Stefan Flow in Char Combustion: A Critical Review of Mass Transfer and Combustion Differences Between Air-Fuel and Oxy-Fuel Conditions

Abstract

Fuel combustion is a crucial process in energy utilization. As a key bulk transport mechanism, Stefan flow significantly affects heat and mass transfer during char combustion. However, its physical nature and engineering implications have long been underestimated, and no systematic review has been conducted. This paper presents a comprehensive review of Stefan flow in char combustion, with a focus on its impact on mass transfer and combustion behavior under both air-fuel and oxy-fuel conditions. It also highlights the critical role of Stefan flow in enhancing energy conversion efficiency and optimizing carbon capture processes. The analysis reveals that Stefan flow has been widely neglected in traditional combustion models, resulting in significant errors in calculated mass transfer coefficients (up to 21% in air-fuel combustion and as high as 74% in oxy-fuel combustion). This long-overlooked deviation severely compromises the accuracy of combustion efficiency predictions and model reliability. In oxy-fuel combustion, the gasification reaction (C + CO₂ = 2CO) induces a much stronger outward Stefan flow, reducing CO₂ transport by up to 74%, weakening local CO₂ enrichment, and substantially increasing the energy cost of carbon capture. In contrast, the oxidation reaction (2C + O₂ = 2CO) results in only an 18% reduction in O₂ transport. Stefan flow hinders the inward mass transfer of O₂ and CO₂ toward the char surface and increases heat loss during combustion, resulting in reduced reaction rates and lower particle temperatures. These effects contribute to incomplete fuel conversion and diminished thermal efficiency. Simulation studies that neglect Stefan flow produce significant errors when predicting combustion characteristics, particularly under oxy-fuel conditions. The impact of Stefan flow on energy balance is more substantial in the kinetic/diffusion-controlled regime than in the diffusion-controlled regime. This review is the first to clearly identify Stefan flow as the fundamental physical mechanism responsible for the differences in combustion behavior between air-fuel and oxy-fuel environments. It addresses a key gap in current research and offers a novel theoretical framework for improving low-carbon combustion models, providing important theoretical support for efficient combustion and clean energy conversion.

1. Introduction

Stefan flow is an important transport mechanism in some physical or chemical processes, such as condensation, evaporation, adsorption, desorption, and gas–solid and gas–liquid reactions. Due to the non-equimolar diffusion of different fluid components, the bulk flow occurs at the interface, which is called Stefan flow. The mechanism in Stefan flow has been explored in many fields. In fluid physics, Stefan flow can change the force on a moving object when evaporation or condensation occurs on the surface of the moving object [1,2,3]. In particle collection, Stefan flow on the evaporation and condensation surface dominates the particle movement near the collecting surface to a certain extent, thus, it has a substantial effect on the collection efficiency of particles [4,5,6]. Stefan flow also occurs when gaseous reactants disappear or gaseous products appear in the carbon combustion [7,8]. According to the gas–solid mass transfer theory, some heterogeneous reactions in the combustion of carbon particles lead to non-equimolar fluxes of gaseous reactants and gaseous products per unit time, making Stefan flow an important transport mechanism for the diffusion of gaseous components.

The Stefan flow correlates closely to the heterogeneous reactions in char combustion. The three most prevalent heterogeneous chemical reactions that occur on the surface of the char particle when the combustion temperature is above 1000 K are as follows in Figure 1 [9,10].

$${\mathrm{C}+\mathrm{O}}_2{=\mathrm{CO}}_2$$

(1)

$${2\mathrm{C}+\mathrm{O}}_2=2\mathrm{CO}$$

(2)

$${\mathrm{C}+\mathrm{CO}}_2=2\mathrm{CO}$$

(3)

When reaction (1) occurs on the surface of the char particle as the primary reaction, the molar flow rate of inward O₂ and outward CO₂ are equal, and it is reasonable to neglect the Stefan flow mechanism [8]. Reactions (2) and (3) show that the molar flow rate of the gaseous product (that is, CO) is twice those of the gaseous reactants (that is, O₂ or CO₂), thus, Stefan flow occurs due to non-equimolar diffusion of gaseous components on the surface of the char particle under the combined action of gas diffusion and chemical reactions. The direction of the Stefan flow depends on the heterogeneous reaction stoichiometry of the gaseous components on the surface of the char particle. For chemical reactions (2) and (3), the number of moles of the gaseous products is greater than that of the gaseous reactants, which initiates an outward Stefan flow perpendicular to the surface of the char particle.

Stefan flow has significant effects on the mass and energy balance in carbon particle combustion [11]. While a large amount of the literature on carbon particles combustion has focused on the surface reaction kinetics and intra-particle diffusion, the researchers have seldom paid attention to the gas diffusion in the boundary layer around the carbon particle [12,13,14]. Generally, a simplified theoretical model is adopted to describe the mass transfer process [15], and the Stefan flow mechanism is usually neglected in some physical models [16,17,18], which results in quite a large calculation error. In the carbon char particle combustion, reaction (2) or (3) that causes non-equimolar diffusion of gaseous components occurs inevitably [19,20]. As a result, the Stefan flow mechanism must exist in char particle combustion.

Based on previous studies, the research progress on the Stefan flow mechanism in carbon combustion is reviewed comprehensively. Firstly, this study introduces the development of the Stefan flow mechanism under different theoretical models in mass transfer calculation. Next, the influence of the Stefan flow on the characteristics of char particle combustion under air-fuel and oxy-fuel conditions is summarized [21]. Finally, considering the differences in char combustion characteristics between air-fuel and oxy-fuel combustion, it is concluded that Stefan flow is an important factor causing differences in combustion characteristics between air-fuel and oxy-fuel combustion. This review is expected to help researchers to understand Stefan flow in carbon combustion easily and comprehensively.

