How to calculate Wagner number Wa current distribution

Calculate Wagner number Wa = (kappa*R_pol)/L ratio kinetic to ohmic resistance in electrochemistry. Wa<<1 primary current ohmic; Wa>>1 secondary uniform. Carl Wagner Germany 1901-1977 + 1951. Tesla + GM + Ford battery + Plug Power fuel cells + Olin chlor-alkali applications.

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Wagner number Wa dimensionless ratio of kinetic (polarization) to ohmic (electrolyte) resistance in electrochemistry determining current distribution on electrode: Wa = (kappa * R_pol) / L. kappa electrolyte conductivity S/m (H2SO4 0.5M 30; KOH 30% 80; Li-ion 1; Nafion PEM 0.1; sea water 5); R_pol polarization resistance Ohm*m^2 = b/(2.303*i) Tafel; L characteristic length m.

Regimes: Wa<0.1 primary current ohmic dominant + non-uniform geometry; 0.1-1 mixed; 1-10 secondary current kinetic uniform; Wa>10 strong kinetic excellent throwing power.

Tafel kinetics: eta = a + b*log(i) Tafel slope 60-120 mV/decade.

Throwing power: (i_min/i_max-1)/(i_avg/i_max-1). History: Carl Wagner (Leipzig Germany 1901 - Gottingen 1977) Universitat Leipzig PhD 1924 + Hamburg 1928-1933 + Jena 1933-1945 + Gottingen 1953-1972 emeritus + Max-Planck-Institut Physikalische Chemie director 1953-1972; 1951 paper Plating Surface Finish 38:997 'Theoretical Analysis of Current Density Distribution' classic Wagner number eponym + 1924 'Thermodynamics of Solids' + Wagner-Schottky defects + 1933 oxidation parabolic + Wagner-Hauffe rule semiconductor + Wagner Award ECS.

Julius Tafel Switzerland 1862-1918; 1905 Tafel equation classic.

John Newman (USA 1938-) Berkeley; 'Electrochemical Systems' 1991 + 4th ed 2021 + Doyle-Fuller-Newman Li-ion battery model 1993.

Allen Bard UT Austin 1933-2024; 'Electrochemical Methods' 1980 with Faulkner classic 850 pages + Wolf Prize 2008 + Priestley Medal 2013. Legacy: Bard-Faulkner 2001 + Newman 2021 + Pletcher-Walsh + Hamann-Hamnett-Vielstich 2007 + Fontana-Greene 'Corrosion Engineering' 4th ed 2018 + ASM Vol 13.

USA research Wagner + electrochemistry + batteries + fuel cells: NIST Gaithersburg MD 3500 + Standard Reference Materials + Battery Test Center; NREL Golden CO 2900 + battery + hydrogen + fuel cells; ORNL Oak Ridge TN 6500 + battery research + nuclear; Sandia Albuquerque NM 14000 + battery + hydrogen; LBNL Berkeley CA 4400 + Battery Lab + Tesla Berkeley collaboration; Argonne IL 3400 + Joint Center for Energy Storage Research JCESR + battery world-leading + Argonne Battery Test Lab; PNNL Richland WA 4500; NETL Pittsburgh + Morgantown 1500 + fuel cells SOFC; EPRI Palo Alto 600; USGS Reston 8000.

USA universities Wagner + battery + fuel cells: MIT Cambridge MA 11700 + Mech Eng + ChemE Yet-Ming Chiang battery world-leading + Donald Sadoway battery + George Crabtree JCESR + Yang Shao-Horn battery; Stanford CA 17000 + Mech Eng Yi Cui battery world-leading + Robert Dunn-Rankin combustion + Will Chueh battery; Caltech Pasadena 2400; UC Berkeley 45000 + ChemE John Newman heritage + 1991 Electrochemical Systems classic + Bryan McCloskey battery + Vassily Vassilakis catalysis; Princeton 8000 + Mech Eng + Frank van Roy + Howard Stone microfluidics; Cornell Ithaca NY 25000 + ChemE Lynden Archer battery + Hector Abruna electrochemistry; Georgia Tech Atlanta 36000 + ChemE; UIUC Urbana 56000 + ChemE; Penn State 50000; UT Austin TX 51000 + ChemE Allen Bard heritage + 1980 Electrochemical Methods classic + John Goodenough Nobel 2019 Li-ion + Arumugam Manthiram battery + Bard-Faulkner; Univ Michigan Ann Arbor 50000 + ChemE Charles Monroe battery + Levi Thompson; Northwestern Evanston IL 22000 + ChemE Mark Hersam + Jiaxing Huang battery; Purdue 47000 + ChemE Vilas Pol battery + Jeff Greeley; Wisconsin-Madison 49000 + ChemE Marc Anderson; Minnesota 51000 + ChemE;

