Enhancing Thermal Performance Investigations of a Methane-Fueled Planar Micro-Combustor with a Counter-Flow Flame Configuration

Abstract

To enhance the performance of combustors in micro thermophotovoltaic systems, this study employs numerical simulations to investigate a planar microscale combustor featuring a counter-flow flame configuration. The analysis begins with an evaluation of the effects of (1) equivalence ratio Φ and (2) inlet flow rate V_i on key thermal and combustion parameters, including the average temperature of the combustor main wall (${\overline{T}}_w$), wall temperature non-uniformity (${\overline{R}}_{Tw}$) and radiation efficiency (${\eta }_r$). The findings indicate that increasing Φ causes these parameters to initially increase and subsequently decrease. Similarly, increasing the inlet flow rate leads to a monotonic decline in ${\eta }_r$, while the ${\overline{T}}_w$ and ${\overline{R}}_{Tw}$ exhibit a rise-then-fall trend. A comparative study between the proposed combustor and a conventional planar combustor reveals that, under identical inlet flow rate and equivalence ratio conditions, the use of the counterflow flame configuration can increase the ${\overline{T}}_w$ while reducing the ${\overline{R}}_{Tw}$. The Nusselt number analysis shows that the counter-flow flame configuration micro-combustor achieves a larger area with positive Nusselt numbers and higher average Nusselt numbers, which highlights improved heat transfer from the fluid to the solid. Furthermore, the comparison of blow-off limits shows that the combustor with counter-flow flame configuration exhibits superior flame stability and a broader flammability range. Overall, this study provides a preliminary investigation into the use of counter-flow flame configurations in microscale combustors.

1. Introduction

In recent years, microelectromechanical systems (MEMS) have found extensive applications across various fields, including aerospace [1,2,3]. However, most of these micro-energy devices currently still rely on battery power, which presents several limitations [4,5]. For example, there is an inherent conflict between the demand for device miniaturization and the low energy density of existing batteries [6]. Furthermore, chemical batteries often fail to meet the requirements for high efficiency, durability, and instantaneous high-power output demanded by MEMS [7]. Additional challenges, such as lengthy charging times, limited charge-discharge cycle life, and environmental pollution, further hinder the advancement of microsystems powered by chemical batteries [8]. In contrast, micro thermal photovoltaic (MTPV) systems [9], which use hydrocarbons as fuel, offer advantages such as higher energy density, extended operating times, and reduced overall mass [10,11,12]. However, unlike combustion in traditional scale combustors, micro-combustors introduce two significant challenges [13]. First, the increased surface-to-volume ratio results in substantial heat loss through the combustor walls, which can lead to flame quenching. Second, reduced residence time makes the flame susceptible to instabilities, including pulsating combustion and flame blowout. These issues significantly restrict the energy conversion efficiency of MTPV systems [14].

Researchers have undertaken extensive efforts to overcome the challenges of achieving efficient and stable combustion in micro-scale combustors. Thermal management techniques are often employed as solutions [15]. For example, heat recirculation combustors not only use the excess enthalpy of exhaust gases to preheat inlet fresh fuel but also create large heat exchange areas to maximize thermal recirculation [16]. This design significantly enhances stability and efficiency in micro-combustors [17,18,19]. Zhao et al. [20] conducted numerical simulations on a heat recirculation micro-combustor fueled by ammonia. The results showed that increasing the length of the heat recirculation channel significantly enhances the wall temperature of the combustor and improves the efficiency of thermophotovoltaic systems. However, it also increases pressure loss and has no significant effect on reducing nitrogen oxide emissions. Tenkolu et al. [21] proposed a recuperative micro-combustor with dual reaction zones, based on the concept of synergistic effects between flow fields and thermal fields. Heat recirculation was adjusted by modifying the cross-sectional areas of the inlet and outlet as well as the length of the baffles. The study revealed that adjusting the baffle length effectively increases flame length, increases wall surface radiative energy, and decreases the standard deviation of the outer wall temperature.