2. Influence of Stefan Flow on Mass Transfer in Char Combustion

Char particle combustion is a complex process involving chemical reactions and multi-component diffusion, and it is difficult to determine the influence of Stefan flow on the mass transfer process. The researchers need to determine the mass transfer coefficient with and without the Stefan flow mechanism by deriving the governing differential equation for the diffusion of gaseous components. In these studies, the limitation of the simplified theoretical model leads to uncertainties in the calculation results. To accurately understand the effect of Stefan flow on mass transfer in char combustion, the research progress on mass transfer involving Stefan flow in char particle combustion is summarized in

Table 1. The summary of research progress on mass transfer involving Stefan flow in char combustion.

Relevant Studies	Models	Dominant Reactions	The Influence Factors of the Mass Transfer
Hayhurst [8]	Single-film Double-film	Single-film model: 2C + O2 = 2CO Double-film model: C + CO2 = 2CO 2CO + O2 = 2CO2 (Homogeneous reaction)	The concentration of gaseous reactants
Peterson [22]	—	2C + O2 = 2CO 2C + CO2 = 2CO	Reaction rate The concentration of gaseous reactants
Fortsch [23]	Single-film	C + O2 = CO2 2C + O2 = 2CO	Reaction rate The proportion of the oxidation reactions Gaseous concentration/composition
Scala [24]	Single-film	C + O2 = CO2 2C + O2 = 2CO	Reaction rate The proportion of the oxidation reactions Gas concentration/composition
Yu [25]	Single-film	2C + O2 = 2CO C + CO2 = 2CO	Reaction rate The number of heterogeneous reactions The ratio of the oxidation reaction to the gasification reaction Gas concentration/composition
Yu [26]	Continuous-film	2C + O2 = 2CO C + CO2 = 2CO 2CO + O2 = 2CO2 (Homogeneous reaction)	Reaction rate The number of heterogeneous reactions The ratio of the oxidation reaction to the gasification reaction Gas concentration/composition The location of the homogeneous reaction of CO

, and the calculation errors caused by neglecting the Stefan flow in different studies are summarized, as shown in Figure 2.

As shown in Table 1, several important combustion models are used to investigate the effect of Stefan flow on the mass transfer in the char combustion, including the single-film, double-film, and continuous-film models. According to mechanism assumptions, in the single-film model, it is assumed that the gaseous products of the oxidation reaction on the surface of the char particle are CO or CO₂, and then the CO formed is oxidized in the free flow. In the double-film model, it is assumed that the homogeneous reaction where CO is converted to CO₂ occurs at a flame sheet in the boundary layer of the char particle and prevents any O₂ from penetrating the particle surface, thus allowing the surface char–CO₂ gasification reaction to control the heterogeneous reaction rate, which represents a limited case. In the continuous-film model, physicochemical processes including surface oxidation, surface gasification, and homogeneous oxidation of CO are considered simultaneously. These different theoretical models can rationally reflect the physical reality of certain char combustion cases.

Hayhurst [8], Fortsch [23], and Scala [24] discussed the effect of Stefan flow on the mass transfer using the single-film and double-film model under air-fuel combustion, as shown in Table 1. Yu [25,26] proposed the mass transfer coefficient considering/neglecting Stefan flow when reactions 2C + O₂ = 2CO and C + CO₂ = 2CO occur in oxy-fuel combustion using the single-film and continuous-film models. The result shows that Stefan flow significantly affects mass transfer characteristics of the char combustion.

2.1. Binary Diffusion Model

The binary diffusion system only considers the two components that consist of the gaseous reactant and product, disregarding the impact of other gaseous components on the mass transfer coefficient. Hayhurst [8] and Paterson [22] in Table 1 analyzed the mass transfer characteristics in the boundary layer of the char particle based on the binary diffusion model. The schematic diagram of a sphere (with diameter d) of carbon burning in stagnant air or oxygen is shown in Figure 3. Reaction (1) is equimolar diffusion, while Reaction (2) is non-equimolar diffusion.

In Figure 3, the radial profiles of the mole fractions of O₂ and CO₂ are illustrated for reaction (1) and reaction (2). With reaction (1) and reaction (2) occurring, the Stefan-Maxwell equation for the inwards flux of O₂ at a radius r is:

$${\varphi }_{{\mathrm{O}}_2}=({\varphi }_{{\mathrm{O}}_2}+{\varphi }_{{\mathrm{CO}}_2})y_{{\mathrm{O}}_2}-{\displaystyle \frac{P{D}_1}{R{T}_m}}{\displaystyle \frac{d{y}_{{\mathrm{O}}_2}}{dr}}$$

(4)

$${\varphi }_{{\mathrm{O}}_2}=({\varphi }_{{\mathrm{O}}_2}+{\varphi }_{\mathrm{CO}})y_{{\mathrm{O}}_2}-{\displaystyle \frac{P{D}_2}{R{T}_m}}{\displaystyle \frac{d{y}_{{\mathrm{O}}_2}}{dr}}$$

(5)

where y denotes the mole fraction of a species, P is the system pressure, T is the system temperature, and D is a pseudo-binary diffusion coefficient for O₂ and CO₂, the rate of consumption of carbon per particle being Q₁ = −4πr² Φ_O2 and Q₂ = −2 × 4πr² Φ_O2, so:

$$Q_1=\pi dS{h}_{\mathrm{EMCD}}\left({\displaystyle \frac{P{D}_1}{R{T}_m}}\right)(y_b-y_s)$$

