How to calculate Bejan number Be dimensionless pressure drop

Calculate Bejan number Be = dP*L^2/(mu*alpha) dimensionless pressure drop forced convection. Be<<1 viscous slow; Be>>1 turbomachinery jet. Adrian Bejan Romania-USA Duke 1948 + constructal theory 1996 + EGM. NIST + ExxonMobil + GE + Pratt&Whitney applications.

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Bejan number Be dimensionless pressure drop in forced convection developed by Bhattacharjee-Grosshandler 1988 + Adrian Bejan Romania-USA Duke 1948 (constructal theory 1996): Be = dP * L^2 / (mu * alpha). dP pressure drop Pa; L characteristic length m; mu dynamic viscosity Pa*s (water 1e-3; oils 0.05; air 2e-5); alpha thermal diffusivity m^2/s (water 1.4e-7; oils 7e-8; air 2.2e-5).

Alternative entropy generation definition: Be = S_gen,heat / (S_gen,heat + S_gen,fluid) Paoletti-Rispoli-Sciubba Sapienza Roma 1989 + Be0.5 heat transfer dominant.

Regimes: Be<1e6 low viscous slow + microchannel; 1e6-1e9 medium HVAC; 1e9-1e12 high industrial; Be>1e12 very high turbomachinery + jet engines + aerospace.

Constructal theory Bejan 1996: flow architecture optimization access-time minimization + tree-shaped networks bifurcating + nature inspired + Bejan 1996 Int J Heat Mass Transfer 40:799.

Entropy generation minimization EGM: S_gen = Q*dT/T^2 + dP*v/T + thermodynamic optimization. History: Adrian Bejan (Galati Romania 1948 - Duke USA emeritus) MIT BSc 1971 + MS 1972 + ScD 1975 + Berkeley 1975-1976 + University Colorado Boulder 1976-1984 + Duke University Durham NC J A Jones Distinguished Professor Mech Eng 1984-2014 emeritus + 30 honorary doctorates Romania Belgium Italy Israel; 1982 textbook 'Entropy Generation Through Heat and Fluid Flow' Wiley classic 248 pages + 1995 'Convection Heat Transfer' Wiley 4th ed 2013 658 pages classic + 1996 paper Int J Heat Mass Transfer 40:799 'Constructal-theory network of conducting paths for cooling a heat generating volume' classic + 1996 'Entropy Generation Minimization' CRC Press 362 pages + 2000 'Shape and Structure from Engineering to Nature' Cambridge classic 324 pages + 2008 'Design with Constructal Theory' Wiley + ASME Heat Transfer Memorial Award 1994.

Sumeet Bhattacharjee + William Grosshandler USA NIST + UMD 1988; 1988 paper Int J Heat Mass Transfer 31:283 'Formation of axial pressure profile in laminar forced convection' classic Bhattacharjee-Grosshandler Bejan number eponym Be = dP*L^2/(mu*alpha).

S Paoletti + F Rispoli + Enrico Sciubba (Sapienza Roma 1955-) Mech Eng + Universita Roma Tre 1989-2018; 1989 ASME AES 10-2:21 'Calculation of exergetic losses in compact heat exchanger passages' classic Paoletti-Rispoli-Sciubba alternative Bejan number entropy ratio definition + Italian heritage Sapienza Sciubba thermoeconomics + exergy.

Sadik Kakac (Ankara Turkey 1932-) METU + Miami University ChemE; 'Heat Exchangers Selection Rating Thermal Design' 3rd ed 2012 CRC Press 1100 pages classic.

Massoud Kaviany (Iran-USA Univ Michigan 1976-2018); 'Principles of Heat Transfer in Porous Media' 2nd ed 1995 Springer 700 pages classic. Legacy: Bejan 'Convection Heat Transfer' 4th ed 2013 Wiley + 'Entropy Generation Minimization' 1996 + Kakac 'Heat Exchangers' 2012 CRC + Kaviany 1995 Springer + Bird-Stewart-Lightfoot 2007 + Incropera-DeWitt-Bergman-Lavine 'Fundamentals' 8th ed 2018 Wiley 1080 pages + Shah-Sekulic 'Fundamentals of Heat Exchanger Design' 2003 Wiley 941 pages classic + Reay-Kew-McGlen 'Heat Pipes' 6th ed 2013.

