From Detailed Description of Chemical Reacting Carbon Particles to Subgrid Models for CFD

Résumé — De la description détaillée des particules de carbone chimiquement réactives aux modèles de sous-maille pour la CFD — Cette étude est consacrée au développement et à la validation d’un sous-modèle pour l’oxydation partielle d’une particule de charbon sphérique se déplaçant dans une atmosphère air / vapeur. Le diamètre de la particule est de 2 mm. La particule de charbon est repré-sentée par du carbone non poreux exempt d’humidité et de cendres, alors que la qualité du charbon est établie en utilisant des expressions du taux de réaction semi-globale extraites de la littérature. Le sous-modèle inclut six espèces chimiques gazeuses (O 2 , CO 2 , CO, H 2 O , H 2 , N 2 ). Trois réactions hétérogènes sont utilisées, ainsi que deux réactions semi-globales homogènes, à savoir l’oxydation du monoxyde de carbone et la réaction du gaz Abstract — From Detailed Description of Chemical Reacting Carbon Particles to Subgrid Models for CFD — This work is devoted to the development and validation of a sub-model for the partial oxidation of a spherical char particle moving in an air / steam atmosphere. The particle diameter is 2 mm. The coal particle is represented by moisture- and ash-free nonporous carbon while the coal rank is implemented using semi-global reaction rate expressions taken from the literature. The submodel includes six gaseous chemical species (O 2 , CO 2 , CO, H 2 O, H 2 , N 2 ). Three heterogeneous reactions are employed, along with two homogeneous semi-global reactions, namely carbon monoxide oxidation and the water-gas-shift reaction. The distinguishing feature of the subgrid model is that it takes into account the inﬂuence of homogeneous reactions on integral characteristics such as carbon combustion rates and particle temperature. The sub-model was validated by comparing its results with a comprehensive CFD-based model resolving the issues of bulk ﬂow and boundary layer around the particle. In this model, the Navier-Stokes equations coupled with the energy and species conservation equations were used to solve the problem by means of the pseudo-steady state approach. At the surface of the particle, the balance of mass, energy and species concentration was applied including the e ﬀ ect of the Stefan ﬂow and heat loss due to radiation at the surface of the particle. Good agreement was achieved between the sub-model and the CFD-based model. Additionally, the CFD-based model was veriﬁed against experimental data published in the literature (Makino et al. (2003) Combust. Flame 132, 743-753). Good agreement was achieved between numerically predicted and experimentally obtained data for input conditions corresponding to the kinetically controlled regime. The maximal discrepancy (10%) between the experiments and the numerical results was observed in the di ﬀ usion-controlled regime. Finally, we discuss the inﬂuence of the Reynolds number, the ambient O 2 mass fraction and the ambient temperature on the char particle behaviour.


INTRODUCTION
Nowadays, due to the continuous increase in CO 2 emissions around the world, traditional power generation using coal as a primary fuel can be considered a waste of carbon. Coal is and will continue to be basically considered as a primary chemical feedstock for the production of gasoline, fertilizers or other chemicals using so-called "coal gasification". For a review of gasification technology development, we refer to the book [2]. Along with coal gasification one other possible way to reduce CO 2 emissions is to use the so-called "oxycombustion" of coal with flue gas recirculation and carbon sequestration. For a review of technological development, we refer to the works [3,4].
In the design of novel combustors or gasifiers working on solid carbonaceous fuels (particles), the important issue is the adequate prediction of the basic characteristics of such devices. Due to the complexity of physical and chemical processes inside gasifiers or combustors, experimental studies are not always capable of characterizing the basic features of all related phenomena. Therefore, computer simulation models such as Euler-Euler or Euler-Lagrange models have become well-established tools for understanding and optimizing fluid-particle flows in combustors and gasifiers. It should be noted that all these models use Computational Fluid Dynamics (CFD) equations and algorithms. For instance, a number of works have been published recently on the numerical modeling of pilot-scale gasifiers, e.g. [5][6][7][8][9], and oxy-fuel combustors, e.g. the review [10]. An analysis of these works shows that the distinguishing feature of all CFD-based simulations of combustors and gasifiers is the use of so-called computational sub-models describing particle-gas interactions on the microscale level, e.g. the burning rate of a particle and the particle temperature.
