Kozeny–Carman equation

The equation is given as:

{{Citation

| last1 = McCabe | first1 = Warren L.

| last2 = Smith | first2 = Julian C.

| last3 = Harriot | first3 = Peter

| title = Unit Operations of Chemical Engineering

| place = New York

| publisher = McGraw-Hill

| year = 2005

| pages = 152–153

| edition = seventh

| isbn = 0-07-284823-5}}

:$\backslash frac\{\backslash Delta\; P\}\{L\}\; =\; \backslash frac\{150\; \backslash mu\}\{\backslash mathit\{\backslash Phi\}\_\backslash mathrm\{s\}^2\; d\_\backslash mathrm\{p\}^2\}\backslash frac\{(1-\backslash varepsilon)^2\}\{\backslash varepsilon^3\}V\_\backslash mathrm\{0\}$

where:

• $\backslash Delta\; P$ is the pressure drop;
• $L$ is the total height of the bed;
• $\backslash mu$ is the viscosity of the fluid;
• $\backslash varepsilon$ is the porosity of the bed ($\backslash simeq\; 0.37$ for randomly packed spheres); {{Cite journal |last1=Wu |first1=Yugong |last2=Fan |first2=Zhigang |last3=Lu |first3=Yuzhu |date=2003-05-01 |title=Bulk and interior packing densities of random close packing of hard spheres |url=https://doi.org/10.1023/A:1023597707363 |journal=Journal of Materials Science |language=en |volume=38 |issue=9 |pages=2019–2025 |doi=10.1023/A:1023597707363 |s2cid=137583828 |issn=1573-4803}}
• $\backslash mathit\{\backslash Phi\}\_\backslash mathrm\{s\}$ is the sphericity of the particles in the packed bed ($\backslash mathit\{\backslash Phi\}\_\backslash mathrm\{s\}$ = 1.0 for spherical particles);
• $d\_\backslash mathrm\{p\}$ is the diameter of the volume equivalent spherical particle;
• $V\_\backslash mathrm\{0\}$ is the superficial or "empty-tower" velocity which is directly proportional to the average volumetric fluid flux in the channels (q), and porosity ($\backslash mathbf\{\backslash varepsilon\}$).

{{Citation

| last1 = McCabe | first1 = Warren L.

| last2 = Smith | first2 = Julian C.

| last3 = Harriot | first3 = Peter

| title = Unit Operations of Chemical Engineering

| place = New York

| publisher = McGraw-Hill

| year = 2005

| pages = 188–189

| edition = seventh

| isbn = 0-07-284823-5}}

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 is a particular case of Darcy's law, with a very specific permeability. Darcy's law states that "flow is proportional to the pressure gradient and inversely proportional to the fluid viscosity" and is given as:

:q $=\; \backslash frac\{\backslash kappa\}\{\backslash mu\}\; \backslash frac\{\backslash Delta\; P\}\{L\}$

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

:$\backslash kappa\; =\; \backslash mathit\{\backslash Phi\}\_\backslash mathrm\{s\}^2\; \backslash frac\; \{\backslash varepsilon^3\; d\_\backslash mathrm\{p\}^2\}\{180(1-\backslash varepsilon)^2\}$

where:

• $\backslash kappa$ is the absolute (i.e., single phase) permeability.

The equation was firstRobert 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) proposed by Kozeny (1927)J. Kozeny, "[http://www.zobodat.at/pdf/SBAWW_136_2a_0271-0306.pdf Ueber kapillare Leitung des Wassers im Boden]." Sitzungsber Akad. Wiss., Wien, 136(2a): 271-306, 1927. and later modified by Carman (1937, 1956).P.C. Carman, "Fluid flow through granular beds." Transactions, Institution of Chemical Engineers, London, 15: 150-166, 1937.P.C. Carman, "Flow of gases through porous media." Butterworths, London, 1956. A similar equation was derived independently by Fair and Hatch in 1933. G.M. Fair, L.P. Hatch, Fundamental factors governing the streamline flow of water through sand, J. AWWA 25 (1933) 1551–1565. A comprehensive review of other equations has been published. 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

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

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Category:Eponymous laws of physics

Category:Equations of fluid dynamics

Category:Unit operations

Category:Porous media