Fuel blending is also recognized as an effective strategy for enhancing thermal performance and improving flame stability under micro combustion conditions [22]. One common approach is to mix high laminar flame speed fuels, like hydrogen, with low flame speed fuels, such as ammonia or hydrocarbons [23]. For instance, Han et al. [24] significantly extended the combustion limits of a microscale combustor by blending hydrogen with ammonia. Similarly, Li et al. [25] examined how adding hydrogen to methane affects flame characteristics in a microscale confined space. Their findings indicated that the flame quenching distance decreases linearly as the proportion of hydrogen increases. This phenomenon is attributed to higher concentrations of H and OH radicals and a lower peak temperature in flames with high hydrogen content. However, utilizing high flame speed fuels or combustion methods in micro-combustors can also cause combustion to occur closer to the inlet, despite broadening the combustion limits. This shift leads to higher upstream temperatures and significant temperature gradients, resulting in wall temperature non-uniformity. Operating under these conditions for extended periods can induce high thermal stress, potentially damaging the combustor. Sui et al. [26] compared the temperature uniformity of premixed combustion in a microscale reactor using different fuels—including methane, carbon monoxide, and hydrogen—with air. Their results showed that hydrogen exhibited the greatest temperature non-uniformity due to its low Lewis number. Additionally, Peng et al. [27] demonstrated that blending methane with hydrogen to reduce laminar flame speed allows the flame to stretch and reposition, thereby enhancing combustor performance.

Structural optimization has been established as an effective strategy for enhancing the efficiency and stability of microscale combustion systems [28]. Incorporating flow-disrupting elements, such as ribs [29,30], baffles [31], or bluff bodies [32,33], into combustors can interfere with fluid flow patterns, while adding expanding features like cavities [34,35] or backward-facing steps induces recirculation zones and regions of low velocity. These modifications effectively prolong the chemical reaction time of the fuel mixture, thereby intensifying combustion reactions and enhancing convective heat transfer. Moreover, such structural optimizations significantly extend the blow-out limits of micro-combustors. For example, Cai et al. conducted numerical simulations to explore the impact of staggered bluff bodies on heat transfer performance in a hydrogen-fueled mesoscale combustor. Their results demonstrated that staggered bluff bodies generated longitudinal vortices, which promoted mixing and substantially improved heat transfer performance. Similarly, Zuo et al. designed a hydrogen-fueled planar micro-combustor equipped with finned bluff bodies and compared its performance to a planar combustor without bluff bodies. The numerical findings revealed that the combustor with finned bluff bodies achieved higher and more uniform wall temperatures, although this came with increased pressure loss. Peng et al. investigated the addition of a front cavity at the inlet of a cylindrical microscale combustor, analyzing how variations in the size and shape of the front cavity influenced flame position and overall combustor performance. In another study, Su et al. [36] performed numerical analyses on dual-chamber combustors and found that retaining the high-temperature flame zone within the downstream cavity resulted in a uniformly high-temperature distribution. Additionally, Faramarzpour et al. [37] examined the effects of backward-facing steps through numerical simulations, concluding that the presence of a step extended the blowout limits and led to higher average wall temperatures.

Previous studies have shown that combustor structural optimization is a simple and effective strategy to enhance thermal performances and flame stabilities of micro-scale combustors. However, many structural modifications proposed in existing research, such as the insertion of bluff bodies or baffles, not only significantly increase pressure losses but also considerably complicate the manufacturing process for micro-scale combustors [38]. Therefore, the aim of this work is to investigate the potential of a counterflow flame configuration, a relatively simple structural modification, to improve combustor performance and flame stability. To the best of the authors’ knowledge, limited studies have investigated the use of counter-flow flame configurations in micro-combustors. Addressing this research gap has been a partial motivation for the present work. This study examines a planar micro-combustor for its potential application in thermophotovoltaic micro-energy systems. The objective is to enhance thermal performance by modifying the conventional design. Specifically, the inlets are repositioned to the two sidewalls to generate a counter-flow flame within the combustor, increasing the reaction zone area and establishing a stagnation plane to prolong residence time. The paper is organized as follows: Section 2 describes the computational domain of the combustor model, the governing equations for the fluid and solid phases, the numerical schemes, the boundary condition configurations, and the performance parameters used to characterize the combustor. Additionally, a grid independence study and model validation are presented. Section 3 analyzes the effects of the equivalence ratio and inlet flow rate. A comparative study is then conducted to evaluate the performance of this burner relative to traditional designs. Finally, Section 4 provides a summary of the main findings of this study.

