How to calculate the Prandtl number (Pr): fundamental dimensionless number in fluid mechanics and heat transfer, ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. Formula: Pr = nu / alpha = (mu * c_p) / k, where mu dynamic viscosity (Pa·s), c_p specific heat at constant pressure (J/kg·K), k thermal conductivity (W/m·K), nu = mu/rho kinematic viscosity (m²/s), alpha = k/(rho*c_p) thermal diffusivity (m²/s). Prandtl number is fluid property (not flow), depends only on composition and temperature. Pr low (Pr momentum, thermal boundary layer thicker than velocity (gases, liquid metals). Pr high (Pr > 1): momentum diffuses faster, velocity boundary layer thicker (viscous oils).

History: Ludwig Prandtl (Germany 1875-1953) Göttingen Universität 1904 boundary layer theory paper III International Mathematical Congress Heidelberg August 1904 'Über Flüssigkeitsbewegung bei sehr kleiner Reibung' fundamental boundary layer concept introduction.

Essential tool for: convective heat transfer engineering shell-and-tube heat exchangers; HVAC design heating ventilation air conditioning; automotive aerodynamics USA Detroit MI Ford GM Tesla; aerospace USA NASA Boeing Lockheed; petrochemical refineries ExxonMobil Chevron; food industry thermal sterilization; nuclear reactors liquid metal coolants; geothermal fluids.

USA heat transfer research: MIT Cambridge MA 1861 mechanical engineering Bird Stewart Lightfoot Transport Phenomena 1960; Stanford Mechanical Palo Alto CA 1925; Caltech Pasadena CA 1891; UC Berkeley CA 1868 Lawrence Berkeley Lab; Princeton NJ 1746; Yale CT 1701; Cornell NY 1865; Carnegie Mellon Pittsburgh PA 1900; Georgia Tech Atlanta GA 1885; Purdue West Lafayette IN 1869.

USA universities heat transfer textbooks: Frank Incropera (USA 1939-) Notre Dame 'Fundamentals of Heat and Mass Transfer' 1981 7 editions classic textbook; Adrian Bejan (Romania-USA 1948-) Duke entropy generation minimization constructal theory; Yunus Cengel (Turkey-USA 1953-) UNR Heat Transfer textbook.

USA companies heat transfer: GE General Electric Boston MA 1892 turbines aerospace 174000 employees; Westinghouse Pittsburgh PA 1886 nuclear power; Bechtel San Francisco CA 1898 engineering construction; Fluor Irving TX 1912 engineering; Honeywell Charlotte NC 1906 industrial controls; Carrier Palm Beach Gardens FL 1915 HVAC 53000 employees founder Willis Carrier; Trane Davidson NC 1885 HVAC; York International York PA 1874 HVAC Johnson Controls.

USA aerospace: NASA Washington DC 1958 17000 employees space; Boeing Chicago IL 1916 100000 employees aerospace; Lockheed Martin Bethesda MD 1995 116000 employees defense F-35 fighter; SpaceX Hawthorne CA 2002 Elon Musk; Northrop Grumman Falls Church VA 1994; Raytheon Technologies Arlington VA 2020 RTX Corporation; Pratt & Whitney East Hartford CT 1925 jet engines.

USA software heat transfer: ANSYS Canonsburg PA 1970 finite element CFD Fluent acquired 2006; COMSOL Stockholm Sweden 1986 multiphysics; OpenFOAM ESI Group open source CFD; Star-CCM Plus Siemens Digital Industries; Aspen Plus Cambridge MA 1981 chemical engineering simulation. Air Pr 25°C = 0.711; water 25°C = 6.13; mercury 25°C = 0.0248; engine oil 100°C = 1500; hydraulic oil = 100; glycerol = 12000; ethanol = 16; benzene = 7.

Prandtl number formula

Fundamental definition: Pr = nu / alpha = (mu * c_p) / k. Dimensionless number (no units). nu kinematic viscosity m²/s = mu/rho where mu Pa·s dynamic viscosity and rho kg/m³ density; alpha thermal diffusivity m²/s = k/(rho*c_p) where k W/m·K thermal conductivity and c_p J/kg·K isobaric specific heat.

Physical meaning: Pr ratio between momentum diffusion rate (transport quantity motion) and heat diffusion rate (transport thermal energy). Pr=1 both processes diffuse equally (Reynolds analogy). Pr<>1 momentum diffuses faster than heat (viscous oils).

Velocity vs thermal boundary layer: relation delta_thermal/delta_velocity ~ Pr^(-1/3) for laminar flows. Pr delta_velocity thermal layer thicker; Pr=1 equal; Pr>1 velocity layer thicker. Important for heat exchanger design (water Pr=7) vs air conditioners (air Pr=0.71).

