Methane pyrolysis in packed bed reactors: Kinetic modeling, numerical simulations, and experimental insights

Pyrolysis of hydrocarbon feeds such as methane (CH 4 ) and natural gas emerges as a pivotal carbon dioxide-free large-scale hydrogen (H 2 ) production process combined with capturing the carbon as solid material. For fundamental understanding and upscaling, the complex kinetics and dynamics of this process in technically relevant reactors such as packed and moving beds still need to be explored, particularly concerning carbon formation and its impact on reactor performance. This study integrates kinetic modeling, numerical simulations, and experimental findings to comprehensively understand CH 4 pyrolysis under industrially relevant conditions and its implications for efficient H 2 production and carbon capture. The investigation covers temperatures from 1273 K to 1873 K, H 2 addition with H 2 :CH 4 ratios of 0 to 4, and hot zone residence time of 1 to 7 s. Two distinct pathways lead to carbon formation: soot formation and carbon deposition. Each pathway originates from different gas-phase precursors. An elementary-step-based gas-phase reaction mechanism is coupled with a soot formation model from polycyclic aromatic hydrocarbon and a newly developed deposition model from light hydrocarbons. Numerical simulations are performed in a packed bed reactor model, incorporating a method of moments for soot formation and a model for carbon deposition. The model is evaluated against experiments and predicts the effects of operating conditions on gas-phase product distribution and carbon formation. It also estimates the change in bed-voidage over operational time. The study reveals that at the temperature 1673 K, CH 4 conversion exceeds 94 %, while both H 2 and solid carbon yields surpass 96 %. The sophisticated modeling and simulation framework presented herein thus provides an enhanced understanding of the CH 4 pyrolysis process and presents a valuable tool for optimizing this process.


Introduction
Hydrogen (H 2 ) is one of the most important raw materials in the petrochemical and chemical industry [1].Its demand is expected to increase further as a modern carbon-free energy carrier [2][3][4][5], where circular storage solutions, i.e. using metals and metal hydrides via oxidation-reduction cycles [6][7][8], will play a key role.Although water electrolysis using renewable energy is at present the only carbon-free pathway to produce high purity H 2 , its high investment costs and energy demands are limiting large scale application [9].The majority of large-scale H 2 production is currently derived from fossil fuels via steam methane reforming, which accounts for almost half of this production [1].Furthermore, dry reforming of methane (DRM) [10][11][12], autothermal reforming of methane (ATR) [13], and catalytic partial oxidation of methane (CPOX) [14,15] are also commercially viable alternatives based on the required syngas H 2 :CO ratio [16].Nonetheless, these processes primarily generate carbon dioxide (CO 2 ) as a byproduct, requiring further procedures for effective carbon capture and storage [17].
In contrast, thermal methane pyrolysis has been studied and reported as a viable alternative to the above mentioned processes [30,31] that in principle allows large-scale H 2 production via CH 4 pyrolysis, for instance as demonstrated by a demo plant that was put into operation by BASF SE in 2021 [32].Additionally, studies have reported that the presence of solid carbon accelerates the carbon deposition [30,33,34].Two different pathways resulting in the formation of solid carbon during pyrolysis of light hydrocarbons can be found in the literature: soot formation from polycyclic aromatic hydrocarbons (PAHs) in an empty tube reactor or flames [35][36][37][38][39][40] and direct deposition through light hydrocarbons on reactor wall or packing material [33,34,[41][42][43][44][45][46][47].Earlier studies on the formation of carbon in the form of soot found that the particle number concentration of soot formed during pyrolysis was directly related to operating conditions, PAH concentrations and temperature [48,49].A series of gas-phase reactions lead to soot nucleation followed by coagulation, sintering, surface growth and condensation [37].Many approaches were proposed to approximate the general equations of dynamics, including the monodisperse model [40,50,51], the method of moments [52][53][54], the sectional method [55][56][57], the stochastic model [58] and the Galerkin model [59].On the other hand, direct carbon deposition from CH 4 , ethylene (C 2 H 4 ), acetylene (C 2 H 2 ), butadiene (C 4 H 6 ), propylene (C 3 H 6 ) and benzene (C 6 H 6 ) was also found to be dependent on temperature, hydrocarbon inlet concentrations, pressure and the available surface area for deposition [41][42][43][44][45][46].Furthermore, Becker et al. [60] reported that the presence of H 2 inhibits the deposition reactions.On the basis of work by Becker et al. [60], Li and Deutschmann [47] elaborated on chemical vapor infiltration (CVI) from CH 4 under sub-atmospheric pressure and coupled a deposition mechanism with reduced gas-phase kinetics while considering H inhibition.
In spite of these promising studies, none of them investigated CH pyrolysis in a packed bed reactor for the formation of H 2 and carbon.Hence, our present study employs a combined approach of lab-scale packed bed reactor experiments and numerical simulations using detailed kinetics to investigate the CH 4 pyrolysis process under industrially relevant conditions.Covering a wide range of operating conditions, namely temperatures above 1273 K, varying H 2 :CH 4 ratios (1, 2, and 4), and residence times (1, 3, 5, and 7 s), the behavior of the system is comprehensively examined.The study incorporates an improved carbon deposition kinetic model derived from light hydrocarbons, as well as a soot formation model.In addition, the developed model incorporates the concept of the filtration mechanism to evaluate and account for the trapping of soot particles within the packed bed material.Furthermore, the detailed gas-phase mechanism proposed by Appel et al. [35] Second moment m  these aspects, the model is evaluated with the aim to provide a more detailed understanding of the carbon deposition and soot formation in CH 4 pyrolysis.This comprehensive approach enables the analysis of interaction between gas-phase reactions, deposition kinetics, and soot particle growth, expanding our knowledge of the overall system behavior and allowing for an improved design of CH 4 pyrolysis processes for large-scale H 2 production.