2. Numerical Methodology

2.1. Computational Domain

A traditional planar combustor is shown in Figure 1a. Different from traditional ones, a planar micro-combustor with two opposed inlets at the center of the combustor side walls is applied, as shown in Figure 1b. The combustor has an overall length of 18 mm, a width of 9 mm, and a height of 4 mm, with a solid wall thickness of 0.5 mm. The cross-sectional view of the combustor at the z = 0 plane is shown in Figure 1c. To ensure consistent inlet areas for the two combustors, the inlet on the sidewall has a length of 4 mm and a height equal to that of the fluid domain, which is 3 mm.

2.2. Numerical Setup

In the present work, numerical simulations are performed via an open-source platform, OpenFOAM. The governing equations for the conservation of mass, momentum, species composition, and energy are expressed as follows:

$${\rho }_g={\displaystyle \frac{p_{th}}{R_g{T}_g}}$$

(1)

$${\displaystyle \frac{\partial {\rho }_g}{\partial t}}+\nabla \cdot \left({\rho }_g\overrightarrow{v}\right)=0$$

(2)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_g\overrightarrow{v}\right)+\nabla \cdot \left({\rho }_g\overrightarrow{v}\overrightarrow{v}\right)=\nabla \cdot \overrightarrow{S}$$

(3)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_{g}h\right)+\nabla \cdot \left({\rho }_g\overrightarrow{v}h\right)=\nabla \cdot \left({\mathsf{\alpha }}_g\nabla T_g\right)-\nabla \cdot \left(\sum _i{h}_i\overrightarrow{J_i}\right)-\sum _i{\mathsf{\Delta }H}_{i,f}^0{R}_i$$

(4)

$${\displaystyle \frac{\partial }{\partial t}}\left({\rho }_g{Y}_i\right)+\nabla \cdot \left({\rho }_g\overrightarrow{v}{Y}_i\right)=-\nabla \overrightarrow{J_i}+R_i$$

(5)

$${\rho }_s{C}_{p,s}{\displaystyle \frac{\partial }{\partial t}}\left(T_s\right)=\nabla \cdot (\nabla \cdot ({\mathsf{\alpha }}_s{T}_s\left)\right)$$

(6)

$$T_s=T_{inter}=T_g$$

(7)

$${\overrightarrow{n}}_i\cdot \left[-\nabla \cdot \left({\mathsf{\alpha }}_s{T}_s\right)\right]={\overrightarrow{n}}_i\cdot -\left({\mathsf{\alpha }}_g\nabla T_g\right)$$

(8)

where the variable $\rho$ symbolizes density, T indicates temperature, $\overrightarrow{v}$ is the velocity vector and R_g represents the universal gas constant. Moreover, t represents time, and $p_{th}$ stands for thermodynamic pressure. In Equation (3), $\overrightarrow{S}=-p_d\overrightarrow{I}+\mu \left(\nabla \overrightarrow{v}+(\nabla {\overrightarrow{v})}^T-{\displaystyle \frac{2}{3}}(\nabla \overrightarrow{v})\overrightarrow{I}\right)$, where $p_d$ is the dynamic pressure, $\overrightarrow{I}$ is the identity tensor, and $\mu$ is the dynamic viscosity. The dynamic viscosity is calculated using Sutherland’s law [39]. The parameter $\mathsf{\alpha }$ represents the thermal conductivity, $H_{i,f}^0$ denotes standard enthalpy, and R_i and Y_i represent the reaction rate and mass fraction of species i, respectively. The diffusion flux $\overrightarrow{J_i}$ is computed using Fick’s law, while $C_p$ refers to the specific heat capacity. Note that the subscript i denotes species i, while the subscript s and g indicate the solid and gas phases, respectively.

The numerical simulation in this study follows the format outlined below. Convection terms are discretized using a second-order upwind scheme, while the diffusion terms are handled with a second-order central differencing scheme. Temporal discretization is carried out with the first-order implicit Euler method. The time step was made adaptive rather than fixed, with an adaptive tuning rule implemented to make the Courant number no greater than 0.3 throughout the whole numerical simulation. Total simulation time is about 10 flow-through cycles, and the presented time-averaged data is based on about 5 flow-through cycles after transients have died away. Pressure and velocity coupling is achieved through the PIMPLE algorithm. Within each time-step, governing equations are solved with three to five outer loops and two inner loops to avoid losing any simulation accuracy. All absolute residuals of governing equations are set to be less than 1 × 10⁻⁷. Since the Reynolds number is relatively low, a laminar model is used [40]. The combustion model is based on the laminar finite rate framework. The chemical mechanism includes 17 species and 25 reversible reactions, a model widely adopted in microscale combustion studies and validated effectively in subsequent research [41].