Heat transfer correlations: Nusselt number Nu = h*L/k depends on Re and Pr. Laminar plate convection: Nu = 0.332 * Re^0.5 * Pr^(1/3); turbulent tube: Dittus-Boelter Nu = 0.023 * Re^0.8 * Pr^n (n=0.4 heating, 0.3 cooling). Sieder-Tate Nu = 0.023 * Re^0.8 * Pr^(1/3) * (mu/mu_w)^0.14 viscosity correction.

Temperature variation: Pr depends strongly on T. Water: Pr=13.6 at 0°C; 7.0 at 20°C; 1.75 at 100°C; 0.86 at 300°C. Air: Pr=0.71 stable 0-1000°C. Engine oil: Pr=10000 at 0°C; 1500 at 100°C; 220 at 200°C. Mercury: Pr=0.025 stable liquid metals.

History and context

Ludwig Prandtl (Germany 1875-1953) Freising Bavaria born, TU München mathematics 1893-1897, Hannover Universität engineering doctorate 1900 'Kipperscheinungen' lateral buckling thin-walled beams. Hannover Hochschule Technische 1900-1904; Göttingen Universität full professor mechanics 1904-1947 Chair Applied Mathematics Mechanics 43 years Director Institut für Angewandte Mathematik und Mechanik.

1904 paper III International Mathematical Congress Heidelberg August 1904 'Über Flüssigkeitsbewegung bei sehr kleiner Reibung' (Fluid Motion at Very Small Viscosity) 8 pages fundamental boundary layer theory revolution fluid mechanics. AVA Aerodynamische Versuchsanstalt Göttingen 1907 founded Prandtl director wind tunnel research + Kaiser-Wilhelm-Institut für Strömungsforschung 1925 Göttingen. Prandtl famous students: Theodore von Kármán (Hungary-USA 1881-1963) Caltech aerodynamics supersonic; Adolf Busemann (Germany 1901-1986) NASA Ames swept wing concept; Albert Betz (Germany 1885-1968) Göttingen Betz limit wind turbine 59.3% maximum 1919; Heinrich Blasius (Germany 1883-1970) TH Hamburg Blasius solution 1908 PhD Göttingen Prandtl supervisor; Hermann Schlichting (Germany 1907-1982) Braunschweig Boundary Layer Theory textbook 1951 classic 8 editions; Walter Tollmien (Germany 1900-1968) Tollmien-Schlichting waves transition turbulence.

Prandtl concepts: boundary layer 1904 separation transition turbulence; mixing length theory 1925; Prandtl-Glauert compressibility correction 1927; Lifting line theory 1918 finite wing aerodynamics; Prandtl-Meyer expansion fan supersonic flow.

Prandtl number nomenclature: Ernst Schmidt (Germany 1892-1975) Munich 1925 paper introduced term 'Prandtl-Zahl' honoring Prandtl.

USA heat transfer pioneers: Theodore von Kármán (Hungary-USA 1881-1963) Caltech aerodynamics fluid mechanics turbulence Karman vortex street von Karman number; William Frank Incropera (USA 1939-) Notre Dame 'Fundamentals of Heat and Mass Transfer' 1981 textbook 7 editions classic; Adrian Bejan (Romania-USA 1948-) Duke entropy generation minimization constructal theory 1996; Yunus Cengel (Turkey-USA 1953-) UNR Heat Transfer textbook; Bird Stewart Lightfoot Transport Phenomena 1960 Wisconsin Madison classic textbook chemical engineering; Hermann Schlichting + Klaus Gersten Boundary Layer Theory 8th edition 2017. USA heat transfer companies: GE General Electric Boston MA 1892 founded Thomas Edison turbines aerospace 174000 employees; Westinghouse Pittsburgh PA 1886 founded George Westinghouse nuclear power 25000 employees; Bechtel San Francisco CA 1898 founded Warren Bechtel engineering construction 55000 employees; Fluor Irving TX 1912 founded John Fluor engineering 41000 employees; Honeywell Charlotte NC 1906 industrial controls 90000 employees; Carrier Palm Beach Gardens FL 1915 founded Willis Carrier HVAC 53000 employees; Trane Davidson NC 1885 founded Bingham Trane HVAC 38000 employees; York International York PA 1874 HVAC subsidiary Johnson Controls Milwaukee WI 1885 100000 employees; Lennox International Richardson TX 1895 HVAC;

