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Capturing the calendering U-shape in lithium-ion electrode thermal conductivity

T0 review · 3 major / 6 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read A compression-indexed contact term on a Knudsen-corrected Zehner–Bauer–Schlünder base reproduces the measured U-shaped through-plane thermal conductivity of calendered lithium-ion electrodes.

desk verdict Solid, honest ZBS extension that finally captures the calendering U-shape with a process-indexed contact term; the fit is real but the coefficients and anode mechanism remain under-determined until texture/SEM arrive. read the letter →

arxiv 2607.11521 v1 pith:DGKDJ7W2 submitted 2026-07-13 cond-mat.mtrl-sci physics.chem-ph

Julius Störk This is my paper · ORCID
classification cond-mat.mtrl-sciphysics.chem-ph
keywords effectivethermalconductivitylithium-ionbatterycalenderingKnudseneffecteffective-mediumtheoryZehner–Bauer–Schlünderinlinequalitycontrolelectrodemicrostructure
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

Battery thermal models need the through-plane conductivity of every porous electrode layer, but that conductivity does not rise steadily as calendering densifies the coating. Measurements show a U-shape: conductivity first falls, then recovers. Porosity-only effective-medium formulas cannot produce that dip. This paper extends the classic Zehner–Bauer–Schlünder packed-bed closure with a Knudsen pore-gas correction and a process-dependent solid-bridge fraction that can damage early and rebuild later under the calender rolls. Across 27 calendering states on graphite anodes and NMC cathodes the model cuts average error from about 31% to 4.5%, near the noise of the reconstructed laser-flash target. The result is that process-driven contact and, for graphite, flake reorientation must be carried explicitly if the measured minimum is to be captured.

What carries the argument

The compression-indexed bridge fraction φ(Π) = max(0, φ₀ + aΠ + bΠ²) multiplying a solid-bridge channel inside the flattened-contact Zehner–Bauer–Schlünder unit cell, always with the Smoluchowski/Knudsen correction on the pore gas; for graphite an optional Hermans orientation factor S(Π) that rotates the anisotropic solid conductivity toward the c-axis floor.

What would settle it

Same-sheet XRD (002) texture and ion-milled SEM contact-area measurements on the calendered graphite sheets: if the Hermans factor does not rise and saturate near the conductivity minimum while NMC shows no conduction-relevant texture change, reorientation is ruled out and the anode minimum collapses to a contact-only account with an unexplained low solid conductivity.

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Extended reading notes

Core claim

The central claim is that a low-dimensional, calendering-aware contact term φ(Π) on a Knudsen-corrected Zehner–Bauer–Schlünder base is sufficient to reproduce the through-plane conductivity U-shape that porosity-only and static-contact closures miss. For NMC cathodes the dip is contact-network shear then interlocking; for graphite anodes the same contact term works, and a bounded flake-reorientation account also fits and explains the low through-plane solid conductivity by driving heat across the weak c-axis. The calibrated quantity is the LFA-derived apparent coating conductivity after foil subtraction, and parameters are reported as grouped bridge conductances and likelihood valleys rather

Load-bearing premise

That a three-coefficient quadratic contact law fitted separately to only six-to-eight calendering states per electrode family is capturing real damage-and-recovery physics rather than residual curve flexibility, even though solid conductivity and as-coated bridge fraction remain partially confounded.

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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 / 6 minor

Summary. The manuscript proposes a calendering-aware extension of the Knudsen-corrected Zehner–Bauer–Schlünder (ZBS) closure for the through-plane effective thermal conductivity of lithium-ion electrode coatings. A compression-indexed contact fraction φ(Π)=max(0, φ0+aΠ+bΠ²) is added to a ZBS base so that the model can reproduce the measured non-monotonic (U-shaped) dependence of λ_eff on calendering across four electrode families (thin/thick graphite anodes and NMC622/NMC811 cathodes; 27 states). Ablation shows MAPE falling from 31.1% (zero-fit M0) to 13.5% (constant contact M1) to 4.5% (process-dependent M2), with leave-one-state-out error 7.8%. For graphite, a secondary reorientation model MR that couples a Hermans factor S(Π) into anisotropic solid conductivity is shown to fit comparably with one fewer parameter, while remaining degenerate with contact-only fits on conductivity data alone; for isotropic NMC, reorientation is inert and contact evolution is required. The authors carefully scope absolute parameters to an LFA-derived, source-subtracted coating target, report grouped bridge conductance and profile-likelihood valleys rather than over-interpreted point estimates, and supply open code, inverse feasibility checks, and a mechanism-discriminating experimental campaign.

