A Simulation-Based Catalyst-Activity-Aware Self-Optimizing Digital Twin for o-Xylene Oxidation to Phthalic Anhydride in a Catalyst-Deactivating Fixed-Bed Reactor

Abstract

Catalyst deactivation shifts the optimal operating region of exothermic fixed-bed reactors, yet most reactor digital twins focus on monitoring rather than catalyst-state-aware operating decisions. This work presents a simulation-based self-optimizing digital-twin prototype integrating a physics-based reactor model, a moving-window constrained activity estimator, and a target-optimization layer for o-xylene oxidation to phthalic anhydride in a vanadia–titania heat-exchanged fixed-bed reactor. Sparse axial temperature and conversion measurements are reconciled to estimate an axial catalyst activity profile; gas and coolant inlet temperatures are then updated subject to a hot-spot safety constraint. The estimator achieved an activity-profile root mean square error (RMSE) of 0.075, an outlet-conversion RMSE of 0.99 percentage points, and an outlet-temperature RMSE of 1.85 K. Under the baseline noisy-measurement scenario, estimated activity optimization raised the mean phthalic anhydride yield from 46.3% under fixed targets to 61.9%, within 0.14 percentage points of the true-activity optimum, while maintaining the maximum reactor temperature below 730 K. In this matched-model simulation study, this corresponds to recovering approximately 99.1% of the yield improvement available with perfect catalyst-state knowledge. The policy remained superior to fixed-target operation across all tested noise levels, sensor configurations, and kinetic pre-exponential perturbations. All results are obtained from synthetic-measurement simulations rather than experimental or plant data, and plant validation is still required to quantify structural model error. The findings demonstrate the value of linking catalyst-state estimation to operating-target adaptation in a reproducible catalytic-reactor digital-twin workflow.

1. Introduction

Catalytic fixed-bed reactors are central to many chemical and petrochemical processes, including oxidation, hydrogenation, reforming, desulfurization, and synthesis reactions. Their performance depends strongly on catalyst activity, temperature distribution, feed composition, heat-transfer behavior, and operating constraints. Catalyst deactivation is a persistent limitation in heterogeneous catalytic processes because activity and selectivity can decline through poisoning, fouling, coking, sintering, phase transformation, or loss of active surface area [1,2,3]. In industrial operation, reactor targets are often selected from design calculations, commissioning studies, offline optimization, or periodic plant tests. These targets may remain fixed for long periods, even though the reactor itself changes continuously as the catalyst deactivates and process conditions drift.

Catalyst deactivation is particularly important in highly exothermic fixed-bed reactors. A reduction in catalyst activity may lower conversion, alter selectivity, shift hot-spot locations, and change the relationship between manipulated variables and product yield. Operators may compensate by increasing inlet temperature, changing coolant conditions, or modifying feed rates [4,5,6]. Without a reliable estimate of the current catalyst condition, however, such adjustments may be conservative, delayed, or based on indirect operating experience rather than systematic optimization.

Digital twins provide a promising route for improving reactor operation because they combine physics-based process models, real-time measurements, and computational decision-making [7,8,9,10]. In the process industries, digital-twin implementations are increasingly discussed for monitoring, prediction, predictive maintenance, process optimization, and operational decision support, but deployment remains challenging because models must remain synchronized with changing plant behavior and uncertain measurements [11,12]. Many existing digital twins are used primarily for monitoring, fault detection, or operator support. A more powerful role emerges when the digital twin can estimate hidden process states and propagate those estimates into operating decisions. For catalytic reactors, one of the most consequential hidden states is the catalyst activity profile. Since catalyst activity is not directly measured during normal operation, it must be inferred from available process measurements such as axial temperature, outlet composition, flow rate, and pressure.

Recent digital-twin research has increasingly incorporated data-driven and hybrid (physics-plus-machine-learning) formulations, in which learned surrogates augment or replace parts of the mechanistic model to improve prediction accuracy or reduce computational cost [7,9,10,12]. At the catalytic-process level, these broader trends include digital-twin-supported self-driving experimentation and autonomous multi-objective reaction optimization [13]. In the catalysis context, these broader ideas are related to catalyst health and condition monitoring, in which measurable process signatures are used to infer progressive activity loss [1,3,14]. The present framework is deliberately physics-based: the hidden catalyst-activity state is inferred by constrained model-based estimation rather than by a learned surrogate, which keeps the inverse problem transparent and the estimated activity physically interpretable. Hybrid or machine-learning-augmented estimators are a natural extension and are noted in the future-work discussion (Section 4.10).

This work develops and evaluates, through simulation, a state-estimation-based self-optimizing digital twin framework for catalyst-deactivating fixed-bed reactors. The key idea is to treat catalyst activity as an estimated hidden state rather than as a fixed model parameter. Process measurements and prior information are reconciled through a constrained moving-window estimation problem with physical bounds on catalyst activity [15,16,17]. The estimated catalyst condition is then passed to an optimization layer that updates reactor operating targets. This structure connects hidden-state estimation with self-optimizing operation, in which operating targets are selected to maintain near-optimal performance despite disturbances and process changes [18,19,20]. The resulting digital twin is therefore evaluated as an adaptive decision-support framework rather than solely as a prediction tool.

The proposed framework is demonstrated using the oxidation of o-xylene to phthalic anhydride (PA) in a heat-exchanged fixed-bed reactor. This system is suitable because catalyst activity affects both the axial temperature profile and outlet conversion, providing measurable information about deactivation. It is also a representative reaction-engineering problem because the desired partial oxidation competes with over-oxidation pathways under strong heat release and hot-spot constraints [5,6,21,22]. The case study combines a reactor model benchmarked against published results, axial activity-profile estimation from sparse synthetic measurements, adaptive thermal-target optimization, and robustness tests under different noise levels, sensor configurations, regularization choices, and kinetic mismatch.

Catalyst deactivation in fixed-bed reactors has been studied from several complementary perspectives. One group of studies focuses on physical or semi-empirical deactivation modeling, where loss of catalyst activity is represented through temperature-, composition-, or time-dependent deactivation laws. For o-xylene oxidation to phthalic anhydride, industrial deactivation has been analyzed using temperature-profile data and detailed reactor models to infer how the catalyst condition changes along the bed [4]. More general activity-profile modeling methods have also been proposed to approximate spatially distributed activity loss using reduced-order representations [23]. These studies are valuable because they clarify how catalyst aging affects reactor behavior, but their primary objective is deactivation description or model reconstruction rather than online operating-target adaptation.

A second group of studies addresses catalyst activity estimation. In this direction, activity profiles are inferred from process measurements such as axial temperature and outlet composition. The work of Cheng et al. is particularly relevant because it demonstrated that temperature, composition, and catalyst activity profiles can be estimated in fixed-bed reactors with decaying catalysts using process measurements and model-based estimation [14]. This establishes the feasibility of reconstructing hidden catalyst states from measurable reactor outputs. However, activity estimation is often used mainly for monitoring, diagnosis, or model updating, and the estimated activity is not necessarily propagated into an optimization layer that changes reactor targets.

A third line of work focuses on reactor optimization, optimal temperature policies, and digital or model-based reactor design. For phthalic anhydride synthesis, reactor optimization studies have investigated how temperature policy and reactor design influence selectivity, yield, and safe operation [5]. More recent model-based digital design concepts for fixed-bed catalytic reactors emphasize the role of mechanistic models in design, scale-up, optimization, and possible online implementation [24]. These optimization studies are important, but they commonly assume that catalyst activity is known, fixed, or updated outside the optimization loop.

Taken together, the literature provides strong foundations for deactivation modeling, catalyst activity estimation, and reactor optimization. However, these elements are commonly studied separately: activity estimation typically terminates in monitoring or model updating, while reactor optimization typically treats catalyst activity as fixed or externally specified. To the authors’ knowledge, limited prior work has closed the complete pathway from sparse reactor measurements, through spatial activity estimation, to catalyst-aware operating-target adaptation. This work addresses that integration gap through a simulation-based proof of concept linking physics-based reactor prediction, moving-window constrained activity estimation, and self-optimizing target updates.

Accordingly, the main research question is whether catalyst activity inferred sequentially from sparse process measurements can be used within a digital twin to recover reactor performance as deactivation progresses, and how sensitive the resulting operating decisions are to measurement quality, sensor configuration, estimator structure, and model–plant mismatch.

The paper makes four principal contributions:

• A general digital-twin workflow is formulated in which an estimated catalyst state is used directly in operating-target optimization.
• The workflow is instantiated for o-xylene oxidation using a literature-benchmarked heat-exchanged fixed-bed reactor model with spatially nonuniform catalyst activity.
• Sparse axial temperatures and outlet conversion are combined in a constrained moving-window estimator, and the propagation of estimation error into optimized operating targets is quantified.
• The workflow is evaluated using repeated noise realizations, sensor-density tests, a uniform-activity comparator, regularization ablation, and kinetic model–plant mismatch.

The fundamental distinction from prior work lies not in the individual components but in the direction of information flow: whereas earlier studies use the estimated catalyst state as a terminal diagnostic for monitoring, diagnosis, or model updating, here it becomes the operative input that selects the operating targets at each step. The novelty therefore lies in the demonstrated integration and evaluation of this end-to-end decision pathway, rather than in proposing a new kinetic model or claiming the first use of catalyst activity estimation. All results reported in this paper are obtained from simulation using synthetic measurements generated from a literature-benchmarked reactor model; no experimental or industrial datasets are used, and the study is presented as a simulation-based proof of concept.

Table 1. Focused literature positioning and distinction of the present work.

Theme	Representative Studies	Main Contribution in Prior Work	Distinction of the Present Work
PA reactor modeling and safe operation	de Lasa [22]; Zuluaga-Botero et al. [6]	Kinetic modeling, pseudoadiabatic behavior, and sensitivity/safe-operation analysis for o-xylene oxidation.	Uses a literature-benchmarked PA reactor model as the physics layer of a self-optimizing digital twin rather than as a stand-alone simulation study.
Catalyst deactivation modeling	Anastasov [4]; activity-profile modeling studies [23]	Quantification or approximation of catalyst deactivation profiles and their effect on reactor behavior.	Represents activity as a sequentially estimated state that is passed to an operating-target optimization layer.
Catalyst activity estimation	Cheng et al. [14]	Estimation of temperature, composition, and activity profiles in fixed-bed reactors with decaying catalysts.	Uses activity estimation for adaptive optimization, not only for monitoring, diagnosis, or model reconstruction.
Digital twins for process operation	Digital-twin reviews and process-industry frameworks [7,8,11]	Real-time model synchronization, monitoring, prediction, and operational decision-support concepts.	Instantiates the digital-twin concept for catalyst-state estimation and catalyst-aware target updates in an exothermic fixed-bed reactor.
Constrained estimation and self-optimizing operation	Moving-horizon/constrained estimation [15,16]; self-optimizing control [18]	Optimization-based state estimation and selection of controlled variables or targets for near-optimal operation.	Connects constrained activity estimation directly to target optimization under catalyst deactivation.
Reactor optimization and digital design	Papageorgiou and Froment [5]; model-based digital fixed-bed reactor design/optimization concepts [24]	Offline or model-based optimization of reactor operation and design, including fixed-bed catalytic reactors.	Connects measurements, activity estimation, and updated operating targets under catalyst deactivation.
This work	Present study	Simulation-based state-estimation and self-optimizing digital-twin prototype for catalyst-deactivating fixed-bed reactors.	Demonstrates the workflow using a literature-benchmarked PA reactor case, estimated axial activity profiles, setpoint optimization, and robustness tests.

positions the present work relative to representative studies across six relevant themes. The contribution is not the reactor model or activity estimator in isolation, but their integration so that an estimated catalyst condition becomes a direct input to an operating-target update layer.

