How to calculate the Biot number (Bi): fundamental dimensionless number transient heat conduction measuring ratio of external convection to internal conduction resistance. Formula: Bi = h*Lc/k_solid where h convection coefficient (W/m²·K), Lc characteristic length (V/A volume per surface area), k_solid SOLID thermal conductivity (W/m·K). Critical: distinguish k_solid vs k_fluid (Nu uses fluid k).

Lumped capacitance criterion: Bi<0.1 uniform T in solid, lumped method valid. T(t) = T_inf + (T_0 - T_inf) * exp(-t/tau), tau = rho*cp*V/(h*A) time constant.

Bi>0.1: significant T gradients, Heisler charts (1947) or one-term approximation required. History: Jean-Baptiste Biot (France 1774-1862) Paris École Polytechnique 1797 Sorbonne professor + Collège de France 1804 paper Bibliothèque Britannique 27:310 'Mémoire sur la propagation de la chaleur' first mathematical heat propagation solids. Joseph Fourier 1822 'Théorie analytique de la chaleur' heat equation. M.P. Heisler 1947 ASME 69:227 charts.

Essential tool for: electronics cooling CPU GPU heat sinks; food industry sterilization pasteurization; metallurgy quenching tempering; automotive brake rotor cooling; aerospace reentry heat shields; buildings thermal mass; nuclear fuel rods UO2 cladding.

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; Princeton NJ 1746; Cornell NY 1865; Carnegie Mellon Pittsburgh PA 1900; Georgia Tech Atlanta GA 1885; Purdue West Lafayette IN 1869; Notre Dame Indiana 1842 Frank Incropera 'Fundamentals of Heat and Mass Transfer' 1981.

USA companies cooling electronics: Intel Santa Clara CA 1968 130000 employees CPU manufacturer; Nvidia Santa Clara CA 1993 30000 GPU AI; AMD Santa Clara CA 1969 26000 CPU GPU; Apple Cupertino CA 1976 165000 M-series chips; Cooler Master Taiwan heat sinks; Noctua Austria CPU coolers.

USA food industry sterilization: Tetra Pak USA Vernon Hills IL aseptic packaging UHT; Dean Foods Dallas TX dairy pasteurization; Conagra Brands Chicago IL 1919 frozen foods; Kraft Heinz Pittsburgh Chicago 2015 39000 employees; General Mills Minneapolis MN 1866 35000 employees; PepsiCo Purchase NY 1965 309000 employees; Coca-Cola Atlanta GA 1892 79000 employees.

USA metallurgy heat treatment: Nucor Charlotte NC 1955 31000 employees largest US steel; US Steel Pittsburgh PA 1901 23000 employees; Cleveland-Cliffs Cleveland OH 1847 21000; Steel Dynamics Fort Wayne IN 1993; Inductotherm Group Rancocas NJ 1953 induction heating equipment; Ipsen Cherry Valley IL 1948 heat treatment furnaces.

USA aerospace reentry: NASA Washington DC 1958 17000 employees space shuttle thermal protection tiles; SpaceX Hawthorne CA 2002 Dragon Starship reentry; Boeing Chicago IL 1916 100000 employees CST-100 Starliner; Lockheed Martin Bethesda MD 1995 116000 Orion capsule; NASA Ames Research Center Moffett Field CA arc-jet testing; NASA Johnson Space Center Houston TX 1961. USA software heat transfer: ANSYS Canonsburg PA 1970 transient FEA; COMSOL Stockholm 1986; Abaqus Dassault 1978; FlowTherm Mentor Graphics; HTRI Navasota TX 1962; OpenFOAM ESI Group.

Biot formula

Definition: Bi = h*Lc/k_solid. Dimensionless. h convection coefficient (W/m²·K), Lc characteristic length (V/A), k_solid SOLID thermal conductivity (W/m·K). Critical distinction: k_solid for conduction inside solid; k_fluid for Nusselt convection.

Physical meaning: Bi ratio external convection resistance (1/h) to internal conduction resistance (Lc/k_solid). Bi<>1: fast convection, slow conduction, important spatial gradients.

Lumped capacitance method: Bi<0.1 valid. T(t) = T_inf + (T_0 - T_inf)*exp(-t/tau) where tau = rho*cp*V/(h*A) = rho*cp*Lc/h time constant. 5*tau ~ 99% completion.

Heisler charts (1947): Bi>0.1 use dimensionless time tau* = alpha*t/Lc² (Fourier number Fo) and dimensionless temperature theta = (T(t)-T_inf)/(T_0-T_inf). Three classic shapes: plane slab, infinite cylinder, sphere. One-term approximation: theta_0 = C*exp(-zeta_1²*Fo) for Fo>0.2.

