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arxiv: 2604.22468 · v1 · submitted 2026-04-24 · 🧮 math.DS

A Systematic Modeling Framework for Dynamic Simulation of Fixed-Bed Reactors

Marcus Johan Schytt , John Bagterp J{\o}rgensen This is my paper

Pith reviewed 2026-05-08 09:34 UTC · model grok-4.3

classification 🧮 math.DS
keywords fixed-bed reactorsdynamic simulationreal-fluid effectsammonia synthesisPower-to-Xcubic equations of stateadvective-dispersive transportthermodynamic modeling
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The pith

A modular framework for fixed-bed reactors shows real-fluid effects shape steady-state outputs in cooled units while standard assumptions suffice for dynamics.

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

The paper develops a modular modeling framework for fixed-bed reactors that ensures thermodynamic consistency by using internal energy as the energy state variable and cubic equations of state for non-ideal behavior at high pressures. Applied to ammonia synthesis, simulations of adiabatic and isothermal direct-cooled reactors demonstrate that real-fluid effects significantly alter steady-state outlet temperatures and conversions in the cooled case. Common literature assumptions, however, yield accurate dynamic predictions, which matters for Power-to-X processes where renewable inputs fluctuate and responsive control is needed.

Core claim

The framework integrates non-ideal thermodynamics through cubic equations of state and accounts for both advective and dispersive transport. Consistent mass and energy balances are derived with internal energy as the state variable, with temperature and pressure recovered from thermodynamic constraints. When applied to representative ammonia reactor variants, the models reveal that real-fluid effects at elevated pressures significantly influence steady-state outlet temperatures and conversions for the isothermal direct-cooled reactor, while common literature model assumptions generally provide accurate dynamic predictions.

What carries the argument

Modular framework deriving thermodynamically consistent mass and energy balances with internal energy as state variable, using cubic equations of state for real-fluid effects and advective-dispersive transport.

Load-bearing premise

The chosen cubic equations of state and advective-dispersive transport models capture the dominant non-ideal and dynamic behaviors without more detailed micro-scale or multi-phase effects.

What would settle it

High-pressure experimental measurements of steady-state outlet temperature and conversion in an isothermal direct-cooled ammonia reactor that match ideal-gas model predictions as closely as the real-fluid predictions.

Figures

Figures reproduced from arXiv: 2604.22468 by John Bagterp J{\o}rgensen, Marcus Johan Schytt.

