How to calculate Graetz number Gz laminar heat transfer pipe

Calculate Graetz number Gz = (D/L)*Re*Pr ratio convective heat transfer to conduction in laminar flow tube thermal entry. Gz<<1 fully developed Nu=3.66; Gz>>1 thermal entry Leveque. Leo Graetz Germany 1856-1941. NIST + ASHRAE + MIT + Stanford + Fresenius dialysis applications.

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Graetz number Gz dimensionless ratio of convective heat transfer to thermal conduction in thermal entry region for laminar flow in tubes: Gz = (D/L) * Re * Pr. D tube diameter m; L tube length m; Re=rho*v*D/mu; Pr=mu*cp/k (air 0.71; water 7; oils 100-1000; mercury 0.025). Equivalent Gz = m_dot*cp/(k*L).

Inverse Graetz: 1/Gz = (alpha*L)/(D^2*v).

Regimes: Gz<1 fully developed Nu=3.66 const T_w + Nu=4.36 const heat flux; 1-10 transition; 10-100 thermal entry Hausen correlation; Gz>100 strong entry Leveque Nu=1.077*Gz^(1/3).

Hausen: Nu_D = 3.66 + (0.0668*Gz)/(1+0.04*Gz^(2/3)).

Sieder-Tate: Nu_D = 1.86*(Re*Pr*D/L)^(1/3)*(mu_b/mu_w)^0.14.

Leveque: Nu = 1.077*Gz^(1/3).

Thermal entry length: L_t/D = 0.05*Re*Pr laminar. History: Leo Graetz (Breslau Silesia Prussia 1856 - Munich Germany 1941) Universitat Strassburg + Munich 1893 Professor; 1885 Annalen Physik 25:337 'Uber die Warmeleitungsfahigkeit von Flussigkeiten' classic Graetz problem + Graetz number eponym + eigenvalue series solution.

Wilhelm Nusselt (Nuremberg Germany 1882 - Munich 1957) TH Munchen 1907-1957; 1916 VDI Z 60:541 Nusselt number eponym + 1925 paper.

Eduard Schmidt (Vohwinkel Germany 1892-1975) TH Karlsruhe + Munich 1937-1962; 1929 paper Schmidt mass transfer analogy + Sc number.

Heinz Hausen (Brunswick Germany 1896-1981) TH Munchen 1949-1968; 1959 paper + 1958 textbook 'Warmeubertragung' classic + Hausen correlation.

Eugene Sieder + Charles Tate USA Sun Oil Co Marcus Hook PA 1936; 1936 Ind Eng Chem 28:1429 'Heat transfer and pressure drop of liquids in tubes' classic Sieder-Tate.

Andre Leveque France 1928; 1928 Annales Mines 13:201 'Les lois de la transmission de chaleur par convection' classic Leveque solution. Legacy: Bird-Stewart-Lightfoot 'Transport Phenomena' 2007 Wiley + Bejan 'Convection Heat Transfer' 4th ed 2013 Wiley + Incropera-DeWitt-Bergman-Lavine 'Fundamentals' 8th ed 2018 Wiley 1080 pages classic + Shah-London 'Laminar Flow Forced Convection in Ducts' 1978 Academic Press 477 pages classic monograph + Kays-Crawford 'Convective Heat and Mass Transfer' 4th ed 2005 McGraw-Hill 558 pages.

USA research Graetz + heat transfer + microchannel + dialysis: NIST Gaithersburg MD 3500 + heat transfer reference + ASHRAE; NREL Golden CO 2900 + thermal storage; ORNL Oak Ridge TN 6500 + nuclear thermal hydraulics + HFIR + ATR; Sandia Albuquerque NM 14000 + Combustion Research Facility CRF Livermore; LBNL Berkeley CA 4400 + Building Technology + thermal management; PNNL Richland WA 4500; Argonne IL 3400; EPRI Palo Alto CA 600; ASHRAE Atlanta GA 1894 56000 standards; NIH National Institutes Health Bethesda MD 1887 18000 + biomedical heat transfer hyperthermia + dialysis; FDA Silver Spring MD 18000 + medical devices dialysis.