The model’s boundary conditions are outlined as follows: The inlet is specified as a typical velocity inlet, with pressure treated as a Zero-Neumann condition, and all other variables are assigned Dirichlet conditions. Fuel and oxidizer are uniformly mixed and injected into the combustion chamber at an inlet temperature of T_i = 300 K, with a uniform velocity distribution which is calculated by the inlet volume flow rate u_i = V_i/A_i, where V_i is the inlet volume flow rate, u_i is the inlet velocity and A_i is the area of combustor inlet. The fuel is pure methane, and the oxidizer is air. The mass fractions of fuel and oxidizer are determined based on the equivalence ratio. The outlet is modeled with a typical pressure outlet condition, where pressure is treated using a Dirichlet condition, and the outlet pressure is set to p₀ = 101,325 Pa, with all other variables using Zero-Neumann conditions. The thermal boundary condition on the inner wall of the combustor is governed by Equations (7) and (8), representing conjugate heat transfer for fluid–solid coupling. The outer wall thermal boundary condition follows a Robin-type formulation, accounting for both natural convection and thermal radiation. The total heat loss from the outer wall to the environment is expressed as:

$$Q_{loss}={\iint }_A\epsilon \sigma {(T_w}^4-{T_0}^4)dA+{\iint }_{A}h(T_w-T_0)dA$$

(9)

where the natural convection coefficient is h = 20 W/(m²·K), and the surface emissivity is ε = 0.85. The material of the solid region within the combustion chamber is 316 stainless steel, with a density of 8000 kg/m³, a thermal conductivity of 16.3 W/(m·K), and a specific heat capacity of 503 J/(kg·K).

2.3. Performance Parameters

To evaluate the thermal performance of the micro-planar combustor, two parameters are introduced: ${\overline{T}}_w$ the average temperature of the main wall surface and ${\overline{R}}_{Tw}$ the average temperature non-uniformity. The mathematical expressions for these parameters are defined as follows [31]:

$${\overline{T}}_w={\iint }_A{T}_w\mathrm{d}A$$

(10)

$${\overline{R}}_{Tw}={\iint }_A{\displaystyle \frac{T_w-{\overline{T}}_w}{{\overline{T}}_w}}\mathrm{d}A\times 100\mathrm{\%}$$

(11)

Here T_w and A represent the local temperature and the area of the outer wall of the combustor, respectively.

The radiation power $Q_r$, radiation efficiency ${\eta }_r$, combustion efficiency ${\eta }_c$ and overall efficiency ${\eta }_o$ of the combustor is defined as follows [42]:

$$Q_r={\iint }_A\epsilon \sigma {(T_w}^4-{T_0}^4)dA$$

(12)

$${\eta }_r={\displaystyle \frac{Q_r}{{\iiint }_{V}qdV}}\times 100\%$$

(13)

$${\eta }_c={\displaystyle \frac{{\iiint }_{V}qdV}{{\dot{m}}_{{CH}_4}\times Q_{LHV}}}\times 100\mathrm{\%}$$

(14)

$${\eta }_o={\displaystyle \frac{Q_r}{{\dot{V}}_{{CH}_4}\times Q_{LHV}}}\times 100\mathrm{\%}={\eta }_r\times {\eta }_c$$

(15)

Here, $T_w$ and $q$ represent the local temperature of the outer wall and the local heat release in the fluid, respectively. The solid wall emissivity is $\epsilon$ = 0.85 and the Stefan–Boltzmann constant is $\sigma$ = 5.67 × 10⁻⁸ W·m⁻²·K⁻⁴. Additionally, ${\dot{V}}_{{CH}_4}$ represents the volumetric flow rate of methane, and $Q_{LHV}$ = 36 MJ/Nm³ denotes the lower heating value (LHV) of methane per unit volume.

To further analyze the heat transfer characteristics within the micro-combustor, the local Nusselt number is introduced. The Nusselt number represents the relative significance of convective heat transfer compared to conductive heat transfer and is calculated as follows:

$$Nu={\displaystyle \frac{hL}{k_g}}$$

(16)

Here, h denotes the convective heat transfer coefficient on the combustor’s inner wall, L is the characteristic length, which in this study is taken as the hydraulic diameter of the combustor outlet, and k_g represents the thermal conductivity of the fluid.

