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REVIEW 3 major objections 5 minor 40 references

On the use of equilibrium models to describe dynamic adsorption regimes

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read PFO kinetics coupled to the Sips isotherm is structurally inconsistent: it yields abrupt breakthrough for α<1 and a rate constant that depends on inlet concentration.

desk verdict Solid analytical work on the PFO-Sips travelling wave model; the empirical indictment of PFO is suggestive but depends on an unverified small-dispersion reduction. read the letter →

arxiv 2607.17941 v1 pith:I6U5NUIQ submitted 2026-07-20 math-ph math.MPphysics.app-phphysics.chem-ph

M. Calvo-Schwarzwalder , A. Valverde , A. Cuesta López , A. Cabrera-Codony , U. Thorat , T.G. Myers This is my paper · ORCID
classification math-phmath.MPphysics.app-phphysics.chem-ph MSC 35C0776S0580A30
keywords adsorptionpseudo-firstordermodelSipsisothermtravellingwavesbreakthroughcurvepackedcolumnkineticmodellinglineardrivingforce
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to settle whether the pseudo-first-order (PFO) kinetic model, a workhorse of adsorption-column modelling for nearly a century, can be trusted as a physical description of breakthrough curves. It shows that when PFO is coupled to the Sips isotherm, the mathematics becomes structurally different from the underlying Sips kinetics: for Sips exponent α<1 the predicted outlet concentration stays exactly zero until a finite breakthrough time and then jumps, whereas the consistent Sips model gives a smooth rise. Fitting both models to experimental data for three different adsorbent–contaminant systems, the paper finds the PFO rate constant kLDF climbs with inlet concentration while the Sips adsorption rate ka stays essentially constant. The conclusion is that apparent good fits of PFO are curve-fitting artefacts, not evidence of a valid mechanism, so parameters extracted with PFO are not transferable across operating conditions.

What carries the argument

The travelling-wave reduction of the advection–dispersion–adsorption system. In scaled coordinates the Sips-PFO kinetic equation becomes dC/dη = γ g_α(C) with g_α(C) = [v C^α − (C^α + v − 1)C] / [v(C^α + v − 1)], where α = m/n is the Sips exponent and v = 1 + κ^{1/n} is the wave speed. This function decides the qualitative behaviour: for α ≤ 1 it has the correct sign on the interval (0,1), so a heteroclinic travelling wave exists; for α < 1 the integral ∫_0^{1/2} ds/g_α(s) converges, producing the finite cut-off η*_α and the abrupt breakthrough; for α > 1 the steady state at C = 0 becomes unstable and an extra root can block the wave entirely. The same function is then fitted to experimental

What would settle it

Run a column experiment in the α<1 regime, such as the Cu(II) system, with high-time-resolution sampling near breakthrough: the Sips-PFO model predicts outlet concentration identically zero until t_b and then a jump, while the Sips model predicts a smooth continuous rise; a measured smooth non-zero tail before the predicted t_b would falsify the structural claim. Alternatively, fix all operating conditions and vary inlet concentration by a factor of ten; if the fitted kLDF does not change, the central inconsistency is absent.

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Extended reading notes

Core claim

The paper's central discovery, on its own terms, is that the pseudo-first-order model combined with the Sips isotherm is not a mild approximation of consistent Sips kinetics but a different dynamical class. The travelling-wave analysis shows that for α=m/n<1 the Sips-PFO equation has a finite cut-off time η*_α: before that time the outlet concentration is exactly zero, so breakthrough is abrupt rather than smooth. Fitting to three datasets then shows an independent inconsistency: the fitted Sips rate ka varies by at most factors of 1.3, 1.1, 1.6 across inlet concentrations, while the fitted PFO rate kLDF increases by factors of 2.5, 2.0, 13.5. The paper concludes that PFO is structurally unf

Load-bearing premise

The analytical derivation assumes the Damköhler and inverse Péclet numbers are so small that accumulation and dispersion terms can be dropped from the mass-balance equation, and the paper does not verify this smallness for the three experimental datasets before applying the analytical breakthrough curves.