USA universities Graetz + heat transfer + microchannel: MIT Cambridge MA 11700 + Mech Eng + ChemE + Bergman-Lavine textbook + Adrian Bejan 1976-2014 Duke + Robert Cohen polymer + Edwin Thomas + Gareth McKinley rheology; Stanford CA 17000 + ChemE Curtis Frank + George Karniadakis CFD; Caltech Pasadena 2400 + GALCIT John Brady microhydrodynamics + Beverley McKeon turbulence; UC Berkeley 45000 + ChemE Van Carey heat transfer + Carlos Fernandez-Pello combustion; Princeton 8000 + MAE + ChemE Howard Stone microfluidics + Sankaran Sundaresan; Cornell Ithaca NY 25000 + ChemE; Georgia Tech Atlanta 36000 + ME Yogendra Joshi microchannel cooling electronics; UIUC Urbana 56000 + ME Suresh Garimella microchannel boiling + Charles Schroeder; Penn State 50000 + ME Stefan Gracias + Daniel Kim; UT Austin TX 51000 + ME; Univ Michigan Ann Arbor 50000 + ME Robert Larson polymer + heat transfer; Northwestern Evanston IL 22000 + ChemE Stephen Davis solidification; Purdue 47000 + ME Sasanka Are Garimella microchannel + heat exchangers; Wisconsin-Madison 49000 + ChemE Bird heritage; Minnesota 51000 + Mech Eng; UCLA 45000 + ME; UC Davis 40000;

RPI Rensselaer 7000 + Bejan heritage + Adrian Bejan Duke 1976-2014 emeritus + Olson; Lehigh Bethlehem PA 7000; Akron OH 18000 Polymer Science; UMass Amherst 30000; Carnegie Mellon Pittsburgh 14000 + ME; Duke 16000 + Bejan emeritus 1976-2014.

USA HVAC heat exchangers industry: Carrier Global Palm Beach Gardens FL 1915 53000 + commercial HVAC chillers heat exchangers; Trane Technologies Davidson NC 39000 + chillers heat exchangers RTU; Lennox Richardson TX 1895 10000; Daikin Applied Plymouth MN 12000; Johnson Controls York PA 105000 + York chillers heat exchangers; SPX Cooling Marley Overland Park KS 4000 + cooling tower heat exchangers; Baltimore Aircoil BAC Jessup MD 1000; Evapco Taneytown MD 4000; Rheem Atlanta GA 4000; AO Smith Milwaukee 12000; Modine Manufacturing Racine WI 1916 11000 + heat transfer products + automotive HVAC.

USA industrial heat exchangers: Alfa Laval USA Richmond VA 1883 + Sweden parent 18000 + plate heat exchangers world-leading + GPHE BPHE + dairy food chemical; Kelvion Visakhapatnam (formerly GEA Heat Exchangers Bochum DE) 4000; SWEP International (Dover Corp) 700 brazed plate; API Heat Transfer 1000; Honeywell UOP Des Plaines IL 5000 + FCC heat integration; Heatcraft Worldwide Refrigeration Stone Mountain GA 4000.

USA microchannel cooling industry: Intel Santa Clara CA 131000 + microchannel cooling CPU electronics + thermal management; NVIDIA Santa Clara CA 1993 30000 + GPU cooling + AI thermal management; AMD Santa Clara CA 1969 26000; HP Hewlett-Packard Palo Alto 51000; Asetek San Jose CA + Aalborg DK 100 + liquid cooling CPU; CoolIT Systems Calgary 250; Vertiv Westerville OH 27000 + data center cooling.

USA dialysis hemodialysis hollow fiber industry: Fresenius Medical Care USA Waltham MA 1996 60000 + hollow fiber dialyzers Polysulfone Helixone + dialysis machines + clinics 4000 USA worldwide leader; DaVita Inc Denver CO 1979 67000 + dialysis clinics 2700 USA; Baxter Deerfield IL 1931 60000 + dialysis + IV bags + medical fluids; NxStage Medical Lawrence MA + Fresenius 2019 1500 + home hemodialysis; NIPRO Medical Bridgewater NJ 1500 + dialysis hollow fiber.

USA medical hyperthermia: BSD Medical Salt Lake City UT 1986 100 + microwave hyperthermia + Pyrexar Medical 50; Pyrexar Medical Salt Lake City UT 50 + BSD-2000 hyperthermia; Magtec Diagnostics + Insightec MR-guided focused ultrasound 200.

