Calculators
HAVA Laboratory · Free engineering calculators
Calculators
Gas-dynamics, rarefied-flow, atmosphere, pipe-flow and heat-transfer calculators for course work and preliminary analysis. Everything is computed in your browser.
Compressible Flow Calculator
Interactive gas-dynamics relations for course work and preliminary analysis. This version includes perfect-gas relations, reacting-air equilibrium, and finite-rate Park chemistry for high-temperature estimates.
Isentropic Flow Relations
Perfect gasSpeed of Sound
a = sqrt(γRT)Normal Shock Relations
Perfect gasReacting Normal Shock
Hansen equilibriumOblique Shock Relations
Perfect gas θ-β-MConical Shock Relations
Perfect gas Taylor-MaccollReacting Oblique Shock
Hansen equilibriumReacting Cone Shock
Hansen equilibrium Taylor-MaccollReacting Air Species
Equilibrium and finite-rate models
Hansen returns an equilibrium state. Park 5, QK 5, Park 11, and MMT integrate finite-rate two-temperature models from the supplied initial state.
Species are reported as mole fraction (X), mass fraction (Y), and concentration (mol/m3).
Special thanks to Prof. William Devenport and the VT Compressible Aerodynamics Calculator for serving as a source of inspiration for this calculator and for Ozgur Tumuklu’s personal and classroom use over the years in perfect-gas calculations.
References
- Standard air calculations: Anderson, J. D., Jr., Modern Compressible Flow: With Historical Perspective, 3rd ed., McGraw-Hill, 2003.
- Hansen model: Hansen, C. F., “Approximations for the Thermodynamic and Transport Properties of High-Temperature Air,” NASA Technical Report R-50, 1959.
- Hansen model: Dunn, A., Ranstead, J., and Ölçmen, S., “Reconstruction of Hansen’s High-Temperature Air Model,” Axioms, Vol. 15, No. 4, Article 283, 2026, doi:10.3390/axioms15040283.
- Park model: Park, C., “Two-temperature interpretation of dissociation rate data,” AIAA Journal, Vol. 29, No. 2, 1991.
- Park model: Park, C., “A review of reaction rates in high-temperature air,” AIAA Journal, Vol. 32, No. 3, 1994.
- QK model: Scanlon, T. J., White, C., Borg, M. K., Palharini, R. C., Farbar, E., Boyd, I. D., Reese, J. M., and Brown, R. E., “Open source Direct Simulation Monte Carlo chemistry modelling for hypersonic flows,” AIAA Journal, Vol. 53, No. 6, 2015.
- MMT model: Torres, J. P., Gross, M., Candler, G. V., and Schwartzentruber, T. E., “Modified Marrone-Treanor Model: Parameterization and Benchmarking for Five-Species Air,” arXiv:2506.20521, 2025.
Knudsen Number Calculator
Mean free path and Knudsen number, Kn = λ/L, for a gas at a standard-atmosphere altitude or at any temperature and pressure. Molecular collisions use Bird's variable hard sphere (VHS) model or the hard-sphere model.
Knudsen Number
λ = 1 / (√2 π d² n)Knudsen number vs altitude
Regimes: continuum Kn < 0.01, slip 0.01–0.1, transition 0.1–10, free molecular Kn > 10. The temperature and number density come from the standard atmosphere; a gas other than air is taken as that pure gas at the same temperature and pressure.
References
- Molecular model and gas data: Bird, G. A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, 1994 (Table A1, eqs. 4.62 and 4.65).
- Atmosphere: U.S. Standard Atmosphere, 1976, NOAA/NASA/USAF, U.S. Government Printing Office, Washington, D.C., 1976.
- Flow regimes: Schaaf, S. A., and Chambre, P. L., “Flow of Rarefied Gases,” in Fundamentals of Gas Dynamics, Princeton University Press, 1958.
Standard Atmosphere Calculator
Properties of the U.S. Standard Atmosphere 1976 from −5 km to 1000 km. Enter an altitude, or a pressure or density to find the pressure or density altitude; the air's composition (mole and mass fractions of each species) is given too. A temperature offset gives a non-standard (ISA + ΔT) day below 86 km. The ratios θ = T/T₀, δ = p/p₀ and σ = ρ/ρ₀ are relative to standard sea level (T₀ = 288.15 K, p₀ = 101325 Pa, ρ₀ = 1.225 kg/m³).
Standard Atmosphere
U.S. Standard Atmosphere 1976Composition
Profile
Table
References
- Model: U.S. Standard Atmosphere, 1976, NOAA/NASA/USAF, U.S. Government Printing Office, Washington, D.C., 1976. Below 86 km the layer equations are evaluated exactly; the kinetic temperature is taken equal to the molecular-scale temperature (they differ by less than 0.05% between 80 and 86 km).