2.4. Model Validation

The number of cells and grid size in the mesh are crucial for balancing computational cost and accuracy. To achieve this balance, a grid independence study was performed. Figure 2 illustrates the centerline temperature distribution along the main wall surface, obtained from numerical simulations using three different meshes containing 324,000, 648,000, and 1,296,000 cells. The results show that the temperature distributions for the 648,000-cell and 1,296,000-cell meshes are nearly identical, whereas the 324,000-cell mesh yields significantly different results. Consequently, a medium-sized mesh with 648,000 cells was selected for this study to achieve high accuracy with reasonable computational effort.

To validate the accuracy of the present solver and computational model, the numerical results are compared with experimental measurements available in the previous literature. Figure 3 presents a comparison between the outer wall temperature distribution obtained in this simulation and the experimental results reported by Tang et al. [43]. The results demonstrate good agreement under two different inlet velocity conditions. Additionally, our previous study provides further details on the validation of the current solver [42]. Therefore, the numerical model employed in this work is deemed sufficiently reliable for further investigation.

3. Results

3.1. Effect of Equivalence Ratio Φ

This section begins by examining and discussing the influence of the equivalence ratio Φ on thermal performance, as it plays a crucial role in determining the heat release rate by regulating the extent of complete combustion. The Φ varies from 0.8 to 1.2, with an increment of 0.1, and the inlet flow rate V_i is fixed at 28.8 mL/s. As shown in Figure 4, the ${\overline{T}}_w$ follows a non-monotonic trend, initially increasing and then decreasing with an increase in Φ, primarily due to the combined effects of chemical energy input and the extent of complete combustion. Conversely, the ${\overline{R}}_{Tw}$ shows the opposite trend, first decreasing and then increasing. Notably, at Φ = 1.2, a sharp drop in ${\overline{T}}_w$ and a sharp rise in ${\overline{R}}_{Tw}$ are observed.

To further analyze the mechanism by which the equivalence ratio affects thermal performances, Figure 5 and Figure 6 show the temperature distributions with isolines at the z = 0 plane and the outer wall of the combustor, respectively. Figure 5 illustrates that, under a constant inlet flow rate, variations in the equivalence ratio within the range of 0.9 to 1.1 have little effect on the flame distribution within the combustor. Fresh premixed gas enters the combustor through two inlets, forming two trapezoidal low-temperature regions near the inlets, while a high-temperature stagnation plane is observed near the symmetry axis at y = 0. The highest temperature regions within the combustor are located near the four corners of the z = 0 plane rectangular cross-section of the burner. When Φ = 0.8, the maximum temperature of the flow field does not exceed 2000 K, and the high-temperature region is located closer to the outlet. In contrast, at an equivalence ratio of Φ = 1.2, the high-temperature regions on either side of the stagnation plane become narrower, while the low-temperature region nearly fills the central area of the combustor and extends toward the outlet. The reduced spacing between the isothermal lines indicates a significant temperature gradient. It is evident that high-temperature regions exceeding 2000 K are not entirely confined within the combustor but reach the outlet, signifying considerable energy loss through exhaust gases. This observation is consistent with the decrease in average temperature and uniformity shown in Figure 4.

Figure 6 depicts the temperature contours of the combustor main wall surface. When Φ = 0.8, most regions exhibit temperatures ranging from 950 K to 1040 K, with an overall uniform distribution. However, the peak temperature on the main wall is relatively low due to the reduced chemical energy at a lower equivalence ratio. When increasing the equivalence ratio to Φ = 0.9, the high-temperature region expands significantly, with two elliptical high-temperature zones (>1100 K) forming near the outlets. When the equivalence ratio reaches Φ = 1.0, there is a slight increase in the maximum temperature, and the high-temperature region on the wall surface (>1120 K) becomes more extensive, with most areas exceeding 1100 K. Regions with temperatures below 1050 K are confined to a small area near the inlets. By further increasing the equivalence ratio to Φ = 1.1, the high-temperature region (>1120 K) slightly decreases, but the area with temperatures above 1100 K is larger compared to the equivalence ratio of 0.9. The distribution of regions below 1050 K is nearly identical to that at 0.9. At an equivalence ratio of 1.2, the overall wall surface temperature decreases significantly, with most areas below 950 K and the emergence of regions with temperatures below 850 K.

