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analogical dictionary

specific impulse (n.)

Specific impulse

Specific impulse (usually abbreviated Isp) is a way to describe the efficiency of rocket and jet engines. It represents the derivative of the impulse with respect to amount of propellant used, i.e., the thrust divided by the amount of propellant used per unit time.[1] If the "amount" of propellant is given in terms of mass (such as kilograms), then specific impulse has units of velocity. If it is given in terms of Earth-weight (such as kiloponds), then specific impulse has units of time. The conversion constant between the two versions of specific impulse is g.[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 is needed for a given delta-v per the Tsiolkovsky rocket equation.

The actual exhaust velocity is the average speed that the exhaust jet actually leaves the vehicle. The effective exhaust velocity is the speed that the propellant burned per second would have to leave the vehicle to give the same thrust. The two are about the same for a rocket working in a vacuum, but are radically different for an air-breathing jet engine that obtains extra thrust by accelerating air. Specific impulse and effective exhaust velocity are proportional.

Specific impulse is a useful value to compare engines, much like miles per gallon or litres per 100 kilometres is used for cars.[3] A propulsion method with a higher specific impulse is more propellant-efficient.[1] 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 effective exhaust velocity.

General considerations

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 be a unit 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 instead, 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 g, the dimensioned constant of gravitational acceleration at the surface of the Earth.

Note that the gain of momentum of a rocket (including fuel) per unit time is not equal to the thrust, because the momentum that the fuel has while in the rocket has to be subtracted to the extent that it is used, i.e., the gain of momentum of a rocket per unit time is equal to the thrust, minus the velocity of the rocket multiplied by the amount of fuel used per unit time. (This gain of momentum of the rocket is the negative of the momentum of the exhaust gas.) See also change of impulse of a variable mass.

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 if the specific impulse is higher. This should not be confused with energy efficiency, which can even decrease as specific impulse increases, since propulsion systems that give high specific impulse require high energy to do so.[4]

In addition it is important that thrust and specific impulse not be confused with one another. The specific impulse is a measure of the impulse per unit of propellant that is 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.[5]

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

Imperial and SI units for various rocket motor performance measurements.

Specific Impulse
(by weight)

Specific Impulse
(by mass)

Effective exhaust velocity
Specific fuel consumption
SI =X seconds =9.8066 X N•s/kg =9.8066 X m/s =(101,972/X) g/kN•s
Imperial units =X seconds =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 English 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 these units and it is also useful for specifying aircraft engine performance.[6]

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 turbopumps. For airbreathing engines the effective exhaust velocity is not physically meaningful, although it can be used for comparison purposes nevertheless.[7]

The N•s/kg is not uncommonly seen, and is 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 lbf/lb•h and is inversely proportional to specific impulse. Specific fuel consumption is used extensively for describing the performance of air-breathing jet engines.[8]

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]:

$\mathrm{F_{\rm thrust}}=I_{\rm sp} \cdot \dot m \cdot g_{\rm 0} \,$

where:

$\mathrm{F_{\rm thrust}}$ is the thrust obtained from the engine, in newtons (or poundals).
$I_{\rm sp}$ is the specific impulse measured in seconds.
$\dot m$ is the mass flow rate in kg/s (lb/s), which is negative the time-rate of change of the vehicle's mass since propellant is being expelled.
$g_{\rm 0}$ is the acceleration at the Earth's surface, in m/s² (or ft/s²).

(When working with English units, it is conventional to divide both sides of the equation by g0 so that the left hand side of the equation has units of lbs rather than expressing it in poundals.)

This Isp in seconds value is somewhat physically meaningful—if an engine's thrust could be adjusted to equal the initial weight of its propellant (measured at one standard gravity), then Isp is the duration the propellant would last.[citation needed]

The advantage that this formulation has is that it may be used for rockets, where all the reaction mass is carried onboard, 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]

$I_{\rm sp}=\frac{v_{\rm e}}{g_{\rm 0}}$[2]

where

Isp is the specific impulse measured in seconds

$v_{\rm e}$ 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. It is therefore most common to see the specific impulse quoted for the vehicle in a vacuum; the lower sea level values are usually indicated in some way (e.g. 'sl').[citation needed]

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:

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

where

$I_{\rm sp} \,$ - is the specific impulse in seconds
$v_{\rm e} \,$ - 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 English units units 32.2 ft/s²).

