Calculate Mach stagnation relations for total or static temperature, pressure, and density using Mach number and specific heat ratio gamma.

Ideal gas with constant ฮณ > 1. Pressure and density use isentropic stagnation conditions; temperatures must be above absolute zero. Gas presets are approximate and depend on temperature.

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Mach Stagnation (Total) Temperature Relation Formula

The following formula is used to calculate the stagnation (total) temperature for a given static temperature, gamma, and Mach number (for isentropic flow of a calorically perfect gas).

Tโ‚€ = T ร— (1 + ((ฮณ - 1) / 2) ร— Mยฒ)

Variables:

  • T0 is the stagnation (total) absolute temperature (kelvin for ratios)
  • T is the static absolute temperature (kelvin for ratios)
  • ฮณ is the ratio of specific heats (gamma)
  • M is the Mach number

To calculate the stagnation (total) temperature, multiply the static temperature by the sum of 1 and the product of ((gamma – 1) / 2) and the square of the Mach number.

What is the Machโ€“Temperature (Stagnation Temperature) Relation?

The Machโ€“temperature relation (more precisely, the stagnation/total temperature relation) connects a flowโ€™s static temperature at Mach number M to the stagnation (total) temperature T0 for isentropic compressible flow. The stagnation temperature is the temperature a moving gas would reach if it were brought to rest adiabatically with no shaft work (ideal โ€œstagnationโ€ process). For a given Mach number, T0 is greater than or equal to the static temperature T. In an isentropic acceleration (Mach number increasing with no heat/work), the static temperature typically decreases while the stagnation temperature remains constant.

How to Calculate Mach Temperature Relation?

The following steps outline how to calculate the Mach temperature relation (stagnation temperature from static temperature and Mach number).

  1. First, determine the static temperature (T), using kelvin in the formula.
  2. Next, determine the ratio of specific heats (ฮณ).
  3. Next, determine the Mach number (M).
  4. Finally, calculate the stagnation (total) temperature using the formula T0 = T * (1 + ((ฮณ – 1) / 2) * Mยฒ).
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Static Temperature (T) = 300 K

Gamma (ฮณ) = 1.4

Mach Number (M) = 2