled vehicle. In the United States, it became the custom to express rocket performance in terms of the measured quantities: thrust and mass flow of the propellants. The thrust divided by the total mass flow of propellant was defined as the specific impulse. Specific impulse is the inverse of specific fuel consumption used in discussing the performance of other types of propulsion systems. In English units, specific impulse is in pounds force per pounds mass per second (Ibf ^{. }sec/Ibm). On seeing pounds in both numerator and denominator, many succumbed to the temptation to cancel them and express specific impulse incorrectly in units of seconds: the two pounds represent different physical phenomena, force and mass, and are connected by the conversion factor 32.2 Ibm ^{.} ft /lbmsec^{2}. In SI, specific impulse is expressed in newtons per kilogram per second or N ^{.} s/kg. English values of specific impulse are converted to SI by multiplying by 9.807, which can be rounded to 10 for approximations. The numerical value of specific impulse and exhaust velocity in SI are the same; only the units are different. Since exhaust velocity is a simple concept to visualize physically and since specific impulse expressed in newtons per kilogram per second is unfamiliar to many, including the author, all performance values in this text have been converted to exhaust velocity in meters/second (m/s). Typical values of exhaust velocity for liquid propellant rockets range from 2000 to 4500 m/s. The V2 had an exhaust velocity of about 2200 m/s, very good for 1944. High energy propellants give exhaust velocities in the range of 3000 to 4500 m/s, and the liquid hydrogenoxygen combination is in the upper part of this range.
Pressure. Expressed as newtons per square meter (N/m^{2}) in SI. One pound per square inch is 6895 N/m^{2}. Rocket combustion pressures used during 19451959 were generally in the range of 300600 lb/in^{2} or 20694137 kN/m^{2}. Rocket combustion [283] pressures were also expressed in atmospheres multiples of sealevel pressure which is the same in any system of units. Since atmospheres are easier grasped than kN/m^{2}, combustion and higher pressures in this text are expressed in atmospheres (atm). One atmosphere is slightly over 100 kN/m^{2}; the 20694137 kN /m^{2} pressure range above becomes 20.440.7 atm.
Nozzle area ratio. The ratio of nozzle exit area to throat area. Area ratio determines the amount of expansion of the exhaust gases through the nozzle and is related to exhaust gas pressures. If a rocket designer is asked to provide a nozzle that is to be operated only on the groundas in the case of an experimental rocket enginehe chooses an area ratio such that the exhaust gas pressure at the nozzle exit is equal to ambient pressure, usually considered as sealevel pressure or 1 atmosphere. If the combustion pressure is 20 atmospheres, the gases undergo an expansion ratio of 20 and this corresponds roughly to a nozzle area ratio of 4. If, for some reason, the designer provides a nozzle area less than that needed for complete expansion, the exhaust gases emerge from the nozzle exit at greater than ambient pressure. In this case the gases are said to be underexpanded, for they have to expand further to reach ambient pressure. On the other hand, if the designer provides a larger area ratio than that needed for complete expansion, the exhaust gases reach a pressure equal to ambient while still in the nozzle. In some cases the gases will continue to follow the nozzle walls and expand to a pressure lower than ambient. In this case the gases are said to be overexpanded. Sooner or later, overexpanded gas must be reconciled with the ambient pressure and nature provides for this adjustment by means of a shock wave. The ideal nozzle is one that provides for complete gas expansion, neither more nor less, for theory shows that this yields maximum exhaust velocity. This poses a problem to the designer of a launch vehicle, because as soon as the vehicle is launched, the ambient pressure begins to fall, approaching zero at very high altitudes. Since the nozzle is fixed, what area ratio should he choose? If he designs for sealevel conditions, he is getting less performance at altitude than could be realized; if he designs for altitude, he suffers some performance loss