From Columns to Heaps: Dimensionless Similarity with PSD-Distributed Damköhler Numbers and Dual-Porosity Flow

This paper establishes a unified dimensionless framework that links particle size distributions and dual-porosity flow structures to Damköhler number distributions, enabling accurate scaling of reacting porous-flow systems from laboratory columns to industrial heaps by accounting for how microscopic heterogeneities break dynamic similarity.

The Gist

Imagine you are trying to figure out how fast a giant pile of crushed rocks will dissolve when you pour a special chemical liquid over it. This is how industries extract metals like copper or gold from ore. The problem is that the giant piles (called "heaps") at the mine are huge, but the tests are done in small columns in the lab.

This paper is like a translation guide that helps engineers understand how to make sure the small lab test accurately predicts what will happen in the giant pile. The author, Juan Segura, argues that simply making the lab column look like a tiny version of the heap isn't enough. You have to match the "personality" of the rocks and the flow of the liquid in a very specific, mathematical way.

Here is the breakdown of the paper's main ideas using simple analogies:

1. The Two Types of Similarity

To get a perfect prediction, you need two things to match:

• The Flow (Macroscopic): How the liquid moves through the pile.
• The Rocks (Microscopic): How the liquid gets inside the individual rocks to dissolve the metal.

The paper says that if you match the flow (like making sure the water moves at the same relative speed in both the lab and the mine), the liquid will spend the same amount of time in the system. However, if the size of the rocks is different, the chemistry breaks down.

2. The "Shrinking Core" Analogy

Imagine each rock is an onion. When the chemical liquid hits it, it starts eating away the outside, leaving a shrinking core of unreacted metal in the middle.

• Small onions get eaten very quickly.
• Big onions take a long time.

In a real pile, you don't just have one size of onion; you have a mix of tiny pebbles, medium rocks, and huge boulders. This mix is called the Particle Size Distribution (PSD).

3. The "Speed Limit" Problem (Film vs. Diffusion)

The paper explains that the speed at which a rock dissolves depends on how the chemical gets to it. There are two main scenarios:

• Scenario A: The "Film Control" (The Traffic Jam at the Door)
  Imagine the chemical has to wait in line to get through a thin film of water surrounding the rock.

  • The Rule: If you double the size of the rock, it takes twice as long to dissolve.
  • The Analogy: It's like a bus stop. If the bus (chemical) is slow to arrive, a big crowd (big rock) takes longer to clear, but it's a linear relationship.

• Scenario B: The "Diffusion Control" (The Maze Inside)
  Imagine the chemical has to squeeze through a tiny maze of pores inside the rock to reach the metal.

  • The Rule: If you double the size of the rock, it takes four times as long to dissolve (because the distance squared matters).
  • The Analogy: This is like a maze. If the maze is twice as wide, the path to the center is much, much longer.
  • The Paper's Big Finding: In this scenario, the tiny rocks (the fine tail of the distribution) act like a turbocharger. They dissolve so fast they dominate the early results, while the huge rocks act like anchors, dragging down the final result for a very long time. The paper shows that if you miss even a few tiny rocks in your lab test, your prediction for the giant pile will be wildly wrong.

4. The "Two-Lane Highway" (Dual-Porosity)

Some ores are like a sponge with two types of holes:

1. Big holes (Mobile): The liquid rushes through these fast.
2. Tiny holes (Immobile): The liquid gets stuck here, moving very slowly or not at all.

The paper introduces a new set of rules to describe how the chemical jumps between the "fast lane" and the "slow lane." If the chemical gets stuck in the slow lane, it can't reach the metal inside the rocks efficiently. The paper provides a way to measure this "stuck-ness" so engineers can account for it.

5. The "Magic Formula" (Dimensionless Groups)

The author creates a set of "magic numbers" (dimensionless groups). Think of these as a universal recipe.

• Instead of saying "Use 5 gallons of water on a 10-foot pile," the recipe says "Use a ratio of water-to-rock that equals X."
• The paper proves that if you match these specific ratios (especially the ones related to rock size and the "maze" inside the rock), you can trust that your small lab test will tell you exactly what will happen in the giant industrial pile.

Summary of the "Takeaway"

The paper warns engineers: Don't just scale up the size of the pile.
If you change the size of the rocks (the PSD) or the internal structure of the ore (the "maze" or dual-porosity) between your lab test and the real mine, the results will be misleading.

• For simple rocks: The size matters a little.
• For complex rocks (diffusion-controlled): The size matters a lot. The smallest rocks and the biggest rocks dictate the entire process.