USA research Bejan + entropy generation + constructal + heat exchangers: NIST Gaithersburg MD 3500 + heat transfer reference + Bhattacharjee-Grosshandler heritage; NREL Golden CO 2900; ORNL Oak Ridge TN 6500; Sandia Albuquerque NM 14000; LBNL Berkeley 4400; PNNL 4500; Argonne 3400; NETL Pittsburgh Morgantown 1500 + biomass; EPRI Palo Alto 600; NASA Glenn Cleveland 1500 + thermal management + microgravity; NASA Marshall Huntsville 6000.

USA universities Bejan + entropy + constructal: MIT Cambridge MA 11700 + Mech Eng Bejan PhD 1975 + ChemE; Duke University Durham NC 16000 + Bejan emeritus 1984-2014 J A Jones Professor + Constructal Law Conference + Adrian Bejan Bibliography world-leading 30 books; UCLA 45000; Stanford CA 17000; Caltech Pasadena 2400; UC Berkeley 45000 Bejan visiting 1975-1976 + Van Carey heat transfer; Princeton 8000 Howard Stone microfluidics; Cornell Ithaca 25000; Georgia Tech Atlanta 36000 + Yogendra Joshi microchannel; UIUC Urbana 56000 + Suresh Garimella; Penn State 50000; UT Austin 51000; Univ Michigan Ann Arbor 50000 + Massoud Kaviany emeritus + Bejan colleague + Bejan honorary doctorate + Robert Larson polymer; Northwestern Evanston IL 22000; Purdue 47000 + Suresh Garimella heritage Heat Transfer Lab world-leading + Constructal world-leading + Bejan colleague; Wisconsin-Madison 49000; Minnesota 51000; UC Davis 40000; Texas A&M 70000; RPI 7000 + Bejan Award + Adrian Bejan ASME Heat Transfer Memorial Award 1994; Lehigh 7000; Univ Colorado Boulder 1976-1984 Bejan + Mech Eng 38000;

Bejan world bibliography 30+ books world-leading thermodynamics; Carnegie Mellon Pittsburgh 14000; Univ Tulsa 4500.

USA HVAC heat exchangers industry Bejan optimization: Carrier Global Palm Beach Gardens FL 53000 + commercial HVAC chillers; Trane Technologies Davidson NC 39000; Lennox Richardson TX 10000; Daikin Applied Plymouth MN 12000; Johnson Controls York PA 105000; Rheem Atlanta GA 4000; AO Smith Milwaukee 12000; Modine Manufacturing Racine WI 11000; Alfa Laval USA Richmond VA 18000; Kelvion 4000; SWEP Dover 700; Heatcraft Atlanta 4000; API Heat Transfer 1000; Honeywell UOP Des Plaines IL 5000 FCC heat integration.

USA petrochemical heat exchangers: ExxonMobil Irving TX 62000 + refineries + heat exchangers; Chevron 45000; Marathon Petroleum 17000; Phillips 66 14000; Valero 9700; Honeywell UOP 5000; BP America 60000; Shell USA 14000; Citgo Petroleum 3500; Air Products 21000; Praxair Linde 27000.

USA turbomachinery jet engines: GE Aerospace Cincinnati OH 50000 + jet engines GE9X GE90 + LEAP CFM; Pratt&Whitney East Hartford CT 38000 + Geared Turbofan PW1000G + V2500 + GP7000; Honeywell Aerospace Phoenix AZ 23000; RTX Raytheon Technologies Waltham MA 174000; Boeing Arlington VA 156000; Lockheed Martin Bethesda MD 116000; Northrop Grumman Falls Church VA 102000; SpaceX Hawthorne CA 13000; Blue Origin Kent WA 11000; Rolls-Royce North America Indianapolis IN 4000.

USA pumps centrifugal industry: Flowserve Irving TX 1997 16000 + pumps centrifugal + Pacific PSP + Worthington; Goulds Pumps Seneca Falls NY 1848 + ITT 8000 + ITT industrial process; Sulzer USA Easley SC 13000; KSB North America 1500; Wilo USA 200; Grundfos USA 6000; Pentair Plymouth MN 11000; Xylem Rye Brook NY 18000 Lowara Goulds; Watts Water Technologies North Andover MA 1500; Roper Technologies Sarasota FL 1981 18000.

USA microchannel cooling Bejan optimization: Intel Santa Clara CA 131000 + microchannel CPU; NVIDIA Santa Clara CA 30000 + GPU AI cooling; AMD 26000; HP Palo Alto 51000; Asetek 100; CoolIT 250; Vertiv Westerville OH 27000; Schneider Electric USA 12000.

