Introduction & Context

The calculation determines the terminal settling velocity of a solid particle immersed in a fluid. At terminal velocity, the gravitational force on the particle is exactly balanced by the aerodynamic drag force. This balance is a fundamental concept in process engineering for designing separators, predicting residence times in reactors, and estimating particle transport in pneumatic conveying systems. The methodology is applicable to any spherical particle moving through a Newtonian fluid under the influence of gravity.

Methodology & Formulas

The procedure follows a systematic conversion of user‑provided practical units to SI, evaluation of geometric properties, computation of the Reynolds number, selection of an appropriate drag‑coefficient correlation, and an iterative solution for the velocity that satisfies the force balance.

1. Unit Conversions

\[ d = d_{\text{mm}} \times 10^{-3}\quad\text{(particle diameter in metres)} \] \[ T = T_{\text{°C}} + 273.15\quad\text{(absolute temperature, K)} \] \[ p = p_{\text{bar}} \times 10^{5}\quad\text{(pressure, Pa)} \] \[ \mu = \mu_{\text{cP}} \times 10^{-3}\quad\text{(dynamic viscosity, Pa·s)} \]

2. Geometry of a Sphere

Projected area: \[ A = \frac{\pi d^{2}}{4} \] Volume: \[ V = \frac{\pi d^{3}}{6} \]

3. Particle Weight

\[ W = \rho_{p}\,V\,g \] where \(\rho_{p}\) is the particle density and \(g\) is the gravitational acceleration.

4. Reynolds Number

\[ \mathrm{Re} = \frac{\rho_{f}\,v\,d}{\mu} \] with \(\rho_{f}\) the fluid density and \(v\) the instantaneous particle velocity.

5. Drag‑Coefficient Correlation

The drag coefficient \(C_{D}\) is selected based on the Reynolds‑number regime. The regimes and corresponding expressions are summarized in the table below.

Reynolds‑Number Range	Drag‑Coefficient Expression
\(\mathrm{Re} \leq \mathrm{Re}_{1}\)	\(C_{D}= \dfrac{24}{\mathrm{Re}}\)
\(\mathrm{Re}_{1} < \mathrm{Re} \leq \mathrm{Re}_{2}\)	\(C_{D}= \dfrac{24}{\mathrm{Re}}\!\left(1 + \alpha\,\mathrm{Re}^{\beta}\right)\)
\(\mathrm{Re}_{2} < \mathrm{Re} \leq \mathrm{Re}_{3}\)	\(C_{D}= C_{1}\)
\(\mathrm{Re}_{3} < \mathrm{Re} \leq \mathrm{Re}_{4}\)	\(C_{D}= C_{2}\)
\(\mathrm{Re} > \mathrm{Re}_{4}\)	\(C_{D}= C_{3}\)

The symbols \(\alpha\), \(\beta\), \(C_{1}\), \(C_{2}\), and \(C_{3}\) represent empirical constants determined from experimental data. The threshold Reynolds numbers \(\mathrm{Re}_{1}\) through \(\mathrm{Re}_{4}\) define the boundaries between laminar, transitional, and turbulent regimes for a smooth sphere.

6. Drag Force

\[ F_{D}= \tfrac{1}{2}\,\rho_{f}\,v^{2}\,C_{D}\,A \]

7. Force Balance and Iterative Solution

The terminal velocity \(v_{t}\) satisfies the equilibrium condition: \[ F_{D}(v_{t}) = W \] Because \(C_{D}\) depends on \(\mathrm{Re}\), which itself depends on \(v\), the equation is nonlinear. A bisection algorithm is employed:

Initialize lower and upper velocity bounds \(v_{\text{low}}\) and \(v_{\text{high}}\).
Compute the midpoint \(v = (v_{\text{low}}+v_{\text{high}})/2\).
Evaluate \(\mathrm{Re}\), select \(C_{D}\) from the table, and calculate \(F_{D}\).
Determine the relative error \(\varepsilon = \bigl|F_{D}-W\bigr|/W\).
If \(\varepsilon\) is less than the prescribed tolerance \(\tau\), the solution has converged; otherwise, replace the bound that yields the smaller force with the current \(v\) and repeat.

The algorithm converges to a velocity that satisfies the force balance within the tolerance \(\tau\).