Figure 7 further analyzes the variations in radiation efficiency ${\eta }_r$, combustion efficiency ${\eta }_c$ and overall efficiency ${\eta }_o$ with changes in the equivalence ratio. Under fuel-lean conditions, ${\eta }_c$ reaches a high level, with nearly 90% of the chemical energy converted into combustion heat. When the equivalence ratio is increased to Φ = 1.0, despite representing the theoretical condition for complete reaction between fuel and air, the short residence time in the microscale combustor leads to a slight reduction in combustion efficiency compared to lean conditions. As the equivalence ratio continues to rise into rich conditions, ${\eta }_c$ declines significantly, dropping to 64.3% at Φ = 1.2. In contrast, ${\eta }_r$ follows a different trend. As Φ increases from 0.8 to 1.2, ${\eta }_r$ initially rises and then decreases, peaking at 28.7% at Φ = 1.0. Furthermore, according to Equation (15), the overall efficiency ${\eta }_o$ is the product of ${\eta }_r$ and ${\eta }_c$. Since ${\eta }_r$ is approximately 20%, significantly lower than the ${\eta }_c$ of around 80%, ${\eta }_o$ is predominantly determined by ${\eta }_r$. Consequently, ${\eta }_o$ also shows an initial increase followed by a decrease as the equivalence ratio changes. Notably, ${\eta }_o$ at equivalence ratios of Φ = 1.0 and 0.9 are nearly identical. Additionally, when combined with the ${\overline{R}}_{Tw}$ shown in Figure 4, it becomes evident that ${\eta }_r$ and ${\eta }_o$ are strongly correlated with ${\overline{R}}_{Tw}$. Therefore, the optimal equivalence ratio range for the combustor investigated in this study is approximately between 0.9 and 1.0.

3.2. Effect of Inlet Flow Rate V_i

In this section, the way in which the inlet flow rate impacts the thermal performance of the combustor will be explored. Figure 8 illustrates the variations in ${\overline{T}}_w$ and ${\overline{R}}_{Tw}$ at different V_i, with the equivalence ratio fixed at Φ = 1. Both ${\overline{T}}_w$ and ${\overline{R}}_{Tw}$ exhibit nonlinear trends as V_i increases. Specifically, as V_i rises from 9.6 to 57.6, ${\overline{T}}_w$ initially increases before declining. The two highest ${\overline{T}}_w$ values are observed at V_i = 28.8 mL/s and 38.4 mL/s, corresponding to 1112 K and 1108 K, respectively. In contrast, ${\overline{R}}_{Tw}$ initially decreases and then increases with higher V_i. The minimum ${\overline{R}}_{Tw}$, observed at V_i = 19.2 mL/s, corresponds to 1.36%.

To clarify the effect of inlet flow rate on thermal performance, Figure 9 displays temperature fields with isolines as a function of V_i on the z = 0 cross-section, with the equivalence ratio fixed at Φ = 1.0. At V_i = 9.6 mL/s, the inlet flow rate is relatively low, and the maximum temperature in the flow field does not exceed 2000 K. Regions with temperatures exceeding 1800 K form two crescent-shaped zones near the inlets. As the inlet flow rate increases, for instance, to V_i = 28.8 mL/s the low-temperature regions near the inlets expand, adopting a trapezoidal shape. Concurrently, the maximum temperature in the flow field rises, and regions exceeding 2000 K emerge, primarily located near the four bottom corners of the trapezoidal low-temperature regions. As the inlet flow rate further increases, the low-temperature regions adjacent to the inlets progressively expand, the high-temperature zones migrate toward the combustor’s four corners, and the central high-temperature area becomes increasingly restricted.

To further analyze the effect of inlet flow rate on wall temperature non-uniformity, we examine the temperature distribution on the main wall surface, as depicted in Figure 10. At an inlet flow rate of V_i = 9.6 mL/s, the high-temperature zone forms a rounded rectangular shape at the center of the main wall. When the flow rate increases to V_i = 19.2 mL/s, the high-temperature zone shifts to a dumbbell-like shape, still centered on the main wall. At this flow rate, the region exceeding 1080 K covers most of the main wall, resulting in relatively high wall temperature uniformity. However, as the flow rate increases beyond 28.8 mL/s, the high-temperature zone splits into two regions located near the outlets, leading to increased wall temperature non-uniformity. When the flow rate further rises to 48 mL/s, the area of the high-temperature zone decreases significantly, and a large low-temperature region emerges between the two inlets. This results in a continued increase in wall temperature non-uniformity.

