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: Set up the force balance on a settling particle.
A spherical inorganic particle of diameter $d$ and density $\rho_s$ falling through still water of density $\rho_w$ and viscosity $\mu$ experiences three forces: gravity pulling it down, buoyancy pushing it up, and viscous drag opposing its motion.

Step 2: Write each force.
Weight of the particle: $W = \frac{\pi}{6}d^3 \rho_s g$.
Buoyant force: $F_B = \frac{\pi}{6}d^3 \rho_w g$.
Viscous drag on a small sphere moving slowly through a fluid (laminar, low Reynolds number regime) is given by Stokes' drag law: $F_D = 3\pi \mu d v$, where $v$ is the settling speed.

Step 3: Apply equilibrium at terminal velocity.
Once the particle reaches its constant (terminal) settling speed, acceleration is zero, so the net force is zero:
$$W - F_B - F_D = 0$$
$$\frac{\pi}{6}d^3(\rho_s - \rho_w)g = 3\pi \mu d v$$
Solving for $v$:
$$v = \frac{g(\rho_s-\rho_w)d^2}{18\mu}$$

Step 4: Identify the governing law and eliminate the other options.
This derived expression is exactly Stokes' law for settling velocity, so it is the law that governs the settling velocity of small discrete inorganic particles in a sedimentation tank. Darcy's law is a seepage-through-soil relation (used for flow in filter media or aquifers), Dupuit's law is a simplifying assumption for unconfined groundwater flow, and Bernoulli's law only tracks energy along a flowing streamline; none of the three involves a drag-versus-weight force balance on an individual particle. So all three can be dropped as unrelated to particle settling.
$$\boxed{\text{Stokes' law governs the settling velocity, option (B).}}$$