Editorial extensions

If this is right

  • If the Sips-PFO model is used in the α<1 regime, the predicted breakthrough curve is discontinuous: it stays at zero until t_b and then jumps, so any dataset with a smooth onset cannot be represented by the model's functional form.
  • The PFO rate constant kLDF extracted at one inlet concentration will not carry over to another concentration; predictions for a new experiment require refitting, so the model is not predictive.
  • The consistent Sips model gives a rate constant ka that remains stable across inlet concentrations, meaning a single ka can describe multiple breakthrough curves, which the PFO model cannot do.
  • For α>1, Sips-PFO travelling waves may fail to exist unless the wave speed is high enough, so the model can fail even to provide a bounded wave solution in some parameter regimes.
  • In the Henry limit (K_S c_in << 1) both models reduce to the same linear form, hiding the structural flaw precisely in the regime where the fitted curves look similar; agreement in that limit should not be read as validation of PFO.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the Sips isotherm is mathematically equivalent to the Hill equation, the same structural objection would transfer to any Hill-type kinetic system approximated by a linear driving force in q with a borrowed equilibrium curve; this extends beyond the adsorption examples tested here.
  • The observed concentration dependence of kLDF could be used as a cheap consistency diagnostic for any proposed kinetic model: fit at two inlet concentrations and check whether the rate constant moves; no single-curve R2 value can substitute for that test.
  • Published studies that report PFO fits for column adsorption could be re-examined to see whether their fitted kLDF values drift with inlet concentration or bed length; if they do, the curves are interpolation rather than physical prediction.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper analyzes a Pseudo-First-Order (PFO) kinetic model coupled with the Sips isotherm (S-PFO) for fixed-bed column adsorption, as an alternative to the thermodynamically consistent Sips kinetic model. Using a traveling-wave ansatz and assuming Da, Pe^{-1} \ll 1, the authors derive implicit analytical breakthrough curves for the S-PFO model for \alpha = 1, 1/2, 1/3, 2/3, and show that for \alpha < 1 the reduced first-order PDE yields a finite breakthrough time \hat{t}_b with an abrupt jump in outlet concentration. They compare these solutions with the Sips traveling-wave results and fit both models to three experimental datasets (toluene, Cu(II), Hg(II)). They report that the S-PFO model gives consistently worse fits and that the fitted k_{LDF} increases monotonically with inlet concentration, while the Sips rate constant k_a remains approximately constant. From this they conclude that the PFO model is 'inherently flawed and structurally unfit' for column adsorption modeling.

Significance. If correct, the paper would be a significant contribution to adsorption modeling: it provides closed-form traveling-wave solutions for a widely used but thermodynamically inconsistent kinetic formulation, and it challenges nearly a century of PFO-based column modeling with a clear, falsifiable observation (the concentration dependence of fitted k_{LDF}). The mathematical derivations in Section 4 and the appendices are careful and reduce correctly to known limits, and the experimental datasets cover three different systems. However, the central empirical and qualitative claims rest on the small-Da and small-Pe^{-1} reduction that is not verified for the experimental conditions. Because the paper's broad conclusion depends on that reduction, the current evidence is not yet sufficient; the analysis is sound in principle but needs either validation of the reduction or a more cautious interpretation of the results.

major comments (3)
  1. [Section 4.2-4.3, Eq. (39)-(41), Figures 4-5] The experimental validation and the central comparison between Sips and S-PFO rely on the reduction from Eq. (12) to the first-order PDE (21), which assumes Da, Pe^{-1} \ll 1. The paper states in Section 3.3 that these parameters 'are typically considered small' but never estimates them for the datasets in Table 1. For dataset 1, using the reported operating conditions and fitted k_a from Table 2, one estimates \ell \approx u\epsilon/(\rho_b q_{max} k_a) \approx 1.1\times 10^{-4} m; with a typical axial dispersion coefficient in the range D \sim 10^{-5}--10^{-4} m^2/s, Pe^{-1}=D/(\ell u) is of order 1--10, not small. If Pe^{-1} is not negligible, Eq. (21), and hence the analytical breakthrough curves (59) and (61), are not applicable to those experiments. The fitted k_{LDF} and k_a values in Table 2 would then be effective parameters that absorb neglected dispersion and accumulation, and
  2. [Section 5.5, Appendix D, Table 2] The predicted finite cut-off time arises from the reduced first-order PDE and would be regularized by dispersion; the structural distinction claim needs qualification.
  3. [Section 5.5, Appendix D, Table 2] The (m,n) selection is partly data-driven; a robustness check is recommended.
minor comments (5)
  1. [Eq. (46)] In the denominator of the expression for g_{1/3}, the term s^{1/2} appears; it should be s^{1/3} to be consistent with the definition of g_\alpha and with the subsequent change of variable u=s^{1/3}.
  2. [Table C.3 vs Table 2] The R^2 values for the S-PFO linear fit in Table C.3 differ substantially from the R^2 values reported in Table 2 for the same fits (e.g., dataset 3 at the highest c_{in}: 0.7176 vs 0.9853). While the text explains that the two tables use different definitions, the large discrepancy may confuse readers. Please state explicitly in the captions which quantity is plotted or tabulated and why the linear-fit R^2 differs from the breakthrough-curve R^2.
  3. [Section 5.1] Typo: 'effet' should be 'effect' in the sentence beginning 'The effet on the travelling wave profiles...'.
  4. [Section 3.3] The statement that Da and Pe^{-1} are 'typically considered small' would benefit from a reference or a concrete order-of-magnitude estimate for the systems of interest. As discussed in the major comments, this is not just a presentation issue for dataset 1.
  5. [References] Reference [12] is listed as 'under review' and reference [38] as 'submitted'; if possible, update these with publication or preprint identifiers. Also, reference [29] contains a typo in the DOI ('0.1016' instead of '10.1016').