The paper provides the mathematical tools to ensure that when you move from the lab to the mine, you aren't just guessing; you are mathematically guaranteed that the "personality" of the reaction remains the same.

Technical Summary

Technical Summary: From Columns to Heaps: Dimensionless Similarity with PSD-Distributed Damköhler Numbers and Dual-Porosity Flow

Problem Statement
Heap leaching remains a critical unit operation in hydrometallurgy, yet scaling up from laboratory columns to industrial heaps relies heavily on engineering judgment and empirical factors. A primary challenge is that while macroscopic hydrodynamic similarity (e.g., matching Peclet numbers) can be achieved between a column and a heap, microscopic similarity is often broken by differences in Particle Size Distribution (PSD) and internal pore structure. Existing mechanistic models are complex, making it difficult to isolate the specific dimensionless groups that govern behavior or to determine precisely what conditions must be matched to ensure meaningful scale-up. Specifically, the interaction between PSD, rate-controlling mechanisms (film vs. diffusion), and dual-porosity hydrology is not fully clarified in a compact, dimensionless framework.

Methodology
The authors develop a unified, dimensionless framework that decouples macroscopic hydrodynamic similarity from microscopic kinetic similarity. The analysis proceeds in three steps:

1. Macroscopic Hydrodynamics: The paper establishes that for geometrically similar heaps, dynamic similarity (matching Reynolds, Froude, and Peclet numbers) ensures identical dimensionless Residence Time Distributions (RTD) for the liquid phase.
2. Microscopic Kinetics (SCM): Using the Shrinking-Core Model (SCM), the work derives explicit mathematical transformations mapping a particle diameter distribution, $f(d_p)$, to a distribution of particle-scale Damköhler numbers, $g(Da)$. This is performed for three regimes:
   • External film control: $Da_{ext} \propto d_p^{-1}$.
   • Intraparticle diffusion control: $Da_{diff} \propto d_p^{-2}$.
   • Mixed control: Modeled as additive resistances, leading to a complex mapping where $Da_{eff} \approx Da_{ext} + Da_{diff}$.
     The analysis utilizes change-of-variable formulas to derive the probability density functions (PDFs) for these Damköhler distributions, highlighting how the PSD tails influence the overall kinetics.
3. Dual-Porosity Coupling: The framework incorporates dual-porosity hydrology (mobile and immobile domains) relevant to stratified ores. This introduces additional dimensionless groups: a capacity ratio ($\omega$) and an interporosity exchange Damköhler number ($Da_{ex}$), which govern the transfer of reagents between advective channels and the stagnant reactive matrix.

A practical workflow is proposed for parameter estimation, separating hydrodynamic calibration (via tracer RTD) from kinetic calibration (via leaching curves) in dimensionless time.

Key Results
Two numerical examples illustrate the framework's utility:

• PSD to Damköhler Mapping: For a lognormal PSD, the resulting Damköhler distribution under diffusion control is significantly more skewed and possesses a heavier upper tail than under film control. This is due to the $d_p^{-2}$ scaling, meaning small particles (the fine tail of the PSD) dominate the mean Damköhler number disproportionately.
• Heap Conversion Sensitivity: Simulations comparing fine ($d_{50}=8$ mm) and coarse ($d_{50}=20$ mm) PSDs show that while both film and diffusion mechanisms are sensitive to PSD, diffusion-controlled systems exhibit a much stronger dependence. Under diffusion control, coarsening the PSD results in a significantly larger absolute penalty in heap-averaged conversion compared to film control. The "fine" tail drives early-time recovery, while the "coarse" tail governs late-time recovery, creating a separation of time scales that is amplified by dual-porosity limitations.

Significance and Claims
The paper claims to provide a "compact, dimensionless framework" rather than a replacement for detailed numerical models. Its primary significance lies in clarifying the conditions required for physically meaningful scale-up in hydrometallurgical heap leaching.

The authors assert that matching only macroscopic hydrodynamic groups (e.g., irrigation rates, overall RTD) is insufficient if the PSD and dual-porosity structures differ. To achieve true similarity, one must also match:

1. The dimensionless PSD (scaled by heap height).
2. The implied distributions of particle-scale Damköhler numbers.
3. The dual-porosity parameters ($\omega$ and $Da_{ex}$).

The work concludes that diffusion-controlled leaching is far more sensitive to PSD tails and dual-porosity structures than film-controlled leaching. Consequently, the identified set of dimensionless groups offers a transparent language for interpreting laboratory testwork, comparing systems with different ore characteristics, and designing sensitivity studies to improve the robustness of scale-up from columns to industrial heaps.