Kozeny–Carman equation

The Kozeny–Carman equation (or Carman-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 and Carman (see ^[1]) 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:^[1][2]

${\displaystyle {\frac {\Delta p}{L}}={\frac {180\mu }{\Phi _{\mathrm {s} }^{2}D_{\mathrm {p} }^{2}}}{\frac {(1-\epsilon )^{2}}{\epsilon ^{3}}}v_{s}}$

where:

• $\Delta p$ is the pressure drop;
• $L$ is the total height of the bed;
• $v_s$ is the superficial or "empty-tower" velocity;
• $\mu$ is the viscosity of the fluid;
• $\epsilon$ is the porosity of the bed;
• ${\displaystyle \Phi _{\mathrm {s} }}$ is the sphericity of the particles in the packed bed;
• ${\displaystyle D_{\mathrm {p} }}$ is the diameter of the related spherical particle.^[3]

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.^[2]

History

The equation was first^[4] proposed by Kozeny (1927)^[5] and later modified by Carman (1937, 1956).^[6][7]

See also

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

References

1. Fluid Mechanics, Tutorial No. 4: Flow through porous passages (PDF)
2. 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
3. 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
4. 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 http://www.polymtl.ca/biblio/epmrt/rapports/rt2003-03.pdf (accessed 2011-02-05)
5. J. Kozeny, "Ueber kapillare Leitung des Wassers im Boden." Sitzungsber Akad. Wiss., Wien, 136(2a): 271-306, 1927.
6. P.C. Carman, "Fluid flow through granular beds." Transactions, Institution of Chemical Engineers, London, 15: 150-166, 1937.
7. P.C. Carman, "Flow of gases through porous media." Butterworths, London, 1956.