Rheem Atlanta GA 1925 HVAC. USA aerospace: NASA National Aeronautics Space Administration Washington DC 1958 founded 17000 employees space agency Goddard Space Flight Center Greenbelt MD; Jet Propulsion Laboratory JPL Pasadena CA Caltech NASA 6000 employees Mars rovers; Johnson Space Center Houston TX 1961; Kennedy Space Center FL 1962; Boeing Chicago IL 1916 100000 employees aerospace 737 747 777 787 aircraft; Lockheed Martin Bethesda MD 1995 116000 employees defense F-35 fighter F-16 F-22; SpaceX Hawthorne CA 2002 Elon Musk Falcon Dragon Starship; Northrop Grumman Falls Church VA 1994; Raytheon Technologies Arlington VA 2020 RTX; Pratt & Whitney East Hartford CT 1925 jet engines GTF F100; GE Aerospace Lynn MA jet engines GE90 GEnx. USA software heat transfer CFD: ANSYS Canonsburg PA 1970 founded John Swanson finite element + Fluent acquired 2006 5500 employees CFD top market; COMSOL Stockholm Sweden 1986 founded multiphysics CFD heat structural; OpenFOAM ESI Group open source CFD foundation; Star-CCM+ Siemens Digital Industries Plano TX; Abaqus Dassault Systèmes Vélizy France 1978; Aspen Plus Cambridge MA 1981 chemical engineering simulation Aspen Technology; SolidWorks Simulation Dassault Systèmes Waltham MA. USA national labs heat transfer: Lawrence Berkeley LBNL CA 1931; Argonne ANL IL 1946; Oak Ridge ORNL TN 1943; Sandia National Laboratories Albuquerque NM Livermore CA 1949 thermal physics combustion; Idaho National Laboratory INL ID 1949 nuclear reactor research.

Pr values table

• Air 0°C Pr: 0.713 standard atmosphere
• Air 25°C Pr: 0.711 ambient
• Air 100°C Pr: 0.703
• Air 500°C Pr: 0.683
• Air 1000°C Pr: 0.731
• Water 0°C Pr: 13.6 high
• Water 25°C Pr: 6.13
• Water 50°C Pr: 3.55
• Water 100°C Pr: 1.75
• Water 300°C Pr: 0.86
• Mercury 25°C Pr: 0.0248 liquid metal
• Sodium 100°C Pr: 0.011 nuclear coolant
• Lead 400°C Pr: 0.024
• Engine oil 0°C Pr: 47000 extreme
• Engine oil 100°C Pr: 1500
• Hydraulic oil 25°C Pr: 100
• Glycerol 25°C Pr: 12500
• Ethanol 25°C Pr: 16
• Benzene 25°C Pr: 7
• Helium 25°C Pr: 0.68
• Hydrogen 25°C Pr: 0.71
• CO2 25°C Pr: 0.77
• R134a refrigerant Pr: 4-5 vapor
• Pr=nu/alpha: ratio diffusivities
• Pr=mu*cp/k: equivalent form
• Reynolds analogy Pr=1: ideal heat-momentum
• Dittus-Boelter Nu: 0.023*Re^0.8*Pr^n
• Sieder-Tate: viscosity correction (mu/muw)^0.14
• Prandtl 1904 paper: Heidelberg congress 8 pages
• Schmidt 1925: nomenclature Prandtl-Zahl
• USA MIT Cambridge MA 1861: BSL Transport Phenomena 1960
• USA Stanford Palo Alto CA 1925: mechanical eng
• USA Caltech Pasadena CA 1891: von Kármán
• USA NASA 1958: 17000 employees aerospace
• USA Boeing Chicago 1916: 100000 aircraft
• USA Carrier 1915: HVAC Willis Carrier founder

Numerical examples

Example 1 - Air 25°C ambient. mu=1.81e-5 Pa·s, cp=1005 J/kg·K, k=0.0257 W/m·K. Pr = (1.81e-5 * 1005) / 0.0257 = 0.708. Near unity: heat and momentum diffuse equally. Thermal boundary ~ velocity. USA HVAC design: Carrier Palm Beach Gardens FL 1915 Willis Carrier; Trane Davidson NC 1885; York Pennsylvania 1874 Johnson Controls.

Example 2 - Water 25°C heat exchanger. mu=8.9e-4 Pa·s, cp=4182 J/kg·K, k=0.598 W/m·K. Pr = (8.9e-4 * 4182) / 0.598 = 6.22. Pr>1: momentum diffuses faster than heat. Velocity boundary thicker. Important shell-and-tube heat exchanger design. USA: GE Boston MA 1892; Westinghouse Pittsburgh PA 1886; Bechtel San Francisco CA 1898.

Example 3 - Mercury 25°C liquid metal. mu=1.526e-3 Pa·s, cp=139.5 J/kg·K, k=8.34 W/m·K. Pr = (1.526e-3 * 139.5) / 8.34 = 0.0255. Pr<<1: heat diffuses much faster than momentum. Liquid metals nuclear reactor coolant fast breeder reactors. USA: Sandia National Labs Albuquerque NM 1949; Idaho National Lab ID 1949 nuclear research.

