Forchheimer Equation
Theory: adding inertia to porous flow
Overview
I calculated a filter's pressure drop with Darcy's law and got only about half the measured value.
Most likely the velocity is beyond Darcy's range. Darcy's law describes slow flow through gaps between grains or fibres where only viscosity matters. As velocity rises the flow accelerates and decelerates through the gaps and separates behind grains, adding a loss proportional to velocity squared. The Forchheimer equation adds exactly that term.
The Forchheimer equation
$u$ is the superficial velocity (flow rate over cross-section), $K$ the permeability [m²] and $\beta$ the Forchheimer (non-Darcy) coefficient [1/m]. The first term is viscous loss and the second inertial. Their ratio measures the departure from Darcy's law:
The Ergun equation for packed beds
For beds of near-spherical particles, the Ergun equation has the same form, so $K$ and $\beta$ can be estimated from particle diameter $d$ and porosity $\varepsilon$. For non-granular media such as fibres or foams, the coefficients must come from test data.
Porous-media science began with fountain water
Darcy's law was an empirical rule found in 1856 by the French engineer Henry Darcy while designing sand filters for the fountains and water supply of Dijon. Nearly half a century later, in 1901, the German engineer Philipp Forchheimer showed that pressure loss exceeds Darcy's prediction at high velocities and added a velocity-squared term. Both equations came from the practical needs of water supply rather than from theory.
Worked example: a bed of beads
Estimating the coefficients
For 3 mm glass beads packed to porosity 0.38, the Ergun equation gives $K \approx 8.57\times10^{-9}$ m² and $\beta \approx 6{,}590$ 1/m. The pressure gradient for water and air was split into the two terms:
| Fluid, velocity | Particle Re | Viscous term | Inertial term | Inertial share |
|---|---|---|---|---|
| Water 1 mm/s | 3 | 117 Pa/m | 7 Pa/m | 5 % |
| Water 5 mm/s | 15 | 584 Pa/m | 164 Pa/m | 22 % |
| Water 10 mm/s | 30 | 1,168 Pa/m | 658 Pa/m | 36 % |
| Water 50 mm/s | 150 | 5,838 Pa/m | 16,445 Pa/m | 74 % |
| Air 0.05 m/s | 10 | 105 Pa/m | 20 Pa/m | 16 % |
| Air 0.5 m/s | 100 | 1,051 Pa/m | 1,977 Pa/m | 65 % |
Even at a particle Reynolds number of only 3 it's off by 5%?
Yes. Darcy's law is usually said to hold strictly only up to a particle Reynolds number of about 1, and real filters and packed beds often run above that. For very slow flows such as groundwater Darcy is enough, but inside equipment it's safer to assume the Forchheimer term is needed. In your filter example, measurement at twice the calculation means inertia is about half the loss.
Fitting the coefficients from test data
Plotting pressure drop divided by velocity against velocity gives a straight line: the intercept is $\mu/K$ and the slope $\beta\rho$. Measure at least four or five points covering the actual operating range. Back-calculating $K$ from a single point folds that velocity's inertial loss into the permeability.
Using it in CFD
Porous-zone input
Porous-zone models in many general-purpose CFD codes take this form, with inputs of viscous resistance $1/\alpha$ and inertial resistance $C_2$. For the bed above, $1/\alpha \approx 1.17\times10^{8}$ 1/m² and $C_2 \approx 1.32\times10^{4}$ 1/m. Whether the inertial term includes the ½ differs between codes, so always check against the manual's equation.
- Superficial or physical velocity: some models work with superficial velocity inside the porous zone, others with the interstitial velocity. When transferring coefficients, make sure both assume the same one.
- Anisotropy: fibrous filters, honeycombs and heat-exchanger fins resist flow very differently by direction. Set large resistances crosswise to channel the flow.
- Near walls: porosity is higher next to walls in packed beds, so flow channels along them (the wall effect). It matters when the container diameter is less than about 10–20 particle diameters.
Is detailed filter modelling worth it?
Meshing filters, catalyst substrates or fin arrays as they are can easily run to hundreds of millions of cells. Usually the goal is flow distribution and pressure drop through the whole device, so replacing them with a porous zone is faster and accurate enough. A two-step approach works well: use a detailed model of a small representative section as a virtual test to obtain the porous coefficients, then pass them to the full model.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Using Darcy's law alone for fast flow | Pressure drop badly underestimated | Add the inertial term |
| Back-calculating K from one point | Wrong at other velocities | Fit a line over several velocities |
| Mixing up the ½ in the CFD coefficient | Inertial resistance off by a factor of 2 | Check the manual's equation |
| Confusing superficial and interstitial velocity | Resistance off by a porosity factor | Use a consistent convention |
| Ignoring anisotropy | Flow leaks sideways | Set large crosswise resistances |
I'd like to try other particle sizes and velocities.
With the Ergun packed-bed pressure drop calculator you can change particle size, porosity and velocity and watch the inertial share. Permeability is explored in the Darcy permeability simulator, and the test method in the constant-head permeability test simulator. Slow flow is covered in Darcy flow, near-wall behaviour in the Brinkman equation, and bed design in packed beds.
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