Significance. If the primary claim holds, the paper supplies the first low-dimensional analytical closure in the tested hierarchy that captures the calendering U-shape rather than a monotonic rise, which matters for process-aware battery thermal models and for treating λ_eff as a process-dependent input rather than a handbook constant. Strengths that raise the contribution above a pure curve fit include: the explicit ablation hierarchy and held-out LOSO; honest method-scoping (LFA vs GHP scale shift 1.8–2.4×); profile-likelihood and Bayesian identifiability analysis that correctly demotes λ_s and φ0 to a valley and elevates the bridge product G0; the cathode inertness test that cleanly requires contact evolution; orthogonal consistency checks (in-plane conductivity via trace conservation, temperature-coefficient sign, XRD texture trend); and a fully open, differentiable JAX implementation with inverse and falsifiable campaign design. These make the work useful even if the quadratic coefficients themselves are not uniquely physical.

major comments (3)
  1. §3.4, Eq. (7) and §5.3/Table 3: The central claim that process-dependent microstructural evolution is necessary is supported by the ablation (M0/M1 cannot produce the dip; M2 does). However, with four free parameters on only 6–8 states per family, the sign-constrained quadratic φ(Π) is still the lowest-order form that can absorb a dip-then-recovery residual. The manuscript already shows low-order polynomials tie on LOSO and that a free spline is unstable (§5.6). The load-bearing claim should be tightened in the abstract/conclusion to “a process-dependent contact term is necessary within the tested hierarchy,” with the specific (a,b) coefficients and G0 values presented as grouped, family-scoped descriptors rather than uniquely identified contact physics, unless the adhesion/SEM campaign of §7 is completed.
  2. §5.4, Table 4 and Eqs. (8)–(9): For graphite, MR and M2 fit equally well (1.8–1.9% thin; 5.2–5.4% thick), and the authors correctly call this a structural degeneracy. The abstract and primary-claim language still present reorientation as the graphite mechanism that “initially reduces favourable through-plane pathways.” That should be demoted consistently to the secondary, bounded claim already stated in the claim-scope paragraph, with the unique identification deferred to same-sheet XRD texture. Leaving the abstract stronger than the body overstates what conductivity data alone establish.
  3. §4.1 and §5.9 / inverse_realdata: The inverse porosity QC is demonstrated at ±0.008 RMSE on synthetic 3%-noise data, but on the real Gandert sheets under a held-out protocol the RMSE is ~0.05 (only 44% of sheets inside ±0.02). The abstract and §5.9 framing still lean on the synthetic figure for “quality control.” Either the real-data inverse performance should be the headline number, or the inverse should be clearly labelled a numerical feasibility study pending the metrology campaign, so that the primary thermal-closure claim is not diluted by an overstated application.
minor comments (6)
  1. Fig. 5 error bars are the reconstruction uncertainty (~5%); state explicitly in the caption that they are not the primary measurement scatter alone, to avoid conflating the four error quantities defined at the start of §5.
  2. Table 1 and Table 3: mark bound-active λ_s values more prominently (already noted with †) and consider moving the full (λ_s, φ0, a, b) tuples to SI, keeping only G0 and sign(a) in the main table to reinforce the grouped-quantity message.
  3. Notation: Π is introduced as compression rate in Eq. (1) but sometimes read as a pressure-like variable; a one-line reminder that Π = 1 − s_co/s_co,0 (thickness-based) would help non-specialist readers.
  4. §5.2 / separator discussion: the compensatory continuum sphere-pack agreement is a useful diagnostic; a short sentence cross-referencing the two-pressure experiment of §7 would tighten the falsifiability link.
  5. References and arXiv date (10.07.2026 / v1 13 Jul 2026) look future-dated relative to typical review timelines; confirm versioning and any dual-submission status for the record.
  6. Minor typography: “calendering U-shape” vs “u-shape” is inconsistent in title/abstract/body; pick one capitalization convention.

