Conceptual Design of a Multitubular Fixed-Bed Reactor for Methanol Ammoxidation to Hydrogen Cyanide over FeMo/SiO₂ Using a Mars–van Krevelen Kinetic Model

Abstract

Methanol ammoxidation over FeMo/SiO₂ has emerged as a promising low-temperature route to hydrogen cyanide (HCN). In this work, an eight-parameter Mars–van Krevelen (MvK) kinetic model, previously established from intrinsic fixed-bed experiments, is embedded in a heterogeneous plug-flow description to design an industrial multitubular reactor with a nominal HCN capacity of 10,000 t∙a⁻¹. The reactor is represented by a bank of isothermal tubes that are operated at 420 °C and a mildly elevated pressure, each packed with spherical FeMo/SiO₂ pellets. Detailed simulations for a 30 mm inner tube diameter and 2 mm pellets, including an Ergun pressure drop and intraparticle diffusion with realistic effective diffusivities, show that a 4 m bed at an outlet pressure of 1.5 bar (abs) achieves an essentially complete methanol conversion with a carbon-based HCN yield of ≈0.95 at a space time of ≈160 gcat∙h∙mol⁻¹. Axial effectiveness factors remain above ≈0.6, indicating moderate but manageable diffusion limitations. Comparison with a 35 mm/3 mm geometry reveals a clear trade-off between pressure drop and HCN selectivity. Parametric studies of space time, feed composition and outlet pressure delineate a broad non-flammable operating window with robust HCN yield and moderate compression duty. The results demonstrate how a mechanistic MvK rate expression can be translated into a practical design framework for FeMo-based multitubular HCN reactors.

1. Introduction

Hydrogen cyanide (HCN) is a key intermediate for the manufacture of acrylonitrile, adiponitrile, methyl methacrylate, cyanuric chloride and various fine and specialty chemicals. Commercial HCN production is still dominated by high-temperature methane-based technologies, notably the Andrussow and Degussa (BMA) processes, which employ Pt/Rh gauze catalysts at 1100–1300 °C in strongly oxidizing mixtures of CH₄, NH₃ and O₂ [1,2,3,4]. These routes are technologically mature but suffer from several drawbacks: severe operating conditions, formation of NO_x, sensitivity to platinum price and gauze aging, and a relatively narrow non-flammable window in the CH₄–NH₃–O₂ system that limits process intensification [1,3,5].

Methanol ammoxidation has emerged as an attractive lower-temperature alternative for HCN production [3,6,7,8]. In this route, HCN is obtained from CH₃OH, NH₃ and O₂ over mixed oxide catalysts at 400–500 °C, with CO and CO₂ as the main deep-oxidation by-products. Because methanol is a liquid, easily stored and increasingly available from coal, natural gas and CO₂-based routes, a methanol-to-HCN process is especially appealing in regions where natural-gas infrastructure is limited [3,6]. In addition, the lower reaction temperature suppresses NO_x formation, and the use of transition-metal oxides instead of noble metals reduces catalyst cost and vulnerability to precious-metal price fluctuations.

Among the non-noble systems, silica-supported Fe–Mo oxides (FeMo/SiO₂) have received considerable attention for methanol ammoxidation [6,7,8,9,10,11]. These catalysts typically contain Fe₂(MoO₄)₃ as the dominant crystalline phase, together with a small amount of α-Fe₂O₃, and exhibit a high methanol conversion with HCN selectivities above 90% over an intermediate temperature window [8,9,10]. Previous studies have examined catalyst composition, preparation method and operating conditions and have demonstrated long-term stability under ammoxidation conditions [6,7,8,9,10,11,12]. However, most reports have focused on catalyst screening and laboratory performance; comparatively few works have attempted to link intrinsic kinetics to the design of industrial-scale reactors.

In a recent study published in Catalysts, we reported a detailed kinetic investigation of methanol ammoxidation to HCN over FeMo/SiO₂ in a fixed-bed reactor operated under conditions free of internal and external transport limitations [8]. Catalyst particles with diameters less than 0.15 mm, combined with Weisz–Prater and Mears criteria, confirmed the intrinsic character of the measured rates. The data were interpreted in terms of a Mars–van Krevelen (MvK) redox mechanism in which lattice oxygen participates explicitly in both the selective ammoxidation and deep-oxidation pathways. Among several mechanistic variants, an eight-parameter rate model (case II) was retained, which expresses the formation rates of HCN, CO and CO₂ as rational functions of the partial pressures of methanol (p_M), ammonia (p_A) and oxygen (p_O2).

While such intrinsic kinetic models are essential for understanding the reaction chemistry, their direct use in industrial reactor design is not straightforward. Commercial HCN plants employ multitubular fixed-bed reactors with tube bundles containing several thousand small-diameter tubes packed with millimeter-sized catalyst pellets. In these systems, intraparticle diffusion, a gas-phase pressure drop and heat-transfer constraints can significantly modify the apparent reaction rates and selectivities relative to those measured in integral laboratory reactors [13,14,15]. Design decisions such as tube diameter, pellet size, tube length and operating pressure must, therefore, be made on the basis of a heterogeneous reactor model that consistently integrates intrinsic kinetics with transport phenomena. To date, however, there has been little systematic work on using mechanistic MvK kinetics for FeMo/SiO₂ to design multitubular reactors for methanol-derived HCN.

The objective of the present contribution is to bridge this gap by translating the intrinsic MvK model for FeMo/SiO₂ into a conceptual design framework for an industrial multitubular reactor producing 10,000 t∙a⁻¹ of HCN. We consider a non-flammable feed composition CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar), in which air supplies are both oxidant and diluent, and additional N₂ is used as an explosion suppressant. A one-dimensional heterogeneous plug-flow model is formulated for a single tube packed with spherical FeMo/SiO₂ pellets. The model combines (i) the intrinsic MvK rate expressions at the pellet scale, (ii) an intraparticle diffusion–reaction description for CH₃OH, NH₃ and O₂, and (iii) gas-phase plug flow with an Ergun pressure drop. Using this framework, we compare alternative tube–particle combinations, analyze the impact of internal diffusion on effectiveness factors and intraparticle concentration profiles, and map the operating window of the industrial reactor as a function of space time, feed composition and outlet pressure. In this way, the study illustrates how a mechanistic MvK kinetic model, originally developed from intrinsic laboratory data, can be deployed as a practical tool for designing and optimizing FeMo-based methanol ammoxidation reactors at an industrial scale.

2. Results and Discussion

2.1. Isothermal Plug-Flow Performance and Pressure Drop at Design Load

To evaluate the hydrodynamic implications at the full industrial capacity, the single-tube model was scaled to a target hydrogen cyanide production rate of 10,000 t·a⁻¹ at 8000 h·a⁻¹, corresponding to ${\dot{n}}_{\mathrm{H}\mathrm{C}\mathrm{N},\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}\approx 4.63\times {10}^4$ mol·h⁻¹. The outlet pressure was fixed at $P\left(L\right)=P_{\mathrm{out}}=1.5$ bar (abs), and the MeOH feed per tube was adjusted so that a bundle of $N_t=4000$ identical tubes produced the required total HCN molar flow. For each tube, the gas phase was modeled as a one-dimensional plug flow with an Ergun-type pressure drop, and the intrinsic Mars–van Krevelen kinetics were implemented in terms of mass-based rates $r_i$ (mol·gcat⁻¹·h⁻¹). The feed composition was kept at CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar) at the reactor inlet.

