Isentropic Flow Calculator: Temperature, Pressure, Density and Area Ratios from Mach Number

Compressible flow ratios for a nozzle or duct.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the Mach number and the ratio of specific heats.

Enter the static temperature and pressure.

The calculator gives the ratios, the stagnation values and the flow speed.

Formulas used

  • \frac{T_0}{T} = 1 + \frac{\gamma - 1}{2}M^2
  • \frac{p_0}{p} = \left(\frac{T_0}{T}\right)^{\gamma/(\gamma - 1)}
  • \frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma + 1}\left(1 + \frac{\gamma - 1}{2}M^2\right)\right]^{\frac{\gamma + 1}{2(\gamma - 1)}}

Worked example

With the default numbers in the calculator above:

  1. Stagnation to static temperature ratio: 1.8
  2. Stagnation to static pressure ratio: 7.82445
  3. Stagnation to static density ratio: 4.34692
  4. Area ratio A over A-star (the area at the throat where M = 1): 1.6875
  5. Stagnation temperature: 518.67 K
  6. Stagnation pressure: 792.81 kPa
  7. Speed of sound at the static temperature (air, R = 287.05): 340.29 m/s
  8. Flow speed: 680.58 m/s

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Mach number 0.5.

  1. Stagnation to static temperature ratio: 1.05
  2. Stagnation to static pressure ratio: 1.18621
  3. Stagnation to static density ratio: 1.12973
  4. Area ratio A over A-star (the area at the throat where M = 1): 1.33984
  5. Stagnation temperature: 302.56 K
  6. Stagnation pressure: 120.19 kPa
  7. Speed of sound at the static temperature (air, R = 287.05): 340.29 m/s
  8. Flow speed: 170.15 m/s

Common questions

What does isentropic mean?

Adiabatic (no heat transfer) and reversible (no friction), so the entropy stays constant.

What is A star?

The throat area where the flow is exactly Mach 1.

When can I ignore compressibility?

Below about Mach 0.3, where the density changes by less than 5 percent.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Any compressible flow or gas dynamics text (isentropic flow relations).

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