8. Output Quantities

The final results reported are:

Terminal velocity \(v_{t}\) (m s\(^{-1}\))
Reynolds number \(\mathrm{Re}\) (dimensionless)
Drag coefficient \(C_{D}\) (dimensionless)
Drag force \(F_{D}\) (N)
Particle weight \(W\) (N)

The drag coefficient (C_D) is a dimensionless number that quantifies the resistance a particle experiences as it moves through a fluid. It links the drag force (F_D) to the fluid’s dynamic pressure, the particle’s projected area, and the flow conditions:

F_D = ½ ρ v² A C_D
ρ = fluid density, v = relative velocity, A = projected area

Knowing C_D allows process engineers to predict settling rates, pressure drops, and mixing efficiency, which are critical for equipment design and scale‑up.

The calculation follows a two‑step procedure:

Determine the Reynolds number (Re) for the particle:
- Re = (ρ v d)/μ
- ρ = fluid density, v = relative velocity, d = particle diameter, μ = fluid viscosity
Apply the appropriate empirical correlation for C_D based on the Re range (see next FAQ).

Commonly used correlations:

Stokes regime (Re < 0.1): C_D = 24/Re
Intermediate regime (0.1 ≤ Re ≤ 1000): C_D = 24/Re · (1 + 0.15 Re^0.687) (Schiller‑Naumann)
Turbulent regime (Re > 1000): C_D ≈ 0.44 (constant for smooth spheres)

Select the correlation that matches the calculated Re for the particle under the specific process conditions.

Adjustments are made through shape factors and roughness corrections:

Introduce a shape factor (k_s) that multiplies the spherical C_D:
- C_D,actual = k_s · C_D,sphere
- Typical values: 1.1–1.3 for mildly elongated particles, up to 2.0 for flat plates.
Apply a roughness correction (k_ε) when the particle surface is not smooth:
- For Reynolds numbers where the boundary layer is turbulent, k_ε ≈ 1 + 0.2 (ε/d), where ε is the average roughness height.
Combine both factors:
- C_D,adjusted = k_s · k_ε · C_D,sphere

Verify the adjusted C_D against experimental data whenever possible, especially for highly irregular particles.

Worked Example: Drag Coefficient for a Small Spherical Particle in Air

A process engineer needs to estimate the drag force on a 2 mm diameter polymer bead that is suspended in a low‑speed airstream inside a laboratory reactor. The air properties are taken at 25 °C and 1.013 bar. The engineer will calculate the Reynolds number, drag coefficient, and resulting drag force for the bead at the estimated terminal velocity.

Knowns

π = 3.141593 (dimensionless)
g = 9.81 m s⁻²
Particle diameter, \(d\) = 2.0 mm = 0.002 m
Particle density, \(\rho_p\) = 1200 kg m⁻³
Air temperature, \(T\) = 25 °C = 298.15 K
Air pressure, \(p\) = 1.013 bar = 101300 Pa
Air density, \(\rho_{air}\) = 1.184 kg m⁻³
Dynamic viscosity of air, \(\mu\) = 0.0181 Pa·s
Velocity bounds from iteration: \(v_{low}\) = 0.1416 m s⁻¹, \(v_{high}\) = 0.1465 m s⁻¹
Tolerance for velocity convergence = 0.01 m s⁻¹

Step‑by‑Step Calculation

Calculate the projected area of the sphere: \[ A = \frac{\pi d^{2}}{4} \] Substituting \(d = 0.002\) m gives \(A = 3.142 \times 10^{-6}\) m².
Calculate the particle volume: \[ V = \frac{4}{3}\pi\left(\frac{d}{2}\right)^{3} \] \(V = 4.189 \times 10^{-9}\) m³.
Determine the particle weight (force due to gravity): \[ W = \rho_p V g \] \(W = 4.93 \times 10^{-5}\) N.
Assume an initial velocity estimate \(v = 0.144\) m s⁻¹ (mid‑point of the bounds).
Compute the Reynolds number: \[ Re = \frac{\rho_{air} v d}{\mu} \] \(Re = 0.0188\) (laminar regime).
For low Reynolds numbers, use the Stokes‑Cunningham correction to obtain an approximate drag coefficient: \[ C_d = \frac{24}{Re}\left(1 + 0.15\,Re^{0.687}\right) \] Substituting \(Re = 0.0188\) yields \(C_d = 1.274 \times 10^{3}\).
Calculate the drag force: \[ F_D = \tfrac{1}{2} C_d \rho_{air} A v^{2} \] \(F_D = 4.91 \times 10^{-5}\) N.
Check convergence: \[ \text{error} = \frac{|F_D - W|}{W} \] \(\text{error} = 0.003\) (0.3 %), which is below the specified tolerance, so the velocity estimate is acceptable.

Final Answer

Drag force on the 2 mm particle at terminal velocity ≈ 4.91 × 10⁻⁵ 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