(6)

$$Q_2=2\pi dS{h}_{\mathrm{EMCD}}\left({\displaystyle \frac{P{D}_2}{R{T}_m}}\right)\ln \left({\displaystyle \frac{1+y_b}{1+y_s}}\right)$$

(7)

where y_s is the mole fraction of a species at the external surface of a reacting particle, y_b is the mole fraction of a species in the bulk gas. Based on Q₁ = πd²k₁·(P/RT) (y_b − y_s), and Q₂ = 2 × πd²k₂·(P/RT) (y_b − y_s), and considering Sh = dk_g/D, therefore, the correction factor considering Stefan flow for mass transfer is:

$${\displaystyle \frac{S{h}_2}{S{h}_1}}={\displaystyle \frac{1}{(y_b-y_s)}}\ln \left({\displaystyle \frac{1+y_b}{1+y_s}}\right)$$

(8)

Equation (8) shows that the mass transfer coefficient depends on the concentration difference of the gaseous reactant between the surface of the char particle and the free flow, and it will rise with the increase in the concentration difference. The results indicate that neglecting the Stefan flow results in an overestimation of the mass transfer coefficient by approximately 10–21% when the oxidation reaction 2C + O₂ = 2CO occurs under static and convective conditions in Figure 2.

Paterson [22] focused on the relationship between the calculation error of mass transfer and heterogeneous reaction rate in the presence of the Stefan flow mechanism. The finding in Table 1 shows that the chemical reaction rate is an important factor affecting the mass transfer coefficient and calculation error. For reactions 2C + O₂ = 2CO or C + CO₂ = 2CO, the calculation error of the mass transfer coefficient adopting the assumption of the equimolar diffusion reaches 17% under slow reaction conditions (the concentration of the gaseous reactants difference between the particle surface and the free flow is approximately zero). Under rapid reaction conditions (the concentration of gaseous reactants on the surface of the char particle tends to zero), the calculation error is as high as 31%. Reducing the concentration of the gaseous reactant in the fluid can reduce the calculation error of mass transfer.

The above studies demonstrate neglecting the Stefan flow results in an overestimation of the mass transfer coefficient in the calculation. Stefan flow must be considered in the binary diffusion system to reduce the significant calculation error, and the reason is that the outward Stefan flow is not conducive to the inward diffusion of gaseous reactants from the free flow to the surface of the char particle. This impact not only depends on the gas properties and the concentration of gaseous components but also correlates closely with the reaction rate on the char particle surface. The effect of Stefan flow on the mass transfer becomes less important with the reduction of reaction rate and the concentration of the gaseous reactants.

The binary diffusion system is relatively simple and not applicable in some combustion cases [27,28,29,30]. The actual char particle combustion system covers the diffusion of various gaseous components, and the interaction of gaseous components will change the diffusion coefficient and mass transfer efficiency, which means the effect of Stefan flow on mass transfer may change in a multi-component diffusion system.

2.2. Multi-Component Diffusion Model

Char combustion involves the diffusion process of various gaseous components in the boundary layer of the char particle, in which the diffusion coefficients of gaseous components differ from one another. The Stefan-Maxwell equation, which covers the diffusion rate of the different gaseous components and the surface reaction rate, can reflect properly this interaction between different gaseous components, and it is usually used to describe the gas diffusion characteristics in the boundary layer of the char particle by choosing the experimental parameters.

2.2.1. In the Absence of Homogeneous Oxidation of CO

Several researchers [23,24,25,26] derived a similar analytical expression of the mass transfer in multi-component systems using the Stefan-Maxwell equation. The assumption that no homogeneous oxidation reaction of CO occurs in the boundary layer of the char particle is adopted in these theoretical studies, which is suitable for the combustion case of small char particles less than 100 μm [31,32]. The schematic diagram of the oxidation of the char particle is displayed in Figure 4a, and the dominant heterogeneous reactions are C + O₂ = CO₂ and 2C + O₂ = 2CO. Fortsch [23] gave the mass transfer correction factor considering Stefan flow using the single-film model:

$${\displaystyle \frac{Sh}{S{h}_{\mathrm{EMCD}}}}={\displaystyle \frac{1}{1+f\cdot y_{\mathrm{b}}}}\cdot {\displaystyle \frac{f\cdot {\varphi }_{{\mathrm{O}}_2}}{1-\exp \left(\frac{f\cdot {\varphi }_{{\mathrm{O}}_2}}{{\phi }_e}\right)}}$$

(9)

where f represents the proportional factor of CO formation in reactions (1) and (2), and φ is diffusion coefficient correction factor in multi-component systems.

Equation (9) suggests that the calculation error of O₂ diffusion resulting from neglecting Stefan flow is related to the proportion of the two oxidation reactions C + O₂ = CO₂ and 2C + O₂ = 2CO as well as the reaction rate and gas concentration/composition. When only the reaction 2C + O₂ = 2CO occurs, the calculation error caused by neglecting the Stefan flow ranges from 9% to 17%. Meanwhile, the occurrence of the reaction C + O₂ = CO₂ with equimolar counter-diffusion also causes the calculation error of 7%, originating from the effect of the difference in the gas diffusion coefficients on the mass transfer efficiency.