Example 4 - Engine oil SAE 30 100°C. mu=0.486 Pa·s (high viscosity), cp=1909 J/kg·K, k=0.144 W/m·K. Pr = (0.486 * 1909) / 0.144 = 6443. Pr>>1: momentum dominates. Velocity boundary very thick. USA automotive: Ford Dearborn MI 1903; GM Detroit MI 1908; Tesla Austin TX 2003 EV.

Example 5 - Helium He light gas. mu=1.99e-5 Pa·s, cp=5193 J/kg·K, k=0.152 W/m·K. Pr = (1.99e-5 * 5193) / 0.152 = 0.680. Pr<1 typical gas. He cryogenic cooling MRI superconductors. USA: Praxair Danbury CT 1907; Air Products Allentown PA 1940 industrial gases.

Limitations

1. Isothermal fluid property: Pr=mu*cp/k at single T, varies significantly with temperature (water 13.6 at 0°C, 1.75 at 100°C).
2. Newtonian fluids: Pr defined assumes mu constant with shear. Non-Newtonian (polymers, blood) apparent viscosity varies.
3. Pressure effects gases: Pr gas relatively constant 0.7 wide pressure/temperature range. Liquids p more relevant.
4. Single phase only: Pr applies single-phase fluids. Phase change (boiling, condensation) Nusselt correlations separate.
5. Compressibility: high-speed gases (Mach>0.3) Pr undergoes Prandtl-Glauert compressibility correction.
6. Turbulence vs laminar: Pr is fluid property. Turbulent Pr_t (eddy thermal/eddy momentum) ~ 0.85-0.95 varies.
7. Multicomponent: liquid/gas mixtures Pr weighted average + interaction effects. Wilke models for gases.
8. Extreme temperatures: cryogenics liquid He Pr~0.5 at 4K; high T plasmas Pr varies notably.
9. Chemical reactivity: combustion reactions Pr varies with composition changing reactant→product.
10. Concentrated solutions: ethylene glycol antifreeze Pr varies 50-200 with concentration vs pure water.

Common errors

Confusing mu (dynamic) and nu (kinematic): mu Pa·s = kg/m·s dynamic viscosity; nu = mu/rho m²/s kinematic. Pr = mu*cp/k uses dynamic; Pr = nu/alpha uses kinematic equivalent.

Specific heat cv vs cp: cp constant pressure (J/kg·K) correct for Pr; cv constant volume. Ideal gases cp = cv + R/M. Liquids cp ≈ cv (incompressible).

Conductivity k vs diffusivity alpha: k W/m·K thermal conduction; alpha = k/(rho*cp) m²/s thermal diffusivity. Frequent confusion. Pr = nu/alpha more direct relation.

Pr at specific temperature: Pr varies a lot with T for liquids. Always specify T reference (e.g. water 25°C Pr=6, no specify implies ambient).

SI vs CGS units: SI Pa·s (kg/m·s) vs cP centipoise (CGS) = 10^-3 Pa·s. Conversion important: water 25°C mu = 0.89 cP = 8.9e-4 Pa·s.

Practical applications

1. Heat exchangers shell-and-tube: Dittus-Boelter Nu=0.023 Re^0.8 Pr^n. USA GE Boston Westinghouse Pittsburgh.
2. HVAC design: air Pr=0.71 standard climatization. Carrier Palm Beach Gardens FL 1915; Trane Davidson NC 1885; Johnson Controls Milwaukee WI 1885.
3. Vehicle aerodynamics: wind tunnel Caltech 1891 von Kármán; Ford GM Tesla Detroit MI; SpaceX Starship Hawthorne CA 2002.
4. Aerospace: leading edge heating reentry; NASA Goddard MD; Boeing Chicago 1916; Lockheed Martin Bethesda MD F-35.
5. Petrochemical refineries: cracking heat transfer Pr oils. ExxonMobil Irving TX; Chevron San Ramon CA; Bechtel San Francisco CA 1898.
6. Food industry: UHT pasteurization sterilization. USA Tetra Pak USA Vernon Hills IL; Dean Foods Dallas TX dairy.
7. Nuclear reactor coolant: liquid metals Hg Na Pr<<1; Sandia Albuquerque NM 1949; Idaho National Lab 1949.
8. Geothermal fluids: USA Geysers California 1960 + Calpine Houston TX geothermal; Iceland geothermal.
9. Cryogenic engineering: He liquid Pr 0.5 superconductors MRI; Praxair Danbury CT 1907; Air Products Allentown PA 1940; LBNL Berkeley CA 1931.
10. Boundary layer aerospace: Prandtl 1904 Göttingen + USA Caltech von Kármán 1881-1963; Boeing 100000 employees Chicago.

Updated July 2026