From Catalyst Aging to Operational Vulnerability: A Benchmark-Validated Framework for Industrial SO₂ Converters

1. Introduction

Sulfuric acid is among the most extensively produced commodity chemicals worldwide, with global output consistently exceeding 200 million tonnes per year, and its consumption is widely regarded as a proxy indicator of industrial economic activity [1]. It serves as a critical raw material in fertilizer manufacturing, petroleum refining, metallurgical ore processing, and chemical synthesis. The contact process is the dominant technology for commercial sulfuric acid production. Within this process, the catalytic oxidation of sulfur dioxide to sulfur trioxide:

$${\mathrm{SO}}_2+\frac{1}{2}{\mathrm{O}}_2\rightleftharpoons {\mathrm{SO}}_3,\Delta H_r=-96.0\mathrm{kJ}{\mathrm{mol}}^{-1},$$

(1)

is the rate-limiting and equilibrium-limited step that governs overall plant conversion, heat recovery, and atmospheric emissions. Because the reaction is reversible and strongly exothermic, achieving high SO₂ conversion requires operating along a temperature path that balances reaction kinetics, which favor higher temperatures, against thermodynamic equilibrium, which favors lower temperatures. Modern industrial installations employ the double-contact double-absorption (DCDA) configuration, in which converter gas passes through three catalytic beds, undergoes an intermediate SO₃ absorption step, and then enters a final bed where residual SO₂ is oxidized. Inter-bed cooling between successive beds manages the exothermic temperature rise and maintains conditions favorable for both kinetics and equilibrium, enabling overall conversions exceeding 99.5% [1,2,3,4].

The active catalyst in industrial SO₂ converters is vanadium pentoxide (V₂O₅) promoted with alkali-metal sulfates and supported on silica or diatomaceous earth. Under process conditions, the active species exists as a liquid vanadium pyrosulfate melt dispersed within the support pores. Although this catalyst exhibits high selectivity and thermal stability, it unavoidably undergoes progressive deactivation during long-term service. Identified deactivation mechanisms include thermal sintering, which reduces the active-phase surface area; poisoning by trace impurities such as arsenic compounds and fluorine species, which irreversibly block active sites; mechanical attrition, which generates fines and increases bed pressure drop; and structural deterioration of the support through pore collapse and surface-area loss [5,6,7,8]. The combined effect is a progressive and largely irreversible reduction in catalyst activity that lowers SO₂ conversion, modifies bed temperature profiles, and increases outlet SO₂ concentrations. Operators must compensate by raising inlet temperatures or accepting declining conversion and rising emissions until catalyst replacement becomes necessary.

Considerable research has addressed modeling and optimization of SO₂ oxidation reactors. Industrial reactor models are generally founded on the Collina/Hougen-Watson kinetic expression, which accounts for the inhibiting adsorption of SO₂ and SO₃ and incorporates an equilibrium correction [2]. Dynamic and steady-state models have been developed to investigate temperature profiles, conversion characteristics, and energy integration in sulfuric acid plants [2,3,4,9,10,11]. Recent work has extended this direction toward multi-objective optimization, plant-connected digital twins, converter inlet-temperature control, and updated vanadium-catalyst modeling [12,13,14,15].

In the field of catalyst deactivation, Froment established a comprehensive framework for modeling activity loss and its coupling with reactor performance [5]. Bartholomew [6] and Moulijn et al. [7] provided authoritative reviews of deactivation mechanisms and predictive strategies applicable across heterogeneous catalyst systems. Martin et al. [16] further unified the vocabulary and classification of deactivation modes across catalytic systems. The specific deactivation behavior of vanadium pyrosulfate SO₂ oxidation catalysts, including active-phase restructuring, alkali-sulfate redistribution, support degradation, and fines generation, has been characterized in mechanistic and industrial spent-catalyst studies [8]. More recently, Pappagallo et al. proposed a pseudo-dynamic methodology in which successive steady-state simulations at progressively reduced activity levels efficiently represent reactor performance loss, demonstrating the value of this approach for fixed-bed and fluidized-bed reactors [17].

Despite these advances, important gaps remain. Previous studies have focused primarily on SO₂ converter behavior under fresh or nominally active catalyst conditions, and most deactivation analyses have addressed conversion decline, hot-spot migration, or catalyst activity evolution at the reactor-performance level. The following industrially important aspects remain insufficiently explored:

• the effect of catalyst activity loss on the two-variable feasible operating region and the degree to which reduced activity restricts permissible conditions;
• activity-loss-induced vulnerability to the simultaneous occurrence of conversion loss and SO₂ slip increase;
• erosion of hotspot margin and migration of the maximum-temperature location as activity declines;
• a quantitative composite index that integrates multiple activity-loss effects into a single scenario-relative operational deterioration indicator;
• illustrative scenario-screening bands derived from the composite index to support transparent comparison of the investigated states.

Therefore, the objective of this work is to apply an activity-loss vulnerability framework to an industrial double-contact SO₂ converter model evaluated at a fresh-catalyst benchmark condition and to develop quantitative indicators for activity-loss-induced operational deterioration. The specific contributions are as follows:

1.

A reactor model evaluated against a fresh-catalyst industrial benchmark is employed to analyze four prescribed catalyst activity-loss scenarios ($a=1.0,0.8,0.6,0.4$) in a four-bed double-contact industrial SO₂ converter.

2.

Bed-wise temperature and SO₂ conversion profiles are characterized under each activity state to quantify thermal and conversion deterioration.

3.

Hotspot evolution, hotspot-location migration, and thermal-margin loss are quantified to connect catalyst activity loss with thermal vulnerability.

4.

Local sensitivity maps are constructed to reveal the combined dependence of converter performance on catalyst activity and bed inlet temperature.

5.

Combined vulnerability and hotspot–conversion tradeoff maps are developed to identify regions where productivity, environmental compliance, and thermal safety are simultaneously at risk.

6.

A two-variable feasible-region analysis quantifies the shrinkage of permissible conditions as catalyst activity declines.

7.

An Operating Efficiency Reduction Index (OERI) is introduced as an illustrative, scenario-relative scalar measure of overall performance deterioration and screening priority.

8.

A compact one-at-a-time parameter sensitivity analysis quantifies the effects of kinetic scale, absorber residual, and feed composition near $a=0.6$, while a separate decision-threshold test examines dependence on the adopted thermal constraint.

The remainder of this paper is organized as follows. Section 2 describes the activity-loss vulnerability framework, performance indicators, feasible-region analysis, scalar indices, and sensitivity screening. Section 3 presents the case study, mathematical model, and fresh-benchmark evaluation. Section 4 presents and discusses the results. Section 5 discusses industrial implications, limitations, and future perspectives. Section 6 summarizes the conclusions.

2. Methodology

2.1. Framework Overview

The activity-loss vulnerability framework developed in this work translates prescribed catalyst activity reductions into quantitative operational and decision-support metrics through five coupled analysis stages, as illustrated in Figure 1.

1.

Stage I — Reactor simulation: The four-bed DCDA SO₂ converter model, evaluated against the fresh industrial benchmark condition, is executed at four prescribed activity levels ($a=1.0,0.8,0.6,0.4$), representing the fresh reference and three increasing activity-loss scenarios. Each simulation yields bed-wise axial temperature and SO₂ conversion profiles together with final outlet flow conditions.

2.

Stage II — Hotspot and thermal safety analysis: The maximum temperature and its axial location are extracted from each bed profile at every activity level to characterize hotspot temperature evolution, hotspot-location migration between beds, and thermal safety-margin consumption.

3.

Stage III — Local sensitivity analysis: The first-bed inlet temperature is systematically varied around the fresh-benchmark operating point at each activity level to generate two-dimensional sensitivity maps of final SO₂ conversion and outlet SO₂ slip, revealing the combined dependence of converter performance on catalyst state and inlet temperature.

4.

Stage IV — Vulnerability and feasible-region assessment: A combined vulnerability map is constructed by overlaying conversion loss and SO₂ slip on a single operating plane. A hotspot–conversion tradeoff map quantifies the competing effects of feed-temperature adjustment on productivity and thermal margin. The feasible region is identified as the intersection of three simultaneous case-study constraints.

5.

Stage V — Scalar index and scenario screening: The Operating Efficiency Reduction Index (OERI) condenses the multi-dimensional activity-loss effects into a single scenario-relative screening measure. Illustrative priority bands support comparison within the investigated scenario set but are not validated maintenance triggers. One-at-a-time perturbations assess local parameter sensitivity near $a=0.6$.

The computational procedure implementing this framework is described in Algorithm 1. The performance indicators, envelope constraints, and index formulations used in each stage are defined in the following subsections.

Algorithm 1:Activity-Loss Vulnerability Framework for Industrial SO2 Converter Assessment.

Require:  Reactor model evaluated at the fresh benchmark condition; activity set $\mathcal{A}=\{1.0,0.8,0.6,0.4\}$; inlet temperature grid $[T_{min},T_{max}]$; constraints $X_{min}=99.5\%$ percentage conversion, $S_{max}=0.10\mathrm{mol}\%$, $T_{\mathrm{safe}}=650 °\mathrm{C}$ Ensure:  Bed profiles, sensitivity maps, vulnerability maps, feasible region, OERI, illustrative scenario-screening bands Ensure:    1: — Stage I: Reactor Simulation — 2: for each $a\in \mathcal{A}$ do 3:     Simulate 4-bed DCDA converter at activity a and fresh-benchmark inlet conditions 4:     Extract bed-wise profiles: $T_b\left(W\right)$, percentage conversion $X_b\left(W\right)$, $F_{{\mathrm{SO}}_2,b}\left(W\right)$ for $b=1,\dots ,4$ 5:     Compute percentage conversion $X_{\mathrm{final}}\left(a\right)$, $S_{S{O}_2}\left(a\right)$, $T_{\mathrm{hotspot}}\left(a\right)$, hotspot bed index 6:     Compute $\Delta T_{\mathrm{margin}}\left(a\right)\leftarrow T_{\mathrm{safe}}-T_{\mathrm{hotspot}}\left(a\right)$ 7: end for 7:   8: — Stage II: Thermal Safety Analysis and Deterioration Normalization — 9: for each $a\in \mathcal{A}$ do 10:     $\Delta X\left(a\right)\leftarrow X_{\mathrm{fresh}}-X\left(a\right)$ in percentage points 11:     $\Delta S\left(a\right)\leftarrow S\left(a\right)-S_{\mathrm{fresh}}$ 12:     $\Delta T_{\mathrm{margin},\mathrm{loss}}\left(a\right)\leftarrow \Delta T_{\mathrm{margin},\mathrm{fresh}}-\Delta T_{\mathrm{margin}}\left(a\right)$ 13: end for 14: Normalize each indicator by its maximum value across $\mathcal{A}$: 14:    $\Delta X_{\mathrm{norm}}\leftarrow \Delta X/\Delta X_{\max }$,   $\Delta S_{\mathrm{norm}}\leftarrow \Delta S/\Delta S_{\max }$,   $\Delta T_{\mathrm{margin},\mathrm{norm}}\leftarrow \Delta T_{\mathrm{margin},\mathrm{loss}}/\Delta T_{\mathrm{margin},\mathrm{loss},\max }$ 14:   15: — Stage III: Local Sensitivity Analysis — 16: for each $(a,T_{\mathrm{in}})\in \mathcal{A}\times [T_{min},T_{max}]$ do 17:     Simulate converter; record percentage conversion X, $S_{S{O}_2}$, $T_{\mathrm{hotspot}}$ 18: end for 19: Construct 2-D sensitivity maps for X and $S_{S{O}_2}$ over the $(a,T_{\mathrm{in}})$ plane 19:   20: — Stage IV: Vulnerability and Feasible-Region Assessment — 21: Build combined vulnerability map: conversion loss (color) + SO2 slip (contours) 22: Build hotspot–conversion tradeoff map 23: for each $(a,T_{\mathrm{in}})$ grid point do 24:     Evaluate constraints: $X\ge X_{min}$,   $S\le S_{max}$,   $T_{\mathrm{hotspot}}\le T_{\mathrm{safe}}$ 25: end for 26: Feasible region ← intersection of all three satisfied constraint regions 26:   27: — Stage V: Scalar Index and Scenario Screening — 28: for each $a\in \mathcal{A}$ do 29:     $\mathrm{OERI}\left(a\right)\leftarrow \frac{1}{3}\left(\Delta X_{\mathrm{norm}}+\Delta S_{\mathrm{norm}}+\Delta T_{\mathrm{margin},\mathrm{norm}}\right)$ 30:     Assign illustrative screening band from Table 1: $[0,0.25)$ Low; $[0.25,0.50)$ Moderate; $[0.50,0.75)$ High; $\ge 0.75$ Critical 31: end for 32: return Activity levels ordered by OERI within the investigated scenario set

Table 1. Illustrative OERI-based scenario-screening bands.

OERI range	Screening level	Illustrative interpretation
$[0,0.25)$	Low	Limited deterioration within the scenario set
$[0.25,0.50)$	Moderate	Intermediate deterioration within the scenario set
$[0.50,0.75)$	High	Substantial deterioration; plant-specific assessment warranted
$\ge 0.75$	Critical	Greatest deterioration within the scenario set
Bands are illustrative scenario-screening aids, not validated maintenance or replacement thresholds.

Table 1. Illustrative OERI-based scenario-screening bands.

OERI range	Screening level	Illustrative interpretation
$[0,0.25)$	Low	Limited deterioration within the scenario set
$[0.25,0.50)$	Moderate	Intermediate deterioration within the scenario set
$[0.50,0.75)$	High	Substantial deterioration; plant-specific assessment warranted
$\ge 0.75$	Critical	Greatest deterioration within the scenario set
Bands are illustrative scenario-screening aids, not validated maintenance or replacement thresholds.

2.2. Performance Indicators

The dimensionless fractional SO₂ conversion used internally in the reactor balances is defined as

$$x_{{\mathrm{SO}}_2}={\displaystyle \frac{F_{{\mathrm{SO}}_2,\mathrm{in}}-F_{{\mathrm{SO}}_2,\mathrm{out}}}{F_{{\mathrm{SO}}_2,\mathrm{in}}}},$$

(2)

where $0\le x_{{\mathrm{SO}}_2}\le 1$. For reporting, plotting, constraint evaluation, and all subsequent performance indicators, percentage conversion is used:

$$X_{{\mathrm{SO}}_2}(\%)=100x_{{\mathrm{SO}}_2}.$$

(3)

Thus, $X_{{\mathrm{SO}}_2}$ denotes percentage conversion throughout the remainder of this paper, whereas the lowercase symbol $x_{{\mathrm{SO}}_2}$ is reserved for dimensionless fractional conversion. The outlet SO₂ slip is expressed on a dry basis as a molar percentage of the total dry outlet stream:

$$S_{S{O}_2}=100{\displaystyle \frac{F_{S{O}_2,out}}{{\displaystyle \sum _i{F}_{i,out}}}}.$$

(4)

For comparison with common gas-emission reporting conventions, the same dry molar fraction is also reported in parts per million by volume:

$$S_{S{O}_2,\mathrm{ppmv}}={10}^4{S}_{S{O}_2},$$

(5)

where $S_{S{O}_2}$ is expressed in dry-basis mol%. The conversion loss and SO₂ slip increase are defined relative to the fresh catalyst reference case ($a=1.0$):

$$\begin{array}{cc}\hfill \Delta X& =X_{\mathrm{fresh}}-X\left(a\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill \Delta S& =S\left(a\right)-S_{\mathrm{fresh}}.\hfill \end{array}$$

(7)

Because X is expressed as percentage conversion, $\Delta X$ is a difference between two percentage values and is therefore reported in percentage points (pp), not as a relative percent change. Similarly, $\Delta S$ is reported in mol%-points unless another unit is stated.

To account explicitly for thermal safety, the hotspot temperature is defined as the global maximum temperature reached anywhere inside the converter:

$$T_{\mathrm{hotspot}}=\underset{b,W}{max}{T}_b\left(W\right),$$

(8)

where $T_b\left(W\right)$ is the axial temperature profile in catalytic bed b as a function of catalyst mass W. An illustrative thermal constraint of $T_{\mathrm{safe}}=650 °\mathrm{C}$ is adopted for the present case study. This value is used as an engineering screening boundary rather than a universal catalyst limit; plant-specific applications should replace it with the applicable catalyst-vendor and operating-policy limit. The hotspot safety margin is defined as

$$\Delta T_{\mathrm{margin}}=T_{\mathrm{safe}}-T_{\mathrm{hotspot}},$$

(9)

with positive values indicating operation within the adopted constraint and negative values indicating exceedance. Loss of margin relative to the fresh catalyst is quantified as

$$\Delta T_{\mathrm{margin},\mathrm{loss}}=\Delta T_{\mathrm{margin},\mathrm{fresh}}-\Delta T_{\mathrm{margin}}\left(a\right).$$

(10)

For use in the scalar deterioration indices, each performance degradation indicator is normalized with respect to its maximum value observed across all activity levels:

$$\Delta X_{\mathrm{norm}}={\displaystyle \frac{\Delta X}{\Delta X_{\max }}},\Delta S_{\mathrm{norm}}={\displaystyle \frac{\Delta S}{\Delta S_{\max }}},\Delta T_{\mathrm{margin},\mathrm{norm}}={\displaystyle \frac{\Delta T_{\mathrm{margin},\mathrm{loss}}}{\Delta T_{\mathrm{margin},\mathrm{loss},\max }}},$$

(11)

where the subscript max denotes the value observed at the lowest activity level ($a=0.4$). By this normalization, all three indicators are bounded in $[0,1]$ across the study range, ensuring comparability across dimensions with different physical units.

These normalized indicators serve as the building blocks for the feasible-region analysis and scalar deterioration indices defined below.

2.3. Two-Variable Feasible Operating Region

The two-variable feasible operating region is defined in the space of first-bed inlet temperature and catalyst activity level by three simultaneous case-study constraints:

$$X_{S{O}_2}\ge 99.5\%,S_{S{O}_2}\le 0.10\mathrm{mol}\%\left(1000\mathrm{ppmv}\right),T_{\mathrm{hotspot}}\le 650 °\mathrm{C}.$$

(12)

The conversion threshold of 99.5% represents the case-study performance target. The SO₂ slip limit of 0.10 mol% (1000 ppmv) is an illustrative process-outlet screening value rather than a jurisdiction-specific stack-emission standard. At standard conditions (273 K, 101.3 kPa), 1 ppmv SO₂≈ 2.86 mg Nm⁻³; the 1000 ppmv threshold therefore corresponds to approximately 2860 mg Nm⁻³ for reference against stack-emission regulations. The temperature value of 650 °C is likewise an adopted case-study constraint.

For each activity level, the first-bed inlet temperature is varied over a representative range and all three conditions are evaluated at each grid point. The feasible region is the intersection of the three individually satisfied regions. Its contraction provides a geometric measure of activity-loss-induced reduction in operational flexibility within this two-variable slice.

2.4. Operating Efficiency Reduction Index and Illustrative Scenario Screening

The Operating Efficiency Reduction Index (OERI) is introduced to quantify the combined operational impact of catalyst activity loss by integrating productivity loss, SO₂-slip deterioration, and thermal-margin loss into a single dimensionless scalar:

$$\mathrm{OERI}={\displaystyle \frac{1}{3}}\left(\Delta X_{\mathrm{norm}}+\Delta S_{\mathrm{norm}}+\Delta T_{\mathrm{margin},\mathrm{norm}}\right).$$

(13)

Equal weights are assigned to the three normalized components as an engineering-judgment baseline in the absence of a universally accepted weighting standard. Although $\Delta X$ and $\Delta S$ represent different reporting perspectives—productivity and process-outlet SO₂ deterioration—they share a common physical root because both are determined by the unreacted SO₂ leaving the converter. Across the four activity states examined here, the Pearson correlation between $\Delta X_{\mathrm{norm}}$ and $\Delta S_{\mathrm{norm}}$ exceeds 0.999, confirming near-perfect collinearity within this scenario set. Their joint inclusion therefore gives the conversion–slip axis greater effective weight than the thermal component and should be regarded as an explicit screening choice rather than evidence of two independent degradation modes. A plant implementation could instead use a fixed-reference slip ratio, e.g., $S\left(a\right)/S_{\mathrm{limit}}$, or recalibrate all components against economic, environmental, and equipment constraints.

By construction, $\mathrm{OERI}\in [0,1]$: the index equals zero for the fresh reference state and unity for the most severely deactivated case in the present scenario set. OERI is therefore a scenario-relative ranking metric; its numerical values are not transferable maintenance thresholds unless the normalization bounds are recalibrated against plant-specific limits, costs, and risk tolerances.

For transparent within-study comparison, illustrative screening bands are assigned from OERI using the following thresholds:

OERI provides a compact deterioration score and ordering of the investigated states. In the present four-scenario set, no investigated case falls within the Moderate band ($[0.25,0.50)$): the activity states map from Low ($a=1.0$ and $a=0.8$, OERI $=0.000$ and $0.209$) to High ($a=0.6$, OERI $=0.558$) and Critical ($a=0.4$, OERI $=1.000$). The value of 1.000 is obtained by construction because $a=0.4$ defines the maximum normalization values; it does not imply a universal failure state. Linear interpolation suggests that the Moderate band would correspond to approximately $a=0.63$–$0.78$ within this particular scenario definition, but a finer activity grid would be required to establish a continuous trajectory.

2.5. Compact Parameter and Decision-Threshold Sensitivity Analysis

A one-at-a-time local sensitivity analysis was performed around the uniform $a=0.6$ activity-loss scenario because its nominal hotspot temperature lies close to the adopted thermal constraint. The kinetic rate scale, inlet SO₂ mole fraction, and inlet O₂ mole fraction were independently perturbed by $\pm 10\%$. For each feed-composition perturbation, the N₂ fraction was adjusted to preserve a normalized mixture and the total molar flow was recalculated at the fixed feed mass flow of 5000 kg h⁻¹.

The nominal intermediate-absorber removal efficiency is 0.999 and therefore cannot be increased by 10% without exceeding its physical upper bound. Consequently, the residual SO₃ fraction, $1-{\eta }_{\mathrm{abs}}$, was perturbed by $\pm 10\%$, corresponding to removal efficiencies of 0.9991 and 0.9989. Separately, the adopted thermal constraint was varied by $\pm 10\%$ (585 and 715 °C) as a decision-threshold sensitivity test. This boundary variation changes the reported margin and screening status but does not represent model-input uncertainty and does not alter reactor outputs. The monitored responses were final conversion, outlet SO₂ slip in ppmv, hotspot temperature, and thermal margin. The parameter screening is a deterministic local sensitivity analysis rather than a probabilistic uncertainty quantification.

2.6. Framework Algorithm

Algorithm 1 presents the complete computational procedure implementing the five-stage activity-loss vulnerability framework.

3. Case Study and Fresh-Benchmark Model Evaluation

3.1. Industrial SO₂ Converter Description

The case study considered in this work is an industrial double-contact double-absorption (DCDA) sulfuric acid converter consisting of four adiabatic fixed-bed catalytic reactors connected in series. Inter-bed heat exchangers are employed to control the exothermic temperature rise and maintain favorable conditions for both reaction kinetics and thermodynamic equilibrium. After the third catalytic bed, sulfur trioxide is removed in an intermediate absorption stage before the gas stream enters the fourth bed for final polishing. This configuration is widely used in modern sulfuric acid plants because it enables overall SO₂ conversions exceeding 99.5%.

The oxidation of sulfur dioxide proceeds according to

$${\mathrm{SO}}_2+{\displaystyle \frac{1}{2}}{\mathrm{O}}_2\rightleftharpoons {\mathrm{SO}}_3,$$

(14)

which is highly exothermic and equilibrium limited. Consequently, the converter must operate along controlled temperature trajectories that balance the opposing influences of reaction kinetics and thermodynamic equilibrium. Inter-bed cooling and intermediate absorption are therefore essential for maintaining favorable operating conditions while maximizing overall SO₂ conversion.

The converter configuration adopted in this study consists of three primary catalytic beds followed by intermediate SO₃ absorption and a final polishing bed. The benchmark operating conditions and catalyst distributions are taken from the industrial four-bed DCDA converter reported by Gómez-García et al. [18] and are used throughout this work.

3.2. Mathematical Model

A one-dimensional pseudo-homogeneous plug-flow model was employed to describe the thermal and compositional behavior of the converter [19,20]. The mathematical model follows the industrial SO₂ converter modeling framework reported by Gosiewski [2] and the sulfuric acid plant model of Kiss et al. [3]. Each catalyst bed was assumed to operate adiabatically, and radial concentration and temperature gradients were neglected. The model consists of coupled species and energy balances combined with a kinetic expression based on the Collina/Hougen–Watson formulation. Intermediate SO₃ removal between the third and fourth beds was represented through a constant absorption efficiency.

The model predicts the evolution of species flow rates and temperature along the catalyst weight coordinate and provides bed-wise temperature profiles, cumulative conversion, and outlet SO₂ concentrations. The system of ordinary differential equations was solved sequentially for each catalytic bed, with inter-bed cooling and intermediate absorption incorporated between successive stages.

3.2.1. Species Balances

The oxidation of sulfur dioxide is described using one-dimensional plug-flow species balances formulated with respect to catalyst weight. The molar flow rates of sulfur dioxide, oxygen, and sulfur trioxide satisfy

$${\displaystyle \frac{d{F}_{S{O}_2}}{dW}}=-r_{S{O}_2},$$

(15)

$${\displaystyle \frac{d{F}_{O_2}}{dW}}=-{\displaystyle \frac{1}{2}}{r}_{S{O}_2},$$

(16)

$${\displaystyle \frac{d{F}_{S{O}_3}}{dW}}=r_{S{O}_2},$$

(17)

where $F_i$ denotes the molar flow rate of species i, W is the catalyst weight, and $r_{S{O}_2}$ is the sulfur dioxide oxidation rate. Nitrogen is treated as an inert diluent; its molar flow rate remains constant throughout each catalytic bed ($d{F}_{{\mathrm{N}}_2}/dW=0$).

3.2.2. Energy Balance

Because each catalyst bed operates adiabatically, the temperature profile is governed by the heat released by the oxidation reaction and the heat capacity of the gas mixture:

$${\displaystyle \frac{dT}{dW}}={\displaystyle \frac{-\Delta H_r{r}_{S{O}_2}}{{\displaystyle \sum _i{F}_i{C}_{p,i}}}},$$

(18)

where $\Delta H_r$ is the heat of reaction and $C_{p,i}$ denotes the heat capacity of species i.

3.2.3. Reaction Kinetics

The oxidation kinetics were represented using the Collina/Hougen–Watson formulation, following previous SO₂ converter modeling and optimization studies [2,12]. This formulation accounts for adsorption effects and equilibrium limitations and has been widely validated for industrial vanadium-based SO₂ oxidation catalysts.

The full rate expression is

$$r_{S{O}_2}=a{\displaystyle \frac{k_1{P}_{S{O}_2}{P}_{O_2}}{22.414{(1+k_2{P}_{S{O}_2}+k_3{P}_{S{O}_3})}^2}}\left(1-{\displaystyle \frac{P_{S{O}_3}}{K_p{P}_{S{O}_2}{P}_{O_2}^{1/2}}}\right),$$

(19)

where a is the catalyst activity factor, $P_i$ denotes the partial pressure of species i in atm, and $r_{S{O}_2}$ is the net SO₂ oxidation rate in kmol ${\mathrm{kg}}_{\mathrm{cat}}^{-1}$ h⁻¹. The temperature-dependent kinetic constants and equilibrium constant are given by

$$\begin{array}{cc}\hfill k_1& =exp\left(12.160-{\displaystyle \frac{5473}{T}}\right),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill k_2& =exp\left(-9.953+{\displaystyle \frac{8619}{T}}\right),\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill k_3& =exp\left(-71.745+{\displaystyle \frac{52596}{T}}\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill K_p& =exp\left({\displaystyle \frac{11300}{T}}-10.68\right),\hfill \end{array}$$

(23)

where T is the absolute temperature in Kelvin. The constants $k_1$, $k_2$, $k_3$ represent the temperature-dependent forward rate constant and adsorption equilibrium constants for SO₂ and SO₃, respectively; $K_p$ is the thermodynamic equilibrium constant for the SO₂ oxidation reaction. These constants are consistent with the published Collina/Hougen–Watson benchmark form widely used in industrial SO₂ converter models [2,12]. The computed rate in kmol ${\mathrm{kg}}_{\mathrm{cat}}^{-1}$ h⁻¹ is converted to mol ${\mathrm{kg}}_{\mathrm{cat}}^{-1}$ s⁻¹ for integration with the energy and species balances.

3.2.4. Intermediate SO₃ Absorption

Following the third catalyst bed, sulfur trioxide is removed in an intermediate absorber. The absorption stage is represented by an absorption efficiency, ${\eta }_{\mathrm{abs}}$, such that

$$F_{S{O}_3,out}=(1-{\eta }_{\mathrm{abs}})F_{S{O}_3,in},$$

(24)

where $F_{S{O}_3,in}$ and $F_{S{O}_3,out}$ denote the SO₃ molar flow rates entering and leaving the absorber, respectively.

3.3. Catalyst Activity-Loss Scenarios

Catalyst activity loss was represented through a dimensionless activity factor, a, defined as

$$a={\displaystyle \frac{r_{\mathrm{aged}}}{r_{\mathrm{fresh}}}},$$

(25)

where $r_{\mathrm{aged}}$ and $r_{\mathrm{fresh}}$ denote the aged and fresh-catalyst reaction rates evaluated under identical operating conditions.

Following the consecutive-steady-state methodology of Pappagallo et al. [17], catalyst activity loss was examined through successive steady-state simulations at prescribed reduced-activity levels. The present model does not calculate activity as a function of time or represent a specific deactivation mechanism; the cases are therefore activity-loss scenarios rather than predictions of catalyst age or service life.

Four representative activity states were considered:

$$a\in \{1.0,0.8,0.6,0.4\},$$

(26)

corresponding to the fresh reference and mild, moderate, and severe relative activity-loss scenarios, respectively.

The selected activity levels constitute a systematic parametric study rather than snapshots calibrated to a specific plant history or catalyst age. Sintering of the silica support, active-phase restructuring, poisoning, and alkali-sulfate redistribution are recognized deactivation mechanisms in supported vanadium SO₂ catalysts, while the resulting activity trajectory depends strongly on feed impurities, temperature history, catalyst formulation, and operating practice [6,7,8,16]. The descriptors fresh, mild, moderate, and severe are therefore used only to distinguish relative activity states within this study and should not be interpreted as specific service-life intervals or replacement criteria.

3.4. Model Parameters and Assumptions

The principal assumptions adopted in this study are summarized below:

1.

One-dimensional plug-flow behavior.

2.

Pseudo-homogeneous reactor model.

3.

Adiabatic operation within each catalyst bed.

4.

Negligible radial concentration and temperature gradients.

5.

Uniform catalyst activity within each bed.

6.

Constant pressure throughout the converter.

7.

Constant intermediate SO₃ absorption efficiency.

8.