Two tube–particle combinations were analyzed (Figure 1):

(A)

D_t = 35 mm, d_p = 3 mm

(B)

D_t = 30 mm, d_p = 2 mm

At the design load, both configurations reach essentially complete methanol conversion over the 4 m packed length. For case (A), the model yields an inlet methanol feed of ${\dot{n}}_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},\mathrm{i}\mathrm{n}}^{\left(\mathrm{tube}\right)}\approx 12.4$ mol·h⁻¹, an outlet HCN production of ${\dot{n}}_{\mathrm{H}\mathrm{C}\mathrm{N},\mathrm{o}\mathrm{u}\mathrm{t}}^{\left(\mathrm{tube}\right)}\approx 11.6$ mol·h⁻¹, and a nearly quantitative methanol conversion of $X_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H}}\approx 1.00$. The corresponding HCN selectivity and carbon-based yields are $S_{\mathrm{H}\mathrm{C}\mathrm{N}}\approx 0.93$ and $Y_{\mathrm{H}\mathrm{C}\mathrm{N}}\approx 0.93$, respectively. With 4000 tubes, the total HCN production matches the plant target.

Figure 1. Predicted axial profiles of methanol conversion X_MeOH and total pressure P along a single tube at the design load for the two tube–pellet combinations: (A) D_t = 35 mm and d_p = 3 mm; (B) D_t = 30 mm and d_p = 2 mm. Conditions: T = 420 °C, P_out = 1.5 bar (abs), CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar), L = 4.0 m.

Figure 1. Predicted axial profiles of methanol conversion X_MeOH and total pressure P along a single tube at the design load for the two tube–pellet combinations: (A) D_t = 35 mm and d_p = 3 mm; (B) D_t = 30 mm and d_p = 2 mm. Conditions: T = 420 °C, P_out = 1.5 bar (abs), CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar), L = 4.0 m.

For case (B), the required per-tube MeOH feed is slightly lower, ${\dot{n}}_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},\mathrm{i}\mathrm{n}}^{\left(\mathrm{tube}\right)}\approx 12.2$ mol·h⁻¹, while the outlet HCN rate per tube is essentially identical ($\approx 11.6$ mol·h⁻¹), again giving $X_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H}}\approx 1.00$. However, the smaller tube and pellet size significantly improve HCN selectivity to $S_{\mathrm{H}\mathrm{C}\mathrm{N}}\approx 0.95$ and $Y_{\mathrm{H}\mathrm{C}\mathrm{N}}\approx 0.95$, reflecting the beneficial impact of reduced external and internal mass-transfer limitations at otherwise identical operating conditions.

The trade-off appears clearly in the pressure drop. For $D_t=35$ mm and $d_p=3$ mm, the Ergun equation predicts a pressure drop of $\Delta P\approx 0.75$ bar over the 4 m bed at the design flow rate, corresponding to an inlet pressure of $P_{\mathrm{in}}\approx 2.25$ bar (abs) for an outlet pressure of 1.5 bar (abs). When the tube diameter is reduced to 30 mm and the pellet size is reduced to 2 mm, the gas velocity and hydraulic resistance both increase, and the pressure drop rises to $\Delta P\approx 1.66$ bar, i.e., $P_{\mathrm{in}}\approx 3.16$ bar (abs). Thus, the 30 mm/2 mm configuration requires a substantially higher inlet pressure—more than twice the pressure drop of the 35 mm/3 mm case—but delivers measurably higher HCN selectivity at essentially complete methanol conversion.

Overall, these results show that both geometries can achieve the required HCN production at a high conversion and selectivity under the chosen isothermal design conditions. The smaller tube diameter and pellet size offer superior intrinsic performance (a higher HCN selectivity and a more favorable basis for radial temperature control and intraparticle diffusion) at the cost of an increased pressure drop; hence, a higher inlet pressure is required. The final choice between the 35 mm/3 mm and 30 mm/2 mm configurations therefore hinges on the acceptable compression duty and mechanical design constraints of the multitubular reactor bundle and downstream purification train.

The associated pressure drop, obtained from the Ergun equation and coupled with the variable-density gas-phase model, is ΔP ≈ 1.66 bar over the 4 m packed length at the design flow rate. Thus, for an outlet pressure of P_out = 1.5 bar (abs), the required inlet pressure is P_in ≈ 3.16 bar (abs). This moderate overpressure is compatible with standard shell-and-tube reactor design and downstream purification units, while providing a sufficiently large driving force to keep the reactor and subsequent separation steps safely above atmospheric pressure.

On this basis, the combination of D_t = 30 mm and d_p = 2 mm is adopted as the base-case geometry for all subsequent simulations, and alternative tube and pellet sizes are evaluated relative to this reference.

At the design load (T = 420 °C, P_out = 1.5 bar (abs) and L = 4.0 m, N_t = 4000), both geometries reach essentially complete methanol conversion, X_MeOH ≈ 1.00. The 30 mm/2 mm base-case tube requires a per-tube MeOH feed of ≈12.2 mol·h⁻¹ and delivers an outlet HCN yield of Y_HCN ≈ 0.95, with a pressure drop of ΔP ≈ 1.7 bar (P_in ≈ 3.2 bar (abs)). The 35 mm/3 mm configuration requires ≈12.4 mol·h⁻¹ MeOH per tube, gives Y_HCN ≈ 0.93, and exhibits a lower pressure drop of ΔP ≈ 0.75 bar (P_in ≈ 2.25 bar (abs)). Thus, the 30 mm/2 mm geometry increases the HCN yield by about two percentage points at the expense of an additional ≈0.9 bar of pressure drop along the 4 m packed length.

From an industrial standpoint, the higher pressure drop in the 30 mm/2 mm tubes implies a larger compressor head upstream of the reactor. At the same throughput, increasing ΔP from ≈0.75 bar (35 mm/3 mm) to ≈1.7 bar (30 mm/2 mm) would roughly double the required compression duty. However, the absolute pressures remain modest (P_in ≈ 3.2 bar (abs) and P_out = 1.5 bar (abs)), so the additional power demand is compatible with standard centrifugal compressors and does not dominate overall operating cost. In this sense, the 30 mm/2 mm geometry trades a moderate increase in compression duty for a robust gain in HCN yield and a more favorable basis for heat management and intraparticle diffusion.

2.2. Comparison with 35 mm Tubes and 3 mm Pellets

To assess the trade-off between transport performance and pressure drop, the same heterogeneous plug-flow model was applied to an alternative geometry with a larger tube diameter and pellet size, D_t = 35 mm and d_p = 3 mm, at otherwise identical operating conditions (Figure 1 and Figure 2). The per-tube methanol feed was again adjusted such that the overall HCN production matches 10,000 t·a⁻¹.

At the design load, the 35 mm/3 mm configuration requires a slightly higher per-tube methanol feed (≈12.4 mol·h⁻¹) to achieve the same per-tube HCN production (≈11.6 mol·h⁻¹) and to essentially complete methanol conversion. The outlet HCN selectivity and carbon-based yield are S_HCN ≈ Y_HCN ≈ 0.93, somewhat lower than the yield for the 30 mm/2 mm base case. This difference is consistent with the stronger intraparticle diffusion limitations predicted for 3 mm pellets: the pellet-scale solution shows more pronounced concentration gradients, with lower effectiveness factors near the tube inlet compared to the 2 mm pellets operated under the same bulk conditions.

Hydrodynamically, the larger tube diameter and pellet size yield a smaller pressure drop. For D_t = 35 mm and d_p = 3 mm, the Ergun equation gives ΔP ≈ 0.75 bar over 4 m at the design flow, corresponding to an inlet pressure P_in ≈ 2.25 bar (abs). The pressure drop is thus reduced by about a factor of two relative to the 30 mm/2 mm configuration. However, this reduction in pressure drop is accompanied by both stronger intraparticle diffusion limitations and a modest loss in HCN selectivity at essentially the same methanol conversion.