Characteristic length: plane slab thickness 2L: Lc = L (half thickness); infinite cylinder radius R: Lc = R/2; sphere radius R: Lc = R/3; irregular geometry Lc = V/A. Distinguish Lc transient (V/A) from L convection correlations (geometric).

History and context

Jean-Baptiste Biot (France 1774-1862) Paris born, École Polytechnique 1794 mathematics 1797 graduate first class, Sorbonne professor mathematics 1800-1809 + Collège de France 1800-1862 long career 62 years, Académie des Sciences 1803 elected. Biot polymath: astronomy (Biot crater Moon), geomagnetism (Biot-Savart law magnetism Felix Savart 1820), polarization light 1812 (Biot-Fresnel law optics), spectroscopy, heat conduction 1804 paper Bibliothèque Britannique 27:310 first mathematical treatment heat propagation solids preceded Fourier 1822. Joseph Fourier (France 1768-1830) Auxerre Paris École Polytechnique 1794 mathematics + Egypt expedition 1798 Napoleon scientific commission 1801-1802 prefect Isère Grenoble, 1822 magnum opus 'Théorie analytique de la chaleur' Paris classic heat equation Fourier law q = -k*grad(T) + Fourier series + Fourier transform. M.P. Heisler 1947 paper Transactions ASME 69:227 charts. Modern textbook references: Frank Incropera (USA 1939-) Notre Dame 'Fundamentals of Heat and Mass Transfer' 1981 textbook 7 editions Wiley classic; Adrian Bejan (Romania-USA 1948-) Duke entropy generation minimization 1996 constructal theory; Yunus Cengel (Turkey-USA 1953-) UNR Heat Transfer textbook; Theodore Bergman + John Lavine 7th-8th edition Incropera. USA heat transfer pioneers: Theodore von Kármán (Hungary-USA 1881-1963) Caltech aerodynamics;

Bird Stewart Lightfoot (USA 1924-2020 + collaborators) University Wisconsin Madison Transport Phenomena 1960 Wiley classic chemical engineering 1500 pages; Hermann Schlichting + Klaus Gersten Boundary Layer Theory 8th edition 2017. USA universities heat transfer: MIT Cambridge MA 1861; Stanford Palo Alto CA 1925; Caltech Pasadena CA 1891; UC Berkeley CA 1868; Princeton NJ 1746; Cornell NY 1865; Carnegie Mellon Pittsburgh PA 1900; Georgia Tech Atlanta GA 1885; Purdue West Lafayette IN 1869; Notre Dame Indiana 1842 Frank Incropera; Illinois Urbana-Champaign 1867; Texas A&M College Station 1876; Michigan Ann Arbor 1817; Northwestern Evanston IL 1851; Penn State University Park 1855; Virginia Tech Blacksburg 1872. USA national labs: Lawrence Berkeley LBNL CA 1931; Argonne ANL IL 1946; Oak Ridge ORNL TN 1943 reactor heat transfer; Sandia National Labs Albuquerque NM Livermore CA 1949 thermal physics combustion; Idaho National Lab INL ID 1949 nuclear reactor research. USA companies cooling electronics: Intel Santa Clara CA 1968 founded 130000 employees CPU manufacturer thermal management; Nvidia Santa Clara CA 1993 founded 30000 GPU AI; AMD Santa Clara CA 1969 26000 CPU GPU; Apple Cupertino CA 1976 165000 M-series chips thermal design; Cooler Master Taiwan heat sinks fans; Noctua Austria CPU coolers premium; Arctic Switzerland; Corsair Milpitas CA 1994. USA food industry sterilization: Tetra Pak USA Vernon Hills IL Lund Sweden parent aseptic packaging UHT;

