Modelling of pressure drop in periodic square-bar packed beds

This study numerically and experimentally investigates fluid flow through periodic square-bar packed beds, revealing that the rotation angle between modules fundamentally alters flow regimes and friction factors, while a proposed module-equivalent diameter successfully collapses the data onto the Ergun correlation for accurate permeability predictions.

The Gist

Imagine you are trying to push a crowd of people through a maze. Sometimes the maze is just a few wide, straight hallways. Other times, it's a chaotic tangle of twisting corridors, dead ends, and sudden squeezes.

This paper is about understanding exactly how hard it is to push that crowd (fluid) through different types of mazes (packed beds), specifically ones made of square bars stacked like a deck of cards.

Here is the breakdown of their research using simple analogies:

1. The Setup: The "Rotating Card Deck"

Most scientists study how fluids move through piles of round marbles (spheres). It's easy to model, but in the real world, things aren't perfect spheres. They are cubes, cylinders, or weird shapes.

The researchers built a special "maze" out of square bars. Imagine a stack of flat, round pancakes. Each pancake has square bars cut into it.

• The Twist: They can rotate each pancake slightly before stacking the next one.
• The Result: By rotating the layers by different amounts (from 0° to 90°), they created 19 different types of mazes, all with the same amount of empty space (porosity), but with very different shapes.

2. The Two Types of Mazes

They discovered that depending on how much they rotated the layers, the maze behaves like one of two things:

• The "Channel" Mode (Small Rotation): If you barely rotate the layers, the gaps line up to form long, winding tunnels. It's like a river flowing through a canyon. The fluid can slip through relatively easily.
• The "Lattice" Mode (Large Rotation): If you rotate the layers more, the gaps get blocked and reconnected in a complex web. It's like a spiderweb or a lattice fence. The fluid has to zigzag, squeeze through tight spots, and swirl around. This creates much more resistance.

3. The "Sweet Spot" of Resistance

The researchers found something surprising: The hardest part to push through isn't always the most twisted path.

• In slow flow (Viscous Regime): When the fluid moves slowly (like honey), the most resistance happens at a 25° rotation. Why? Because at this specific angle, the bars create a "pinch point"—a severe bottleneck that squeezes the fluid tight, creating maximum friction.
• In fast flow (Inertial Regime): When the fluid moves fast (like water in a fire hose), the most resistance happens at a 60° rotation. Why? Because the fluid is moving so fast it can't follow the curves. It crashes into the bars, separates, and creates chaotic swirls (eddies) behind them. It's like a car speeding around a sharp corner and losing control.

4. The "Magic Ruler"

Scientists use formulas to predict how hard it is to push fluid through a maze. The most famous one is the Ergun equation. However, this equation usually fails for weird shapes because it doesn't know how to measure the "size" of the obstacles.

The team tried two ways to measure the "size" of their square bars:

1. The "Single Bar" Ruler: Measuring just one bar. This failed because it didn't account for how the bars touch each other when rotated.
2. The "Module" Ruler: Measuring the whole stack as a single unit, accounting for how much surface area the fluid actually touches.

The Winner: The "Module" ruler worked like magic. It allowed them to collapse all their complex data onto the standard Ergun equation. It's like realizing that to predict traffic, you shouldn't just count the cars, but measure the total width of the road they are trying to squeeze through.

5. When Does Chaos Begin?

They also looked at when the flow stops being smooth and starts getting chaotic (turbulent).

• For most of their mazes, the flow started getting "bumpy" and inertial (where speed matters more than stickiness) around a specific speed (Reynolds number of ~7.5).
• However, at a 55° rotation, the chaos started even earlier. The specific shape of the holes at this angle created little whirlpools very quickly.

Why Does This Matter?

This isn't just about square bars. This research helps engineers design better:

• Catalytic converters in cars (to clean exhaust).
• Chemical reactors (to mix chemicals efficiently).
• Filters (to clean water or air).

By understanding how the shape and rotation of the packing material changes the flow, engineers can design reactors that are smaller, cheaper, and more efficient, rather than just guessing with round marbles.

In a nutshell: They proved that the "twist" in your packing material changes the physics of the flow entirely. Sometimes a slight twist creates a bottleneck; sometimes a big twist creates a whirlpool. And to predict it, you have to measure the whole puzzle, not just the pieces.

Technical Summary

1. Problem Statement

Packed-bed reactors are critical in chemical and process industries, yet most existing flow models rely on spherical particles. Real-world industrial packings often involve irregular, non-spherical geometries (e.g., cylinders, polyhedra), which induce complex flow patterns that standard correlations (like the Ergun equation) fail to capture accurately.

The specific challenge addressed in this study is the lack of accurate models for irregular interstitial geometries where the macroscopic properties (porosity, cross-section) remain constant, but the microscopic flow paths change drastically due to geometric arrangement. The authors aim to quantify how varying the rotation angle of stacked square-bar modules affects:

• Pressure drop (friction factor).
• Permeability and the Forchheimer coefficient.
• The transition from viscous (Darcy) to inertial (Forchheimer) flow regimes.

2. Methodology

Geometry and Experimental Setup:

• Configuration: The study utilizes a modular packed-bed reactor consisting of disk-shaped modules containing square bars. Each module can be rotated relative to the one below it by an angle $\alpha$.
• Parameters: The study covers 19 rotation angles ($\alpha = 0^\circ$ to $90^\circ$ in $5^\circ$ increments) at a fixed porosity ($\phi = 0.322$).
• Flow Conditions: Simulations span a pore Reynolds number range of $Re_p = 0.1$ to $200$ (based on bar size $B$).
• Validation: Numerical results were validated against Particle Image Velocimetry (PIV) measurements from Velten et al. (2026) for $\alpha = 30^\circ$ at $Re_p = 100$ and $200$.