Experimental
Experiments were performed in a tubular flow reactor designed for kinetic studies at high temperature (up to 2073 K) and high pressure (up to 10 bar) as shown in the schematic in Fig. 1.The flow diagram of the experimental setup is provided in Fig. S1.More details about the setup are described in a previous publication by Angeli et al. [61].The α-Al 2 O 3 tubular reactor of 0.02 m inner diameter and 1.0 m length was placed in a stainless-steel vessel equipped with heating elements and insulation, which results in an isothermal zone of 0.4 m.The packed bed (length of 0.12 m) consisting of acetylene coke particles (approx.0.002 m diameter) was kept in a graphite foil container, which is open at the top and has holes at the bottom to allow the effluent gas stream exiting the packed bed.After placing the container at the beginning of isothermal zone, the reaction gas mixture was fed into the reactor via mass flow controllers while controlling the operating conditions by a LabViewbased software.The product gas stream was continuously analyzed with an HPR-20 mass spectrometer (Hiden Analytical), particularly quantifying H 2 , CH 4 , C 2 H 2 , C 2 H 4 , C 2 H 6 and C 6 H 6 .The operating conditions chosen for the present study are summarized in Table 1.

Modeling approach and numerical simulation
In order to fully understand the CH 4 pyrolysis process, it is essential to develop a comprehensive model that considers the various phenomena occurring within the laboratory reactor as depicted in Fig. 2.This model should include elements such as gas-phase reactions, solid carbon deposition, soot formation and growth, and its interactions with the packed bed.The subsequent section provides details on the kinetic models as well as strategies for their effective coupling.

Gas-phase reactions
Since the CH 4 pyrolysis reaction starts with gas-phase species, a reliable gas-phase elementary-step based mechanism is critical.For the present work, a detailed gas-phase mechanism by Appel et al. [35] that was originally developed for describing the combustion of C 2 hydrocarbons, hereinafter referred to as ABF2000, was employed.To adapt it to pyrolysis conditions, all oxygen-related species and reactions were removed, leading to a shortened mechanism comprising 247 reactions and 76 species.The choice of ABF2000 was driven by the presence of polycyclic aromatic hydrocarbons (PAHs) formation in this mechanism, which are considered to be a crucial factor for coupling the soot formation model to gas-phase reactions.When PAHs species collide with each other (pyrene in case of ABF2000), dimers are assumed to form that result in the formation of soot clusters [36,62].In particular, pyrene helps to overcome the thermodynamic barrier between two and three rings, and the dimer is responsible for nucleating soot.To validate this assumption, Schuetz and Frenklach [62] studied the formation of pyrene dimers by using molecular dynamics simulations, which showed that at a temperature of 1600 K, 15 % of the collisions resulted in stable dimers with a lifetime exceeding 10 ps.These findings suggest that pyrene dimers may act as soot nuclei under certain conditions.The nucleation rate is assumed to be the collision rate of two C 16 H 10 (pyrene) molecules in the free-molecular region, with a van der Waals enhancement factor of 2.2 being taken into account [63][64][65][66].To incorporate this information in the soot model, the chemical source term utilized is the net production rate of the dimers.The chemical source term for the soot model is determined by integrating the pyrene dimer formation reaction and rate constant into the ABF2000 mechanism shown in Eqs. ( 1) and (2), respectively [67,68], C 16 H 10 + C 16 H 10 →Dimer (1) where, E d is enhancement factor, d d is effective dimer diameter, k B is Boltzmann constant, T the temperature, and M d is reduced mass of the dimer.

Soot surface growth kinetics
Soot surface growth is a crucial aspect of the CH 4 pyrolysis process, and several mechanisms have been proposed to explain the growth dynamics [35,36,56,69].One widely accepted model is the HACA (Habstraction C 2 H 2 -addition) surface growth mechanism suggested by Appel et al. [35].The HACA mechanism has been extensively studied and validated through experimental and theoretical studies, providing a detailed understanding of the surface kinetics of soot growth [35][36][37]58].For the current study, the HACA surface growth mechanism was modified by removing all oxygen-related reactions, resulting in the modified growth rate depicted in Eq. (3).The rate constants for this modified growth rate are presented in Arrhenius format in Table 2.For the present study, both the surface site density and the fraction of surface sites available for the reaction are obtained from the literature [36].
where, α is fraction of surface site available, and χ CsootH is sites per unit surface area of the particles.

Carbon deposition and hydrogen inhibition kinetics
The chemical vapor deposition (CVD) from light hydrocarbons is considered to be a complex surface reaction where solid carbon is deposited from gaseous light hydrocarbons [33,47,60,70,71].As mentioned above, gas-phase reactions are described using the ABF2000 mechanism.In a next step, the heterogeneous surface reactions were then coupled to the deposition precursors from ABF2000.The reaction mechanism of CVD from gas-phase hydrocarbon was first proposed by Becker et al. [60] and later employed by Li and Deutschmann [47] for describing chemical vapor infiltration (CVI).CH 4 pyrolysis in the current study takes place in a flow reactor and at atmospheric pressure.Therefore, the work by Li and Deutschmann [47] was extended by selecting additional deposition precursors as illustrated in Fig. 3 based on a gas-phase mechanistic study on CH 4 pyrolysis using the ABF2000 mechanism [72].While activation energy values were taken from the work by Tesner [70], pre-exponential factors were fitted to the packed bed experiments, which further improved the H 2 inhibition functions.The resulting updated kinetic parameters used in the present study are summarized in Table 3.

