Specific impulse

Specific impulse (usually abbreviated Isp) is a measure of the efficiency of rocket and jet engines. By definition, it is the impulse delivered per unit of propellant consumed, and is dimensionally equivalent to the thrust generated per unit propellant flow rate.[1] If mass (kilogram or slug) is used as the unit of propellant, then specific impulse has units of velocity. If weight (newton or pound) is used instead, then specific impulse has units of time (seconds). The conversion constant between these two versions is the standard gravitational acceleration constant (g0).[2] The higher the specific impulse, the lower the propellant flow rate required for a given thrust, and in the case of a rocket, the less propellant needed for a given delta-v, per the Tsiolkovsky rocket equation.

Specific impulse is a useful value to compare engines, much like miles per gallon or liters per 100 kilometers is used for cars.[3] A propulsion method and system with a higher specific impulse is more propellant-efficient.[1][4] While the unit of seconds can seem confusing to laypeople, it is fairly simple to understand as "hover-time": how long a rocket can "hover" before running out of fuel, given the weight of that propellant/fuel. Of course, the weight of the rocket has to be taken out of consideration and so does the reduction in fuel weight as it's expended; the basic idea is "how long can any given amount of x hold itself up". Obviously that must mean "...against Earth's gravity", which means nothing in non-Earth conditions; hence Isp being given in velocity when propellant is measured in mass rather than weight, and the question becomes "how fast can any given amount of x accelerate itself?"

Note that Isp describes efficiency in terms of amount of propellant, and does not include the engine, structure or power source. Higher Isp means less propellant needed to impart a given momentum. Some systems with very high Isp (cf. ion thrusters) may have relatively very heavy/massive power generators, and produce thrust over a long period; thus, while they are "efficient" in terms of propellant mass carried, they may actually be quite poor at delivering high thrust as compared to "less efficient" engine/propellant designs.

Another number that measures the same thing, usually used for air breathing jet engines, is specific fuel consumption. Specific fuel consumption is inversely proportional to specific impulse and the effective exhaust velocity. The actual exhaust velocity is the average speed of the exhaust jet, which includes fuel combustion products, nitrogen, and argon, as it leaves an air breathing engine. The effective exhaust velocity is the exhaust velocity that the combusted fuel and atmospheric oxygen only would need to produce the same thrust. The two are identical for an ideal rocket working in vacuum, but are radically different for an air-breathing jet engine that obtains extra thrust by accelerating the non-combustible components of the air. Specific impulse and effective exhaust velocity are proportional.

General considerations

The amount of propellant is normally measured either in units of mass or weight. If mass is used, specific impulse is an impulse per unit mass, which dimensional analysis shows to have units of speed, and so specific impulses are often measured in meters per second and are often termed effective exhaust velocity. However, if propellant weight is used, an impulse divided by a force (weight) turns out to be a unit of time, and so specific impulses are measured in seconds. These two formulations are both widely used and differ from each other by a factor of g0, the dimensioned constant of gravitational acceleration at the surface of the Earth.

Note that the rate of change of momentum of a rocket (including its propellant) per unit time is equal to the thrust.

The higher the specific impulse, the less propellant is needed to produce a given thrust during a given time. In this regard a propellant is more efficient the greater its specific impulse. This should not be confused with energy efficiency, which can decrease as specific impulse increases, since propulsion systems that give high specific impulse require high energy to do so.[5]

It is important that thrust and specific impulse not be confused. The specific impulse is a measure of the impulse produced per unit of propellant expended, while thrust is a measure of the momentary or peak force supplied by a particular engine. In many cases, propulsion systems with very high specific impulses—some ion thrusters reach 10,000 seconds—produce low thrusts.[6]

When calculating specific impulse, only propellant that is carried with the vehicle before use is counted. For a chemical rocket the propellant mass therefore would include both fuel and oxidizer; for air-breathing engines only the mass of the fuel is counted, not the mass of air passing through the engine.

