How to calculate the Sherwood number Sh mass transfer

Calculate Sherwood number Sh = h_m*L/D convective mass transfer coefficient analog Nusselt heat. Dittus-Boelter Sh=0.023*Re^0.8*Sc^(1/3). Frössling sphere 2+0.6*Re^0.5*Sc^(1/3). Thomas Sherwood MIT 1937 textbook absorption extraction.

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How to calculate the Sherwood number (Sh): dimensionless number convective mass transfer ratio. Formula: Sh = h_m * L / D. Sh=2 sphere lower bound diffusion-only; Sh>>1 convection dominates.

Correlations: Dittus-Boelter mass analog Sh = 0.023*Re^0.8*Sc^(1/3); Pohlhausen plate Sh = 0.332*Re^0.5*Sc^(1/3); Frössling 1938 sphere Sh = 2 + 0.6*Re^0.5*Sc^(1/3) classic droplet evaporation; Ranz-Marshall 1952 confirmed. Heat-mass analogy Chilton-Colburn 1934 j-factor: Sh/(Re*Sc^(1/3)) = Nu/(Re*Pr^(1/3)) = f/8.

Thomas K. Sherwood (USA 1903-1976) MIT Cambridge MA 1937 textbook 'Absorption and Extraction' McGraw-Hill founder mass transfer engineering. MIT undergraduate ChE 1924, PhD 1929 Warren Lewis advisor, faculty 1929-1969 Dean School Engineering 1946-1952. Sherwood-Pigford 1952 + Sherwood-Pigford-Wilke 1975 'Mass Transfer' definitive textbook.

Essential tool for: distillation; gas absorption CO2 capture; spray drying food; chromatography; bioreactor fermentation; pharma drug delivery MDI inhalers; microfluidics lab-on-chip; combustion spray engines; electrochemistry fuel cells.

USA chemical engineering research: MIT Cambridge MA 1861 Sherwood legacy mass transfer founder + Bird Stewart Lightfoot Wisconsin 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; UPenn Philadelphia 1740; UDel Newark DE 1743 CCEI Catalysis; UMinnesota Minneapolis 1851; Texas A&M College Station 1876; UWisconsin Madison 1848 Bird Stewart Lightfoot legacy; UTexas Austin 1883; Northwestern Evanston IL 1851; UMichigan Ann Arbor 1817.

USA petrochemical refineries: ExxonMobil Irving TX 1999 70000 oil gas largest US oil; Chevron San Ramon CA 1879 47000; ConocoPhillips Houston TX 2002; Phillips 66 Houston TX 1917; Marathon Petroleum Findlay OH 1887; Valero San Antonio TX 1980; Dow Chemical Midland MI 1897 36000; DuPont Wilmington DE 1802 22000; 3M St Paul MN 1902 95000; Eastman Chemical Kingsport TN 1920; Honeywell UOP Des Plaines IL 1914 catalysts adsorbents distillation.

USA pharma drug delivery: Pfizer NYC 1849 88000; Merck Rahway NJ 1891 71000; Johnson & Johnson New Brunswick NJ 1886 130000; Eli Lilly Indianapolis IN 1876 41000; Catalent Somerset NJ 2007 18000 drug delivery; 3M St Paul MN 1902 MDI metered dose inhalers historical Solvay; Becton Dickinson Franklin Lakes NJ 1897 75000 syringes.

USA bioreactor fermentation: Anheuser-Busch InBev St Louis MO 1852 beer; Genentech South San Francisco CA 1976 first biotech 12000; Amgen Thousand Oaks CA 1980 24000 biotech; Biogen Cambridge MA 1978; Regeneron Tarrytown NY 1988; Moderna Cambridge MA 2010 mRNA.

USA microfluidics: Whitesides Lab Harvard PDMS pioneer 1990s; Quake Lab Stanford; MIT Microsystems; Sandia National Labs Albuquerque NM 1949.