USA nuclear cooling heat exchangers: Westinghouse Cranberry Township PA 9500 + AP1000 + heat exchangers passive cooling; GE Hitachi Nuclear Wilmington NC 5000; BWX Technologies Lynchburg VA 7000; NuScale Power Tigard OR 600 + SMR Small Modular Reactor; TerraPower Bellevue WA 600 + Bill Gates + sodium cooled fast reactor.

Graetz formula

Definition: Gz = (D/L)*Re*Pr. Equivalent Gz = m_dot*cp/(k*L).

Hausen: Nu_D = 3.66 + (0.0668*Gz)/(1+0.04*Gz^(2/3)).

Sieder-Tate: Nu_D = 1.86*(Re*Pr*D/L)^(1/3)*(mu_b/mu_w)^0.14.

Leveque: Nu = 1.077*Gz^(1/3) thin BL.

Thermal entry: L_t/D = 0.05*Re*Pr laminar.

History

Leo Graetz (Breslau Silesia Prussia 1856 - Munich 1941) Strassburg + Munich 1893 + 1885 Annalen Physik 25:337 classic Graetz problem + eigenvalue series. Wilhelm Nusselt (Nuremberg 1882 - Munich 1957) TH Munchen 1907-1957 + 1916 VDI Z 60:541 + Nusselt number. Eduard Schmidt (Vohwinkel 1892-1975) TH Karlsruhe + Munich 1937-1962 + 1929 + Schmidt mass transfer + Sc number. Heinz Hausen (Brunswick 1896-1981) TH Munchen 1949-1968 + 1959 + 1958 textbook 'Warmeubertragung' classic. Sieder-Tate USA Sun Oil 1936 Marcus Hook PA. Andre Leveque France 1928 Annales Mines 13:201 classic. Bird-Stewart-Lightfoot 2007 + Bejan 2013 + Incropera 8th 2018 + Shah-London 1978 + Kays-Crawford 2005.

USA research: NIST 3500; NREL 2900; ORNL 6500; Sandia 14000; LBNL 4400; PNNL 4500; Argonne 3400; EPRI 600; ASHRAE 56000; NIH 18000 hyperthermia dialysis; FDA 18000.

USA universities: MIT 11700 Bergman-Lavine + Bejan emeritus + Cohen + McKinley; Stanford 17000 Frank Karniadakis; Caltech 2400 Brady McKeon; UC Berkeley 45000 Carey heat transfer + Fernandez-Pello; Princeton 8000 Stone Sundaresan; Cornell 25000; Georgia Tech 36000 Joshi microchannel electronics; UIUC 56000 Garimella microchannel boiling; Penn State 50000; UT Austin 51000; Michigan 50000 Larson; Northwestern 22000 Davis; Purdue 47000 Garimella microchannel; Wisconsin-Madison 49000 Bird heritage; Minnesota 51000; UCLA 45000; RPI 7000; Lehigh 7000; Akron 18000; UMass Amherst 30000; Carnegie Mellon 14000; Duke 16000 Bejan emeritus 1976-2014.

USA HVAC: Carrier 53000; Trane 39000; Lennox 10000; Johnson Controls 105000; SPX Marley 4000; BAC 1000; Evapco 4000; Rheem 4000; AO Smith 12000; Modine Racine 11000.

USA industrial HX: Alfa Laval USA 18000 plate HE world-leading; Kelvion 4000 ex-GEA; SWEP Dover 700; API HT 1000; Honeywell UOP 5000; Heatcraft Atlanta 4000.

USA microchannel: Intel 131000; NVIDIA 30000 GPU cooling AI; AMD 26000; HP 51000; Asetek 100; CoolIT 250; Vertiv 27000.

USA dialysis: Fresenius Medical USA Waltham 60000 hollow fiber dialyzers world-leading 4000 clinics; DaVita Denver 67000 2700 clinics; Baxter Deerfield 60000; NxStage Lawrence MA 1500 home; NIPRO 1500.

USA hyperthermia: BSD Medical Salt Lake 100; Pyrexar 50 BSD-2000; Insightec 200 MR-guided focused ultrasound.

USA nuclear: Westinghouse Cranberry 9500 AP1000; GE Hitachi Nuclear 5000; BWX 7000; NuScale Tigard 600 SMR; TerraPower Bellevue 600 Bill Gates sodium fast reactor.