Units

Various equivalent rocket motor performance measurements, in SI and English Engineering units
Specific impulse, by weight Specific impulse, by mass Effective exhaust velocity Specific fuel consumption
SI = X s = 9.8066·X N·s/kg = 9.8066·X m/s = 101,972/X g/kN·s
English Engineering units = X s = X lbf·s/lb = 32.16·X ft/s = 3,600/X lb/lbf·h

By far the most common unit used for specific impulse today is the second, and this is used both in the SI world as well as where Imperial units are used. Its chief advantages are that its units and numerical value are identical everywhere, and essentially everyone understands it. Nearly all manufacturers quote their engine performance in seconds, and it is also useful for specifying aircraft engine performance.[7]

The effective exhaust velocity in units of m/s is also in reasonably common usage. For rocket engines it is reasonably intuitive, although for many rocket engines the effective exhaust speed is considerably different from the actual exhaust speed due to, for example, fuel and oxidizer that is dumped overboard after powering turbo-pumps. For air-breathing engines the effective exhaust velocity is not physically meaningful, although it can be used for comparison purposes nevertheless.[8]

The values expressed in N·s/kg are not uncommonly seen and are numerically equal to the effective exhaust velocity in m/s (from Newton's second law and the definition of the newton.)

Another equivalent unit is specific fuel consumption. This has units of g/(kN·s) or lb/(lbf·h) and is inversely proportional to specific impulse. Specific fuel consumption is used extensively for describing the performance of air-breathing jet engines.[9]

Specific impulse in seconds

General definition

For all vehicles specific impulse (impulse per unit weight-on-Earth of propellant) in seconds can be defined by the following equation:[3]

$F_\text{thrust} = g_0 \cdot I_\text{sp} \cdot \dot m,$

where:

$F_\text{thrust}$ is the thrust obtained from the engine, in newtons (or pounds force),
$g_0$ is the acceleration at the Earth's surface, in m/s2 (or ft/s2),
$I_\text{sp}$ is the specific impulse measured in seconds,
$\dot m$ is the mass flow rate in kg/s (or slugs/s), which is negative the time-rate of change of the vehicle's mass (since propellant is being expelled).

The English unit pound mass is more commonly used than the slug, and when using pounds/sec for mass flow rate, the conversion constant g0 becomes unnecessary, because the slug is dimensionally equivalent to pounds divided by g0:

$F_\text{thrust} = I_\text{sp} \cdot \dot m.$

Isp in seconds is equal to the amount of time a rocket must be fired to use a quantity of propellant with weight (measured at one standard gravity) equal to its thrust.

The advantage of this formulation is that it may be used for rockets, where all the reaction mass is carried on board, as well as aeroplanes, where most of the reaction mass is taken from the atmosphere. In addition, it gives a result that is independent of units used (provided the unit of time used is the second).

The specific impulse of various jet engines

Rocketry

In rocketry, where the only reaction mass is the propellant, an equivalent way of calculating the specific impulse in seconds is also frequently used. In this sense, specific impulse is defined as the thrust integrated over time per unit weight-on-Earth of the propellant:[2]

$I_{\rm sp}=\frac{v_\text{e}}{g_0},$

where

Isp is the specific impulse measured in seconds,
$v_\text{e}$ is the average exhaust speed along the axis of the engine (in ft/s or m/s),
g0 is the acceleration at the Earth's surface (in ft/s2 or m/s2).

In rockets, due to atmospheric effects, the specific impulse varies with altitude, reaching a maximum in a vacuum. This is because the exhaust velocity isn't simply a function of the chamber pressure, but is a function of the difference between the interior and exterior of the combustion chamber. It is therefore important to note whether the specific impulse refers to operation in a vacuum or at sea level. Values are usually indicated with or near the units of specific impulse (e.g. "sl", "vac").