Figure 11 illustrates the variation of $Q_r$ and ${\eta }_r$ under different inlet flow rates, with the equivalence ratio fixed at Φ = 1. According to Equation (13), $Q_r$ is directly proportional to the fourth power of temperature. As a result, $Q_r$ initially increases and then decreases with rising inlet flow rates. However, within the range of flow rates studied in this paper, ${\eta }_r$ decreases monotonically as the flow rate increases. It is evident from Figure 8 that at high inlet flow rates, the region of highest temperature in the flow field shifts closer to the outlet. This leads to a significant portion of high-temperature gas being expelled without sufficient time to transfer heat to the solid wall, resulting in energy conversion losses. These findings suggest that MEMS must balance power demand and energy conversion efficiency. Optimizing combustor design is essential to minimize efficiency losses caused by instantaneous high-power demands.

3.3. Comparative Analysis

In this section, the combustion characteristics and thermal performance of the counter-flow flame combustor utilized in this study are contrasted with those of a traditional combustor, as illustrated in Figure 1c. Additionally, the effect of the application of the counter-flow configuration on thermal performance and the underlying physical mechanisms is investigated. Figure 12 presents a quantitative comparison of ${\overline{T}}_w$ and ${\overline{R}}_{Tw}$ between the conventional combustor and the counter-flow flame combustor under different equivalence ratios, with the inlet flow rate fixed at V_i = 14.4 mL/s. For equivalence ratios of Φ = 0.9, 1.0, and 1.1, the ${\overline{T}}_w$ of the counter-flow combustor is observed to be 37.82 K, 32.27 K, and 58.54 K higher, respectively, than that of the conventional combustor. In contrast, the ${\overline{R}}_{Tw}$ value of the counter-flow flame combustor is 3.98%, 2.87%, and 6.34% lower, respectively, than that of the conventional combustor. These findings demonstrate that the counter-flow flame configuration improves both the mean value and uniformity of the wall temperature across all equivalence ratios, with the enhancements being especially significant under fuel-rich conditions. This indicates that adopting a counter-flow flame configuration is an effective strategy for enhancing wall temperature and achieving a more uniform wall temperature distribution.

Figure 13 presents a comparative analysis of ${\eta }_r$, ${\eta }_c$ and ${\eta }_o$ for the two combustor configurations. It can be observed that the differences in ${\eta }_c$ between the two combustors are minimal and primarily governed by the equivalence ratio. The counter-flow flame configuration primarily enhances ${\eta }_r$, which in turn affects ${\eta }_o$. At equivalence ratios of 0.9, 1.0, and 1.1, the counter-flow flame configuration increases the value of ${\eta }_r$ by 5.00%, 4.32%, and 7.23%, respectively.

To further explore the underlying physical mechanisms driving the above parameter variations, Figure 14 and Figure 15 compare the temperature distributions in the flow field and on the walls of the two combustor configurations under various equivalence ratios. Figure 14 demonstrates that changes in the equivalence ratio within the range of 0.9–1.1 do not fundamentally alter the temperature distribution in the flow field for either combustor. As shown in Figure 14b, the blue low-temperature regions in the counter-flow flame combustor are confined to a small area near the inlet, while the high-temperature regions form two back-to-back crescent shapes on either side of the combustor’s y-axis. In contrast, the high-temperature region in the conventional combustor exhibits an inverted V-shape, located further downstream, with a large low-temperature region visible near the inlet. Figure 15b illustrates that the temperature distribution on the main wall of the counter-flow flame combustor exhibits symmetrical characteristics, with the high-temperature region located at the center of the combustor, transferring heat outward to the surrounding areas. In contrast, as shown in Figure 15a, the high-temperature region on the wall of the conventional combustor appears in an inverted teardrop shape, located downstream. The upstream wall exhibits significantly lower temperatures due to the distance from the high-temperature region and the influence of the low-temperature fluid near the inlet.

