Introduction & Context

Terminal velocity represents the constant speed attained by a particle falling through a fluid when the gravitational force is exactly balanced by the sum of the buoyancy and drag forces. In process engineering, this calculation is fundamental to the design and operation of separation equipment, such as gravity settlers, hydrocyclones, and sedimentation tanks. Accurate determination of settling rates is critical for optimizing particle-fluid separation, ensuring product quality in food processing (e.g., starch recovery), and managing solid-liquid suspensions in chemical reactors.

Methodology & Formulas

The calculation relies on Stokes' Law, which describes the drag force on a spherical particle moving through a viscous fluid at low Reynolds numbers. The terminal velocity v_t is derived by equating the gravitational force, buoyancy force, and drag force.

First, the density difference between the particle and the fluid is defined as:

\[ \Delta\rho = \rho_s - \rho_l \]

The terminal velocity is then calculated using the following expression:

\[ v_t = \frac{g \cdot d^2 \cdot (\rho_s - \rho_l)}{18 \cdot \mu} \]

To ensure the validity of the Stokes' Law assumption, the Reynolds number (Re) must be calculated to confirm the flow regime:

\[ Re = \frac{\rho_l \cdot v_t \cdot d}{\mu} \]

Reynolds Number (Re)	Flow Regime	Validity Status
Re < 0.1	Stokes' Law Regime	Highly Accurate
0.1 ≤ Re < 2.0	Stokes' Law Regime	Acceptable (Increasing Error)
Re ≥ 2.0	Transition/Turbulent	Invalid (Stokes' Law not applicable)

The choice of correlation depends on the particle Reynolds number (Re), which characterizes the flow regime around the settling particle:

Use Stokes' Law for laminar flow conditions where Re is less than 0.1. This is typical for very small particles in high-viscosity fluids.
Use the Intermediate or Transition regime for Re between 0.1 and 1000, often requiring iterative calculations or empirical drag coefficient correlations.
Use Newton's Law for turbulent flow conditions where Re is greater than 1000, where inertial forces dominate over viscous forces.

Standard terminal velocity equations assume perfect spheres. For non-spherical particles, you must apply a correction factor known as the sphericity (ψ). The impact includes:

Increased drag force due to higher surface area-to-volume ratios.
A reduction in the effective settling velocity compared to a sphere of equal volume.
The need to adjust the drag coefficient (C_d) based on the specific geometry of the particle.

In industrial slurries, the standard single-particle settling model often fails due to the following factors:

Hindered settling effects, where high particle concentrations cause collisions and fluid displacement that slow down the settling rate.
Wall effects, which occur when the particle diameter is a significant fraction of the vessel diameter, increasing drag.
Non-Newtonian fluid behavior, where the apparent viscosity of the carrier fluid changes based on shear rate.
Particle-particle interactions and potential flocculation that change the effective diameter of the settling mass.

Worked Example: Settling Velocity of Fine Silica Particles

In a water treatment facility, a process engineer needs to determine the settling velocity of fine silica particles in a sedimentation tank to optimize residence time. The particles are assumed to be spherical and settling under laminar flow conditions (Stokes' Law regime).

Knowns:

Particle diameter (d): 20.0 microns (2.0e-05 m)
Particle density (ρ_s): 1500.0 kg/m³
Fluid density (ρ_l): 998.0 kg/m³
Fluid dynamic viscosity (μ): 0.001 Pa·s
Gravitational acceleration (g): 9.81 m/s²

Step-by-Step Calculation:

Calculate the density difference between the particle and the fluid:
\[ \Delta\rho = \rho_s - \rho_l = 1500.0 - 998.0 = 502.0 \text{ kg/m}^3 \]
Apply Stokes' Law to calculate the terminal settling velocity (v_t):
\[ v_t = \frac{g \cdot d^2 \cdot \Delta\rho}{18 \cdot \mu} \] \[ v_t = \frac{9.81 \cdot (2.0 \times 10^{-5})^2 \cdot 502.0}{18 \cdot 0.001} = 0.000109 \text{ m/s} \]
Verify the Reynolds number (Re) to ensure laminar flow conditions (Re < 1):
\[ Re = \frac{\rho_l \cdot v_t \cdot d}{\mu} = \frac{998.0 \cdot 0.000109 \cdot 2.0 \times 10^{-5}}{0.001} = 0.002 \]

Since Re is 0.002, the assumption of Stokes' Law is valid.
Convert the velocity to millimeters per second:
\[ v_{t,mms} = 0.000109 \text{ m/s} \cdot 1000 \text{ mm/m} = 0.109 \text{ mm/s} \]

Final Answer: The terminal settling velocity of the silica particles is 0.109 mm/s.

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