It should be noted that in spite of significant progress in the development of macroscale models for particulate flows and their numerical implementation in many commercial codes (ANSYS-Fluent, -CFX, e.g. see the work [11]), and open-source codes (openFOAM, e.g. see the work [12]), the sub-models which are used in the macroscale simulations, correspond to the models developed in the early 1970s, see the works [13,14]. One of the first subgrid models introduced by Baum and Street [13] and Smith [14] were based on the so-called "circuit analog" where kinetically controlled and diffusion-controlled reaction rates represent fictitious electric resistance placed sequentially in one electrical circuit. This class of models, which is sometimes called a kinetic/diffusion rate model, is based on the assumption that the heterogeneous reactions occur at the particle surface. Until now the well-known Baum and Street model, which is used for example as a default submodel in the commercial CFD-software Fluent, has referred to so-called "surface reaction models". These models were developed and are still used for modeling coal particle combustion taking into account the basic carbon-oxygen reaction, C + O 2 . Furthermore, the mass fractions of gaseous species at the particle surface are not taken into account in this model and gasification reactions are largely not considered. Moreover, the influence of particle velocity on the particle combustion rate is not well introduced. It should be noted that one of the first submodels for a moving chemically reacting char particle was developed by Tu et al. [15], Parker and Hottel [16], who took into the acount the Reynolds number in their mathematical description of the burning rates of char particles. However, these findings were not directly utilized in kinetic/diffusion-based submodels.
Until now, however, in spite of their many disadvantages, kinetic/diffusion-based models tuned using additional coefficients have showed surprisingly good performances during numerical simulations of pulverized coal gasification in entrained-flow gasifiers in respect to thier agreement with experimental data, e.g. see the works [8] and [9,17], where the commercial software Fluent6. 3. and ANSYS-Fluent12 were utilized, respectively. In the case of work [17], the kinetic/diffusion rate model was modified using an additional term characterising a moving flame front. In work [9], where a kinetic/diffusion model was used, the values of the pre-exponential factor and the activation energy for some chemical reactions were "modified slightly in accordance with the comparison between numerical simulations and experimental data" (citation).
Sijercic and Hanjalic [6] adopted the kinetic/diffusion rate model for modelling pulverized coal gasification taking into account the so-called gasification reactions C + CO 2 and C + H 2 O and the influence of particle velocity on the oxidation rate of a char particle. However, the surface concentrations of O 2 , CO 2 and H 2 O were eliminated from their considerations when calculating carbon consumption rates. An advanced sub-model was developed by Hayhurst [18], who reconsidered simple models for the burning of a porous coal char particle taking into account the influence of convection on the interfacial mass transfer in one-film and two-film model approaches. However, the model was not validated against experimental or CFD-based numerical simulations. For recent reviews of the common char oxidation sub-models, we refer to the works [10,19]. According to these reviews, the correct prediction of the particle burning rate and the particle temperature is an essential part of successfully modelling gasifiers/combustors. Finally, it should be emphasized that the development of accurate submodels for chemically reacting char particles depends on the understanding of interfacial phenomena occurring near and on the particle surface. This understanding can be achieved by carrying out experiments and detailed numerical simulations, which have to be validated against experiments.
Recently, parallel to the advanced experimental studies of char particle combustion, e.g. see [20,21], CFD-based particle-resolved numerical modelling of such processes has become a standard tool for understanding and predicting coal particle behavior in a hot gaseous environment. In particular, with significant progress in computational methods and computational hardware, a rapid increase can be detected in publications devoted to CFD-based numerical simulations a chemically reacting coal particles, e.g. see [22][23][24][25][26]. For instance, Lee et al. [26] carried out onedimensional fully transient numerical simulations of the chemically reacting flow field on and around the surface of an isolated carbon particle. Computations were performed for a spherically symmetric system. It was shown that for small particles (< 200 μm) transient gasification rates can deviate greatly from those predicted under the Pseudo-Steady-State approach (PSS). However, it should be noted that the authors used the so-called isobaric flow assumption, where the convection effect was represented by the Stefan flow only.
More advanced studies about the influence of convection on the oxidation of carbon particles have been performed recently. In particular, Blake [22] presented numerical simulations for the steady combustion of a spherical carbon particle in the slow viscous flow of an oxidizing ambient. In contrast to a work by Lee et al. [26], Blake [22] coupled the ambient convection, characterized by a small Reynolds number, with a strong radial convection, associated with the Stefan flow due to the particle mass loss. He found out, that in general, ambient convection increases the particle mass loss rate but has no effect in the kinetic control limit. The next significant contribution to the understanding of char particle behavior in a mixed convective environment came from Higuera [23]. In particular, he carried out a comprehensive CFD-based numerical study (in the framework of a steady-state approach coupled with a simple chemistry) on the influences of the size and velocity of the coal char particle, the temperature and the gas composition on the burning rate, the particle temperature and the extinction of the flame. It was shown that the Reynolds number of the particle plays a significant role in establishing the combustion or gasification regimes. However, no systematic analysis was carried out of the influence of the inflow temperature on the behavior of the particle with a fixed Reynolds number. Furthermore, there was no information in the paper about the grid resolution and the size of the computational domain and no code validation.