USA cogeneration entropy minimization: NextEra Energy Juno Beach FL 14000; Duke Energy Charlotte NC 28000; Southern Company Atlanta GA 28000; Exelon Corp Chicago IL 31000; American Electric Power AEP Columbus OH 17000; Dominion Energy Richmond VA 17000; Babcock&Wilcox Akron OH 5300; GE Power Schenectady NY 30000.

Bejan formula

Pressure drop: Be = dP*L^2/(mu*alpha) Bhattacharjee-Grosshandler 1988.

Entropy ratio: Be = S_gen,heat/(S_gen,heat+S_gen,fluid) Paoletti-Rispoli-Sciubba 1989.

Constructal theory Bejan 1996: flow architecture optimization access-time + tree-shaped networks.

EGM: S_gen = Q*dT/T^2 + dP*v/T Bejan thermodynamic optimization.

History

Adrian Bejan (Galati Romania 1948 - Duke USA emeritus) MIT 1971-1975 + Berkeley 1975-1976 + Colorado Boulder 1976-1984 + Duke 1984-2014 J A Jones Professor + 30 doctorates + 1982 'Entropy Generation Through Heat and Fluid Flow' Wiley + 1995 'Convection Heat Transfer' 4th 2013 + 1996 constructal Int J HMT 40:799 + 'Shape and Structure' 2000 Cambridge + ASME Heat Transfer Memorial Award 1994. Bhattacharjee-Grosshandler USA NIST UMD 1988 Int J HMT 31:283 + Bejan number eponym Be = dP*L^2/(mu*alpha). Paoletti-Rispoli-Sciubba Sapienza Roma 1989 ASME AES 10-2:21 alternative entropy ratio. Kakac Turkey-Miami 'Heat Exchangers' 2012 CRC. Kaviany Michigan 'Heat Transfer in Porous Media' 1995 Springer. Bird-Stewart-Lightfoot 2007 + Incropera 2018 + Shah-Sekulic 2003.

USA research: NIST 3500 Bhattacharjee-Grosshandler heritage; NREL 2900; ORNL 6500; Sandia 14000; LBNL 4400; PNNL 4500; Argonne 3400; NETL 1500; EPRI 600; NASA Glenn 1500; NASA Marshall 6000.

USA universities: MIT 11700 Bejan PhD 1975; Duke 16000 Bejan emeritus 1984-2014 J A Jones Professor world-leading; Stanford 17000; Caltech 2400; UC Berkeley 45000 Bejan visiting + Carey; Princeton 8000 Stone microfluidics; Cornell 25000; Georgia Tech 36000 Joshi microchannel; UIUC 56000 Garimella; Penn State 50000; UT Austin 51000; Michigan 50000 Kaviany emeritus + Larson; Northwestern 22000; Purdue 47000 Garimella Heat Transfer Lab world-leading; Wisconsin-Madison 49000; Minnesota 51000; RPI 7000; Colorado Boulder 38000 Bejan 1976-1984; Carnegie Mellon 14000.

USA HVAC: Carrier 53000; Trane 39000; Lennox 10000; Daikin Applied 12000; Johnson Controls 105000; Rheem 4000; AO Smith 12000; Modine Racine 11000; Alfa Laval USA 18000; Kelvion 4000; Heatcraft 4000; Honeywell UOP 5000.

USA petrochemical: ExxonMobil 62000; Chevron 45000; Marathon Petroleum 17000; Phillips 66 14000; Valero 9700; Honeywell UOP 5000; Air Products 21000; Praxair Linde 27000.

USA turbomachinery: GE Aerospace 50000 GE9X GE90 LEAP CFM; Pratt&Whitney 38000 Geared Turbofan PW1000G; Honeywell Aerospace 23000; RTX 174000; Boeing 156000; Lockheed Martin 116000; Northrop Grumman 102000; SpaceX 13000; Blue Origin 11000; Rolls-Royce NA 4000.

USA pumps: Flowserve Irving 16000; Goulds ITT 8000; Sulzer USA 13000; Pentair 11000; Xylem 18000 Lowara Goulds; Roper Tech 18000; Watts Water 1500.

USA microchannel: Intel 131000; NVIDIA 30000; AMD 26000; HP 51000; Asetek 100; CoolIT 250; Vertiv 27000; Schneider Electric 12000.

USA cogeneration: NextEra 14000; Duke Energy 28000; Southern 28000; Exelon 31000; AEP 17000; Dominion 17000; Babcock&Wilcox 5300; GE Power 30000.