Overall, the comparison indicates that the 30 mm/2 mm design represents a favorable compromise for the FeMo/SiO₂-catalyzed methanol ammoxidation to HCN. The smaller tube diameter facilitates radial temperature control and justifies the isothermal wall assumption, while 2 mm pellets keep intraparticle diffusion limitations at a moderate level and allow the intrinsic Mars–van Krevelen kinetics to be expressed with high HCN selectivity. Although the 35 mm/3 mm geometry is attractive from a purely hydraulic perspective, its lower HCN selectivity and stronger pellet-scale gradients make it less suitable as a base-case design. In the remainder of this work, all detailed parametric studies are therefore carried out for the 30 mm/2 mm configuration, and the 35 mm/3 mm case is only used as a reference for sensitivity analysis.

2.3. Intraparticle Diffusion Assessment in the 30 mm/2 mm Base-Case Tube

For the base-case geometry (tube inner diameter 30 mm and pellet diameter 2 mm), the heterogeneous plug-flow model with spherical pellet diffusion–reaction was evaluated at the full design load. The methanol feed per tube was adjusted to F_MeOH,in = 12.200 mol·h⁻¹, corresponding to $F_{{\mathrm{N}\mathrm{H}}_3,\mathrm{i}\mathrm{n}}$ = 13.420 mol·h⁻¹, $F_{{\mathrm{O}}_2,\mathrm{i}\mathrm{n}}$ = 25.620 mol·h⁻¹ and $F_{{\mathrm{N}}_2,\mathrm{i}\mathrm{n}}$ = 133.285 mol·h⁻¹ for the feed composition CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar). At an outlet pressure of P_out = 1.5 bar (abs), the Ergun equation predicts an inlet pressure of P_in = 3.22 bar (abs), i.e., a pressure drop of about ΔP = 1.72 bar over the 4 m packed length. Under these conditions the reactor achieves essentially complete methanol conversion, X_MeOH ≈ 1.00, with an outlet HCN selectivity and carbon-based yield of S_HCN ≈ Y_HCN ≈ 0.947, which is consistent with the intrinsic Mars–van Krevelen kinetics established in the kinetic study.

Figure 3 shows the axial profile of methanol conversion in the 30 mm/2 mm tube. Methanol is consumed rapidly: X_MeOH reaches about 0.8 at z ≈ 1.0 m and exceeds 0.95 at z ≈ 1.5 m, approaching unity around z ≈ 2.0 m. Beyond this point the bed operates on a lean methanol tail gas and mainly serves to polish residual CH₃OH and complete the ammoxidation to HCN.

Although Figure 3 shows that more than 95% of methanol is converted within the first 1.5 m of the bed at the nominal design conditions, we retain a tube length of L = 4.0 m in the base-case design. The additional downstream section acts as a polishing zone that ensures essentially complete methanol removal and stabilizes the HCN yield over the broader operating window of W/F_MeOH ≈ 80–220 gcat·h·mol⁻¹ that is considered in Section 2.4. This extra length also provides robustness with respect to uncertainties in the kinetic parameters, effective diffusivities, heat-transfer coefficients and potential catalyst deactivation or flow maldistribution at an industrial scale. The associated penalty in pressure drop remains modest: extending the bed from ≈2 m to 4 m only increases the total ΔP to about 1.7 bar in the 30 mm/2 mm geometry, which is acceptable in the context of the overall compression duty of the plant.

The corresponding axial effectiveness factors for methanol, ammonia and oxygen are plotted in Figure 4. At the tube inlet, all three effectiveness factors are slightly below unity, ${\eta }_{\mathrm{M}}$ ≈ ${\eta }_{\mathrm{A}}$ ≈ ${\eta }_{{\mathrm{O}}_2}$≈ 0.96, indicating mild intraparticle diffusion limitations where the reactant partial pressures are still close to their feed values. Moving downstream, the effectiveness factors decrease and reach a minimum of about ${\eta }_{\mathrm{m}\mathrm{i}\mathrm{n}}\approx 0.61$ at $z/L\approx 0.5$. For orientation, if one maps this numerically obtained effectiveness factor onto the analytical expression for a first-order reaction in a spherical pellet, it corresponds to an effective Thiele modulus with an order of $\mathsf{\Phi }\approx 3–4$, i.e., a regime of moderate diffusion control. This mapping is only used as an approximate diagnostic: the actual pellet calculations always employ the full non-linear MvK kinetics rather than a first-order rate law. Further along the reactor, as methanol and ammonia are depleted and the intrinsic reaction rates decrease, the Thiele modulus is reduced and the effectiveness factors increase again, reaching η ≈ 0.68 near the tube outlet. The three curves remain close to one another because the effective diffusivities of CH₃OH, NH₃ and O₂ differ only moderately, and their source terms are governed by the same MvK rate expression.

Figure 5 presents the intraparticle concentration profiles of CH₃OH, NH₃ and O₂ at four axial positions (z = 0.0, 0.5, 1.0 and 2.0 m). At the reactor inlet (z = 0.0 m), the pellets operate with relatively high reactant concentrations and exhibit noticeable but moderate radial gradients: the methanol concentration increases from about 3.0 mol·m⁻³ at the pellet center to about 3.7 mol·m⁻³ at the external surface, while ammonia and oxygen show similar variations of roughly 10–15%. At z = 0.5 m the overall concentrations are lower, and the gradients become more pronounced, indicating the onset of internal diffusion limitations as the intrinsic reaction rate increases. Around z = 1.0–2.0 m, where the bed is most strongly reacting, methanol is almost completely depleted in the pellet core and attains only a small concentration even at the surface, whereas ammonia and oxygen still exhibit finite concentrations. This confirms that methanol is the locally limiting reactant in this region and that the pellets operate in a regime of moderate diffusion control.

Overall, these results show that, for 2 mm FeMo/SiO₂ pellets under the design conditions, intraparticle diffusion reduces the accessible intrinsic activity by roughly 30–40% in the most strongly reacting section of the bed but does not drive the system into a severely diffusion-limited regime. The combination of moderate effectiveness factors (η_min ≈ 0.61), nearly complete methanol conversion and high HCN yield supports the use of 2 mm pellets as a suitable compromise between kinetic performance and the pressure drop in the 30 mm tube geometry.

2.4. Operating Window and Parametric Sensitivity

While the previous sections focused on the base-case design at the nominal load (T = 420 °C, P_out = 1.5 bar (abs), D_t = 30 mm, and d_p = 2 mm), an industrial reactor must operate robustly over a wider range of conditions. In this subsection, the heterogeneous plug-flow model is used to delineate a practical operating window for the 30 mm/2 mm geometry, with emphasis on the influence of space time, feed composition and pressure level on methanol conversion, HCN yield and pressure drop.

2.4.1. Effects of Space Time

At 400 °C and atmospheric pressure, the constant-pressure plug-flow model with spherical pellets was evaluated for the laboratory feed composition NH₃/O₂/CH₃OH = 1.1/2.7/1.0 that was used in the kinetic study. Figure 6 shows the predicted outlet methanol conversion as a function of the space time W/F_MeOH for 2 mm FeMo/SiO₂ pellets with the effective diffusivities D_e,MeOH = 2.2 × 10⁻⁶ m² s⁻¹, $D_{e, \mathrm{N}{\mathrm{H}}_3}$ = 3.1 × 10⁻⁶ m² s⁻¹ and $D_{e, {\mathrm{O}}_2}$ = 2.4 × 10⁻⁶ m² s⁻¹. Methanol conversion increases monotonically with W/F_MeOH, from X_MeOH ≈ 0.2 at 25 gcat·h mol⁻¹ to X_MeOH ≈ 0.93 at 150 gcat h mol⁻¹. The corresponding HCN yield (Figure 7) follows the same trend, rising from Y_HCN ≈ 0.18 to ≈0.81 over the same range.