In conclusion, it should be noted that recent publications devoted to the CFD-based modelling of chemically reactive coal char particles moving in a hot oxidizing atmosphere (see the works [24,25,27]), basically describe the influence of input parameters such as the particle velocity, ambient temperature or particle diameter for single cases. However, no efforts have been made to transfer new knowledge gained in these new works to existing or novel surface reaction-based submodels. Motivated by this fact, in this work, we develop a sub-model for the partial oxidation of a spherical nonporous char particle moving in an air/steam atmosphere. The model is validated against a comprehensive CFD-based model, where the Navier-Stokes equations coupled with the energy and species conservation equations were used to solve the problem by means of the pseudosteady state approach. Only the oxidation of the residual carbon, or coke, is considered. Following pioneering experiments by Gudmundsen and Smith [28] the time required to burn the fixed carbon is approximately 90 per cent of the total burning time for the coal particles. From this point of view, the char combustion/gasification sub-model is of great importance in the successful modeling of large-scale combustors or gasifiers.

PROBLEM FORMULATION
Before we proceed with the formulation of a sub-model, we next describe the computational setup used for CFD-based simulations. In particular, we consider a single spherical coal char particle with a diameter of D = 2 mm placed in a)  a stationary position in a hot oxidizing environment with the main gas flow passing around it. The inflow velocity is assumed to be uniform and is determined by means of the Reynolds number calculated as: where ρ ∞ and μ ∞ are the density and molecular viscosity, respectively, corresponding to the inflow temperature T in and the gas composition. Four Reynolds numbers are considered: 0, 10, 50 and 100, corresponding to the laminar flow regime. Two cases with different inflow gas compositions are considered: the first case corresponds to the so-called dry air atmosphere with 0.233 mass fraction (Y) of O 2 , 0.001 mass fraction of H 2 O and with the rest of N 2 , while the second case refers to the so-called reduced oxidation condition with For the fixed composition of the inflow gas, the inflow gas temperature T in was varied from 1 000 K to 3 000 K. All calculations were accomplished at a total pressure of 1 bar. The modelling configuration and the size of the domain are illustrated in Figure 1 and Table 1. It can be seen that the ambient gas phase consists of O 2 , CO 2 , CO, H 2 O, H 2 and N 2 .
The chemistry is modelled using semi-global homogeneous and heterogeneous reactions written as follows [29]: Heterogeneous (surface) reactions: Homogeneous (gas phase) reactions: In reaction 5, steam H 2 O plays the role of a catalyst [29,30]. It should be noted that semi-global heterogeneous and homogeneous chemical reactions are widely used to model industrial combustors or gasifiers using computational fluid dynamics software, e.g. see [31]. However, at the same time it is a well-known fact that global reaction rates are often only valid in a certain range of conditions and should be used very cautiously, see [32,33]. Finally, it should be noticed that the H 2 -oxidation reaction (H 2 + O 2 ) was not taken into account in this work. For the dry air atmosphere, it is an acceptable assumption because under low steam concentration (Y H 2 O = 0.001) the water-gas-shift reaction does not play any important role in the balance of species. For the reduced oxidation condition (Y H 2 O = 0.074) at higher ambient temperatures the oxygen concentration near the carbon particle (within flame radius) is zero, thus H 2 (which is produced the reactions 3 and 5 on the particle surface and near the particle surface, respectively) does not react with O 2 . However, at lower temperatures, when the oxygen concentration near the particle is not zero, an H 2 -oxidation reaction can occur, thought the rate of reactions 3 and 5 at lower temperatures are low. In particular, estimated simulations carried out for the reduced oxidation condition at T in = 1 200 K showed that the discrepancy inṁ C and T s predicted with and without H 2 -oxidation reaction does not exceed 4% or 3%, respectively.

CFD-BASED MODEL
To proceed with the description of the mathematical CFDbased model, several assumptions are introduced in order to solve the problem. In this work, we utilize the PSS approach, see [34], due to the fact that the consumption time of the particle is always large compared to the convective and diffusion time scales for the gas phase. The gas flow is treated as an incompressible ideal gas following the model described in [35].
Before we proceed with the mathematical formulation of a model, several assumptions are introduced: -the particle shape is spherical and the gas flow is symmetrical; -the porosity of the particle is not taken into account, thus the intraparticle diffusion is neglected. The surface reaction model is used to simulate the interaction between the reacting solid surface and the gas phase; -the particle consists of carbon only. In particular, the drying and devolatilization of the particle is not included due to the steady-state character of the model; -the radiation of the gas phase is not taken into account. It should be noted that this assumption is a rough approximation, which can lead to the overestimation of the particle-surface temperature; -the buoyancy effect is neglected.
Before we proceed with the formulation of the mathematical models, we would like to discuss some limitations, which follow from the assumptions used in this work. First, our basic assumption is that heterogeneous reactions occur on the particle surface only. Thus, the carbon consumption is only correlated to the outer surface of the particle. However, it is a well-known fact that partial oxidation of a char particle at lower ambient temperatures T in < 1 000 K (in a gasification regime) occurs inside the particle, namely in pores. To overcome this limitation a random pore model [36] can be applied. Accordingly, in this case the so-called intrinsic kinetics for all heterogeneous reaction are required. So far, however, there has been little discussions in the literature about CFD-based models describing char conversion under the influence of an ambient gas flow where intrinsic kinetic used. Basic models, e.g. the so-called shrinking reacted core model, consider only the diffusion-controlled oxidation of char [37,38] taking into account intraparticle diffusion and intrinsic reactivity under isothermal conditions. The next problem is that interplay between porosity and the gas flow should be taken into account. Recently, Witting et al. [39] showed numerically that in some cases the gas flow can penetrate the porous particle at moderate Re numbers and high porosity values. With this in mind, as a first step we assume that the heterogeneous reactions occur only on the particle surface.
In conclusion, the neglecting of drying and devolatilization is explained by the fact that subgrid models for coal conversion are basically split on three submodels, namely, drying, devolatilization and char conversion [8,17]. In order to validate a sub-model for the partial oxidation of a spherical nonporous char particle moving in an air/steam atmosphere, we use a comprehensive CFD-based model where the oxidation of the residual carbon, or coke, is considered.
Taking into account these assumptions, the complete set of governing equations takes the following form:  (3), ⊗ denotes the dyadic product of two vectors. Equation (7) depicts the species net production rate R i , where i stands for the ith species and r for the rth reaction. It is computed as the sum of the Arrhenius reaction sources over the N R reactions in which the species are involved: Here, the Arrhenius molar rate of creation/destruction R i,r is calculated as: for a homogeneous reaction, where ν i,r is the stoichiometric coefficient for reactant i in reaction r and ν i,r is the stoichiometric coefficient for product i in reaction r. η j,r and η j,r are the forward and backward rate exponents for each reactant and product species j in reaction r, respectively.

Boundary Conditions
At the surface of the particle, the balance of mass, energy and species concentration is applied including the effect of the Stefan flow and the heat loss due to radiation at the particle surface. Due to the fact that heterogeneous reactions affect the mass and energy balance at the interface, they have an important influence on the boundary conditions for the gas species and the temperature. The convective and diffusive mass fluxes of the gas-phase species at the surface are balanced by the production/destruction rates of gas-phase species caused by surface reactions (see [40]): where R i,s (kmol/m 2 s) is the production rate of species i due to the surface reaction,ṁ C is the net mass flux between the surface and the gas, the index gas refers to the gas side at the wall and n is the vector normal to the wall. Here S is the particle emissivity, which conforms to the emissivity of a black body ( S = 1). It should be noted that the heat flux into the solid in Equation (11) was neglected. This is an acceptable condition while λ s /λ gas 1. Basically, on a chemically nonreacting solid surface the fluid velocity on a solid wall is zero, which corresponds to the well-known no-slip boundary condition. However, if a heterogeneous chemical reaction occurs on the solid surface, then the velocity can be nonzero. This flow is called the Stefan flow and characterizes the net mass flux between the surface and the gas. Hence, the heterogeneous reactioninduced Stefan velocity takes the following form:

Reaction Kinetics and Transport Properties
The kinetic coefficients k r of chemical reactions (2) through (7) are calculated using the extended Arrhenius expression: where A r is the pre-exponential factor, n T is the temperature exponent and E A is the activation energy. The values for A r , n T and E A with corresponding units are given in Table 2 [29,30,[41][42][43].
Notice that the CO oxidation-reaction order (reaction 4) is not directly related to the stoichiometric coefficient of the reaction due to the global character of this reaction. Finally, we note that the reaction C + 2H 2 was not included in considerations because its rate is much lower than those of the other heterogeneous reactions.
The heat capacity of the mixture was calculated as follows: where c p,i is the specific heat capacity of the single species at constant pressure. The polynomial coefficients are taken from [44]. The values for λ, μ and D are calculated using kinetic theory. In particular, the dynamic viscosity for the single species μ i is computed as follows: Here, Ω μ,i is the viscous collision integral with T * i = T ( /k B ) i . The thermal conductivity for the particular species λ i is defined as a function of μ i and c p,i : For the calculation of μ and λ for the mixture, we use kinetic theory and the ideal gas law as follows: Here, χ is the mole fraction, Φ corresponds to μ or λ. The diffusion coefficient D i,m for the species i in the mixture m is computed taking into account the local mole fractions of the individual species of the mixture: Here, D i j is the diffusion coefficient for a binary mixture, Ω D is the diffusive collision integral, which describes the interaction between molecules and σ i j is the average collision diameter for the binary mixture. The values for σ i and /k B were taken from [45] as default values.

SUB-MODEL
Due to the multiscale character of chemically reacting particulate flows in gasifiers/combustors, very often so-called zero-dimensional (0D) heat and mass transfer models are used to calculate the dynamics of carbon consumptions rates for each particle. In this class of models, basically, no transport equations in the form of Equations (4) and (5) are solved. In spite of that so-called semi-empirical relations are used to evaluate the phenomena occurring inside each cell of the grid, e.g. see the book [46]. In particular, to calculate the heat and mass transfer between coal particles and the bulk gas, so-called semi-empirical Nu-based relations are used. Next, we enhance the well-known 0-D heat transfer model to predict the char particle behaviour in a hot gaseous environment. The distinguishing feature of this subgrid model is that it takes into account the influence of homogeneous reactions on integral characteristics such as the carbon consumption rates and the particle temperature. In particular, to take into account the influence of homogeneous reactions on the carbon consumption rates, we introduce a virtual flamesheet, which has a volume V F , near the particle where homogeneous reactions occur, see Figure 2. Next, we present a mathematical formulation of the submodel. The subgrid energy-balance equation takes the following form: The subgrid species balance equations: where the particle surface, A p , and the 'flame' volume, V F , are calculated as follows: Here, r S describes the particle radius and r F is the external flame radius. The molar concentration c can be calculated depending on the mass fraction in the following way:  Figure 2 Principal scheme of reactions zones used in subgrid model.
Thus, Equation (20) can be written as follows: Here, β is the mass transport coefficient and, c ∞ and c S are the species molar concentrations in the ambient atmosphere and at the particle surface, respectively. To simplify the model, we assumed that ρ S ≈ ρ ∞ . The heat transfer coefficient α and the mass transfer coefficient β are calculated using the Nusselt and Sherwood numbers, respectively: where λ is the thermal conductivity and D Le is the diffusion coefficient calculated from the following relation: Here, we assumed that the Lewis number Le is equal to unity, which is a frequently used assumption for modeling coal combustion using one-film and two-film models, e.g. see [29]. The Prandtl number Pr and the Schmidt number Sc are derived as the following: where ν is the kinematic viscosity, a is the thermal diffusivity and μ is the dynamic viscosity. Following the assumption that the Lewis number is unity, the Schmidt and Prandtl numbers are equal, leading to Nu = S h. The Nusselt number is calculated using modified Ranz and Marshall relation [47,48]: where Re is calculated using Equation (1).
The comparison of equations representing boundary conditions (see Eq. 9-11) with Equations (19) and (23) reveal similarity in some terms. The only difference is the introduction of a 'virtual flamesheet' consisting of homogeneous chemical reactions.
The carbon mass flow can be calculated using the surface area of the particle and interfacial species concentrations which are used in heterogeneous reactions: It should be noted that in contrast to the basic subgrid model developed for the modelling of pulverized coal gasification/combustion (e.g. see [6,13,14]), in this model, we use interfacial values of the species mass fractions in stead of bulk values in Equation (28). However, this condition requires the whole system of balance equations to be solved iteratively. A detailed description of each balance equation is given in Appendix A. We note that the values of T in used in the CFD-based model correspond to the values of T ∞ = T in utilized in the subgrid model.

NUMERICS AND VALIDATION
Commercial software [45] was adopted to solve the problem under consideration. In particular, the governing equations (2-5) were solved following an implicit finite-volume technique. For pressure-velocity coupling the SIMPLE algorithm was used [49]. The convective terms in all equations were discretized by means of the QUICK scheme [50]. The proper size of the domain and the grid resolution were chosen by calculating the drag coefficient C D and comparing it with tabulated data published in the literature. Good agreement was observed between our simulations and published data. The details are published elsewhere [25] and not repeated here. Finally, it should be noted that in order to resolve the flame sheet and the thermal and chemical species boundary layers properly we had to use grid refinement near the particle surface. In particular, Figure 3 shows the grids used in the simulations, including a zoomed view near the particle surface, where a and b are the nondimensional size of the control volumes near the particle surface scaled with the particle diameter D. In particular, in the case of diffusion and non-diffusion the grid cells near the particle surface take the following values: a = δz/D = 1.5 × 10 −3 , b = δl/D = 1.5 × 10 −2 and a = 3.3 × 10 −3 , b = 2 × 10 −2 , respectively.
