Particulate matter at an altitude of 8 km in the upper atmosphere has an average diameter of 3

step 1 For a tiny particle, Reynolds number is very small. So Stokes law works. Then time is equal to d

Particulate matter at an altitude of 8 km in the upper...

Particulate matter at an altitude of $8 \mathrm{~km}$ in the upper atmosphere has an average diameter of $3 \mu \mathrm{m}$. If a particle has a mass of $42.5\left(10^{-12}\right) \mathrm{g},$ determine the time needed for it to settle to the earth. Assume gravity is constant, and for air, $\rho=1.202 \mathrm{~kg} / \mathrm{m}^{3}$ and $\mu=18.1\left(10^{-6}\right) \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$.

Key Concepts

Terminal Velocity

Terminal velocity is the constant speed that a particle reaches when the net force acting on it is zero, meaning that the gravitational force is exactly balanced by drag forces. In the context of settling particles, terminal velocity is critical because it determines the rate at which particles fall through a fluid under gravity when resistance forces do not change with time.

Stokes' Law

Stokes' Law provides the formula to calculate the drag force on small, spherical particles moving at low Reynolds numbers in a viscous fluid. The law, expressed as F_drag = 6??rv, where ? is the dynamic viscosity, r is the particle radius, and v is the velocity of the particle, is fundamental for determining terminal velocity in systems where inertial effects are negligible compared to viscous effects.

Force Balance for Particle Motion

A key concept in determining the settling time is the force balance between gravity and viscous drag. For a particle falling in a fluid, the gravitational force (adjusted for buoyancy if necessary) is balanced by the drag force when terminal velocity is reached. This balance allows the derivation of expressions for terminal velocity and is essential in predicting the behavior of particles in a fluid medium under constant gravitational acceleration.

Settling Time Calculation

Settling time refers to the duration required for a particle to fall a given distance through a fluid once it reaches its terminal velocity. This calculation is made by dividing the distance by the terminal velocity. Understanding settling time is important for applications in atmospheric science, environmental engineering, and other fields dealing with particulate matter behavior in fluids.

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