- 86 – 1000 km: temperature from the 1976 equations; pressure and density from the curve fits to the 1976 tables by R. A. Braeunig, Atmospheric Models.
- Composition: below 86 km the 1976 sea-level fractions of dry air; from 86 to 1000 km number densities of N₂, O, O₂, Ar, He and H from the 1976 diffusion equations, integrated as in the open-source ussa1976 package (Y. Nollet).
- Transport properties: Sutherland's law for viscosity and the 1976 standard's thermal-conductivity relation, defined up to 86 km.
- Inspiration: the Digital Dutch Standard Atmosphere Calculator.
Frictional Flows Calculator
Wall friction in pipe and duct flow: the Darcy friction factor from the Moody chart (Colebrook equation) for incompressible flow, and Fanno flow for compressible, adiabatic flow with friction, either as ratio tables or for a real pipe with its roughness.
Moody Chart
1/√f = −2 log₁₀(ε/3.7D + 2.51/(Re√f))Moody chart
The highlighted curve is your ε/D; the dot is the operating point. The dashed line marks the start of fully rough flow, Re·√f·(ε/D) = 200, beyond which f no longer depends on Re. The shaded band is the uncertain transitional range, 2300 < Re < 4000.
Fanno Flow Relations
Perfect gas, adiabatic constant-area duct* is the sonic (choking) state reached after a further duct length L*. f is the Fanning friction factor averaged over L*; with the Darcy factor from the Moody chart, 4f = fD.
Fanno Flow in a Pipe
Friction factor from the Moody chartThe Darcy friction factor fD = 4f comes from the Colebrook equation (64/Re if laminar) at the Reynolds number for the mean of the inlet and exit temperatures; in a constant-area duct ρV is constant, so Re changes only through μ(T). The duct is adiabatic, and 4f̄L/D = fDL/D is subtracted from the inlet's 4fL*/D to find the exit Mach number. If L is longer than the choking length L*, the given inlet state cannot be maintained.
References
- Moody chart: Moody, L. F., “Friction Factors for Pipe Flow,” Transactions of the ASME, Vol. 66, No. 8, 1944, pp. 671–684.
- Colebrook equation: Colebrook, C. F., “Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws,” Journal of the Institution of Civil Engineers, Vol. 11, No. 4, 1939, pp. 133–156.
- Explicit correlations: Haaland, S. E., Journal of Fluids Engineering, Vol. 105, 1983, pp. 89–90; Swamee, P. K., and Jain, A. K., Journal of the Hydraulics Division (ASCE), Vol. 102, 1976, pp. 657–664; Churchill, S. W., Chemical Engineering, Vol. 84, No. 24, 1977, pp. 91–92.
- Fanno flow: Anderson, J. D., Jr., Modern Compressible Flow: With Historical Perspective, 3rd ed., McGraw-Hill, 2003; Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1, Ronald Press, 1953.
- Roughness and fluid data: Çengel, Y. A., and Cimbala, J. M., Fluid Mechanics: Fundamentals and Applications, McGraw-Hill.
Heat Transfer Calculator
Heat addition in compressible duct flow (Rayleigh flow), and forced convection in a pipe. Rayleigh flow is frictionless flow in a constant-area duct with heat added or removed; heating drives the flow toward M = 1, where it chokes.
Theory and equations (PDF): assumptions, derivations, flow behaviour, choking and the correlations behind each tab.
Rayleigh Flow Relations
Perfect gas, frictionless constant-area duct* is the sonic (choking) state reached by adding heat. The static temperature is largest at M = 1/√γ, so a T/T* value has one solution below T_max (lower Mach number) and one above it.
Rayleigh Flow with Heat Addition
q = cp(T₀₂ − T₀₁)Heat added raises T₀ by q/cp; the exit Mach number follows from T₀₂/T₀* on the same (subsonic or supersonic) branch as the inlet. Negative q is cooling.
Forced Convection in a Pipe
h = Nu k / DTurbulent flow uses the Gnielinski correlation with the Darcy friction factor from the Colebrook equation for the pipe's roughness (the same as the Frictional Flows calculator); laminar flow uses the fully developed values Nu = 3.66 (uniform wall temperature) or 4.36 (uniform heat flux).
References
- Rayleigh flow: Anderson, J. D., Jr., Modern Compressible Flow: With Historical Perspective, 3rd ed., McGraw-Hill, 2003; Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, Vol. 1, Ronald Press, 1953.
- Gnielinski correlation: Gnielinski, V., “New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow,” International Chemical Engineering, Vol. 16, No. 2, 1976, pp. 359–368.
- Dittus–Boelter correlation: Dittus, F. W., and Boelter, L. M. K., University of California Publications in Engineering, Vol. 2, 1930, p. 443.
- Fluid properties and laminar values: Çengel, Y. A., and Ghajar, A. J., Heat and Mass Transfer: Fundamentals and Applications, McGraw-Hill.