Bejan values table

  • Be = dP*L^2/(mu*alpha): Bhattacharjee-Grosshandler 1988
  • Be = S_gen,heat/(total): Paoletti-Rispoli-Sciubba 1989
  • Be<1e6 low: microchannel
  • 1e6-1e9 medium: HVAC
  • 1e9-1e12 high: industrial
  • Be>1e12 very high: turbomachinery
  • Constructal Bejan 1996
  • EGM thermodynamic
  • Bejan Romania-USA Duke 1948: 30 doctorates
  • Bejan 1982 'Entropy Gen' Wiley
  • Bejan 1995 'Convection Heat Transfer': 4th 2013
  • Bejan 1996 constructal
  • Bhattacharjee-Grosshandler 1988
  • Paoletti-Rispoli-Sciubba 1989 Sapienza: Italian heritage
  • USA NIST 3500: Bhattacharjee-Grosshandler
  • USA MIT Bejan 1971-1975
  • USA Duke 16000: Bejan emeritus 1984-2014
  • USA Carrier 53000 + Trane 39000: HVAC
  • USA Alfa Laval USA 18000: plate world-leading
  • USA GE Aerospace 50000: GE9X GE90
  • USA Pratt&Whitney 38000: PW1000G
  • USA Flowserve 16000 + Xylem 18000: pumps
  • USA Intel 131000 + NVIDIA 30000: microchannel
  • USA Purdue Garimella: Heat Transfer Lab world-leading
  • USA Michigan Kaviany emeritus: porous media textbook

Numerical examples

Example 1 - Microchannel water. Be=7e6 medium. USA: Intel + NVIDIA + AMD chip cooling.

Example 2 - HVAC chiller. Be=9e8. USA: Carrier 53000 + Trane 39000 + Daikin Applied + Alfa Laval USA plate world-leading.

Example 3 - Industrial heat exchanger. Be=9e14 very high. USA: ExxonMobil + Chevron + Honeywell UOP FCC + Marathon Petroleum.

Example 4 - Jet engine. Be=2e14. USA: GE Aerospace 50000 + Pratt&Whitney 38000 PW1000G Geared Turbofan + Honeywell Aerospace + RTX.

Example 5 - Oil pipeline. Be=3e17. USA: ExxonMobil + Chevron + Halliburton + Schlumberger pipelines.

Limits and considerations

  1. Two definitions: pressure drop vs entropy ratio.
  2. Reynolds vs Bejan complementary.
  3. Constructal theory paradigm.
  4. EGM thermodynamic vs economic.
  5. Microchannel surface effects Knudsen.
  6. Compressibility high speed.
  7. Buoyancy mixed convection.
  8. Variable properties Pr(T).
  9. Multi-component diffusion.
  10. Turbulence Re>2300.

Common errors

Confusing Be definitions: pressure drop vs entropy.

Constructal theory: flow architecture vs traditional.

EGM minimization: thermodynamic vs economic.

Microchannel scaling: surface effects.

Italian heritage Sapienza: Sciubba Paoletti Rispoli 1989.

Practical applications

  1. HVAC heat exchangers: USA Carrier 53000 + Trane 39000 + Daikin Applied + Alfa Laval USA 18000 plate world-leading + Kelvion + Heatcraft + Modine Racine 11000.
  2. Petrochemical heat exchangers: USA ExxonMobil 62000 + Chevron 45000 + Marathon 17000 + Phillips 66 + Valero + Honeywell UOP 5000 FCC + BP America + Shell USA + Air Products 21000.
  3. Turbomachinery jet engines: USA GE Aerospace 50000 GE9X GE90 LEAP + Pratt&Whitney 38000 PW1000G + Honeywell Aerospace + RTX 174000 + Boeing 156000 + Lockheed Martin + Northrop Grumman + SpaceX + Blue Origin + Rolls-Royce NA.
  4. Pumps centrifugal: USA Flowserve Irving 16000 + Goulds ITT 8000 + Sulzer USA 13000 + Xylem 18000 Lowara Goulds + Pentair 11000 + Roper Tech 18000.
  5. Microchannel CPU GPU cooling: USA Intel 131000 + NVIDIA 30000 GPU AI + AMD 26000 + HP + Asetek + CoolIT + Vertiv 27000 data center.
  6. Constructal theory: USA Duke Bejan emeritus 30 books world-leading + MIT + Purdue Garimella + UC Berkeley.
  7. EGM optimization: USA NIST + NREL + EPRI + ASHRAE + Bejan textbook.
  8. Cogeneration power plants: USA NextEra 14000 + Duke Energy 28000 + Southern 28000 + Exelon 31000 + AEP + Dominion + Babcock&Wilcox + GE Power.
  9. Solar thermal CSP: USA BrightSource Ivanpah + SolarReserve Crescent Dunes + Abengoa Solana + NREL.
  10. Universities heat transfer Bejan school: Duke + MIT + Purdue Garimella + Michigan Kaviany + UIUC + Georgia Tech + Stanford + UC Berkeley + Caltech.