The remainder of the paper is organized as follows. Section 2 presents the proposed methodology, including the measurement–estimation–optimization decision workflow, activity estimation problem, self-optimizing operating layer, and robustness evaluation procedure. Section 3 describes the o-xylene oxidation case study, reactor model, kinetic expressions, assumptions, validation basis, and catalyst activity representation. Section 4 presents and discusses the results, including activity observability, estimation performance, self-optimizing operation, robustness tests, novelty positioning, and limitations. Section 5 summarizes the conclusions.

2. Methodology

2.1. General Self-Optimizing Digital Twin Framework

The proposed framework provides a general structure for catalyst-deactivating fixed-bed reactors. The central idea is to combine process measurements, a physics-based reactor digital twin, a sequential catalyst activity estimator, and a self-optimizing operating layer in a catalyst-state-aware decision workflow. Figure 1 illustrates the intended closed-loop architecture: process measurements are compared with model predictions to estimate the hidden catalyst activity profile, and the estimate is then used to update reactor operating targets. In the present proof-of-concept implementation, estimation is first evaluated along a prescribed deactivation trajectory at fixed nominal inputs, after which the estimated activity sequence is supplied to the target-optimization layer. Thus, the study demonstrates the complete measurement–estimation–optimization decision pathway, but it does not simulate actuator dynamics or feed each optimized target back into the synthetic plant measurement generator at the next sampling instant. In digital-twin terminology, the present realization is therefore best described as a simulation-based digital-twin prototype (or emulator): it exercises the full decision pathway against a synthetic plant but does not maintain continuous synchronization with a physical reactor, which is the defining feature of a fully deployed digital twin. Accordingly, “self-optimizing” here refers to the target-adaptation logic rather than to a physically closed feedback loop, and the “closed-loop” architecture of Figure 1 denotes the intended deployment configuration rather than the loop actuated in this study.

The framework is applicable to catalytic reactor configurations where catalyst degradation affects measurable outputs such as temperature, conversion, selectivity, or yield. Representative examples include multi-bed adiabatic reactors, heat-exchanged fixed-bed reactors, and staged catalytic systems. The framework consists of four interacting layers:

1.

Physical reactor and measurement system: provides process measurements such as axial temperature, outlet composition, flow rate, and pressure.

2.

Physics-based digital twin: predicts reactor states from a mechanistic model given current inputs and the estimated catalyst activity.

3.

Sequential catalyst activity estimator: uses the mismatch between measured and predicted outputs to infer the hidden catalyst activity profile at each sampling step.

4.

Self-optimizing operating layer: uses the estimated activity to compute updated reactor operating targets subject to process constraints.

The intended plant-connected closed-loop workflow is

$$\begin{array}{l}\mathrm{reactor}\to \mathrm{measurements}\to \mathrm{digital}\mathrm{twin}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\to \mathrm{state}\mathrm{estimation}\to \mathrm{self}-\mathrm{optimizing}\mathrm{layer}\to \mathrm{updated}\mathrm{targets}.\end{array}$$

(1)

This structure enables adaptive operation because the optimization problem is solved using the current estimated catalyst condition rather than assuming fixed or fresh catalyst activity.

2.1.1. Physics-Based Reactor Model

For a general catalytic reaction network containing $N_r$ reactions, the following equations establish the framework notation for reaction stoichiometry and plug-flow material balances. The case-specific mole, energy, and coolant balances used in this study are given and attributed in Section 3.4. Reaction q may be written as

$${\displaystyle \sum _{j=1}^{N_c}}{\nu }_{jq}^{-}{A}_j\to {\displaystyle \sum _{j=1}^{N_c}}{\nu }_{jq}^{+}{A}_j,$$

(2)

where $A_j$ denotes chemical species j, $N_c$ is the number of components, and ${\nu }_{jq}^{-}$ and ${\nu }_{jq}^{+}$ are the reactant and product stoichiometric coefficients. Defining the signed coefficient ${\nu }_{jq}={\nu }_{jq}^{+}-{\nu }_{jq}^{-}$, the balance for component j in catalyst bed or reactor segment i may be expressed in catalyst-weight coordinates as

$${\displaystyle \frac{d{F}_{j,i}}{d{W}_i}}={\displaystyle \sum _{q=1}^{N_r}}{\nu }_{jq}{r}_{q,i}.$$

(3)

For an adiabatic catalyst bed, the energy balance can be written as

$${\displaystyle \frac{d{T}_i}{d{W}_i}}={\displaystyle \frac{-{\sum }_{q=1}^{N_r}\Delta H_q\left(T_i\right)r_{q,i}}{{\sum }_{j=1}^{N_c}{F}_{j,i}{C}_{p,j}\left(T_i\right)}}.$$

(4)

For heat-exchanged reactors, the energy balance is modified by adding a heat-removal term that depends on the heat-transfer coefficient, heat-transfer area, and coolant temperature.

For a key reactant A, the bed conversion is

$$X_i={\displaystyle \frac{F_{A,in,i}-F_{A,out,i}}{F_{A,in,i}}},$$

(5)

and the overall reactor conversion is

$$X_{total}={\displaystyle \frac{F_{A,0}-F_{A,out}}{F_{A,0}}}.$$

(6)

For reversible systems, equilibrium relations may provide additional feasibility constraints. The present o-xylene oxidation case is modeled using the irreversible reaction network described in Section 3.3, so no equilibrium-conversion constraint is imposed.

2.1.2. Catalyst Activity-Modified Kinetics

Catalyst activity is introduced as a slowly varying state that modifies the apparent reaction rates. For reaction q in reactor segment i, the rate is written as

$$r_{q,i}\left(t\right)=a_i\left(t\right)r_{q,i}^{int}(T_i,{\mathit{C}}_i),$$

(7)

where $r_{q,i}^{int}$ is the intrinsic rate under reference (fresh-catalyst) conditions, ${\mathit{C}}_i$ is the concentration vector, and $a_i\left(t\right)\in (0,1]$ is the local catalyst activity factor. In general, the activity factor satisfies

$$0<a_i\left(t\right)\le 1,$$

(8)

where $a_i=1$ corresponds to fresh catalyst and $a_i\to 0$ represents complete local deactivation. In practice a small positive lower bound is imposed to maintain numerical conditioning of the reactor model; the value used in the present case study is given in Section 2.3. For spatially distributed catalyst beds, activity may be represented as a continuous axial profile $a(z,t)$ or approximated by a finite number of activity nodes:

$$\mathit{a}\left(t\right)={[a_1\left(t\right),a_2\left(t\right),\dots ,a_m\left(t\right)]}^T.$$

(9)

2.1.3. Augmented State Estimation

A general process state vector may be defined as

$$\mathit{x}={[\mathit{T},\mathit{F}]}^T,$$

(10)

where $\mathit{T}$ collects the axial temperature profile and $\mathit{F}$ the species molar flow rates; derived quantities such as conversion, selectivity, and reaction rates are outputs of the model rather than independent states. To include catalyst activity, the augmented state vector is

$${\mathit{x}}_a={[\mathit{x},\mathit{a}]}^T.$$

(11)

For a dynamic implementation, the digital twin may predict process evolution using

$${\widehat{\mathit{x}}}_{k+1}=f({\widehat{\mathit{x}}}_k,{\mathit{u}}_k,{\widehat{\mathit{a}}}_k),$$

(12)

and the measurement model is

$${\widehat{\mathit{y}}}_k=h({\widehat{\mathit{x}}}_k,{\mathit{u}}_k,{\widehat{\mathit{a}}}_k).$$

(13)

For reactors where deactivation is slow relative to the process time constants, the state propagation simplifies to a quasi-steady solve ${\widehat{\mathit{x}}}_k=f({\mathit{u}}_k,{\widehat{\mathit{a}}}_k)$ at each estimation instant. This simplification is applied in the present case study and justified in Section 3.5.

2.2. Decision-Workflow Algorithm

Algorithm 1 summarizes the framework-level computational workflow. The same logic can be applied to other catalyst-deactivating reactors when the activity state is observable from available measurements. In the present PA case study, the measured outputs are sparse axial temperatures and outlet conversion, the estimated hidden state is the axial catalyst activity profile, and the optimized targets are the gas and coolant inlet temperatures. As noted above, the current implementation evaluates the estimation sequence at nominal inputs and subsequently evaluates the target updates; Step 6 represents the feedback action required in a future plant-connected or fully coupled closed-loop implementation.

Algorithm 1 State-estimation-based self-optimizing digital twin workflow

Require:  Physics-based reactor model $f\left(·\right)$ and measurement model $h\left(·\right)$  Require:  Initial prior activity estimate ${\widehat{\mathit{a}}}_{-1}$, input bounds ${\mathit{u}}_{min},{\mathit{u}}_{max}$, and safety limit $T_{max}^{safe}$  Require:  Measurement covariance $\mathit{R}$, regularization weights ${\lambda }_s,{\lambda }_p$, and sampling index $k=0,1,\dots ,N-1$ 1: for each sampling instant k do 2:     Collect process measurements ${\mathit{y}}_k$, including sparse reactor temperatures and outlet conversion. 3:     Simulate the digital twin using current inputs ${\mathit{u}}_k$ and a trial activity profile $\widehat{\mathit{a}}$. 4:     Estimate the catalyst activity profile by solving $${\widehat{\mathit{a}}}_k=arg\underset{0.05\le \widehat{\mathit{a}}\le 1}{min}{∥{\mathit{y}}_k-h({\widehat{\mathit{x}}}_k,{\mathit{u}}_k,\widehat{\mathit{a}})∥}_{{\mathit{R}}^{-1}}^2+{\lambda }_s{∥\mathit{D}\widehat{\mathit{a}}∥}_2^2+{\lambda }_p{∥\widehat{\mathit{a}}-{\widehat{\mathit{a}}}_{k-1}∥}_2^2.$$ 5:     Solve the catalyst-aware target optimization problem $${\mathit{u}}_{k+1}^{*}=arg\underset{{\mathit{u}}_{min}\le \mathit{u}\le {\mathit{u}}_{max}}{max}J({\widehat{\mathit{x}}}_k,{\widehat{\mathit{a}}}_k,\mathit{u}),$$ where J maximizes PA yield with move penalties, and any candidate violating $T_{max}>T_{max}^{safe}$ is excluded (Section 2.4). 6:     Apply or recommend the updated operating targets ${\mathit{u}}_{k+1}^{*}$; in the present study, record them for performance evaluation. 7:     Store ${\widehat{\mathit{a}}}_k$, ${\mathit{u}}_{k+1}^{*}$, predicted outputs, and performance indicators for monitoring. 8:  end for

In the present implementation, $\mathit{R}=\mathrm{diag}({\sigma }_T^2{\mathit{I}}_s,{\sigma }_X^2)$ is the diagonal measurement-noise covariance, where s is the number of temperature sensors. The detailed residual construction and regularization terms are given in Section 2.3.2.

2.3. Moving-Window Constrained Activity Estimation

2.3.1. Measurement Vector

The estimator uses a measurement vector consisting of sparse axial gas temperatures and outlet conversion:

$${\mathit{y}}_k={[T(z_1,t_k),T(z_2,t_k),\dots ,T(z_s,t_k),X_{out}\left(t_k\right)]}^T,$$

(14)

where s is the number of temperature sensors. In the base case, temperature measurements are taken at four axial locations: $z=[0.5,1.0,1.5,2.0]$ m. For the robustness scenarios, the denser six-sensor layout uses $z=[0.25,0.50,0.85,1.20,1.60,2.00]$ m and the sparser three-sensor layout uses $z=[0.5,1.25,2.0]$ m. All three layouts include an outlet-temperature measurement at $z=2.0$ m, while outlet conversion is included as a separate scalar measurement. Measurement noise is added as

$$\begin{array}{cc}\hfill T_{meas}(z_i,t_k)& =T(z_i,t_k)+{ϵ}_{T,i,k},{ϵ}_{T,i,k}\sim \mathcal{N}(0,{\sigma }_T^2),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill X_{meas}\left(t_k\right)& =X_{out}\left(t_k\right)+{ϵ}_{X,k},{ϵ}_{X,k}\sim \mathcal{N}(0,{\sigma }_X^2),\hfill \end{array}$$

(16)

where ${\sigma }_T$ and ${\sigma }_X$ are the standard deviations of the temperature and conversion measurement noise, respectively; their numerical values for the base and stress-test cases are defined in Section 2.5.

