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Specific impulse

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The specific impulse (commonly abbreviated Isp) of a propulsion system is the impulse (change in momentum) per unit of propellant.

Depending on whether the amount of propellant is expressed in mass or by weight (conventionally sea level weight on the Earth) the dimension of specific impulse is that of speed or time, respectively, differing by a factor of g, the gravitational acceleration at the surface of the Earth.

Contents

General considerations

Essentially, the higher the specific impulse, the less propellant is needed to gain a given amount of momentum. In this regard a propulsion method is more propellant-efficient if the specific impulse is higher. This should not in any way be confused with energy-efficiency, which can even decrease as specific impulse increases, since many propulsion systems that give high specific impulse require high energy to do so.

In addition it is important that thrust and specific impulse not be confused with one another. The specific impulse is a measure of the thrust per unit of propellant that is expended, while thrust is a measure of the momentary or peak force supplied by a particular engine. In fact, propulsion systems with very high specific impulses (such as ion thrusters: 3,000 seconds) are power limited to producing low thrusts, due to the relatively high weight of power generators.

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.

Examples

Specific impulse of various propulsion technologies
Engine "Ve" eff. exhaust velocity
(N·s/kg or m/s) 		Specific impulse
(s) 		Fuel mass
(kg) 		Energy expended
(GJ) 		Energy per kg
(MJ/kg)
Jet engine
30,000 3,000 50,000 2135 43
Solid rocket
2,000 200 190,000 95 1
Bipropellant rocket
4,500 450 8,200 103 13
Ion thruster 30,000 3,000 620 775 1250
VASIMR 300,000 30,000 100 4,500 45,000

An example of a specific impulse measured in time is 453 seconds, or, equivalently, an effective exhaust velocity of 4500 m/s, for the Space Shuttle Main Engines when operating in vacuum.

An air-breathing engine typically has a much larger specific impulse than a rocket: a jet engine may have a specific impulse of 2000-3000 seconds or more at sea level.

In some ways, comparing specific impulse seems unfair in the case of jet engines and rockets. However in rocket or jet powered aircraft, specific impulse is approximately proportional to range, and rockets do indeed perform much worse than jets below approximately 85000 feet (~25 km) in that regard.

The highest specific impulse for a chemical propellant ever test-fired in a rocket engine was lithium, fluorine, and hydrogen (a tripropellant): 542 seconds (5320 m/s). However, this combination is impractical, see rocket fuel.

Nuclear thermal rocket engines differ from conventional rocket engines in that thrust is created strictly through thermodynamic phenomena, with no chemical reaction. The nuclear rocket typically operates by passing hydrogen gas over a superheated nuclear core. [Testing in the 1960s] yielded specific impulses of about 850 seconds (8340 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 1640 s (16100 m/s) but a maximum thrust of only 68 millinewtons. The hypothetical Variable specific impulse magnetoplasma rocket (VASIMR) propulsion should yield a minimum of 10,000-300,000 m/s but will probably require a great deal of heavy machinery to confine even relatively diffuse plasmas, so they will be unusable for very-high-thrust applications such as launch from planetary surfaces.

Specific impulse in seconds

For all vehicles specific impulse (impulse per unit weight-on-Earth of propellant) in seconds can be defined by the following equationRocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz:

[\mathrm}=I_ \cdot \frac \cdot g_ \,]
where:

Fthrust is the thrust obtained from the engine, in newtons (or poundals).
Isp is the specific impulse measured in seconds.
[\frac ] is the mass flow rate in kg/s (or lb/s), which is minus the time-rate of change of the vehicle's mass, since propellant is being expelled.
g0 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 becomes the thrust in lbf rather than 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. In practice, the specific impulses of real engines vary somewhat with both altitude and thrust; nevertherless, Isp is a useful value to compare engines; much like 'miles per gallon' is used for cars.

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 hydrocarbon fuelled jet engines
Enlarge
The specific impulse of various hydrocarbon fuelled jet engines

Rocketry - specific impulse in seconds

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 change in momentum per unit weight-on-Earth of the propellant:

[I_=\frac}}]
where

Isp is the specific impulse measured in seconds

[v_] 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').

Interpretations

When expressed in units of seconds, the specific impulse can be interpreted in the following ways:

e.g. for hydrogen/oxygen, with a specific impulse of 460 seconds (4500 m/s): A very simplified example can make this point clear: Lets look at a hydrogen based engine:

The ideal reaction is: 2H2 + O2[\rightarrow]2H2O +467 kJ/mol. 

If the O2 came from a tank in a rocket the specific gives (again over-simplified)



[\frac =467 \ \mathrm],  where the mass is 18 g  (2H + O, 2 g/mol + 16 g/mol).



Solving for v, we get: 5093 m/s, about 5000 m/s under ideal conditions (ejection temperature 0 K).


If somehow we were to arrange that we wouldn't have to carry the oxygen the mass is now 2 g, and magically the energy was still 467 kJ, we would get: 15,280 m/s.

We can improve that by pushing great amounts of non-combustion air. 
This is possible because the energy is proportional to the square of the ejection speed but the “force” is proportional to the speed (due to simple momentum). The presence of nitrogen makes things even better. If we see the diagrams of big, efficient turbofans we will see that this is important part of the optimization guides. (http://anirudh.net/seminar/ge90.pdf by example)


So the reason why the specific impulse of a turbofan is so large is partly because the atmosphere provides the oxidant, so the plane does not carry it. But more importantly, the air is used as reaction mass, and the fuel is mainly used as an energy source.



Rocketry - 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 terms of thrust per unit mass flow of propellant (instead of per unit weight flow).  This is an equally valid (and 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, ve.  The two definitions of specific impulse are proportional to one another, and related to each other by:



[v_ = g_0 I_ \,]



where 


Isp is the specific impulse in seconds


ve is the specific impulse measured in metres per second (in the U.S. feet/second), which is the same as the effective exhaust velocity measured in metres per second


g0 is the Earth's gravitational constant, 9.81 metres per second per second (in English units 32.2 ft/s²).


(Note that different symbols are sometimes used; for example, c is also sometimes seen for exhaust velocity.  While the symbol Isp 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}=v_ \cdot \frac   \,]



where 


[\frac   ] is the mass flow rate, which is minus the time-rate of change of the vehicle's mass, since propellant is being expelled. 


Interpretations


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. Using Newton's laws of motion it is not difficult to verify that for a fixed mass of fuel, the total change in velocity (in fact, momentum) it can accomplish can only be increased by increasing 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.


Due to the law of conservation of momentum, to change the speed of the spacecraft by an amount equal to 1% of the effective exhaust speed, approximately requires an exhaust mass equal to 1% of the mass of the spacecraft, including the fuel that has not yet been spent.


A useful rule of thumb is that the Δv that can be produced with a propellant mass of 63.2 % of the initial total mass is equal to the effective exhaust velocity (see Rocket equation.)


The speed is also approximately twice the power per unit thrust


For a Δv that is much smaller than the specific impulse, the fuel required is approximately proportional to the Δv. For a Δv that is larger than the specific impulse, this requirement of carrying the fuel and spending much of the fuel on accelerating the propellant (the burned fuel and oxidizer), gives rise to an exponential increase in fuel requirement (and larger tanks which also add to the mass).
See spacecraft propulsion calculations and Tsiolkovsky rocket equation for details.


e.g for hydrogen/oxygen, with a specific impulse of 4500 m/s (460 seconds):



References





See also























 





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