Specific impulse

The specific impulse (usually abbreviated as ISP) defines the efficiency of an engine. It is thrust per the rate of fuel consumption. Or equivalently, it is change in momentum per amount of fuel consumed. It can be expressed as a duration or velocity (typically seconds and meters per second), depending whether fuel is measured by it's mass, or by it's weight on the surface of Earth (specifically, weight in a gravitational field exerting acceleration of exactly 9.80665 meters per second squared). Note that gravity on Kerbin is slightly different at 9.81 meters per second squared.

If an engine with specific impulse $I_{sp,g_{0}}$ (in seconds) is generating $F$ newtons of thrust, the fuel it burns every second will weigh ${\frac {F}{I_{sp,g_{0}}}}$ newtons on Earth (the planet whose $g_{0}$ is used to convert the unit of specific impulse into “seconds”). If the specific impulse is given as $I_{sp}$ (in meters/second) then it will burn ${\frac {F}{I_{sp}}}$ kilograms of fuel every second.

Formulas

Definition

$I_{sp}={\frac {F_{T}}{\dot {m}}}$

Where:

• $I_{sp}$ is the specific impulse in meters per second
• $F_{T}$ the thrust in newtons
• ${\dot {m}}$ the fuel consumption in kg/s

By multiplying this value with g₀ it is possible to change the unit to only seconds avoiding conversion issues between the SI and customary units. The value g₀ behaves like a conversion factor and doesn't change when the gravity for the craft is changing. Usually both values are called specific impulse and are abbreviated by I_sp. The name I_sp,g0 is used here only to clarify that both values aren't the same. This value is sometimes called weight specific impulse.

$I_{sp,g_{0}}={\frac {F_{T}}{{\dot {m}}\cdot g_{0}}}={\frac {I_{sp}}{g_{0}}}$

Where:

• $I_{sp,g_{0}}$ is the specific impulse in seconds
• $g_{0}$ is the surface gravity in the required unit (exactly 9.80665 meters per second squared)

The formula using the it (e.g. for Δv) has to specify what unit it does expect and if the value is defined in the other one it has to be converted.

Multiple engines

The combined specific impulse of multiple engines is calculated by the following formula:

$I_{sp}={\frac {\sum \limits _{i}F_{T_{i}}}{\sum \limits _{i}{\dot {m}}_{i}}}={\frac {\sum \limits _{i}F_{T_{i}}}{\sum \limits _{i}{\frac {F_{T_{i}}}{I_{sp_{i}}}}}}$

Where:

• $I_{sp}$ is the specific impulse in meters per second
• $I_{sp_{i}}$ the specific impulse of each engine in meters per second
• $F_{T_{i}}$ the thrust of each engine in newton
• ${\dot {m}}$ the fuel consumption in kg/s

When the fuel consumption is not used in this formula, it is only important that all thrust values have the same unit (e.g. kilonewtons) and the specific impulse have all the same unit (e.g. seconds). The result is then in the same unit as the specific impulses of the engines. If all engines have the same specific impulse the resulting specific impulse will be the same.

The result is equivalent to the weighted harmonic mean of the engines' specific impulses, weighted by each engine's thrust.

Relation with altitude

In most cases, the specific impulse is linearly dependent of the pressure (in atm units) in KSP:

Isp(P) = Isp(vac) + (Isp(atm) - Isp(vac)) * P

And if the fuel consumption is constant, it allows us to write the same thing for the thrust:

T(P) = T(vac) + (T(atm) - T(vac)) * P

Physical background

Before 1.0 in KSP the fuel consumption on most engines depended on the atmospheric pressure with the lowest consumption (and thus highest specific impulse) in vacuum. In the real world this is usually reversed: The fuel consumption stays always the same but the thrust is increasing over time, because it is easier to have a steady fuel flow. This is how KSP behaves since 1.0, as well. Because a higher thrust with the same fuel consumption it is more efficient the specific impulse rises. The specific impulse can only be calculated using this method for reaction engines and not jet engines as those work on another principle.

Although the unit of the specific impulse is a velocity it is lower than the exhaust speed usually, because some of the fuel consumed isn't used for propelling directly, but runs the turbopumps to fuel the engine.

Conversion factor

To convert the specific impulse between the handy weight specific impulse and the physical usable specific impulse it had to be converted with g₀.

Example

The Kerbal X has six LV-T45 Liquid Fuel Engines with a specific impulse of 320 s in atmosphere and one Rockomax "Mainsail" Liquid Engine with a specific impulse of 280 s. The average specific impulse of all engines is then:

$I_{sp}={\frac {6\cdot 200{\text{kN}}+1500{\text{kN}}}{6\cdot {\frac {200{\text{kN}}}{320{\text{s}}}}+{\frac {1500{\text{kN}}}{280{\text{s}}}}}}=296.47{\text{s}}$

To convert this value into a physical usable value.

