Specific impulse

Specific impulse 

I was many times using the term specific impulse. So, let's speak about it in more detail.   

Specific impulse (usually abbreviated I_sp) is a measure of how efficiently a reaction mass engine, such as a rocket using propellant or a jet engine using fuel, generates thrust. A propulsion system which has a higher specific impulse uses the mass of the propellant more efficiently than a propulsion system with lower specific impulse. Now, in the case of a rocket, this means less propellant is needed for a given delta-v, so that the vehicle attached to the engine can more efficiently gain altitude and velocity.

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

Thrust is force, the thrust obtained from the engine (newtons or pounds force),

g₀ is the standard gravity, which is nominally the gravity at Earth's surface (m/s2),

I_sp is the specific impulse measured (seconds),

dm/dt is the mass flow rate of the expended propellant (kg/s)

I_sp in seconds is the amount of time a rocket engine can generate thrust, given a quantity of propellant whose weight is equal to the engine's thrust.

In Atmosphere

For engines like cold gas thrusters whose reaction mass is only the fuel they carry, specific impulse is exactly proportional to the effective exhaust gas velocity.

Note: A cold gas thruster is a type of rocket engine which uses the expansion of a pressurized gas to generate the thrust. In contrast with the traditional rocket engines, a cold gas thruster does not involve any combustion and therefore has lower thrust and efficiency compared to conventional monopropellant and bipropellant rocket engines. Cold gas thrusters are generally the simplest version of a rocket engine because their design consists only of a fuel tank, a regulating valve, a propelling nozzle, and the little of plumbing. They are the cheapest, simplest, and most reliable propulsion systems available for orbital maintenance, maneuvering and attitude control.

In the context of an atmosphere, the specific impulse can include the contribution to impulse provided by the mass of external air that is accelerated by the engine, such as by fuel combustion or by external propeller. Jet engines and turbofans use external air for both combustion and bypass, and therefore have a much higher specific impulse than rocket engines.

For air-breathing engines, we count only the fuel mass. We do not count the mass of air passing through the engine. Air resistance and the engine's inability to keep a high specific impulse at a fast burn rate are limiting factors to the propellant consumption rate. If it were not for air resistance and the reduction of propellant during flight, specific impulse would be a direct measure of the engine's effectiveness in converting propellant mass into forward momentum.

The specific impulse in terms of propellant mass spent has units of distance per time, which is a notional velocity called the effective exhaust velocity. This is higher than the actual exhaust velocity because the mass of the combustion air is not being accounted for. Actual and effective exhaust velocity are the same in rocket engines operating in a vacuum.

The amount of propellant can be measured either in units of mass or weight. If the mass is used, the specific impulse is an impulse per unit of mass. Dimensional analysis will show us that those are units of speed, specifically the effective exhaust velocity. As the SI system is mass-based, this type of analysis is usually done in meters per second. If a force-based unit system is used, impulse is divided by propellant weight, which is a measure of force, resulting in units of time (seconds). These two formulations differ from each other by the standard gravitational acceleration (g₀) at the surface of the earth.

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 is means that less propellant is needed to produce a given thrust for a given time and the more efficient the propellant is. 

Thrust and specific impulse should not be mixed. Thrust is the force supplied by the engine and depends on the amount of reaction mass travelling through the engine. 

Specific impulse measures the impulse which is produced per unit of propellant and it is proportional to the exhaust velocity. Thrust and specific impulse are related by the design and propellants of the engine, but this relationship is weak. 

When we  are calculating specific impulse, we count only propellant carried with the vehicle before use. In case of rockets, a heavier engine with a higher specific impulse may not be as effective in gaining altitude, distance, or velocity as a lighter engine with a lower specific impulse. Also it is a reason, why so many rocket designs have multiple stages. 

The first stage is usually optimized for high thrust to boost the following stages, which have higher specific impulse, into higher altitudes where these stages can perform more efficiently.

The specific impulse of various jet engines (SSME is the Space Shuttle Main Engine); source: wiki

In rocketry

In rocketry, the only reaction mass is the propellant, so the specific impulse is calculated using an alternative method, giving results with units of seconds. Specific impulse is defined as the thrust integrated over time per unit weight-on-Earth of the propellant:

where

I_sp is the specific impulse measured in seconds,

v_e is the average exhaust speed along the axis of the engine (in m/s),

g₀ is the standard gravity (in 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. Values are usually given for operation at sea level ("sl") or in a vacuum ("vac").

Comparison between different types of engine designs. Merlin and Raptor are built by SpaceX, RD-180 is Russian engine, The F-1 is a rocket engine developed by Rocketdyne, BE-4 is built by Blue Origin, and the last one is RS-25, also known as the Space Shuttle Main Engine (SSME); the source is https://everydayastronaut.com/.