Circularity Check

2 steps flagged · score 3.0 of 10

Mild fitted-form circularity on the U-shape: the quadratic φ(Π) is chosen as the lowest-order dip-capable ansatz and fitted to the same series; necessity via ablation remains independent content.

  1. fitted input called prediction [Sec. 3.4 Eq. (7); Sec. 5.3 ablation / Fig. 5]
    "The lowest-order polynomial able to express a dip followed by a recovery is a quadratic, so we set φ(Π)=max(0,φ0+aΠ+bΠ²)… Making the contact fraction process-dependent (M2, Eq.(7)) reaches 4.5%… and reproduces the u-shape (Fig. 5), where the dip reflects early bridge damage (a<0 in all four families)"

    The functional form is chosen specifically because it can produce a non-monotonic dip-then-recovery; a is free to be negative and b free to recover. Fitting (φ0,a,b) to the same 6–8 calendering states that exhibit the U-shape then reporting that M2 “reproduces the u-shape” is largely by construction of ansatz capacity. M0/M1 failure still shows process dependence is needed, but the existence of the dip under M2 is not an independent prediction—it is the fit of a dip-capable form to dip-containing data.

  2. fitted input called prediction [Sec. 5.9 Inverse feasibility; Fig. 8]
    "On a synthetic production batch of 200 calendered sheets (true porosities drawn from N(0.30,0.012)… with 3% relative measurement noise), Newton inversion (Eq.(10)) recovers porosity with a root-mean-square error of 0.0078 absolute and zero bias"

    Synthetic porosities are inverted through the same closure that generates λeff(ε); recovery to ±0.008 is a self-consistency / numerical feasibility check, not an external prediction. The paper labels it a feasibility study and reports coarser ~0.05 RMSE on real held-out sheets, so the circularity is minor and disclosed rather than load-bearing for the primary claim.

full rationale

The paper does not claim a zero-shot first-principles derivation of the calendering U-shape from independent contact-area or texture data. Its primary result is a calibrated hierarchy: Knudsen-corrected ZBS (M0) and constant-contact (M1) cannot produce the measured non-monotonicity, while a sign-constrained quadratic φ(Π) fitted per family does, cutting MAPE 31.1%→4.5%. That necessity argument is not circular—it is an ablation against nested baselines on the same target. The soft circularity is narrower: Sec. 3.4 explicitly selects the quadratic as the lowest-order polynomial able to express damage-then-recovery, then Sec. 5.3 reports that the fitted form “reproduces the u-shape.” With a free to be negative and b free to recover, non-monotonicity is capacity of the ansatz, not an independent prediction of the dip’s existence. The paper largely owns this (phenomenological, not derived; grouped G0; structural degeneracy of MR vs M2; method-scoped LFA scale), so the central claim still has independent content. No self-citation load-bearing chain, uniqueness theorem, or renaming of a known result was found. Orthogonal checks (in-plane via trace conservation, temperature-coefficient sign, external XRD trend overlays) are not forced by the through-plane fit. Score 3 reflects one clear fitted-input-as-reproduction step without collapse of the necessity claim.

Assumptions & free parameters 7 free parameters · 7 assumptions · 2 invented entities

The central U-shape claim rests on the classical ZBS unit cell plus Knudsen gas physics (prior literature), plus a low-order phenomenological contact law whose coefficients are fitted per family, plus optional graphite orientation kinematics. Free parameters outnumber independent states modestly; invented structure is the process-indexed bridge fraction and the Hermans-coupled through-plane solid conductivity, both given external handles (adhesion/SEM, XRD) that are proposed but not measured on these sheets.