For comparison, the intrinsic kinetic study reported complete methanol conversion, an HCN yield of 92.5% at 400 °C and a space time of 125 gcat·h·mol⁻¹ in a fixed-bed reactor operated under conditions free of internal and external transport limitations. In the present pellet-scale model, at the same nominal space time the predicted methanol conversion and HCN yield are X_MeOH ≈ 0.86 and Y_HCN ≈ 0.79, respectively. The shift in the conversion and yield curves towards higher space times reflects the impact of intraparticle diffusion in the 2 mm pellets, which reduces the effective reaction rate relative to the intrinsic Mars–van Krevelen kinetics that are measured on crushed catalyst particles. Overall, the constant-pressure PFR with the diffusion–reaction pellet model reproduces the qualitative dependence of conversion and selectivity on the space time observed in the kinetic study, while providing a physically consistent estimate of the additional catalyst requirements associated with internal diffusion in industrially relevant catalyst shapes.

This comparison confirms that the same intrinsic MvK parameter set, calibrated at 1 bar [8], provides a consistent description of the dependence of conversion and selectivity on space time at 400 °C and can be reasonably extended to the industrial regime (420 °C and 2–3 bar) as a first-principles basis for conceptual reactor design.

2.4.2. Effect of Space Time on the Industrial Design Conditions (420 °C, 30 mm/2 mm, with Pressure Drop)

In a second step, the influence of space time was re-evaluated under the industrial design conditions, i.e., for the 30 mm/2 mm tube–pellet combination including the Ergun pressure drop. The outlet pressure was fixed at 1.5 bar (abs), while the inlet pressure was determined self-consistently from the Ergun equation as a function of flow rate. The feed composition was kept at CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar), and the wall temperature was fixed at 420 °C.

A series of simulations was carried out by varying the per-tube methanol feed to F_MeOH, as such the corresponding space time covered the range W/F_MeOH ≈ 80–220 gcat· h·mol⁻¹. For the chosen tube geometry (D_t = 30 mm, L = 4.0 m, ε_b = 0.40, and ρ_b ≈ 7.0 × 10⁵ gcat· m⁻³), the range of W/F_MeOH corresponds to per-tube methanol feed rates of approximately 25–9 mol h⁻¹ and inlet pressures between about 2.3 and 3.6 bar (abs). The resulting outlet methanol conversion and HCN yield are summarized in Figure 8 and Figure 9.

Figure 8 shows the outlet methanol conversion as a function of space time. Even at the lowest value considered, W/F_MeOH ≈ 80 gcat·h·mol⁻¹, the model predicts X_MeOH,out > 0.99, and the conversion rapidly approaches unity as W/F_MeOH is increased further. In the design region around W/F_MeOH ≈160 gcat·h·mol⁻¹, the reactor operates at essentially complete methanol conversion. This behavior reflects the combined effect of the higher temperature and the higher average pressure in the industrial tube relative to the kinetic experiments. At 420 °C and mean pressures of 2–3 bar, the intrinsic Mars–van Krevelen rates are significantly larger than at 400 °C and 1 bar, so that complete conversion is achieved at space times well below those required in the laboratory reactor, even in the presence of moderate intraparticle diffusion limitations.

The corresponding outlet HCN yield and selectivity are plotted in Figure 9. Unlike the conversion, the HCN yield exhibits a shallow but systematic decrease with increasing space time. At W/F_MeOH ≈ 80 gcat·h·mol⁻¹ the carbon-based HCN yield is Y_HCN,out ≈ 0.97, while at W/F_MeOH ≈ 220 gcat·h·mol⁻¹ it drops to Y_HCN,out ≈ 0.94. The design point, highlighted in Figure 8 and Figure 9, corresponds to W/F_MeOH ≈160 gcat·h·mol⁻¹, an inlet pressure of P_in ≈ 3.2 bar (abs) for P_out = 1.5 bar (abs), and an outlet performance of X_MeOH,out ≈ 1.00 and Y_HCN,out ≈ 0.95. This point represents a practical compromise: the space time is large enough to guarantee essentially complete methanol conversion and a robust safety margin, yet not so large that excessive bed length and pressure drop are invested in a tail section where HCN yield is slightly eroded by deep oxidation to CO and CO₂.

Taken together with the constant-pressure simulations at 400 °C (Section 2.4.1), which reproduce the dependence of conversion and selectivity on W/F_MeOH measured in the intrinsic kinetic study, the present results show that under industrial conditions the combination of higher temperature, moderate overpressure and 2 mm FeMo/SiO₂ pellets lead to high intrinsic activity. As a consequence, the apparent space time required for essentially complete methanol conversion is significantly less than in the laboratory reactor, even though intraparticle diffusion reduces the local effectiveness factors to values as low as η ≈ 0.6 in the most strongly reacting section of the bed. This confirms that the 30 mm/2 mm base-case design can meet the target HCN capacity at high yield with a comparatively compact catalyst inventory.

2.4.3. Effect of Feed Composition (NH₃/MeOH and O₂/MeOH Ratios)

The influence of the feed composition around the base-case design point was examined by varying the NH₃/MeOH and O₂/MeOH molar ratios at fixed tube geometry (30 mm/2 mm), outlet pressure (1.5 bar abs), wall temperature (420 °C) and space time (W/F_MeOH ≈ 160 gcat·h·mol⁻¹). Under these conditions the reference feed composition CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 corresponds to NH₃/MeOH = 1.1 and O₂/MeOH ≈ 2.1 and yields essentially complete methanol conversion with an outlet HCN yield of about 0.95.

Effect of NH₃/MeOH Ratio

Figure 10 (left and right panels) shows the effect of the NH₃/MeOH molar ratio in the range 0.9–1.3 at constant O₂/MeOH ≈ 2.1. Over this range the outlet methanol conversion remains virtually complete (X_MeOH,out > 0.999 for all cases), so the impact of the NH₃/MeOH ratio is reflected almost entirely in the product distribution. At NH₃/MeOH = 0.9 the HCN yield is limited to Y_HCN,out ≈ 0.89, indicating a shortage of ammonia relative to methanol and an increased fraction of deep oxidation to CO and CO₂. Increasing the ratio to 1.0 and 1.1 markedly improves the HCN yield, which rises to approximately 0.93 and 0.95, respectively. A further increase to NH₃/MeOH = 1.2–1.3 provides only a modest additional gain (Y_HCN,out ≈ 0.96–0.965), suggesting that the catalyst surface becomes saturated with ammonia and that the system enters a regime of diminishing returns with respect to NH₃ excess.

From a process design perspective, NH₃/MeOH ≈ 1.1 therefore represents a suitable compromise: it is sufficiently above stoichiometric to achieve high HCN yield and to suppress deep oxidation, while avoiding an unnecessarily large ammonia surplus that would increase raw-material consumption and the load on downstream neutralization and scrubbing units. This conclusion is also consistent with the intrinsic kinetic study, where an NH₃/CH₃OH ratio of 1:1 was found to provide high HCN selectivity at moderate space time.

Effect of O₂/MeOH Ratio

The effect of the oxidant level was explored by varying the O₂/MeOH ratio between 1.5 and 2.7 at fixed NH₃/MeOH = 1.1 (Figure 11). As in the base case, methanol conversion is essentially complete for all compositions, indicating that the reaction is not limited by oxygen over the range considered. In contrast, the HCN yield exhibits a weak but systematic decrease with increasing O₂/MeOH. At the lowest oxygen level studied (O₂/MeOH = 1.5) the model predicts Y_HCN,out ≈ 0.949. As the ratio is increased to 1.8, 2.1, 2.4 and 2.7, the HCN yield gradually declines to roughly 0.948, 0.947, 0.946 and 0.945, respectively. This trend reflects the fact that higher oxygen partial pressure promotes deep oxidation to CO and CO₂, which is in line with the experimental observation that increasing the O₂/methanol ratio mainly enhances the formation of CO_x while only having a minor effect on the conversion to HCN.

Very low O₂/MeOH ratios are undesirable because they move the process closer to the flammability limits of the CH₃OH–NH₃–air system and reduce the safety margin with respect to incomplete catalyst reoxidation. On the other hand, excessively high O₂/MeOH ratios waste oxidants and erode the HCN yield. The base-case value O₂/MeOH ≈ 2.1, corresponding to an air/MeOH ratio of 10.0 at NH₃/MeOH = 1.1, thus emerges as a reasonable compromise between safety, conversion and selectivity.

Overall, the composition sweeps confirm that, under industrial design conditions, the reactor performance is highly robust: methanol is fully converted across a broad range of NH₃ and O₂ excess, and the HCN yield only varies within a relatively narrow band (≈0.89–0.97). Within this window, the chosen base-case composition CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 lies close to the plateau of the high HCN yield, while preserving a conservative safety margin against both ammonia deficiency and flammability constraints.

2.4.4. Implications for Safe and Flexible Operation

Finally, the impact of the overall pressure level was assessed by varying the outlet pressure in the range P_out =1.3–2.0 bar (abs) at fixed tube geometry and plant capacity. For each outlet pressure, the per-tube methanol feed MeOH (in tube) was adjusted so that a bundle of N_t = 4000 identical tubes produced the target HCN rate of 4.63 × 10⁴ mol (10,000 t·a⁻¹ at 8000 h·a⁻¹ on stream). The feed composition and temperature were kept at the base-case values, CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025 (molar) and T = 420 °C, while the tube and pellet sizes were fixed at 30 mm and 2 mm, respectively.

Figure 12 shows the resulting inlet pressure as a function of P_out. As expected from the Ergun equation, increasing the outlet pressure shifts the entire pressure profile upwards in an almost linear fashion. When P_out is raised from 1.3 to 2.0 bar (abs), the calculated inlet pressure increases only moderately from about 3.1 to 3.5 bar (abs). The corresponding pressure drop, ΔP = P_in − P_out, is plotted in Figure 13. Over the range considered, ΔP decreases from approximately 1.8 bar at P_out = 1.3 bar (abs) to about 1.5 bar at P_out = 2.0 bar (abs), remaining in a narrow window of 1.5–1.8 bar for the 4 m packed length. Thus, even at elevated outlet pressure, the required compression duty and mechanical design pressure for the reactor tubes remain modest.

The effect of P_out on reactor performance is summarized in Figure 14, which reports the outlet HCN yield and selectivity as functions of the outlet pressure. For all cases, the methanol conversion is essentially complete (X_MeOH,out ≈ 1.00), and the HCN yield only varies slightly with pressure, from Y_HCN,out ≈ 0.945 at P_out = 1.3 bar (abs) to ≈ 0.951 at P_out = 2.0 bar (abs). The selectivity follows the same trend. This weak dependence confirms that, under isothermal operation and at fixed feed composition, the intrinsic Mars–van Krevelen kinetics are governed primarily by the relative reactant partial pressures rather than by the absolute pressure level. The moderate increase in average pressure slightly enhances the intrinsic rates but has no qualitative impact on conversion or selectivity.

From a design standpoint, these results indicate that operating the multitubular reactor at a mildly elevated outlet pressure, P_out ≥ 1.5 bar (abs), is advantageous. The reactor and downstream purification train remain safely above atmospheric pressure, which simplifies leak detection and prevents air ingress, while the total pressure drop across the 30 mm/2 mm tubes stays within 1.5–2.0 bar. Within this window, the plant can be operated flexibly between partial load and overload conditions by adjusting the per-tube feed without sacrificing high HCN yield and without approaching the flammability limits of the CH₃OH–NH₃–air–N₂ system that were defined in Section 2.

3. Reaction Network and Mars–van Krevelen Kinetics

3.1. Overall Reaction Network

The ammoxidation of methanol over FeMo/SiO₂ predominantly proceeds via three overall stoichiometric reactions: one selective ammoxidation route to hydrogen cyanide and two deep-oxidation routes to CO and CO₂. Methanol, ammonia and oxygen are treated as reactants, while nitrogen is considered inert and acts only as a diluent. The overall reactions are

HCN formation

$$\mathrm{C}{\mathrm{H}}_3\mathrm{O}\mathrm{H}+\mathrm{N}{\mathrm{H}}_3+{\mathrm{O}}_2\to \mathrm{H}\mathrm{C}\mathrm{N}+3{\mathrm{H}}_2\mathrm{O}$$

(1)

CO formation

$$\mathrm{C}{\mathrm{H}}_3\mathrm{O}\mathrm{H}+{\mathrm{O}}_2\to \mathrm{C}\mathrm{O}+2{\mathrm{H}}_2\mathrm{O}$$

(2)

CO₂ formation

$$\mathrm{C}{\mathrm{H}}_3\mathrm{O}\mathrm{H}+{\displaystyle \frac{3}{2}}{\mathrm{O}}_2\to \mathrm{C}{\mathrm{O}}_2+2{\mathrm{H}}_2\mathrm{O}$$

(3)

Equilibrium calculations show that, under the conditions of interest (360–460 °C and near-atmospheric pressure), all three reactions are strongly shifted towards products. At 400 °C, the equilibrium constants are approximately

K_HCN ≈ 5.7 × 10³³, K_CO ≈ 3.4 × 10³⁷ and K_CO2 ≈ 8.7 × 10⁵⁴,

so reverse rates can safely be neglected, and the reactions can be treated as irreversible.

3.2. Mars–van Krevelen Rate Expressions

Intrinsic kinetics for methanol ammoxidation over FeMo/SiO₂ were established in our recent study using a differential fixed-bed reactor operated under conditions that were free of internal and external transport limitations. Catalyst particle sizes below 0.15 mm, together with Weisz–Prater and Mears criteria for mass and heat transfer, confirmed that the measured rates are intrinsic and are not affected by diffusion or heat-transfer artifacts.

The experimental data were interpreted in terms of a Mars–van Krevelen (MvK) redox mechanism, in which lattice oxygen explicitly participates in the surface reactions. Among several mechanistic variants, an eight-parameter “case II” model was retained on the basis of statistical quality of fit, parameter identifiability and physical plausibility. In this model, lattice oxygen is represented by an effective oxidizing surface species (O₂*), which is formed by dissociative adsorption of gas-phase O₂ and consumed in the selective ammoxidation and deep-oxidation steps.

Assuming a quasi-steady-state coverage of O₂*, the intrinsic formation rates of HCN, CO and CO₂ per unit catalyst mass follow the rational MvK rate expressions

$$\mathrm{D}\mathrm{e}\mathrm{n}=k_1{p}_{O_2}+k_2{p}_A{p}_M+k_3{p}_M+1.5k_4{p}_M$$

(4)

$$r_{\mathrm{H}\mathrm{C}\mathrm{N}}={\displaystyle \frac{k_1{k}_2\hspace{0.17em}{p}_{O_2}\hspace{0.17em}{p}_M\hspace{0.17em}{p}_A}{\mathrm{D}\mathrm{e}\mathrm{n}}}$$

(5)

$$r_{\mathrm{C}\mathrm{O}}={\displaystyle \frac{k_1{k}_3\hspace{0.17em}{p}_{O_2}\hspace{0.17em}{p}_M}{\mathrm{D}\mathrm{e}\mathrm{n}}}$$

(6)

$$r_{\mathrm{C}{\mathrm{O}}_2}={\displaystyle \frac{k_1{k}_4\hspace{0.17em}{p}_{O_2}\hspace{0.17em}{p}_M}{\mathrm{D}\mathrm{e}\mathrm{n}}}$$

(7)

where $p_{O_2}$, $p_A$ and $p_M$ are the partial pressures of O₂, NH₃ and CH₃OH, respectively (in bar), and $r_i$ are intrinsic rates in mol·gcat⁻¹·h⁻¹.

The temperature dependence of the kinetic coefficients is described by Arrhenius relations

$$k_{\mathrm{j}}(T)=A_{\mathrm{j}}\mathrm{e}\mathrm{x}\mathrm{p}(-{\displaystyle \frac{E_{\mathrm{j}}}{RT}}\hspace{0.17em}), \mathrm{j}=1,\dots ,4$$

(8)

with the pre-exponential factors A_j and the activation energies E_j taken from the regression of the intrinsic kinetic data (retained case-II model):

$$A_1=6.87401\times {10}^{-1}, E_1=1.34061\times {10}^4\mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$$

$$A_2=6.03154\times {10}^6, E_2=6.70289\times {10}^4 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$$

$$A_3=5.72650\times {10}^0, E_3=2.57893\times {10}^4 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$$

$$A_4=2.10023\times {10}^3, E_4=5.35679\times {10}^4 \mathrm{J}·{\mathrm{m}\mathrm{o}\mathrm{l}}^{-1}$$

These parameters reproduce the measured conversions and selectivities over 340–440 °C and space times between 20 and 100 gcat·h·mol ⁻¹, with HCN selectivities above 90% at 400–420 °C.

For reactor-scale simulations, the mass-based rates are converted to volumetric source terms using the skeletal catalyst density ρ_cat (kg·m⁻³). The volumetric rates for methanol (M), ammonia (A), oxygen and the three products are then written as

Reactants

$$R_{\mathrm{M}}={\rho }_{cat}\left(r_{HCN}+r_{CO}+r_{{CO}_2}\right)$$

(9)

$$R_{\mathrm{A}}={\rho }_{cat}{r}_{HCN}$$

(10)

$$R_{O_2}={-\rho }_{cat}\left(r_{HCN}+r_{CO}+{1.5r}_{{CO}_2}\right)$$

(11)

Products

$$R_{\mathrm{H}\mathrm{C}\mathrm{N}}={\rho }_{cat}{r}_{HCN}$$

(12)

$$R_{\mathrm{C}\mathrm{O}}={\rho }_{cat}{r}_{CO}$$

(13)

$$R_{{CO}_2}={\rho }_{cat}{r}_{{CO}_2}$$

(14)

These volumetric source terms enter directly into the intraparticle diffusion–reaction model (Section 3.2) and, via the interphase fluxes, into the heterogeneous plug-flow model of the reactor tubes.

In the companion kinetic study [8], the MvK parameters were estimated from experiments performed at 1.0 bar and 360–460 °C, for NH₃/MeOH and O₂/MeOH ratios chosen so that the resulting partial pressures span approximately $p_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H}}=0.01–0.12 \mathrm{bar}$, $p_{\mathrm{N}{\mathrm{H}}_3}=0.01–0.14 \mathrm{bar},$ and $p_{{\mathrm{O}}_2}=0.02–0.30 \mathrm{bar}$. Under the industrial feed composition considered here (CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025), the local partial pressures of methanol, ammonia and oxygen along the tubes remain within, or very close to, this envelope, so that the MvK model is not extrapolated to extreme composition or temperature regions. Moreover, the rate expressions are formulated explicitly in terms of partial pressures, and the gas mixture can be treated as ideal in the modest pressure range investigated (1–3 bar). The use of the same intrinsic parameter set at slightly elevated total pressures therefore corresponds with the application of the model at similar partial-pressure ratios and coverage regimes, which is a common and reasonable approximation in preliminary reactor design.

3.3. Feed Composition and Operating Conditions

For the industrial design, the feed composition is specified on a molar basis as

$$\mathrm{C}{\mathrm{H}}_3\mathrm{O}\mathrm{H}:\mathrm{N}{\mathrm{H}}_3:\mathrm{a}\mathrm{i}\mathrm{r}:{\mathrm{N}}_2\left(\mathrm{a}\mathrm{d}\mathrm{d}\right)=1:1.1:10.0:3.025$$

(15)

where air is taken as 21% O₂ and 79% N₂ (molar), and N₂(add) only denotes the nitrogen that is added explicitly as an explosion suppressant, not the nitrogen that is already present in the air stream. This composition lies in the non-flammable region of the CH₃OH–NH₃–air system while providing sufficiently high partial pressures of methanol and ammonia for efficient ammoxidation.

For a given per-tube methanol feed rate $F_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}^{\left(\mathrm{tube}\right)}$ (mol·h⁻¹), the corresponding inlet molar flow rates of the key species in a single tube are

$$F_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}=F_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}^{\left(\mathrm{tube}\right)}$$

(16)

$$F_{\mathrm{N}{\mathrm{H}}_3,0}=1.1\hspace{0.17em}{F}_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}$$

(17)

$$F_{\mathrm{air},0}=10.0\hspace{0.17em}{F}_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}$$

(18)

$$F_{{\mathrm{O}}_2,0}=0.21\hspace{0.17em}{F}_{\mathrm{air},0}$$

(19)

$$F_{{\mathrm{N}}_2,\mathrm{a}\mathrm{i}\mathrm{r},0}=0.79\hspace{0.17em}{F}_{\mathrm{air},0}$$

(20)

$$F_{{\mathrm{N}}_2,\mathrm{a}\mathrm{d}\mathrm{d},0}=3.0\hspace{0.17em}{F}_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H},0}$$

(21)

$$F_{{\mathrm{N}}_2,0}=F_{{\mathrm{N}}_2,\mathrm{a}\mathrm{i}\mathrm{r},0}+F_{{\mathrm{N}}_2,\mathrm{a}\mathrm{d}\mathrm{d},0}$$

(22)

For the industrial design, the feed composition is specified on a molar basis as CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.0,where air is taken as 21% O₂ and 79% N₂ (molar), and N₂(add) only denotes the nitrogen that is added explicitly as an explosion suppressant, not the nitrogen that is already present in the air stream. According to published flammability maps for CH₃OH–NH₃–air mixtures near atmospheric pressure [3,4,14], mixtures with air/MeOH ≥ 8 and NH₃/MeOH ≈ 1.0–1.2 lie outside the reported flammable envelope over a broad temperature range. Therefore, the present composition, with air/MeOH = 10.0 and NH₃/MeOH = 1.1, resides in the non-flammable region, while the additional N₂(add) = 3.0 further dilutes the mixture and increases the margin to both lower and upper flammability limits without excessively reducing the reactant partial pressures of CH₃OH and NH₃. This composition lies in the non-flammable region of the CH₃OH–NH₃–air system while providing sufficiently high partial pressures of methanol and ammonia for efficient ammoxidation.

Unless stated otherwise, all reactor simulations are performed with an isothermal tube-wall temperature of 693 K (420 °C). The outlet pressure P_out is taken between 1.3 and 2.0 bar (abs) depending on the scenario, while the inlet pressure P_in is obtained from the Ergun equation for the chosen tube geometry and total flow rate.

Methanol ammoxidation to hydrogen cyanide is a strong exothermic process. The literature data indicate a standard reaction enthalpy of approximately ΔH°₍₂₉₈₎ ≈ −350 kJ mol⁻¹ for the selective ammoxidation step CH₃OH + NH₃ + O₂ → HCN + 3 H₂O, while full combustion of methanol to CO₂ and H₂O is even more exothermic (ΔH°₍₂₉₈₎ ≈ −600 kJ mol⁻¹). At the nominal design load of the 30 mm/2 mm base-case tube ($F_{\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H}}$ ≈ 12.2 mol h⁻¹, Y_HCN ≈ 0.95), the corresponding heat release is in the order of 1–1.5 kW per tube. For a 4.0 m tube length this heat must be removed through an external area of ≈0.38 m². Using typical overall heat-transfer coefficients for gas-cooled multitubular reactors (U ≈ 300–500 W m⁻² K⁻¹), the required mean temperature driving force between the tube-side gas and the coolant is therefore only about 8–15 K. This estimate suggests that, in a properly designed shell-and-tube reactor, the axial temperature rise and hot-spot elevation in the tubes remain limited (≲20 K) and that the assumption of nearly isothermal operation around 420 °C is a reasonable first-order approximation. A fully non-isothermal treatment including detailed shell-side heat-transfer design would be a valuable extension, but it lies outside of the scope of the present conceptual study.

3.4. Multitubular Fixed-Bed Configuration

The industrial reactor is modeled as a vertical shell-and-tube vessel containing N_t identical tubes of inner diameter D_t and packed-bed length L_bed. Each tube is filled with spherical FeMo/SiO₂ catalyst pellets with the diameter d_p. Based on the intrinsic kinetic study and preliminary transport calculations, tube diameters in the range D_t = 30–35 mm and pellet diameters d_p = 2–3 mm are considered, with the 30 mm/2 mm combination later adopted as the base-case geometry (Section 4).

The packed-bed porosity is denoted ε_b, and the skeletal density of the catalyst pellets is ρ_p. The corresponding bulk catalyst density in the bed is

$${\rho }_b=\left(1-{\epsilon }_b\right)\hspace{0.17em}{\rho }_p$$

(23)

which, for ε_b ≈ 0.40 and ρ_p ≈1.2 × 10⁶ gcat·m⁻³, gives ρ_b ≈ 7.0 × 10⁵ gcat·m⁻³.

The tube cross-sectional area is

A_t = π D_t²/4

(24)

and the mass of catalyst loaded into a single tube is

$$W_{\mathrm{tube}}={\rho }_b{A}_t{L}_{\mathrm{bed}}$$

(25)

The total catalyst inventory in the multitubular reactor is then

$$W_{\mathrm{tot}}=N_{\mathrm{t}}{W}_{\mathrm{tube}}$$

(26)

The base-case design (D_t = 30 mm, L_bed ≈ 4.0 m, d_p = 2 mm, and ε_b = 0.40) corresponds to a catalyst mass per tube of approximately 2.0 kg and a total inventory on the order of 8 × 10³ kg for N_t = 4000 tubes. Shell-side cooling is used to maintain the tubes close to isothermal operation at 420 °C and is consistent with standard industrial practice for strong exothermic reactions in multitubular fixed-bed reactors.

4. Reactor Model

After specifying the overall reactor configuration, a detailed heterogeneous model is formulated for a single representative tube. The model consists of a one-dimensional gas-phase plug-flow description coupled with a spherical pellet diffusion–reaction model.

4.1. Gas-Phase Plug-Flow Model

The gas phase in each tube is described as a one-dimensional plug flow along axial coordinate z (0 ≤ z ≤ L_bed), with negligible radial gradients in temperature and composition. At 420 °C and near-atmospheric pressures, the gas mixture can be treated as ideal.

Let C_i(z) denotes the bulk gas-phase concentration of species i (mol·m⁻³) and u_g(z) the superficial gas velocity (m·h⁻¹). For the reacting species CH₃OH (M), NH₃ (A), O₂, HCN, CO and CO₂, the axial mass balances read

$$u_{\mathrm{g}}\hspace{0.17em}{\displaystyle \frac{\mathrm{d}{C}_i}{\mathrm{d}z}}=a_{\mathrm{s}}\hspace{0.17em}{N}_i\left(i={\mathrm{M}, \mathrm{A}, \mathrm{O}}_2, {\mathrm{HCN}, \mathrm{CO}, \mathrm{CO}}_2\right),$$

(27)

where $u_{\mathrm{g}}$ is the superficial gas velocity in the tube, $C_i$ is the bulk gas-phase concentration of species $i$ (mol·m⁻³), $a_{\mathrm{s}}$ is the external catalyst surface area per unit reactor volume (m²·m⁻³), and $N_i$ is the molar flux of species $i$ from the bulk gas to the catalyst surface (mol·m⁻²·h⁻¹). Inert N₂ is included in the total concentration and pressure but does not participate in the reaction.

In the numerical implementation, the intraparticle diffusion–reaction equations are solved in the radial coordinate, and the net molar flux $N_i$ entering the gas-phase balances is obtained from the volume-averaged intraparticle reaction rate (i.e., by integrating the radial profiles over the pellet volume), so that the non-uniform concentration profile inside the pellet is fully accounted for and no additional one-point diffusion approximation is introduced.

The interphase molar flux is described by a standard gas-film relation,

$$N_{\mathrm{i}}=k_{g,\mathrm{i}}\hspace{0.17em}\left(C_{\mathrm{i}}-C_{\mathrm{i},s}\right),$$

(28)

where $k_{g,i}$ is the gas-film mass-transfer coefficient for species $i$ (m·h⁻¹), and $C_{i,s}$ is the concentration at the external pellet surface. The coefficients $k_{g,i}$ are evaluated from Sherwood-type correlations for flow over spheres,

$${\mathrm{Sh}}_{\mathrm{i}}={\displaystyle \frac{k_{g,i}\hspace{0.17em}{d}_p}{D_{i,\mathrm{m}}}}=2+1.1\hspace{0.17em}{\mathrm{Re}}_{\mathrm{p}}^{0.6}\hspace{0.17em}{\mathrm{Sc}}_{\mathrm{i}}^{1/3},$$

(29)

where d_p is the pellet diameter, D_i,m is the molecular diffusion coefficient of species i in the gas mixture, Re_p is the particle Reynolds number and Sc_i is the Schmidt number.

Pressure drop along the packed bed is computed from the Ergun equation and coupled with the mass balances. Because the intrinsic kinetics are expressed in terms of partial pressures, the local total pressure P(z) is used to evaluate

p_i(z) = y_i(z) P(z)

(30)

where y_i is the gas-phase mole fraction of species i. The gas-phase balances are therefore solved with a variable pressure profile rather than assuming constant pressure.

At the reactor inlet (z = 0), the bulk gas concentrations are determined by the chosen feed composition and the total pressure:

C_i(z = 0) = C_i,in, i = M, A, O₂, N₂

(31)

HCN, CO and CO₂ concentrations are zero at the inlet.

4.2. Intraparticle Diffusion–Reaction Model

Transport and reaction inside the spherical FeMo/SiO₂ pellets are described by a non-isothermal-but-here-assumed-isothermal diffusion–reaction model in the radial coordinate $r$(0 ≤ $r$ ≤ $R_p$ with $R_p=d_p/2$). For each reacting gas-phase species, CH₃OH, NH₃ and O₂, a steady-state balance between diffusion and reaction is written as

$${\displaystyle \frac{1}{r^2}}\hspace{0.17em}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}r}}\left(r^2{D}_{\mathrm{e},\mathrm{j}}\hspace{0.17em}{\displaystyle \frac{\mathrm{d}{C}_j}{\mathrm{d}r}}\right)+R_{\mathrm{j}}\left(C_{\mathrm{M}},C_{\mathrm{A}},C_{{\mathrm{O}}_2}\right)=0,\left(\mathrm{j}={\mathrm{M}, \mathrm{A}, \mathrm{O}}_2\right),$$

(32)

where $D_{\mathrm{e},\mathrm{j}}$ is the effective intraparticle diffusivity of species $\mathrm{j}$ (m²·h⁻¹) and $R_{\mathrm{j}}$ is the local volumetric reaction rate of species $\mathrm{j}$ (mol·m⁻³·h⁻¹) as defined in Equations (9)–(11).

In this work, the effective intraparticle diffusivities are taken as $D_{\mathrm{e},\mathrm{M}\mathrm{e}\mathrm{O}\mathrm{H}}\approx 2.2\times {10}^{-6}\hspace{0.17em}{\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$, $D_{\mathrm{e},\mathrm{N}{\mathrm{H}}_3}\approx 3.1\times {10}^{-6}\hspace{0.17em}{\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$ and $D_{\mathrm{e},{\mathrm{O}}_2}\approx 2.4\times {10}^{-6}\hspace{0.17em}{\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$. These values were obtained from gas-phase binary diffusion coefficients at 400–420 °C (of order ${10}^{-5}\hspace{0.17em}{\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$) and corrected for pellet porosity and tortuosity via the standard relation $D_{\mathrm{e},i}=({\epsilon }_p/\tau )\hspace{0.17em}{D}_{i,\mathrm{g}\mathrm{a}\mathrm{s}}$, with ${\epsilon }_p\approx 0.6$ and $\tau \approx 3$. The resulting $D_{\mathrm{e},i}$ values in the ${10}^{-6}\hspace{0.17em}{\mathrm{m}}^2\hspace{0.17em}{\mathrm{s}}^{-1}$ range are typical for reactant-sized molecules in silica-based porous oxides at these temperatures. In the present model the $D_{\mathrm{e},i}$ are used as species-wise, mixture-averaged Fick diffusivities and are treated as constant along the bed; the pressure only changes from about 1.5 to 3.2 bar, so the associated variation in gas-phase diffusivities is small compared to the uncertainty in ${\epsilon }_p/\tau$ and does not affect the conclusion that intraparticle diffusion limitations remain moderate. At the pellet center, symmetry implies

dC_j/dr|(r = 0) = 0

(33)

At the external surface r = R_p, continuity of flux between pellet and gas film gives

−D_e,jdC_j/dr|(r = R_p) = k_g,j (C_j,bulk − C_j,s)

(34)

The local rate expressions R_j (C_M, C_A, $C_{{\mathrm{O}}_2}$) are obtained by evaluating the MvK kinetic model (Equations (4)–(14)) at the pellet temperature and local gas concentrations via

$$p_{\mathrm{j}}=C_{\mathrm{j}}RT,\left(\mathrm{j}={\mathrm{M}, \mathrm{A}, \mathrm{O}}_2\right),$$

(35)

The pellet-scale model therefore remains fully consistent with the intrinsic kinetics measured under the transport-free conditions.

4.3. Numerical Solution

The coupled gas-phase and intraparticle equations are solved by nested integration in the axial coordinate $\mathit{z}$. For the given axial position $\mathit{z}$, the spherical pellet diffusion–reaction problem (Equations (32)–(35)) is formulated as a two-point boundary value problem in the radial coordinate $\mathit{r}$ and solved on a one-dimensional grid, typically with ${\mathit{N}}_{\mathit{r}}=31$–41 nodes using the collocation solver bvp4c in MATLAB R2025a. The iterations are continued until the maximum relative change in the concentration profile between successive mesh refinements falls below ${10}^{-6}$. The volume-averaged reaction rates and effectiveness factors are then obtained by numerical integration of the converged radial profiles over the pellet volume.

The gas-phase plug-flow equations (Equation (27)), coupled with the Ergun pressure-drop relation, are integrated in $\mathit{z}$ using a fourth-order Runge–Kutta scheme on a uniform axial grid. For the 4.0 m tubes considered here, ${\mathit{N}}_{\mathit{z}}=200$ axial steps are used, corresponding to a step size $\Delta \mathit{z}\approx 0.02$ m. Grid independence was verified by repeating selected simulations with ${\mathit{N}}_{\mathit{z}}=400$ and doubling ${\mathit{N}}_{\mathit{r}}$; the resulting changes in outlet methanol conversion, HCN yield and axial effectiveness factors were below 0.1%, confirming that the chosen discretization is sufficient for the present design study.

In all simulations the reactor tubes are assumed to operate isothermally at 420 °C. This is justified by the strongly diluted feed composition (CH₃OH:NH₃:air:N₂(add) = 1:1.1:10.0:3.025), which limits the adiabatic temperature rise to a few tens of kelvin, and by the use of small-diameter tubes with intensive shell-side cooling in industrial multitubular reactors. Simple enthalpy estimates based on the overall stoichiometry and standard reaction heats indicate an adiabatic temperature rise below ≈30–40 K at the base-case composition and conversion, and additional runs with ±10–20 K perturbations of the tube temperature showed only minor changes in the required space time and HCN yield. The isothermal plug-flow assumption should therefore be regarded as an engineering simplification that is appropriate for conceptual design; a fully non-isothermal tube-bundle model with detailed heat-transfer design is beyond the scope of this work and will be addressed in future studies.

5. Conclusions and Outlook

A conceptual design of a multitubular fixed-bed reactor for methanol ammoxidation to hydrogen cyanide over FeMo/SiO₂ catalysts has been developed on the basis of a mechanistic Mars–van Krevelen (MvK) kinetic model. The main findings can be summarized as follows:

• The reduced eight-parameter MvK model established in the companion kinetic study accurately describes methanol conversion and product selectivities over 360–460 °C, providing a consistent representation of the competing ammoxidation and deep-oxidation pathways.
• Scaling this kinetic model to an industrial multitubular configuration targeting 10,000 t·a⁻¹ of HCN shows that both 35 mm/3 mm and 30 mm/2 mm tube–particle combinations can achieve essentially complete methanol conversion at 420 °C, with HCN selectivities above approximately 0.93, provided that appropriate tube lengths and total tube counts are selected.
• The smaller 30 mm/2 mm geometry offers distinct advantages. At the design load (L = 4 m, N_t = 4000, and P_out = 1.5 bar (abs)), it delivers an outlet HCN selectivity and carbon-based yield of approximately S_HCN ≈ Y_HCN ≈ 0.95 at essentially complete methanol conversion, albeit at the expense of a higher pressure drop (ΔP ≈ 1.7 bar) and therefore a higher inlet pressure (P_in ≈ 3.2 bar (abs)).
• A detailed pellet-scale analysis reveals that intraparticle diffusion in 2 mm FeMo/SiO₂ pellets leads to effectiveness factors as low as η_min ≈ 0.6 in the most strongly reacting section of the bed, corresponding to an effective Thiele modulus of Φ ≈ 3–4. Nevertheless, diffusion limitations remain moderate: the effectiveness factors increase downstream as the reactants are depleted, and the pellets approach kinetic control near the tube outlet.
• The resulting design—combining 30 mm tubes, 2 mm pellets, an outlet pressure of 1.5 bar (abs), and a tube length of 4 m—represents a favorable compromise between HCN yield, pressure drop, and radial temperature control. The reactor can be operated safely within a non-flammable composition window, with moderate compression duty and a robust margin against incomplete conversion.

Overall, the present work demonstrates how a mechanistic MvK rate expression can be embedded in a heterogeneous plug-flow model to support the conceptual design of FeMo/SiO₂-based HCN reactors. The framework can be extended in future studies to incorporate non-isothermal effects, dynamic operation (start-up and load changes), catalyst deactivation, and more detailed representations of the downstream separation train. Such extensions will further strengthen the link between intrinsic kinetics and industrial reactor design for low-temperature ammoxidation routes to hydrogen cyanide.