| γ | M | T0/T | p0/p | ρ0/ρ | A/A* | T0 (K) | p0 (kPa) | T (K) | p (kPa) | V (m/s) |
|---|---|---|---|---|---|---|---|---|---|---|
| 1.4 | 2.0 | 1.800000 | 7.824449 | 4.346916 | 1.687500 | 300 | 101.325 | 166.666667 | 12.949794 | 517.558370 |
Example assumes ideal-gas air (R≈287 J/kg·K) and isentropic conditions.
The calculator uses standard perfect-gas isentropic relations with heat capacity ratio γ and Mach number M.
- Total-to-static temperature:
T0/T = 1 + (γ−1)/2 · M² - Total-to-static pressure:
p0/p = (T0/T)^(γ/(γ−1)) - Total-to-static density:
ρ0/ρ = (T0/T)^(1/(γ−1)) - Area–Mach relation:
A/A* = (1/M) · [ (2/(γ+1))·(1+(γ−1)/2·M²) ]^((γ+1)/(2(γ−1)))
When you solve from A/A*, a numerical bisection method is used (subsonic or supersonic branch).
If T0, p0, and R are provided, the tool also computes:
T = T0/(T0/T), p = p0/(p0/p),
ρ = p/(R·T), a = √(γRT), and V = M·a.
- Select what you know: Mach, a ratio, or the area ratio.
- Enter
γand the chosen input value. - Optionally add
T0,p0, andRto get velocities and static properties. - Press Calculate to display results above the form.
- Use Download CSV or Download PDF to save outputs.
Isentropic Assumptions and Limits
Isentropic relations describe steady, adiabatic, reversible flow of an ideal gas. In practice, they are often a first‑order baseline for nozzles, diffusers, inlets, and wind‑tunnel contractions. The calculator converts between Mach number and key ratios, so you can quickly sanity‑check a design point before detailed loss models. Treat results as “best‑case” performance; friction, heat transfer, and shocks reduce pressure recovery and change effective area.
Choosing Gamma and Gas Constant
Heat capacity ratio γ and gas constant R control how strongly temperature and pressure vary with Mach number. For dry air near room temperature, γ≈1.4 and R≈287 J/kg·K are common. Steam, CO₂, and combustion products can differ, so use values consistent with your thermodynamic state. Small changes in γ shift the pressure ratio notably at high Mach, influencing predicted nozzle thrust and compressor inlet conditions.
Mach, Area, and Choking
The area–Mach relation links geometry to compressibility. At the sonic condition (M=1), the flow is choked and the local area equals A*. For a given A/A* greater than one, two mathematical solutions exist: a subsonic branch and a supersonic branch. The calculator solves either branch numerically, which is useful when you know throat size and need the exit Mach for a converging–diverging nozzle or a measured area ratio.
From Total to Static Properties
Total (stagnation) conditions T0 and p0 remain constant along an isentropic streamline. Using T0/T and p0/p, the tool recovers static temperature and pressure, then computes density from p=ρRT. With γ, R, and T, it also finds the speed of sound a=√(γRT) and velocity V=M·a. These outputs support quick estimates of dynamic pressure, Reynolds number trends, and instrumentation ranges.
Engineering Interpretation and Checks
Use the results to compare operating points, not to replace full system analysis. If predicted p0/p implies very low static pressure, check for condensation, real‑gas effects, or cavitation risk in cryogenic lines. For high supersonic Mach, validate that downstream back pressure will not force a normal shock inside the nozzle. Pair isentropic baselines with efficiency, loss coefficients, or CFD to finalize performance.
1) What does A* mean in the results?
A* is the critical area where Mach number equals 1. At this throat condition the mass flow is maximized for given stagnation properties, and the flow is said to be choked.
2) Why are there subsonic and supersonic solutions for A/A*?
For any area ratio greater than one, the isentropic area–Mach equation has two valid roots: one with M<1 and one with M>1. Geometry alone does not select the branch; boundary conditions do.
3) Which γ and R values should I enter?
Use values matching your gas and temperature range. Air is commonly γ≈1.4 and R≈287 J/kg·K near ambient conditions. For steam or hot combustion products, compute γ and R from property tables or a real‑gas model.
4) Does the calculator include shocks or friction losses?
No. It assumes ideal, reversible, adiabatic flow. If shocks, wall friction, or heat transfer are expected, treat the results as an upper‑bound baseline and apply loss models or CFD for final design.
5) Can I compute velocity without entering T0 and p0?
You can still obtain the non‑dimensional ratios and A/A*. To compute velocity and static properties in physical units, provide T0 and either p0 with R (for density) or at least T0, γ, and R (for speed of sound).
6) What unit system should I use for inputs?
Any consistent units work. Enter T0 in kelvin, p0 in kPa (or Pa), and R in J/kg·K (or kJ/kg·K with matching pressure units). The calculator labels outputs to help keep consistency.