free parameters (7)
  • λs (through-plane solid conductivity per family)
    Fitted in M2 within anisotropy or NMC bounds; often bound-active and confounded with φ0; reported only via valleys/grouped quantities.
  • φ0 (as-coated bridge area fraction)
    Fitted per family; product G0=φ0λb is the robust contact descriptor.
  • a (shear-damage rate in φ(Π))
    Fitted with a constrained negative; carries the early-calendering downswing.
  • b (interlocking recovery in φ(Π))
    Fitted with b≥0; zero when calendering window never reaches interlocking (e.g. NMC811).
  • S0, s (Hermans as-coated factor and alignment rate, MR)
    Fitted for graphite reorientation branch; absolute S0 sensitive to literature λa.
  • β (gas–surface jump coefficient)
    Held at 1.64; varied 1.5–2.0 moves electrode predictions ≤1.7% but separator more; not fitted to primary data.
  • λa, λc, λb (constituent conductivities)
    Literature-anchored constants (e.g. λa=150, λc=6 W m−1 K−1; bridge values ~130 anode / ~24 cathode) that set contrast and G0 scale.
assumptions (7)
  • domain assumption Zehner–Bauer–Schlünder packed-bed unit cell (fluid channel + particle-core channel with deformation parameter B) is the correct morphology for low-additive electrodes.
    Sec. 3.2; justified by morphology diagnostics vs Maxwell–Eucken and Bruggeman in Sec. 5.2.
  • domain assumption Smoluchowski/Knudsen correction λg,eff=λg/(1+2β Kn) with single hydraulic pore size applies to dry electrode pores under helium purge.
    Sec. 3.3; credited to Oehler; always-on, no free fit in primary hierarchy.
  • ad hoc to paper Calendering contact evolution is representable by the lowest-order quadratic φ(Π)=max(0,φ0+aΠ+bΠ²) with a<0, b≥0.
    Sec. 3.4; motivated by adhesion dip/recovery but not derived from measured contact area; benchmarked against higher-order forms in Sec. 5.6.
  • domain assumption Steady diffusive conduction only; radiation and pore convection negligible at ambient coating scale (A1).
    Sec. 3.1; ambient scoping; temperature dependence treated only as a sign check.
  • domain assumption A single lumped bridge fraction φ absorbs intra-coating bridges and residual coating–collector interface left by foil subtraction (A4).
    Sec. 3.1; acknowledged as inseparable without paired GHP/adhesion/SEM metrology.
  • ad hoc to paper Graphite through-plane solid conductivity follows Hermans orientation: λ⊥s(S)=(1−S)(2λa+λc)/3 + S λc with S=clip(S0+sΠ,0,1).
    Sec. 3.5; crystallographic, bounded, inert for isotropic NMC; degenerate with contact-only fit on conductivity alone.
  • domain assumption Calibration target is LFA-derived apparent coating conductivity after series foil subtraction under helium.
    Sec. 4.1; absolute parameters method-scoped (1.8–2.4× GHP shift).
invented entities (2)
  • Compression-indexed contact fraction φ(Π) (quadratic damage–recovery law)
    purpose: Inject process-dependent bridge-network evolution so the closure can produce a non-monotonic λ_eff(Π) U-shape.
    Not measured contact area; coefficients fitted thermally. Independent handles proposed (pull-off adhesion, ion-milled SEM) but not on these sheets.
  • Calendering-driven Hermans factor S(Π) coupled into anisotropic solid conductivity for graphite
    purpose: Explain anomalously low fitted through-plane λs and provide a parsimonious anode-specific mechanism for the dip.
    Trend consistent with external XRD texture literature (Baade, Malifarge) and in-plane prediction vs Loges, but same-sheet texture not measured; structurally degenerate with φ(Π).