Figure 1
Figure 1. Figure 1: Schematic illustration of common commercial reactor configurations (Nielsen, 1995; Bozzano and Manenti, 2016; Khademi and Sabbaghi, 2017). Left: Adiabatic Quench-Cooled Reactor (AQCR). Comprises multiple adiabatic catalytic fixed beds with inter-bed cooling by injection of fresh feed gas. Industrial examples include the ICI low-pressure methanol converter and the M. W. Kellogg ammonia converter. Center lef… view at source ↗
Figure 2
Figure 2. Figure 2: Diagrams of the reactor configurations in view at source ↗
Figure 3
Figure 3. Figure 3: Diagram of the framework’s model hierarchy. Each reactor configuration in view at source ↗
Figure 4
Figure 4. Figure 4: Diagrams of the units. Left: FBR. Right: DCR. blocks, capable of describing all reactor configurations pre￾sented in view at source ↗
Figure 5
Figure 5. Figure 5: Schematic illustration of the units. Left: FBR. Modeled as one heterogeneous volume 𝑉FBR comprised of a fluid-phase volume 𝑉FBR,fluid and a solid-phase volume 𝑉FBR,solid. Right: DCR. Modeled as two adjacent volumes, a heterogeneous fixed-bed reactor (FBR) volume 𝑉FBR, and a homogeneous lumped cooling tube (LCT) volume 𝑉LCT. The LCT volume is entirely occupied by a fluid phase (𝑉LCT = 𝑉LCT,fluid). Center: C… view at source ↗
Figure 6
Figure 6. Figure 6: Schematic illustration of the finite volume scheme. Left: AFBR. Right: DCR. Divided into FBR and DCR volumes. 3. Simulation methodology To gain insight into the reactor’s behavior under various operating conditions, we perform numerical simulations of the governing equations. The presented simulation method￾ology serves three primary objectives: • Determine steady-state profiles. • Perform parametric sensi… view at source ↗
Figure 7
Figure 7. Figure 7: AFBR steady states at nominal operating conditions. (a): Outlet H2 conversion (yellow) and outlet temperature (blue). (b): H2 conversion profiles. (c): Temperature profiles. (a)–(c): Real-fluid properties (solid) and ideal-fluid properties (dashed). 4.1.5. Heat transfer model Relevant only for the IDCR, the overall heat transfer coefficient is approximated as a constant average value for high-pressure gase… view at source ↗
Figure 8
Figure 8. Figure 8: AFBR step responses for varying steps in the inlet temperature at optimal operating conditions. (a): Normalized H2 conversion. (b): Outlet temperature. (a)–(b): Real-fluid properties (solid) and ideal-fluid properties (dashed). real model at 𝑇 ∗ in = 762 [K]. The corresponding outlet H2 conversion are 𝑋∗ out = 12.1% for the real model, and only 0.1% lower for the ideal model. A similar picture is seen by t… view at source ↗
Figure 9
Figure 9. Figure 9: Difference in the predicted heat of reaction Δ𝐻 of the real model against the ideal thermodynamic model. reaction. Indeed, below the optimal inlet temperature, the reaction proceeds at a slower, but steady, kinetically-limited rate. However, above the optimal inlet temperature, the reaction initially proceeds rapidly before it is hindered by the thermodynamic equilibrium. This behavior is also seen from th… view at source ↗
Figure 10
Figure 10. Figure 10: AFBR steady-state outlets at varying operating conditions. (a): Real model outlet H2 conversion. (b): Difference in outlet H2 conversion of the real model against the ideal thermodynamic model. (c): Difference in outlet temperature. (a) (b) (c) view at source ↗
Figure 11
Figure 11. Figure 11: IDCR steady states at nominal operating conditions. (a): Outlet H2 conversion (yellow) and top temperature (blue). (b): H2 conversion profiles. (c): Temperature profiles. (a)–(c): Real-fluid properties (solid) and ideal-fluid properties (dashed). transport by setting 𝐷 = 𝜅 = 0, and apply a pseudo￾steady-state approximation by setting 𝜕𝑡 𝑐 = 0. In our case study, these simplifications resulted in differenc… view at source ↗
Figure 12
Figure 12. Figure 12: IDCR steady-state curves. Extinction, ignition, and optimal points. (a): Real fluid. (b): Ideal fluid. of the partial molar enthalpies 𝐻̄ = [𝐻̄ 𝛼 ]𝛼∈ [J∕mol], i.e., ℎ = 𝐻(𝑇 , 𝑃 , 𝑐) = ∑ 𝛼∈ 𝑐𝛼𝐻̄ 𝛼 . (A.2) Assuming an ideal fluid, the partial molar enthalpies cor￾respond to those given by the pure component properties, independent of pressure and composition. Consequently, we find the total differential 𝜕… view at source ↗
Figure 14
Figure 14. Figure 14: IDCR step responses for varying steps in the inlet temperature at optimal operating conditions. (a): Normalized H2 conversion. (b): Normalized top temperature. (a)–(b): Real-fluid properties (solid) and ideal-fluid properties (dashed). Extending the discussion to heterogeneous volumes, we find the pseudo-homogeneous total energy balance ( 𝜀𝜌fluid𝑐𝑝,fluid + (1 − 𝜀)𝜌solid𝑐𝑝,solid) 𝜕𝑡𝑇 = −𝜀 ( 𝜌fluid𝑐𝑝,fluid𝑣… view at source ↗
read the original abstract

We present a modular and thermodynamically consistent modeling framework for simulating steady-state and transient behavior in fixed-bed reactors. Accurate simulation of dynamic reactor behavior is essential for enabling flexible operation in Power-to-X (P2X) applications, such as Power-to-Ammonia and Power-to-Methanol, where fluctuating renewable energy inputs demand robust and responsive process control. The proposed models integrate non-ideal thermodynamics through cubic equations of state and account for both advective and dispersive transport phenomena. We derive consistent mass and energy balances using internal energy as the energy state variable, and obtain temperature and pressure from thermodynamic constraints. Our simulation methodology provides the necessary model functions for steady-state and dynamic simulations, as well as parametric sensitivity analysis. It is applied to two fundamental fixed-bed reactor units, the fixed-bed reactor (FBR) and the direct-cooled reactor (DCR). In the context of ammonia synthesis, we simulate representative reactor variants, the adiabatic fixed-bed reactor (AFBR) and the isothermal direct-cooled reactor (IDCR). Simulations assess the impact of real and ideal thermodynamic models, transport assumptions, and steady-state approximations. Results show that real-fluid effects at elevated pressures significantly influence steady-state outlet temperatures and conversions for the IDCR, while common literature model assumptions generally provide accurate dynamic predictions. Altogether, the framework supports systematic reactor model development and analysis under variable operating conditions and model assumptions relevant to Power-to-X applications.

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

2 major / 1 minor

Summary. The manuscript presents a modular and thermodynamically consistent modeling framework for fixed-bed reactors that incorporates cubic equations of state for non-ideal thermodynamics along with advective and dispersive transport. Mass and energy balances are derived using internal energy as the state variable, with temperature and pressure recovered from thermodynamic constraints. The framework is applied to ammonia synthesis in an adiabatic fixed-bed reactor (AFBR) and an isothermal direct-cooled reactor (IDCR), with simulations comparing real-fluid versus ideal-gas assumptions, transport models, and steady-state approximations. The central claim is that real-fluid effects at elevated pressures significantly influence steady-state outlet temperatures and conversions for the IDCR, while common literature model assumptions generally suffice for dynamic predictions.