Updated July 2026

Frequently Asked Questions

What is the Bejan number?

Be = dP*L^2/(mu*alpha) dimensionless pressure drop in forced convection Bhattacharjee-Grosshandler 1988. Alternative Be = S_gen,heat/(S_gen,heat+S_gen,fluid) Paoletti-Rispoli-Sciubba Sapienza Roma 1989 Italian heritage. Be<1e6 low microchannel; 1e6-1e9 medium HVAC; Be>1e12 very high turbomachinery jet. Adrian Bejan Romania-USA Duke 1948 + constructal theory 1996 + EGM. USA: NIST 3500 + Duke 16000 Bejan emeritus + Carrier 53000 + GE Aerospace 50000 + Intel 131000.

How do I calculate Bejan?

Procedure: (1) Be = dP*L^2/(mu*alpha); (2) dP Pa pressure drop; (3) L characteristic m; (4) mu Pa*s (water 1e-3); (5) alpha m^2/s (water 1.4e-7); (6) regime; (7) entropy ratio alternative. Microchannel: Be=7e6. Industrial HX: Be=9e14. USA: Duke Bejan emeritus + MIT + Purdue Garimella Heat Transfer Lab world-leading + Michigan Kaviany porous media.

What is constructal theory?

Constructal theory Adrian Bejan Duke 1996 paper Int J Heat Mass Transfer 40:799 + 1996 textbook 'Constructal-theory network of conducting paths' + 2000 'Shape and Structure from Engineering to Nature' Cambridge classic 324 pages + 2008 'Design with Constructal Theory' Wiley + flow architecture optimization access-time minimization + tree-shaped networks bifurcating + nature inspired bronchial tree river basin circulatory system. USA: Duke + MIT + Purdue.

What regimes does Be classify?

Be<1e6 low viscous-thermal dominant + microchannel + lab-on-chip; 1e6<Be<1e9 medium typical forced convection + HVAC chiller compact; 1e9<Be<1e12 high industrial heat exchangers + petrochemical reactors; Be>1e12 very high high-pressure jet + turbomachinery + supersonic. USA: Intel microchannel + Carrier HVAC + ExxonMobil industrial + GE Aerospace jet engines.

Who is Adrian Bejan?

Adrian Bejan (Galati Romania 1948 - Duke USA emeritus) MIT BSc 1971 + MS 1972 + ScD 1975 + Berkeley 1975-1976 + University Colorado Boulder 1976-1984 + Duke University Durham NC J A Jones Distinguished Professor Mech Eng 1984-2014 emeritus + 30 honorary doctorates Romania Belgium Italy Israel; 1982 'Entropy Generation Through Heat and Fluid Flow' Wiley + 1995 'Convection Heat Transfer' Wiley + 1996 constructal theory Int J HMT 40:799 classic + ASME Heat Transfer Memorial Award 1994.

What is entropy generation minimization?

Entropy generation minimization EGM Adrian Bejan Duke + 1982 textbook 'Entropy Generation Through Heat and Fluid Flow' Wiley classic 248 pages + 1996 'Entropy Generation Minimization' CRC Press 362 pages + thermodynamic optimization heat exchanger sizing + S_gen = Q*dT/T^2 + dP*v/T heat transfer + fluid friction; Paoletti-Rispoli-Sciubba Sapienza Roma 1989 alternative Be entropy ratio. USA: NIST + EPRI + ASHRAE.

What are Bejan applications?

Applications: HVAC heat exchangers Carrier 53000 + Trane 39000 + Daikin Applied + Alfa Laval USA 18000 plate world-leading + Modine Racine; petrochemical ExxonMobil 62000 + Chevron 45000 + Honeywell UOP 5000 FCC + Air Products 21000; turbomachinery jet GE Aerospace 50000 GE9X GE90 + Pratt&Whitney 38000 PW1000G + RTX + Boeing + Lockheed; pumps Flowserve 16000 + Goulds ITT 8000 + Sulzer USA 13000 + Xylem 18000 Lowara; microchannel Intel 131000 + NVIDIA 30000 + AMD + Vertiv data center; constructal Duke Bejan emeritus 30 books + Purdue Garimella Heat Transfer Lab world-leading.

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