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Pith. "Pith review of Capturing the calendering U-shape in lithium-ion electrode thermal conductivity." pith.science (2026). https://pith.science/paper/DGKDJ7W2

@misc{pith2026260711521,
  author       = {Pith},
  title        = {Pith review of: Capturing the calendering U-shape in lithium-ion electrode thermal conductivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DGKDJ7W2}},
  note         = {Machine review of arXiv:2607.11521}
}
read the original abstract

Calendering is a key manufacturing step in lithium-ion electrode production, increasing volumetric energy density by reducing electrode porosity. Its effect on through-plane effective thermal conductivity, however, can be non-monotonic: measurements of graphite-based anodes show an initial decrease in thermal conductivity during early calendering followed by recovery at higher compaction. Conventional porosity-based effective-medium closures cannot reproduce this U-shaped behaviour. We develop a calendering-aware extension of the Zehner--Bauer--Schl\"under model that combines a Knudsen-corrected porous-medium baseline with a compression-indexed contact contribution. For graphite electrodes, the model represents the competing effects of increasing particle contact and calendering-induced reorientation of anisotropic graphite particles, which initially reduces favourable through-plane heat-transport pathways. For quasi-isotropic NMC cathodes, the observed response is instead captured through process-dependent contact-network evolution. Across 27 calendering states spanning thin and thick graphite anodes and NMC622 and NMC811 cathodes, the proposed closure reduces the mean absolute percentage error from 31.1% for the zero-fit reference model to 4.5%. The result shows that incorporating process-dependent microstructural evolution is necessary to capture the measured conductivity minimum. Validation across additional electrode formulations, thicknesses, and chemistries remains necessary to assess transferability.

Figures

Figures reproduced from arXiv: 2607.11521 by the authors.

Figure 1
Figure 1. Physical picture of the closure. Through-plane heat crosses a packed coating by two parallel routes: through particle cores (λs) and across inter-particle bridge/contact zones (φλb), with the pore phase (gas or electrolyte, Knudsen-corrected) in between (a). Calendering (Π ↑) compacts the bed and reshapes the contact network, captured by the process-dependent contact term φ(Π) for every family (b). For the graphite … view at source ↗
Figure 2
Figure 2. What is measured and what is fitted. Top: the calibration target is the apparent through-plane coating conductivity λco obtained from a coated-stack laser-flash (LFA) measurement by source subtraction of the collector foil; it is method-scoped to LFA. Bottom: the model hierarchy, the ZBS base M0, plus the always-on Knudsen pore-gas correction, plus a constant contact fraction (M1), plus the process-dependent contact… view at source ↗
Figure 3
Figure 3. Retained fraction of pore-gas conduction, Eq. (4), at 293 K and 1 atm for air (solid) and helium (dashed). Markers: hydraulic pore sizes of the three reference systems (air curve). Shaded band: measured polyolefin-separator pore diameters; inside it, the air mean free path exceeds the pore size. 0.2 0.3 0.4 0.5 porosity 0 1 2 3 4 eff [W m 1 K 1 ] (a) anode (graphite) electrolyte-filled dry, continuum dry, Knudsen 0.… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Zero-fit λeff(ε) for (a) graphite anode, (b) NMC cathode, (c) PE separator: electrolyte-filled, dry￾continuum, and dry-Knudsen variants. Green band: typical post-calendering porosity window. 0.24 to 32W m−1 K−1 . More informative is that the structure-specific baseline…
Figure 5
Figure 5. Figure 5: In-sample calibration against the LFA-derived target. Calibration curves for all 27 calendering states. M0 is the zero-fit ZBS+Kn reference; M1 adds a fitted constant contact fraction; M2 adds the process-dependent φ(Π) term. The constant-contact curve removes most of …
Figure 6
Figure 6. Figure 6: The anode mechanism competition. (a) The reorientation closure MR (solid) and the contact quadratic M2 (dashed) describe the graphite u-shape equally well and are not distinguished by conductivity data; MR is the more parsimonious member (one fewer parameter, physicall…
Figure 7
Figure 7. Figure 7: Trend-level consistency of the reorientation mechanism with measured graphite texture (Baade et al. [9], Table I). Measured degree of preferred (002) orientation (right axis, squares) is high as-coated and rises modestly with calendering; the model Hermans factor S (le…
Figure 8
Figure 8. Figure 8: Inverse porosity QC on a synthetic 200-sheet batch at 3% measurement noise: (a) parity with predicted ±σε bars and the ±0.02 specification window; (b) recovery-error histogram against the first-order uncertainty prediction of Eq. (11). Calender / semi-dry extrusion (ro…
Figure 9
Figure 9. Figure 9: The differentiable closure as a closed-loop inline monitor for continuous (semi-dry, granulate-based) electrode production. A fast through-plane thermal reading is inverted in microseconds (Newton on the JAX closure, Eq. (10)) to porosity ε, the as-coated bridge conduc…
Figure 10
Figure 10. Figure 10: External / orthogonal consistency checks (not calibration tiers). Calibrate once, check three inde￾pendent ways. The closure is calibrated only on room-temperature, through-plane laser-flash data, yet retains the relevant trend and ordering for three quantities it nev…
Figure 11
Figure 11. Figure 11: The calendering setpoint as a three-way design decision, all driven by the reorientation state S(Π). (a) One setpoint, three normalised outcomes: energy density rises monotonically, rate 1/τ⊥ falls, and heat removal kcell traces the u-shape, with the thermal-dip band …