Dean Foods Dallas TX dairy pasteurization; Conagra Brands Chicago IL 1919 frozen foods Birds Eye Healthy Choice Marie Callender's; Kraft Heinz Pittsburgh Chicago 2015 fusion 39000 employees; General Mills Minneapolis MN 1866 35000 employees Cheerios Pillsbury Häagen-Dazs; PepsiCo Purchase NY 1965 309000 employees Pepsi Frito-Lay Quaker; Coca-Cola Atlanta GA 1892 79000 employees soft drinks; Mars Inc McLean VA 1911 candy chocolate; Hershey Hershey PA 1894 chocolate. USA metallurgy heat treatment: Nucor Charlotte NC 1955 31000 employees largest US steel electric arc furnace; US Steel Pittsburgh PA 1901 founded JP Morgan Andrew Carnegie 23000 employees integrated; Cleveland-Cliffs Cleveland OH 1847 21000 employees acquired AK Steel 2020; Steel Dynamics Fort Wayne IN 1993 11000 employees mini-mills; Inductotherm Group Rancocas NJ 1953 induction heating equipment; Ipsen Cherry Valley IL 1948 heat treatment furnaces vacuum atmosphere; ASM International Materials Park OH 1913 metallurgy society. USA aerospace reentry thermal protection: NASA Washington DC 1958 17000 employees Space Shuttle thermal tiles 1981-2011 24000 silica HRSI HRSI tiles; SpaceX Hawthorne CA 2002 Dragon capsule PICA-X heat shield ablative + Starship stainless steel + tiles; Boeing Chicago IL 1916 CST-100 Starliner; Lockheed Martin Bethesda MD 1995 Orion capsule Avcoat heat shield; NASA Ames Research Center Moffett Field CA arc-jet testing high-enthalpy; NASA Johnson Space Center Houston TX 1961;

NASA Kennedy Space Center FL 1962; ATK Northrop Grumman Innovation Systems propulsion. USA software heat transfer: ANSYS Canonsburg PA 1970 founded John Swanson finite element + Fluent + Mechanical 5500 employees market leader; Abaqus Dassault Systèmes 1978; COMSOL Stockholm 1986 multiphysics; FlowTherm Mentor Graphics Wilsonville OR 1981 (acquired Siemens 2017); HTRI Heat Transfer Research Inc Navasota TX 1962 industry standard heat exchanger; OpenFOAM ESI Group open source CFD foundation; LS-DYNA Livermore CA crash thermal; Solidworks Simulation Dassault Waltham MA.

Biot values table

• Bi = h*Lc/k_solid: definition
• Bi < 0.01: very low, lumped highly accurate
• Bi < 0.1: low, lumped capacitance valid <5%
• Bi 0.1-1: moderate, Heisler charts
• Bi 1-10: high, conduction-dominated
• Bi > 100: extreme, T_surface = T_fluid
• Lumped tau: rho*cp*V/(h*A)
• Slab Lc: half thickness L
• Cylinder Lc: R/2
• Sphere Lc: R/3
• Steel k: 50 W/m·K
• Copper k: 400 W/m·K
• Aluminum k: 240 W/m·K
• Brick k: 0.7 W/m·K
• Food k: 0.4-0.6 W/m·K
• h air free: 5-25 W/m²K
• h air forced: 25-250 W/m²K
• h water free: 100-1000 W/m²K
• h water forced: 500-15000 W/m²K
• h boiling: 2500-50000 W/m²K
• h condensation: 5000-100000 W/m²K
• Biot 1804 paper: Bibliothèque Britannique 27:310
• Fourier 1822: Théorie analytique chaleur
• Heisler 1947: ASME 69:227 charts
• Incropera 1981 textbook: Notre Dame 7 editions
• USA Intel Santa Clara 1968: 130000 CPU
• USA Nvidia Santa Clara 1993: 30000 GPU
• USA Tetra Pak Vernon Hills: aseptic UHT
• USA Conagra Chicago 1919: frozen foods
• USA Kraft Heinz 2015: 39000 employees
• USA Nucor Charlotte 1955: 31000 steel largest
• USA Inductotherm NJ 1953: induction heating
• USA Ipsen IL 1948: heat treatment furnaces
• USA NASA 1958: 17000 thermal protection
• USA SpaceX 2002: PICA-X heat shield
• USA ANSYS 1970: FEA market leader

Numerical examples

Example 1 - Steel sphere cooling air. Steel sphere R=0.05 m air-cooled. h=25 W/m²K natural convection, k_steel=50 W/m·K. Lc = R/3 = 0.017 m. Bi = 25*0.017/50 = 0.0083 < 0.01 very low. Lumped highly accurate <2%. USA: heat treatment Inductotherm NJ 1953; Ipsen IL 1948 furnaces.

Example 2 - Brick oven water. Brick Lc=0.1 m water boiling forced convection h=500 W/m²K, k_brick=0.7 W/m·K. Bi = 500*0.1/0.7 = 71.4 high. Large gradients, T_surface ~ 100°C, interior cools slowly. Heisler charts required. USA: brick industry; ASTM C90 Standard Specification CMU concrete masonry units.

