Step 1: Understanding the Question:
The question asks which equation the theory of filtration is based on.
Step 2: Key Formula or Approach:
Filtration is the process of forcing a liquid through a porous filter medium, a cake, cloth, or membrane, under an applied pressure difference. Darcy's law describes exactly this kind of flow through a porous medium.
\[ Q = \frac{k A \, \Delta P}{\mu L} \]
Step 3: Detailed Explanation:
Here \(Q\) is the volumetric flow rate of filtrate, \(k\) is the permeability of the filter medium, \(A\) is the filtering area, \(\Delta P\) is the pressure difference driving flow, \(\mu\) is the viscosity of the liquid, and \(L\) is the thickness of the filter bed or cake. As filtration proceeds and the cake builds up, \(L\) increases and the flow rate falls, which is why filtration equipment either needs increasing pressure over time or is designed for constant-rate operation. This direct link between pressure, permeability, and flow rate is what makes Darcy's law the foundation of filtration theory, unlike Dalton's law for gas pressures, the BET equation for surface area from gas adsorption, or Stokes' law for particle settling, none of which describe pressure-driven flow through a filter medium.
Step 4: Final Answer:
The theory of filtration is based on the Darcy equation.