USA food spray drying: General Mills Minneapolis MN 1866 35000; PepsiCo Purchase NY 1965 309000; Kraft Heinz Pittsburgh Chicago 2015 39000; Tetra Pak USA Vernon Hills IL; SPX Flow Charlotte NC 2015 spray dryers; GEA Group Hudson WI Chicago IL equipment.

Sherwood formula

Sh = h_m*L/D. Dimensionless. Heat-mass analogy: Nu(heat)→Sh(mass), Pr→Sc, k_fluid→D. USA: MIT 1861 Sherwood legacy + BSL textbook 1960; Notre Dame 1842 Incropera 1981; Wisconsin Madison 1848 Bird Stewart Lightfoot.

History and context

Thomas K. Sherwood USA 1903-1976 MIT 1937 textbook 'Absorption and Extraction' founder mass transfer. Sherwood-Pigford 1952 + Sherwood-Pigford-Wilke 1975 'Mass Transfer' McGraw-Hill 685 pages definitive. Adolf Fick 1855 Marburg Fick law. Albert Einstein 1905 Stokes-Einstein. Frössling Sweden 1938 droplet correlation. Ranz-Marshall 1952 Chemical Engineering Progress 48:141 confirmed sphere. Chilton-Colburn 1934 Industrial Engineering Chemistry 26:1183 j-factor heat-mass-friction analogy. Wilke-Chang 1955 liquid diffusivity correlation. USA universities chemical engineering: MIT Cambridge MA 1861 Sherwood faculty 1929-1969 Dean School Engineering 1946-1952; 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; UPenn Philadelphia 1740; UDel Newark DE 1743; UMinnesota Minneapolis 1851; Texas A&M College Station 1876; UWisconsin Madison 1848 Bird Stewart Lightfoot legacy; UTexas Austin 1883; Northwestern Evanston IL 1851; UMichigan Ann Arbor 1817; UCSB Santa Barbara CA 1909. USA petrochemical refineries distillation: ExxonMobil Irving TX 1999 70000; Chevron San Ramon CA 1879 47000; ConocoPhillips Houston TX 2002; Phillips 66 Houston TX 1917; Marathon Petroleum Findlay OH 1887; Valero Energy San Antonio TX 1980; Dow Chemical Midland MI 1897 36000;

DuPont Wilmington DE 1802 22000 Tyvek Kevlar Teflon; 3M St Paul MN 1902 95000 Post-it Scotch; Eastman Chemical Kingsport TN 1920; Honeywell UOP Des Plaines IL 1914 catalysts adsorbents distillation. USA pharma drug delivery MDI: Pfizer NYC 1849 88000; Merck Rahway NJ 1891 71000; Johnson & Johnson New Brunswick NJ 1886 130000; Eli Lilly Indianapolis IN 1876 41000 diabetes; Bristol Myers Squibb NYC 1989 32000; AbbVie North Chicago IL 2013 Humira; Catalent Somerset NJ 2007 18000 drug delivery; 3M St Paul MN 1902 95000 MDI metered dose inhalers historical Solvay; Becton Dickinson Franklin Lakes NJ 1897 75000 syringes; Medtronic Minneapolis MN 1949 95000 medical devices. USA bioreactor fermentation: Anheuser-Busch InBev St Louis MO 1852 16000 beer Budweiser; Coors Brewing Golden CO 1873 + Miller MillerCoors; Boston Beer Boston MA 1984 Sam Adams; Sierra Nevada Chico CA 1979; Genentech South San Francisco CA 1976 12000 first biotech; Amgen Thousand Oaks CA 1980 24000; Biogen Cambridge MA 1978 multiple sclerosis; Regeneron Tarrytown NY 1988 11000; Moderna Cambridge MA 2010 mRNA Spikevax; Vertex Pharmaceuticals Boston MA 1989 cystic fibrosis. USA microfluidics: George Whitesides Harvard PDMS soft lithography pioneer 1990s; Stephen Quake Stanford microfluidic devices; David Beebe Wisconsin BME microfluidics; Don Ingber Harvard organ-on-chip; Daniel Chiu Washington microfluidics; MIT Microsystems Technology Lab; Sandia National Labs Albuquerque NM 1949.

