Final answer:
The settling velocity of a 100 micron diameter particle falling through water at 20°C is approximately 0.000002 m/s. The Reynolds number for this particle can be calculated to be approximately 0.1996.
Step-by-step explanation:
According to Stokes' law, the settling velocity of a particle in a viscous fluid can be calculated using the formula:
v = (2/9) * (g * r^2 * (ρp - ρf)) / η
Where:
- v is the settling velocity
- g is the acceleration due to gravity
- r is the radius of the particle
- ρp is the density of the particle
- ρf is the density of the fluid
- η is the viscosity of the fluid
In this case, the diameter of the particle is given as 100 microns, which is equivalent to a radius of 50 microns or 0.05 mm. The density of the particle is 1.75 g/m³. Assuming water at 20°C as the fluid, the density of water at this temperature is approximately 998 kg/m³ and the viscosity of water is approximately 0.001 kg/(m·s). Plugging these values into the formula, we get:
v = (2/9) * (9.8 * (0.05)^2 * (1.75 - 998)) / 0.001
Simplifying the equation gives us the settling velocity:
v ≈ 0.000002 m/s
To calculate the Reynolds number, we use the formula:
Re = (ρvD) / η
Where:
- Re is the Reynolds number
- ρ is the density of the fluid
- v is the velocity of the particle
- D is the diameter of the particle
- η is the viscosity of the fluid
Plugging in the values for density, settling velocity, and diameter, we get:
Re ≈ (998 * 0.000002 * 0.0001) / 0.001
Simplifying the equation gives us the Reynolds number:
Re ≈ 0.1996