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

### Single particle settling and settling in suspension (hindered settling)

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)

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

### 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)

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

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

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

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

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