Scala [24] considered the change of different gaseous components with the location in the boundary of the char particle based on the study of Fortsch [23] and gave the value of the correction factor:

$$\begin{array}{l}{\displaystyle \frac{Sh}{{Sh}_{\mathrm{EMCD}}}}={\varphi }_{{\mathrm{O}}_2}\cdot \{\left({\displaystyle \frac{y_{{\mathrm{N}}_2}^{\mathrm{b}}+y_{\mathrm{CO}}^{\mathrm{b}}+2f\cdot y_{{\mathrm{O}}_2}^{\mathrm{b}}}{1-2f}}\right)\left[1-\exp \left(-{\varphi }_{{\mathrm{O}}_2}\left({\displaystyle \frac{f-1}{{\phi }_{{\mathrm{CO}}_2}}}+1-2f\right)\right)\right]+\\ \left({\displaystyle \frac{1}{{\phi }_{{\mathrm{H}}_2\mathrm{O}}}}-{\displaystyle \frac{1}{{\phi }_{{\mathrm{CO}}_2}}}\right)y_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{b}}\left[{\displaystyle \frac{1-\exp \left(-{\varphi }_{{\mathrm{O}}_2}\left(f-1+\frac{1-2f}{{\phi }_{{\mathrm{H}}_2\mathrm{O}}}\right)\right)}{f-1+\frac{1-2f}{{\phi }_{{\mathrm{H}}_2\mathrm{O}}}+\frac{f}{{\phi }_{{\mathrm{CO}}_2}}}}\right]+\\ \left({\displaystyle \frac{1}{1-2f}}\right)y_{{\mathrm{CO}}_2}^{\mathrm{b}}\left[{\displaystyle \frac{1-f}{f}}-{\displaystyle \frac{(1-f)\left(1-\frac{1}{{\phi }_{{\mathrm{CO}}_2}}\right)y_{{\mathrm{H}}_2\mathrm{O}}^{\mathrm{b}}}{f-1+\frac{1-2f}{{\phi }_{{\mathrm{H}}_2\mathrm{O}}}+\frac{f}{{\phi }_{{\mathrm{CO}}_2}}}}\right]\times {\left[1-\exp \left(-{\varphi }_{{\mathrm{O}}_2}\cdot {\displaystyle \frac{f}{{\phi }_{{\mathrm{CO}}_2}}}\right)\right]}^{-1}\}\end{array}$$

(10)

The analysis results of Equation (10) show that the calculation error of the mass transfer coefficient increases with an increase in the dimensionless flow rate of O₂ (the ratio of O₂ flow rate to total mass flow rate) and the concentration of O₂ in the free flow in air-fuel combustion, and the calculation error can reach 10% in the O₂-rich conditions of the air-fuel combustion as shown in Figure 1. The mass transfer coefficient was also calculated in oxy-fuel combustion with low-oxygen content, and the calculation error of mass transfer is minimal and more than 3%.

In the above-mentioned studies, only the mass transfer characteristics with heterogeneous oxidation reactions C + O₂ = CO₂ and 2C + O₂ = 2CO was investigated in the mass transfer calculation, and the gasification reaction C + CO₂ = 2CO was not involved. However, in oxy-fuel combustion full of O₂ and CO₂, the proportion of CO₂ is 60–80% [33,34,35]. CO₂ not only acts as a diluent but also reacts with the char particle, and the gasification reaction C + CO₂ = 2CO is also one of the dominant reactions in oxy-fuel combustion. Therefore, the influence of Stefan flow on mass transfer characteristics in oxy-fuel combustion is different from that in air-fuel combustion.

Yu [25] derived the mass transfer coefficients when reactions (1) and (2) occur in oxy-fuel combustion as shown in Figure 4b. The result illustrates that the calculation error of mass transfer rises and the inward mass flow rates of CO₂ and O₂ decrease when reactions 2C + O₂ = 2CO and C + CO₂ = 2CO are considered simultaneously. This is because the more intense outward Stefan flow hinders the mass transfer of gaseous reactants to the surface of the char particle due to the increase in the overall reaction rate. Moreover, Stefan flow has a greater impact on the mass transfer in gasification reaction C + CO₂ = 2CO than that in oxidation reaction 2C + O₂ = 2CO. Under general conditions of oxy-fuel combustion, the Stefan flow caused by the gasification reaction C + CO₂ = 2CO decreases the mass flow rate of CO₂ by up to 74% as shown in Figure 1, while the oxidation reaction 2C + O₂ = 2CO is only 18% for O₂.

2.2.2. In the Presence of Homogeneous Oxidation of CO

Compared with air-fuel combustion, an atmosphere rich in O₂ and CO₂ in oxy-fuel combustion contributes to the homogeneous oxidation reaction of CO in the boundary layer of the char particle [36,37,38]. The homogeneous reaction of CO, which releases a large amount of heat, enhances the temperature and consumption rate of the char particles [39,40]. Moreover, the existence of the homogeneous oxidation reaction of CO also changes the mass transfer behaviour including the Stefan flow, resulting in a change in the distribution of gaseous components [41,42], as shown in Figure 5. The governing differential equation for the diffusion of gaseous components needs to be constructed to predict the location of the CO homogeneous flame front and then explore the influence of Stefan flow on the mass transfer characteristics in high-temperature oxy-fuel combustion.