Beyond the qualitative analysis of flame temperature distribution previously discussed, it is imperative to evaluate the heat transfer between the fluid phase and the solid phase. This evaluation is conducted by calculating the Nusselt number, which serves as an indicator of the ratio between convective and conductive heat transfer, thereby characterizing the heat transfer performance of the counter-flow flame combustor. Importantly, the direction of heat flux determines the sign of the Nusselt number. In this context, a positive Nusselt number signifies heat transfer from the fluid to the solid, whereas a negative value indicates heat transfer in the reverse direction. As depicted in Figure 16a, the blue regions representing negative Nusselt numbers occupy a large area in the conventional combustor, particularly near the inlet, indicating strong heat absorption by the fresh mixture from the solid wall in this region. High Nusselt number regions are confined to narrow areas near the inverted V-shaped flame. In contrast, the Nusselt number distribution in the counter-flow flame combustor is entirely different, displaying a more uniform overall distribution. For both positive and negative Nusselt numbers, the maximum absolute values are smaller than those in the conventional combustor, indicating more uniform heat transfer. Furthermore, the larger area of positive Nusselt numbers in the counter-flow flame combustor suggests that more heat is transferred from the fluid to the solid rather than being carried away by exhaust gases. Figure 16b illustrates the average Nusselt number, which further highlights this characteristic. Under all three equivalence ratio conditions, the counter-flow flame configuration significantly increases the average Nusselt number compared to the conventional combustor.

The flammability limit has long been a significant challenge in microscale combustors. Figure 17 compares the blow-out limits of the two combustor configurations, demonstrating that the application of the counter-flow flame configuration significantly extends the operational range of the combustor. For instance, at an equivalence ratio of 1.0, the blow-out limit increases from 21.6 mL/s to 84 mL/s. Previous studies have shown that, in conventional combustors, as the inlet flow rate increases, the flame tends to become asymmetric or oscillating before eventually being blown out due to insufficient chemical reaction time caused by shortened residence time [44]. In contrast, the counter-flow flame configuration, with two opposing inlets aligned along the y-axis, creates a stagnation plane that stabilizes the flame, thus improving performance under higher flow rates.

4. Conclusions

In this study, a micro planar combustor with a counter-flow flame configuration, fueled by premixed methane, is introduced. Numerical simulations were carried out using a low-Mach-number reacting flow solver integrated with fluid–solid conjugate heat transfer, implemented on the OpenFOAM platform. Initial steps included model validation and a grid independence analysis. Following this, the effects of the equivalence ratio and inlet flow rate were analyzed. Detailed investigations were conducted into the performance enhancements and the fundamental physical mechanisms of the counter-flow flame configuration in comparison to traditional combustors. The key findings include the evaluation of the average temperature of the combustor’s main wall (${\overline{T}}_w$), wall temperature non-uniformity (${\overline{R}}_{Tw}$), radiation efficiency (${\eta }_r$), and combustion efficiency (${\eta }_c$).

• The average temperature of the combustor main wall (${\overline{T}}_w$), wall temperature non-uniformity (${\overline{R}}_{Tw}$) and radiation efficiency ${\eta }_r$ vary nonmonotonically with the equivalence ratio (Φ). However, increasing Φ results in a monotonic decrease in combustion efficiency (${\eta }_c$). The overall efficiency ${\eta }_o$ of the combustor is primarily determined by ${\eta }_r$ and follows a similar trend, reaching its maximum at an equivalence ratio of 1.0.
• ${\overline{T}}_w$ initially increases and then decreases as the inlet flow rate V_i rises, while ${\overline{R}}_{Tw}$ decreases first and then increases. Notably, the highest ${\overline{T}}_w$ and lowest ${\overline{R}}_{Tw}$ do not occur under the same V_i conditions. Both ${\eta }_r$ and ${\eta }_o$ decline monotonically with increasing V_i. Excessively high V_i results in reductions in thermal and heat transfer performances. Thus, selecting the inlet flow rate requires balancing power demand and energy conversion efficiency.
• Comparative studies demonstrate that combustors with a counter-flow flame configuration achieve higher ${\eta }_c$ and ${\eta }_r$, as well as a ${\overline{T}}_w$ under identical operating conditions. The configuration alters the shape and location of high-temperature zones, centralizing them on the main wall, improving wall temperature uniformity. Nusselt number analysis shows higher average values and smaller negative local areas, indicating superior heat transfer characteristics. Moreover, this design significantly enhances flame stability, expanding the combustor’s operational range and blow-off limit.

Overall, these findings emphasize that adapting micro-planar combustors with a counter-flow flame configuration is a straightforward and effective approach to enhancing thermal performance. Furthermore, the insights provided in this study are valuable for guiding experimental investigations and informing the design of combustors to optimize energy conversion efficiency in systems related to propulsion and power generation. In future work, further optimization of the inlet size and arrangement of the counter-flow flame combustor could be explored to enhance the performance of the micro-combustor.