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

The correct option is Stokes’ law.

Step-by-step Explanation:

In a water treatment plant, sedimentation is a physical water treatment process using gravity to remove suspended solids from water. For discrete, non-flocculating inorganic particles settling individually under laminar flow conditions (where the Reynolds number is less than 1), the settling velocity is governed by Stokes' law.

Stokes' law is derived by balancing the three forces acting on a settling spherical particle: the gravitational force acting downwards, the buoyant force acting upwards, and the drag force acting upwards resisting the motion.

When the particle reaches terminal settling velocity, the net force acting on it is zero:
$$F_g=F_b+F_d$$
where:
- $F_g$ is the gravitational force: $F_g={\rho }_{p}gV$
- $F_b$ is the buoyant force: $F_b={\rho }_{f}gV$
- $F_d$ is the drag force (from Stokes' drag): $F_d=3\pi \mu d{v}_s$
- $V$ is the volume of the spherical particle: $V=\frac{\pi }{6}{d}^3$

Substituting the volume into the force balance equation and solving for the settling velocity ($v_s$) yields the Stokes' law equation:
$$v_s=\frac{g\left({\rho }_p-{\rho }_f\right)d^2}{18\mu }$$
Where:
- $v_s$ is the settling velocity of the particle
- $g$ is the acceleration due to gravity
- ${\rho }_p$ is the mass density of the particle
- ${\rho }_f$ is the mass density of the fluid (water)
- $d$ is the diameter of the particle
- $\mu$ is the dynamic viscosity of the fluid

Analysis of other options:
- Darcy's law: Describes the flow of a fluid through a porous medium.
- Dupuit's law: Used in groundwater hydrology to estimate steady-state flow to a well.
- Bernoulli's law: Relates pressure, velocity, and elevation in a continuous, inviscid fluid flow.