Mixing of miscible liquids: Dimensionless scaling for intermediate-to-large density differences in a stirred tank

This study utilizes numerical simulations of a stirred tank with miscible liquids to demonstrate that while mixing time correlates positively with the Richardson number, a derived exponential scaling law based on Power, Froude, and Richardson numbers successfully collapses all data onto a single master curve for intermediate-to-large density differences.

The Gist

Imagine you are trying to mix a giant vat of two different liquids: a heavy, thick syrup at the bottom and a lighter, thinner juice on top. You drop in a giant spinning paddle (an impeller) to stir them together.

In the real world, this is a common task in factories making everything from medicine to wastewater treatment. But here's the catch: because the liquids have different weights (densities), the heavy one wants to stay at the bottom and the light one wants to float on top. This creates a "struggle" between the spinning paddle trying to mix them and gravity trying to keep them separated.

This paper is like a detective story where scientists used powerful computer simulations to figure out exactly how long it takes to get these two liquids perfectly mixed, and how to predict that time without having to build a physical tank and run expensive tests every time.

The Setup: A Digital Test Kitchen

The researchers built a virtual version of a standard industrial mixing tank.

• The Tank: It's a big cylinder with walls and four vertical fins (baffles) to stop the liquid from just spinning in a circle like a lazy river.
• The Paddle: A spinning blade in the middle.
• The Liquids: They simulated a 50/50 mix of a heavy liquid and a lighter liquid. They didn't use real chemicals; they just treated them as "heavy" and "light" fluids with the same thickness (viscosity).
• The Method: Instead of using standard math equations, they used a clever trick called the Lattice Boltzmann Method. Think of this as simulating the liquid not as a continuous blob, but as billions of tiny, invisible billiard balls bouncing around and colliding. This allowed them to see exactly how the turbulence (the chaotic swirling) behaved.

The Big Question: How Fast Can We Mix?

The main goal was to find a "magic formula" to predict the mixing time.

• The Variables: They changed two main things:
  1. How fast the paddle spins (Reynolds number): Faster spinning usually means more turbulence and faster mixing.
  2. How different the weights are (Richardson number): If the liquids are almost the same weight, they mix easily. If one is much heavier, gravity fights the mixing, creating layers that are hard to break.

The Discovery: The "Gravity vs. Spin" Battle

The researchers found some interesting patterns:

1. When Gravity Doesn't Matter (Same Weight):
   If the two liquids have the exact same weight, the mixing time is surprisingly consistent. No matter how fast you spin the paddle (within a certain range), the "dimensionless mixing time" (a fancy way of saying "how many turns of the paddle it takes") stays constant at about 20 turns. It's like a rule of nature: once the water is churning enough, spinning it faster doesn't make it mix any quicker in terms of paddle turns.

2. When Gravity Fights Back (Different Weights):
   As soon as the liquids have different weights, the heavy one wants to stay at the bottom. The heavier the difference, the harder it is to mix.

   • The Trend: The more different the weights are, the longer it takes to mix.
   • The Surprising Twist: If you keep the "weight difference" constant and just spin the paddle faster, the mixing time doesn't always go down. Sometimes, spinning faster actually makes it take longer to reach a specific point of mixing.
   • Why? Imagine the heavy liquid is like a thick blanket. If you spin the paddle too fast, you create a lot of energy, but the heavy liquid forms a stable "cap" that the lighter liquid can't penetrate. The energy is wasted on swirling the top layer while the bottom layer stays sealed off. It's like trying to stir a pot of soup where the heavy vegetables have settled into a solid block at the bottom; spinning the spoon faster just splashes the broth on top without breaking the vegetable block.

The Solution: A New "Master Curve"

The team's biggest achievement was creating a single, simple formula that combines all these factors. They found that if you look at the mixing time through the lens of three specific numbers (Power, Froude, and Richardson), all their messy data points collapse onto one smooth, exponential curve.

Think of it like this: Before, engineers had to guess or run hundreds of tests to see how a new liquid would mix. Now, they have a "recipe." If you tell them the weight difference and how fast you plan to spin, this formula predicts the mixing time with high accuracy.

The Bottom Line

The paper concludes that for these specific industrial tanks:

• Turbulence is key: Once the liquid is fully churning, the mixing behavior is predictable.
• Gravity is the boss: If the liquids have different densities, gravity creates a "stratification" (layering) that resists mixing.
• Faster isn't always better: In systems with heavy density differences, simply cranking up the motor speed doesn't guarantee faster mixing; sometimes it just creates a more stable separation.

The authors provide this new formula to help engineers design better mixing processes without needing to build expensive prototypes first. They plan to test this formula on different tank shapes and paddle types in the future, but for now, it works perfectly for the standard tanks they simulated.