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PFO travelling-wave results and the model comparison are derived from the stated equations, not from fitted outputs or self-citations.

full rationale

The central qualitative claims are derived in a self-contained way. The travelling-wave existence condition (alpha <= 1) and the finite cut-off time for alpha < 1 follow from the sign analysis of g_alpha in Eq. (27) and from the convergence of Phi_alpha(0+) in Appendix B; these are mathematical consequences of the stated PFO kinetic equation coupled to the Sips isotherm, not of fitted parameters. The Sips benchmark is not imported as a black box: the paper derives the Sips isotherm from F=0 in Eq. (6), and the comparison in Figs. 5-6 uses the explicit ODEs (26) and (27). The experimental section fits only the rate constants k_a and k_LDF after fixing isotherm parameters from equilibrium data; the reported variation of k_LDF with c_in is a fitted-parameter comparison, not a prediction forced by construction, and the same (m,n) configuration is used for both models, so the model-order selection does not bias the relative comparison. The main weakness noted in Section 3.4 is that Da, Pe^{-1} << 1 is assumed but not verified for the experimental datasets; however, this is an unverified approximation condition and a correctness risk, not a circularity, because Eqs. (21)/(59) are not defined in terms of the experimental outcomes they are used to judge. The self-citations ([11], [21], [25]) are to prior published derivations and datasets used as benchmarks or supporting context; none substitutes for the derivation performed in this paper. No exhibited equation reduces to another by construction, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard math plus domain assumptions: trace transport (u constant), negligible dispersion (Pe^-1 small), slow kinetics relative to flow (Da small), and a traveling wave ansatz. The free parameters are qmax, KS, ka, kLDF fitted to isotherm and breakthrough data; the (m,n) choice is made by chemical reasoning and fit quality.

free parameters (5)
  • qmax = 4.059 (dataset 1), 0.1208 (dataset 2), 0.3789 (dataset 3) mol/kg
    Maximum adsorption capacity, fitted to isotherm data (Table D.5).
  • KS = 406.09, 4.0016, 0.2688 (m^3/mol)^α
    Sips equilibrium constant, fitted to isotherm data (Table D.5).
  • ka = Varies per dataset and cin; e.g., 1.12e-1 to 1.07e-1 m^3/(mol s) for dataset 1
    Adsorption rate constant in Sips model, fitted to breakthrough curves (Table 2).
  • kLDF = Varies per dataset and cin; e.g., 6.05e-4 to 1.53e-3 s^-1 for dataset 1
    Mass transfer coefficient in PFO model, fitted to breakthrough curves (Table 2).
  • (m,n) = (1,1), (1,2), (1,3) for datasets 1-3
    Stoichiometric coefficients chosen by chemical mechanism and fit quality; not fitted continuously but selected.
assumptions (5)
  • domain assumption Trace adsorbate assumption: fluid velocity u is constant along the column (Section 2.1).
    Used to write Eq. (1) without a momentum equation.
  • domain assumption Da, Pe^-1 ≪ 1: dispersion and accumulation terms negligible relative to advection/adsorption (Section 3.4).
    Necessary to reduce the PDE to Eq. (21) and to use traveling wave solutions.
  • ad hoc to paper Traveling wave ansatz: solutions depend on x and t through η = x - L - v(t - th) (Section 4.1).
    Permits analytical progress; standard in the field.
  • domain assumption Sips kinetic model (3) is the physically consistent model; the isotherm is its steady state (Section 2.3.1).
    The paper's benchmark model; the equivalence to Hill kinetics is cited.
  • domain assumption Isotherm consistency requirement: a kinetic model should reduce to an isotherm at steady state (Section 1).
    Basis for dismissing PFO as inconsistent.

