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The stagnation temperature is a function of the altitude and
 Mach number of the object. Real gas effects introduce a change
 at higher Mach numbers

As an object moves through a gas, the gas molecules near the object are disturbed and move around the object. Aerodynamic forces are generated between the gas and the object and the magnitude of these forces depend on many factors associated with the object and the gas. The speed of the object relative to the gas introduces many significant effects. We characterize the speed of the object by a non-dimensional number called the Mach number; the Mach number is the ratio of the speed of the object to the speed of sound in the gas. The speed of "sound" is actually the speed of transmission for small, isentropic disturbances in the gas. The physical state of the gas depends on the Mach number of the object. In our discussions, we will use the Mach number of the object and the Mach number of the flow interchangeably. If we travel with the object as it moves through the air, the air moves past the object at the speed of the object. So, the Mach number of the object and the Mach number of the flow are the same number.

For a moving flow of gas, there are several different values for the temperature of the gas. The static temperature is the temperature of the gas if it had no ordered motion and was not flowing. From kinetic theory, static temperature is related to the average kinetic energy of the random motion of the molecules of the gas. The value of the static temperature of air depends on the altitude. For a moving flow, there is a dynamic temperature associated with the kinetic energy of ordered motion of the flow in the same way that static temperature is related to the average kinetic energy of the random motion of the molecules. The total temperature is the sum of the static temperature and the dynamic temperature. and the value of total temperature depends on the Mach number of the flow. If the moving flow is isentropically brought to a halt on the body, we measure the stagnation temperature. The stagnation temperature is important because it is the temperature that occurs at a stagnation point on the object. Because the total temperature does not change through a shock wave, the stagnation temperature and and the total temperature have the same value at a stagnation point.

In the process of slowing the flow, the gas is heated due to the kinetic energy of flow. The amount of the heating depends on the specific heat capacity of the gas. If the specific heat capacity is a constant value, the gas is said to be calorically perfect and if the specific heat capacity changes, the gas is said to be calorically imperfect. At subsonic and low supersonic Mach numbers, air is calorically perfect. But under low hypersonic conditions, air is calorically imperfect. Derived flow variables, like the speed of sound and the isentropic flow relations are slightly different for a calorically imperfect gas than the conditions predicted for a calorically perfect gas because some of the energy of the flow excites the vibrational modes of the diatomic molecules of nitrogen and oxygen in the air.

On the figure, we have plotted the value of stagnation temperature for a standard day atmosphere as a function of altitude and Mach number. There are two sets of lines on the figure because of the inclusion of real gas effects. The solid line is the computed stagnation temperature for a calorically perfect gas and the dashed line is the computed stagnation temperature for a calorically imperfect gas. At the lower Mach numbers, below Mach 3, the values of stagnation temperature are the same, because the temperature is not high enough to excite the vibrational modes. But beginning around Mach 3, real gas effects become increasingly important with increasing Mach number.

For the perfect gas, the stagnation temperature is derived from the isentropic total temperature equation:

Tt = T * [1 + M^2 * (gamma-1)/2]

where Tt is the total temperature, T is the static temperature at a given altitude, M is the Mach number, and gamma is the ratio of specific heats for a calorically perfect gas and has a constant value of 1.4.

For the calorically imperfect gas, the ratio of specific heats is not a constant but a function of the static temperature. Mathematical models based on a simple harmonic vibrator have been developed. The details of the analysis were given by Eggars in NACA Report 959. A synopsis of the report is included in NACA Report 1135. Solving the integrated energy equation with the assumption of variable specific heat, one obtains an equation for the Mach number in terms of the ratio of the static and total (stagnation) temperature:

M^2 = (2 (Tt/T) / gam) * [(gamma/(gamma-1) * (1 - T/Tt) + (theta/Tt) * (1/(e^theta/Tt -1) - 1/(e^theta/T -1)]

where gamma is the ratio of specific heats for a perfect gas, theta is a thermal constant equal to 5500 degrees Rankine, and gam is the ratio of specific heats including a correction for the vibrational modes:

gam = 1 + (gamma - 1) / ( 1 + (gamma-1) * [(theta/T)^2 * e^(theta/T) /(e^(theta/T) -1)^2])

This equation must be solved iteratively to obtain a value for the total temperature. Here is a Java calculator that solves the calorically imperfect gas equation:

To change input values, click on the input box (black on white), backspace over the input value, type in your new value. Then hit the red COMPUTE button to send your new value to the program. You will see the output boxes (yellow on black) change value. You can use either Imperial or Metric units and you can input either the Mach number or the speed by using the menu buttons. Just click on the menu button and click on your selection. If you are an experienced user of this calculator, you can use a sleek version of the program which loads faster on your computer and does not include these instructions. You can also download your own copy of the program to run off-line by clicking on this button:

Button to Download a Copy of the Program

As a test computation, set the Mach number to 5 and the altitude to 50,000 feet. Notice that the calculated total temperature for the calorically imperfect gas is less than the value for the perfect gas. The difference in temperature is caused by the excitation of the vibrational modes; some of the energy of the flow has been used to alter the state of the molecules.


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Editor: Tom Benson
NASA Official: Tom Benson
Last Updated: May 07 2021

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