Gas Injector Design: Critical Velocity and Speed of Sound, Reduced Isentropic Velocity and Mach Number

The design of a gas injector involves many physical-property parameters and related calculations. Critical velocity is one of the commonly used parameters.

In some cases, we often see the terms supercritical jetting or subcritical jetting, and may assume that these concepts are related to supersonic or subsonic velocity. Let us look at what the relationship actually is.

The difference is easy to see from the definitions of the two concepts.

The speed of sound is defined as the propagation speed of sound in a given medium, expressed as:

That is, the speed of sound is related to the given medium and its temperature; more specifically, it is related to the molar mass, adiabatic index and temperature of the medium. For a medium that can be regarded as an ideal gas, the adiabatic index is only a function of temperature, so the speed of sound depends only on molar mass and temperature.

Critical velocity means that the flow velocity of the medium reaches the speed of sound in that medium under the corresponding state. For gases, critical velocity usually appears in nozzle structures. Parameters such as pressure and temperature in a nozzle are usually expressed as stagnation parameters. For simplicity, the stagnation parameters are usually taken as the nozzle inlet parameters and marked with the subscript “0”. The critical velocity expressed by stagnation parameters is:

When the critical velocity is equal to the local speed of sound, then:

This is the relative-temperature expression when the reduced isentropic velocity of the gas reaches the critical velocity.

The ratio of the gas isentropic velocity to the local speed of sound is the Mach number, usually denoted by M; the ratio of the gas isentropic velocity to the critical velocity is the reduced isentropic velocity, usually denoted by λ. When the gas stagnation temperature is equal to the ambient temperature, it is clear that the critical velocity is less than the local speed of sound in the medium.

The relationship between Mach number and reduced isentropic velocity is as follows:

According to the relationship above, the following conclusions can be obtained:

1) When the gas isentropic velocity is zero, the Mach number is also zero; that is, the gas is stationary.

2) When the reduced isentropic velocity is 1, the Mach number is also 1. When critical jetting occurs at the nozzle outlet, the flow velocity is exactly equal to the speed of sound under the stagnation state. A reduced isentropic velocity of the gas in the nozzle less than 1 corresponds to subcritical jetting, while a value greater than 1 corresponds to supercritical jetting;

3) The greater the isentropic velocity, the greater the corresponding Mach number. Note that the maximum value of the Mach number is infinity, while the maximum value of the reduced isentropic velocity is: ${\lambda }_{\max }=\sqrt{\frac{k+1}{k-1}}$. At room temperature, the maximum reduced isentropic velocity of air is 2.45, while its corresponding Mach number is infinity.