Is it true that Non-Newtonian effects are always significant for particle settling.?

For particle Reynolds numbers above 1, non-Newtonian effects on friction drag can often be ignored in favor of pressure drag, simplifying calculations.

Is it true that All particles settle at the same terminal velocity in a given fluid.?

Terminal velocity is highly dependent on particle size, shape, density, and fluid properties, as well as flow regime and container effects.

Particle Motion in Fluid: A Process Engineering Cheat Sheet

Understanding particle motion in fluids is crucial for designing chemical engineering unit operations like sedimentation tanks and cyclones, by analyzing forces, flow regimes, and particle properties.

Core Principles

Particle motion in fluids is governed by gravity, buoyancy, and drag forces, with drag being dependent on flow regime (laminar, transitional, turbulent) and particle Reynolds number.
Terminal velocity (UT) is reached when drag force equals the net gravitational force, and its calculation is simplified for small particles in laminar flow.
Non-Newtonian fluids and particle characteristics (size, shape, orientation, container boundaries) significantly impact settling velocity, requiring specific correlations for accurate prediction.

Action Steps

1. Identify the forces acting on a particle (gravity, buoyancy, drag) to determine its motion.
2. Calculate the particle Reynolds number (Rep) to determine the flow regime (creeping, transitional, turbulent).
3. For Newtonian fluids in creeping flow (Rep < 1), calculate terminal velocity (UT) using Stokes' Law and appropriate drag coefficient (CD).
4. For non-Newtonian fluids, assess if the fluid has yield stress and calculate the dimensionless number Y to determine if motion occurs.
5. For irregular particle shapes, use volume equivalent diameter (dv) and sphericity (ψ) or shape factor (χ) to adjust terminal velocity calculations.
6. Account for container boundaries using the wall factor (fw) if particle diameter exceeds 10% of the container diameter.
7. For cyclones, calculate the critical diameter (dcrit) to determine separation efficiency for different particle sizes.

Formulas

Drag Force (Stokes' Law): $F_D = 3\pi\mu dU$
Terminal Velocity (Newtonian, Creeping Flow): $U_T = \frac{4gd(\rho_p - \rho_f)}{3C_D\rho_f}$
Particle Reynolds Number: $Re_p = \frac{\rho_f U d}{\mu}$
Volume Equivalent Diameter: $d_v = \sqrt[3]{\frac{6V}{\pi}}$
Sphericity: $\\psi = \frac{\pi d_v^2}{A_p}$
Shape Factor: $\\chi = \frac{A}{A'}$
Wall Factor: $f_w = \frac{U_{CT}}{U_T}$
Critical Cyclone Diameter: $d_{crit} = \sqrt{\frac{9\mu D_c}{4\pi N \nu (\rho_p - \rho_f)}}$
Cyclone Grade Efficiency: $\\eta = \frac{1}{1 + (\frac{d_{crit}}{d})^2}$

Key Terms

Terminal Velocity (UT): The maximum constant velocity a freely falling particle reaches when the drag force equals its net weight.
Reynolds Number (Rep): A dimensionless number characterizing the flow regime (laminar, transitional, turbulent) around a particle, based on fluid properties, particle velocity, and diameter.
Sphericity (ψ): A dimensionless measure of a particle's shape, defined as the ratio of the surface area of a sphere of the same volume to the particle's actual surface area.
Wall Factor (fw): The ratio of a particle's terminal velocity in a confined vessel to its terminal velocity in an infinite expanse of fluid, accounting for wall effects.
Grade Efficiency (η): The percentage of particles of a specific size that a cyclone will separate at 50% efficiency.

Pro Tips

For small particles settling in Newtonian fluids, the initial acceleration phase can be neglected as terminal velocity is reached very quickly.
In non-Newtonian fluids with yield stress, if the applied stress is less than the yield stress, the particle will remain stationary.
For cyclone design, the grade efficiency curve (η vs. d) is essential for predicting separation performance across a range of particle sizes.

Pitfalls to Avoid

Assuming Newtonian fluid behavior for non-Newtonian fluids leads to inaccurate terminal velocity calculations.
Ignoring particle shape and orientation can lead to significant errors in drag force and terminal velocity predictions.
Neglecting the effect of container walls can overestimate particle settling velocity in confined systems.

Myth vs Reality

Non-Newtonian effects are always significant for particle settling.: For particle Reynolds numbers above 1, non-Newtonian effects on friction drag can often be ignored in favor of pressure drag, simplifying calculations.
All particles settle at the same terminal velocity in a given fluid.: Terminal velocity is highly dependent on particle size, shape, density, and fluid properties, as well as flow regime and container effects.

Real World Examples

Settling of sand particles in a small-scale wastewater treatment plant.: Batch sedimentation tanks are used, with particle size > 100 µm, requiring understanding of settling behavior and zone formation.
Catalytic reactors for oil cracking.: Requires understanding particle motion in fluid to design efficient reactor systems.
Separating particles between 10 µm and 100 µm from air.: Cyclones are commonly used due to their simple design and low manufacturing cost.

Statistics

Particle size range for cyclones: 10 µm to 100 µm
Wall effect significant if particle diameter > X% of container diameter: 10%
Drag coefficient for turbulent flow (approx.): 0.44

Timeline

1851: George Gabriel Stokes derived the expression for total drag force on a sphere, known as Stokes' law.
1933: Schiller, L., & Neumann published work on particle motion, likely contributing to drag coefficient correlations.
1940: Lapple, C. E., & Shepherd published research on Industrial & Engineering Chemistry, relevant to particle technology.
1950: Brown et al. published research showing drag curves for varying sphericities.
1988: S. M. Walas published 'Chemical Process Equipment Selection and Design', including cyclone critical diameter calculations.
1991: Kunii, D., & Levenspiel published 'Fluidization Engineering'.
2008: Chhabra, R.P. and Richardson, J.F. published 'Non-Newtonian Flow and Applied Rheology'.
2021: Goodfellow, H. D., & Wang published 'Industrial Ventilation Design Guidebook Volume 2'.
2024: Rhodes, M. J. published 'Introduction to Particle Technology (3rd ed.)'.

People

George Gabriel Stokes: Developed the expression for total drag force on a sphere (Stokes' Law).
S. M. Walas: Authored 'Chemical Process Equipment Selection and Design', providing formulas for cyclone critical diameter.

Quiz

What is the primary force resisting particle motion in a fluid?: Drag
Which flow regime is characterized by a particle Reynolds number (Rep) between 0 and 1?: Creeping Flow
What dimensionless number is used to quantify the effect of container walls on particle settling velocity?: Wall Factor