USA food spray drying: General Mills Minneapolis MN 1866 35000 cereals milk powder; PepsiCo Purchase NY 1965 309000 spray drying instant; Kraft Heinz Pittsburgh Chicago 2015 39000; Tetra Pak USA Vernon Hills IL Lund Sweden parent; SPX Flow Charlotte NC 2015 spray dryers food beverage; GEA Group Hudson WI Chicago IL spray drying equipment industrial.

Sherwood values table

  • Sh = h_m*L/D: definition
  • Sh = 2: sphere lower bound
  • Sh > 10: convection dominates
  • Tube laminar C_wall constant: Sh = 3.66
  • Tube laminar j_wall constant: Sh = 4.36
  • Dittus-Boelter mass: 0.023*Re^0.8*Sc^(1/3)
  • Sieder-Tate viscosity: (mu/muw)^0.14
  • Plate laminar local: 0.332*Re^0.5*Sc^(1/3)
  • Plate laminar average: 0.664*Re^0.5*Sc^(1/3)
  • Frössling sphere: 2 + 0.6*Re^0.5*Sc^(1/3)
  • Ranz-Marshall sphere: 2 + 0.6*Re^0.5*Sc^(1/3)
  • j_M = Sh/(Re*Sc^(1/3)): Chilton-Colburn
  • j_M = j_H = f/8: heat-mass-friction analogy
  • Sherwood 1937 textbook: 'Absorption and Extraction'
  • Sherwood-Pigford 1952: 2nd edition
  • Sherwood-Pigford-Wilke 1975: 'Mass Transfer'
  • Frössling 1938: droplet evaporation
  • Ranz-Marshall 1952: confirmed
  • Chilton-Colburn 1934: j-factor
  • Wilke-Chang 1955: liquid diffusivity
  • USA MIT 1861: Sherwood faculty 1929-1969
  • USA Wisconsin Madison 1848: BSL legacy
  • USA UDel Newark 1743: CCEI Catalysis
  • USA Notre Dame 1842: Incropera
  • USA ExxonMobil 1999: 70000 oil largest
  • USA Dow Chemical 1897: 36000
  • USA DuPont 1802: 22000
  • USA Honeywell UOP 1914: catalysts
  • USA Pfizer 1849: 88000
  • USA Merck 1891: 71000
  • USA 3M St Paul 1902: 95000 MDI inhalers
  • USA Catalent 2007: 18000 drug delivery
  • USA Genentech 1976: 12000 first biotech
  • USA Amgen 1980: 24000 biotech
  • USA Whitesides Harvard: PDMS microfluidics
  • USA SPX Flow 2015: Charlotte NC spray dryers
  • USA GEA Group Hudson WI: spray drying equipment

Numerical examples

Example 1 - Petroleum distillation tube. D=0.05m Re=50000 Sc=600. Dittus-Boelter Sh=1115. h_m=2.2e-5 m/s. USA: ExxonMobil Irving TX 1999 70000; Chevron San Ramon 1879; Honeywell UOP Des Plaines IL 1914.

Example 2 - Coffee spray drying drop. D=100 µm Re=100 Sc=0.6 air-water vapor. Frössling Sh=2+0.6*10*0.843=7.06. USA: PepsiCo 1965 309000 spray drying; General Mills 1866; SPX Flow 2015.

Example 3 - Plate evaporation water. L=1m Re_x=5000 Sc=0.6. Pohlhausen Sh_x=0.332*70.7*0.843=19.8 local. Average 39.6. USA: Tetra Pak Vernon Hills IL packaging; GEA Group Hudson WI spray dryers.

Example 4 - Combustion fuel droplet. D=200 µm Re=1000 Sc=2. Frössling Sh=2+0.6*31.6*1.26=25.9 high. USA: Cummins Columbus IN 1919 75000 diesel; BorgWarner Auburn Hills MI 1880 50000 fuel injection; Caterpillar Peoria IL 1925 110000.