Yu et al. took the homogeneous reaction of CO into account in their study of the effect of Stefan flow on the mass transfer process of char particle combustion and gave a schematic diagram of the mass transfer process as shown in Figure 6. The correction factor for oxygen diffusion in a stagnant gas neglecting Stefan flow and multi-component diffusion film is obtained. The results shows that the combination of the CO homogeneous flame front and the Stefan flow has a complex effect on mass transfer under the high-temperature oxy-fuel combustion, resulting in totally different effects on the transport of O₂ and CO₂ [26,43]. For oxidation reaction 2C + O₂ = 2CO, the heat released by the CO homogeneous reaction in the boundary layer increases the combustion reaction rate of the char particle, and the reinforced outward Stefan flow increases the resistance of the inward transport of O₂ to the char particle surface, hence, neglecting Stefan flow results in the large calculation error of mass transfer. In contrast, for the gasification reaction C + CO₂ = 2CO, a high concentration of CO₂ is produced by the homogeneous reaction of CO in the reaction front. When the flame front approaches the char surface, the high concentration gradient of CO₂ between the reaction front and the surface of the char particle weakens the blocking effect of Stefan flow on the inward diffusion of CO₂, which reduces the calculation error caused by neglecting Stefan flow. The result also illustrates that even if the homogeneous reaction of CO exists, the influence of Stefan flow on the transport of CO₂ is greater than that of O₂ when both the oxidation reaction 2C + O₂ = 2CO and the gasification reaction C + CO₂ = 2CO are considered at the same time, which is consistent with findings above.

2.3. Summary

Stefan flow is an indispensable transport mechanism in the combustion model of char particles, and neglecting Stefan flow will lead to the inaccurate calculation result of mass transfer. The calculation error becomes larger with the increase in the intensity of Stefan flow, and the reason is that the more intense Stefan flow increases the resistance of the inward transport of gaseous reactants from the free flow to the surface of char particles significantly. The calculation error of mass transfer caused by neglecting Stefan flow correlates to model and parametric selection including physical properties, the reaction rate, gas concentration/composition, the oxidation/gasification reactions, and homogeneous oxidation of CO. In contrast, it is concluded that the influence of Stefan flow on mass transfer in oxy-fuel combustion is more significant than that of air-fuel combustion, especially as Stefan flow caused by the gasification reaction C + CO₂ = 2CO exerts a more important impact on the mass transfer process.

Although Stefan flow plays a critical role in mass transfer during char combustion, most existing studies have focused on theoretical models, such as single-film, double-film, and continuous-film approaches, which face significant limitations in capturing real combustion behavior. These models often rely on simplified assumptions, overlooking key factors such as complex surface reaction kinetics, the coupling of multi-component diffusion, and the variation of diffusion coefficients with gas composition. For example, the single-film model assumes that CO produced at the char surface is instantly oxidized in the bulk gas, while the double-film and continuous-film models idealize CO oxidation as occurring at a fixed flame interface. In practice, the location, intensity, and interaction of these reactions with surrounding gas diffusion are far more dynamic and spatially variable. Additionally, most current studies remain theoretical, lacking experimental validation under practical combustion conditions. To improve model fidelity, future research should critically examine model assumptions and limitations, integrate model predictions with high-quality experimental data, and investigate the interaction mechanisms within multi-component diffusion systems. Developing reliable experimental validation methods and calibration strategies will be essential for improving model accuracy and better assessing the role of Stefan flow in real combustion environments.

3. Influence of Stefan Flow on Combustion Characteristics

The char combustion process involves heat and mass transfer as well as chemical reactions [44,45,46,47]. Stefan flow in char combustion acts as a convective term for heat and mass transfer and affects the combustion characteristics of the char particle significantly, including ignition, char reactivity, burnout time, and combustion performance. It is unrealistic to study the influence of Stefan flow on combustion characteristics by experiment, but the numerical simulation can be used to properly reflect the characteristics of heat and mass transfer and reaction kinetics to evaluate the influence of Stefan flow on the combustion characteristics [48,49,50].

Theoretical studies of char combustion involving Stefan flow in air-fuel and oxy-fuel combustion are summarized in

Table 2. The summary of the simulation studies of char combustion involving Stefan flow.

Relevant Studies	Models	Combustion Atmosphere	Dominant Reactions	The Mentioned Influence Factors of the Stefan Flow	Research Conclusions
Kalinchak [51]	-	O2/N2	C + O2 = CO2 2C + O2 = 2CO	The concentration of O2 Particle temperature Particle size	(a) Stefan flow reduces the mass flow rate of O2 and enhances heat removal. (b) The influence of Stefan flow on the combustion characteristics can be neglected when the combustion temperature is lower than 1000 K. (c) The intensity of Stefan flow increases as char particle temperature and the concentration of O2 rise.
Gonzalo-Tirado [52]	Single-film	O2/N2	2C + O2 = 2CO	The concentration of O2	(a) Stefan flow leads to the reduction of mass and heat transfer rate. (b) The effect of the Stefan flow is almost negligible for low O2 concentrations and it gradually becomes significant when the content of O2 is above 20%.
Vorobiev [53]	Single-film	O2/N2	C + O2 = CO2 2C + O2 = 2CO	Reaction rate Particle size/shape Particle temperature	(a) Stefan flow leads to the reduction of the reaction rate. (b) The intensity of Stefan flow depends on particle size and shape, and the combustion characteristics of small particles or particles with a high aspect ratio is most affected by Stefan flow. (c) Stefan flow has a more noticeable impact on particle temperature with an increase in combustion rate.
Maffei [54]	Single-film	O2/N2 O2/CO2	Char oxidation Char–CO2 gasification	The concentration of O2	Stefan flow has an unimportant effect on the combustion temperature and burnout time of the char particle.
Hecht [55]	Single-film	O2/CO2	5-step mechanism	Reaction rate The concentration of O2	(a) Stefan flow leads to the reduction of char consumption rate and combustion temperature. (b) With the increase in the concentration of O2 and reaction rate, the effect of Stefan flow on the combustion characteristics becomes more significant.
Ou [56]	Single-film	O2/CO2	2C + O2 = 2CO C + CO2 = 2CO	The concentration of O2 Particle size Ambient temperature	(a) Stefan flow accelerates the heat loss away from char particles so that higher content of O2 is required for the ignition of carbon particles. (b) Stefan flow leads to the reduction of reaction rate, combustion temperature, and burnout rate.
Yu [43]	Continuous-film	O2/CO2	2C + O2 = 2CO C + CO2 = 2CO 2CO + O2 = 2CO2 (Homogeneous reaction)	The concentration of O2 Particle size Ambient temperature Kinetic parameters	(a) Stefan flow leads to the reduction of combustion temperature and the increase in burnout time. (b) In the kinetic/diffusion-controlled region, the influence of Stefan flow on the combustion characteristics is the most significant.

