Washburn equation

The Washburn equation (after Edward W. Washburn , who derived it in 1921) describes in physics the capillary flow in porous materials in simplified form as:

${\displaystyle L={\sqrt {\frac {\gamma \cdot D\cdot t\cdot \cos \phi }{4\cdot \eta }}}}$

With

• the depth of penetration into which a liquid ${\displaystyle L}$
• the viscosity and ${\displaystyle \eta }$
• the surface tension penetrates ${\displaystyle \gamma }$
• within time ${\displaystyle t}$

into a fully wettable material

• with the average pore diameter and ${\displaystyle D}$
• the contact angle between liquid and material. ${\displaystyle \phi }$

This equation gained popularity in England through the physicist Len Fisher of the University of Bristol . He demonstrated the application of the equation using a cookie dipping experiment to make the science of physics more accessible by describing everyday problems.

Derivation

The Hagen-Poiseuille law

${\displaystyle {\frac {dV}{dt}}={\frac {\pi \cdot r^{4}}{8\cdot \eta }}{\frac {\Delta p}{l}}}$

is applied to the capillary flow of a liquid in a cylindrical tube without the influence of an external gravitational field .

After inserting the printout

${\displaystyle dV=\pi r^{2}dl}$

for a differential volume, which is defined by the differential length of a liquid in a pipe, the following equation is obtained: ${\displaystyle dl}$

${\displaystyle \Rightarrow {\frac {\delta l}{\delta t}}={\frac {\sum p}{8r^{2}\eta l}}(r^{4}+4\epsilon r^{3}).}$

In it is

• ${\displaystyle \sum p=p_{a}+p_{h}+p_{c}}$the sum of all acting pressures , including:
  • the atmospheric pressure ${\displaystyle p_{a}}$
  • the hydrostatic pressure and ${\displaystyle p_{h}}$
  • the pressure equivalent due to capillary forces, ${\displaystyle p_{c}}$
• ${\displaystyle \epsilon }$the coefficient of sliding friction , which becomes 0 for wettable materials,
• ${\displaystyle r}$the radius of the capillary .

The individual pressure components can be expressed as follows:

${\displaystyle p_{h}=\rho \cdot g\cdot h-\rho \cdot g\cdot l\sin \psi ,}$

${\displaystyle p_{c}={\frac {2\cdot \gamma }{r}}\cdot \cos \phi .}$

With

• the density of the liquid ${\displaystyle \rho }$
• the alignment angle of the pipe with respect to a horizontal axis. ${\displaystyle \psi }$

Inserting these equations for the individual pressures leads to a first order differential equation that describes the depth of penetration of the liquid into the pipe: ${\displaystyle l}$

${\displaystyle \Rightarrow {\frac {\delta l}{\delta t}}={\frac {[p_{a}+g\rho (h-l\sin \psi )+{\frac {2\gamma }{r}}\cos \phi ](r^{4}+4\epsilon r^{3})}{8r^{2}\eta l}}.}$

Individual proof

1. ^ Edward W. Washburn: The Dynamics of Capillary Flow . In: Physical Review . tape 17 , no. 3 , 1921, pp. 273-283 , doi : 10.1103 / PhysRev.17.273 .

Web links

• Derivation, development and application of the Washburn equation