Example 5 - Bioreactor O2 fermentation beer. Bubble D=1 mm Re=500 Sc=480. Frössling Sh=2+0.6*22.4*7.83=107 high. h_m=2.25e-4 m/s. USA: Anheuser-Busch St Louis 1852 16000; Genentech SSF 1976 12000; Amgen Thousand Oaks 1980 24000.

Limitations

  1. Specific geometry: each correlation Sh valid for geometry. Errors outside.
  2. Re Sc validity range: Dittus-Boelter Re 1e4-1.6e5 Sc 0.7-2500; Frössling Re 1-200.
  3. Boundary conditions: C_wall constant vs flux constant give different Sh.
  4. Entry effects: Sh varies L/D fully developed.
  5. Property variation: D varies with T (Arrhenius) and composition (Stefan-Maxwell).
  6. Diluted solute: dilute. Concentrated requires Stefan-Maxwell.
  7. Phase change: evaporation condensation separate Sherwood (Higbie penetration).
  8. Roughness effects: rough surfaces enhance 10-50%.
  9. Compressibility: high-speed Mach>0.3 corrections.
  10. Free stream turbulence: Tu>5% impacts laminar plate.

Common errors

Sh local vs average: Sh_x local; Sh_L average integrated. Plate laminar Sh_L = 2*Sh_L (constant).

Characteristic length L: tubes D hydraulic diameter; plate L flow direction; spheres D diameter; bubble D diameter.

D mass diffusivity vs k thermal: Sh uses D mass species. NOT k thermal conductivity.

Sc 1/3 exponent: Dittus-Boelter mass standard Sc^(1/3); Nusselt heat uses Pr^0.4 heating or Pr^0.3 cooling. Different.

Sphere Sh=2 limit: Re→0 Sh=2 lower bound diffusion-only. Sh<2 physically impossible.

Practical applications

  1. Distillation: USA ExxonMobil 70000 + Chevron 47000 + ConocoPhillips Houston TX + Honeywell UOP Des Plaines IL.
  2. CO2 capture: Sandia Albuquerque NM 1949; LBNL Berkeley CA 1931; Argonne IL 1946.
  3. Chromatography: USA Waters Corp Milford MA 1958 HPLC; Agilent Santa Clara CA 1999.
  4. Spray drying food: USA General Mills 1866 + PepsiCo 1965 + Kraft Heinz 2015 + SPX Flow Charlotte NC 2015 + GEA Group Hudson WI.
  5. Bioreactor: Anheuser-Busch 1852 + Genentech SSF 1976 12000 + Amgen 1980 24000 + Biogen 1978 + Regeneron 1988 + Moderna 2010.
  6. Pharma MDI: 3M St Paul MN 1902 95000 + Catalent Somerset NJ 2007 + Pfizer NYC 1849 + Merck Rahway NJ 1891 71000.
  7. Microfluidics: Whitesides Lab Harvard PDMS pioneer; Quake Stanford; MIT Microsystems.
  8. Combustion fuel injectors: Cummins Columbus IN 1919 75000 + BorgWarner Auburn Hills MI 1880 + Caterpillar Peoria IL 1925.
  9. Electrochemistry fuel cells: Tesla Austin TX 2003; Plug Power Latham NY 1997; Bloom Energy San Jose CA 2001.
  10. Cosmetics emulsions: Procter & Gamble Cincinnati OH 1837 99000; Estée Lauder NYC 1946.

Updated July 2026

Frequently Asked Questions

What is Sherwood number?

Dimensionless number convective mass transfer ratio: Sh = h_m*L/D. Sh=2 sphere lower bound; Sh>>1 convection dominates. Correlations: Dittus-Boelter Sh=0.023*Re^0.8*Sc^(1/3); Frössling sphere Sh=2+0.6*Re^0.5*Sc^(1/3). Thomas Sherwood 1937 MIT textbook 'Absorption and Extraction' founder mass transfer. USA: MIT Cambridge MA 1861 Sherwood legacy 1929-1969; Wisconsin Madison 1848 BSL textbook 1960; Notre Dame 1842 Incropera.

