Question

The settling velocity of inorganic particles in the sedimentation tank of a water treatment plant is governed by

• A. Darcy's law
• B. Stokes' law
• C. Dupuit's law
• D. Bernoulli's law

Hint:

Balance weight, buoyancy, and viscous drag on a small settling particle; the resulting terminal velocity expression is Stokes' law.

Answer 1

The Correct Option is B

Step 1: Understanding the Question.
We need to identify which physical law decides the settling (terminal) velocity of small inorganic (discrete, non-flocculating) particles as they sink through water in a sedimentation tank of a water treatment plant.

Step 2: Key Formula or Approach.
When a small particle settles in a fluid under gravity, three forces act on it: its weight, the buoyant force, and the viscous drag force from the fluid. At low Reynolds number (laminar settling, \(Re < 1\), which is the usual case for the fine inorganic particles removed in a sedimentation tank), the drag force follows Stokes' law, \(F_D = 3\pi\mu d v\), where \(\mu\) is the dynamic viscosity, \(d\) is the particle diameter, and \(v\) is the settling velocity. Balancing weight, buoyancy, and drag gives the terminal (Stokes') settling velocity:
\[ v_s = \frac{g(\rho_s - \rho_w)d^2}{18\mu} \]
where \(\rho_s\) is the density of the particle and \(\rho_w\) is the density of water.

Step 3: Detailed Explanation.
This equation, known as Stokes' law, directly connects the particle diameter, the density difference between particle and fluid, and the fluid viscosity to the settling velocity, and is the basis of surface loading (overflow rate) design for sedimentation tanks in water treatment.
Darcy's law describes seepage flow of water through a porous granular medium (as in filter beds or groundwater aquifers) and relates flow velocity to hydraulic gradient, not to a single particle falling through a fluid, so it does not apply here.
Dupuit's law is an approximation used for unconfined groundwater flow, assuming the hydraulic gradient equals the slope of the water table and the flow is horizontal; it deals with flow through soil, not particle settling.
Bernoulli's law (the energy equation) relates pressure head, velocity head, and elevation head along a streamline for a flowing fluid; it is an energy conservation statement and does not describe the drag-controlled sinking of a discrete particle.
None of these three alternatives model the balance of gravity, buoyancy, and viscous drag that fixes a particle's settling speed, so they can be ruled out.

Step 4: Final Answer.
The settling velocity of inorganic (discrete) particles in a sedimentation tank is governed by Stokes' law.
\[ \boxed{\text{Option (B): Stokes' law}} \]