Technical Summary

Technical Summary: Dimensionless Scaling for Intermediate-to-Large Density Differences in a Stirred Tank

Problem Statement
The mixing of miscible liquids is a critical unit operation in industrial processes, ranging from pharmaceutical production to wastewater treatment. While turbulent mixing in single-phase, homogeneous systems is well-characterized, the presence of density differences between fluids introduces buoyancy-driven stratification. This phenomenon creates vertically layered regions where gravitational forces counteract convective transport, potentially suppressing vertical mixing and increasing the time required to achieve homogeneity. Although empirical correlations exist for single-phase tracer mixing and specific stratified cases, a generally valid framework that consistently accounts for gravitational, inertial, viscous, and buoyancy effects across different flow regimes and geometries remains elusive. This study addresses the challenge of predicting mixing times in baffled stirred tanks containing two miscible liquids with intermediate-to-large density differences.

Methodology
The authors employed Computational Fluid Dynamics (CFD) using the Lattice Boltzmann Method (LBM) to simulate turbulent mixing in a baffled stirred tank. The system consisted of a 50/50 volume ratio of two miscible liquids with identical dynamic viscosities but varying densities ($\rho_l$ and $\rho_h$), where the lighter fluid was initially stratified above the heavier fluid.

• Numerical Approach: The simulations solved the incompressible Navier-Stokes equations using a D3Q19 lattice scheme. To handle high Reynolds numbers ($Re > 10^5$), a Smagorinsky-Lilly Large Eddy Simulation (S-L LES) model was implemented to model sub-grid turbulence.
• Scalar Transport: A mass-flux-based scalar transport model was adopted to track the concentration of the lighter liquid. This model coupled the scalar field to the momentum equation via a concentration-dependent gravitational body force, allowing for the simulation of buoyancy effects arising from local density variations.
• Simulation Space: A systematic parameter study was conducted by varying the impeller rotational speed ($N$) and the density of the lighter phase ($\rho_l$). This resulted in a broad range of dimensionless numbers: Reynolds numbers ($Re$) from 50,000 to 125,000 and Richardson numbers ($Ri$) from 0 to 14.
• Validation: The numerical accuracy was verified through a Grid Convergence Index (GCI) analysis, which estimated numerical uncertainty at approximately 2.39% on the production grid. Additionally, simulation results were validated against experimental tracer data from literature, showing good agreement in probe response and mixing time definitions.

Key Results
The study quantified mixing time ($t_m$), defined as the time required to reach 90% homogeneity, and expressed it in dimensionless form ($t^*_m = N \cdot t_m$).

1. Homogeneous Baseline ($Ri = 0$): In the absence of density differences, the dimensionless mixing time collapsed to a constant value of $t^*_m \approx 20$ across the entire investigated Reynolds number range. This confirms that the flow is fully turbulent and that inertial mixing dominates, consistent with established literature for baffled tanks.
2. Effect of Stratification ($Ri > 0$): As the Richardson number increased, indicating larger density differences, the dimensionless mixing time increased systematically. The data followed a clear exponential trend across all Reynolds numbers.
3. Proposed Scaling Law: The authors derived a master curve that collapses all simulation data onto a single exponential scaling relation:
   $$t^*_m = 20 \cdot e^{0.58 \cdot N_p \cdot Fr \cdot Ri}$$
   Where $N_p$ is the Power number, $Fr$ is the Froude number, and $Ri$ is the Richardson number. The pre-factor of 20 represents the inertial baseline, while the exponential term captures the stabilizing effect of buoyancy.
4. Non-Monotonic Behavior: When analyzing dimensional mixing times ($t_m$) at constant $Ri$, the study observed a non-monotonic dependence on $Re$ for higher Richardson numbers ($Ri \geq 8$). While increasing $Re$ (and thus impeller speed) initially reduces mixing time by enhancing turbulence, further increases lead to stronger density stratification (due to the coupling of parameters in the simulation setup), which suppresses vertical transport. This interplay results in a minimum mixing time at intermediate Reynolds numbers.
5. Flow Structure: Velocity field analysis revealed that at high $Ri$, the lighter phase can become "sealed off" from the impeller's power input, with vertical structures appearing primarily in the denser liquid. Power Spectral Density (PSD) analysis indicated that while buoyancy affects the spatial organization of the flow, the impeller-driven dynamics still dominate the energy cascade.

Significance and Claims
The paper claims to provide a compact, predictive correlation for turbulent mixing times in initially density-stratified systems within the fully turbulent regime. The derived scaling law, based on the combined influence of the Power, Froude, and Richardson numbers, offers a simple analytical expression for engineers and application experts.

The authors note that, to their knowledge, no comparable correlation for stratified systems in the fully turbulent regime has been previously reported. The significance of this work lies in its ability to streamline mixing process design by reducing the reliance on expensive real-world testing. The proposed model allows for the estimation of mixing times in systems where buoyancy effects are significant, capturing the complex competition between enhanced turbulence and strengthened stratification. The authors modestly suggest that future work will involve testing this correlation on different tank geometries and impeller types, including angled stirrers, to evaluate the robustness of the proposed scaling.