from overexpansion at sealevel. For rocket engines in upper stages, he doesn't have this problem for the ambient pressure is close to zero at ignition. In rocket engines for all stages, however, he must balance the gain in performance from providing a larger area ratio nozzle against the added weight and cooling requirements of the larger nozzle and must also stay within special limits of the vehicle. A typical rocket engine for an upper stage will have an area ratio of about 40. In the text, you will seldom encounter area ratio, but performance at sealevel and altitude is used and in each case it is implied that a nozzle is provided for complete, or nearly complete, expansion of the exhaust gases.
Heat transfer rate. Heat flow per unit time per unit area. In SI, joules/second ^{.} meter^{2} (J/s ^{.} m^{2}). One Btu/s ^{. }in^{2} equals 1.637 J/s ^{.} m^{2}. Rocket combustion temperatures. from 2500 to 4500 K, and pressures, from 20 to 40 atmospheres, produce very high heat transfer rateson the order of 20 to 200 times greater than those produced in boilers and superheaters in a steam plant, for example, and much higher than in other types of internal combustion engines. Typical values of rocket heat transfer rates range from 1 to 20 J/s^{ .} m^{2} , but can be higher in local areas.
[284] Mass ratio. The ratio of the gross (total) mass of a rocket vehicle to its empty mass (M_{o}/ M_{e}). The difference between the two is the mass of propellant expended during the operation. The empty mass includes the rocket engine, the structure, controls, and the payload. For the last stage of a multistage launch vehicle, the payload is the spacecraft. For other stages, the payload is the loaded stages above it. Mass ratios range from about 3 to 10. The V2 had a mass ratio of about 3.
Vehicle velocity. Expressed in meters per second (m/s). Vehicle velocity is a function of the rocket exhaust gas velocity, mass ratio of the vehicle, aerodynamic drag during flight through the atmosphere, gravitational pull expressed in terms of burning time of the rocket, and the trajectory. In 1903 Tsiolkovskiy derived the velocity for a rocket in vertical flight, disregarding drag and gravitational pull. This yielded the fundamental rocket flight equation named after him and so identified in the text. The equation is:
where V is vehicle velocity, V_{e} is exhaust gas velocity, M_{o} is gross mass, M_{e} is empty mass, and ln is the natural logarithm. The, direct relationship between vehicle velocity and exhaust gas velocity accounts for the interest in high energy propellants that yield high exhaust velocities.
Multistage rocket. A vehicle composed of two or more complete rocket systems, each including propellant and associated controls and structure. The last (upper) stage carries the spacecraft as the payload. The payload of the first stage is the upper stages and the spacecraft. Investigators learned a very long time ago that rocket vehicle velocity could be increased by having one rocket unit riding piggyback on another until the first expends all of its propellant. The dead weight of the first stage is normally jettisoned. The second rocket unit or stage begins operating at the velocity given it by the first stage and adds its own velocity increase, for a much higher final velocity. If the first two stages have identical exhaust velocities and mass ratios, for example, the final velocity will be twice that of either operating singly.
Other parameters and units. Horsepower is given in its equivalent kilowatts (kW), and revolutions per minute (rpm) was retained. The conversion of feet to meters and pounds to kilograms is obvious.
Parameter 
English 
SI 
This text 

Thrust 
lb 
N 
N (lb) 
Propellant flow 
lbm/s 
kg/s 
kg/s 
Mixture ratio 
various 
 
various 
Specific impulse 
lbf ^{. }s/lbm 
N ^{.} s/kg 
not used 
Exhaust velocity 
ft/s 
m/s 
m/s 
Pressure 
lbf/in^{2} 
N/m^{2} 
kN/m^{2} or atm 
Nozzle area ratio 
A_{e}/A_{t} 
 
A_{e}/A_{t} 
Heat transfer rate 
Btu/s ^{.} in^{2} 
J/s^{ .} m^{2} 
J/s^{ .} m^{2} 
Mass ratio 
M_{o}/M_{e} 
 
M_{o}/M_{e} 
Vehicle velocity 
ft/s 
m/s 
m/s 
Power 
hp 
W 
kW 
Length 
ft 
m 
m 
Mass 
lbm 
kg 
kg 



 
N 

newtons 
s 

seconds 
kg 

kilograms 
m 
 meter 
kW 
 kilowatts 
J 
 joule 
c 
 centi (10^{2}) 
k 
 kilo (10^{3}) 
M 
 mega (10^{6}) 