A spherical particle settles in a liquid contained in a narrow vessel. Upon what variables would you expect the falling velocity of the particle to depend? Obtain the relevant dimensionless groups.
For particles of a given density settling in a vessel of large diameter, the settling velocity is found to be inversely proportional to the viscosity of the liquid. How would this depend on particle size?

Question

Accepted Answer

This problem is very similar to Problem 1.13, although, in this case, the liquid through which the particle settles is contained in a narrow vessel. This introduces another variable, D, the vessel diameter and hence the settling velocity of the particle is given by: [latex]u =f\left(d, ho, \mu, D, ho_s, g ight)[/latex]. The dimensions of each variable are: [latex]u= L / T , d= L , ho= M / L ^3, \mu= M / L T , D= L , ho_s= M / L ^3[/latex] and g = L/T². With 7 variables and 3 fundamental dimensions, there will be [latex](7-3)=4[/latex] dimensionless groups. Taking [latex]d, ho[/latex] and μ as the recurring set, then:

[latex]d \equiv L[/latex], L = d

[latex] ho \equiv M / L ^3, \quad M = ho L ^3= ho d^3[/latex]

[latex]\mu \equiv M / L T , \quad T = M / L \mu= ho d^3 / d \mu= ho d^2 / \mu[/latex]

Thus: dimensionless group 1: [latex]u T / L =u ho d^2 /(\mu d)=d u ho / \mu[/latex]

dimensionless group 2: [latex]D / L =D / d[/latex]
dimensionless group 3: [latex] ho_{ s } L ^3 / M = ho_{ s } d^3 /\left( ho d^3 ight)= ho_{ s } / ho[/latex]

and dimensionless group 4: [latex]g T ^2 / L =g ho^2 d^4 /\left(\mu^2 d ight)=g ho^2 d^3 / \mu^2[/latex]

Thus: [latex]\underline{\underline{(d u ho / \mu)= f \left((D / d)\left( ho_s / ho ight)\left(g ho^2 d^3 / \mu^2 ight) ight)}}[/latex]

In particular, [latex](d u ho / \mu)=K\left(g ho^2 d^3 / \mu^2 ight)^n[/latex] where K is a constant.

For particles settling in a vessel of large diameter, [latex]u \propto 1 / \mu[/latex]. But [latex](u / \mu) \propto\left(1 / \mu^2 ight)^n[/latex] and, when [latex]n=1, n \propto 1 / \mu[/latex]. In this case:

[latex](d u ho / \mu)=K\left(g ho^2 d^3 / \mu^2 ight)[/latex]

or: [latex]d u \propto d^3[/latex] and [latex]u \propto d^2[/latex]

[latex]\underline{ ext{Thus the settling velocity is proportional to the square of the particle size.}}[/latex]

Question 1.P.15: A spherical particle settles in a liquid contained in a narr...

Related Answered Questions

Compare the costs of electricity at 1 p per kWh and gas at 15 p per therm. ...

Verified Answer:

It is found experimentally that the terminal settling velocity u0 of a spherical particle in a fluid is a function of the following quantities: particle diameter, d; buoyant weight of particle (weight of particle _ weight of displaced fluid), W; fluid density, ρ, and fluid viscosity, μ. Obtain ...

Verified Answer:

Verified Answer:

Upon what variables would the rate of filtration of a suspension of fine solid particles be expected to depend? Consider the flow through unit area of filter medium and express the variables in the form of dimensionless groups. It is found that the filtration rate is doubled if the pressure ...

Verified Answer:

Verified Answer:

Verified Answer:

A liquid is in steady state flow in an open trough of rectangular cross section inclined at an angle θ to the horizontal. On what variables would you expect the mass flow per unit time to depend? Obtain the dimensionless groups which are applicable to this problem. ...

Verified Answer:

Heat is transferred from condensing steam to a vertical surface and the resistance to heat transfer is attributable to the thermal resistance of the condensate layer on the surface. What variables are expected to affect the film thickness at a point? Obtain the relevant dimensionless groups. For ...

Verified Answer:

Obtain by dimensional analysis a functional relationship for the wall heat transfer coefficient for a fluid flowing through a straight pipe of circular cross-section. Assume that the effects of natural convection may be neglected in comparison with those of forced convection. It is found by ...

Verified Answer:

Liquid flows down an inclined surface as a film. On what variables will the thickness of the liquid film depend? Obtain the relevant dimensionless groups. It may be assumed that the surface is sufficiently wide for edge effects to be negligible. ...

Verified Answer: