Computational study on the flame and extinction behavior of a high enthalpy air slab

A promising type of an air-breathing engine for supersonic flight is the ramjet engine. A ramjet engine relies on ram compression, a phenomenon that uses the forward motion of the engine to compress the incoming air, eliminating the need for turbomachinery and minimizes moving parts, producing a theoretically simpler design.¹ Ramjet engines can further be simplified with the inclusion of a solid fuel, so beneficial due to its larger specific energy content compared to a liquid fuel.² Thus, solid fuel ramjet engines (SFRJs) are heavily used in missile and artillery systems for cost efficiency and a simplistic design when high speed, long range, and high specific impulse are desired.³ Although SFRJs are simple with the lack of moving parts, the details of the combustion progress are complicated due to the multiphase component. Specifically, the regression rate and combustion dynamics of the solid fuel are critical. As such, much work has focused on various aspects of the solid fuel and combustor flow field. An important aspect to consider is the heated air entering a SFRJ due to the shockwave compression. This high enthalpy air flows over the solid fuel grain heating it. Heat transfer pyrolyzes the solid fuel so that gaseous fuel mixes and reacts with the incoming air in a diffusion flame.

One of the most important parameters is the regression rate of the solid fuel. Numerous studies have investigated the conditions and parameters that could affect fuel regression. These studies highlighted how swirling intensity,^4,5 mass flow,⁴ and inlet temperatures⁴ enhance regression rates in solid fuels. The higher regression rates occur due to the higher degree of swirl at the fuel surface, causing an increase in local heat and mass transport, which can enhance flame holding. Additional studies have examined the various heat transfer mechanisms across the fuel surface and identified convection as the dominant mode of heat transfer for a SFRJ.⁶ Studies have used thermal analysis,^7,8 radiation-induced pyrolysis⁹ and combustion,¹⁰ counter-flow burners,¹¹ and other methods of characterizing the fuel itself including quantifying its pyrolysis products.⁷ However, there are limitations to how well these experiments recreate the fluid dynamic conditions in SFRJ combustors, particularly with geometries needed to improve flame holding.

Previous studies have found enhanced combustion and flame holding capabilities with incorporating a backward facing step.^12,13 Step inclusion is shown to induce a degree of turbulence for improved oxidizer and vaporized solid fuel mixing, while creating a re-circulation region of a high temperature air–product mixture for promoting self-ignition and flame holding downstream.^14–16 Further studies have identified the performance of various step heights and angles and their impact on the re-circulation zone and flow reattachment properties.^12,14 Schulte¹⁷ examined various inlet to flame holder port area ratios to identify the operation envelop of SFRJ. Further work expanded to observe how air temperatures affected the inlet to flame holder area ratios and indicated extended operational envelopes with increased air temperature.^16,17 Additional studies examined how the flame holder port to inlet operational area ratios were affected by port to throat area ratios, which found minimum area ratios needed for sustained combustion.¹³ 

The focus of previous works on the interaction between the solid fuel and the flow field provides important details of how the SFRJs operate. However, there has been limited work, understanding the flame holding from a fundamental gas-phase combustion perspective. The pyrolysis products from the solid fuel create what can be described as a surface jet cross-flow instead of the classic single jet in a cross flow. Furthermore, the high enthalpy air and its influence on the combustion process (i.e., flame extinction) have not been studied in a diffusion flame. As such, the goal of this work is to understand how the incoming air temperature influences reaction pathways and their influence on flame holding/extinction. We investigate high enthalpy air over a gaseous H₂ surface jet in a two-dimensional slab burner configuration. Realistically, the pyrolysis products will be a mixture of smaller hydrocarbons, which depends on the initial fuel and heating rate.¹⁸ Since the oxidation of hydrogen is common among hydrocarbon combustion mechanisms, the use of H₂ in this work enables applicability to all solid (and liquid) hydrocarbon fuels. The reactions for the hydrogen combustion provided in the mechanism are listed in  Appendix B for completeness. Furthermore, this simplified geometry intends to minimize the influence of the geometry on a surface jet cross-flow system, strictly allowing a focus on the fluid dynamics and combustion aspects.