The under-relaxation factors for T and Y i variables were set at 0.7 due to the strong coupling between the species and the energy conservation equations. The iterations were stopped when the maximal normalized residual for all equations was less than 10 −7 . In order to reach the convergence 10 5 , iterations were necessary. CFD-based model: computational grid used in simulations for Re > 0 and zoomed view near the particle surface. The grid cell near the particle surface takes the following values: Figure 4 Schematic drawing of the experimental setup. Reproduced from [1].
The validation of the model and software against the analytic two-film model is reported in [25]. Next, we validate the software and the model we use against experiments by Makino et al. [1], where a graphite cylinder reacting with hot air was studied under laminar and turbulent flow conditions. The geometry and boundary conditions used in the validation are taken from the benchmark experiment, see Figure 4. In particular, to repeat the benchmark experiment Carbon consumption rates in dependence on the rod surface temperature predicted numerically and experimentally [1] at T ∞ = 320 K and a = 3 300 s −1 corresponding to Re =  Figure 4. In particular, a 2D rod with a diameter D = 5 mm is placed (at coordinates 30D and 40D) in the domain with a length of 130D and a height of 80D. The ambient airflow temperature T ∞ is set to 320 K. The rod surface temperature T s is varied between 1 200-2 500 K. In experiments by Makino et al. [1], the carbon combustion rate was measured in the forwardstagnation region, see Figure 4. The velocity gradient a u (stretch rate, for details see the work [1]) in the forward stagnation point was set to 3 300 s −1 corresponding to the range of the Reynolds number from 36 to 167 in dependence on T s . The flow was laminar. It should be noted that the critical Reynolds numbers did not correspond to the well-known characteristic values for isothermal flows past a cylinder. Due to the high temperature gradients, the gas viscosity and density are changed significantly within the domain, which leads to a relaminisation effect. An analysis of results applied to combustion rates reveals relatively good agreement between numerical and experimental data, see Figure 5. In particular, as expected, nonlinear dependency can be detected between T s and carbon consumption. The non-linearity characterizes two different combustion regimes. The first regime is a so-called kinetically controlled regime governed by kinetics. This regimes is defined by a rapid increase in carbon consumption rates with an increase in T s . We note that for this regime very good agreement was achieved between the numerical results and experiments. The second regime is characterized by combustion rates approaching constant values. This regimes is called a diffusion-limited regime and is governed by convection-diffusion processes. For this a) b) Figure 6 a) Contour plots of the temperature, b) CO 2 mass fraction, predicted numerically for laminar flow regime for a = 3 300 s −1 .
Here, T ∞ = 320 K and T s = 2 000 K. regime, the disagreement between experimental and numerical data reaches about 10%. We explain this disagreement by the failure to take into account the porosity of the cylinder surface, which was reported in experiment [1]. In particular, our recent three-dimensional simulations of a porous char particle reacting in a stream of hot air showed that in a diffusion-limited regime CO 2 diffuses inside the pores and reacts with carbon leading to the increase in carbon consumption [51]. In conclusion, an illustration of this regime is shown in Figure 6, which depicts contour plots of the temperature and CO 2 mass fraction predicted numerically for a laminar flow regime at a u = 3 300 s −1 , T ∞ = 320 K and T s = 2 000 K. The so-called flame sheet can be detected, characterized by higher values of T and CO 2 around the rod. In particular, the CO oxidation in the boundary layer leads to the formation of a thin flame sheet where the CO 2 concentration and the temperature have their maxima. It should be noted that the basic features of the flame sheet can be described well using the two-film model, e.g. see the book [29].

RESULTS AND DISCUSSION
Below follows a description of the results validating the subgrid model developed for the partial oxidation of a char particle moving in a hot oxidizing environment. First, some a) words should be said about the key parameters characterizing particle-gas interaction. In particular, the so-called specific gasification/combustion rateṁ C (see Eq. 28) and the difference between the ambient gas temperature and the particle surface temperature ΔT = (T s − T ∞ ) are the most important parameters governing the interaction between particles and gas, including the mass and the heat transfer.