UCLA 45000 + Yang Yang battery + Bruce Dunn battery; UC San Diego 41000 + Mech Eng Y Shirley Meng battery world-leading + Sustainable Power and Energy Center; UC Davis 40000 + Battery Lab Davis; Texas A&M 70000 + Mech Eng + Choongho Yu battery; Univ Pittsburgh 28000 + ChemE Spencer Pringle + electrochemistry; Univ Pennsylvania 25000 + ChemE; Carnegie Mellon Pittsburgh 14000 + ChemE Venkat Viswanathan battery + Mech Eng; Drexel Philadelphia PA 24000 + Materials Science Yury Gogotsi battery world-leading + Drexel Nanomaterials Institute.

USA battery industry Li-ion + Tesla + Ford + GM: Tesla Inc Palo Alto CA + Austin TX 1903 127000 + Tesla Energy + Powerwall Megapack + Gigafactory Nevada Sparks + Texas + Berlin + battery world-leading; Ford Motor Dearborn MI 1903 175000 + Ford F-150 Lightning + Mustang Mach-E + BlueOval City + Marshall MI battery plant; General Motors GM Detroit MI 1908 167000 + GM Ultium battery + Hummer EV + Cadillac Lyriq + LG Energy Solution joint venture Ultium Cells; Stellantis North America Auburn Hills MI 50000 + Jeep Ram + Dodge + Chrysler + Pacifica BEV; Rivian Plymouth MI 2009 14000 + Rivian R1S + R1T + Amazon Delivery Van EDV; Lucid Motors Newark CA 2007 8000 + Lucid Air + Gravity SUV; Polestar Sweden-USA 1000; Fisker Manhattan Beach CA 1500; QuantumScape San Jose CA 2010 1000 + solid state battery world-leading + Volkswagen partnership; Solid Power Louisville CO 2011 200 + solid state batteries Ford BMW partnerships; Sila Nanotechnologies Alameda CA 2011 600 + silicon anode + BMW partnership; Group14 Technologies Woodinville WA 2015 100 + silicon anode SCC55 + Porsche partnership.

USA Li-ion battery cell manufacturers: LG Energy Solution USA 5000 + GM Ultium Cells joint venture + Honda joint venture + Stellantis joint venture + Lordstown Ohio + Spring Hill Tennessee + Lansing Michigan; Panasonic Energy Reno NV + Kansas City KS 4000 + Tesla joint venture + Gigafactory Nevada Sparks; Samsung SDI USA 2000 + Stellantis joint venture + GM joint venture; SK On USA Atlanta GA + Hyundai 4000 + Hyundai joint venture + Ford joint venture BlueOval; CATL Pomona CA office; BYD USA Lancaster CA 1500 + electric buses; EnerSys Reading PA 1888 11000; Duracell Bethel CT 1924 + Berkshire Hathaway 7000.

USA fuel cells PEM SOFC industry: Plug Power Latham NY 1997 2400 + PEM fuel cells + green hydrogen + GenDrive forklift + Amazon Walmart customers; Ballard Power Systems Vancouver BC + Bend OR 1500 + PEM fuel cells; Bloom Energy San Jose CA 2001 2000 + SOFC fuel cells + Bloom Server; FuelCell Energy Danbury CT 1969 600 + MCFC molten carbonate; 3M St Paul MN 92000 + Nafion membranes ion-exchange world-leader; Chemours Wilmington DE 6000 + Nafion DuPont spin-off; Cummins Cellcentric Columbus IN 73000 + fuel cells trucks Daimler joint venture; Nuvera Fuel Cells Billerica MA 200.

USA chlor-alkali electrolysis industry: Olin Corporation Clayton MO 1892 8500 + chlor-alkali + Niagara Falls NY + Freeport TX + winchester ammunition; Westlake Chemical Houston TX 1986 16000 + chlor-alkali; Occidental Petroleum Houston TX 1920 13000 + chlor-alkali OxyChem; Tronox Stamford CT 1916 4500 + chlor-alkali; Formosa Plastics USA Point Comfort TX 4000 + chlor-alkali; SABIC Innovative Plastics 4000.

