Introduction & Context

The calculation of the drag coefficient for immersed particles is a fundamental task in process engineering, particularly in the design of fluid-solid separation equipment, pneumatic conveying systems, and chemical reactors. When a particle moves through a fluid, it experiences a resistive force known as drag, which opposes its motion. Accurately determining this force is critical for predicting particle settling velocities, residence times in reactors, and the efficiency of cyclone separators or scrubbers. This reference sheet provides the methodology for calculating the drag force based on the dimensionless Reynolds number, which characterizes the flow regime surrounding the particle.

Methodology & Formulas

The calculation follows a sequential approach, beginning with the determination of the flow regime and concluding with the calculation of the total drag force.

First, the Reynolds number (Re) is calculated to define the ratio of inertial forces to viscous forces:

\[ Re = \frac{\rho \cdot V \cdot D}{\mu} \]

Where ρ is the fluid density, V is the relative velocity, D is the particle diameter, and μ is the dynamic viscosity.

For particles within the intermediate flow regime, the drag coefficient (C_D) is determined using the empirical Schiller-Naumann correlation:

\[ C_D = \frac{24}{Re} \cdot (1 + 0.15 \cdot Re^{0.687}) \]

The projected cross-sectional area (A) of a spherical particle is calculated as:

\[ A = \pi \cdot \left( \frac{D}{2} \right)^2 \]

Finally, the total drag force (F_D) exerted on the particle is derived from the drag equation:

\[ F_D = C_D \cdot A \cdot \frac{1}{2} \cdot \rho \cdot V^2 \]

Reynolds Number (Re) Range	Flow Regime	Applicability Note
Re < 1	Stokes' Law Regime	Correlation may be inaccurate; use Stokes' Law.
2 ≤ Re ≤ 800	Intermediate Regime	Correlation is accurate for this range.
800 < Re < 1000	Transition Zone	Results are approximate.
1000 ≤ Re ≤ 100,000	Newtonian Regime	C_D approaches constant value (~0.44); correlation may be inaccurate.

To select the correct correlation, you must first calculate the particle Reynolds number (Re). Once Re is established, apply the following guidelines:

For Re < 1 (Stokes Law regime), use the linear relationship C_D = 24/Re.
For 1 < Re < 1000 (Intermediate regime), utilize empirical correlations such as the Schiller-Naumann equation.
For Re > 1000 (Newtonian regime), assume a constant drag coefficient, typically C_D ≈ 0.44 for spheres.

The drag coefficient is highly sensitive to the sphericity of the particle. If your particles are non-spherical, you must adjust the calculation using the following factors:

Calculate the particle sphericity (ψ), defined as the ratio of the surface area of a sphere with the same volume to the actual surface area of the particle.
Apply a shape factor correction to the standard drag coefficient equation.
Account for increased pressure drag caused by flow separation points that differ from those of a perfect sphere.

In dense systems, the presence of neighboring particles creates a hindered settling effect, which increases the effective drag. To account for this, consider these adjustments:

Incorporate a voidage function, such as the Richardson-Zaki correlation, to modify the terminal velocity.
Adjust the drag coefficient based on the local volume fraction of the solid phase.
Recognize that particle-particle interactions significantly increase the apparent viscosity of the fluid medium.

Worked Example: Drag Force on a Spherical Particle

In a pneumatic conveying system, a spherical catalyst particle with a diameter of 10 mm is transported through an air stream. To ensure the system operates within the desired efficiency parameters, we must calculate the drag force exerted on the particle by the fluid flow.

Knowns:

Particle diameter (D): 0.01 m
Fluid velocity (V): 2.0 m/s
Fluid density (ρ): 1.204 kg/m³
Fluid dynamic viscosity (μ): 1.82e-05 Pa·s

Step-by-Step Calculation:

Calculate the Reynolds number (Re) to characterize the flow regime: \[ Re = \frac{\rho \cdot V \cdot D}{\mu} = \frac{1.204 \cdot 2.0 \cdot 0.01}{1.82 \times 10^{-5}} \approx 1323.077 \]
Calculate the projected frontal area (A) of the spherical particle: \[ A = \frac{\pi \cdot D^2}{4} = \frac{3.142 \cdot (0.01)^2}{4} \approx 7.854 \times 10^{-5} \text{ m}^2 \]
Determine the drag coefficient (C_D) using the standard correlation for a sphere in this flow regime: \[ C_D \approx 0.398 \]
Calculate the drag force (F_D) using the drag equation: \[ F_D = C_D \cdot A \cdot \frac{\rho \cdot V^2}{2} = 0.398 \cdot 7.854 \times 10^{-5} \cdot \frac{1.204 \cdot (2.0)^2}{2} \approx 7.521 \times 10^{-5} \text{ N} \]

Final Answer: The drag force exerted on the particle is 7.521e-05 N.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle