Math 2.0

Fluid mechanics Diffusion, convection and porous media Fact

Kozeny–Carman equation

Builds on Darcy's lawViscosity

Statement

Consider slow, steady, laminar flow of a fluid with dynamic viscosity μ\mu through a packed bed of solid particles of length LL. Let ε∈(0,1)\varepsilon \in (0, 1) be the porosity of the bed, DpD_p the particle diameter and Φs∈(0,1]\Phi_s \in (0, 1] the sphericity of the particles. Let usu_s be the superficial velocity, the volume flow rate per unit cross-sectional area of the bed. The Kozeny–Carman equation states that the pressure drop Δp\Delta p across the bed is ΔpL=180 μΦs2Dp2 (1−ε)2ε3 us.\frac{\Delta p}{L} = \frac{180 \, \mu}{\Phi_s^2 D_p^2} \, \frac{(1 - \varepsilon)^2}{\varepsilon^3} \, u_s. Equivalently, the bed obeys Darcy's law us=(k/μ)(Δp/L)u_s = (k / \mu)(\Delta p / L) with permeability k=Φs2Dp2 ε3180 (1−ε)2.k = \frac{\Phi_s^2 D_p^2 \, \varepsilon^3}{180 \, (1 - \varepsilon)^2}.

Remarks

The relation is semi-empirical. Its form comes from modelling the pore space as a bundle of tortuous capillary tubes whose hydraulic diameter is proportional to εDp/(1−ε)\varepsilon D_p / (1 - \varepsilon), and applying the Hagen–Poiseuille law in each tube. The constant 180180 is fitted to experiments on beds of near-spherical particles.

Example: for a bed of spheres (Φs=1\Phi_s = 1) with Dp=1D_p = 1 mm and ε=0.4\varepsilon = 0.4, the permeability is k=10−6⋅0.064/(180⋅0.36)≈9.9×10−10k = 10^{-6} \cdot 0.064 / (180 \cdot 0.36) \approx 9.9 \times 10^{-10} m2^2.

The equation needs laminar flow through the pores. At larger pore Reynolds numbers inertial losses grow like us2u_s^2, and the Ergun equation adds such a term.

Going deeper

For a student

Darcy's law says that flow through a porous medium is proportional to the pressure gradient, but it does not say what the permeability is. The Kozeny–Carman relation supplies the permeability of a packed bed from its geometry. The factor ε3/(1−ε)2\varepsilon^3 / (1 - \varepsilon)^2 shows the strong dependence on porosity: halving the porosity from 0.40.4 to 0.20.2 lowers the permeability by a factor of about 1414. It is used in filtration, chemical reactors, soil mechanics and models of compaction.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptStokes flow, concept, no evidence levelStokes flowconceptViscosity, concept, no evidence levelViscosityconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptDarcy's law, statement, no evidence levelDarcy's lawstatementKozeny–Carman equation, statement, no evidence levelKozeny–Carman equationstatement
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  1. Euclidean space object
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  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
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  13. Darcy's law statement
  14. Kozeny–Carman equation statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Kozeny%E2%80%93Carman_equation
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AboutDarcy's law
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Changed3 November 2026

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