2.3.2. Objective Function

At each time step, the unknown activity profile is represented by a vector of activity nodes,

$${\widehat{\mathit{a}}}_k={[{\widehat{a}}_{1,k},{\widehat{a}}_{2,k},\dots ,{\widehat{a}}_{m,k}]}^T.$$

(17)

For the present study, the activity profile is estimated by solving the following constrained nonlinear least-squares problem:

$$\begin{array}{cc}\hfill {\Phi }_k\left(\widehat{\mathit{a}}\right)& ={∥{\displaystyle \frac{{\mathit{T}}_{pred}\left(\widehat{\mathit{a}}\right)-{\mathit{T}}_{meas,k}}{{\sigma }_T}}∥}_2^2+{∥{\displaystyle \frac{X_{pred}\left(\widehat{\mathit{a}}\right)-X_{meas,k}}{{\sigma }_X}}∥}_2^2\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\lambda }_s{∥\mathit{D}\widehat{\mathit{a}}∥}_2^2+{\lambda }_p{∥\widehat{\mathit{a}}-{\widehat{\mathit{a}}}_{k-1}∥}_2^2,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\widehat{\mathit{a}}}_k& =arg\underset{0.05\le \widehat{\mathit{a}}\le 1}{min}{\Phi }_k\left(\widehat{\mathit{a}}\right).\hfill \end{array}$$

(19)

Here $\mathit{D}\in {\mathbb{R}}^{(m-1)\times m}$ is the first-order finite-difference matrix that penalizes axial gradients in the estimated activity profile,

$$\mathit{D}=\left[\begin{array}{ccccc}-1& 1& 0& \dots & 0\\ 0& -1& 1& \dots & 0\\ ⋮& & \ddots & \ddots & ⋮\\ 0& \dots & 0& -1& 1\end{array}\right].$$

(20)

The first two terms fit the measured outputs; the third term discourages spatially oscillatory activity estimates; and the fourth anchors the trial profile to the previous estimate, reflecting the slow timescale of catalyst deactivation relative to the measurement sampling interval. The lower activity bound of 0.05 maintains numerical conditioning while allowing severe deactivation to be represented. The case-study weights are ${\lambda }_s=0.7$ and ${\lambda }_p=15$. These are algorithmic regularization parameters rather than covariance-derived quantities: ${\lambda }_s$ controls axial smoothness and ${\lambda }_p$ controls step-to-step persistence. They were selected by ablation-guided tuning rather than by a formal hyperparameter search (for example, an L-curve or cross-validation procedure), which is noted as future work; their individual contribution is examined through the ablation study in Section 4.5. The constrained least-squares problem is solved using the Trust-Region Reflective algorithm in SciPy 1.17.1 (open-source scientific-computing library; https://scipy.org) with ${ϵ}_x={ϵ}_f={ϵ}_g={10}^{-8}$, a maximum of 250 function evaluations, and a warm start from the preceding estimate. A constrained moving-window least-squares estimator was chosen in preference to an extended or unscented Kalman filter because it enforces the hard physical activity bounds ($0.05\le a_i\le 1$) natively, is robust for the strongly nonlinear quasi-steady reactor map without requiring covariance or sigma-point propagation, and keeps the inverse problem transparent and reproducible for a proof of concept; its relationship to a full nonlinear moving-horizon estimator, and the rationale for adopting the latter in future work, are discussed in Section 4.10.

2.3.3. Estimator Performance Metrics

Estimator performance is evaluated using three scalar metrics. The activity-profile root mean square error,

$${\mathrm{RMSE}}_a=\sqrt{{\displaystyle \frac{1}{mN}}\sum _{k=0}^{N-1}\sum _{i=1}^m{\left({\widehat{a}}_{i,k}-a_{i,k}\right)}^2},$$

(21)

measures the overall accuracy of the reconstructed activity field across all nodes and time steps. The outlet-conversion prediction error,

$${\mathrm{RMSE}}_X=\sqrt{{\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{\left({\widehat{X}}_{out,k}-X_{out,k}\right)}^2},$$

(22)

and the outlet-temperature prediction error,

$${\mathrm{RMSE}}_T=\sqrt{{\displaystyle \frac{1}{N}}\sum _{k=0}^{N-1}{\left({\widehat{T}}_{out,k}-T_{out,k}\right)}^2},$$

(23)

quantify how well the estimated activity profile reproduces the corresponding noise-free synthetic plant outputs. Conversion errors are reported in percentage points and temperature errors in kelvin.

2.3.4. Uniform-Activity Comparator

To test whether spatial temperature measurements add useful information beyond a bulk activity estimate, a conversion-only comparator is evaluated in every noise-and-sensor scenario. The comparator assumes uniform catalyst activity, ${\widehat{\mathit{a}}}_{unif,k}={\widehat{a}}_{unif,k}\mathit{1}$, and estimates the scalar activity from outlet conversion:

$${\widehat{a}}_{unif,k}=arg\underset{0.05\le \widehat{a}\le 1}{min}{\left({\displaystyle \frac{X_{pred}\left(\widehat{a}\mathit{1}\right)-X_{meas,k}}{{\sigma }_X}}\right)}^2+{\lambda }_p{\left(\widehat{a}-{\widehat{a}}_{unif,k-1}\right)}^2.$$

(24)

This comparator uses no axial temperature measurements and cannot reconstruct a spatial activity gradient. It therefore isolates the additional information provided by the proposed spatial estimator rather than serving as a no-estimation control.

2.4. Self-Optimizing Operating Layer

The self-optimizing layer updates reactor operating targets based on the estimated catalyst condition. For the heat-exchanged fixed-bed reactor in the present case study, the decision vector consists of the inlet gas temperature and the coolant inlet temperature:

$$\mathit{u}={[T_{in},T_{c,in}]}^T.$$

(25)

The optimization objective formulated for the present study maximizes phthalic anhydride yield while penalizing large target moves away from the nominal operating point:

$$J=Y_{PA}+w_S{S}_{PA,out}-{\lambda }_u{\parallel \mathit{u}-{\mathit{u}}_{ref}\parallel }_2^2,$$

(26)

where $Y_{PA}=X_{out}{S}_{PA,out}$ is the phthalic anhydride yield, $w_S>0$ is a small weight on outlet selectivity, and ${\lambda }_u>0$ penalizes movement away from the nominal reference targets. Although selectivity already enters implicitly through yield, the secondary term $w_S{S}_{PA,out}$ provides an additional optimization gradient at high-conversion conditions where yield is relatively flat but selectivity still varies; $w_S$ is kept small so that yield remains the dominant objective. The nominal fixed-target reference is ${\mathit{u}}_{ref}={[625,625]}^T$ K, which is also the baseline for fixed-target operation (Section 4.3). The hot-spot safety limit

$$T_{max}^{safe}=730\mathrm{K}$$

(27)

is enforced by excluding any grid-search candidate with $T_{max}>T_{max}^{safe}$ via a large infeasibility penalty, so the constraint is effectively treated as a hard bound within the search space. The case-study values are $w_S=0.01$ and ${\lambda }_u=2\times {10}^{-4}$ K⁻², and the thermal-target search covers $T_{in},T_{c,in}\in [620,632]$ K at 1-K resolution (169 candidate points total). The inlet gas and coolant inlet temperatures were chosen as the decision variables because they are the most direct and commonly used levers for the hot-spot/selectivity trade-off in heat-exchanged o-xylene oxidation and act on the dominant physics of heat generation and removal; restricting the problem to these two variables also keeps it interpretable and permits the exhaustive grid search and two-dimensional operating-map visualization used later. For this low-dimensional, bounded problem, an exhaustive grid search guarantees the global optimum among the specified 1-K candidate points and makes the hot-spot feasibility screening trivial to apply and audit. Additional industrially relevant variables (feed composition and loading, total flow, oxygen ratio, and pressure) and the associated safety constraints, together with a scalable gradient-based or model-predictive-control optimizer to replace the grid search, are deferred to future work (Section 4.10), since the grid-point count grows exponentially with the number of manipulated variables.

Self-optimized operation calculates updated values of $T_{in}$ and $T_{c,in}$ from the estimated activity profile using a two-dimensional grid search over the allowable thermal-target range:

$$[T_{in}^{*}\left(t\right),T_{c,in}^{*}\left(t\right)]=arg\underset{\mathit{u}\in \mathcal{U}}{max}J\left(\widehat{a}(z,t),T_{in},T_{c,in}\right),$$

(28)

where $\mathcal{U}$ denotes the feasible set of inlet temperature targets.

2.5. Performance and Robustness Evaluation Procedure

The primary performance endpoint is the optimization-value gap $\Delta Y_{PA}$, measuring the yield shortfall of the estimated activity policy relative to the true-activity upper bound; secondary metrics are ${\mathrm{RMSE}}_a$, ${\mathrm{RMSE}}_X$, and ${\mathrm{RMSE}}_T$. The main estimation and target-update demonstration uses $N=15$ sampling steps. Each robustness scenario uses $N=10$ sampling steps; this shorter sequence reduces computational cost while retaining the full prescribed progression from fresh to final deactivated activity. Because no plant historian data are used, the noise levels are sensitivity-analysis assumptions rather than plant-specific instrument specifications. The baseline case uses ${\sigma }_T=4$ K and ${\sigma }_X=1$ percentage point (pp); lower- and higher-noise cases use $({\sigma }_T,{\sigma }_X)=(2\mathrm{K},0.5\mathrm{pp})$ and $(6\mathrm{K},1.5\mathrm{pp})$, respectively. The number of axial temperature sensors is varied across $s=3,4,6$. Each scenario is repeated using 10 independent Gaussian-noise realizations (seeds 11, 22, …, 110), and the reported values are means and sample standard deviations across those trials.

Additional tests (i) remove each regularization term individually (ablation study); (ii) compare performance against a conversion-only uniform-activity estimator; (iii) perturb all plant kinetic pre-exponential factors by $-5\%$, $+5\%$, and $+10\%$ while retaining nominal kinetics in the estimator (model–plant mismatch study); (iv) apply a fixed $+5$ K bias to one temperature sensor and, separately, remove one temperature sensor (sensor-fault stress tests); and (v) vary the estimator activity-node count and placement and the first-step initialization (discretisation and initialization sensitivity). The corresponding results are reported in Section 4.4, Section 4.5, Section 4.6, Section 4.7 and Section 4.8.

The optimization-value gap quantifies the performance loss attributable to estimation error:

$$\Delta Y_{PA}=Y_{PA}^{\mathrm{true}-\mathrm{opt}}-Y_{PA}^{\mathrm{est}-\mathrm{opt}},$$

(29)

where $Y_{PA}^{\mathrm{true}-\mathrm{opt}}$ is the phthalic anhydride yield obtained by optimizing with the true (known) activity profile, serving as the ideal upper bound, and $Y_{PA}^{\mathrm{est}-\mathrm{opt}}$ is the yield achieved by the practical digital-twin policy that uses only the estimated activity profile.

3. Case Study and Reactor Model

The proposed framework is demonstrated using the oxidation of o-xylene to phthalic anhydride (PA) in a heat-exchanged fixed-bed reactor. Phthalic anhydride is one of the most important aromatic intermediates in the chemical industry, with global production exceeding one million tonnes per year, primarily used in the manufacture of plasticizers, dyes, and resins [6,21]. The reaction system is strongly exothermic and has been widely used as a benchmark for fixed-bed catalytic reactor modeling, temperature-profile prediction, and safe-operation studies [5,6,22]. The desired reaction converts o-xylene to phthalic anhydride, while undesired reactions lead to over-oxidation and carbon oxide formation. Reactor performance is strongly affected by temperature, catalyst activity, and heat removal.