Graetz values table

  • Gz = (D/L)*Re*Pr
  • Gz = m_dot*cp/(k*L)
  • 1/Gz = (alpha*L)/(D^2*v)
  • Gz<1 fully developed: Nu=3.66
  • 1-10 transition
  • 10-100 thermal entry: Hausen
  • Gz>100 strong entry: Leveque
  • Hausen Nu = 3.66 + (0.0668*Gz)/(1+0.04*Gz^(2/3))
  • Sieder-Tate Nu = 1.86*(Re*Pr*D/L)^(1/3)
  • Leveque Nu = 1.077*Gz^(1/3)
  • Nu=4.36 const heat flux
  • L_t/D = 0.05*Re*Pr laminar
  • Pr air 0.71 water 7
  • Graetz Germany 1885
  • Nusselt 1916
  • Schmidt 1929
  • Hausen 1959
  • Sieder-Tate 1936
  • Leveque 1928
  • USA NIST 3500
  • USA Carrier 53000 + Trane 39000: HVAC
  • USA Alfa Laval 18000: plate HE world-leading
  • USA Fresenius Medical 60000: dialysis world-leading 4000 clinics
  • USA DaVita 67000: 2700 clinics
  • USA Intel 131000 + NVIDIA 30000: microchannel cooling
  • USA Westinghouse 9500: AP1000 nuclear

Numerical examples

Example 1 - Long tube fully developed. Gz=0.5 Nu=3.66. USA: Fresenius Medical hollow fiber dialysis.

Example 2 - Compact heat exchanger. Gz=35 thermal entry. USA: Carrier + Trane + Alfa Laval plate HE.

Example 3 - Microchannel cooling. Gz=0.5 fully developed. USA: Intel + NVIDIA + AMD + Asetek liquid CPU cooling.

Example 4 - Dialysis hollow fiber. Gz=0.21. USA: Fresenius Medical Waltham 60000 + DaVita Denver 67000 + Baxter Deerfield + NxStage home.

Example 5 - Strong entry. Gz=3500 Leveque. USA: ExxonMobil + Chevron + Honeywell UOP FCC heat exchangers + Westinghouse AP1000.

Limits and considerations

  1. Laminar Re<2300 + Dittus-Boelter turbulent.
  2. Constant properties + Sieder-Tate correction.
  3. Const wall T vs heat flux 3.66 vs 4.36.
  4. Thermal vs hydrodynamic entry simultaneous developing.
  5. Compressibility high speed gas.
  6. Buoyancy mixed convection.
  7. Variable cross-section helical Dean.
  8. Surface roughness.
  9. Slip flow microchannel Knudsen.
  10. Non-Newtonian power-law Carreau.

Common errors

Confusing Gz and Pe: Gz=(D/L)*Pe.

Const wall T vs heat flux: 3.66 vs 4.36.

Hausen vs Sieder-Tate vs Leveque: range Gz.

Thermal entry length: laminar vs turbulent.

Hydrodynamic vs thermal entry: simultaneous developing.

Practical applications

  1. HVAC heat exchangers: USA Carrier 53000 + Trane 39000 + Lennox 10000 + Johnson Controls 105000 + SPX Marley + BAC + Modine Racine 11000.
  2. Industrial heat exchangers: USA Alfa Laval 18000 plate HE world-leading + Kelvion 4000 + SWEP Dover + API Heat Transfer + Honeywell UOP 5000 FCC.
  3. Microchannel CPU cooling electronics: USA Intel 131000 + NVIDIA 30000 GPU AI + AMD 26000 + HP 51000 + Asetek + CoolIT + Vertiv data center 27000.
  4. Dialysis hollow fiber medical: USA Fresenius Medical Care Waltham 60000 + DaVita Denver 67000 2700 clinics + Baxter Deerfield 60000 + NxStage home 1500 + NIPRO 1500.
  5. Medical hyperthermia oncology: USA BSD Medical 100 + Pyrexar Salt Lake 50 BSD-2000 + Insightec 200 MR-guided focused ultrasound.
  6. Nuclear cooling heat exchangers: USA Westinghouse Cranberry 9500 AP1000 passive + GE Hitachi Nuclear 5000 + BWX 7000 + NuScale Tigard SMR + TerraPower Bellevue sodium.
  7. Solar thermal CSP heat exchangers: USA BrightSource Ivanpah Mojave 392 MW + SolarReserve Crescent Dunes Tonopah NV + Abengoa Solana Mojave.
  8. Petrochemical heat exchangers FCC: USA ExxonMobil 62000 + Chevron 45000 + Marathon 17000 + Phillips 66 + Valero + Honeywell UOP 5000.
  9. Universities heat transfer research: MIT Bergman-Lavine Bejan + Stanford + Caltech Brady McKeon + UC Berkeley Carey + Princeton Stone + Georgia Tech Joshi microchannel + UIUC Garimella + Purdue Garimella + Duke Bejan emeritus.
  10. Cogeneration biomass heat exchangers: USA NextEra + Duke Energy + Southern Company + Babcock&Wilcox CFB.