How to calculate Sherwood?

Sh = h_m*L/D. Procedure: (1) h_m mass transfer coefficient (m/s); (2) L characteristic length; (3) D diffusivity solvent-solute pair. Correlations: Dittus-Boelter Sh=0.023*Re^0.8*Sc^(1/3); Frössling sphere Sh=2+0.6*Re^0.5*Sc^(1/3). Example: tube Re=50000 Sc=600 → Sh=1115. USA: MIT Cambridge MA 1861; ExxonMobil Irving TX 1999 70000; Honeywell UOP Des Plaines IL 1914.

Who defined Sherwood?

Thomas K. Sherwood (USA 1903-1976) MIT Cambridge MA 1937 textbook 'Absorption and Extraction' McGraw-Hill founder mass transfer engineering. MIT undergraduate 1924, PhD 1929 Warren Lewis advisor, faculty 1929-1969 Dean School Engineering 1946-1952. Sherwood-Pigford 1952 + Sherwood-Pigford-Wilke 1975 'Mass Transfer' definitive textbook 685 pages. Frössling Sweden 1938 sphere; Ranz-Marshall 1952 confirmed; Chilton-Colburn 1934 j-factor. USA: MIT 1861; Wisconsin Madison BSL 1960.

What is Frössling correlation?

Classic sphere correlation droplet evaporation: Sh = 2 + 0.6 * Re^0.5 * Sc^(1/3). Nils Frössling Sweden 1908-1991 1938 paper Beiträge zur Geophysik 52:170 droplet evaporation. Walter Ranz + W.R. Marshall 1952 Chemical Engineering Progress 48:141 confirmed Ranz-Marshall. Sh=2 lower bound Re→0 sphere diffusion-only. Example Re=100 Sc=1: Sh=8. USA spray drying: PepsiCo Purchase NY 1965 309000 instant; General Mills Minneapolis MN 1866; SPX Flow Charlotte NC 2015 spray dryers.

Difference Sherwood vs Nusselt?

Sherwood Sh = h_m*L/D mass transfer (analog Nusselt heat). Nusselt Nu = h*L/k_fluid heat transfer. Heat-mass analogy Chilton-Colburn 1934 j-factor: Sh/(Re*Sc^(1/3)) = Nu/(Re*Pr^(1/3)) = f/8. Sh substitutes Nu, Sc substitutes Pr in correlations. USA: MIT Cambridge MA 1861; Stanford Palo Alto CA 1925; Caltech 1891; UC Berkeley 1868; Notre Dame 1842 Incropera 1981.

What are Sherwood applications?

Applications: distillation USA ExxonMobil Irving TX 1999 70000 + Chevron 1879 47000 + Honeywell UOP IL 1914; spray drying food General Mills 1866 35000 + PepsiCo 1965 309000 + SPX Flow Charlotte NC 2015 + GEA Group Hudson WI; pharma MDI 3M St Paul 1902 95000 + Catalent Somerset NJ 2007 + Pfizer 1849 88000; bioreactor Anheuser-Busch 1852 + Genentech 1976 12000 + Amgen 1980 24000; microfluidics Whitesides Harvard + Quake Stanford.

How to choose Sh correlation?

Choose: (1) flow regime laminar/turbulent; (2) geometry tube/plate/sphere; (3) Re Sc range; (4) boundary conditions C_wall vs flux constant; (5) entry effects L/D. Standard: Dittus-Boelter Re 1e4-1.6e5 Sc 0.7-2500 turbulent tube; Pohlhausen Sh=0.332*Re^0.5*Sc^(1/3) plate laminar; Frössling Sh=2+0.6*Re^0.5*Sc^(1/3) sphere. USA: MIT Cambridge MA 1861 Sherwood faculty; Wisconsin Madison BSL textbook 1960; Notre Dame 1842 Incropera 7 editions.

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