USA semiconductor Cu plating BEOL: Intel Santa Clara CA 1968 131000 + Cu plating BEOL Back-End-of-Line + 8/22/28 nm; TSMC USA Phoenix AZ 11000 + 5/3 nm fabs Cu plating; GlobalFoundries Malta NY 13000; Micron Technology Boise ID 1978 49000 + DRAM NAND + Cu plating; Texas Instruments Dallas TX 33000; Analog Devices 24000; NXP Semiconductors USA 31000; Onsemi Phoenix AZ 25000; Applied Materials Santa Clara CA 1967 33000 + Cu plating equipment + Endura Producer; Lam Research Fremont CA 1980 18000 + Cu plating tool ECD Sabre.

USA cathodic protection oil&gas: ExxonMobil Irving TX 62000 + offshore + cathodic protection; Chevron 45000; Halliburton Houston TX 50000; Schlumberger 90000; Baker Hughes 60000; NOV 36000; Williams Companies Tulsa OK 1908 5500 + natural gas pipelines + cathodic protection; Kinder Morgan Houston TX 11000 + pipelines.

Wagner formula

Definition: Wa = (kappa*R_pol)/L.

Tafel R_pol: R_pol = b/(2.303*i) + Tafel 60-120 mV/decade.

Throwing power: (i_min/i_max-1)/(i_avg/i_max-1).

History

Carl Wagner (Leipzig 1901 - Gottingen 1977) Leipzig PhD 1924 + Hamburg 1928-1933 + Jena + Gottingen 1953-1972 emeritus + MPI Physikalische Chemie director + 1951 Plating Surface Finish 38:997 + Wagner-Schottky + 1933 oxidation + Wagner-Hauffe + Wagner Award ECS. Tafel Switzerland 1862-1918 + 1905 classic. Newman Berkeley 1991 'Electrochemical Systems' + Doyle-Fuller-Newman Li-ion 1993. Bard UT Austin 1933-2024 + 1980 'Electrochemical Methods' + Wolf Prize 2008. Bard-Faulkner 2001 + Newman 2021 + Fontana-Greene 2018.

USA research: NIST 3500; NREL 2900; ORNL 6500; Sandia 14000; LBNL 4400; Argonne 3400 JCESR battery world-leading; PNNL 4500; NETL 1500; EPRI 600; USGS 8000.

USA universities: MIT 11700 Chiang battery world-leading + Sadoway + Crabtree JCESR + Shao-Horn; Stanford 17000 Cui battery world-leading + Chueh; Caltech 2400; UC Berkeley 45000 Newman heritage 1991 classic + McCloskey; Princeton 8000; Cornell 25000 Archer + Abruna; Georgia Tech 36000; UIUC 56000; Penn State 50000; UT Austin 51000 Bard heritage 1980 + John Goodenough Nobel 2019 Li-ion + Manthiram + Bard-Faulkner; Michigan 50000 Monroe; Northwestern 22000 Hersam Huang battery; Purdue 47000 Pol Greeley; Wisconsin-Madison 49000; Minnesota 51000; UCLA 45000 Yang battery + Dunn; UC San Diego 41000 Meng battery world-leading; UC Davis 40000; Texas A&M 70000 Yu battery; Carnegie Mellon 14000 Viswanathan; Drexel 24000 Gogotsi battery world-leading.

USA battery: Tesla 127000 Powerwall Megapack Gigafactory; Ford 175000 F-150 Lightning Mustang Mach-E; GM 167000 Ultium Hummer EV; Stellantis NA 50000; Rivian 14000; Lucid 8000; QuantumScape 1000 solid state world-leading + VW partnership; Solid Power 200 Ford BMW; Sila 600 silicon anode BMW; Group14 100 Porsche; LG Energy Solution USA 5000 GM Honda Stellantis; Panasonic Reno 4000 Tesla Gigafactory; Samsung SDI 2000; SK On 4000 Hyundai Ford BlueOval; EnerSys 11000; Duracell Berkshire 7000.

USA fuel cells: Plug Power 2400 PEM Amazon Walmart; Ballard 1500; Bloom Energy 2000 SOFC; FuelCell Energy 600 MCFC; 3M 92000 Nafion world-leader; Chemours 6000 Nafion DuPont; Cummins Cellcentric 73000 Daimler; Nuvera 200.

USA chlor-alkali: Olin 8500; Westlake 16000; Occidental OxyChem 13000; Tronox 4500; Formosa 4000.

USA semiconductor Cu plating: Intel 131000; TSMC 11000; GlobalFoundries 13000; Micron 49000; TI 33000; Applied Materials 33000 Cu plating tool; Lam Research 18000 ECD Sabre.

USA cathodic protection: ExxonMobil 62000; Chevron 45000; Halliburton 50000; Schlumberger 90000; Baker Hughes 60000; NOV 36000; Williams 5500; Kinder Morgan 11000.

Wagner values table

  • Wa = (kappa*R_pol)/L
  • R_pol = b/(2.303*i): Tafel
  • Tafel slope 60-120 mV/decade
  • Wa<0.1 primary: ohmic
  • 1-10 secondary: kinetic uniform
  • Wa>10 strong: throwing power
  • H2SO4 0.5M kappa=30
  • KOH 30% kappa=80
  • Li-ion kappa=1
  • Nafion PEM 0.1
  • Wagner Germany 1901-1977: MPI
  • Wagner 1951 Plating Surface Finish
  • Tafel 1905
  • Newman Berkeley 1991
  • Bard Austin 1980: Wolf Prize 2008
  • USA NIST 3500
  • USA Argonne 3400: JCESR world-leading
  • USA Tesla 127000: Gigafactory
  • USA Ford 175000: F-150 Lightning
  • USA GM Ultium 167000
  • USA QuantumScape solid state world-leading
  • USA Plug Power 2400: PEM
  • USA Bloom Energy 2000: SOFC
  • USA Olin 8500: chlor-alkali
  • USA Intel 131000: Cu plating BEOL
  • USA UT Austin Goodenough Nobel 2019 Li-ion

Numerical examples

Example 1 - Chlor-alkali DSA. Wa=3 secondary uniform. USA: Olin 8500 + Westlake + Occidental OxyChem.

Example 2 - Li-ion battery. Wa=100 strong kinetic. USA: Tesla 127000 Gigafactory + Panasonic Reno + GM Ultium + Ford F-150.

Example 3 - PEM fuel cell. Wa=0.5 mixed. USA: Plug Power 2400 + Ballard + Bloom Energy SOFC + 3M Nafion + Chemours.

Example 4 - Cu plating BEOL. Wa=0.5. USA: Intel 131000 + TSMC Phoenix + Applied Materials Lam Research Cu plating tools.

Example 5 - Cathodic protection pipeline. Wa=5000. USA: ExxonMobil 62000 + Williams 5500 pipelines + Kinder Morgan 11000.

Limits and considerations

  1. Linearized Butler-Volmer.
  2. Tertiary mass transfer.
  3. Convection-diffusion Levich.
  4. Multiple reactions side.
  5. Geometry corner.
  6. Ionic non-uniform Nernst-Planck.
  7. Bubble void fraction.
  8. Membrane Nafion bipolar.
  9. Time-dependent EIS.
  10. Temperature Arrhenius.

Common errors

Wa vs Damkohler.

Primary vs secondary tertiary.

R_pol Tafel definition.

Throwing power.

Kappa ionic vs electronic.

Practical applications

  1. Chlor-alkali DSA: USA Olin 8500 + Westlake 16000 + Occidental OxyChem + Tronox + Formosa Point Comfort.
  2. Li-ion battery Gigafactory: USA Tesla 127000 Powerwall Megapack Gigafactory Nevada Texas Berlin + Ford 175000 F-150 Lightning Mustang Mach-E BlueOval City + GM Ultium 167000 Hummer Cadillac + Stellantis NA + Rivian + Lucid + Polestar + QuantumScape solid state + Solid Power + Sila silicon anode + Group14 Porsche.
  3. Battery cell manufacturers: USA LG Energy Solution 5000 GM Ultium Cells Honda Stellantis + Panasonic Reno 4000 Tesla + Samsung SDI 2000 + SK On 4000 Hyundai Ford BlueOval + EnerSys 11000 + Duracell 7000.
  4. Fuel cells PEM SOFC: USA Plug Power 2400 Amazon Walmart + Ballard 1500 + Bloom Energy 2000 SOFC + FuelCell Energy 600 MCFC + 3M 92000 Nafion world-leader + Chemours Nafion + Cummins Cellcentric Daimler.
  5. Electroplating Cu BEOL semiconductor: USA Intel 131000 + TSMC Phoenix 11000 + GlobalFoundries Malta NY + Micron Boise 49000 + Applied Materials Santa Clara 33000 Cu plating Endura Producer + Lam Research 18000 ECD Sabre.
  6. Cathodic protection oil&gas: USA ExxonMobil 62000 offshore + Chevron + Halliburton 50000 + Schlumberger + Baker Hughes + NOV + Williams 5500 pipelines + Kinder Morgan 11000.
  7. Solar storage Tesla Powerwall: USA Tesla Energy + Sunrun + SunPower + Enphase Energy + Generac residential.
  8. Universities battery research: MIT Chiang Sadoway Shao-Horn + Stanford Cui Chueh + Berkeley Newman heritage + Cornell Archer Abruna + UT Austin Bard Goodenough Nobel + Drexel Gogotsi + Northwestern Hersam + UCSD Meng + UCLA Yang Dunn + Carnegie Mellon Viswanathan + Argonne JCESR.
  9. EV charging infrastructure: USA Tesla Supercharger + Electrify America Volkswagen + ChargePoint + EVgo + Blink Charging.
  10. Hydrogen production Olin chlor-alkali co-product: USA Olin Westlake Occidental + green hydrogen Plug Power.