The main operating variables which are basically used as input parameters in subgrid models are the particle Reynolds number (see Eq. 1), the ambient gas temperature T in = T ∞ and the ambient species mass fractions Y i, ∞ .
To proceed, first we discuss the results obtained for the first case corresponding to the so-called dry air atmosphere with 0.233 mass fraction of O 2 and 0.001 mass fraction of H 2 O. In particular, to study the impact of the Reynolds number Re and the ambient temperature T in on the particle oxidation behavior, several sets of simulations were performed. In every set, the Reynolds number was fixed and the ambient temperature T in was varied between 1 000 K and 3 000 K in order to cover both kinetically-controlled and diffusion-controlled regimes. The results of simulations in the form of integral characteristics and spatial distributions obtained for the dry air case are shown in Figure 7 and Figures 8, 9, respectively. For instance, Figure 7 depicts the carbon mass fluxṁ C and the temperature difference ΔT s predicted numerically using a CFD-based model and a subgrid model. We note that the valuesṁ C and ΔT s predicted using the CFD-based model correspond to the surface-averaged values. It can be seen that an increase in the particle velocity increases the particle mass loss rate. This statement is in good agreement with simulations carried out by Blake [22] and Higuera [23]. Theṁ C values obtained using the subgrid model are very close to the data predicted utilizing CFD-based model. This close agreement was achieved due to parametric fitting of r F , the values of which are given in Table 3 in the form r F /r S . It can be seen that an increase in the Re number increases r F . This effect can be explained physically due to the enhancement of species transport near the particle surface as Re increases. Thus the role of homogeneous reactions near the surface increases. An analysis of Figure 7b shows that, similarly to the very good agreement between the subgrid model predictions and the CFD-based model calculations regarding carbon consumption rates, the subgrid model produces a fairly accurate description of the values of ΔT s calculated using the CFD-based model. Both models reveal that the temperature difference ΔT s reaches its maxima in the interval between 1 200 K and 1 250 K. We found out that ΔT s decreases as T in increases, due to the flame detachment from the particle surface. To illustrate this effect, Figures 8 and 9 depict a zoomed view of the temperature contour plots calculated for the different ambient temperatures at constant Re equal to 10 and 100 for a particle size of D = 2 mm, respectively. It can be seen that in dependence on T in three basic regimes can be generally identified. The definition of the regimes is based on the "visual" analysis of contour plots, e.g. distribution of the temperature and CO 2 around the particle. In particular, it can be seen that at the highest values of T in (T in > 2 000 K) (see short comment (1) on the impact of Re on the regime change) the so-called envelope flame exists around a moving spherical particle, see Figures 8e and 9e. The flame shape covers the particle and is elongated in the direction of the flow. This well-known effect has been illustrated numerically and experimentally before by many authors. To link in with these effects, we refere to the works of Higuera [23] and Raghavan et al. [52], respectively.
A decrease in the ambient temperature leads to the transition from the envelope-flame regime to the so called surfaceattached flame, see seen that the CO 2 mass fraction is at a maximum at the particle surface. This effect is explained by the fact that CO oxidation occurs very close to the particle surface. At the same time, an analysis of a zoomed view of the temperature contour plots shows that the temperature does not reach its maximum value on the particle surface. Instead, a maximum T is located very close to the particle surface. This effect is attributed to the heterogeneous endothermic reaction 2 which leads to a decrease in T on the particle surface. It should be noted that in the front part of the particle surface the temperature is higher in comparison to the temperature in the rear part of the particle. This effect is related to the higher values of CO 2 mass fraction in the rear part of the particle. This disbalance in CO 2 is explained by the next regime.
A further decrease in the ambient temperature leads to the transition from the surface-attached flame regime to the so-called wake flame, see Figure 8a and Figure 9a. It can be seen that there is almost no CO oxidation in the front part of the particle surface. The flame, which can be detected at maximum CO 2 and T , is established only in the rear part of the particle (wake region). The increase in the particle Re number promotes the formation of the wake flame regime. This effect can be seen by comparing Figure 8a Figure 9a, which show a zoomed view of Y CO 2 contour plots near the carbon particle calculated for Re = 10 and Re = 100 for the ambient temperature T in = 1 200 K, respectively. Following findings by Raghavan et al. [52], the effect of the wake flame at high Re numbers is attributed to the flow separation in the recirculation zone, e.g. see Figure 10c. In particular, the extinction of the flame in the front part of the particle is attributed to the lower flow residence time in comparison to the reaction time. However, in the wake region, the recirculation flow increases the residence time for reactants. In the case of low particle Re numbers flows, on the other hand, where the recirculation does not exist, a similar wake flame regime can be observed, see Figure 10b. This effect is explained by the Stefan flow, which modifies the boundary layer thickness around the particle for low Re number flow regimes. In particular, Figure 10a shows that the isothermal flow past a sphere at Re = 10 does not have a recirculation zone. At the same time, in the case of chemically reacting particles the Stefan flow appears (see Eq. 12), changing the near-particle flow field in the downstream direction, see Figure 10b. Finally, this effect leads to the wake flame regime.