$I_{sp}=g_{I_{sp}}\cdot 296.47{\text{s}}=2911.34{\frac {\text{m}}{\text{s}}}$

This value can then be used to calculate the fuel consumption: (Due to migration errors, LaTeX can't be changed. Corrections: 2911.34 is 2907.38, 927.4 is 928.7, 0.185 and 185 are 0.186 and 186.)

${\begin{aligned}I_{sp}&={\frac {6\cdot 200{\text{kN}}+1500{\text{kN}}}{\dot {m}}}=2911.34{\frac {\text{m}}{\text{s}}}\\{\dot {m}}&={\frac {6\cdot 200000{\text{N}}+1500000{\text{N}}}{2911.34{\frac {\text{m}}{\text{s}}}}}\\&=927.4{\frac {\text{kg}}{\text{s}}}\end{aligned}}$

Because the engines use liquid fuel and oxidizer with a density of 5000 kg/m³ it is possible to calculate the volume consumed.

${\dot {V}}={\frac {\dot {m}}{\rho }}={\frac {927.4{\frac {\text{kg}}{\text{s}}}}{5000{\frac {\text{kg}}{{\text{m}}^{3}}}}}=0.185{\frac {{\text{m}}^{3}}{\text{s}}}=185{\frac {\text{l}}{\text{s}}}$

Of course these values of the craft are valid for Kerbin's atmosphere. Because of the staging pattern engines will be dropped until reaching vacuum and thus changing the efficiency again. This time the efficiency lowers, because the higher efficient engines drop first leaving only the Mainsail engine with the lowest efficiency of all engines on the craft.

Comparing engines

Because different engines have different masses, simply comparing their specific impulse is sometimes not enough.

However, we can improve the comparison by using the total energy (work) that the engine produces.

• Definition: $F=I_{sp}{\dot {m_{f}}}$
• total burn time ($m_{f}$ is the total fuel mass): $t_{max}=m_{f}{\frac {I_{sp}}{F}}$
• Third newton's law: ${\frac {dv}{dt}}m=F$, where the mass has now three components: payload mass + engine mass + fuel mass ($m_{p}+m_{e}+m_{f}$), which we can integrate to obtain:

$v(t)=I_{sp}\log \left({\frac {\left(m_{p}+m_{e}+m_{f}\right)}{\left(m_{p}+m_{e}+m_{f}\right)-{\frac {F}{I_{sp}}}t}}\right)\,\,0<t\leq m_{f}{\frac {I_{sp}}{F}}$ (notice that we are considering no gravitational field here) (this is a version of the so called ideal rocket equation)

With the above, we can compute the total work done as

$W=\int _{x_{0}}^{x_{1}}F\cdot dr=\int _{0}^{t_{max}}F{\frac {dr\left(t\right)}{dt}}dt=F\int _{0}^{t_{max}}v\left(t\right)dt$

or

With this formula we can compare engines. The higher the W, the better since the work can be converted to either potential energy (i.e. distance from the gravitational body) or kinetic energy (i.e. speed).

For example, for the same vehicle with a given mass $m_{p}$, we can compare a single Poodle engine ($I_{sp}=390$, $m_{e}=2$, $F=220$) against four nuclear engines ($I_{sp}=800$, $m_{e}=2.25*4=9$, $F=4*60=240$). For a small vehicle (say $m_{p}=1,m_{f}=10$), the nuclear engine is better (i.e. the ratio of the works W) by a factor of 1.9. With more fuel $m_{p}=1,m_{f}=20$, it increases to 2.2. Increasing the payload (e.g. m_p = 10) increases this ratio to 2.8 (i.e. almost 3 times better). A more interesting case is the 48-7S ($I_{sp}=350$, $m_{e}=0.1$, $F=30$) vs the LV-909 ($I_{sp}=390$, $m_{e}=0.5$, $F=50$), two small engines. Even with a higher $I_{sp}$, the LV-909 is less efficient than the 48-7S. This is because its mass is 5 times higher than the 48-7S, which is not compensated by the increase of only ~10% of $I_{sp}$. Of course, when leaving a gravitational body, the thrust F must be able to compensate the gravity pull. This naturally adds a constraint to the choice of engines (i.e. the PB-ION Electric Propulsion System is the most efficient thruster but it is not able to lift itself, let alone fuel and a payload, off of Kerbin).

See also

• Terminology
• Specific impulse on Wikipedia