Specific impulse as a speed (effective exhaust velocity)

Because of the geocentric factor of g0 in the equation for specific impulse, many prefer to define the specific impulse of a rocket (in particular) in terms of thrust per unit mass flow of propellant (instead of per unit weight flow). This is an equally valid (and in some ways somewhat simpler) way of defining the effectiveness of a rocket propellant. For a rocket, the specific impulse defined in this way is simply the effective exhaust velocity relative to the rocket, ve. The two definitions of specific impulse are proportional to one another, and related to each other by:

$v_\text{e} = g_0 I_\text{sp},$

where

$I_\text{sp}$ is the specific impulse in seconds,
$v_\text{e}$ is the specific impulse measured in m/s, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s2),
$g_0$ is the acceleration due to gravity at the Earth's surface, 9.81 m/s2 (in Imperial units 32.2 ft/s2).

This equation is also valid for air-breathing jet engines, but is rarely used in practice.

(Note that different symbols are sometimes used; for example, c is also sometimes seen for exhaust velocity. While the symbol $I_\text{sp}$ might logically be used for specific impulse in units of N·s/kg; to avoid confusion, it is desirable to reserve this for specific impulse measured in seconds.)

It is related to the thrust, or forward force on the rocket by the equation:[10]

$F_\text{thrust} = v_\text{e} \cdot \dot m,$

where $\dot m$ is the propellant mass flow rate, which is the rate of decrease of the vehicle's mass.

A rocket must carry all its fuel with it, so the mass of the unburned fuel must be accelerated along with the rocket itself. Minimizing the mass of fuel required to achieve a given push is crucial to building effective rockets. The Tsiolkovsky rocket equation shows that for a rocket with a given empty mass and a given amount of fuel, the total change in velocity it can accomplish is proportional to the effective exhaust velocity.

A spacecraft without propulsion follows an orbit determined by its trajectory and any gravitational field. Deviations from the corresponding velocity pattern (these are called Δv) are achieved by sending exhaust mass in the direction opposite to that of the desired velocity change.

Actual exhaust speed versus effective exhaust speed

Note that effective exhaust velocity and actual exhaust velocity can be significantly different, for example when a rocket is run within the atmosphere, atmospheric pressure on the outside of the engine causes a retarding force that reduces the specific impulse, and the effective exhaust velocity goes down, whereas the actual exhaust velocity is largely unaffected. Also, sometimes rocket engines have a separate nozzle for the turbo-pump turbine gas, and then calculating the effective exhaust velocity requires averaging the two mass flows as well as accounting for any atmospheric pressure.[citation needed]

For air-breathing jet engines, particularly turbofans, the actual exhaust velocity and the effective exhaust velocity are different by orders of magnitude. This is because a good deal of additional momentum is obtained by using air as reaction mass. This allows a better match between the airspeed and the exhaust speed, which saves energy/propellant and enormously increases the effective exhaust velocity while reducing the actual exhaust velocity.[citation needed]

Energy efficiency

Rockets

For rockets and rocket-like engines such as ion-drives a higher $I_{sp}$ implies lower energy efficiency: the power needed to run the engine is simply:

$\frac {dm} {dt} \frac { v_e^2 } {2}$

where ve is the actual jet velocity.

whereas from momentum considerations the thrust generated is:

$\frac {dm} {dt} v_e$

Dividing the power by the thrust to obtain the specific power requirements we get:

$\frac {v_e} {2}$

Hence the power needed is proportional to the exhaust velocity, with higher velocities needing higher power for the same thrust, causing less energy efficiency per unit thrust.

However, the total energy for a mission depends on total propellant use, as well as how much energy is needed per unit of propellant. For low exhaust velocity with respect to the mission delta-v, enormous amounts of reaction mass is needed. In fact a very low exhaust velocity is not energy efficient at all for this reason; but it turns out that neither are very high exhaust velocities.

Theoretically, for a given delta-v, in space, among all fixed values for the exhaust speed the value $v_\text{e}=0.6275 \Delta v$ is the most energy efficient for a specified (fixed) final mass, see energy in spacecraft propulsion.