Packed bed reactor model
In this study, a one-dimensional packed bed reactor model was employed to simulate CH 4 pyrolysis experiments conducted in a laboratory reactor.The model simulates the chemically reacting flow of a gas mixture in axial direction under steady-state conditions, and solves the continuity equation, species conservation equation, pressure drop, and equation of state using a differential-algebraic solver (LIMEX) [73,74].
The equation of continuity, represented by Eq. ( 4), is used to calculate the mass flux in a system.
where, ρ is gas phase density, u is superficial velocity of gas, z is reactor axial coordinate, a v is surface area to volume ratio, M i is molar mass of species i, ṡ is rate of surface reaction, and S g is set of gas-phase species.Along with Eq. ( 4), the species conservation Eq. ( 5) is solved.
where, Y i is mass faction of species i, ωi is rate of formation of gas-phase species i, and ε bed is packed bed-voidage.
Furthermore, the pressure drop in the packed bed reactor is calculated based on Eq. ( 6) where, p is pressure, f is friction factor and d p is bed particle diameter.The friction factor f is calculated based on the Ergun Equation [75].
Fig. 2. Methane pyrolysis in a packed bed reactor: gas-phase reactions, soot formation and carbon deposition.
Soot surface growth kinetic data in Arrhenius format Rate constant  Furthermore, in this study, the energy balance equation was not explicitly solved.Instead, the experimentally measured temperature profiles within the reactor were used as an input parameter.By directly incorporating the measured temperature profile, the model can focus on simulating the chemical reactions and species transport within the system while maintaining consistency with the experimental conditions.
The DETCHEM software package [74,[76][77][78][79][80] was used to perform packed bed reactor (DETCHEM PBR ) simulations where a transient wrapper (DETCHEM PBR_transient ) calculates carbon deposition as a function of operation time.This simulation assumes that the time scale for deposition reactions is much longer than the residence time of the gas phase.The transient wrapper calls the DETCHEM PBR simulation iteratively at each time step, performing a quasi-steady-state simulation to calculate the concentration profiles of the gas-phase species, while keeping the concentrations of the deposition species fixed.However, coverage varies depending on the location within the reactor.The locally resolved gas-phase concentrations are then transferred back to the transient wrapper for a time integration step for the transient species Eq. ( 7).The simulation process is completed by repeating the steadystate packed bed reactor simulation with updated inlet conditions and the newly updated state of the surface.
where, c deposition is concentration of deposited species, c gas is gas-phase concentration, and c surface is concentration of non-deposited surface species.

Method of moment model
The modeling of soot formation plays a significant role in comprehending and optimizing the CH 4 pyrolysis process.The method of moments (MOM) is a widely used approach to model the time evolution of the size and number density of soot particles through a set of moment equations [63,69,[81][82][83][84][85][86].
These equations consider the key phenomena of nucleation, growth, and coagulation that govern the behavior of soot particles.Based on the assumption that soot particles are spheres and homogeneous, the MOM model can provide insight into how temperature, pressure, and reactant concentration affect nucleation, growth, and coagulation.
For the present work, instead of solving for the entire particle size distribution (PSD), we focus on the first three moments (M 0 , M 1 and M 2 ), hereby capturing the most important aspects of the process.The governing equations for the lognormal moment model were written in a dimensionless form in terms of moment change rates.The zeroth moment M 0 represents the total particle number concentration N and is affected by nucleation and coagulation, as shown in Eq. (8).
where, N ′ is dimensionless particle number concentration, Θ is dimensionless residence time, I ′ is dimensionless nucleation rate, and ξ is dimensionless coagulation coefficient.
For the present study, the dimensionless coagulation coefficient ξ are the harmonic average of coefficients in the free molecular ξ FM and continuum regimes ξ C (see SI, Equations (S1)-(S5)).The first moment M 1 represents the particle volume concentration V 1 and is affected by both nucleation and surface growth, as shown in Eq. (9).
where, V ′ 1 is dimensionless particle volume concentration, n * m is dimensionless monomer concentration, and W ′ is dimensionless soot surface growth rate.
The second moment M 2 represents the particle volume V 2 and is affected by nucleation, surface growth, and coagulation, as shown in Eq. (10).
where, V ′ 2 is dimensionless second aerosol volume moment, and ζ is dimensionless coagulation coefficient, calculated as harmonic average of coefficients in the free molecular ζ FM and continuum regimes ζ C (see SI, Equations (S6)-(S9)).
Assuming spherical particles, the mean particle diameter d p,soot can be calculated using Eq.(11).
where, d p,soot is soot particle diameter, m m is initial monomer mass, M 1 is first moment, ρ soot is soot density, N A is Avogadro number, and M 0 is zeroth moment.
The PSD is assumed to follow a lognormal distribution, which allows the geometric mean volume (v g , Eq. ( 12)) and geometric standard deviation (σ g , Eq. ( 13)) to be expressed in terms of the first three moments (M 0 , M 1 and M 2 ) of the PSD.
where, v g is geometric mean volume, and σ g is geometric standard deviation.

Soot filtration model
The deposition of soot on the carbon particles of the packed bed is a critical aspect that significantly impacts the system's overall performance.To address this phenomenon, the modeling approach has been expanded to incorporate the concept of a filtration mechanism.The presented method, illustrated in Fig. 4, effectively captures the behavior Fig. 4. Soot deposition mechanisms on packed bed particle.
of soot deposition in a packed bed by considering various mechanisms, including Brownian diffusion, interception, and inertia.
The efficiency of soot capture in 0a packed bed is determined by breaking the bed down into several zones and by introduction of a zone efficiency E i (Eq.( 14)), which is based on the local porosity and pore size at any given time [87,88].
Herein, d c is the diameter of the spherical collectors constituting a packed bed and calculated as shown in Eq. (15).Furthermore, total mass m d and density ρ d takes both, carbon accumulated due to deposition reactions and entrapped soot particles.
The initial collector diameter d c,0 is determined from the clean packed bed's voidage ε bed,0 and pore diameter d pore,0 .
The E i is calculated using a single bed particle efficiency η i according to Eq. ( 17).η i is calculated using a mass balance of particles and the pore velocity as a characteristic velocity for the particles [89].
where, η Diffusion is single sphere Brownian efficiency, η Interception is single sphere interception efficiency, and η Inertial is single sphere inertial efficiency.

Inertial deposition
Inertial deposition occurs when particles in a fluid flow are unable to change their trajectory as the flow approaches the surface of a bed particle.This leads to larger particles being deposited on the bed particle through impaction.The efficiency of inertial deposition for a single soot particle can be described by the Stokes number in Eq. ( 18) and calculated using Eqs.( 19)-( 21) [90,91].
where, St is Stokes number, SCF is Stokes-Cunningham factor, u pore is interstitial or pore velocity, μ is dynamic gas viscosity, d c is collector unit diameter, and Kn is Knudsen number.

Brownian diffusion deposition
At low flow velocities, Brownian diffusion [87,92] is the main mechanism for depositing small particles on a surface.As the size of soot particles decreases, their motion is increasingly affected by Brownian motion, causing them to diffuse away from the streamlines and be deposited on a carbon particle of the packed bed.By accounting for the Peclet number and Kuwabara's hydrodynamic factor (K), the efficiency of Brownian diffusion deposition for a single sphere can be calculated using the provided Eqs. ( 22)- (25).

Interception deposition
The interception mechanism is a way for bed particles to collect particles whose diameter is larger and whose trajectory does not deviate from the streamline [87,93].This occurs when the streamline brings the particle close enough to the surface of the bed particle for it to get stuck.The efficiency of this mechanism depends on the relative size of the soot particle and the bed particle.The efficiency of interception deposition for a bed particle in a packed bed can be calculated using the Eq.(26).
The efficiency of soot capture in each zone is calculated by evaluating the impact of soot particle buildup on the microstructure of the packed bed.

Impact of temperature and hydrogen dilution
Based on an experimental assessment of the reaction conditions [30] as well as the analysis of the modified ABF2000 gas-phase mechanism, temperature and H 2 :CH 4 molar feed ratio emerged as critical process parameters during CH 4 pyrolysis.This section examines the impact of these parameters after coupling the gas-phase ABF2000 model with the deposition and soot models, hereby covering both homogeneous and heterogeneous reactions.

Gas-phase species
In a first step, an end-of-pipe analysis was performed to compare experimental results with simulations.Fig. 5 illustrates end-of-pipe CH 4 conversions, H 2 yields and by-product mole fractions for both experiments and simulations.Fig. 5 (a) and (b) shows the gas-phase product distribution for major and minor species at different temperatures and H 2 :CH 4 molar feed ratio of 1.Under these conditions, the good agreement is obtained for major species, whereas minor species shows visible deviations from experimental data.Additionally, simulations were conducted to assess the current coupled model (gas-phase, carbon deposition, and soot formation) and the gas-phase mechanism (ABF2000) against experimental data.The evaluation was conducted using the root mean square error (RMSE) method, and the results can be found in Table S1.A RMSE of above 10 % was found major species when using the ABF2000 mechanism, whereas the current model results in an average RMSE of less than 5 %.In comparison to ABF2000, the current model performed better in terms of alignment with the experimental data.Furthermore, Fig. 5(c), CH 4 conversion in the experiments for a H 2 : CH 4 molar feed ratio of 2 and a residence time of 5 s begins at 1273 K and rises rapidly to 84 % at 1573 K before essentially reaching full conversion at 1873 K. H 2 yield correlates closely with CH 4 conversion and reaches 100 % at 1873 K. Throughout all conditions evaluated, the combined yield of all the by-products remains below 5 %.Nevertheless, the mole fractions of the main by-products at these conditions, namely C 2 H 2 and C 6 H 6 , are shown in Fig. 5(d).Both the by-products reach a peak value at intermediate temperatures, however, do not survive at temperatures above 1673 K.At 1473 K and a H 2 :CH 4 feed ratio of 1, a CH 4 conversion of 76 % is obtained in the experiments, which reduces to 40 % when the ratio is increased to 4 (Fig. 5(a) and (e)).If a higher temperature of 1673 K is chosen, the effect of the H 2 :CH 4 ratio on CH 4 conversion and H 2 yield becomes negligible and the CH 4 pyrolysis is primarily governed by the reactor temperature.Simulations and end-ofpipe measurements are in good agreement with regard to CH 4 conversion and H 2 yield as well as for the by-products above 1473 K for which, however, there is a noticeable discrepancy between 1273 and 1473 K (Fig. 5