The remainder of the paper is structured as follows: Sec. II briefly summarizes the governing equations of a reacting flow system. Section III describes the computational approach and the parameters studied and analyzed. Section IV illustrates the numerical modeling performed on the proposed procedure, and present results concerning the blow-off mass flux (Sec. IV A), normalized flame height (Sec. IV B), conductive heat flux analysis (Sec. IV C), convective heat flux (Sec. IV D), and combustion chemistry (Sec. IV E).

The fluid and chemical behavior of the slab burner can be described by the reacting Navier–Stokes equations: continuity [Eq. (1)], momentum [Eq. (2)], energy [Eq. (3)], and species conservation [Eq. (4)] equation. Due to the time dependence of chemical rates and species production, the unsteady reacting Navier–Stokes are presented and numerically solved,

where ρ, U, and p represent the density, velocity, and pressure, respectively. In the momentum equation and energy equation, τ, μ, k, and h represent the stress tensor, dynamic viscosity, kinetic energy, and enthalpy, respectively. α and $Ṙheat$ represent the effective thermal diffusivity and heat generation due to reactions. In the species conservation equation, Y_k, $Ṙk$⁠, and μ represent the species mass fraction, chemical source term, and effective dynamic viscosity.

A useful device in conducing SFRJ experiments is a slab burner, a simplified combustion chamber. The simplicity of the chamber geometry is beneficial in simplifying the fluid dynamics and minimizing the geometry effect on the reaction process, primarily allowing a focus and analysis on the combustion process. The current study utilizes a rectangular prism slab burner, with a dimension of $120.65×19.05 mm2$ (Fig. 1).

To conduct the computational analysis, the open-source computational fluid dynamics (CFD) software OpenFOAM^© was utilized to numerically solve and model the reacting Navier–Stokes equations for reacting flows.¹⁹ The slab burner domain was modeled with a mesh size of $356×20$⁠, selected to achieve mesh-independency. The turbulence model selected for this project is the industry standard $k−ϵ$ model. The study utilized the University of California San Diego (UCSD) mechanism containing 244 elementary reactions and 50 species and thermodynamic polynomial coefficients to obtain the thermodynamic properties in relation to temperature.²⁰ 

In the current study, the oxidizer and fuel are air (0.23O₂/0.77N₂) and hydrogen. It is important to note that although hydrogen is not a solid fuel, overall selection of hydrogen provides a simpler mechanism allowing for an easier analysis on the fluid and chemical properties, as a first step toward expanding and creating a model for solid fuel combustion. The air inlet temperatures are chosen to begin at atmospheric 300 K, with increase in increments of 100 till 1000 K. This range of temperatures is expected based upon flight operation (e.g., altitude and Mach number). The temperature of the fuel was sustained at 300 K throughout all the air temperatures tested, with the chamber pressure set to 1 atm. Additionally, for each air temperature condition, the air velocities tested began at 1 m/s, an increase in increments of 5 m/s till flame extinction occurred. The velocity of the fuel was calculated as to sustain an effective stoichiometric air to fuel mass flow rates. The ratio of hydrogen and air was 34.32 by mass.

The model utilizes the pseudo-transient local-time stepping (LTS) technique. LTS is a method of time step manipulation of each individual cell that allows local time-steps to be as high as possible based on a Courant–Friedrichs–Lewy (CFL) value to accelerate a solution to steady state.²¹ This method creates a computed time step field, with model calculations utilizing this local cell time step rather than a global time step. The LTS method was selected due to the chemical rates and species productions dependence on time, of which a full steady-state simulation could not be conducted.

