Darcy law yields extreme speed for gas flow through packed spheres

The Darcy law is as follows: $$u=-\frac{k}{\mu}\nabla p.$$

Assume we have a gas, then $$\mu$$ is about $$10^{-5}$$. For packed spheres a few $$\rm mm$$ in diameter, $$k$$ is of order $$10^{-8}\ {\rm m^2}$$. Say the pressure difference across a $$0.1\ \rm m$$ bed is $$10\ \rm Pa$$. Then we get $$u= 0.1\ \rm m/s$$, which seems really high for such a small pressure difference. I mean, tens of pascals is a variation which is comparable to random pressure fluctuations. Is this reasonable?

• It's difficult to assess how reasonable your figures are unless you specify units for all of them. This will also help to check that you've done the right calculation. Commented Jun 20, 2012 at 20:18
• Everything in SI.
– tiam
Commented Jun 27, 2012 at 13:37
• What makes you say that the speed is unreasonably high? Commented Jun 27, 2012 at 18:10
• Because I gave a hard time imagining 10 Pa pressure fluctuations cause a 10 cm/s speed in a packed bed. I am modelling combustion in porous media and it is very sensitive to velocity: $u=0.05$ m/s and $u=0.25$ m/s can yeild quite different results. And accoring to Darcy's law it seems that velocities would fluctate randomly in those value ranges because it is a matter of a few Pa fluctuations in pressure.
– tiam
Commented Jun 28, 2012 at 13:05

For the parameters provided, together with a density of $$1\ \rm kg/m^3$$, and substituting a flow length scale of 0.001 m, the Reynolds number reaches a value around 10. This means that the flow in the pore space is non-laminar and inertial effects can not be ignored. A straightforward engineering approach is to apply so-called Forchheimer corrections.