Steady-state operation for each activity level.

The operating conditions, feed composition, catalyst inventories, kinetic parameters, and thermodynamic properties employed in the simulations are summarized in Table 2 and Table 3.

3.5. Fresh-Benchmark Model Evaluation

The mathematical model employed in this work was configured and evaluated against the industrial four-bed SO₂ converter data reported by Gómez-García et al. [18]. The configuration consisted of matching the inter-bed temperature policy (bed inlet temperatures at Beds 2, 3, and 4) to the values reported in that reference and using the published catalyst volumes and bulk density directly as the catalyst inventory. No adjustment was made to the published Collina/Hougen–Watson kinetic constants; the rate expression in Equations (19) – (23) was used without modification. The published industrial operating data thus serve as the fresh-condition benchmark against which the model outlet temperatures and overall conversion are compared.

The benchmark evaluation focused on reproducing bed outlet temperatures and overall SO₂ conversion under the fresh reference operating condition. Good agreement was obtained, with relative errors below 0.21% across all bed outlet temperatures and overall SO₂ conversion, supporting use of the model as a basis for the subsequent parametric activity-loss analysis. This comparison does not constitute direct validation of the reduced-activity cases, for which corresponding industrial measurements were unavailable.

Representative benchmark-comparison results are summarized in Table 4. The evaluated operating condition was subsequently used as the fresh-catalyst reference state ($a=1.0$). The vulnerability maps, feasible-region analysis, and scenario ordering are model predictions generated by varying catalyst activity from this reference condition.

Progressive reductions in catalyst activity were then imposed to represent different activity-loss scenarios. The resulting changes in temperature profiles, percentage conversion, process-outlet SO₂, and operating flexibility were analyzed using the methodology presented in Section 2.

3.6. Physical Self-Consistency of the Deactivated Simulation Cases

Direct experimental or industrial validation of the deactivated cases ($a=0.8$, $0.6$, $0.4$) is not possible in the absence of published operational data from aged SO₂ converters. The physical self-consistency of these cases was therefore demonstrated through three complementary arguments.

Energy balance consistency. The activity factor a appears exclusively as a multiplicative pre-factor on the reaction rate term in the kinetic expression (Equation 19). The adiabatic energy balance,

$${\displaystyle \frac{dT}{dW}}={\displaystyle \frac{|\Delta H_r|r_{S{O}_2}(a,T,P_i)}{{\displaystyle \sum _i{F}_i{C}_{p,i}}}},$$

(27)

is integrated numerically at each activity level. Because a scales only the rate magnitude, the local slope of the temperature–conversion trajectory with respect to dimensionless fractional conversion within any adiabatic bed,

$${\displaystyle \frac{dT}{d{x}_{{\mathrm{SO}}_2}}}={\displaystyle \frac{|\Delta H_r|F_{S{O}_2,0}}{{\displaystyle \sum _i{F}_i{C}_{p,i}}}},$$

(28)

is independent of a and determined solely by the heat of reaction, initial feed molar flow rate, and gas mixture heat capacity. Equation (28) uses the fractional conversion $x_{{\mathrm{SO}}_2}$ defined in Equation (2); a slope expressed per percentage point of conversion is smaller by a factor of 100. As catalyst activity decreases, less SO₂ is converted per unit bed weight, so the reactor traverses a shorter arc of the same adiabatic line. The shape of the temperature–conversion trajectory is preserved across all four activity levels; only the endpoint changes. This relationship was verified by confirming that the ratio of temperature rise to fractional SO₂ conversion increment within each bed remained consistent with the value predicted by Equation (28) at all activity levels, with deviations below the numerical integration tolerance.

Thermodynamic equilibrium constraint. The Collina/Hougen–Watson rate expression (Equation 19) contains the equilibrium correction factor

$$\left(1-{\displaystyle \frac{P_{S{O}_3}}{K_p{P}_{S{O}_2}{P}_{O_2}^{1/2}}}\right),$$

(29)

which approaches zero as the system approaches thermodynamic equilibrium. Because this factor multiplies the net rate, the model cannot predict conversion exceeding the local equilibrium value at any axial position or activity level, regardless of the value of a. Thermodynamic self-consistency is therefore enforced by the rate expression structure itself, not by any post-hoc correction.

Qualitative consistency with published observations. The aging trends predicted by the model — progressive reduction in overall SO₂ conversion, increase in outlet SO₂ slip, rise in hotspot temperature, and migration of the peak temperature from Bed 1 to Bed 2 as activity declines — are qualitatively consistent with published descriptions of aging behavior in multi-bed industrial SO₂ converters [2,3,4]. The redistribution of the exothermic heat release from the first to downstream beds as early-bed activity is lost is a recognized characteristic of catalyst aging in this reactor type [9].

These three arguments collectively demonstrate that the deactivated cases are physically self-consistent. While they cannot substitute for quantitative experimental validation against industrial aged-catalyst data, they confirm that the model generates thermodynamically admissible and qualitatively credible results across the full range of studied activity levels. Quantitative validation of deactivated converter predictions against industrial measurements from aged plants remains an important priority for future work.

4. Results and Discussion

The proposed activity-loss framework was applied to a double-contact SO₂ converter model evaluated at the fresh reference condition. Four catalyst activity levels were investigated: the fresh reference ($a=1.0$) and mild ($a=0.8$), moderate ($a=0.6$), and severe ($a=0.4$) relative activity-loss scenarios. The reduced-activity cases are deterministic parametric predictions rather than independently validated deactivation data.

4.1. X–T Trajectory Evolution Under Activity Loss

Figure 2 compares the fresh catalyst case ($a=1.0$) with the severe deactivation case ($a=0.4$) using temperature–conversion trajectories, where X denotes percentage conversion. In an SO₂ converter, equilibrium conversion decreases with increasing temperature, whereas the kinetic rate generally benefits from higher temperature over the relevant operating range. Within an adiabatic bed, Equation (28) shows that the local temperature rise per fractional-conversion increment is governed by reaction enthalpy and mixture heat capacity and is not directly changed by the activity multiplier. Deactivation therefore does not fundamentally alter the adiabatic temperature–conversion slope; instead, the aged catalyst traverses a shorter portion of the admissible trajectory over the available catalyst inventory and exits each bed at a lower percentage conversion. The cumulative endpoint deficit increases from bed to bed, and inter-bed cooling cannot recover the reaction progress achieved by the fresh catalyst under the same inlet-temperature policy.

Reduced activity decreases the conversion achieved within the available catalyst inventory, and the cumulative shortfall becomes the primary driver of total conversion loss in the severe activity-loss scenario.

4.2. Bed-Wise Temperature Profiles

Figure 3 compares the outlet temperatures of the four catalyst beds across all activity levels. As activity decreases from $a=1.0$ to $a=0.4$, the heat-release load redistributes from Bed 1 to downstream beds. The Bed 1 outlet temperature falls progressively from 614 °C to 431 °C, reflecting reduced SO₂ conversion in the first stage. Simultaneously, the Bed 2 outlet temperature rises to 651 °C at $a=0.6$ and 660 °C at $a=0.4$, exceeding the adopted 650 °C constraint. This redistribution concentrates the exothermic load in Bed 2. The effect extends to Bed 4, where outlet temperatures increase as a larger fraction of the remaining conversion shifts to later stages.

Bed-wise temperature profiles therefore serve as a clear and measurable indicator of catalyst deterioration, with the Bed 2 spike providing an early warning of thermal-limit breach under progressive deactivation.

4.3. Hotspot Evolution and Thermal Safety

In addition to bed outlet temperatures, the maximum temperature reached inside the converter provides a direct measure of thermal safety. The hotspot temperature, $T_{\mathrm{hotspot}}$, was extracted from the axial temperature profile of each catalytic bed for all catalyst activity levels. The hotspot safety margin was defined as

$$\Delta T_{\mathrm{margin}}=T_{\mathrm{safe}}-T_{\mathrm{hotspot}},$$

(30)

where $T_{\mathrm{safe}}=650 °\mathrm{C}$ was used as the illustrative thermal constraint for this case study.

Figure 4 summarizes the three principal hotspot diagnostics. Panel (a) shows that the predicted hotspot rises from 613.7°C for the fresh reference to 660.3°C at $a=0.4$. Panel (b) shows migration of the maximum-temperature location from Bed 1 to Bed 2 as activity declines, indicating downstream redistribution of the exothermic load. Panel (c) shows the corresponding reduction in margin relative to the adopted 650 °C constraint, from $+36.3$ K to $-10.3$ K.

Table 5 summarizes the corresponding hotspot location, hotspot temperature, margin, and position relative to the adopted constraint. The progression from operation within the constraint to operation near or above it shows that decreasing activity progressively consumes thermal margin.

Figure 5 presents a hotspot risk map as a function of catalyst activity and first-bed feed temperature. The color scale represents hotspot temperature, while the contour line corresponds to $T_{\mathrm{hotspot}}=650 °$C. This figure shows that increasing feed temperature can improve reaction performance, but it also increases the risk of exceeding the thermal limit, especially under aged catalyst conditions.

The hotspot analysis strengthens the interpretation of feasible-region shrinkage. Activity loss not only reduces conversion and increases SO₂ slip; it also shifts the location of maximum heat release and reduces the available thermal margin.

The reduction in hotspot safety margin has direct implications for the OERI. In this work, the normalized loss of hotspot safety margin is used as the thermal component of OERI alongside the normalized conversion loss and SO₂ slip increase. Consequently, OERI provides a compact scenario-relative measure of catalyst deterioration by simultaneously accounting for productivity loss, process-outlet SO₂ deterioration, and thermal-robustness degradation.

With the thermal behavior of the reduced-activity converter established, the following subsections examine how prescribed activity loss translates into measurable conversion losses across the individual catalyst beds and across the full operating space.

4.4. Bed-Wise Conversion Profiles

Figure 6 shows the cumulative SO₂ conversion profiles across the four catalyst beds. The fresh case reaches a final conversion of 99.758%, whereas the severe deactivation case reaches 96.812%. The bed-wise profiles show that aging effects accumulate along the converter, making downstream conversion particularly sensitive to upstream activity loss. The difference becomes increasingly evident in the later beds because the cumulative penalty of lower reaction rates grows from bed to bed.

Conversion deterioration is monotonic with declining catalyst activity, and bed-wise conversion measurements are highly sensitive indicators of aging state.

4.5. Local Sensitivity Maps: Conversion and SO₂ Slip

Figure 7a presents the local sensitivity of final SO₂ conversion to catalyst activity and first-bed feed temperature. Higher inlet temperatures improve conversion by increasing reaction rates, whereas lower catalyst activities reduce achievable conversion throughout the explored range. Catalyst activity exerts a stronger influence on conversion than moderate temperature variations; the validated operating condition lies within the high-conversion region for fresh catalysts but migrates toward lower-performance regions as aging progresses. Feed temperature can only partially compensate for activity loss and cannot recover fresh-catalyst performance under severe deactivation.

Figure 7b shows the outlet SO₂ slip under the same operating plane. The fresh case produces 0.02296 mol% (230 ppmv) at the validated point, which rises to 0.29597 mol% (2960 ppmv) in the $a=0.4$ activity-loss scenario. Taken together, the two maps confirm that catalyst activity strongly influences both conversion and process-outlet SO₂, and motivate the combined vulnerability representation presented in Section 4.7.

4.6. Conversion–Thermal Safety Tradeoff

The hotspot analysis also reveals a fundamental operating tradeoff. Increasing the first-bed feed temperature can partially recover SO₂ conversion under reduced activity, but it simultaneously increases hotspot temperature and reduces the available thermal margin. Figure 8a visualizes this tradeoff; the red contour identifies the adopted 650 °C boundary.

Figure 8b extends the analysis by combining conversion, SO₂ slip, and the adopted thermal constraint in one operating map. At high activity and moderate temperature, all three case-study constraints are satisfied. At higher temperature, conversion and slip may remain within their adopted constraints while the predicted hotspot exceeds the adopted boundary. In the lower-activity portion of the map, conversion and slip constraints are increasingly violated, confirming that temperature manipulation alone cannot recover fresh-reference performance. The convergence of constraint boundaries at intermediate activity identifies a region where the predicted operating window is narrow and multiple performance objectives are simultaneously at risk.

4.7. Combined Activity-Loss Vulnerability Map

Figure 9 combines conversion loss and outlet SO₂ slip in a single map. The color scale represents conversion loss relative to the fresh reference case, while the contour lines represent SO₂ slip levels. This visualization simultaneously identifies performance deterioration, process-outlet SO₂ penalties, and regions of heightened operational vulnerability. Regions with high conversion loss also correspond to elevated SO₂ slip, demonstrating that prescribed activity loss creates a coupled performance–outlet-slip vulnerability that cannot be adequately captured by monitoring either indicator alone.

The combined map provides a compact representation of the operational consequences of prescribed catalyst activity loss and supports the central argument of this study: reduced activity should be treated as an operational vulnerability problem, not merely as a reduction in reaction rate.

4.8. Two-Variable Feasible Operating Region

Figure 10 shows the two-variable feasible operating region obtained from the simultaneous overlap of three case-study constraints:

$$X_{S{O}_2}\ge 99.5\%,S_{S{O}_2}\le 0.10\mathrm{mol}\%,T_{\mathrm{hotspot}}\le 650 °\mathrm{C}.$$

(31)

The feasible region shrinks progressively as catalyst activity decreases, and the mechanism of shrinkage changes across activity-loss severity. For the fresh catalyst ($a=1.0$), a finite inlet-temperature window satisfies all three case-study constraints. At $a=0.8$, the thermal constraint removes part of the higher-temperature region. At $a=0.6$, the adopted thermal boundary is exceeded at the reference inlet temperature and the predicted conversion of 98.476% is below the 99.5% target. At $a=0.4$, the conversion constraint eliminates feasibility throughout the investigated inlet-temperature range; at the reference temperature, conversion is 96.812%. Catalyst activity loss therefore reduces operating flexibility through thermal-boundary tightening and loss of attainable conversion within the investigated domain.

4.9. Operating Efficiency Reduction Index and Illustrative Scenario Screening

To quantify the combined impact of catalyst activity loss, the OERI introduced in Section 2.4 is evaluated at each activity level using normalized conversion loss, SO₂ slip increase, and hotspot-margin loss (Equation (11)). Figure 11a decomposes OERI into its three contributors. The thermal component rises most steeply between $a=1.0$ and $a=0.6$, while conversion and slip components grow steadily. The index reaches 1.0 at $a=0.4$ by construction because that case defines the scenario-relative normalization maximum.

Applying the illustrative screening bands from Table 1, the four activity states are ordered as follows: $a=1.0$ (OERI = 0.000) is Low; $a=0.8$ (OERI = 0.209) is Low; $a=0.6$ (OERI = 0.558) is High; and $a=0.4$ (OERI = 1.000) is Critical. Figure 11b visualizes this scenario-relative ordering. These labels compare only the investigated states and do not prescribe maintenance action.

Unlike conversion alone, OERI combines conversion loss, process-outlet SO₂ deterioration, and hotspot safety-margin loss into a single scenario-relative scalar. In its present form it is suitable only for comparative screening of the investigated scenarios; plant-level monitoring or maintenance decisions require independent normalization bounds, weights, and validation against operational and maintenance data.

Table 6 consolidates the key performance deterioration indicators across all four catalyst activity levels.

Table 6. Performance deterioration summary across catalyst activity levels.

Activity	Percentage conversion (%)	$\mathit{\Delta }\mathit{X}$ (pp)	SO2 Slip (mol%)	SO2 Slip (ppmv)	Hotspot (°C)	Margin (K)	OERI	Priority
1.0	99.758	0.000	0.02296	230	613.7	+36.3	0.000	Low
0.8	99.307	0.451	0.06534	653	628.6	+21.4	0.209	Low
0.6	98.476	1.281	0.14264	1426	651.0	−1.0	0.558	High
0.4	96.812	2.946	0.29597	2960	660.3	−10.3	1.000	Critical
pp: percentage points; ppmv values are dry molar fractions; Margin is relative to $T_{\mathrm{safe}}=650 °$C; Priority from Table 1.

Table 6. Performance deterioration summary across catalyst activity levels.

Activity	Percentage conversion (%)	$\mathit{\Delta }\mathit{X}$ (pp)	SO2 Slip (mol%)	SO2 Slip (ppmv)	Hotspot (°C)	Margin (K)	OERI	Priority
1.0	99.758	0.000	0.02296	230	613.7	+36.3	0.000	Low
0.8	99.307	0.451	0.06534	653	628.6	+21.4	0.209	Low
0.6	98.476	1.281	0.14264	1426	651.0	−1.0	0.558	High
0.4	96.812	2.946	0.29597	2960	660.3	−10.3	1.000	Critical
pp: percentage points; ppmv values are dry molar fractions; Margin is relative to $T_{\mathrm{safe}}=650 °$C; Priority from Table 1.

Table 7. OERI-based scenario ordering and illustrative screening guidance.

Activity	OERI	Screening level	Rank	Interpretation
1.0	0.000	Low	4	Normal operation
0.8	0.209	Low	3	Review trends and increase monitoring
0.6	0.558	High	2	Substantial deterioration in the investigated set
0.4	1.000	Critical	1	Maximum normalized deterioration in the investigated set
Screening level assigned from Table 1; Rank 1 = greatest modeled deterioration, not a validated maintenance trigger.

Table 7. OERI-based scenario ordering and illustrative screening guidance.

Activity	OERI	Screening level	Rank	Interpretation
1.0	0.000	Low	4	Normal operation
0.8	0.209	Low	3	Review trends and increase monitoring
0.6	0.558	High	2	Substantial deterioration in the investigated set
0.4	1.000	Critical	1	Maximum normalized deterioration in the investigated set
Screening level assigned from Table 1; Rank 1 = greatest modeled deterioration, not a validated maintenance trigger.

4.10. Activity-Loss Indicator Summary

Figure 11c consolidates four key deterioration indicators: conversion loss, outlet SO₂ slip increase, OERI, and hotspot safety-margin loss. Together the panels provide a compact view of the productivity, process-outlet, composite-screening, and thermal dimensions of the investigated activity-loss scenarios.

4.11. Sensitivity to Non-Uniform Catalyst Activity Distribution

The main analysis assumed uniform catalyst activity applied equally to all four beds. In practice, Bed 1 operates at the highest temperature and highest SO₂ partial pressure, and typically deactivates faster than downstream beds [6,7]. To assess how this spatial non-uniformity influences converter performance, three non-uniform activity profiles were simulated and compared against the uniform case at a similar average activity. Table 8 summarises the results.

Two findings from Table 8 are particularly significant. First, the non-uniform profile $[0.4,0.4,0.8,0.8]$ has the same average activity ($\overline{a}=0.60$) as the uniform case $a=0.6$, yet produces a markedly different thermal outcome: the Bed 2 hotspot temperature rises to 660.3 °C compared with 651.0 °C under uniform activity, and the safety margin deteriorates from $-1.0$ K to $-10.3$ K. Second, despite the more severe thermal condition, the non-uniform case achieves higher overall conversion (99.078% versus 98.476%), because the more active downstream beds compensate for the conversion shortfall of the deactivated first two beds. This decoupling of conversion and thermal safety under spatially non-uniform deactivation highlights a fundamental limitation of the uniform-activity assumption: models that track only average activity may underestimate thermal risk while simultaneously overestimating it from a conversion perspective. Furthermore, the non-uniform profile $[0.4,0.6,0.8,1.0]$ with a higher average activity ($\overline{a}=0.70$) still exhibits a hotspot 9.4 K above the uniform-$a=0.6$ case, confirming that spatial distribution of deactivation dominates the thermal response more than average activity alone. These findings underscore the importance of bed-resolved activity monitoring and motivate future extensions of the framework toward spatially distributed deactivation models.

4.12. Compact Parameter and Decision-Threshold Sensitivity Analysis

Figure 12 summarizes the one-at-a-time perturbations around the uniform $a=0.6$ scenario. The nominal prediction was 98.476% conversion, 1426 ppmv outlet SO₂, and a hotspot of 651.0 °C, corresponding to a margin of $-1.0$ K relative to the adopted constraint. Feed composition was the dominant source of local output variability. A $+10\%$ perturbation in inlet SO₂ reduced conversion by 0.705 percentage points, increased outlet slip by 891 ppmv, and raised the hotspot by 9.93 K. A $-10\%$ perturbation in O₂ reduced conversion by 0.861 percentage points and increased slip by 798 ppmv. Conversely, reducing SO₂ or increasing O₂ moved the predicted condition toward higher conversion and lower outlet slip.

The kinetic scale had a smaller but still material influence: a $\pm 10\%$ perturbation produced conversion changes of $-0.397$ to $+0.316$ percentage points, slip changes of $+367$ to $-294$ ppmv, and hotspot changes of $+3.54$ to $-4.54$ K. Changing the residual SO₃ fraction after the absorber by $\pm 10\%$ changed conversion by less than 0.001 percentage points and slip by less than 0.5 ppmv because the nominal removal efficiency is already 99.9%. In the separate decision-threshold test, varying the adopted thermal boundary by $\pm 10\%$ did not alter reactor predictions but changed the reported margin by $\pm 65$ K. This result concerns classification dependence on the selected plant-specific boundary rather than physical model sensitivity.

4.13. Comparison with Previous Studies

Previous SO₂ converter studies have primarily focused on kinetic modeling, reactor simulation, X–T diagrams, and optimization under nominal or fresh-catalyst conditions [2,3,10,12]. These works provide important foundations for understanding sulfur dioxide oxidation and optimizing converter operation. However, they generally do not quantify how prescribed activity loss changes the feasible operating region or scenario-level operational vulnerability.

General catalyst-deactivation studies have shown that catalyst activity loss affects reactor conversion and temperature profiles [5,6,7]. The pseudo-dynamic approach of Pappagallo et al. [17] is particularly relevant because it uses successive steady-state simulations at decreasing catalyst activity to represent reactor performance loss. The present work follows the same general philosophy of activity-based scenario analysis but extends it in a different direction: instead of stopping at conversion loss or temperature-profile evolution, the framework translates prescribed activity loss into vulnerability maps, feasible-region shrinkage, OERI, and scenario-screening indicators.

The novelty of this work therefore lies not in SO₂ converter modeling alone, nor in the general concept of catalyst activity, but in converting fresh-benchmark-anchored activity-loss simulations into operational vulnerability and scenario-screening metrics for an industrial SO₂ converter. Table 9 positions the present work relative to representative prior studies.

4.14. Summary of Principal Findings

Table 10 consolidates the principal findings of the study into a single reference, linking each quantitative result to its industrial implication.

5. Industrial Implications, Limitations, and Future Perspectives

5.1. Industrial Implications

The simulation results suggest that catalyst activity loss should be monitored not only through conversion loss but also through SO₂ slip, hotspot migration, margin consumption, and feasible-region shrinkage. The model is evaluated at the fresh reference condition, whereas the reduced-activity results are parametric predictions. Their applicability to a specific plant therefore depends on calibration against measured activity proxies and applicable plant constraints. Within the present scenario set, $a=0.6$ marks a transition at which the predicted hotspot exceeds the adopted 650 °C boundary by 1.0 K and the scenario-relative OERI exceeds 0.5, placing it in the illustrative High screening band per Table 1. This observation is useful for comparing the investigated states but is not, by itself, a validated maintenance trigger.

5.2. Limitations

Although the proposed activity-loss framework provides a physically interpretable methodology for assessing reduced catalyst performance in industrial SO₂ converters, several limitations should be acknowledged.

First, the primary analysis assumed uniform catalyst activity applied equally to all four beds. In practice, activity loss may be spatially non-uniform because bed conditions and contaminant exposure differ. The sensitivity study in Section 4.11 demonstrated that even at the same average activity, a front-loaded profile produced a hotspot temperature 9.3 °C higher than the uniform $a=0.6$ case while achieving higher conversion. The uniform assumption may therefore underestimate thermal risk when early-bed activity loss dominates.

Second, the present study used prescribed steady-state activity-loss scenarios and did not calculate deactivation rate, catalyst age, or time on stream. The reported conversion decline, hotspot growth, and feasible-region shrinkage should therefore be interpreted as scenario predictions after the process has settled. Transient startup, shutdown, load disturbances, and rapid deactivation events remain outside the model scope.

Third, the feasible operating region is only a two-variable slice in activity–first-bed-temperature space. Other relevant variables, including downstream inlet temperatures, flow, pressure, and absorber performance, were held fixed. The adopted conversion, SO₂ slip, and temperature constraints are illustrative and should be replaced with plant-specific requirements.

Fourth, OERI uses scenario-relative normalization and equal illustrative weights. Its numerical values are not transferable plant thresholds, and its normalization should be recalibrated using economic penalties, applicable regulations, catalyst-vendor limits, and maintenance history. Furthermore, as noted in Section 2.4, the conversion-loss and SO₂-slip components of OERI are near-perfectly correlated ($r>0.999$ across this scenario set) because both derive from the same unreacted SO₂ flow. Their joint inclusion implicitly weights the conversion-emission axis more heavily than the thermal component. The correlation should be explicitly acknowledged when OERI is used as a screening tool, and a fixed-reference slip ratio $S\left(a\right)/S_{\mathrm{limit}}$ should be substituted when a fully independent emission component is required.

Fifth, the reduced-activity cases ($a=0.8$, $0.6$, $0.4$) represent physically screened parametric states but have not been validated against industrial measurements with independently estimated catalyst activity. The present work is therefore a scenario analysis rather than a calibrated prediction of a specific plant’s aging trajectory.

Sixth, the compact parameter study is local and one-at-a-time. It identifies influential inputs near $a=0.6$ but does not capture parameter interactions or probability distributions. A global uncertainty analysis should follow when parameter covariance and industrial measurement uncertainty become available.

5.3. Future Perspectives

The results of this study motivate several concrete research extensions, ordered from immediate modeling refinements to broader system-level integration.

First, and most directly motivated by the non-uniform activity sensitivity analysis (Section 4.11), future work should extend the discrete profiles considered here to continuous axial activity distributions and experimentally calibrated bed-specific activity estimates. The finding that a front-loaded profile at the same average activity produces a 9.3 °C higher hotspot temperature demonstrates the importance of bed-resolved information for reliable thermal-risk assessment.

Second, the framework could be integrated with online measurements through state-estimation and emission-forecasting techniques to enable real-time catalyst health monitoring [21,22]. Such implementation would require plant-specific calibration of the OERI normalization, weights, and action thresholds. Specific targets include bed outlet temperature sensors, pressure drop, and periodic conversion measurements as activity proxies.

Third, the feasible-region and OERI framework could be embedded within a digital twin or model-predictive optimization platform [13,14,23] to support adaptive inlet-temperature scheduling while preserving plant-specific hotspot margins.

Fourth, the feasible-region shrinkage results show that even moderate deactivation ($a=0.6$–$0.8$) substantially restricts operating flexibility. This motivates plant-wide extensions that include intermediate absorber aging and interactions between converter and absorption stages, since the SO₃ slip from an aging absorber compounds the converter’s emission challenge.

Finally, the sustainability dimension of spent catalyst management represents an additional axis of practical relevance. Recent hydrometallurgical studies demonstrate technically viable routes for recovering vanadium, cesium, and aluminum from spent sulfuric-acid catalysts [24,25,26]. These material-recovery pathways could be integrated with the scenario-ranking framework to jointly evaluate replacement timing and catalyst valorization.

6. Conclusions

This study applied an activity-loss vulnerability framework to an industrial double-contact SO₂ converter model evaluated against a fresh-catalyst benchmark condition. Four prescribed activity levels were investigated: a fresh reference and three progressively reduced-activity scenarios. The analysis linked activity decline to temperature profiles, hotspot evolution, SO₂ conversion, outlet SO₂ slip, a two-variable feasible operating region, and scenario-relative deterioration indicators.

The main conclusions are as follows:

1.

Prescribed catalyst activity loss produced clear and measurable changes in converter response. As activity decreased from $a=1.0$ to $a=0.4$, final SO₂ conversion decreased from 99.758% to 96.812%, while outlet SO₂ slip increased from 230 to 2960 ppmv. Bed-wise profiles showed upstream activity loss cascading into downstream conversion deficits.

2.

Hotspot temperature increased from 613.7 °C at $a=1.0$ to 660.3 °C at $a=0.4$, with the margin changing from $+36.3$ K (within the adopted constraint) to $-10.3$ K (exceeding the adopted constraint). Concurrently, the hotspot migrated from Bed 1 to Bed 2.

3.

Local sensitivity and tradeoff maps showed that catalyst activity exerts a dominant influence on conversion, SO₂ slip, and hotspot temperature. Although feed-temperature adjustment can partially compensate for activity loss, it simultaneously increases hotspot temperature and reduces the available safety margin, creating a fundamental performance–thermal safety tradeoff that cannot be resolved by temperature adjustment alone under severe deactivation.

4.

The combined vulnerability map demonstrated that conversion loss and SO₂ slip increase occur simultaneously in reduced-activity operating regions, highlighting coupled productivity and process-outlet consequences that cannot be captured by monitoring either indicator in isolation.

5.

The two-variable feasible-region analysis showed that activity loss reduces operating flexibility within the investigated $(a,T_{\mathrm{in}})$ domain.

6.

The compact parameter sensitivity analysis showed that feed SO₂/O₂ composition and kinetic scale dominate local output variability at $a=0.6$. A $+10\%$ SO₂ perturbation increased outlet slip by 891 ppmv and hotspot temperature by 9.93 K, whereas a $\pm 10\%$ change in absorber residual changed slip by less than 0.5 ppmv. Thermal-status classifications depended strongly on the adopted plant-specific boundary.

7.

The scenario-relative OERI condensed conversion loss, SO₂ slip increase, and hotspot-margin loss into a compact within-study screening score. Because its components are equally weighted, conversion loss and SO₂ slip are strongly correlated, and the worst scenario defines the normalization maximum, its numerical values are not validated maintenance thresholds and require plant-specific reformulation before operational use.

8.

A sensitivity study with non-uniform bed activity distributions revealed that spatial deactivation patterns significantly alter the thermal response even at the same average activity. A front-loaded profile ($a=[0.4,0.4,0.8,0.8]$, $\overline{a}=0.60$) produced a hotspot temperature of 660.3 °C and a safety margin of $-10.3$ K, compared with 651.0 °C and $-1.0$ K for uniform $a=0.6$, while simultaneously achieving higher overall conversion (99.08% versus 98.48%). This decoupling of conversion and thermal risk under non-uniform deactivation demonstrates that average activity alone is an insufficient descriptor of converter safety, and that bed-resolved activity monitoring is required for reliable risk assessment.

9.

Overall, the results demonstrate that catalyst activity loss in industrial SO₂ converters should be treated as an operational-vulnerability problem, not merely as a reduction in kinetic rate. The proposed framework converts fresh-benchmark-anchored reactor-model outputs into scenario-relative operational indicators.

Future work should extend the analysis to continuous axial and experimentally calibrated bed-specific activity profiles, mechanistic deactivation kinetics, absorber deterioration, plant-wide dynamics, and online activity estimation using industrial measurements.