Exit Velocity Calculator

The exit velocity is the speed at which exhaust gases leave a jet engine, gas turbine or rocket through a nozzle. This converts hot pressurised gas into thrust. The faster the gas exits, the greater the momentum carried away and the more powerful the forward propulsive force on the vehicle.

This calculator calculates this velocity from the state of the gas at the nozzle inlet. As the equation is directly manipulable, missing inputs can be calculated backwards as well.

What is exit velocity?

As hot gas flows through a nozzle and expands, pressure and temperature decrease while velocity increases. The thermal energy and pressure energy is converted into directed kinetic energy. The exit velocity is the speed of the gases at the end of the nozzle. In a jet engine it determines thrust and together with flight speed affects fuel consumption. For rockets it is an important performance indicator that should be maximized for the entire propulsion system.

The gas accelerates because it can expand into a region of lower pressure. The greater the difference between the inlet and ambient pressures, the more the gas will expand and the faster it will ultimately move.

How to use this calculator:

Enter the gas adiabatic heat transfer coefficient, inlet temperature and inlet pressure, ambient pressure, gamma and nozzle efficiency and the calculator will return the exit velocity. No mode selection is required. Leave a field blank to have it calculated for you so that you can not only calculate the exit velocity but also work backwards from a measured velocity to determine the nozzle efficiency or the required inlet temperature for a target velocity. Enable the thrust option and enter mass flow rate and flight speed to display total thrust, net thrust, engine power and propulsion efficiency.

Formula for exit velocity:

When an ideal gas expands through a nozzle in a steady and adiabatic process the enthalpy drop is converted to kinetic energy by the conservation of energy. To account for actual losses, an efficiency $\eta_n$ is introduced for the nozzle, which gives the exit velocity as follows:

Here $C_p$ is the specific heat capacity at constant pressure, $T_0$ is the absolute inlet temperature, $P_{in}$ and $P_{amb}$ are respectively the inlet pressure and ambient pressure, $\gamma$ is the ratio of specific heats, and $\eta_n$ is the nozzle efficiency. The values in square brackets represent the isentropic expansion fraction and give the proportion of energy from the expanding gas that is usable.

Consider an example of a small gas turbine. The gas enters the nozzle at 1000 K and 200 kPa, and expands to sea level pressure of 101.325 kPa. The isobaric specific heat capacity is 1150 J/(kg·K), the ratio of specific heats is 1.33, and the nozzle efficiency is 95 percent.

First consider the term for the pressure ratio:

Therefore, gas gives off about 15.5 percent of its thermal energy. If this value is put into the main equation, it results in:

The result is approximately 582 m/s, which is a typical value for small turbines with a moderate pressure ratio. The following table shows the individual symbols and their numerical values in this example.

Determine any values

As each equation always has exactly one unique unknown, you can calculate not only the exit velocity but also any single parameter. If you leave a field blank, the calculation tool will select the corresponding format. Two particularly useful formats are calculating nozzle performance from measured speed and finding inlet temperature for desired exit velocity.

The value of gamma can also be back-calculated. Since it is contained in an exponent, it can be solved using logarithms. Thus this tool can not only be used for one-off calculations but also design problems such as determining the required pressure ratio that a nozzle needs to achieve a given exit velocity.

Ideal and actual nozzle performance

A perfect nozzle converts all of the available enthalpy drop into kinetic energy. In real nozzles some energy is lost due to friction and turbulence, so this ideal value cannot be achieved. The nozzle efficiency is the ratio of actual to ideal kinetic energy, and the exit velocity changes according to its square root. The calculation tool shows both the ideal and actual exit velocities, so that the difference between them represents a visible source of loss.

The lost energy is not destroyed but remains in the gas as heat so that lower efficiency nozzles produce somewhat higher temperature and lower velocity exhaust. A simple convergent nozzle will typically have an efficiency of about 90 percent to 95 percent, while a well-designed convergent-divergent nozzle can achieve efficiencies of 95 percent to 98 percent.

Outlet temperature, sound speed, mach number

Gases cool as they expand and accelerate. The calculator gives the static temperature at outlet, local speed of sound, and Mach number at outlet. The Mach number at outlet is the ratio of exit velocity to the speed of sound at that point. Mach number is an important factor in nozzle design. In a normal convergent nozzle, flow can only be accelerated up to a Mach number of 1. Flow breaks down there and the velocity does not increase further.

In order to achieve a velocity greater than the speed of sound, a convergent-divergent nozzle is required, such as those found on rockets or military afterburning aircraft (also known as de Laval nozzles). If your calculator shows an outlet Mach number greater than 1 then this means that a simple nozzle will not be able to achieve this velocity and a de Laval shape is needed.

Thrust and driving power

If you enable the Thrust option, enter mass flow and flight speed. The total thrust is the momentum leaving the nozzle, which is equal to the product of mass flow and exit velocity. Net thrust is calculated by subtracting the momentum that the engine already has from forward motion, which is equal to the product of mass flow and the difference between exit velocity and flight speed.

Propulsive efficiency measures the proportion of energy in the jet that is actually used to move the aircraft forward as opposed to the portion which serves to set the air behind the aircraft into motion. The closer the exit velocity is to the flight speed, the higher the propulsive efficiency. This is the fundamental reason why high bypass turbofan engines are more fuel efficient than very high exhaust velocity jet engines even though they produce the same thrust since they expel a large volume of air at low speed.

Representative values that can be used as a starting point:

These two gas properties, namely the isentropic specific heat ratio and gamma, depend on the type of gas and temperature. These values can be used as a good starting point.

Areas of application:

Commercial passenger jets fly at speeds near Mach 0.85 and keep the turbofan engine's exit velocity in the range of 300 to 400 m/s. This is an ideal region for fuel economy, as a large volume of air is flowing at moderate speed and matches well with flight speed.

On the other hand, military jet aircraft that use afterburners are the exact opposite. They accelerate to supersonic speeds with an exhaust velocity of over 1,000 m/s but because the exhaust trails far behind the vehicle, the propulsion efficiency is reduced. Rockets are an extreme example where the exhaust velocity can be between 3,000 and 4,500 m/s. This is ideal in a vacuum but results in very large losses in the atmosphere. The same equation describes all three cases with only the input values changing.

Tips for accurate results:

Instead, at the inlet you should use total enthalpy (or stagnation temperature) and total pressure. This equation converts both thermal and pressure energy of the incoming gas, with total enthalpy containing a portion of kinetic energy as well. Temperature must be in absolute values. Either give it in Kelvin or let a calculation tool convert from Celsius. Make sure that ambient pressure is lower than inlet pressure since the gas needs a lower pressure zone to expand into. Also pick the value for gamma and constant specific heat which matches the actual temperature of the gas.