The boundary conditions for the simulations run are listed in Table I. The walls are defined as adiabatic with a no-slip condition applied. The pressure within the chamber was set to 1 atm with zero-gradients at the inlets and wall boundaries. The air and fuel inlet conditions are applied as mass fractions.

Figures 2 and 3 depict general velocity and temperature profiles seen in the simulations performed, respectively. The region above the high temperature zone is primarily air, and the region below is primarily hydrogen. Due to the structure of the diffusion flame, the high temperature zone had the greatest product mole fractions. The combustion process occurs at the region of the flame zone, which is identified by the high temperature region. Because the combustion process occurs at the flame zone region, the product species showcase a similar profile to the temperature profile where typically the maximum mole fraction of species is located at the higher temperatures.

The air mass flux blow-off limit identifies when the flame is no longer sustained. The blow-off air mass flux for each air inlet temperature is presented in Table II. The results indicated two temperature regimes, based on the blow-off mass air flux. The temperature regime 300–500 K had a blow-off mass flux of about $≈14.5 kg/sm2$⁠, whereas the temperature regime 600–1000 K had a blow-off mass flux of about $≈22.5 kg/sm2$ with a peak of $≈24.277 kg/sm2$ occurring at 800 K. The result indicated that within each regime, the blow-off mass flux is nearly independent of air inlet temperature. Additionally, the blow-off velocity limit at each temperature is shown in Table II. The results indicate an increase in blow-off velocity with inlet temperatures. Observing the near constant blow-off mass flux for the two temperature regimes and understanding that air density decreases with temperature, the blow-off velocity increases to compensate for the density mass loss to sustain the near constant mass flux.

Data of the flame height provide information regarding the proximity of the flame to the fuel surface, which is important in understanding the heat transfer process to sustain combustion. The maximum temperature along the outlet was used to determine the flame height, which was then normalized to the chamber height of 19.05 mm. As shown in Eq. (5), Re number scales to temperature by $T−3/2$⁠, with density (ρ) scaling to temperature via ideal gas law and kinematic viscosity (μ) scaling to temperature via the kinetic theory of gasses without the temperature dependence of collision integral,²² 

Employing temperature scaling as a ratio of the fuel and air temperature and multiplying it with the flow Reynolds number, the following relation is obtained:

The modified Re vs normalized flame height is shown in Fig. 4. The use of modified Re shows that the normalized flame height data depict similarity solution behavior, which highlights the effects that the air and fuel inlet temperatures simultaneously have on flame height.

The results indicated all air inlet temperatures exhibited a decreasing normalized flame height with an increasing modified Re until Re $∼104$⁠. For a given temperature inlet, increasing the Re, which would indicate purely an increase in air velocity, led to a decrease in normalized flame height. The decreasing normalized flame height with the increase in modified Re occurs due to the increasing air momentum caused by increasing the air mass flow rate. The increased amount of air momentum in the axial direction pushes the flame toward the fuel surface.

The results show that with an increase in Re and as the blow-off mass flux is approached for each inlet air temperature, the normalized flame height approaches a value of $≈0.303$⁠. The constant value of the normalized flame height can be explained by the usage of hydrogen fuel. The hydrogen gas properties are near constant due to a constant 300 K temperature and 1 atm chamber pressure at the fuel inlet. While the air momentum is increasing with the increase in inlet air velocity, the compressibility limit of hydrogen gas might be reached, where the hydrogen cannot be further compressed at the temperature and pressure condition leading to a layer of hydrogen gas above the fuel inlet causing the constant normalized flame height at high Re.

In locations where the flame height is closer to the fuel inlet, higher heat fluxes are expected because of the proximity and larger temperature gradients between flame temperature and fuel inlet. It is important to note that while the present study uses a gaseous fuel source, the examined results would nonetheless provide observations into the general heat transfer process to understand the characteristics of solid fuel modeling/regression and provide insight into the flame holding and attachment. In the present study, heat flux data at the fuel edge inlet were observed since this location is the anchoring point of the flame, location highlighted in the inset of Fig. 5. Additionally, the conductive heat flux is also expected to be the largest at this location since this is where the flame height is closest to the fuel surface.

