Thrust Calculator

Thrust is the force that moves an aircraft forward. A jet engine takes in air, heats it with burnt fuel and then ejects the exhaust gases at high speed backwards, creating thrust. Newton's third law explains the rest: as the gas flows in one direction, the engine and the plane move in the opposite direction. This tool uses a common formula to calculate thrust that engineers use to size engines, check performance at altitude and compare different designs. By entering an engine's parameters, net thrust can be calculated, or if a field is left blank, this parameter can be derived from it.

What is thrust?

Thrust is a force that is measured in Newtons. According to Newton's second law of motion, force is the product of mass and acceleration. In a jet engine this mass refers to the air stream which is constantly being accelerated. A general formula divides total thrust into two parts:

One part is the impulse thrust which comes from changing the speed of the air stream. The other part is pressure thrust which only occurs when the exhaust gases leaving the nozzle are at a different pressure than the surrounding air. When a properly designed nozzle operates in flight conditions, these two pressures will be nearly equal so that the pressure term usually goes away and the result is dominated by the impulse term.

The formula for thrust

In particular, net thrust is the sum of the impulse term and the pressure term.

where F_n is the net thrust, m-dot is the mass flow rate of air, V_e is the exhaust velocity, V_f is the flight speed, P_e is the pressure at the nozzle exit, P_a is ambient pressure and A_e is the area at the nozzle exit. The first term is the product of the mass flow rate and change in velocity while the second term is the product of the pressure difference and outlet area.

When the nozzle is fully expanded, the pressure at the outlet is equal to ambient and the second term disappears. The momentum term becomes the total thrust. As this is the default setting of this tool, the pressure fields are only shown when they are turned on.

Since this relationship is linear for each quantity, the formula can be rearranged to solve for any unknown quantity. If you input thrust, velocity and mass flow rate, you will get the amount of air that must exit the engine. If you input thrust and flow rate, you will get the exhaust velocity.

Instructions for using the calculator:

First enter the mass flow rate of air, which is the amount of air per second that the engine ejects. A small turbojet will eject about 50 kilograms of air per second. The GE9X engine on the Boeing 777X will eject over 1300 kilograms of air per second. Next enter the exhaust velocity, which is the speed at which the gas leaves the nozzle. For high bypass ratio turbofans this is usually between 300 and 600 meters per second, while for turbojets it's typically between 600 and 1200 meters per second.

The airspeed is the speed at which the aircraft moves through the air. When calculating thrust on a runway with no wind, leave this value as zero. In this case, the thrust will be maximum. Check the net thrust in the appropriate field. The default unit is kilonewtons but you can change it to newtons or pounds if desired. Each field has its own units menu so for example you could enter flow rate in pounds per second and speed in miles per hour while displaying thrust in kilonewtons.

Example calculation:

Assume a motor is pumping out 100 kilograms of air per second, flying at 200 meters per second and exhaust gases are coming out at 700. The pressure term is assumed to be zero because the nozzle is fully expanded.

First, we calculate the change in velocity.

Then we multiply this by the mass flow.

The result is 50 kilonewtons. The reason for this is simple: the more air that is expelled or the faster the air is expelled backwards, the greater the thrust produced. Turbofan engines increase efficiency by expelling large amounts of air at a moderate speed. In contrast, turbojet engines sacrifice this efficiency and expel smaller amounts of air at higher speeds. Therefore, turbojets are used when speed performance is prioritized over fuel efficiency.

Impulse thrust and pressure thrust.

These two terms differ in their behavior. The momentum thrust is almost always the main part of the total thrust and can be represented by a simplified form. The pressure thrust is a corrective term that matters only when the nozzle is not fully expanded.

If the exhaust pressure at the nozzle is still higher than ambient pressure, then the nozzle is in a state of under expansion. This often occurs at high altitudes where the outside air is thinner and in this case the pressure term increases thrust. If the exhaust pressure falls below ambient pressure, then the nozzle is in a state of over expansion which reduces thrust. By enabling the Thrust with Pressure option and entering the exit pressure, ambient pressure and nozzle size, the split between the two thrusts will be shown. The same pressure term is also why rocket nozzles can produce additional thrust in space.

Static thrust and dynamic thrust

The thrust varies with flight speed. When the aircraft is on the ground, the difference in speeds equals the exhaust velocity itself and the thrust is at its maximum. As the plane accelerates, this difference decreases, and so does net thrust.

The aircraft continues to accelerate until the decreasing thrust is equalized by increasing drag. At this point there is no further available thrust even if the engines are operating at full power. This is one of the reasons jet aircraft have a finite top speed. The static and dynamic thrust curves show the complete curve of the engine from its maximum on the runway to zero when flight speed approaches exhaust velocity.

Thrust-to-weight ratio

Once the thrust is known, the thrust to weight ratio can be calculated by dividing the thrust by the weight of the aircraft. This single value gives a good indication of the maneuverability of an aircraft. A ratio that exceeds 1 means that the thrust exceeds the weight, which will cause the nose of the aircraft to pitch up and accelerate. Fighter jets are designed this way.

The thrust-to-weight ratio of commercial airliners is much lower and is around 0.25 to 0.35. This is enough for the plane to take off easily from the ground, but not enough for a vertical climb. A higher thrust-to-weight ratio allows for greater rate of climb and acceleration, but requires larger and more fuel-intensive engines. If you enter a weight, the Thrust-to-Weight Calculator will calculate the corresponding value.

Areas of application:

The Aerospace team uses this formula early in the engine selection process and again later to ensure that the selected engine will meet its rated performance. High bypass turbofan engines such as the GE9X or Trent XWB move a large amount of air (more than 1,500 kg per second) at an appropriate exit velocity to allow high speed on long range flights but with some compromise in fuel burn.

Turbojet engines work in the opposite manner; they take in less air but have a much higher exhaust velocity, which allows them to produce high thrust at high altitude, overcoming drag. Engines such as those on the F-15 and the ones that once powered Concorde operate this way. This general formula makes these trade-offs explicit and allows for quantification of them.

Typical thrust values for various engines.

The following table shows the approximate thrust values for various engines at sea level start conditions. It includes both small turbojet engines used in business jets and the largest turbofan engines ever operated in commercial service.

Important notes:

Please check if the input values are correct. The ambient pressure decreases with increasing altitude, from about 101325 pascals at sea level to approximately 26500 pascals (at an altitude of 10,000 meters). Therefore, for calculations related to flight, the pressure values for this state must be used. In reality, thrust is reduced by factors such as installation effects, inlet pressure recovery and nozzle efficiency, but these factors are not taken into account in this formula.

At a fully expanded nozzle the pressure term disappears completely and the impulse term dominates the result. At operating conditions outside of design range or supersonic conditions, manufacturer data or full engine cycle models must be used. This calculation tool should be considered as a simple approximation tool and not a replacement for test stand.