Question

A water flow transports spherical particles (diameter = 2mm; density = 3g/cm³) in suspension mode. If additional particles of density 2g/cm³ are added into the flow, then the diameter of the particles that can be transported without a change in terminal fall velocity, using Stokes law, is _____mm. (Round off to two decimal places) (Use density of water = 1g/cm³)

Answer 1

Correct Answer: 2.8

To determine the diameter of particles that can still be transported without changing terminal fall velocity, we apply Stokes' Law for the terminal velocity (v) given by:
v = (2/9) * (r²(ρ_p - ρ_w)/η) * g 
where ρ_p is the density of the particle, ρ_w is the density of the water, η is the dynamic viscosity of the fluid, g is the acceleration due to gravity (assumed constant), and r is the radius of the particle.

We need the new particle diameter for the same terminal velocity with ρ_p = 2 g/cm³.
Since v remains constant and g, η, and ρ_w do not change, equate velocities:
((2/9) * (r₁²(ρ₁ - ρ_w))) = ((2/9) * (r₂²(ρ₂ - ρ_w)))
where ρ₁, ρ₂ are densities, and r₁, r₂ the respective radii.

Given:

• ρ₁ = 3 g/cm³
• ρ₂ = 2 g/cm³
• d₁ = 2 mm => r₁ = 1 mm (since diameter = 2*radius)

By inserting known values:
(1² * (3 - 1)) = (r₂² * (2 - 1))
Solving: 1² * 2 = r₂² * 1
Thus, r₂² = 2 → r₂ = √2 mm.
So, the diameter d₂ is 2√2 mm.
Approximating: d₂ = 2.83 mm.