(b), (d), (f)).
As demonstrated in Fig. 5 with the end-of-pipe measurements, examining the axial profiles is equally crucial for a comprehensive understanding of CH 4 pyrolysis.The simulated axial profiles of H 2 formation for different temperatures and H 2 :CH 4 molar feed ratios are shown in Fig. 6 and compared with end-of-pipe experimental measurements.For temperatures higher than 1573 K, the H 2 formation starts already in the pre-heating zone in front of the packed bed.However, a further rapid increase in H 2 formation can be seen at the beginning of the bed for all temperatures and H 2 :CH 4 feed ratios.Apart from the case of a H 2 :CH 4 feed ratio of 4 and a temperature of 1473 K, where the simulation tends to over predict the H 2 formation as depicted in Fig. (c), the simulations conducted after coupling the ABF2000 gas phase mechanism with the deposition and soot model predict the experimental data with high accuracy.
For the conditions investigated in this study, the H 2 yield is always comparably high, while the formation of gaseous by-products occurs in minimal quantities.This is seen in the end-of-pipe experimental results in Fig. 7 and Fig. 8 for C 2 H 2 and C 6 H 6 , respectively.In line with previous studies [30,31,60], C 2 H 2 is identified as a species of particular relevance during gas phase CH 4 pyrolysis.Specifically, at a temperature of 1673 K and a H 2 :CH 4 feed ratio of 1 (Fig. 7(a)), C 2 H 2 reaches a maximum of mol-% at the inlet of the packed bed, before it is quickly consumed within the bed reaching a negligible amount at the reactor outlet.Our simulations suggest similar trends also for 1573 and 1673 K for all the H 2 :CH 4 ratios (Fig. 7(b) and (c)), whereas at 1473 K, the C 2 H 2 mole fraction continues to increase downstream of the packed bed before sharply decreasing approx.0.25 m before the reactor outlet.Furthermore, increasing the H 2 :CH 4 feed ratio decreases the peak mole fractions, which suggests a suppression of C 2 H 2 formation as gaseous byproduct.
Axial profiles of C 6 H 6 are shown in Fig. 8(a), (b) and (c).At 1673 K, the simulations suggest a peak mole fraction even prior to the packed bed.At 1573 K, the peak appears just at the entrance of bed, whereas there is no peak at 1473 K but rather a steady increase along the reactor length.However, an increase of the H 2 :CH 4 feed ratio suppresses C 6 H formation.Notably, even though the mole fractions of by-products are in very small quantities, model results agree well to the experiments underlining the accuracy.
Overall, the model accuracy remains particularly high between hot zone temperatures of 1473 K and 1873 K.For H 2 :CH 4 feed ratio of 4 and 1473 K, the model tends to over predict H 2 formation while under predicting C 2 H 2 and C 6 H 6 .It can be attributed to stronger H 2 inhibition of deposition reactions in experiments discussed in section 4.1.3.Furthermore, gaseous by-products are formed within the reactor in the pre-heating zone as well as in the packed bed, but typically decline towards negligible values by the reactor outlet that are close to the calibration limits of the analytical instruments.Note, that for such data a generous estimation of the measurement errors may sum up to ± 5 % of the absolute value, which only allows to derive some general trends for the by-product concentrations.The mole fraction trend of the most important by-product C 2 H 2 at 1473 K, that strongly differs from those predicted for 1573 K and 1673 K (Fig. 7) can be explained by a trade-off between C 2 H 2 formation in the gas-phase and C 2 H 2 consumption in the carbon particle packed bed due to carbon deposition reactions.Although C 2 H 2 is still formed in the gas-phase downstream of the packed bed since CH 4 conversion still takes place, only the reactor walls remain available for deposition; therefore, the deposition reaction rate is smaller than in the packed bed with its substantially higher surface area.As a consequence, the trade-off results in net C 2 H 2 production (reactor length 0.5 m to 0.75 m).Antes et al. [94] reported a similar effect on the deposition reaction by varying the area/volume ratio of the packing material.The further drop in the molar fraction of C 2 H 2 that is observed from a reactor position of 0.75 m onwards occurs due to reverse reactions in the gasphase [72].Analogous to C 2 H 2 , also C 2 H 4 forms prior to the packed bed.Since according to the gas-phase reaction sequence C 2 H 4 forms prior to C 2 H 2 (Fig. 3), the peak concentration of C 2 H 4 is observed earlier along the reactor axis, as illustrated in Fig. S2(a), (b) and (c).The peak concentration of C 6 H 6 just in front of the packed bed can be explained by the trade-off between slow C 6 H 6 formation in the gas-phase and a quick consumption due to deposition.At 1473 K, C 6 H 6 deposition is slower than higher temperatures.This leads to a steady increase in C 6 H 6 along the reactor axis at 1473 K. Furthermore, increasing the H 2 :CH 4 ratio  suppresses peak mole fraction of C 6 H 6 (Fig. 8) and also of C 2 H 2 (Fig. 7).This can be attributed to hydrogen inhibition phenomena.Therefore, temperature and H 2 :CH 4 feed ratio are the critical process parameters to achieve high CH 4 conversion and to simultaneously control by-product formation.Furthermore, the availability of surface area and the fact that reverse reactions can take place also influence gas-phase species.

Soot formation
The exploration of temperature and H 2 :CH 4 feed ratio effects on the primary formation of soot is presented here, pivoting focus on simulation results due to the typically low quantity of soot observed during the experimental measurement campaign.
Fig. 9 demonstrates the influence of H 2 :CH 4 ratio and temperature on the particle number concentration and clearly shows that soot formation ahead of the bed across all temperatures and feed molar ratios takes place.Rising temperatures lead to an earlier and accelerated soot formation primarily due to the formation of pyrene dimers, which are favored at higher temperatures (Fig. 9(a)).A peak in the soot particle concentration emerges when the pyrene dimer formation rate aligns with the coagulation rate, and this peak becomes more prominent with increasing temperature.Inside the bed, the particle number concentration decreases sharply at elevated temperatures and H 2 :CH 4 ratios, attributed to the increased coagulation rate due to the limited space for soot particles, resulting in more collisions (Fig. 9(b)).Moving to the post-hot zone downstream the packed bed, the collision rate decreases, causing a slower reduction in soot particle concentration.Towards the end of the bed, the soot particle number concentration becomes relatively stable, primarily due to the formation of more pyrene dimers via reversible reactions in the gas-phase at lower temperatures, thereby escalating soot nucleation rates (Fig. 9(c)).This effect is particularly significant at 1473 K, leading to a surge in particle numbers, a gradual decrease, and ultimately a consistent soot particle number towards the bed's end.On the other hand, the H 2 :CH 4 ratio exerts minimal influence on the soot particle number concentration within the reactor at 1573 K and 1673 K (Fig. 9).At 1473 K, however, gas-phase chemistry takes the forefront, significantly driving soot formation.At this temperature, an increased H 2 :CH 4 ratio results in a slower growth of the particle number concentration in the bed due to the gradual formation of the pyrene dimer that competes with the coagulation rate.In the post-hot zone, the particle number concentration diminishes as the H 2 :CH 4 ratio increases due to reversible gas-phase reactions.This shift is also marked by a rightward movement of the peak of soot particle number, indicating a slower soot nucleation process (Fig. 9(c and temperature on the soot primary particle diameter along the reactor axis.According to the simulation data in Fig. 10(a), the highest primary particle diameter of 248 nm is found at an H 2 :CH 4 ratio of 2 at 1673 K, whereas the smallest particle diameter of 21 nm is predicted at an H 2 : CH 4 ratio of 4 at 1473 K (Fig. 10(c)).The axial profiles of the soot primary particle diameter reveal a gradual initial growth, followed by a more dramatic surge towards the reactor outlet.This sharp increase in primary particle diameter can be explained by an increase in collision rate at higher temperatures.The slow yet continuous rise of the nucleation rate, which diminishes the collision effect with an increasing H 2 : CH 4 ratio, contributes to a reduction of the primary soot particle diameter.
In summary, the exploration of the interplay between temperatures, H 2 :CH 4 ratio, and their effects on soot formation provides several information.In particular, an increasing temperature results in a decrease of the soot particle number concentration and a corresponding increase of the soot particle diameter.Conversely, an increasing H 2 :CH 4 ratio correlates with a rising particle number concentration and a decreasing primary particle diameter.These trends provide a deeper understanding of the mechanisms involved in soot formation, emphasizing the key roles temperature and H 2 :CH 4 ratio play in influencing the particle number concentration and primary particle diameter.

Deposition
Similar to soot formation, the quantification of carbon depositions is only possible after finishing an experiment and in addition, the experimental setup design does not allow to gain data with spatial resolution.Therefore, the present section focuses on the quantitative analysis of deposited carbon by means of simulations.Fig. 11 shows the impact of temperature and H 2 :CH 4 feed ratio on axial carbon deposition profiles along the reactor length.Deposition begins on the reactor wall already before the reaction gases reach the carbon particles packed bed and raises rapidly upon entering the bed.Herein, the temperature exerts a significant influence on both the deposition profile and the quantity of deposition.At 1673 K and 1573 K most of the deposition takes place at the beginning of the bed, regardless of the H 2 :CH 4 feed ratio (Fig. 11(a),  (b), (c)), whereas at 1473 K a more uniform deposition across the packed bed is achieved.A total of 0.91 g carbon deposition is estimated by simulations at 1473 K and a H 2 :CH 4 feed ratio of 2, which increases to 1.21 g at 1573 K and to 1.23 g at 1673 K.Although the amount of CH 4 in the feed is lower by default with an increase of the H 2 :CH 4 feed ratio from 2 to 4 at 1473 K, decreases the carbon deposition by around 10 % (in relation to the respective amount of CH 4 in the feed).It is therefore important to simultaneously compare the effect of the H 2 :CH 4 feed ratio on both CH 4 conversion and the carbon deposition amount.At 1673 and 1573 K, CH 4 conversion is only slightly reduced if the H 2 content in the feed increases, therefore the deposition reduction is related to the lower carbon level in the feed.On the other hand, increasing the H 2 :CH 4 feed ratio from 1 to 4 at 1473 K reduces CH 4 conversion from 74 % to 59 %, which goes along with a reduction of the deposited carbon amount from 1.48 g to 0.46 g.
Since there are two surface mechanisms considered in the current study, i.e. soot formation and carbon deposition, it is important to compare solid carbon formation estimates from different mechanisms.For a temperature of 1673 K and a H 2 :CH 4 feed ratio of 2, solid carbon formed due to soot is 6.6 mg compared to 1.23 g due to surface  deposition.Carbon formed due to soot remains below 20 mg for the process conditions considered in this study.
Overall, deposition is the dominant mechanism resulting in solid carbon formation compared to soot formation.This can be explained by the large surface area that is available for deposition, whereas only limited space and residence time are available for gas-phase reactions resulting in PAHs and then soot particles formation.Since gas-phase deposition precursors (C 2 H 2 , C 6 H 6 , C 2 H 4 , and C 2 H 6 ) are also formed in the pre-heating zone, deposition starts already on the pre-heating zone wall (Fig. 11).However, the rapid increase in the deposition at the entrance of the bed can be attributed to a much larger surface area in the packed bed compared to the empty tube reactor walls.
Furthermore, the temperature exerts a profound influence on both, the deposition profile and the deposition quantity.At 1473 K deposition occurs uniformly throughout the bed.This uniform deposition distribution is desirable, as excessive deposition at the beginning of the reactor bed can lead to reactor clogging.Nevertheless, it comes at the cost of substantially lower CH 4 conversion at 1473 K compared to the higher temperatures.The H 2 :CH 4 feed ratio is a vital reaction control parameter for temperatures below 1573 K.At 1473 K CH 4 conversion is reduced by 15 % when H 2 :CH 4 feed ratio is increased from 1 to 4, which can be attributed to H 2 inhibition that is less pronounced for higher temperatures [70].