Updated July 2026

Frequently Asked Questions

What is the Wagner number?

Wa = (kappa*R_pol)/L dimensionless ratio kinetic to ohmic resistance in electrochemistry determining current distribution. Wa<0.1 primary ohmic geometry; 1-10 secondary kinetic uniform; Wa>10 strong excellent throwing power. Carl Wagner Germany 1901-1977 + 1951 paper Plating Surface Finish 38:997 + Wagner-Schottky defects + Wagner-Hauffe rule. USA: NIST 3500 + Argonne JCESR battery world-leading + Tesla 127000 + Ford F-150 Lightning + GM Ultium + Olin chlor-alkali.

How do I calculate Wagner?

Procedure: (1) Wa = (kappa*R_pol)/L; (2) kappa S/m (H2SO4 30; KOH 80; Li-ion 1; Nafion 0.1); (3) R_pol Ohm*m^2 = b/(2.303*i) Tafel slope 60-120 mV/decade; (4) L characteristic m; (5) regime. Chlor-alkali: Wa=3 secondary. Li-ion: Wa=100 strong kinetic. USA: MIT Chiang + Stanford Cui + Berkeley Newman heritage + UT Austin Bard Goodenough Nobel.

What is primary vs secondary distribution?

Primary distribution Wa->0 ohmic dominant + Laplace + geometry only + corner overpotential infinite + sharp edges. Secondary distribution Wa intermediate linearized Butler-Volmer kinetic + smooth bulk effects. Tertiary Wa large + mass transfer Nernst diffusion + concentration polarization. Carl Wagner 1951 + John Newman Berkeley 'Electrochemical Systems' classic 1991. USA: Olin chlor-alkali + Tesla Gigafactory + Plug Power PEM.

What regimes does Wa classify?

Wa<0.1 primary current ohmic geometry + non-uniform sharp edges; 0.1-1 mixed; 1-10 secondary current kinetic uniform good throwing; Wa>10 strong kinetic nearly uniform excellent throwing power. USA: Olin chlor-alkali + Tesla Li-ion + Plug Power PEM + Intel Cu plating BEOL + Williams pipelines cathodic protection.

Who was Carl Wagner?

Carl Wagner (Leipzig Germany 1901 - Gottingen 1977) Universitat Leipzig PhD 1924 + Hamburg 1928-1933 + Jena 1933-1945 + Gottingen 1953-1972 emeritus + Max-Planck-Institut Physikalische Chemie director 1953-1972 + Wagner Award ECS; 1951 paper Plating Surface Finish 38:997 Wagner number eponym + 1924 'Thermodynamics of Solids' + Wagner-Schottky defects + 1933 oxidation parabolic + Wagner-Hauffe semiconductor doping rule.

What is Tafel kinetics?

Tafel kinetics Julius Tafel Switzerland 1862-1918 Universitat Zurich Wurzburg + 1905 paper Z Phys Chem 50:641 'Uber die Polarisation bei kathodischer Wasserstoffentwicklung' classic + eta = a + b*log(i) + Tafel slope b ~ 60 mV/decade fast + 120 mV/decade slow + Butler-Volmer linearized + i_0 + R_pol = b/(2.303*i). USA: UT Austin Bard Goodenough Nobel 2019 + Berkeley Newman.

What are Wagner applications?

Applications: chlor-alkali DSA USA Olin 8500 + Westlake 16000 + Occidental OxyChem + Tronox + Formosa; Li-ion battery Gigafactory Tesla 127000 + Ford 175000 F-150 Lightning + GM Ultium 167000 + LG Energy Solution + Panasonic Reno + Samsung SDI + SK On + QuantumScape solid state world-leading; fuel cells PEM SOFC Plug Power 2400 + Ballard 1500 + Bloom Energy 2000 + 3M Nafion world-leader + Cummins Cellcentric; electroplating BEOL Intel 131000 + TSMC + Applied Materials + Lam Research; cathodic protection ExxonMobil + Williams 5500 + Kinder Morgan 11000.

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