# Average shear rate of of particle sureface

• hizen_14
In summary, the question asks for the average shear rate at the particle surface, given a diffusion coefficient of 100 nm particles with a density of 1050 kg/m3 in a viscous solvent at 25 C. The equation for this is D = kT/6πηa. Another equation, Pe = a2γ/D, is also mentioned, but it is unclear if it should be used. The student has completed parts a-c of the problem and their answer to part c was 0.05 m/s, but upon further analysis, they have changed the radius of the particle from 100µm to 50µm and have arrived at a more reasonable answer of 0.149 m/s. They also provide
hizen_14

## Homework Statement

Diffusion coefficient (D) of 100 nm particles ( density = 1050 kg/m3) in a viscous solvent at 25 C is 2.2 x 10-13 m3/s

What is the average shear rate at the particle surface?

## Homework Equations

D = kT/ 6πηa (1)
where k is Boltzman constant, T = temperature, ηsusp = viscosity and a = radius

Pe = a2γ /D (2)

where γ = shear rate

## The Attempt at a Solution

Assumed particles are in water

To find shear rate of particle surface I am not too sure that eq2 is the right equation to use or not. Can we assume that Pe = 1 ??

Pe = 1 means the solvent in the viscous state ?

Thank you

Zen

This can't be the complete problem statement. Please provide the complete problem statement, with its exact wording.

Chet

hizen_14
Hi Chet, This is the full question

Thanks

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If I understand correctly, you have completed parts a, b, and c, and now you want want to determine the answer to part d, correct? If so, what was your answer to part c.

Chet

hizen_14
Hi Chet !

Yes you are right I have done part a-c. My answer on C was 0.05 m/s

Thanks

Zen

hizen_14 said:
Hi Chet !

Yes you are right I have done part a-c. My answer on C was 0.05 m/s

Thanks

Zen
Are you sure about that average velocity of the particles? It seems awfully large.

Chet

hizen_14
Thanks Chet

I have changed the radius of particle from 100µm to 50µm but the velocity was larger than 50 µm. it doesn't make sense. :(

Vav = ((3kT)/m)1/2

Volume of particle = 4/3 πr3 = 4/3 π (50x10-9)3 = 5.236 x10-22 m3
Mass of particle = ρV = 1050 x 5.236x 10-22 = 5.498 x 10-19 kg

Vav = ((3x1.38 x 10-23 x 298 )/ 5.498 x 10-19 )1/2 = 0.149 m/s

I think this one is more reasonable ?

v = (2r2 Δρg) / 9η
= ( 2x( 50x10-9 )2 x (1050-1000) x 9.8) / (9 x 8.9x10-4)
= 3.059 x 10-10 m/s

THanks

hizen_14 said:
I think this one is more reasonable ?

v = (2r2 Δρg) / 9η
= ( 2x( 50x10-9 )2 x (1050-1000) x 9.8) / (9 x 8.9x10-4)
= 3.059 x 10-10 m/s

THanks
This is much better, if analyzed correctly. I can't vouch for that part of the calculation (because my experience is not in that area), but I can help you get the average shear rate at the particle surface once the particle velocity is established. Sometimes the shear rate is taken as the radial velocity gradient at the particle surface, and some times it is taken as the square root of the second invariant of the rate of deformation tensor. How is it defined in your situation?

Do you know the equations for the components in spherical coordinates of the velocity vector in Stokes flow past a moving sphere? If so, please write them down.

Chet

hizen_14
Hi Chet.

Chestermiller said:
Sometimes the shear rate is taken as the radial velocity gradient at the particle surface, and some times it is taken as the square root of the second invariant of the rate of deformation tensor. How is it defined in your situation?

I am not really sure about but I can see as the radial velocity gradient at the particle surface (dv/dx)

Chestermiller said:
Do you know the equations for the components in spherical coordinates of the velocity vector in Stokes flow past a moving sphere? If so, please write them down.

Fg-Fb-Fd = 0 at Terminal Velocity
where Fg = gravity force, Fb = buoyancy force and Fd = drag force.

Fg = (Vρpg)
Fb = (Vρfg)
Fd = 9πηrv

v = (2r2 Δρg) / 9η

I am not really sure this is what you asked for.

Thanks

hizen_14 said:
Hi Chet.
I am not really sure about but I can see as the radial velocity gradient at the particle surface (dv/dx)
Fg-Fb-Fd = 0 at Terminal Velocity
where Fg = gravity force, Fb = buoyancy force and Fd = drag force.

Fg = (Vρpg)
Fb = (Vρfg)
Fd = 9πηrv

v = (2r2 Δρg) / 9η

I am not really sure this is what you asked for.

Thanks
I'm looking the equations for the components as a function of r and θ.

Chet

hizen_14

?

hizen_14 said:
I'm more interested in the tangential velocity.

Chet

## What is the definition of average shear rate?

The average shear rate of a particle surface is the measure of the rate at which a particle surface shears or cuts through a fluid. It is usually expressed in units of inverse seconds (s-1).

## How is average shear rate calculated?

The average shear rate is calculated by dividing the velocity gradient of the fluid by the distance between two parallel surfaces. This can be represented by the equation γ = (∂u/∂y), where γ is the average shear rate, u is the velocity gradient, and y is the distance between surfaces.

## Why is average shear rate important in particle surface analysis?

Average shear rate is important in particle surface analysis because it helps to determine the amount of force and stress applied to the particles, which can affect their behavior and properties. It is also used to calculate the viscosity and flow properties of the fluid surrounding the particles.

## How does average shear rate affect particle size and shape?

The average shear rate can affect particle size and shape by influencing the forces acting on the particles. A higher shear rate can cause particles to break apart or deform, while a lower shear rate may allow particles to aggregate or settle. This can impact the overall size and shape distribution of particles in a system.

## What factors can influence the average shear rate of a particle surface?

The average shear rate can be influenced by a number of factors, including the velocity and flow rate of the fluid, the geometry of the particle surface, and the presence of other particles or obstacles in the system. Temperature, pressure, and viscosity of the fluid can also affect the average shear rate.

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