Impact of residence time
The impact of residence time (τ) in the hot zone on the CH 4 pyrolysis process is studied in this section.Simulations are performed to analyze the behavior of gas-phase species, soot formation, and deposition.

Gas-phase species
Fig. 12 illustrates the influence of τ on H 2 production at 1473, 1573 and 1673 K.The impact of τ on H 2 formation is particularly pronounced at 1473 K, where increasing τ from 1 s to 3 s results in a significant rise in H 2 production (Fig. 12 (a)).In contrast, the subsequent increase in H 2 formation is relatively small when τ is increased from 5 s to 7 s.This trend persists as the temperature rises to 1573 K and 1673 K, as shown in Fig. 12(b) and (c), respectively.
Axial profiles of the main gas-phase by-product C 2 H 2 are shown in Fig. 13.At 1473 K, a peak of C 2 H 2 is obtained at the entrance of the bed for τ of 5 s and 7 s, whereas for 1 s and 3 s a steady increase is observed (Fig. 13(a)).At a temperature of 1573 K, the axial profiles of H 2 formation exhibit distinct characteristics.In the case of 1 s residence time, an increase in H 2 formation is observed along the length of the bed.However, for residence times of 3 s, 5 s, and 7 s, the profiles display peaks at the beginning of the bed, as illustrated in Fig. 13(b).A further increase in temperature to 1673 K leads to nearly similar C 2 H 2 profiles for all residence times (Fig. 13(c)).Notably, considerably higher C 2 H 2 levels form at a residence time of 1 s compared to 7 s.This trend remains true for all the temperatures presented.
Overall, residence time has significant impact on the CH 4 pyrolysis product distribution.The impact of τ on H 2 production is particularly pronounced at a temperature of 1473 K, where H 2 production decreases sharply when τ is below 3 s.As τ decreases, C 2 H 2 share in the byproducts increases, and the highest C 2 H 2 formation occurs at τ of 1 s.When τ is shorter, gas-phase C-C coupling reactions are limited in their progression, leading to the formation of only light hydrocarbons.Additionally, the contact time between the gas-phase and the particles in the packed bed is minimal, resulting in the escape of C 2 H 2 and unreacted CH 4 , as depicted in Fig. 13.It is consistent with previous studies [95,96] which have reported increased selectivity towards C 2 H 2 and other C hydrocarbons for residence times below 1 s.

Soot formation
It is consensus that the formation of soot strongly depends on the generation of PAHs in the gas-phase [35,49,57].A longer τ provides opportunity for gas-phase reactions to advance towards the production of PAHs, consequently resulting in elevated soot formation and particle growth.These findings emphasize the crucial role of τ in shaping the intricate relationship between PAHs and soot formation.Consequently, longer residence time provides more time for the growth and accumulation of PAHs, leading to increased soot formation.Fig. 14 illustrates the influence of τ and temperature on the primary particle diamter.An increase in temperature promotes the formation of pyrene dimers, which in turn leads to a higher primary particle diameter at a residence time of 5 s (c.f.Fig. 10).While at 1473 K (Fig. 14(a)), the residence time barely influences the primary particle diameter, the effect becomes more pronounced with increasing temperature, as depicted in Fig. 14(b) for K and intensifies even further at 1673 K, as shown in Fig. 14(c).At K and a τ of 1 s, the primary particle diameter remains below 50 nm; it more than doubles at 3 s and finally primary particle diameters of nm and 205 nm are obtained at 5 s and 7 s, respectively.It is noteworthy that the growth of primary particle diameter slows down after a residence time of 5 s.These findings underscore the intricate interplay between residence time, temperature, and primary particle diameter, providing valuable insights into the behavior of the reaction system.