3.1. Process Description

A dilute o-xylene feed is mixed with air, where oxygen is present in large excess and nitrogen is treated as an inert carrier. The gas mixture enters a bank of catalyst-filled tubes surrounded by molten-salt coolant. The catalyst is a vanadia–titania formulation commonly used for selective oxidation of o-xylene to phthalic anhydride. Heat generated by the strongly exothermic reaction network is removed through the tube wall to the coolant, and the reactor is operated under co-current coolant-flow conditions in the present model.

The process measurements assumed available to the digital twin are sparse axial gas temperatures and outlet conversion. These measurements are representative of the type of information used in model-based fixed-bed reactor studies and activity-estimation work [6,14]. The hidden state of interest is the axial catalyst activity profile $a(z,t)$, which modifies the apparent kinetic rates and is not measured directly. The self-optimizing layer uses the estimated activity profile to update the inlet gas temperature and coolant inlet temperature while respecting a hot-spot temperature constraint.

Figure 2 shows the reduced reaction network used in the digital-twin model. The network is intentionally simpler than the full industrial oxidation chemistry, but it preserves the key reaction-engineering tradeoff emphasized in previous o-xylene oxidation studies: the desired partial oxidation pathway competes with over-oxidation routes that reduce selectivity and increase heat release [6,22].

The simplified reaction network used in this study consists of three reactions:

$$\begin{array}{cc}\hfill A+3{\mathrm{O}}_2& \to B+3{\mathrm{H}}_2\mathrm{O},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill B+7.5{\mathrm{O}}_2& \to 8{\mathrm{CO}}_2+2{\mathrm{H}}_2\mathrm{O},\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill A+10.5{\mathrm{O}}_2& \to 8{\mathrm{CO}}_2+5{\mathrm{H}}_2\mathrm{O},\hfill \end{array}$$

(32)

where A denotes o-xylene and B denotes phthalic anhydride. Reaction (30) is the desired partial oxidation; reactions (31) and (32) represent undesired over-oxidation of PA and direct combustion of o-xylene, respectively.

3.2. Reactor Configuration

The case-study reactor is an industrial multi-tube fixed-bed reactor for the partial oxidation of o-xylene to phthalic anhydride over a V₂O₅/TiO₂ catalyst [4,25]. The reactor tubes are surrounded by molten-salt coolant, which removes the heat generated by the strongly exothermic oxidation reactions. The model represents one representative tube, while coolant heat transfer is scaled using the total number of tubes. The parameter values used in the simulation are summarized in

Table 2. Process and model parameters used for the phthalic anhydride reactor case study.

Parameter	Symbol	Value	Unit	Comment
Reactor geometry
Tube inner diameter	$D_t$	0.025	m	Representative fixed-bed tube
Tube length	L	2.0	m	Axial model domain
Number of tubes	$t_n$	3000	–	Used to scale coolant heat removal
Catalyst bed
Catalyst bulk density	${\rho }_b$	1300	kg m−3	Packed-bed catalyst density
Bed void fraction	$ϵ$	0.50	–	Used in pressure-drop calculation
Particle diameter	$D_p$	0.022	m	Effective catalyst-particle diameter
Gas phase
Gas density	${\rho }_g$	1.293	kg m−3	Constant-property approximation
Superficial gas velocity	u	3600	m h−1	Nominal inlet velocity
Gas heat capacity	$C_{p,g}$	0.2498	kcal kg−1 K−1	Constant heat capacity
Gas molecular weight	$M_g$	29.48	kg kmol−1	Used for inlet molar-flow calculation
Heat removal and coolant
Overall heat-transfer coefficient	U	82.7	kcal m−2 h−1 K−1	Gas-to-coolant heat transfer
Coolant heat capacity	$C_{p,c}$	0.3105	kcal kg−1 K−1	Molten-salt coolant
Coolant mass flow rate	$w_c$	72,000	kg h−1	Total coolant flow rate
Feed and nominal operation
Inlet o-xylene partial pressure	$P_{A,0}$	0.9322	kPa	Dilute o-xylene feed
Inlet gas temperature	$T_{in}$	625	K	Nominal fixed target
Coolant inlet temperature	$T_{c,in}$	625	K	Nominal fixed target
Total pressure	$P_0$	101.325	kPa	Atmospheric-pressure operation
Reaction heats
Desired oxidation heat magnitude	$|\Delta H_{21}|$	307,122	kcal kmol−1	o-Xylene to phthalic anhydride
PA over-oxidation heat magnitude	$|\Delta H_{22}|$	783,700	kcal kmol−1	Obtained from Hess’s law
Direct combustion heat magnitude	$|\Delta H_{23}|$	1,090,822	kcal kmol−1	o-Xylene to carbon oxides

Italicised rows are group sub-headings that divide the table into parameter categories.

; they follow the benchmark PA reactor model and kinetic data reported in the literature [6,22].

The reactor is modeled as a one-dimensional pseudo-homogeneous plug-flow reactor. The axial coordinate is denoted by z, with $z=0$ at the reactor inlet and $z=L$ at the reactor outlet. The ordinary differential equation (ODE) state vector integrated along the reactor is

$$\mathit{s}\left(z\right)={[F_A,F_B,F_C,F_N,T,T_c,P]}^T,$$

(33)

where $F_A$, $F_B$, $F_C$, and $F_N$ are the molar flow rates of o-xylene, phthalic anhydride, carbon oxides, and inert nitrogen, respectively; T is the gas-phase temperature; $T_c$ is the coolant temperature; and P is the total pressure. The symbol $\mathit{s}$ is used here to distinguish the case-study ODE reactor state from the general framework state vector $\mathit{x}$ introduced in Section 2.1.3.

The total inlet molar flow rate per representative tube is calculated from the superficial velocity and tube cross-sectional area:

$$F_{T,0}={\displaystyle \frac{u_s{\rho }_g{A}_{tube}}{M_g}},$$

(34)

where $u_s$ (m h⁻¹) is the superficial gas velocity at inlet conditions and $A_{tube}=\pi D_t^2/4$ is the tube cross-sectional area. The inlet o-xylene and inert molar flow rates are initialized as

$$\begin{array}{cc}\hfill F_{A,0}& ={\displaystyle \frac{P_{A,0}}{P_0}}{F}_{T,0},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill F_{N,0}& =\left(1-{\displaystyle \frac{P_{A,0}}{P_0}}\right)F_{T,0},\hfill \end{array}$$

(36)

with initial conditions $F_B\left(0\right)=F_C\left(0\right)=0$, $T\left(0\right)=T_{in}$, $T_c\left(0\right)=T_{c,in}$, and $P\left(0\right)=P_0$.

3.3. Kinetic Model

The rate constants follow the de Lasa kinetic form [6,22]:

$$\begin{array}{cc}\hfill k_{21}\left(T\right)& =21.07exp\left(10.59-{\displaystyle \frac{13,587.68}{T}}\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill k_{22}\left(T\right)& =21.07exp\left(11.62-{\displaystyle \frac{15,801.97}{T}}\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill k_{23}\left(T\right)& =21.07exp\left(9.73-{\displaystyle \frac{14,392.88}{T}}\right),\hfill \end{array}$$

(39)

where the subscript notation $k_{2i}$ follows de Lasa [22]: the first digit (2) identifies the reaction family (selective oxidation of o-xylene), and the second digit ($i=1,2,3$) indexes the individual reactions corresponding to (30)–(32), respectively. All rate constants have units of kmol/(kg_cat h kPa). Catalyst activity modifies the apparent rate for each reaction directly through the activity factor $a(z,t)$, so no separate effective rate-constant definition is required; the activity enters the rate expressions shown below.

The local organic partial pressures are computed from molar flows:

$$\begin{array}{cc}\hfill F_T& =F_A+F_B+F_C+F_N,\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill P_j& ={\displaystyle \frac{F_j}{F_T}}P.\hfill \end{array}$$

(41)

The activity-modified reaction-rate expressions per unit reactor length are

$$\begin{array}{cc}\hfill r_A& =-{\rho }_{L}a(z,t)\left[k_{21}\left(T\right)+k_{23}\left(T\right)\right]P_A,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill r_B& ={\rho }_{L}a(z,t)\left[k_{21}\left(T\right)P_A-k_{22}\left(T\right)P_B\right],\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill r_C& =8{\rho }_{L}a(z,t)\left[k_{23}\left(T\right)P_A+k_{22}\left(T\right)P_B\right],\hfill \end{array}$$

(44)

where ${\rho }_L={\rho }_b{A}_{tube}$ is the linear catalyst density.

3.4. Mole, Energy, Coolant, and Pressure Balances

The following mole, energy, coolant, and pressure balances follow the standard one-dimensional pseudo-homogeneous plug-flow formulation used in the benchmark PA reactor models [6,22], specialized here by the activity factor $a(z,t)$. The reactor mole balances are

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{F}_A}{dz}}& =r_A,\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{F}_B}{dz}}& =r_B,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{F}_C}{dz}}& =r_C,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{F}_N}{dz}}& =0.\hfill \end{array}$$

(48)

The o-xylene conversion is calculated as

$$X_A\left(z\right)={\displaystyle \frac{F_{A,0}-F_A\left(z\right)}{F_{A,0}}},$$

(49)

and the phthalic anhydride selectivity is calculated as

$$S_{PA}\left(z\right)={\displaystyle \frac{F_B\left(z\right)}{F_{A,0}-F_A\left(z\right)}}.$$

(50)

The gas-phase energy balance includes heat generation by the exothermic reactions and heat removal to the molten salt coolant:

$${\displaystyle \frac{dT}{dz}}={\displaystyle \frac{Q_{gen}-Q_{cool}}{u{\rho }_g{C}_{p,g}}},$$

(51)

where

$$Q_{gen}={\rho }_{b}a(z,t)\left[k_{21}{P}_A|\Delta H_{21}|+k_{22}{P}_B|\Delta H_{22}|+k_{23}{P}_A|\Delta H_{23}|\right]$$

(52)

and

$$Q_{cool}={\displaystyle \frac{4U(T-T_c)}{D_t}}.$$

(53)

The heats of reaction are $\Delta H_{21}=-307,122$ kcal/kmol, $\Delta H_{22}=-783,700$ kcal/kmol, and $\Delta H_{23}=-1,090,822$ kcal/kmol [22]. The value of $\Delta H_{22}$ is confirmed from Hess’s law: $\Delta H_{22}=\Delta H_{23}-\Delta H_{21}=(-1,090,822)-(-307,122)=-783,700$ kcal/kmol, consistent with the stoichiometric relationship between reactions (30)–(32).

For co-current operation, the coolant energy balance is

$${\displaystyle \frac{d{T}_c}{dz}}={\displaystyle \frac{t_n\pi D_{t}U(T-T_c)}{w_c{C}_{p,c}}}.$$

(54)

The pressure drop is represented using an Ergun-type expression [26]:

$${\displaystyle \frac{dP}{dz}}=-{\displaystyle \frac{G^2(1-ϵ)}{{\rho }_g{D}_p{ϵ}^3}}\left[{\displaystyle \frac{150(1-ϵ)}{R{e}_p}}+1.75\right].$$

(55)

3.5. Model Assumptions and Validation

The main assumptions are: one-dimensional plug flow, negligible axial dispersion and radial gradients, pseudo-homogeneous catalyst bed, a pseudo-first-order kinetic reduction in which oxygen does not appear explicitly in the rate expressions, constant gas properties, slow catalyst deactivation relative to the reactor residence time, and quasi-steady reactor behavior for any fixed activity profile. The timescale separation between deactivation and residence time is well satisfied in the present case: the reactor residence time is on the order of seconds, whereas the deactivation sequence used for estimation spans tens of time steps representing hours to days of operation, so the quasi-steady assumption is appropriate.

Three of these assumptions warrant explicit qualification. First, the feed contains air at $P_{A,0}=0.9322$ kPa, corresponding to an inlet oxygen partial pressure of approximately 21.08 kPa. A stoichiometric post-calculation using the three modeled reaction rates gives 18.2% oxygen consumption for the fresh-catalyst benchmark case (an approximate outlet oxygen partial pressure of 17.25 kPa). Thus, this calculation does not establish a strictly constant oxygen concentration. The pseudo-first-order organic-partial-pressure rate form is retained because it is the published reduced kinetic form of de Lasa [22] used for the benchmark, but its error due to omitted oxygen-state dynamics was not quantified here. An explicit oxygen balance and oxygen-dependent kinetics are needed for predictive use outside this benchmark scope. Second, constant gas properties are used because the same dilute feed makes the mixture predominantly nitrogen, whose bulk thermophysical properties vary only weakly over the operating temperature window; this is the standard simplification adopted in the benchmark PA reactor models [6,22], although its quantitative effect on temperature predictions was not separately evaluated here. A temperature- and composition-dependent property model is a useful higher-fidelity extension for assessing plant-level prediction accuracy. Third, pressure is retained for physical completeness and generality; the reaction rates depend on species partial pressures $P_j=(F_j/F_T)P$, and the Ergun term keeps the model transferable to configurations (longer beds, smaller particles, or higher throughput) in which pressure drop is significant. Under the present dilute-feed benchmark conditions, the computed pressure drop is 1.11 kPa (1.10% of inlet pressure). Removing this pressure drop in a benchmark sensitivity calculation changes $T_{max}$ by 0.50 K, outlet conversion by 0.50 percentage points, and PA selectivity by $-0.29$ percentage points; pressure therefore has a modest, but not dominant, influence on the reported benchmark outputs.

Forward-Model Validation

The model was benchmarked under fresh-catalyst conditions with $T_{in}=T_{c,in}=625$ K, $P_{A,0}=0.9322$ kPa, and co-current coolant flow, matching the conditions reported in [6,22]. The predictions for maximum temperature, outlet conversion, and PA selectivity agree satisfactorily with the published benchmark values. This literature comparison verifies the implementation of the reference model, but it is not independent experimental or industrial validation.

Table 3. Forward-model validation under fresh-catalyst benchmark conditions.

Quantity	Symbol	Model Value	Benchmark/Interpretation
Hot-spot temperature	$T_{max}$	676.9 K	677 K
Hot-spot location	$z\left(T_{max}\right)$	2.00 m	2.0 m, monotonic
Outlet conversion	$X_{out}$	88.4%	88–95%
Outlet PA selectivity	$S_{PA,out}$	77.9%	70–75%
Outlet temperature	$T_{out}$	676.9 K	677 K
Pressure drop	$\Delta P$	1.11 kPa	small at atmospheric operation

summarizes the benchmark comparison.

3.6. Catalyst Activity-Profile Representation

Catalyst deactivation in fixed-bed reactors is often spatially nonuniform, with activity loss concentrated near the inlet or outlet depending on the deactivation mechanism. To capture spatial variation while keeping the estimation problem tractable, the continuous activity field is approximated by piecewise-linear interpolation between a finite number of axial activity nodes:

$$a(z,t)\approx {\mathcal{I}}_{\mathrm{lin}}\left[a_1\left(t\right),a_2\left(t\right),\dots ,a_m\left(t\right)\right],$$

(56)

where m is the number of activity nodes, ${\mathcal{I}}_{\mathrm{lin}}$ denotes piecewise-linear interpolation along the reactor axis, and the nodes are placed at uniformly spaced axial positions $z_i=(i-1)L/(m-1)$ for $i=1,\dots ,m$. In the present case study $m=4$, giving nodes at $z=[0,0.67,1.33,2.0]$ m; this choice provides a balanced representation of inlet-to-outlet activity gradients while keeping the inverse problem tractable with the five available measurements ($s=4$ temperatures plus one conversion). The sensitivity study in Section 4.6 compares three, four, and five estimator nodes and a shifted four-node layout. It shows that the three-node representation is slightly smoother and has the lowest dense-grid profile RMSE for this synthetic trajectory, the two four-node layouts perform similarly, and the five-node representation modestly increases RMSE. Four uniform nodes are therefore retained as a transparent resolution compromise, rather than claimed as a unique optimum; a formal local-identifiability check for this configuration is reported in Section 4.1. The activity vector is

$$\mathit{a}\left(t\right)={[a_1\left(t\right),a_2\left(t\right),\dots ,a_m\left(t\right)]}^T,$$

(57)

with estimation bounds

$$0.05\le a_i\left(t\right)\le 1.0,$$

(58)

where the lower bound of 0.05 prevents numerical singularity while permitting severe local deactivation to be represented, consistent with Section 2.3.

To characterize the effect of activity loss on reactor behavior and to verify observability before running the estimator, four representative axial activity-profile cases are examined: (i) fresh catalyst ($a_i=1$ everywhere), (ii) uniformly deactivated catalyst, (iii) linear outlet-side deactivation, and (iv) localized outlet-side activity loss. The forward-model responses for these cases are shown in Figure 3 and Figure 4. For the main estimation and self-optimization study, a synthetic time-evolving deactivation sequence of $N=15$ steps is used. The activity profile advances monotonically from fresh catalyst ($\mathit{a}={[1,1,1,1]}^T$) to a final deactivated state ${\mathit{a}}_{final}={[0.90,0.75,0.60,0.45]}^T$ (stronger deactivation toward the outlet, representing cumulative thermal and product exposure), with a nonlinear severity progression: at step k, ${\mathit{a}}_k=\mathit{1}+{(k/14)}^{1.35}({\mathit{a}}_{final}-\mathit{1})$. This synthetic trajectory provides a controlled severe-deactivation scenario for evaluating the estimator across a broad activity range; it does not represent a calibrated physical deactivation law. The framework treats catalyst activity as an estimated hidden state that multiplies the apparent rates and is by design agnostic to the mechanism that produced the activity loss, so it does not require a specific coking, poisoning, sintering, or phase-transformation law to operate; prescribing a monotonic, outlet-weighted trajectory therefore lets the estimator be exercised over a broad, repeatable activity range without confounding it with the uncertainties of a particular deactivation kinetic. The outlet-weighted final profile $[0.90,0.75,0.60,0.45]$ is an intentionally severe, physically plausible spatial-gradient scenario chosen to test recovery of outlet-side activity loss; real o-xylene oxidation catalysts may instead deactivate near the inlet (poisoning or fouling) or within the hot zone (sintering or phase change) depending on mechanism and history, and the alternative uniform, linear, and localized patterns examined in the observability study confirm that the method is not tied to this single profile. Coupling the estimator to a mechanistic deactivation model, so that activity is both estimated and forward-predicted, is identified as future work.

4. Results and Discussion

All results in this section are obtained from a sequential simulation study using synthetic measurements generated from the benchmark forward model. The nominal estimator and target-update calculations use the same model structure, while additional kinetic-mismatch tests perturb the synthetic plant to probe sensitivity to model–plant error. The reported estimation and optimization figures should therefore be interpreted as simulation-based proof-of-concept results rather than plant-validated performance; factors that would reduce performance in deployment are discussed in Section 4.10. Because the nominal estimator and target optimizer share the reactor structure used to generate the synthetic measurements, the matched-model results are subject to the well-known “inverse crime” of simulation-based estimation and should be read as an optimistic upper reference. The kinetic model–plant mismatch study (Section 4.8) partially relaxes this matched-model condition by perturbing the plant pre-exponential factors while retaining nominal estimator kinetics. Additive measurement noise and varying sensor density assess robustness to measurement quality and information availability, respectively, but do not remove the shared-model limitation. Full avoidance of the inverse crime requires validation against data from a higher-fidelity or real system, which is identified as the primary future step in Section 4.10.

This section presents and discusses the simulation results in six parts. Section 4.1 reports the forward-model validation and confirms that catalyst deactivation is observable from the available measurements. Section 4.2 presents the activity-profile estimation results. Section 4.3 demonstrates the self-optimizing operation and compares fixed-target, true-activity, and estimated activity policies. Section 4.4 examines robustness to measurement noise and sensor density. Section 4.9 discusses novelty positioning, and Section 4.10 discusses limitations and future work.

4.1. Forward-Model Validation and Activity Observability

Under benchmark fresh-catalyst conditions, the model predicted $T_{max}=676.9$ K, $X_{out}=88.4\%$, and $S_{PA}=77.9\%$ (Table 3). These values are consistent with the benchmark results reported in the literature [6,22], confirming that the model is suitable as the physics engine of the digital twin. The temperature profile increases monotonically from inlet to exit under the present configuration (dilute feed, co-current molten-salt coolant, $T_{in}=T_{c,in}=625$ K), so the maximum temperature coincides with the reactor outlet rather than a mid-bed hot spot; this behavior is consistent with the benchmark model of de Lasa [22] at these operating conditions. Figure 5 shows the predicted axial temperature profile, confirming the monotonic rise from inlet to outlet and the absence of a mid-bed hot spot under these benchmark conditions.

Uniform catalyst deactivation was simulated at $a\in \{1.0,0.8,0.6,0.4\}$. As activity decreased, both the reactor temperature rise and outlet conversion decreased systematically, as shown in Figure 3. At $a=0.4$, the model predicted $T_{max}=639.8$ K and $X_{out}=26.9\%$, corresponding to a maximum-temperature reduction of approximately 37 K and a conversion reduction of approximately 61.5 percentage points relative to the fresh-catalyst case (

Table 4. Effect of uniform catalyst activity on observability of deactivation.

a	${\mathit{T}}_{max}$ (K)	${\mathit{X}}_{\mathit{out}}$ (%)	${\mathit{S}}_{\mathit{PA},\mathit{out}}$ (%)	${\mathit{T}}_{\mathit{out}}$ (K)
1.00	676.9	88.4	77.9	676.9
0.80	663.4	66.4	84.3	663.4
0.60	650.3	44.8	86.9	650.3
0.40	639.8	26.9	88.2	639.8

). These results confirm that catalyst deactivation produces large, monotonic, and readily distinguishable changes in both thermal and compositional outputs across all four deactivation levels, establishing that the hidden activity state is strongly observable from the available process measurements. Figure 4 further shows that the representative axial activity profiles considered; uniform, linear outlet-side, and localized outlet-side deactivation; produce distinct axial temperature signatures, confirming that spatial activity gradients along the bed are also recoverable from the sparse temperature measurements used in the estimator.

To complement this qualitative evidence with a formal local-identifiability statement, the sensitivity (Jacobian) matrix of the five modeled outputs (four axial temperatures and outlet conversion) with respect to the four activity nodes was evaluated at the final deactivated operating point by finite differences. The Jacobian has full column rank (four nonzero singular values, $84.4$, $24.6$, $15.5$, and $10.4$ in the mixed temperature/conversion units used) with a modest condition number of approximately $8.1$, confirming that the four-node activity profile is locally identifiable from the available measurements and that the inverse problem is well-conditioned at the operating point. Global uniqueness is not guaranteed for a nonlinear model; distinct activity profiles can, in principle, produce nearly indistinguishable outputs, which is precisely why the spatial-smoothness and temporal-prior terms are included: they select a preferred regularized solution among near-equivalent candidates. The ablation study in Section 4.5 shows that the temporal prior is the dominant regularizer, reducing activity-profile RMSE substantially for the tested noise level.

4.2. Activity-Profile Estimation Results

The activity estimator was evaluated across 10 independent noise realisations (seeds 11–110) using a 15-step deactivation sequence under the baseline noise level (${\sigma }_T=4$ K, ${\sigma }_X=1$ percentage point). The estimator represented the activity profile using four axial nodes at $z=[0,0.67,1.33,2.0]$ m. The final true activity profile was

$${\mathit{a}}_{true}=[0.90,0.75,0.60,0.45],$$

(59)

representing stronger outlet-side deactivation consistent with greater thermal exposure near the bed exit. The mean final estimated profile across the 10 seeds was $\widehat{\mathit{a}}\approx [0.874,0.773,0.543,0.531]$. The full estimation performance metrics are reported in

Table 5. Catalyst activity-profile estimation performance (means over 10 independent noise realizations, baseline scenario: ${\sigma }_T=4$ K, ${\sigma }_X=1$ pp, 4 axial sensors).

Metric	Value	Unit
Activity-node RMSE (mean, all steps)	0.075	dimensionless
Final activity-profile RMSE (mean)	0.104	dimensionless
Outlet conversion RMSE (mean)	0.99	percentage points
Outlet temperature RMSE (mean)	1.85	K

. The baseline activity-profile RMSE averaged across all 10 seeds and 15 steps was 0.075; the ${\mathrm{RMSE}}_X$ was 0.99 percentage points, and the ${\mathrm{RMSE}}_T$ was 1.85 K, confirming that the estimated activity profile reproduces both outlet measurements with good fidelity despite the sparse sensor configuration. These results are important because the optimizer never receives the true catalyst activity profile; it receives only the estimated state reconstructed from four axial temperature measurements and one outlet-conversion measurement. In this sense, the estimator converts routinely available process measurements into actionable catalyst-state information, avoiding reliance on direct catalyst characterization during normal operation. The ensemble-mean profile snapshots in Figure 6 show the progressive deepening of the axial deactivation gradient: by step 14, the mean estimated profile declines from $\widehat{a}\approx 0.87$ at the inlet to $\widehat{a}\approx 0.53$ at the outlet. Thus, the estimator captures the main inlet-to-outlet deactivation trend, but the pointwise activity estimates remain noisy and partially biased because only sparse axial temperatures and one outlet-conversion measurement constrain the four-node activity profile. The measured and predicted outlet temperature and conversion fits are shown in Figure 7; both outputs track the simulated measurements closely with no systematic bias. The final smoothed profile in Figure 8 provides the operating-relevant activity distribution passed to the optimizer.

4.3. Self-Optimizing Operation Results

Under fixed operation, the reactor was maintained at $T_{in}=T_{c,in}=625$ K throughout the deactivation sequence. As catalyst activity declined to the final profile $[0.90,0.75,0.60,0.45]$, fixed-target operation progressively lost performance; the final phthalic anhydride yield under fixed targets dropped to approximately 46.3%.

The self-optimizing control results are from an illustrative single run (seed 44). Using the estimated activity profile, the optimizer updated the thermal targets to $T_{in}^{*}=632$ K and $T_{c,in}^{*}=632$ K over the deactivation sequence (Figure 9), recovering the final PA yield to approximately 62.1% while maintaining the maximum reactor temperature below the 730 K safety limit. Figure 10 compares the resulting PA yield and outlet conversion with fixed-target operation across the deactivation sequence, and Figure 11 shows the corresponding maximum reactor temperature under both policies. The overall performance outcomes are summarized in

Table 6. Performance comparison between fixed-target operation and self-optimized operation under catalyst deactivation (single illustrative run, seed 44).

Quantity	Value	Unit
Initial fixed PA yield	68.85	%
Final fixed PA yield	46.34	%
Final self-optimized PA yield	62.09	%
Final fixed conversion	53.87	%
Final self-optimized conversion	75.56	%
Final optimized Tin	632.00	K
Final optimized Tc,in	632.00	K
Final optimized Tmax	670.01	K
Maximum optimized Tmax	690.19	K

.

Table 7. Final deactivated-state comparison of fixed operation, ideal true-activity optimization, and estimated activity optimization under the baseline noisy measurement scenario (${\sigma }_T=4$ K, ${\sigma }_X=1$ percentage point). The true-activity optimum is deterministic; other values are means over $n=10$ independent noise realisations.

Policy	Activity Information	Final ${\mathit{Y}}_{\mathit{PA}}$ (%)	Gain Over Fixed (pp)	Gap to True Optimum (pp)
Fixed-target operation	No optimization; $T_{in}=T_{c,in}=625$ K	46.34	0.00	15.75
True-activity optimization	True activity profile $a(z,t)$	62.09	15.75	0.00
Estimated activity optimization	Estimated activity profile $\widehat{a}(z,t)$	61.95	15.61	0.14 ± 0.20

compares three policies at the final deactivated state under the baseline noisy measurement scenario (means over 10 noise realisations): (i) fixed-target operation (≈$46.3\%$), (ii) ideal optimization using the true activity profile (upper bound, ≈$62.1\%$), and (iii) practical optimization using the estimated activity profile (≈$61.9\%$). The estimated activity optimization falls within $0.14\pm 0.20$ percentage points of the true-activity optimum on average. Expressed as recovery efficiency, the true-activity optimum provides 15.75 percentage points of recoverable yield improvement over fixed-target operation, while the estimated activity policy recovers 15.61 percentage points, or approximately 99.1% of the available performance recovery. This indicates that estimation uncertainty has little practical effect on the optimized operating target in the baseline scenario.

It is worth assessing directly whether an outlet-conversion RMSE of 0.99 percentage points is acceptable for industrial operating decisions. The optimizer selects a temperature target rather than the conversion value itself, and the yield surface is locally flat near the optimum. In the baseline simulation, the resulting estimated activity policy is within 0.14 pp of the true-activity optimum; this supports adequacy of the observed estimation quality for the target-selection task studied here, but does not establish a general causal conversion-error-to-yield relationship. Acceptability for tighter product-quality or emission specifications, where a 1 pp conversion error could matter directly, would need to be re-evaluated against plant data (Section 4.10).

Regarding online feasibility, a representative estimation-and-optimization cycle: the constrained least-squares activity solve (up to 250 reactor evaluations) followed by the 169-point target grid search; executed in approximately 1.1 s (about 0.4 s for estimation and 0.7 s for optimization, with each reactor solve taking roughly 6 ms). The timing was obtained on an Intel Core Ultra 7 255H workstation (Intel Corporation, Santa Clara, CA, USA) using Python 3.14.3 (Python Software Foundation, Wilmington, DE, USA) and SciPy 1.17.1. This is several orders of magnitude shorter than the deactivation time scale (hours to days), confirming that the workflow is compatible with online target updates; the grid search is moreover trivially parallelizable and would be replaced by a faster gradient-based optimizer in a scaled implementation with additional manipulated variables.

To connect the self-optimizing target update with safe-operation analysis, Figure 12 shows the two-dimensional operating map of the final deactivated reactor state as a function of gas and coolant inlet temperatures. The colour field and white contours represent PA yield, while the dashed red contour marks the 730 K hot-spot safety boundary, and the gray region is infeasible. The step-by-step self-optimizing control trajectory is overlaid as a blue line, starting from the fixed nominal point (625 K, 625 K) and ending at the final self-optimized target. The map confirms that the trajectory moves into a region of substantially higher PA yield while remaining below the 730 K constraint throughout. This visualization follows the operating-window logic used in parametric sensitivity studies of o-xylene oxidation [6], but here it is generated from the deactivated catalyst state as estimated by the digital twin rather than from the fresh-catalyst model.

The final self-optimized target (632 K, 632 K) lies on the upper edge of the imposed thermal-target search box, but the 730 K hot-spot constraint is not active in the demonstrated case: Table 6 reports a maximum optimized temperature of 690.19 K, leaving a 39.81 K model-predicted margin. Thus, the reported optimum is limited by the selected search range rather than by the hot-spot constraint. Nevertheless, deployment feasibility remains sensitive to activity estimation and structural model error because a biased activity estimate could under-predict the true peak temperature. For plant use, a safety back-off should constrain the model prediction to $T_{max}\le 730-\Delta$, where $\Delta$ incorporates both measurement-fit error and a validated allowance for structural model error. A formally verified probabilistic (chance-constrained) margin is identified as future work (Section 4.10).

4.4. Robustness to Measurement Noise and Sensor Density

The stress tests showed that estimator and optimizer performance vary with measurement quality and sensor density (

Table 8. Effect of measurement noise and sensor configuration on activity estimation and optimized performance.

Scenario	Temp. Sensors	${\mathit{\sigma }}_{\mathit{T}}$ (K)	${\mathit{\sigma }}_{\mathit{X}}$ (pp)	Activity-Node ${\mathit{RMSE}}_{\mathit{a}}$	Est.-Opt. ${\mathit{Y}}_{\mathit{PA}}$ (%)	Gap (pp)
Baseline (4T, moderate noise)	4	4.0	1.0	0.075	61.95	0.14 ± 0.20
Lower noise (4T)	4	2.0	0.5	0.055	62.03	0.06 ± 0.12
Denser sensing (6T)	6	4.0	1.0	0.070	61.98	0.11 ± 0.15
Sparser sensing (3T)	3	4.0	1.0	0.080	61.92	0.17 ± 0.24
Higher noise (4T)	4	6.0	1.5	0.082	61.98	0.11 ± 0.20

Note: Values in the “Gap” column are mean ± standard deviation over 10 independent noise realisations. Each scenario uses one outlet-conversion measurement in addition to the listed axial temperature sensors. ${\sigma }_T$ is the temperature-noise standard deviation, ${\sigma }_X$ is the outlet-conversion-noise standard deviation in percentage points, and pp denotes percentage points. The yield gap is measured relative to the true-activity optimum.

; Figure 13, Figure 14 and Figure 15). All scenarios were evaluated over 10 sampling steps (seeds 11–110) using the estimator settings (${\lambda }_s=0.7$, ${\lambda }_p=15$) and optimizer settings ($w_S=0.01$, ${\lambda }_u=2\times {10}^{-4}$ K⁻²) from the main demo; reported values are means ± standard deviations across 10 independent noise realizations. Regarding noise sensitivity: in the low-noise case (${\sigma }_T=2$ K, ${\sigma }_X=0.5$ pp), the ${\mathrm{RMSE}}_a$ was $0.055\pm 0.009$ and the yield gap was $0.057\pm 0.12$ percentage points; in the baseline case (${\sigma }_T=4$ K, ${\sigma }_X=1$ pp), the yield gap was $0.14\pm 0.20$ percentage points; and in the high-noise case (${\sigma }_T=6$ K, ${\sigma }_X=1.5$ pp), the yield gap was $0.11\pm 0.20$ percentage points. The self-optimizing benefit over fixed-target operation is preserved across all three noise scenarios.

Regarding sensor density: the dense six-sensor configuration ($s=6$) achieved ${\mathrm{RMSE}}_a=0.070\pm 0.011$ with a yield gap of $0.11\pm 0.15$ percentage points, closely approaching the ideal true-activity optimum. The sparse three-sensor case ($s=3$) yielded ${\mathrm{RMSE}}_a=0.080\pm 0.012$ and a yield gap of $0.17\pm 0.24$ percentage points. The estimated activity policy remained superior to fixed-target operation across all tested noise levels and sensor configurations.

Regarding the minimum sensor requirement for stable estimation, three axial temperature sensors plus the outlet-conversion measurement (the $s=3$ layout) were sufficient, within the tested simulation scenarios, to maintain stable spatial estimation and preserve the benefit over fixed-target operation; four sensors plus outlet conversion is the recommended configuration for the four-node activity representation because it provides five available measurements, while adding a sixth sensor produced only diminishing returns. The number of resolvable activity nodes is thus limited by the number of independent measurements, which is why the base case uses $m=4$. These stress tests cover random measurement noise, sensor density, one fixed sensor-bias case, one single-sensor-loss case, and (in Section 4.8) kinetic pre-exponential mismatch. Missing measurements, sensor drift, communication delays, outliers, and kinetic errors beyond $+10\%$ were not quantitatively evaluated, so no performance claim is made for those conditions. A fixed sensor bias propagates into an activity-estimation bias, because the estimator cannot distinguish a biased reading from genuine deactivation, whereas a single sensor loss reduces the effective sensor count toward the three-sensor case shown above. A paired ten-seed, ten-step check found that applying a $+5$ K bias to one sensor increased the activity-profile RMSE from $0.075\pm 0.011$ to $0.083\pm 0.013$, while removing one sensor increased it to $0.080\pm 0.012$. Thus, both tested disturbances modestly degrade profile recovery, with the fixed bias producing the larger increase; these limited tests do not establish robustness to arbitrary sensor failures or larger kinetic uncertainty. Robust handling of bias, missing data, drift, delays, and outliers (for example, bias augmentation within a full moving-horizon estimator, missing-data logic, or outlier-robust residuals) is identified as future work (Section 4.10).

4.5. Regularisation Ablation Study

To isolate the contribution of each regularisation term in the estimator objective, three ablation scenarios were evaluated at the baseline sensor configuration (4 axial temperature sensors, ${\sigma }_T=4$ K): (i) no spatial smoothing (${\lambda }_s=0$, ${\lambda }_p=15$), (ii) no temporal prior (${\lambda }_s=0.7$, ${\lambda }_p=0$), and (iii) no regularisation (${\lambda }_s=0$, ${\lambda }_p=0$). All 10 noise seeds were used per scenario.

Table 9. Regularisation ablation study: effect of removing spatial smoothing (${\lambda }_s=0.7$) and temporal prior (${\lambda }_p=15$) on activity estimation and optimizer performance ($n=10$ trials, 4 axial sensors, ${\sigma }_T=4$ K, nominal kinetics).

Regularization Configuration	Activity-Node ${\mathit{RMSE}}_{\mathit{a}}$	Est.-Opt. ${\mathit{Y}}_{\mathit{PA}}$ (%)	Gap to True Optimum (pp)
Full regularization (${\lambda }_s=0.7$, ${\lambda }_p=15$)	0.075 ± 0.011	61.95 ± 0.20	0.14 ± 0.20
No spatial smoothing (${\lambda }_s=0$, ${\lambda }_p=15$)	0.077 ± 0.012	61.95 ± 0.20	0.14 ± 0.20
No temporal prior (${\lambda }_s=0.7$, ${\lambda }_p=0$)	0.114 ± 0.024	61.98 ± 0.15	0.11 ± 0.15
No regularization (${\lambda }_s=0$, ${\lambda }_p=0$)	0.120 ± 0.026	62.01 ± 0.14	0.09 ± 0.14

Note: All scenarios use the baseline sensor configuration (4 axial temperature sensors, ${\sigma }_T=4$ K, ${\sigma }_X=1$ pp, nominal kinetics). Values are mean ± standard deviation over 10 independent noise realizations. The full-regularization row reproduces the baseline result from Table 8 for direct comparison.

reports the results.

The ablation results demonstrate a strong contribution from the temporal prior (${\lambda }_p=15$): removing it raises the activity-profile RMSE from 0.075 to 0.114 (a 52% increase), while removing only spatial smoothing causes a minor increase to 0.077. Removing both terms gives RMSE $=0.120$ (60% above the full-regularisation baseline). The temporal prior is the dominant regulariser because the activity state evolves gradually relative to the measurement noise; without it, the estimator fits more of the step-to-step measurement variation, producing large jumps. Spatial smoothing provides a smaller benefit because the piecewise-linear profile interpolation already partially couples adjacent nodes. The yield gap is essentially unchanged across all four configurations, consistent with the finding in Section 4.7 that the 1 K grid optimizer does not translate incremental accuracy improvements into measurable yield gains.

4.6. Activity-Node Discretisation and Initialization Sensitivity

For clarity, two error definitions appear in this work. The estimator performance reported in Section 4.2, Section 4.3, Section 4.4, Section 4.5, Section 4.6, Section 4.7 and Section 4.8 uses the activity-node ${\mathrm{RMSE}}_a$ (the activity-profile RMSE evaluated at the four estimated activity nodes; baseline $\approx 0.075$), whereas the discretisation and initialization studies in this subsection evaluate reconstruction accuracy on a common dense axial grid using the dense-grid profile ${\mathrm{RMSE}}_a$ (baseline $\approx 0.052$), so that estimators with different node counts and placements can be compared on an equal footing. Because they are computed on different grids, the two metrics are not directly comparable in magnitude.

The activity-node representation and the first-step initialization were tested using the baseline four-temperature-sensor configuration and 10 independent noise realizations. In the discretisation test, the same continuous synthetic plant trajectory was retained, and only the estimator representation was changed.

Table 10. Sensitivity of activity estimation to the number and axial placement of estimator activity nodes (10 noise realizations, 4 axial temperature sensors, ${\sigma }_T=4$ K, ${\sigma }_X=1$ pp).

Estimator Activity Nodes	m	Node Positions (m)	Dense-Grid Profile ${\mathit{RMSE}}_{\mathit{a}}$	Conversion RMSE (pp)	Outlet-Temperature RMSE (K)
Three uniform	3	0.00; 1.00; 2.00	0.043 ± 0.007	0.98 ± 0.19	1.76 ± 0.41
Four uniform (baseline)	4	0.00; 0.67; 1.33; 2.00	0.052 ± 0.006	0.99 ± 0.18	1.85 ± 0.43
Four shifted	4	0.00; 0.50; 1.25; 2.00	0.051 ± 0.006	0.98 ± 0.18	1.88 ± 0.47
Five uniform	5	0.00; 0.50; 1.00; 1.50; 2.00	0.058 ± 0.008	0.98 ± 0.18	1.86 ± 0.43

Note: The synthetic plant activity is represented by the same four-node continuous reference trajectory in every case. Only the estimator discretisation is changed. Values are mean ± standard deviation over 10 noise realisations (four temperature sensors, ${\sigma }_T=4$ K, ${\sigma }_X=1$ pp). Errors are evaluated on a common dense axial grid.

shows that three uniformly spaced nodes gave the lowest dense-grid profile RMSE ($0.043\pm 0.007$), while four uniform and shifted four-node layouts performed similarly ($0.052\pm 0.006$ and $0.051\pm 0.006$, respectively). Five nodes increased the RMSE to $0.058\pm 0.008$, consistent with the limited information content of five measurements. The baseline four-node layout is retained because it resolves four axial activity values and remains locally identifiable, while the sensitivity results make clear that it is a modeling-resolution compromise rather than a uniquely optimal discretisation.

Table 11. Sensitivity of activity estimation to the first-step numerical initial guess (10 noise realizations, baseline four-node estimator).

First-Step Activity Guess	Dense-Grid Profile ${\mathit{RMSE}}_{\mathit{a}}$	Conversion RMSE (pp)	Outlet-Temperature RMSE (K)
Uniform $a_0=0.20$	0.051 ± 0.006	0.99 ± 0.18	1.85 ± 0.43
Uniform $a_0=0.60$	0.052 ± 0.006	0.99 ± 0.18	1.85 ± 0.43
Uniform $a_0=0.98$	0.052 ± 0.006	0.99 ± 0.18	1.85 ± 0.43

Note: All cases use the four uniformly spaced estimator nodes and the same arrival prior ($a=0.98$ at the first step). Values are mean ± standard deviation over 10 noise realisations.

tests first-step uniform guesses of 0.20, 0.60, and 0.98 while holding the physical arrival prior at 0.98. The resulting metrics are indistinguishable at the reported precision, showing that the bounded least-squares solve and subsequent warm start remove sensitivity to the numerical initial guess in this tested range. This result does not establish robustness to a misspecified arrival prior, which remains an important extension for plant data.

4.7. Comparison with Uniform-Activity Baseline

Table 12. Comparison of moving-window estimator versus uniform-activity baseline comparator (10 trials per scenario).

Scenario	Activity-Node ${\mathit{RMSE}}_{\mathit{a}}$		Yield Gap (pp)
	Uniform (Baseline)	Moving-Window	Uniform (Baseline)	Moving-Window
Baseline (4T)	0.092 ± 0.000	0.075 ± 0.011	0.57 ± 0.00	0.14 ± 0.20
Lower noise (4T)	0.092 ± 0.000	0.055 ± 0.009	0.57 ± 0.00	0.06 ± 0.12
Denser sensing (6T)	0.092 ± 0.000	0.070 ± 0.011	0.57 ± 0.00	0.11 ± 0.15
Sparser sensing (3T)	0.092 ± 0.000	0.080 ± 0.012	0.57 ± 0.00	0.17 ± 0.24
Higher noise (4T)	0.093 ± 0.000	0.082 ± 0.014	0.57 ± 0.00	0.11 ± 0.20

Note: The uniform baseline estimator uses only the outlet-conversion measurement and assumes spatially uniform deactivation. The moving-window estimator additionally uses sparse axial temperature measurements to resolve the spatial activity profile. Values are mean ± standard deviation over 10 independent noise realizations. Yield gap is measured relative to the true-activity optimum.

compares the moving-window estimator against the uniform-activity baseline in every noise-and-sensor scenario. The uniform baseline uses only the outlet-conversion measurement and assumes spatially uniform deactivation; the moving-window estimator additionally incorporates sparse axial temperature measurements to resolve the spatial activity profile.

The activity RMSE of the uniform baseline was approximately 0.092 across all noise-and-sensor scenarios, reflecting the fact that uniform deactivation cannot be distinguished from the mean deactivation level of the true profile using a single outlet measurement. The moving-window estimator achieved lower activity RMSE than the uniform baseline across all tested scenarios because the additional axial temperature measurements allowed it to resolve the spatial gradient more accurately than a single-output uniform estimator. Despite this, the moving-window estimator provides spatial information that the uniform baseline cannot: it identifies which section of the bed is most active, which is required for hot-spot safety monitoring and localized inlet-condition adjustments. In the optimization trials, the two-dimensional 1 K-resolution grid search was not fine enough to translate the spatial activity advantage into a consistent yield improvement; a finer-resolution or gradient-based optimizer would be expected to reveal larger gains from the spatial estimate.

4.8. Model-Mismatch Sensitivity

The nominal results reported above use a matched-model synthetic plant: the estimator and target optimizer use the same model structure that generates the nominal noisy measurements. To assess sensitivity to model–plant mismatch, three additional scenarios were run in which all reaction pre-exponential factors in the synthetic plant were perturbed by $-5\%$, $+5\%$, and $+10\%$ while the estimator and optimizer continued to use nominal kinetics.

Table 13. Effect of kinetic model–plant mismatch on estimator and optimizer performance (10 trials per scenario, ${\sigma }_T=4$ K, four axial sensors).

Pre-Exponential Perturbation	Activity-Node ${\mathit{RMSE}}_{\mathit{a}}$	Est.-Opt. ${\mathit{Y}}_{\mathit{PA}}$ (%)	Gap to True Optimum (pp)
Nominal (0%)	0.075 ± 0.011	61.95 ± 0.20	0.14 ± 0.20
$+5\%$	0.078 ± 0.011	65.49 ± 0.22	0.16 ± 0.22
$+10\%$	0.094 ± 0.010	67.66 ± 0.33	0.45 ± 0.33
$-5\%$	0.095 ± 0.013	58.17 ± 0.20	0.11 ± 0.20

Note: The perturbation is applied uniformly to all three reaction pre-exponential factors in the plant simulation while the estimator uses nominal kinetics. Values are mean ± standard deviation over 10 independent noise realizations (4 axial temperature sensors, ${\sigma }_T=4$ K).

summarises the results.

Under nominal conditions (0% perturbation), the yield gap was $0.14\pm 0.20$ pp, consistent with the stress-test baseline. With $+5\%$ perturbation, the gap was $0.16\pm 0.22$ pp, essentially unchanged. With $+10\%$ perturbation, the gap increased to $0.45\pm 0.33$ pp; the higher absolute yield levels in this scenario (≈67.7% vs. $61.9\%$ nominal) reflect the more reactive perturbed plant rather than improved estimation. The $-5\%$ perturbation produced a gap of $0.11\pm 0.20$ pp, slightly below nominal. These results indicate that the estimated activity policy remains within less than 0.5 pp of the true-activity optimum across the full tested mismatch range, evaluated on the same perturbed plant in each case. The result does not eliminate the need for plant validation, but it suggests that the target-update layer is relatively insensitive to moderate uniform kinetic pre-exponential-factor mismatch; its value comes from repeatedly adapting operating targets using the inferred catalyst state rather than from a single fixed nominal design calculation.

