Menu

Terminal velocity of a particle in a fluid : formula, step by step calculation guide

Single particle settling and settling in suspension (hindered settling)

Follow us on Twitter
Question, remark ? Contact us at contact@myengineeringtools.com


1. Introduction
2. Terminal velocity step by step calculation
3. Hindered settling step by step calculation
4. Excel calculator

1. Introduction

What is the terminal velocity ?

The terminal velocity of a particle in a fluid is the maximum speed that can reach a particle free falling when the gravity forces and the drag forces + the upthrust (Archimedes principle) equal. The calculation of the terminal velocity of a particle is of interest in many unit operations such as the sedimentation of a slurry after liquid-solid mixing, the separation in a cyclone or in a fluid bed.

It should be noted that the terminal velocity of a single particle is different than the terminal velocity of a particle among many others as it is the case in suspensions for example, the presence of other particles indeed hinders the settling of the particle and thus reduces its terminal velocity. This settling velocity in the presence of dense concentration of other particles is called the hindered velocity and can be used to determined at which rate will sediment a suspension.

2. Terminal velocity step by step calculation

How do you find the terminal velocity of a particle ?

  • The particles considered must have a spherical shape, it could be solids particles, but also liquid in a gas or even a heavier liquid drop in another liquid. However, some limitations apply (Rep < 150000 for a solid in a gas, Rep < 500 for a liquid in another liquid, Rep < 100 for a liquid in a gas)
  • In any case, the particle must be heavier than the fluid
  • To be noted that a precision of +/-25% is to be expected, results must therefore be interpreted with care

2.1 STEP 1 : Calculate the drag coefficient

The drag coefficient K can be calculated thanks to the following formula [Mc Cabe]:

K = dp.[(g*ρf*(ρpf))/μ2](1/3)

Drag coefficient K calculation
With :
K = drag coefficient
dp = particle diameter (m)
g = gravity acceleration = 9.81 (m.s-2)
ρf = fluid density (kg/m3)
ρp = particle (material) apparent density (kg/m3)
μ = fluid viscosity (Pa.s)

2.2 STEP 2 : Determine the coefficients b and n for settling velocity calculation

The coefficients are tabulated according to the value of K. Please refer to the table below to determine b and n.

Flow Regime
K
b
n
Stokes K<3.3 24 1
Intermediate 3.3<K<43.6 18.5 0.6
Newton 43.6<K<2360 0.44 0

2.3 STEP 3 : Calculate the terminal velocity

The terminal velocity (or settling velocity) can be calculated thanks to the following equation :

Ut = [(4*g*dp(1+n)*(ρpf))/(3*b*μnf(1-n))]1/(2-n)

Calculation formula for particle terminal velocity

With :
Ut = terminal velocity of single particle (not hindered) (m/s)
b and n = coefficient determined at step 3

2.4 STEP 4 : Check validity of the correlation

To check the validity, calculate the Reynolds particle :

The particle Reynolds number can be calculated with the following formula :

Rep = dp.Utf

Calculation of Reynolds particle
With :
Rep = Reynolds particle (-)
dp = particle diameter (m)
Ut = terminal velocity of the single particle (not hindered) (m/s)
ρf = fluid density (most probably a liquid) (kg/m3)
μ = fluid viscosity (Pa.s)

For a solid in a fluid : Rep must be < 150000
For a liquid in another liquid : Rep must be < 500
For a liquid in a gas : Rep must be < 100

3. Hindered settling step by step calculation

What is the terminal settling velocity of a suspension ?

A high density of particles settling is influencing the terminal velocity of particles, for example in the case of a liquid-solid suspension. The velocity is then called hindered settling velocity. It is possible to calculate it by 1st determining the volume fraction of solids in suspension, then calculating a Reynolds number for the settling particles.

3.1 STEP 1 : calculate the Reynolds number of a settling particle

The particle Reynolds number can be calculated with the following formula :

Rep = dp.Utf

Calculation of Reynolds particle

With :
Rep = Reynolds particle (-)
dp = particle diameter (m)
Ut = terminal velocity of the single particle (not hindered) (m/s)
ρf = fluid density (most probably a liquid) (kg/m3)
μ = fluid viscosity (Pa.s)

3.2 STEP 2 : determine the parameter m

The parameter m can be determined depending on the value of Rep.

Rep m
Rep < 0.5 4.65
0.5 < Rep < 1300 4.375*Rep-0.0875
Rep > 1300 2.33

3.3 STEP 3 : calculate the hindered settling velocity

The hindered settling velocity can then be calculated, knowing the volumic fraction ε of solid in the suspension and using the following formula to correct the terminal velocity of a single isolated particle [Perry] :

Ut' = Ut*(1-ε)m

Calculation of hindered settling velocity in suspensions

With :
Ut' = hindered settling velocity of particles in the suspension (m/s)
Ut = settling velocity of a single isolated particle calculated at paragraph 2.3
ε = volume fraction of solids in suspension (-)
m = determined at STEP 2

4. Excel terminal velocity calculator tool

Please access to this page to download the free xls calculation tool for Churchill correlation.

Source

[McCabe] Unit Operations of Chemical Engineering 7th edition, page 136

[Perry] Perry's Chemical Engineer's Handbook, Section 6 Fluid and Particle Dynamics, page 6-53, McGraw-Hill, 2008



Copyright www.MyEngineeringTools.com

The content of MyEngineeringTools.com is copyrighted but no warranty nor liability is ensured. The content of this site is to be seen as a help and important information and calculation must always be double checked by the user through the quality procedure of his organization or by checking another source. The user must always respect all applicable regulation. The use of the information is at the user and its organization own risk and own cost.

About privacy and cookies on this site

Our site uses cookies and other technologies from 3rd party companies. Access our Privacy Policy in the "About" page to know more about those cookies and technologies . Further use of this site will be considered consent.

www.myengineeringtools.com is secured by SSL encryption

Follow us on Twitter