. In these studies, different model assumptions are adopted to simulate the combustion process, and generally, the physical aspects and operating parameters in combustion models that affect the char combustion characteristics mainly involve particle size/shape, the concentration of O₂, particle temperature, and reaction rate, and ambient temperature. Hence drawing a uniform comment on the theoretical results is impossible. Next, the influence of Stefan flow on combustion characteristics in air-fuel and oxy-fuel combustion will be discussed below.

3.1. Influence of Stefan Flow on Combustion Characteristics in Air-Fuel Combustion

Kalinchak et al. [51,57,58] performed an analysis of the heat and mass transfer of a char particle involving heterogeneous reactions C + O₂ = CO₂ and 2C + O₂ = 2CO in air-fuel combustion. The study preliminarily concluded that Stefan flow significantly reduces the oxygen concentration in the boundary layer by impeding the inward diffusion of oxidizing species toward the particle surface, thereby lowering the local reaction rate. This suppression becomes particularly prominent at high temperatures (T > 2000 K) and for small particles (d < 1 mm). Numerical results indicate that Stefan flow can decrease the surface oxygen concentration by approximately 15–30% and reduce the local heat release density by up to 40%. Additionally, the outward mass flux associated with Stefan flow extracts part of the reaction heat from the particle surface, diminishing convective heat transfer to the surrounding gas, reducing the particle heating rate, and altering the surface temperature profile. In the intermediate temperature range (1600 K–2400 K), Stefan flow exacerbates mass transfer resistance and facilitates the endothermic reaction (3), leading to a reduction in heat release density and a transition into a transitional combustion zone. At higher temperatures (T > 2400 K), the process becomes predominantly diffusion-controlled.

The above-mentioned method cannot accurately reflect the interaction and competition between different heterogeneous chemical reactions on the surface of the char particle, and the combustion model can be used to describe the char combustion characteristics. In these combustion models as shown in Table 2, a modified version of the Peclet number was introduced to consider the effect of the Stefan flow in the energy balance. The formula of ${Pe}_m$ is expressed as follows:

$${Pe}_m={\displaystyle \frac{-\mathrm{q}{\mathrm{d}}_{\mathit{p}}{\sum }_i{v}_i{c}_{g,i}}{\lambda }}$$

(11)

Here, $\mathit{q}$ is the overall burning rate per unit char particle external surface area, ${\mathit{d}}_{\mathrm{p}}$ is the diameter of the char particle, $v_i$ are analogous to stoichiometric coefficients of different heterogeneous chemical reactions. $c_{g,i}$ is heat capacity for different gaseous components and $\lambda$ is the thermal conductivity of gas around the char particle. The ${Pe}_m$ represents the ratio of convection heat transfer to heat conduction. The method can reflect the Stefan flow on the combustion characteristics by introducing the enthalpy values of various gaseous components in the boundary layer of the char particle [33,50,51].

Gonzalo [52] used the single-film model to explore the influence of the Stefan flow on the combustion characteristics with oxidation reaction 2C + O₂ = 2CO. The finding shows that the influence of the Stefan flow on the combustion characteristics is negligible in low-O₂ concentration conditions. The effect of Stefan flow on heat and mass transfer and the combustion characteristics becomes more significant with the increase in the concentration of O₂ and reaction rate, and the error of heat and mass transfer caused by neglecting Stefan flow is less than 10% at 20% O₂ content. Similarly, Vorobiev et al. [53] investigated the effect of Stefan flow on the energy balance at the kinetic/diffusion-controlled region of char combustion. The effect of Stefan flow on reaction rate is small in air-fuel combustion, and neglecting Stefan flow will bring about a 4–5% difference in reaction rate for a particle size of 100 μm at 2300 K. This effect becomes more pronounced in high-oxygen environments, and neglecting Stefan flow may lead to a serious deviation of burnout time. Moreover, the impact of Stefan flow is influenced by particle shape, size, and the combustion environment. Studies have shown that small particles and non-spherical particles with a high aspect ratio, represented by the ratio of length to diameter, are more susceptible to Stefan flow limitations in heat and mass transfer. For example, at the same reaction rate, cylindrical particles require higher temperatures than ellipsoidal ones, while spherical particles are the least affected by Stefan flow. As combustion progresses, biomass particles tend to become more spherical, a process known as spheroidization, which further alters the influence of Stefan flow on their combustion behavior.

A comprehensive multi-scale single-film model of char particle combustion was used by Maffei et al. [54], which involves pyrolysis and volatilization reactions and homogeneous and heterogeneous reactions. The results in Table 2 indicate that Stefan flow has a minor effect on the combustion temperature and burnout time, and the deviation caused by neglecting Stefan flow will not exceed 100 K at the highest reactivity with the combustion temperature of 3000 K. Murphy [59] examined quantitatively the effect of Stefan flow on the temperatures of char particles, and the result shows that the temperature deviation is less than 50 K in air-fuel combustion. Based on the above-mentioned conclusion, it is concluded that the influence of Stefan flow on the combustion characteristics of the char particle is not significant in air-fuel combustion.

