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arxiv: 2605.06144 · v1 · submitted 2026-05-07 · ⚛️ physics.flu-dyn

Recognition: unknown

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

Christian Witz, Johannes Khinast, Johan Remmelgas, Manuela Dubacher, Michael Dekner, Michael R. Wagner, Nikoletta Patsaki, Peter Varun Dsouza, Philipp Eibl, Stefan Reimann-Zitz

Authors on Pith no claims yet

Pith reviewed 2026-05-08 05:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords mixing timestirred tankmiscible liquidsdimensionless scalingRichardson numberdensity differencenumerical simulationmaster curve
0
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The pith

An exponential scaling using Power, Froude, and Richardson numbers collapses dimensionless mixing times for miscible liquids with varying densities in stirred tanks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Numerical simulations of a stirred tank with equal parts of two miscible liquids systematically varied the Reynolds and Richardson numbers to cover intermediate to large density differences. Mixing time showed a clear increase with Richardson number while Reynolds number effects were weaker and less consistent. An exponential scaling relation built from the Power, Froude, and Richardson numbers was derived and applied to the dimensionless mixing time. This relation brought all simulation results onto a single master curve, offering a compact description of mixing behavior under buoyancy-influenced conditions.

Core claim

In numerical simulations of a stirred tank mixing two miscible liquids at 50/50 ratio, the dimensionless mixing time follows an exponential scaling with the Power number, Froude number, and Richardson number that collapses data across the studied range of Reynolds and Richardson numbers onto one master curve.

What carries the argument

Exponential scaling relation constructed from the Power, Froude, and Richardson numbers to normalize dimensionless mixing time.

If this is right

  • Mixing time increases systematically with Richardson number due to buoyancy stratification.
  • The chosen dimensionless groups suffice to unify results without separate Reynolds corrections in the examined regime.
  • The master curve supplies a practical predictor for homogenization time once power input, impeller speed, and density difference are known.
  • Industrial stirred-tank design can use the scaling to estimate performance across moderate to large density contrasts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scaling may guide selection of impeller speed or power to achieve target mixing times when densities differ.
  • It could reduce reliance on full CFD runs for preliminary process design if the groups remain dominant.
  • Testing the same groups in tanks with different aspect ratios or impeller types would check broader applicability.
  • Continuous-flow or non-batch mixers might follow analogous collapse if buoyancy and inertia are similarly parameterized.

Load-bearing premise

Reynolds number effects remain secondary and the Power, Froude, and Richardson numbers together capture the dominant physics without needing extra terms across the studied range of density differences.

What would settle it

A new set of simulations or experiments at Reynolds numbers well outside the original range that produce dimensionless mixing times falling off the master curve.

Figures

Figures reproduced from arXiv: 2605.06144 by Christian Witz, Johannes Khinast, Johan Remmelgas, Manuela Dubacher, Michael Dekner, Michael R. Wagner, Nikoletta Patsaki, Peter Varun Dsouza, Philipp Eibl, Stefan Reimann-Zitz.

Figure 1
Figure 1. Figure 1: Schematic representation of the stirred tank system with characteristic dimensions view at source ↗
Figure 2
Figure 2. Figure 2: LBM Scheme: D3Q19 left, D3Q27 right Method To establish mixing times for the system depicted in view at source ↗
Figure 3
Figure 3. Figure 3: Temporal evolution of substrate concentration during mixing. The impeller induces progressive view at source ↗
Figure 4
Figure 4. Figure 4: Left: Dimensionless mixing times t ∗ m as a function of Ri. Different symbols depict different Re values. Right: t ∗ m at Ri=0 for different Re values. As a visual guide, a dashed line is shown at t ∗ m = 20. with g being the gravitational acceleration g = |g| = 9.81 m s 2 . Results The numerical accuracy and reliability of the simulation were evaluated prior to analysing the mixing times. Like any other C… view at source ↗
Figure 5
Figure 5. Figure 5: Plot of the best fit B values as a function of Np F r for different Re values. The dashed line indicates Eq. (26). value is consistent with literature data. Bouwmans et al. [8] report a dimensionless 95% mixing time of t ∗ m,95 = 26. Assuming an exponential decay of concentration fluctuations during the final homogenization stage, the mixing time corresponding to different homogenization criteria scales as… view at source ↗
Figure 6
Figure 6. Figure 6: t ∗ m data plot against our model Eq. (27 using A = 20 and B = 0.58 · Np · F r. With this scaling relation established, the behavior of the dimensional mixing times shown in view at source ↗
Figure 7
Figure 7. Figure 7: Dimensionless mixing time t ∗ m over Ri. Our data is depicted with symbols, the full lines are our model view at source ↗
Figure 8
Figure 8. Figure 8: Mixing time tm as a function of Re with different symbols for different constant Ri. 10 view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of reactor velocity cross-sections at view at source ↗
Figure 10
Figure 10. Figure 10: Power spectral analysis of the velocity field for the data shown in Fig. 9(a), (b) and (c). With view at source ↗
Figure 11
Figure 11. Figure 11: Power number Np versus spatial resolution 1/h (lattice nodes per meter). Based on a three-level GCI analysis following ASME V&V20, the numerical uncertainty on the production grid was estimated as unum,21 = 2.39%. The dashed line represents the reference power number for a Rushton turbine reported by Rushton et al. [51]. and the extrapolated relative error e 21 ext = view at source ↗
Figure 12
Figure 12. Figure 12: Mixing time validation based on probe response. Experimental data (solid lines) and simulation view at source ↗
Figure 13
Figure 13. Figure 13: Mixing time validation for probe 4. Experimental response (solid line) and simulation results view at source ↗
Figure 14
Figure 14. Figure 14: Comparison between the normalized probe response and the relative mixing index (RMI) for view at source ↗
Figure 15
Figure 15. Figure 15: Instantaneous snapshot of the volume-of-fluid (VOF) simulation at view at source ↗
read the original abstract

Mixing of miscible liquids is an essential process in multiple industrial settings, usually with the intent to homogenize the product. This seemingly simple process is in fact a complex hydrodynamic problem that has a direct impact on the product quality. In this study, numerical simulations of a stirred tank were performed with a 50/50 ratio of liquids and systematically varied the Reynolds and Richardson numbers. A positive correlation between the mixing time and the Richardson number was observed, as reported in the literature. The influence of the Reynolds number was not as pronounced and clear. Based on the Power, Froude and Richardson numbers, we were able to derive an exponential scaling for the dimensionless mixing time that collapsed all our data onto one master curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript reports numerical simulations of mixing two miscible liquids at a 50/50 volume ratio in a stirred tank, with systematic variation of Reynolds and Richardson numbers. It observes a positive correlation of mixing time with Richardson number (consistent with prior literature) but finds Reynolds number effects less pronounced. Using the Power, Froude, and Richardson numbers, the authors derive an exponential scaling for the dimensionless mixing time that collapses all simulation data onto a single master curve.

Significance. If the data collapse is robust, the scaling offers a practical engineering correlation for mixing times under intermediate-to-large density differences, where many existing models assume small density contrasts. The empirical collapse onto Power-Froude-Richardson groups is a concrete strength that could guide industrial stirred-tank design, though its value is limited by the absence of independent validation or theoretical derivation.

major comments (3)
  1. [Methods] Methods section: The manuscript provides no information on grid resolution, grid-independence tests, numerical scheme details, or quantitative error estimates for the computed mixing times. These omissions are load-bearing because the central claim rests on the reliability of the simulation data used to fit and collapse the exponential scaling.
  2. [Results] Results and scaling section: The exponential scaling is presented as derived from the Power, Froude, and Richardson numbers, yet the exact functional form (including fitted coefficients) and the precise definitions of these groups under variable density are not stated. Without these, it is impossible to assess whether the collapse is a genuine prediction or a post-hoc fit, directly affecting the claim's generality.
  3. [Results] Validation: No comparison of the simulated mixing times against experimental data or established benchmarks is reported, even for the limiting case of equal densities. This weakens confidence in the data collapse, especially given the noted uncertainty in Reynolds-number effects.
minor comments (2)
  1. [Abstract] The abstract states that Reynolds-number influence 'was not as pronounced and clear' but does not quantify this (e.g., via partial derivatives or additional figures at fixed Ri). Adding such quantification would clarify the decision to omit Re from the scaling.
  2. [Results] Notation for the dimensionless mixing time and the exact exponential expression should be introduced explicitly with an equation number in the main text to improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on our manuscript. We have carefully considered each comment and revised the manuscript to improve clarity and completeness. Below, we provide point-by-point responses to the major comments.

