Stokes' Law is essential for sedimentation basin design. What key parameter does this law primarily relate?

Stokes' Law primarily relates the drag force experienced by a sphere moving through a viscous fluid to the sphere's velocity. Specifically, it quantifies the force required to keep a small sphere moving at a constant velocity within a fluid. Let's break down what that means.

First, 'drag force' is the resistance a fluid exerts on an object moving through it. Think of sticking your hand out of a car window – the force pushing your hand backward is drag. Stokes' Law provides a mathematical equation to calculate this drag force under specific conditions. The equation is F = 6πηrv, where F is the drag force, η (eta) is the dynamic viscosity of the fluid, r is the radius of the sphere, and v is the velocity of the sphere.

'Viscous fluid' refers to a fluid that resists flow. Honey is a highly viscous fluid; water is less viscous. Viscosity is a measure of this resistance – a higher viscosity means greater resistance to flow. Dynamic viscosity (η) is a specific measure of this resistance, often expressed in units of Pascal-seconds (Pa·s) or Poise (P).

'Sphere' is crucial. Stokes' Law is derived assuming the object is a perfect sphere. While it can be applied approximately to other shapes, its accuracy decreases significantly with deviations from a spherical shape.

'Velocity' is the speed and direction of the sphere's movement. Stokes' Law applies when the sphere is moving at a constant velocity, meaning it's not accelerating.

Crucially, Stokes' Law is valid only under certain conditions: the sphere must be small (typically less than 1 mm in diameter), the fluid must be Newtonian (meaning its viscosity is constant regardless of shear rate), the flow must be laminar (smooth and layered, not turbulent), and the sphere must be moving slowly enough that the Reynolds number (a dimensionless quantity that predicts flow patterns) is less than 1. A Reynolds number less than 1 indicates laminar flow.

In sedimentation basins, Stokes' Law is used to predict the settling velocity of particles. Knowing the particle size (related to the sphere's radius), density, and the density and viscosity of the water, engineers can calculate how quickly particles will settle out of the water. This is essential for designing basins of appropriate size and configuration to achieve effective solids removal.