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Pith. "Pith review of On the use of equilibrium models to describe dynamic adsorption regimes." pith.science (2026). https://pith.science/paper/I6U5NUIQ

@misc{pith2026260717941,
  author       = {Pith},
  title        = {Pith review of: On the use of equilibrium models to describe dynamic adsorption regimes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6U5NUIQ}},
  note         = {Machine review of arXiv:2607.17941}
}
read the original abstract

We present a column adsorption model that couples a Pseudo-First-Order (PFO) kinetic formulation with the Sips isotherm framework. Using a traveling wave approximation, we derive analytical solutions for specific operating conditions. Qualitatively, these solutions deviate significantly from their pure Sips counterparts: instead of a smooth, continuous increase in concentration at the column outlet, the PFO-Sips model predicts an abrupt, sudden breakthrough. We validate these analytical solutions against diverse experimental datasets from the literature. The results reveal that the PFO-based model consistently underperforms compared to the original Sips formulation. Furthermore, this validation exposes fundamental inconsistencies within the PFO framework. We demonstrate that despite its widespread use in the literature for almost a century, the PFO model is inherently flawed and structurally unfit for describing column adsorption dynamics.

Figures

Figures reproduced from arXiv: 2607.17941 by the authors.

Figure 1
Figure 1. Schematic of the experimental setup. is measured until the end of the experiment. A schematic of this is provided in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Phase portraits of Eq. (27) for different values of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. (a) Phase portrait of Eq. (27) for α = 1.5. (b) Solutions of Eq. (27) for α = 1.5 with vˆ = 2 and vˆ = 4. In line with the analogous analysis done in [11], we will proceed to determine Φα for particular cases of interest. 4.3.1. Case α = 1 In this case we have m = n. In particular, for m = n = 1 this reduces to the L-PFO model. The function g1 is g1 (s) = vsˆ − (s + ˆv − 1)s vˆ(s + ˆv − 1) = s(1 − s) vˆ(s + ˆv − 1),… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Dependence of (a) η ∗ α and (b) tˆh − tˆb as functions of the travelling wave velocity vˆ. In both panels we have fixed γ = 1. 5. Results 5.1. Cut-off value η ∗ α Contrary to the case α = 1 or the Sips model for m ≤ n with m, n ≥ 1, the breakthrough curve for α < 1 pre…
Figure 6
Figure 6. Figure 6: Let us discuss to cases separately. The parameter γ works as a scaling factor. For γ ≪ 1 the breakthrough curves of S-PFO and Sips models coincide towards the end of the process, whereas for γ ≫ 1 the curves are similar after the initial breakthrough, but diverge as tˆ…
Figure 5
Figure 5. Figure 5: Travelling wave solutions for (a) α = 1/2, (b) α = 1/3, (c) α = 2/3. The limiting case with no desorption (vˆ = 1), where all the models collapse to the same profile (Eq. (56)) is shown for comparison. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the solutions to Eqs. (26) and (27), with [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Breakthrough curves for the limit KScin ≪ 1, computed numerically. Dashed lines correspond to the S-PFO model with m = n = 1, whereas solid lines correspond to the Henry model, Eq. (63). In both panels, we have set Da = 10−5 , Pe−1 = 0.01, γ = κ and (a) κ = 9 and (b) κ…
Figure 8
Figure 8. Figure 8: Linear fitting of Ψmn vs t − th for dataset 2 [7] (operating conditions in [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Breakthrough curves obtained for both Sips and S-PFO models with [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Breakthrough curves obtained for both Sips and S-PFO models with [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Breakthrough curves obtained for both Sips and S-PFO models with [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Concentration dependence of the S-PFO model parameter [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.