Determine the velocity of the aerosol solid particles when t=10 μs, if when t=0 they leave the c

Determine the velocity of the aerosol solid particles when...

Determine the velocity of the aerosol solid particles when $t=10 \mu \mathrm{s}$, if when $t=0$ they leave the can with a horizontal velocity of $30 \mathrm{~m} / \mathrm{s}$. Assume the average diameter of the particles is $0.4 \mu \mathrm{m}$ and each has a mass of $0.4\left(10^{-12}\right) \mathrm{g}$. The air is at $20^{\circ} \mathrm{C}$. Neglect the vertical component of the velocity.

Key Concepts

Newton’s Second Law in Fluid Dynamics

Newton’s second law, which relates the net force on an object to its mass and acceleration, is fundamental when studying particle motion in fluids. It allows for the formulation of equations of motion that include both the particle’s inertia and the opposing drag forces, providing a framework for calculating changes in velocity over time.

Particle Relaxation Time

The particle relaxation time is the characteristic time over which a particle adjusts to changes in its motion due to the balance between its inertia and the drag force acting on it. It is determined by the ratio of the particle’s mass to the drag coefficient and is crucial in transient analyses of particle motion.

Aerodynamic Drag

Aerodynamic drag is the force exerted by a fluid on an object moving through it. In the context of particles moving in air, this drag force opposes the motion of the particles and is a key factor in determining their deceleration and eventual terminal velocity.

Stokes’ Law

Stokes’ law describes the drag force experienced by small spherical particles moving at low velocities in a viscous fluid. It states that the drag force is proportional to the particle’s diameter, the fluid’s viscosity, and the particle’s velocity, making it essential for analyzing the behavior of aerosol particles in air.

Transcript

00:01 To consider the um find the velocity of an aerosol particle and it shot out at 30 meters per second we're given our village that's our initial velocity of the right we're given the diameter of the particles which i converted to meters we're given the mass particle which i converted to kilometers and we're given our final time which i converted from microseconds to seconds so we know that some of our forces will be m a and that's all to this way here's my diagram i'm given in the book fd will be equal to cd times ad of my particle times one half times the density times u squared.

00:43 Okay, and we know that our reynolds number will be equal to density ud over this.

00:53 Okay.

00:55 When the reynolds number is less than or equal to one, it's satisfied that our drag coefficient is.

01:02 Then we know that cd is equal 24 divided by our reynolds number.

01:21 We can look at a graph.

01:23 And the frame of reference is going to be as follows.

01:31 Okay.

01:33 So our final velocity, let's figure a few other things out.

01:37 Our density of air at 20 degrees is 1 .20, i believe, and our dynamic viscosity of air at that same temperature is 18 .1 times 10 to the minus 6th newton seconds per meter squared.

02:03 Just look those values up and let's do a free body diagram.

02:08 Do i have room for a free body diagram on my last page? so i've got the particle and i've got this mg pulling it down and this is my fd in this direction.

02:25 I should have gone.

02:26 That's my drag in the opposite direction.

02:28 Okay.

02:28 So now looking at, let's look at our summation of my forces and the x equal to negative force of drag will equal to mass times a x.

02:46 Let's figure this out here.

02:47 So i can do d vx.

02:55 I can substitute that in.

03:06 It should have been minus or not that should have been equals.

03:27 And that will give me negative fd equals m d vx over d t okay so far so good and we know that this can be flowing aerostatic so we know that this is equal to this and my drag force will be density of my air u square and u equals v so this will be equal to cd okay, and we know that our reynolds number around this kind of piece will be density of air times you.

04:28 And i'm going to go since you equals v, v times d over our fluid viscosity...

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