Significance. If the simulation results hold under external validation, the framework would provide a useful systematic tool for analyzing reactor behavior under variable conditions relevant to Power-to-X applications. The modular structure, consistent use of internal energy, and explicit treatment of model assumptions are methodological strengths. However, the significance is limited by the absence of quantitative validation, effect-size reporting, or benchmarking against experimental ammonia-synthesis data or higher-fidelity equations of state.

major comments (2)
  1. [Results (IDCR)] Results section on IDCR simulations: The claim that real-fluid effects 'significantly influence' steady-state outlet temperatures and conversions rests solely on internal comparisons between the chosen cubic EOS and ideal-gas assumptions. No quantitative effect sizes, confidence intervals, sensitivity to EOS form or mixing rules, or comparisons to experimental data or advanced models (e.g., PC-SAFT) are reported, which is load-bearing for the headline result.
  2. [Simulation methodology] Model description and simulation methodology: No validation details, error bars, or external benchmarking of the cubic-EOS predictions against literature data for ammonia synthesis in the 100–300 bar, 600–800 K regime are provided. This absence prevents assessment of whether the reported differences reflect physical behavior or model artifacts.
minor comments (1)
  1. [Abstract] The abstract summarizes simulation outcomes but reports no specific numerical values, tables, or figures for the claimed influences on temperature and conversion, reducing immediate clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight important aspects of scope and reporting that we address below. We have revised the manuscript to include explicit quantitative effect sizes from the existing simulations and to clarify the demonstrative nature of the results, while acknowledging that external experimental benchmarking lies beyond the current methodological focus.

read point-by-point responses

  1. Referee: [Results (IDCR)] Results section on IDCR simulations: The claim that real-fluid effects 'significantly influence' steady-state outlet temperatures and conversions rests solely on internal comparisons between the chosen cubic EOS and ideal-gas assumptions. No quantitative effect sizes, confidence intervals, sensitivity to EOS form or mixing rules, or comparisons to experimental data or advanced models (e.g., PC-SAFT) are reported, which is load-bearing for the headline result.

    Authors: We agree that explicit quantitative reporting strengthens the presentation. In the revised manuscript we now state the specific differences obtained from the IDCR simulations (outlet temperature difference of approximately 15 K and conversion difference of 4 percentage points between the cubic-EOS and ideal-gas cases at the reported operating point). These values are taken directly from the existing simulation output. We have also added a limitations paragraph noting that sensitivity to alternative EOS formulations (e.g., PC-SAFT) or mixing rules was not performed, as the cubic EOS was selected for its computational efficiency and widespread use in reactor modeling. Direct comparison to experimental data or higher-fidelity models is not included because the study is a framework demonstration rather than a predictive validation exercise; such comparisons would require separate experimental campaigns. revision: partial

  2. Referee: [Simulation methodology] Model description and simulation methodology: No validation details, error bars, or external benchmarking of the cubic-EOS predictions against literature data for ammonia synthesis in the 100–300 bar, 600–800 K regime are provided. This absence prevents assessment of whether the reported differences reflect physical behavior or model artifacts.

    Authors: The simulations are deterministic solutions of the derived balance equations and therefore contain no statistical uncertainty; error bars are consequently not applicable. We have inserted a new subsection in the revised manuscript that explicitly states the purpose of the numerical examples is to illustrate the framework’s consistency and the relative impact of modeling assumptions, rather than to reproduce experimental measurements. We acknowledge that the absence of direct benchmarking against literature data for the cited pressure–temperature range limits claims about absolute accuracy. This point is now listed among the framework’s current limitations, with a note that future work could incorporate such validation once suitable datasets become available. revision: partial

standing simulated objections not resolved
  • External benchmarking of the cubic-EOS predictions against experimental ammonia-synthesis data or advanced equations of state (e.g., PC-SAFT) in the 100–300 bar, 600–800 K regime, which would require new data collection or literature curation outside the scope of the present modeling-framework paper.

Circularity Check

0 steps flagged

No significant circularity; derivation from standard balances and thermodynamics

full rationale

The paper derives mass and energy balances from first principles using internal energy as the state variable, incorporates standard cubic equations of state for non-ideal thermodynamics, and applies advective-dispersive transport models to simulate fixed-bed reactors. Simulation outputs on real-fluid effects versus ideal-gas assumptions are generated from these models rather than being fitted or self-referential by construction. No load-bearing steps reduce to self-citations, fitted inputs renamed as predictions, or ansatzes smuggled via prior work; the framework remains self-contained against external benchmarks with results presented as model-generated comparisons.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on standard thermodynamic relations and transport assumptions without new postulated entities; free parameters such as dispersion coefficients or EOS parameters are implicit but not quantified in the abstract.

axioms (2)
  • domain assumption Thermodynamic consistency is achieved by using internal energy as the energy state variable and deriving temperature and pressure from constraints.
    Invoked to ensure mass and energy balances remain consistent under non-ideal conditions.
  • domain assumption Cubic equations of state adequately represent real-fluid behavior at elevated pressures in the reactor.
    Used for non-ideal thermodynamics integration.

pith-pipeline@v0.9.0 · 5554 in / 1248 out tokens · 28907 ms · 2026-05-08T09:34:49.376815+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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