read point-by-point responses

  1. Referee: [Methods] Methods section: The manuscript provides no information on grid resolution, grid-independence tests, numerical scheme details, or quantitative error estimates for the computed mixing times. These omissions are load-bearing because the central claim rests on the reliability of the simulation data used to fit and collapse the exponential scaling.

    Authors: We fully agree that these details are essential for assessing the reliability of our results. In the revised manuscript, we have expanded the Methods section to include information on the computational grid resolution, the outcomes of grid-independence studies demonstrating that mixing times are converged to within acceptable tolerances, the specific numerical schemes used for solving the governing equations, and quantitative estimates of numerical errors based on our sensitivity analyses. revision: yes

  2. Referee: [Results] Results and scaling section: The exponential scaling is presented as derived from the Power, Froude, and Richardson numbers, yet the exact functional form (including fitted coefficients) and the precise definitions of these groups under variable density are not stated. Without these, it is impossible to assess whether the collapse is a genuine prediction or a post-hoc fit, directly affecting the claim's generality.

    Authors: We appreciate this observation. The scaling relation was obtained by fitting the simulation data, and we have now explicitly stated the functional form in the revised Results section, including the fitted coefficients. We have also provided the precise definitions of the Power, Froude, and Richardson numbers as applied to the variable-density flows in our simulations, clarifying how density variations are accounted for in each group. revision: yes

  3. Referee: [Results] Validation: No comparison of the simulated mixing times against experimental data or established benchmarks is reported, even for the limiting case of equal densities. This weakens confidence in the data collapse, especially given the noted uncertainty in Reynolds-number effects.

    Authors: We acknowledge the value of validation against experiments. Our study is numerical in nature, and direct experimental comparisons were not included in the original manuscript. In the revision, we have added a paragraph discussing our results in the context of existing literature for the equal-density limit, where our findings align with known trends for mixing times at high Reynolds numbers. For cases with significant density differences, we note that experimental data are limited, which motivated our numerical approach. We have also highlighted the less pronounced Reynolds number effects as an area requiring further study. revision: partial

Circularity Check

1 steps flagged

Exponential scaling fitted to collapse simulation data onto master curve

specific steps
  1. fitted input called prediction [Abstract]
    "Based on the Power, Froude and Richardson numbers, we were able to derive an exponential scaling for the dimensionless mixing time that collapsed all our data onto one master curve."

    The scaling is obtained by choosing an exponential form in Po, Fr and Ri that forces collapse of the simulation results (varying Re and Ri at fixed 50/50 ratio). The 'derivation' is therefore the fitting procedure itself; the collapsed master curve is guaranteed once the functional dependence is selected to match the data trends.

full rationale

The central result is an empirical exponential scaling for dimensionless mixing time constructed from Power, Froude and Richardson numbers to collapse the authors' own simulation dataset. While the paper presents this as a derivation, the explicit statement that it collapses 'all our data' indicates the functional form and coefficients were selected to match the observed trends rather than predicted independently from first principles. No self-citation chain or definitional loop is present, but the 'prediction' reduces to a post-hoc fit of the input data. The assumption that Re effects are secondary is stated directly from the same dataset, supporting the choice of groups without external validation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim relies on the assumption that these dimensionless numbers suffice and that the scaling parameters are general rather than specific to the simulated cases.

free parameters (1)
  • exponential scaling coefficients
    The exponential form likely includes fitted constants to achieve the data collapse on the master curve.
axioms (1)
  • domain assumption Mixing time depends primarily on Power, Froude, and Richardson numbers for the studied conditions
    Invoked when deriving the scaling from the simulation results.

pith-pipeline@v0.9.0 · 5464 in / 1266 out tokens · 40652 ms · 2026-05-08T05:54:48.327463+00:00 · methodology

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