3.2. Influence of Stefan Flow on Combustion Characteristics in Oxy-Fuel Combustion

The influence of Stefan flow on combustion characteristics in oxy-fuel combustion is different from that in air-fuel combustion due to different combustion atmospheres [60]. Hecht et al. [55] used the single-film model involving intra-particle diffusion to predict the combustion temperature and the char consumption rate in oxy-fuel combustion. The study finds the combustion temperature and char consumption rate are significantly reduced by Stefan flow, and the prediction result neglecting Stefan flow is more inaccurate with the increase in the concentration of O₂ and reaction rate. For a 100 um char particle burning in 60% fraction of O₂, the particle temperature is overestimated by 270 K and the char consumption rate is also over-predicted by up to 17% when Stefan flow is neglected.

As shown in Table 2, Ou et al. [56] and Yu et al. [43] adopted the single-film model and continuous-film model to investigate the effect of Stefan flow on the heat balance of a char particle, respectively. The result illustrates that the outward Stefan flow hinders the diffusion of O₂ and CO₂ to the surface of the char particle and accelerates the heat loss from the char particle to the free flow, which has a negative effect on char particle combustion. In these simulation studies, for example, in a 30% O₂/70% CO₂ atmosphere, neglecting Stefan flow results in an overestimation of the surface temperature by 101–145 K (approximately 7–10%) and of the combustion rate by 18–25%. Although incorporating the Peclet correction improves the prediction to some extent, it still fails to fully capture the actual impact of Stefan flow. The conclusion also indicates that the Stefan flow mechanism is indispensable in the simulation study of char particle combustion in oxy-fuel combustion, and Stefan flow exerts a more significant effect on the char combustion in oxy-fuel combustion than that in oxy-fuel combustion based on the studies mentioned above.

The impact of Stefan flow on the char combustion characteristics correlates closely to the combustion-controlled region in oxy-fuel combustion. In the kinetic-controlled region with a relatively low combustion temperature, the char combustion rate depends on the oxidation reaction without diffusion limitation, and hence the effect of Stefan flow as the bulk flow on combustion characteristics is negligible. With the increase in the combustion temperature and char particle size, the kinetic/diffusion-controlled and diffusion-controlled regions with high temperatures are reached. Char combustion is gradually controlled by gas diffusion under these combustion conditions, and the proportion of heat flux entrained by Stefan flow is up to 45.19% in high temperatures [61]. As a result, Stefan flow becomes important and dominates the heat and mass transfer process in char combustion [62].

The combined effect of Stefan flow and CO oxidation has an important influence on the combustion process of char particles in an oxy-fuel environment. The reverse mass flux induced by Stefan flow inhibits oxygen transport into the boundary layer, while CO oxidation within this region releases additional heat, altering both the local temperature field and mass transfer structure. These coupled effects are most significant in the kinetic/diffusion-controlled region. Although their influence diminishes under diffusion-controlled region, they still lead to noticeable deviations in particle burnout rate and surface temperature distribution [43]. The reason is that Stefan flow comes from the non-equimolar diffusion generated by heterogeneous reactions on the surface of the char particle, which is affected by the co-action of the reaction kinetics and gas diffusion, and thus Stefan flow is governed by both heterogeneous reactions and gas diffusion in the kinetic/diffusion-controlled region [43,52]. In contrast, the Stefan flow in the diffusion-controlled region is mainly affected by gas diffusion, therefore, Stefan flow affects char combustion characteristics significantly in the kinetic/diffusion-controlled region.

3.3. Summary

Stefan flow plays a pivotal role in heat and mass transfer during char combustion, particularly under oxygen-enriched or CO₂-rich conditions, where its influence on combustion dynamics becomes more pronounced. The outward mass flux induced by Stefan flow hinders the inward diffusion of oxygen, while the associated convective heat loss leads to reduced reaction rates, lower particle temperatures, and extended burnout times. This influence becomes increasingly significant with stronger Stefan flow. In the kinetic/diffusion coupling region, Stefan flow has a dominant impact on the local energy balance, far exceeding its role in the diffusion-controlled region. Additionally, CO oxidation within the boundary layer introduces supplementary heat release, further modifying the thermal field and reaction environment. Numerical simulations consistently show that omitting Stefan flow from combustion models results in the overestimation of temperature and reaction rates, particularly at high temperatures and for small particle sizes, where such deviations are amplified. Accurate modeling of char combustion under oxy-fuel or CO₂-diluted conditions therefore requires explicit consideration of Stefan flow and its coupled boundary layer effects.