Next, we study the influence of the so-called reduced oxidation condition (which corresponds to the ambient gas composition with Y O 2 = 0.11 and Y H 2 O = 0.074) on the char particle behavior using the CFD-based model and subgrid models. The results of simulations using integral characteristics and spatial distributions are illustrated in Figure 11 and Figures 12, 13, respectively. In particular, Figure 11 depicts a comparison between the carbon mass fluxṁ C and the temperature difference ΔT s calculated using the CFDbased model and subgrid model. It can be seen that similarly to the previous dry air case the increase in the particle velocity increases the particle mass loss rate. However, due to the reduced oxygen concentration in the ambient gas the absolute values ofṁ C are lower compared to similar values predicted for the dry air atmosphere, see Figure 7. An analysis of Figure 11b shows that the ignition interval, where the temperature difference ΔT s reaches a maxima, is shifted to the higher temperatures between 1 200 K and 1 400 K (depending on Re), which is logical due to the reduced ambient concentration of O 2 . Overall the increase inṁ C and ΔT s in dependence on T in is not as steep as the curves shown in Figure 7.  forṁ C obtained using the subgrid model are very close to the data predicted utilizing a CFD-based model. This close agreement was achieved due to parametric fitting r F , the values of which are given in Table 3  Contour plots of temperature (left) and CO 2 mass fraction (right) near to carbon particle with the D = 2 mm, Re = 100 and a) T in = 1 200 K, b) T in = 1 400 K, c) T in = 1 600 K, d) T in = 2 000 K, e) T in = 2 600 K. The ambient mass fractions of O 2 and H 2 O are 0.11 and 0.074, respectively. regimes in dependence on the ambient temperature and Re numbers.
To demonstrate the relevance of homogeneous reactions in the accurate prediction of the gasification/combustion rates for a single char particle moving in a hot environment, we introduce Figure 14, which repeats Figure 11 to some extent. However, the distinguishing feature of this figure is additional thin curves representing the results of the subgrid model, where the thickness of the virtual zone for homogeneous reactions was set to zero. It can be seen that failing to take into consideration homogeneous reactions in a subgrid model leads to the overestimation of carbon consumption rates and underestimation of the particle temperature. This is quite an interesting result considering that mainstream kinetic-diffusion-based models, e.g. see the works [13,14], do not take this effect into account. To compare the performances of the subgrid model, we have developed with a kinetic-diffusion-based model, in Figure 15 we plot the carbon mass flux calculated for a particle with a diameter of 2 mm using different models. In particular, the kinetic-diffusion-based model is introduced in Appendix B. As expected, the kinetic-diffusion-based model shows poor performance for large Re numbers in comparison to the submodel developed in this work. This is explained by the fact that it does not take into account the convection effects or consider the interfacial species mass fraction when calculating the carbon consumption rate.

CONCLUSIONS
In this work, we developed and validated a sub-model for the partial oxidation of a spherical char particle with a diameter of 2 mm moving in an air/steam atmosphere. The submodel includes six gaseous chemical species (O 2 , CO 2 , 1 000 1 500 2 000 2 500 3 000 T in (K) Carbon mass fluxṁ C calculated using different models for a char particle with a diameter of 2 mm at Y O 2 ,∞ = 0.11 and Y H 2 O,∞ = 0.074.
CO, H 2 O, H 2 , N 2 ). Three semi-global heterogeneous reactions are employed, along with two semi-global homogeneous reactions, namely carbon monoxide oxidation and the water-gas shift reaction. The subgrid model takes into account the influence of homogeneous reactions on integral characteristics such as carbon gasification/combustion rates and the temperature difference between the particle surface temperature and the ambient temperature. Failing to take into consideration homogeneous reactions leads to the overestimation of carbon consumption rates and the underestimation of the particle temperature. The submodel was validated by comparing its results with a comprehensive CFD-based model, which was verified against experimental data published in the literature. The submodel allows insights into the details of the moving char particle's behaviour without performing sophisticated and time-consuming CFD-based simulations. Additionally, this model allows us to study the relative influence of various design and operating parameters.