With the normalized flame height, the fuel inlet edge conductive heat flux followed a similarity solution behavior. Thus, the modified Re was employed from the temperature ratio scaling via Eqs. (5) and (6) with results showing the conductive heat flux data collapsing onto itself with the modified Re.

Examining the conductive heat flux at the fuel inlet edge, two distinctive trends occur: a large heat flux drop at lower temperatures and a nearly constant heat flux at high temperatures. Figure 6 highlights the large heat flux drop for an inlet air temperature of 300 K, specifically occurring between Re = 95 882 (air speed of 8 m/s) and Re = 11 026 (air speed of 9 m/s). Examining the temperature profiles at the fuel inlet air for both Re values, a decrease in local temperature is observed as shown in Fig. 7. The decrease in temperature at the fuel inlet edge for Re of 95 882 and 11 026 indicates a detaching flame, suggesting that the chemical timescales are not decreasing as fast as the transport timescales. The flame detachment is similarly seen at air inlet temperatures 400 and 500 K, indicating that this behavior occurs at lower air inlet temperatures. In contrast, the conductive heat flux results for temperatures between 600 and 1000 K indicated no large decrease. The heat flux at the fuel inlet edge for the temperatures 600 to 1000 K has an initial increase until a peak, with a slight decrease and an almost near constant heat flux value with an increase in Re. The lack of large drop in conductive heat flux for this temperature range is a strong indication of an attached and anchored flame throughout the Re ranges investigated in this study.

The convective heat flux was calculated using the Nusselt number, defined as the ratio of convective and conductive heat flux, via Re and Pr correlations for laminar [Eq. (7)]²³ and turbulent flows [Eq. (8)].²³ Extracting the Pr values for the simulated data showed a near constant value of Pr $≈0.71$ at the fuel inlet edge. Overall, the convective heat flux values were larger than the conductive heat flux, which indicates that convection is the dominant method of heat flux at the fuel inlet edge. With all air inlet temperatures that have been studied, the convective heat flux increased with the increase in Re. This increase in convective heat flux is expected considering how the bulk fluid motion increases with higher velocities, which in turn increases the heat flux rate between the higher temperature air to the fuel surface. As with the case with the conductive heat flux, two temperature regimes are observed as shown in Fig. 8. Temperatures between 300 and 500 K do not experience any sudden changes in convective heat flux. In fact, the convective heat flux appears to increase at a constant rate. However, at air inlet temperatures between 600 and 1000 K, a large increase in convective heat flux occurs. The increase is occurring as the flow approaches a fully turbulent regime (Re $≈10 000$⁠). The increase in convective heat flux as a turbulent flow is approached could be a result of the higher bulk motion and energy coupled with the higher degree of energy mixing between fluid layers in turbulent flows. Higher air inlet temperatures have steeper rises in convective heat flux results at high Re and is likely occurring due to chemical pathway changes not present at lower temperature ranges,

The observation of large changes to the heat transfer and attachment point described in Sec. IV D suggests that the combustion chemistry is changing in each regime. It is necessary to identify how the chemical pathway changes.

The Damköhler (Da) number represents the ratio of the fluid timescale, defined as reciprocal of the strain rate, and the reaction timescale, calculated via Eq. (9) for bimolecular reactions¹⁸ and Eq. (10) for trimolecular reaction,¹⁸ where molecules A and B are the reactants and k is the Arrhenius reaction rate coefficient,

Analyzing the Damköhler number can provide an insight on the reaction rates. The Da was calculated at the location of greatest heat flux to the surface since the detachment of the flame since extinction only occurred after flame detachment. The Da as a function of Re at various air inlet temperatures for several reactions is displayed in Figs. 9–11 with the remaining presented in the  Appendix for completeness. For all cases and most reactions, Da is very large at low Re, indicating that the chemical timescales are much shorter than the transport timescales. This observation is expected in a diffusion limited flame. The Da decreases for most reactions as the extinction conditions are approached.

