Kozeny–Carman equation

The Kozeny–Carman equation (or Carman–Kozeny equation or Kozeny equation) is a relation used in the field of fluid dynamics to calculate the pressure drop of a fluid flowing through a packed bed of solids. It is named after Josef Kozeny and Philip C. Carman. The equation is only valid for laminar flow. The equation was derived by Kozeny (1927)[1] and Carman (1937, 1956)[2][3][4] from a starting point of (a) modelling fluid flow in a packed bed as laminar fluid flow in a collection of curving passages/tubes crossing the packed bed and (b) Poiseuille's law describing laminar fluid flow in straight, circular section pipes.

Equation

The equation is given as:[4][5]

${\displaystyle {\frac {\Delta p}{L}}=-{\frac {150\mu }{{\mathit {\Phi }}_{\mathrm {s} }^{2}D_{\mathrm {p} }^{2}}}{\frac {(1-\epsilon )^{2}}{\epsilon ^{3}}}v_{\mathrm {s} }}$

where:

• ${\displaystyle \Delta p}$ is the pressure drop;
• ${\displaystyle L}$ is the total height of the bed;
• ${\displaystyle v_{\mathrm {s} }}$ is the superficial or "empty-tower" velocity;
• ${\displaystyle \mu }$ is the viscosity of the fluid;
• ${\displaystyle \epsilon }$ is the porosity of the bed;
• ${\displaystyle {\mathit {\Phi }}_{\mathrm {s} }}$ is the sphericity of the particles in the packed bed;
• ${\displaystyle D_{\mathrm {p} }}$ is the diameter of the volume equivalent spherical particle.[6]

This equation holds for flow through packed beds with particle Reynolds numbers up to approximately 1.0, after which point frequent shifting of flow channels in the bed causes considerable kinetic energy losses.

This equation can be expressed as "flow is proportional to the pressure drop and inversely proportional to the fluid viscosity", which is known as Darcy's law.[5]

${\displaystyle v_{\mathrm {s} }=-{\frac {\kappa }{\mu }}{\frac {\Delta p}{L}}}$

Combining these equations gives the final Kozeny equation for absolute (single phase) permeability

${\displaystyle \kappa =a{\frac {\epsilon ^{3}D_{\mathrm {p} }^{2}}{(1-\epsilon )^{2}}}}$

• ${\displaystyle \epsilon }$ is the porosity of the bed (or core plug) [fraction]
• ${\displaystyle D_{\mathrm {p} }}$ is average diameter of sand grains [mm = millimeter]
• ${\displaystyle \kappa }$ is absolute (i.e. single phase) permeability [mD = millidarcy]
• ${\displaystyle a}$ is the proportionality and unity factor [mD / mm²]

The combined proportionality and unity factor ${\displaystyle a}$ has typically average value of 0.8E6 /1.0135 from measuring many naturally occurring core plug samples, ranging from high to low clay content, but it may reach a value of 3.2E6 /1.0135 for clean sand. The denominator is included explicitly to remind us that permeability is defined using [atm] as pressure unit while reservoir engineering calculations and reservoir simulations typically use [bar] as pressure unit.

History

The equation was first[7] proposed by Kozeny (1927)[1] and later modified by Carman (1937, 1956).[2][3] A similar equation was derived independently by Fair and Hatch in 1933 [8]. A comprehensive review of other equations has been published [9]

See also

• Fractionating column
• Random close pack
• Raschig ring
• Ergun equation

References

1. J. Kozeny, "Ueber kapillare Leitung des Wassers im Boden." Sitzungsber Akad. Wiss., Wien, 136(2a): 271-306, 1927.
2. P.C. Carman, "Fluid flow through granular beds." Transactions, Institution of Chemical Engineers, London, 15: 150-166, 1937.
3. P.C. Carman, "Flow of gases through porous media." Butterworths, London, 1956.
4. Fluid Mechanics, Tutorial No. 4: Flow through porous passages (PDF)
5. McCabe, Warren L.; Smith, Julian C.; Harriot, Peter (2005), Unit Operations of Chemical Engineering (seventh ed.), New York: McGraw-Hill, pp. 163–165, ISBN 0-07-284823-5
6. McCabe, Warren L.; Smith, Julian C.; Harriot, Peter (2005), Unit Operations of Chemical Engineering (seventh ed.), New York: McGraw-Hill, pp. 188–189, ISBN 0-07-284823-5
7. Robert P. Chapuis and Michel Aubertin, "PREDICTING THE COEFFICIENT OF PERMEABILITY OF SOILS USING THE KOZENY-CARMAN EQUATION", Report EPM–RT–2003-03, Département des génies civil, géologique et des mines; École Polytechnique de Montréal, January 2003 https://publications.polymtl.ca/2605/1/EPM-RT-2003-03_Chapuis.pdf (accessed 2011-02-05)
8. G.M. Fair, L.P. Hatch, Fundamental factors governing the streamline flow of water through sand, J. AWWA 25 (1933) 1551–1565.
9. E. Erdim, Ö. Akgiray and İ. Demir, A revisit of pressure drop-flow rate correlations for packed beds of spheres, Powder Technology Volume 283, October 2015, Pages 488-504