Deposition
This section analyzes the impact of τ and temperature on the deposition.As depicted in Fig. 15 Overall, the residence time has a significant influence on the deposition of carbon.The increasing carbon deposition at low residence times results in very high changes in bed-voidage with operating time, e.g. as illustrated in Fig. 16(a) for τ = 1 s, that can ultimately cause clogging of the reactor.Therefore, it is crucial to obtain reliable simulation data that predict the operation time before the reactor has to be stopped due to clogging.The simulations conducted in this study using a comprehensive reactor model show good agreement with the end-of-pipe gas-phase measurements, highlighting its reliability and robustness.The RMSE analysis demonstrates good agreement between experiments and simulations.Additionally, the model presented is highly efficient in terms of computation; it can be executed on a single core in just a few minutes, thereby enabling rapid analysis of both the chemical kinetics and the system as a whole.This combination of high accuracy and swift processing significantly enhances the model's utility in practical applications and make it particularly attractive for upscaling.The evaluated model can be utilized to obtain axially resolved data of gas-phase species, both major and minor, as well as carbon deposition.This allows for the assessment and optimization of H 2 yield and CH 4 conversion while minimizing by-products formation.Avoiding the formatting of byproducts is crucial because their presence in the effluent product gas stream would necessitate additional downstream separation processes.Furthermore, the model's capability to provide axially resolved profiles for deposited carbon and bed-voidage enables the implementation of preventive measures to avoid clogging in the reactor.However, further enhancements are required to improve the model's accuracy specifically within the low temperature from 1273 to 1473 K and high H 2 :CH 4 feed ratio.To achieve this, it is crucial to move beyond the global kinetic approach employed in this work and incorporate a microkinetic description of heterogeneous deposition reactions [97,98].Such advancements in model development can contribute significantly to the field, paving the way for more efficient and optimized CH 4 pyrolysis processes in industrial applications.

Conclusions
A packed bed reactor model was expanded to incorporate soot formation via the method of moment model.Furthermore, a detailed gasphase mechanism describing the pyrolysis of CH 4 was coupled with a newly developed deposition mechanism.
Numerical simulations, which integrate both homogeneous and heterogeneous chemistry, shows a good agreement with packed bed reactor experiments at temperatures exceeding 1473 K. Key process parameters of CH 4 pyrolysis, namely temperature, H 2 :CH 4 ratio, and residence time, are highlighted.Notably, temperatures above 1673 K result in over 94 % CH 4 conversion and nearly 100 % H 2 yield.With rising temperature, the model predicts an increasing primary particle diameter of soot, but a decreasing particle number concentration.Furthermore, the residence time plays a crucial role, particularly at higher residence time, which favors the formation of pyrene dimers in the gas phase and consequently results in an increasing soot particle size.Among the carbon deposition and soot formation processes investigated, deposition was found to be the dominant phenomenon.Temperature notably affects axial deposition profiles; at 1473 K, deposition is uniformly distributed across the bed, whereas at elevated temperatures, such as 1573 K and 1673 K, significant deposition primarily occurs at the bed entrance.Additionally, shorter residence times, correlated with higher inlet mass flow rates, lead to increased deposition at the bed entrance, potentially reducing bed-voidage significantly and leading to reactor clogging.The H 2 :CH 4 molar feed ratio's influence on CH 4 conversion become particularly evident at temperatures below 1573 K. Above this temperature, the effect of temperature supersedes the inhibition effect of hydrogen.Therefore, controlling the H 2 content in the feed, along with temperature and residence time, can be utilized to regulate the reaction and mitigate carbon deposition.
In conclusion, the simulations presented herein that rely on a novel model integrating gas-phase, soot formation, and carbon deposition, offer a comprehensive understanding of CH 4 pyrolysis.The model does not only provide insights that are of academic relevance, but it is also valuable resource for reactor scale-up and process development in this field.Beyond predicting simple end-of-pipe, the model also provides spatially resolved information on deposition and bed-voidage, which are challenging to obtain experimentally.Notably, computational costs are low and the simulations can be run on a desktop computer within reasonable time frames.Moreover, the model presented herein, for the first time, serves as a universal tool to describe a variety of processes, including flame pyrolysis, diesel particulate filters, coke formation, and chemical vapor deposition, whenever deposition is relevant.

η
Single bed particle efficiencyη Diffusion Single sphere Brownian efficiencyη Inertial Single sphere inertial efficiencyη Interception Single sphere interception efficiency -Θ Dimensionless residence timeλ Gas mean free path m μ Dynamic gas viscosity kg m − 1 s − 1 ξandζ Dimensionless coagulation coefficientsρ Gas phase density kg m − 3 ρ soot Soot density kg m − 3 σ g geometric standard deviationχ CsootH sites per unit surface area of the particles site m − 2 ω Rate of gas phase reaction mol m − 3 s − 1

Fig. 7 .
Fig. 7. Axial profiles of main gas phase by-product C 2 H 2 along with experimental comparison; (a) H 2 :CH 4 molar feed ratio of 1, (b) ratio of 2, (c) ratio of 4. Lines represent simulation results while points represent end-of-pipe experimental measurements.

Fig. 6 .
Fig. 6.Elucidation of main product H 2 along reactor length for different temperatures along with experimental comparison; (a) H 2 :CH 4 molar feed ratio of 1, (b) ratio of 2, (c) ratio of 4. Lines represent simulation results while markers represent end-of-pipe experimental measurements.

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.Mokashi et al.

Fig. 14 .Fig. 15 .
Fig. 14.Primary particle diameter as a function of the reactor length for different residence times, while keeping a H 2 :CH 4 feed ratio 2 and a p = 1 bar; (a) 1473 K, (b) 1573 K, and (c) 1673 K.
is coupled with the deposition and soot model.By integrating

Table 1
Experimental conditions during reactor operation.

Table 3
Kinetic parameters for deposition reactions with H 2 inhibition function.