Updated July 2026

Frequently Asked Questions

What is the Graetz number?

Gz = (D/L)*Re*Pr dimensionless ratio convective heat transfer to thermal conduction in thermal entry region for laminar flow tube. Gz<1 fully developed Nu=3.66 const T_w; 10-100 thermal entry Hausen; Gz>100 strong entry Leveque Nu=1.077*Gz^(1/3). Leo Graetz Germany 1856-1941 + 1885 paper. USA: NIST 3500 + Carrier 53000 + Alfa Laval 18000 + Fresenius Medical 60000 + Intel microchannel 131000.

How do I calculate Graetz?

Procedure: (1) Gz = (D/L)*Re*Pr; (2) D m diameter; (3) L m length; (4) Re=rho*v*D/mu; (5) Pr=mu*cp/k (water 7); (6) Hausen Nu = 3.66 + (0.0668*Gz)/(1+0.04*Gz^(2/3)); (7) regime. Dialysis: Gz=0.21 fully developed. Compact HX: Gz=35. USA: MIT Bergman-Lavine textbook + UIUC Garimella microchannel + Purdue.

What is the Graetz problem?

Graetz problem Leo Graetz Germany 1885 paper Annalen Physik 25:337 'Uber die Warmeleitungsfahigkeit von Flussigkeiten' classic + laminar Hagen-Poiseuille flow + analytical eigenvalue series Graetz solution + thermal entry developing + Nu_inf=3.66 const T_w + Nu=4.36 const heat flux. USA: MIT Bejan + UIUC Garimella + Purdue Garimella microchannel + Georgia Tech Joshi.

What regimes does Gz classify?

Gz<1 fully developed Nu=3.66 const T_w; 1-10 transition; 10-100 thermal entry Hausen Nu=3.66+(0.0668*Gz)/(1+0.04*Gz^(2/3)); Gz>100 strong entry Leveque Nu=1.077*Gz^(1/3). USA: ExxonMobil FCC + Carrier compact HX + Fresenius dialysis + Intel microchannel + NuScale SMR.

Who was Leo Graetz?

Leo Graetz (Breslau Silesia Prussia 1856 - Munich Germany 1941) Universitat Strassburg + Universitat Munchen 1893 Professor; 1885 paper Annalen Physik 25:337 'Uber die Warmeleitungsfahigkeit von Flussigkeiten' classic Graetz problem laminar pipe heat transfer + Graetz number eponym + analytical eigenvalue series solution + 1900 'Lehrbuch Theoretischen Physik' + 1922 'Die Atomtheorie' Munich.

What is the Hausen correlation?

Hausen correlation Heinz Hausen Germany TH Munchen 1949-1968 + 1959 paper + 1958 textbook 'Warmeubertragung im Gegenstrom' classic + Nu_D = 3.66 + (0.0668*Gz)/(1+0.04*Gz^(2/3)) + valid Re0.5 + isothermal wall + thermal entry developing laminar. USA: Carrier + Trane + Alfa Laval plate heat exchangers + Modine Racine.

What are Graetz applications?

Applications: HVAC heat exchangers Carrier 53000 + Trane 39000 + SPX Marley + Modine 11000; industrial HX Alfa Laval 18000 plate world-leading + Kelvion + Honeywell UOP 5000 FCC; microchannel CPU cooling Intel 131000 + NVIDIA 30000 + AMD + Asetek + Vertiv data center; dialysis hollow fiber Fresenius Medical 60000 4000 clinics worldwide leader + DaVita 67000 + Baxter; hyperthermia oncology BSD + Pyrexar BSD-2000 + Insightec MR-FUS; nuclear cooling Westinghouse 9500 AP1000 + NuScale SMR + TerraPower Bill Gates.

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