Example 3 - Copper component oil. Copper component Lc=0.01 m oil-cooled h=200 W/m²K, k_copper=400 W/m·K. Bi = 200*0.01/400 = 0.005 very low. Lumped <2%. USA: cooling electronics Intel Santa Clara CA 1968 130000 + Nvidia 1993 30000 GPU + AMD 1969 26000.

Example 4 - Food sterilization steam. Cylindrical can Lc=0.02 m saturated steam h=5000 W/m²K, k_food=0.5 W/m·K. Bi = 5000*0.02/0.5 = 200 extreme high. T_surface = 121°C immediately; interior cooks slowly Fourier series. USA: food industry Conagra Chicago 1919; Kraft Heinz 2015 39000; Tetra Pak Vernon Hills.

Example 5 - Aluminum plate water time constant. Aluminum plate Lc=0.005 m water cooled h=100, k=200, rho*cp=3.5e6 J/m³K. Bi = 100*0.005/200 = 0.0025 very low. tau = 3.5e6*0.005/100 = 175 s = 2.9 min. 5*tau = 14.6 min 99% completion. USA: heat sinks Intel CPU Apple M-series chips; Cooler Master Noctua Corsair.

Limitations

1. Lumped Bi<0.1 only: applicable Bi<0.1 (error <5%); above requires Heisler charts or Fourier series.
2. Simple geometry: Heisler charts only plane slab, infinite cylinder, sphere. Irregular shapes need FEA.
3. Single-phase solid: homogeneous isotropic. Composites multi-phase separate treatment.
4. Constant h: assumes h convection coefficient time/space constant. T variations require iterative.
5. No phase change: melting solidification require Stefan number + latent heat.
6. Anisotropy: anisotropic solids (wood, composites) require tensor k.
7. Variable natural convection: complex geometries h varies with position.
8. Radiation effects: T>500°C radiation contribution h_rad combined with h_conv.
9. Steady state vs transient: Bi for transient only. Steady state uses thermal resistance R = L/(k*A).
10. Internal heat generation: solid generating heat (Joule, nuclear) requires Q_gen + Bi separate.

Common errors

k_solid vs k_fluid: Bi uses solid k (steel=50, copper=400, aluminum=240, brick=0.7); Nu uses fluid k. Common error confuses.

Characteristic length: slab Lc = half thickness L; cylinder Lc = R/2; sphere Lc = R/3; irregular Lc = V/A. Error using full R instead of R/2 or R/3.

Lumped Bi<0.1: lumped criterion Bi<0.1 error <5%; above Heisler charts. Error applying lumped Bi=0.5 gives 30-50% errors.

Time constant tau: tau = rho*cp*Lc/h. Time to reach 63.2% final. 5*tau ~ 99%. Error confusing tau with total time.

T_inf vs T_initial: T(t) = T_inf + (T_0 - T_inf)*exp(-t/tau). T_inf ambient fluid (asymptotic); T_0 initial solid. Error inverts the two.

Practical applications

1. Electronics cooling: CPU GPU heat sinks Bi<0.1 lumped. USA Intel Santa Clara 1968 130000; Nvidia Santa Clara 1993 30000; AMD 1969; Apple Cupertino 1976 M-series.
2. Food industry sterilization: USA Conagra Chicago 1919; Kraft Heinz 2015 39000; General Mills 1866 35000; Tetra Pak Vernon Hills aseptic UHT.
3. Metallurgy quenching: USA Nucor Charlotte 1955 31000 largest; US Steel Pittsburgh 1901; Cleveland-Cliffs 1847; Inductotherm NJ 1953 induction; Ipsen IL 1948 furnaces.
4. Automotive brake: USA Ford Dearborn MI 1903; GM Detroit MI 1908; Tesla Austin TX 2003 EV brake disc cooling.
5. Aerospace reentry heat shield: NASA Washington DC 1958 17000 Space Shuttle tiles; SpaceX Hawthorne CA 2002 PICA-X Starship; Boeing Chicago 1916 Starliner; Lockheed Martin Bethesda MD Orion.
6. Buildings thermal mass: USA passive cooling; ASHRAE Atlanta GA 1894 building science; LEED certification.
7. Body cooling hypothermia: Newton's law of cooling lumped capacitance human body; CDC Atlanta GA 1946 health.
8. Glass tempering: tempered glass quenching Bi>0.1; USA Owens-Illinois Perrysburg OH 1903.
9. Nuclear fuel rod: UO2 ceramic + cladding. USA Westinghouse Pittsburgh PA 1886; Idaho National Lab 1949; Sandia Albuquerque NM 1949.
10. Dental implants: titanium heat treatment. USA Zimmer Biomet Warsaw IN 1927.

Updated July 2026