This equation is also valid for airbreathing 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 $I_{sp}$ 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:

$\mathrm{F_{\rm thrust}}=v_{\rm e} \cdot \dot m \,$[9]

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 the 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 turbopump 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 airbreathing 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 for 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 Tsiolkovsky rocket equation.

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.[10] (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

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)
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
Dual Stage Four Grid Electrostatic Ion Thruster[11] 210,000 21,400 22,500
VASIMR[12][13][14] 30,000-120,000 3,000-12,000 1,400

For a more complete list see: Spacecraft propulsion#Table of methods

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.[15] 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.[16]

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).[17] 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.[18]

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.[19]

Nuclear thermal rocket engines differ from conventional rocket engines in that thrust is created strictly through thermodynamic phenomena, with no chemical reaction.[20] 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.[21] The Variable specific impulse magnetoplasma rocket (VASIMR) engine currently in development will theoretically yield 10,000−300,000 m/s but will require a large electricity source and a great deal of heavy machinery to confine even relatively diffuse plasmas, and so will be unusable for high-thrust applications such as launch from planetary surfaces.[22]

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 larger rocket 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[23] 3,240
SSME rocket engine Space shuttle vacuum 7.95 225 453[24] 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[25] 33.8 3,012 29,553
CF6-80C2B1F turbofan Boeing 747-400 cruise 0.605[25] 17.1 5,950 58,400
General Electric CF6 turbofan Sea level 0.307[25] 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:[26] 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 .0370 67.49
Estes A8-3 2.5 .0306 81.76
Estes B4-2 5.0 .0816 61.25
Estes B6-4 5.0 .0612 81.76
Estes C6-3 10 .1223 81.76
Estes C11-5 10 .1078 92.76
Estes D12-3 20 .2443 81.86
Estes E9-6 30 .3508 85.51

References

1. ^ a b "What is specific impulse?". Qualitative Reasoning Group. Retrieved 22 December 2009.
2. ^ a b c Benson, Tom (11 July 2008). "Specific impulse". NASA. Retrieved 22 December 2009.
3. ^ a b Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz
4. ^ http://www.geoffreylandis.com/laser_ion_pres.htp
5. ^ "Mission Overview". exploreMarsnow. Retrieved 23 December 2009.
6. ^ http://www.grc.nasa.gov/WWW/k-12/airplane/specimp.html
7. ^ http://www.qrg.northwestern.edu/projects/vss/docs/propulsion/3-what-is-specific-impulse.html
8. ^ http://www.grc.nasa.gov/WWW/k-12/airplane/sfc.html
9. ^ Aerospace Propulsion Systems By Thomas A. Ward
10. ^ Note that this limits the speed of the rocket to the maximum exhaust speed.
11. ^ http://www.esa.int/esaCP/SEMOSTG23IE_index_0.html
12. ^ http://vasimr.net/TimSTAIF2005.pdf
14. ^ http://spacefellowship.com/news/art24083/vasimr-vx-200-meets-full-power-efficiency-milestone.html
15. ^ http://www.astronautix.com/engines/ssme.htm
16. ^ http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node85.html
17. ^ http://www.dunnspace.com/isp.htm
18. ^ http://www.britannica.com/EBchecked/topic/198045/effective-exhaust-velocity
19. ^ 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, Jun 10-14, 1968.
20. ^ http://trajectory.grc.nasa.gov/projects/ntp/index.shtml
21. ^ http://www.mendeley.com/research/characterization-of-a-high-specific-impulse-xenon-hall-effect-thruster/
22. ^ http://www.nasa.gov/vision/space/travelinginspace/future_propulsion.html
23. ^ Astronautix NK33
24. ^ Astronautix SSME
25. ^ a b c "Data on Large Turbofan Engines". Aircraft Aerodynamics and Design Group. Stanford University. Retrieved 22 December 2009.
26. ^ Estes 2011 Catalog www.acsupplyco.com/estes/estes_cat_2011.pdf