# Specific impulse

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Specific impulse (usually abbreviated Isp) is a way to describe the efficiency of rocket and jet engines. It represents the impulse (change in momentum) per unit of propellant. The higher the specific impulse, the less propellant is needed to gain a given amount of momentum. Isp is a useful value to compare engines, much like "miles per gallon" is used for cars. A propulsion method with a higher specific impulse is more propellant-efficient.

Propellant is normally measured either in units of mass, or in units of weight at sea level on the Earth. 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.

## 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 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 (discussed later).

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.

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" effective exhaust velocity
(m/s, N·s/kg)
Specific impulse
(s)
Energy per kg
(MJ/kg)
Jet engine
29 000 (actual is ~300) 3000 43
Solid rocket
2500 250 3.0
Bipropellant liquid rocket
4400 450 9.7
Ion thruster 29 000 3000 430
VASIMR 290 000 30 000 43 000

(For a more complete list see: Spacecraft propulsion#Table of methods and their specific impulse)

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 jet 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 whereas a rocket would be around 200-400 seconds. Note also that an air-breathing engine is more efficient; this is because the actual exhaust speed is much lower, because air is used as reaction mass, particularly with turbofan jets. Since the actual, physical exhaust velocity is lower, the kinetic energy it carries away is lower and thus the jet engine uses far less energy to generate thrust, even allowing for the fact that more air is exhausted at lower speeds to get the same thrust as a smaller amount of mass at higher speeds.

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.

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 suborbital rockets do indeed perform much worse than jets 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 through 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 (16 100 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.

## Units

Specific Impulse (by weight) Specific Impulse (by mass) Effective exhaust velocity Specific Fuel Consumption =X seconds =9.8066 X N·s/kg =9.8066 X m/s =(101972/X) g/kN·s =X seconds =X lbf·s/lb =32.16 X ft/s =(3600/X) lb/lbf·h

By far the most common units 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 is 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.

The effective exhaust velocity 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 not precisely the same as the actual exhaust speed due to, for example, fuel and oxidizer that is dumped overboard after powering turbopumps. For airbreathing engines it is not physically meaningful although can be used for comparison purposes nevertherless.

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.)

The units of ft/s were used by NASA during Apollo, but seems to have fallen into disuse, and NASA are moving towards using SI units wherever possible.

The lbf·s/lb unit sees little use but is covered in some textbooks.

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 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[1]:

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

where:

Fthrust is the thrust obtained from the engine, in newtons (or poundals).
Isp is the specific impulse measured in seconds.
$\frac {\Delta m} {\Delta t}$ is the mass flow rate in kg/s (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 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.

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).

File:Specific-impulse-kk-20050824.png
The specific impulse of various hydrocarbon fuelled 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 change in momentum per unit weight-on-Earth of the propellant:

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

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').

## 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, 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 metres per second, which is the same as the effective exhaust velocity measured in metres per second (former usage in the U.S. was feet/second, though that is now obsolete)
$g_0 \,$ - is the acceleration due to gravity at the Earth's surface, 9.81 m/s² (in English 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 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:

$\mathrm{F_{\rm thrust}}=v_{\rm e} \cdot \frac {\Delta m} {\Delta t} \,$

where

$\frac {\Delta m} {\Delta t}$ is the mass flow rate, which is minus the time-rate of change of the vehicle's mass, since propellant is being expelled.

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.

### Actual exhaust speed versus effective exhaust speed

Note that effective exhaust velocity and actual exhaust velocity are very often 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 the effective exhaust velocity is an average of the two mass flows. Finally, for airbreathing jet engines particularly turbofans, the actual exhaust velocity and the effective exhaust velocity are different by orders of magnitude, since additional momentum is obtained by using air as reaction mass which saves propellant and enormously increases the effective exhaust velocity while reducing the actual exhaust velocity.

## Energy efficiency

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

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

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, and thus are less energy efficient.

However, this is not true in general, for example, 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 $I_{sp}$.

## References

1. Rocket Propulsion Elements, 7th Edition by George P. Sutton, Oscar Biblarz