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    Sample matrix.Six calendering states fromΠ = 0to the production maximum (single batch, single-side coatings); duplicate as-coated sheets for theφ0 re-anchor; each state measured dry and electrolyte-soaked

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    Porosity ground truth.Areal mass; flat-anvil micrometer thickness at≤0.3 N mm−2(not a ball-tip dial gauge, which compresses the coating [13]); helium-pycnometric solid density; GUM uncertainty budget

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    Conductivity.Through-plane laser flash with penetration-model evaluation [46], recording the purge gas (it is the pore gas), or a guarded hot plate;20◦C; n = 4repeats per sheet, bringing3% single-shot noise to1.5%

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    most informative for separators

    Two-pressure Knudsen discrimination(most informative for separators; recommended for the most porous electrode state): repeat the dry measurement at about1000and at or below100mbar chamber pressure. By Eq.(4), pore-gas conduction depends on pressure throughKn∝1/p, while skelet...

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    Equation(7) claims the conductivity-relevant contact fraction tracks interface damage and interlocking; thenφ(Π)and adhesion must share sign structure and dip location

    Adhesion correlation.Pull-off adhesion strength at every calendering state. Equation(7) claims the conductivity-relevant contact fraction tracks interface damage and interlocking; thenφ(Π)and adhesion must share sign structure and dip location

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    Cross-section image quantification.Ion-milled SEM cross-sections at each calendering state, with the contact and bridge areas quantified by image segmentation. This yields a direct, microscopy-based estimate ofφ(Π)that is independent of any thermal measurement, and therefore t...

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    (8) directly; the same ion-milled SEM cross-sections give an independent flake- angle distribution

    Flake-orientation texture (the reorientation test).For graphite anodes, X-ray diffraction texture analysis (the graphite(002)pole figure) at each calendering state gives the Hermans orientation factor S(Π)of Eq. (8) directly; the same ion-milled SEM cross-sections give an inde...

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    The joint closure of Sec

    Joint multi-property calibration (thermal, ionic, electronic).On thesamecalendered sheets, add a through-plane ionic measurement (symmetric-cell electrochemical impedance or limiting-current polarization, giving the through-plane tortuosityτ⊥(Π)) and a through-plane electronic...

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    Multi-recipe extension and informative design.Once the first recipe is complete, the same matrix should be repeated across recipes that vary one factor at a time (binder system, conductive- additive type and content, particle size) and across processing route (wet-slurry versu...

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    Closed-loop extrusion control (GranuGoIn deliverable).Implement the differentiable soft sensor on a continuous semi-dry granule-extrusion setup (TU Braunschweig / VARTA) to actively control the calendering line loads based on upstream solid-content variations, validating set-p...

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    Scope and limitations

    Acceptance criteria.(i) Calibrated residuals at or below the measurement uncertainty at allΠ; (ii) the pressure split consistent with Eq.(4) within theβband; (iii) theφ(Π)–adhesion correlation as predicted; (iv) the image-derived contact fraction consistent with the thermally ...

    2023

Pith tools

Reviewed July 14, 2026 · model on record in the stance chip above.