4. Discussion

This section focuses on the differences in the combustion characteristics of char particles between air-fuel combustion and oxy-fuel combustion. Air-fuel combustion refers to the conventional approach of burning fuel with atmospheric air, which consists of approximately 21% oxygen and 78% nitrogen. Although nitrogen is chemically inert, it absorbs heat during combustion and reduces the flame temperature, thereby lowering thermal efficiency and facilitating the formation of pollutants such as NO_x and SO_x. The combustion products are relatively complex, comprising CO₂, H₂O, and a large volume of N₂, which complicates post-combustion gas treatment. In contrast, oxy-fuel combustion utilizes pure or oxygen-enriched gases instead of air, often with recycled CO₂ introduced to moderate the elevated flame temperature. The absence of nitrogen in this environment significantly suppresses NO_x generation, enhances thermal efficiency, and simplifies carbon capture, as the resulting flue gas primarily contains CO₂ and H₂O [63,64,65,66]. From a reaction pathway perspective, air-fuel combustion (O₂/N₂) is dominated by complete oxidation reaction (1) due to the relatively low diffusional resistance of oxygen in nitrogen. However, in oxy-fuel combustion (O₂/CO₂), the high concentration of CO₂ promotes reaction (3), an endothermic process that shifts the system toward a kinetic/diffusion coupling region. The denser CO₂ atmosphere also increases mass transfer resistance compared to N₂, leading to slower diffusion of oxygen toward the char surface. Moreover, mass transfer effects become more pronounced under oxy-fuel conditions, suppressing inward O₂ diffusion and enhancing outward convective fluxes of heat and mass from the particle surface. These factors collectively influence the ignition characteristics, combustion rate, and burnout behavior of solid fuels under oxy-fuel conditions [34,67,68,69,70].

A series of studies have compared the combustion characteristics between oxy-fuel combustion and conventional air-fuel combustion as shown in Table S1 in the Supplementary Materials, and the results show the combustion characteristics are similar in air-fuel and oxy-fuel combustion in the TGA [71,72] in which the combustion reaction is kinetically controlled. However, DTF and DFC experiment [73,74,75,76,77,78,79] results show the substitution of CO₂ with N₂ leads to changes in char combustion characteristics, and these changes in combustion characteristics include the decrease in char combustion temperature and the increase in burnout rate and combustion time at the same content of O₂ in which the combustion reaction is kinetic/diffusion-controlled and diffusion-controlled.

The differences in the char combustion characteristics between air-fuel and oxy-fuel combustion mainly result from the influence of the thermophysical properties of the background gas on heat and mass transfer and reaction kinetics, and are shown in the following aspects:

• The diffusivity of O₂ in CO₂ is 0.78 times that in N₂, and the lower diffusion flux of O₂ in oxy-fuel combustion [75,76,80,81,82].
• The heat capacity and molecular weight of CO₂ are higher than N₂ [73,74,75,76,77].
• The char–CO₂ gasification reaction becomes more prominent in oxy-fuel combustion; the endothermicity (172 kJ/mol C) of the gasification reaction is strong, which lowers the char combustion temperature and thus reduces the oxidation rate.

According to the findings in Section 3, we suspect the differences in the char combustion characteristics between air-fuel and oxy-fuel combustion relate to the Stefan flow mechanism. In the kinetic/diffusion-controlled and diffusion-controlled regions, the Stefan flow as the bulk flow has a more significant impact on the energy balance of char particle combustion in oxy-fuel combustion than that in air-fuel combustion. More combustion heat is transferred away from the char particles in the oxy-fuel combustion under the combined action of the Stefan flow and the gaseous components with high diffusivity and heat capacity, which results in the reduction of the combustion reactivity. Therefore, this process is also one of the reasons for the difference in combustion characteristics between air-fuel and oxy-fuel combustion. In contrast, in TGA experiments where combustion reaction is in the kinetic-controlled region, the effect of Stefan flow on combustion characteristics is negligible, and thus Stefan flow is not responsible for the differences in combustion characteristics under these circumstances.

The char–CO₂ gasification reaction is highly endothermic, which absorbs part of the combustion and radiation heat and leads to a lower char temperature [83,84]. The contribution of char–CO₂ gasification reaction to the total char consumption rate can exceed 10% at high temperatures in oxy-fuel combustion [32,85], which is much higher than that in air-fuel combustion. According to Section 2, the Stefan flow generated by the reaction C + CO₂ = 2CO has a stronger effect on the diffusion of CO₂, and thus the Stefan flow generated by a high proportion of char–CO₂ gasification reaction has a more significant impact on the char combustion characteristics in oxy-fuel combustion. In a word, the mechanism of Stefan flow is also one of the important sources for the differences in combustion characteristics between the two atmospheres, which has not been studied in previous relevant studies.

5. Conclusions

Stefan flow is a noteworthy transport mechanism in the mass transfer system of char particle combustion. A correct and comprehensive understanding of the Stefan flow is essential for improving the mass transfer theory of char particle combustion. In this study, Stefan flow theory in char combustion is reviewed thoroughly. The basic theory of the Stefan flow generated by the heterogeneous combustion reactions and the calculation errors of mass transfer caused by neglecting the Stefan flow under different theoretical models are introduced. Furthermore, the effects of Stefan flow on the combustion characteristics in air-fuel and oxy-fuel combustion are summarized.

In the mass transfer calculation, Stefan flow is the key factor to predicting the heat transfer coefficient correctly, previous studies have shown that the calculation errors of mass transfer caused by neglecting the Stefan flow range from 1 to 74%, which depends on the model and parametric selection including physical properties, the reaction rate, the ratio of oxidation to gasification reaction, and the location of the homogeneous reaction of CO. The effect of Stefan flow on mass transfer in oxy-fuel combustion is more significant than that in air-fuel combustion.

Stefan flow not only affects the mass transfer behaviour in the boundary layer of the char particle but also has an important impact on the char combustion characteristics. It accelerates heat loss and reduces the reactivity of char particles, which results in the reduction of reaction rate, combustion temperature, and char burnout rate; this phenomenon is more remarkable in the kinetic/diffusion-controlled region. Stefan flow affects combustion characteristics in oxy-fuel combustion more significantly than that in air-fuel combustion, which is an important reason for explaining the difference in combustion characteristics of the char particle in high-temperature conditions between air-fuel and oxy-fuel combustion.