However, a variable exhaust speed can be more energy efficient still. For example, if a rocket is accelerated from some positive initial speed using an exhaust speed equal to the speed of the rocket no energy is lost as kinetic energy of reaction mass, since it becomes stationary.[11] (Theoretically, by making this initial speed low and using another method of obtaining this small speed, the energy efficiency approaches 100%, but requires a large initial mass.) In this case the rocket keeps the same momentum, so its speed is inversely proportional to its remaining mass. During such a flight the kinetic energy of the rocket is proportional to its speed and, correspondingly, inversely proportional to its remaining mass. The power needed per unit acceleration is constant throughout the flight; the reaction mass to be expelled per unit time to produce a given acceleration is proportional to the square of the rocket's remaining mass.

Also it is advantageous to expel reaction mass at a location where the gravity potential is low, see Oberth effect.

Air breathing

Air-breathing engines such as turbojets increase the momentum generated from their propellant by using it to power the acceleration of inert air rearwards. It turns out that the amount of energy needed to generate a particular amount of thrust is inversely proportional to the amount of air propelled rearwards, thus increasing the mass of air (as with a turbofan) both improves energy efficiency as well as $I_{sp}$.

Examples

For a more complete list see: Spacecraft propulsion#Table of methods
Specific impulse of various propulsion technologies
Engine Effective exhaust velocity
(m/s, kg·m/(s·kg))
Specific impulse
(s)
Energy per kg of exhaust
(MJ/kg)
Turbofan jet engine
(actual V is ~300 m/s)
29,000 3,000 ~0.05
Solid rocket
2,500 250 3
Bipropellant liquid rocket
4,400 450 9.7
Ion thruster 29,000 3,000 430
VASIMR[12][13][14] 30,000–120,000 3,000–12,000 1,400
Dual-stage 4-grid electrostatic ion thruster[15] 210,000 21,400 22,500

An example of a specific impulse measured in time is 453 seconds, which is equivalent to an effective exhaust velocity of 4,440 m/s, for the Space Shuttle Main Engines when operating in a vacuum.[16] An air-breathing jet engine typically has a much larger specific impulse than a rocket; for example a turbofan jet engine may have a specific impulse of 6,000 seconds or more at sea level whereas a rocket would be around 200–400 seconds.[17]

An air-breathing engine is thus much more propellant efficient than a rocket engine, because the actual exhaust speed is much lower, the air provides an oxidizer, and air is used as reaction mass. Since the physical exhaust velocity is lower, the kinetic energy the exhaust carries away is lower and thus the jet engine uses far less energy to generate thrust (at subsonic speeds).[18] While the actual exhaust velocity is lower for air-breathing engines, the effective exhaust velocity is very high for jet engines. This is because the effective exhaust velocity calculation essentially assumes that the propellant is providing all the thrust, and hence is not physically meaningful for air-breathing engines; nevertheless, it is useful for comparison with other types of engines.[19]

The highest specific impulse for a chemical propellant ever test-fired in a rocket engine was 542 seconds (5,320 m/s) with a tripropellant of lithium, fluorine, and hydrogen. However, this combination is impractical; see rocket fuel.[20]

Nuclear thermal rocket engines differ from conventional rocket engines in that thrust is created strictly through thermodynamic phenomena, with no chemical reaction.[21] The nuclear rocket typically operates by passing hydrogen gas through a superheated nuclear core. Testing in the 1960s yielded specific impulses of about 850 seconds (8,340 m/s), about twice that of the Space Shuttle engines.

A variety of other non-rocket propulsion methods, such as ion thrusters, give much higher specific impulse but with much lower thrust; for example the Hall effect thruster on the SMART-1 satellite has a specific impulse of 1,640 s (16,100 m/s) but a maximum thrust of only 68 millinewtons.[22] The Variable specific impulse magnetoplasma rocket (VASIMR) engine currently in development will theoretically yield 20,000−300,000 m/s, and a maximum thrust of 5.7 newtons.[23]

Larger engines

Here are some example numbers for larger jet and rocket engines:

Specific fuel consumption (SFC), specific impulse, and effective exhaust velocity numbers for various rocket and jet engines.
Engine type Scenario SFC in lb/(lbf·h) SFC in g/(kN·s) Specific impulse (s) Effective exhaust velocity (m/s)
NK-33 rocket engine Vacuum 10.9 309 331[24] 3,240
SSME rocket engine Space shuttle vacuum 7.95 225 453[25] 4,423
Ramjet Mach 1 4.5 127 800 7,877
J-58 turbojet SR-71 at Mach 3.2 (Wet) 1.9 53.8 1,900 18,587
Rolls-Royce/Snecma Olympus 593 Concorde Mach 2 cruise (Dry) 1.195[26] 33.8 3,012 29,553
CF6-80C2B1F turbofan Boeing 747-400 cruise 0.605[26] 17.1 5,950 58,400
General Electric CF6 turbofan Sea level 0.307[26] 8.696 11,700 115,000

Model rocketry

Specific impulse is also used to measure performance in model rocket motors. Following are some of Estes' claimed values for specific impulses for several of their rocket motors:[27] Estes Industries is a large, well-known American seller of model rocket components. The specific impulse for these model rocket motors is much lower than for many other rocket motors because the manufacturer uses black powder propellant and emphasizes safety rather than maximum performance. The burn rate and hence chamber pressure and maximum thrust of model rocket motors is also tightly controlled.

Specific impulses for several commercially available Estes rocket motors.
Engine Total impulse (Ns) Fuel weight (N) Specific impulse (s)
Estes A10-3T 2.5 0.0370 67.49
Estes A8-3 2.5 0.0306 81.76
Estes B4-2 5.0 0.0816 61.25
Estes B6-4 5.0 0.0612 81.76
Estes C6-3 10 0.1223 81.76
Estes C11-5 10 0.1078 92.76
Estes D12-3 20 0.2443 81.86
Estes E9-6 30 0.3508 85.51

References

1. "What is specific impulse?". Qualitative Reasoning Group. Retrieved 22 December 2009.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
2. Benson, Tom (11 July 2008). "Specific impulse". NASA. Retrieved 22 December 2009.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
3. Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz
4. Hutchinson, Lee (14 April 2013). "New F-1B rocket engine upgrades Apollo-era design with 1.8M lbs of thrust". Ars Technica. Retrieved 15 April 2013. The measure of a rocket's fuel efficiency is called its specific impulse (abbreviated as 'ISP'—or more properly Isp).... 'Mass specific impulse...describes the thrust-producing efficiency of a chemical reaction and it is most easily thought of as the amount of thrust force produced by each pound (mass) of fuel and oxidizer propellant burned in a unit of time. It is kind of like a measure of miles per gallon (mpg) for rockets.'<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
5. http://www.geoffreylandis.com/laser_ion_pres.htp
6. "Mission Overview". exploreMarsnow. Retrieved 23 December 2009.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
7. http://www.grc.nasa.gov/WWW/k-12/airplane/specimp.html
8. http://www.qrg.northwestern.edu/projects/vss/docs/propulsion/3-what-is-specific-impulse.html
9. http://www.grc.nasa.gov/WWW/k-12/airplane/sfc.html
10. Aerospace Propulsion Systems By Thomas A. Ward
11. Note that this limits the speed of the rocket to the maximum exhaust speed.
12. http://vasimr.net/TimSTAIF2005.pdf
14. http://spacefellowship.com/news/art24083/vasimr-vx-200-meets-full-power-efficiency-milestone.html
15. http://www.esa.int/esaCP/SEMOSTG23IE_index_0.html
16. http://www.astronautix.com/engines/ssme.htm
17. http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node85.html
18. http://www.dunnspace.com/isp.htm
19. http://www.britannica.com/EBchecked/topic/198045/effective-exhaust-velocity
20. ARBIT, H. A., CLAPP, S. D., DICKERSON, R. A., NAGAI, C. K., Combustion characteristics of the fluorine-lithium/hydrogen tripropellant combination. AMERICAN INST OF AERONAUTICS AND ASTRONAUTICS, PROPULSION JOINT SPECIALIST CONFERENCE, 4TH, CLEVELAND, OHIO, June 10–14, 1968.
21. http://trajectory.grc.nasa.gov/projects/ntp/index.shtml
22. http://www.mendeley.com/research/characterization-of-a-high-specific-impulse-xenon-hall-effect-thruster/