4.9. Novelty Positioning Against Existing Literature

As summarized in Table 1, catalyst activity estimation in fixed-bed reactors has been studied previously, including studies where temperature measurements are used to infer spatial activity profiles [14]. Reactor optimization and model-based digital design have also been reported for catalytic reactors, including the PA system [5,24]. The novelty of the present work is therefore not the introduction of a new kinetic model, nor the isolated concept of catalyst activity estimation. Instead, the contribution lies in demonstrating, in a simulation-based proof of concept, how an estimated catalyst activity profile is propagated directly into a self-optimizing target-update layer, linking sparse measurements to hidden-state estimation and updated operating targets. This distinction is important because measurements alone reveal that reactor performance has deteriorated, but they do not identify how the operating targets should be adjusted; the estimated catalyst state supplies the physical information needed to select catalyst-specific temperature targets.

In this framework, the estimated catalyst activity serves as the operative input to the optimization problem at each time step, not merely as a diagnostic output. This transforms the digital twin from a passive monitoring tool into an adaptive decision-support system that adjusts operating targets as the catalyst degrades. From an industrial perspective, maintaining operation near the catalyst-specific optimum could help delay conservative replacement decisions, reduce the economic penalty of progressive deactivation, and preserve yield without requiring extensive sensor retrofitting. The present work is scoped as a computational methodology contribution using a literature-benchmarked reactor model and synthetic measurements; plant deployment would require further validation against experimental or industrial data, as discussed in Section 4.10.

Although demonstrated for o-xylene oxidation, the framework is not specific to this chemistry: it applies to any catalyst-deactivating reactor in which activity loss produces a measurable thermal or compositional signature, such as methanol-to-olefins with coking, steam reforming with sintering, or other selective-oxidation systems subject to poisoning. Because the reactor is solved quasi-steadily at each sampling instant, the approach is also agnostic to the absolute deactivation time scale: the sampling interval need only be short relative to the deactivation time constant, so faster (hours) or slower (months) deactivation is accommodated by rescaling the sampling interval and retuning the temporal-prior weight ${\lambda }_p$, which encodes the expected step-to-step activity change.

4.10. Limitations and Future Work

The five limitations listed below are ranked by their effect on confidence in the reported performance numbers. The most important is the simulation-only validation basis (Limitation 1), which means the reported RMSE and yield figures cannot be directly extrapolated to real plant conditions. The second most important is the simplified 1D model (Limitation 2), which constrains the practical scope. The remaining three (Limitation 3: coarse activity discretisation; Limitation 4: simplified estimator; Limitation 5: simplified optimizer) constrain the deployment pathway but do not affect the internal consistency of the simulation results.

Limitation 1: Simulation-only validation. The study is a simulation-based proof of concept. The nominal synthetic measurements are generated from the same benchmark model structure used by the estimator, with additive Gaussian noise superimposed, and the kinetic-mismatch study perturbs only the reaction pre-exponential factors. Therefore, the results do not cover all forms of structural model error that would be present in plant deployment. Heat-transfer degradation, non-Gaussian sensor noise, sensor bias, catalyst-batch variability, and structural kinetic uncertainty could each reduce performance relative to the reported values. In particular, error in the lumped overall heat-transfer coefficient U is expected to have the largest effect, because it sets the hot-spot temperature that defines both the 730 K constraint and the location of the optimum; kinetic activation-energy and mechanism error, and mismatch in the true (mechanistic) deactivation pattern, would similarly bias the estimated activity, since the estimator attributes all model–data mismatch to activity. The kinetic pre-exponential mismatch study (Section 4.8) already shows the yield gap growing from 0.14 to 0.45 pp under a $+10\%$ perturbation, illustrating that structural errors erode the reported ≈99.1% recovery; within the tested range, however, they do not eliminate it. Larger or systematic (non-random) errors, especially in U and in the deactivation pattern, would reduce it further, and similar quantitative estimation performance should therefore not be assumed for real industrial measurements. Plant data, independent experimental measurements, or pilot-scale tests would be required to quantify these effects and assess deployment feasibility. Practical deployment would also require reliable measurement data infrastructure: calibrated axial-temperature and conversion measurements, timestamp synchronization across data streams, screening and treatment of missing, delayed, or outlying data, and integration with the plant historian and control system. Sensor locations should be selected from an observability and sensitivity analysis across the anticipated operating envelope, rather than copied directly from the simulated layout. A deployed system would also need periodic parameter reconciliation against plant data, automated quality checks, a conservative safety back-off, and a fallback operating policy when measurements or model-health checks are unreliable.

The numerical mismatch experiment quantifies only uniform pre-exponential-factor perturbations; it does not separately quantify uncertainty in U, activation energies, reaction mechanism, or the deactivation law. These effects can be partially absorbed by the activity estimate and thereby bias the inferred profile and the resulting targets. Consequently, the reported activity RMSE and yield-recovery metrics must not be extrapolated to industrial measurements until they are evaluated against independent pilot-scale or plant data.

Limitation 2: One-dimensional pseudo-homogeneous reactor model. The reactor model is a one-dimensional pseudo-homogeneous model. This level of model detail is appropriate for repeated estimation and optimization because it is computationally efficient and reproduces the benchmark reactor behavior used in this study. However, it neglects radial temperature gradients, intraparticle diffusion limitations, detailed catalyst morphology, possible nonuniform coolant-side effects, and explicit oxygen-state dynamics. A more detailed two-dimensional heterogeneous model with an oxygen component balance and oxygen-dependent kinetics would be required for industrial reactor design, safety certification, or predictive operation beyond the benchmark scope.

Limitation 3: Low-dimensional activity discretization. The catalyst activity profile is represented by a small number of axial nodes. This low-dimensional representation improves numerical conditioning and avoids overfitting sparse measurements, but it cannot resolve fine-scale deactivation patterns. The smoother shape of the estimated profile relative to the true profile (Section 4.2) is an estimator artifact caused by spatial regularization and limited sensor coverage, rather than a physical feature of the catalyst state. The estimated profile should therefore be interpreted as an effective activity distribution that supports operating decisions, not as a direct microscopic measurement of the catalyst state. These choices limit geometric fidelity and prevent fine-scale deactivation reconstruction.

Limitation 4: Single-step moving-window estimator. The estimator is a moving-window constrained activity estimator with spatial smoothing and temporal regularization. It is deliberately described as a moving-window estimator rather than a full nonlinear moving-horizon estimation (MHE) implementation [17,27]: the current formulation estimates activity independently at each time step using the previous estimate as a regularization anchor, rather than optimizing over a sliding window of past measurements while enforcing dynamic process constraints simultaneously. A rigorous MHE formulation would improve estimation consistency and enable principled tuning of the horizon length.

Limitation 5: Simplified steady-state optimizer. The optimization layer is a simplified steady-state operating-target update, not a plant-ready economic model predictive controller or real-time optimizer [27,28,29]. Only two manipulated variables, inlet gas temperature and coolant inlet temperature, are optimized via a coarse two-dimensional grid search. Practical deployment would additionally require dynamic actuator models, ramp-rate limits, feed-flow constraints, coolant-system hydraulics, economic cost terms, catalyst-life penalty functions, and formally verified plant safety margins. Taken together, these five limitations define the scope of the present work as a simulation-based proof-of-concept computational framework.

Future work can be organized into two tiers. Near-term methodological extensions include: (i) replacing the current step-by-step estimator with a rigorous sliding-window nonlinear moving-horizon estimation (MHE) formulation to improve estimation consistency and enable principled horizon-length tuning; directly addressing Limitation 4; (ii) replacing the grid-search optimizer with a model predictive control (MPC) layer that handles dynamic actuator constraints and multiple manipulated variables; addressing Limitation 5; and (iii) testing finer activity discretizations ($m>4$) to assess the resolution limit imposed by available sensor configurations. Longer-term priorities include: (iv) validation against pilot-scale or industrial plant data to assess robustness to structural model mismatch, sensor bias, and catalyst-batch variability; the primary validation gap identified in Limitation 1; (v) extension of the optimization objective to a yield-versus-catalyst-lifetime multi-objective formulation. The estimated activity and hot-spot temperature provide relevant inputs for this extension, but a validated mechanism-specific deactivation model and economic catalyst-life trade-offs are required before catalyst lifetime can be predicted or optimized; and (vi) a systematic parametric-uncertainty study (for example, Monte Carlo sampling of the heat-transfer coefficient, pressure-drop parameters, and feed composition) coupled with chance-constrained optimization, to quantify how larger operating-parameter uncertainties propagate into the reported optimization performance and the hot-spot feasibility margin.

5. Conclusions

Catalyst deactivation progressively degrades fixed-bed reactor performance, yet most reactor digital twins lack the ability to estimate the current catalyst condition and use it to adapt operating targets. This work addressed that gap by developing a simulation-based catalyst-activity-aware self-optimizing digital-twin framework for catalyst-deactivating fixed-bed reactors. The central idea is to treat catalyst activity as a hidden reactor state, estimate it sequentially from sparse process measurements, and use the estimated condition to update reactor operating targets rather than holding them fixed.

The framework was demonstrated using the oxidation of o-xylene to phthalic anhydride over a vanadia–titania catalyst in a heat-exchanged fixed-bed reactor. A one-dimensional pseudo-homogeneous reactor model was benchmarked under fresh-catalyst conditions ($T_{in}=T_{c,in}=625$ K), reproducing the literature benchmark values of $T_{max}$, $X_{out}$, and $S_{PA}$ reported by de Lasa [22] and Zuluaga-Botero et al. [6]; this comparison verifies reference-model implementation rather than providing independent validation. Catalyst activity was incorporated as an axial activity profile modifying the apparent reaction rates, directly linking catalyst-state loss to outlet conversion, selectivity, heat release, and hot-spot location.

Deactivation to 40% activity reduced the maximum temperature by approximately 37 K and outlet conversion by approximately 61.5 percentage points, demonstrating clear measurement sensitivity to activity loss, and a formal local-identifiability check confirmed that the four-node activity profile is well-conditioned from the available measurements. The moving-window constrained estimator recovered the inlet-to-outlet activity gradient with an activity-profile ${\mathrm{RMSE}}_a=0.075$, an outlet-conversion ${\mathrm{RMSE}}_X=0.99$ percentage points, and an outlet-temperature ${\mathrm{RMSE}}_T=1.85$ K. Supplying the estimated activity to the self-optimizing layer raised the final phthalic anhydride yield from 46.3% under fixed-target operation to 61.9%, placing it within 0.14 percentage points of the true-activity optimum and recovering approximately 99.1% of the available improvement, while maintaining the maximum reactor temperature below the 730 K safety limit. The benefit persisted across noise levels (${\sigma }_T=2$–6 K), sensor counts ($s=3$–6), a regularization ablation (the temporal prior being the dominant regularizer), and kinetic pre-exponential mismatch ($-5\%$ to $+10\%$, evaluated on the perturbed plant).

All reported values were obtained from simulation with synthetic measurements and are not plant-validated; because the nominal estimator and plant share the reactor model, they represent an optimistic matched-model reference, and structural model error, sensor bias, non-Gaussian disturbances, and heat-transfer degradation remain to be quantified through experimental or pilot-scale testing. Within this scope, the contribution is a simulation-based proof of concept that integrates constrained catalyst-activity estimation with operating-target optimization in a measurement–estimation–optimization decision workflow, converting sparse process measurements into catalyst-specific operating recommendations. Priorities before deployment are a rigorous sliding-window moving-horizon estimator, a model-predictive-control layer replacing the grid search, additional manipulated variables, a validated mechanism-specific catalyst-lifetime objective, systematic parametric-uncertainty analysis, and validation against pilot-scale or plant data to assess robustness to structural model mismatch.