Numerical Approach:

• Solver: Open-source CFD solver OpenFOAM-12 was used.
• Governing Equations: Incompressible Navier-Stokes and continuity equations were solved using Particle-Resolved Direct Numerical Simulation (PRDNS).
• Meshing: Unstructured meshes generated with Gmsh. A mesh sensitivity analysis determined that $\approx 1$ million cells per module provided sufficient accuracy.
• Boundary Conditions: Periodic boundary conditions were applied to replicate an infinitely repeating porous structure. Non-conformal coupling (NCC) interfaces were used between layers.
• Averaging: A "periodic averaging" technique was employed to average flow fields across identical modules, significantly reducing data storage requirements and improving statistical convergence.

Key Geometric Definitions:

• Channel-like: $\alpha \le 10^\circ$ (aligned bars, low resistance).
• Lattice-like: $\alpha \ge 15^\circ$ (interconnected voids, complex paths).
• Characteristic Lengths: The study compares two diameter definitions:
  1. Single-bar diameter ($d_{sb}$): Constant for all angles.
  2. Module-equivalent diameter ($d_{eq}$): Angle-dependent, derived from the wetted surface area of the entire module.

3. Key Contributions

1. Geometric Classification: The study establishes a quantitative classification of packed beds into channel-like and lattice-like regimes based on the bed-to-particle diameter ratio ($D/d_{eq}$) and hydraulic tortuosity ($\tau$).
   • Channel-like: $D/d_{eq} < 3$, $\tau < 1.1$.
   • Lattice-like: $D/d_{eq} > 3$, $\tau > 1.1$.
2. Superior Characteristic Length Scale: The authors demonstrate that the module-equivalent diameter ($d_{eq}$), which accounts for the angle-dependent wetted surface area, is far superior to the constant single-bar diameter ($d_{sb}$) for correlating friction factors and permeability in non-spherical, periodic packings.
3. Regime Transition Mapping: The study maps the transition from viscous to inertial flow, identifying that the critical Reynolds number ($Re_{crit}$) for the onset of inertial effects is generally $\approx 7.5$, but varies significantly with geometry (e.g., earlier onset at $\alpha = 55^\circ$).
4. Validation of PRDNS: Provides a high-fidelity numerical database validated against experimental PIV data for a complex, non-spherical geometry, a rare achievement in the field.

4. Key Results

Flow Structure and Friction Factor:

• Viscous Regime ($Re_p < 1$): The friction factor peaks at $\alpha = 25^\circ$. This is attributed to severe flow path constriction caused by the specific alignment of bars, creating high local shear stress.
• Inertial Regime ($Re_p > 50$): The angle yielding maximum friction shifts to $\alpha = 60^\circ$. Here, the geometry creates contraction-expansion sequences that induce flow separation, recirculation, and form drag, which dominate over viscous losses.
• Channel-like vs. Lattice-like: Channel-like geometries ($\alpha \le 10^\circ$) exhibit high permeability and low friction. Lattice-like geometries show heterogeneous velocity fields with strong recirculation zones.

Permeability and Drag Models:

• Ergun Correlation: Using the module-equivalent diameter ($d_{eq}$), the friction factor data for lattice-like geometries collapses well onto the Ergun correlation.
• Permeability Models:
  • The Blake-Kozeny model using $d_{eq}$ achieved the best agreement with DNS data for lattice-like geometries (Mean Absolute Percentage Error, MAPE = 12.3%).
  • Models using the constant single-bar diameter ($d_{sb}$) failed significantly (MAPE = 80.2%) as they could not capture the angle-dependent changes in wetted surface area.
  • All models underestimated permeability for channel-like geometries, as they are designed for multiply connected pore structures.

Forchheimer Coefficient ($C_F$):

• $C_F$ varies strongly with $\alpha$, peaking at $\alpha = 60^\circ$.
• Channel-like and $\alpha = 90^\circ$ configurations show delayed transition to the inertial regime, resulting in lower $C_F$ values within the studied $Re_p$ range.

Transition to Inertia:

• The transition to the inertial regime (defined as inertial contribution > 5% of total pressure drop) occurs at $Re_p \approx 7.5$ for most lattice-like geometries.
• The $\alpha = 55^\circ$ case showed the earliest transition ($Re_p \approx 5$), likely due to unique void-space connectivity promoting early localized recirculation.

5. Significance

This research provides a critical bridge between idealized spherical packing models and complex industrial realities. By isolating the effect of geometric arrangement (rotation angle) while keeping porosity constant, the study proves that macroscopic correlations (like Ergun) are insufficient without appropriate geometric scaling.

The findings are significant for:

• Reactor Design: Enabling more accurate prediction of pressure drops in reactors using non-spherical catalysts or structured packings.
• Model Development: Validating that using an angle-dependent equivalent diameter ($d_{eq}$) is essential for generalizing drag laws to non-spherical media.
• Operational Safety: Providing precise data on when inertial effects become dominant, which is crucial for preventing channeling or excessive pressure drops in high-flow applications.

The study concludes that while the Ergun equation can be adapted for these geometries using $d_{eq}$, future work is needed to develop geometry-specific models for the Forchheimer coefficient and to explore fully turbulent regimes at higher Reynolds numbers.