The results indicate that the third body reactions do not play a significant role in the chemical process since the largest Da for those reaction is on the order of 1. A Da on the order of 1, which occurs at 300 K, suggests the transport, and chemistry rates are similar despite the flame being transport limited and likely do not contribute. Furthermore, the slower reaction rates of third body reactions are expected considering the lower pressures/densities and, therefore, lower probability of three-body reactions occurring.

It is also shown that the Da values for the reactions involving the H-abstraction of H₂, such as in Fig. 9, do not appear to a have as much air inlet temperature dependence as other reactions, such as chain branching reactions in Figs. 10 and 11. A change in the reaction rate of the reactions involving the breakdown of H₂ is not expected when the fuel is at a constant 300 K, far from the dissociation temperatures of hydrogen. The lack of hydrogen breakdown is likely due to the faster diffusion speeds of H₂ not allowing adequate time for transportation of energy from the air stream to reach the fuel inlet. Figure 9 shows that regardless of air inlet temperature, the Da approaches the order of $10−1$ as extinction approaches. This result suggests that the radical attack on molecular hydrogen becomes slow and leads to extinction.

Since H₂ concentrations are high at the fuel inlet, the radical concentration/production is likely the cause of increased timescale for the H₂ + OH reaction. Figures 10 and 11 and Table III summarize the several important chain branching reactions. Higher air inlet temperatures affect the production of O and OH radicals and are observed with the increase in Da for reactions H + O₂ and H₂O + O. Specifically, at higher air inlet temperatures, the chemical timescales for these reactions are very fast (high Da). Conversely, the chemical timescales decrease sufficiently at higher Re at lower enthalpy flows (300–500 K) that extinction occurs due to a lack of radical production. The relative change in extinction Da for these reactions is displayed in Table III. The greatest change from 300 to 1000 K at extinction was H₂O + O. This chain branching reaction is endothermic. Thus, this chemical pathway only occurs when the enthalpy of the flow is high enough.

Further analysis of the reactions highlights the importance on the OH radical, specifically reactions involving HO₂ or H₂O₂. In these reactions, OH radicals play an important role by providing the necessary hydrogen abstraction for H₂O production. The occurrence of these reactions is beneficial due to the highly exothermic formation of H₂O and the potentiality of H₂O dissociation to 2OH. The formation of 2OH from H₂O dissociation is precisely occurring at higher air inlet temperatures, as seen with the increase in the Da value. In essence, at higher temperatures, HO₂ and H₂O₂ species have a pathway to convert to OH radicals via H₂O dissociation and increase the overall OH production. This increase in overall OH production additionally enhances the reactions rates of OH and HO₂ or H₂O₂ reactions, as shown with the a magnitude change of 10⁴ and 10⁶, respectively.

The process described above establishes the methodology for the lack of flame detachment and large heat flux drop for air inlet temperatures between 600 and 1000 K. The extra contributions in OH radical production from dissociating H₂O are the driving force behind the sustained flame at higher Re for higher air inlet temperatures. This extra OH production develops a larger radical pool allowing the reaction to keep pace with the transport timescales, which sustains the combustion process.

The effect of air inlet temperatures on the combustion process and flame holding properties of a slab burner was presented in this study. It was found that the air mass flux, conductive flux, and convective flux behave differently between two temperature regimes: 300 to 500 K and 600 to 1000 K. For each respective temperature regime, the blow-off air mass flux was found to be near constant. Additionally, the blow-off air velocity increased with an increase in air inlet temperatures allowing for higher strain rates. The normalized flame height data indicated a similarity solution behavior, with the application of a Re temperature scaling showing the data collapse onto itself. For all air inlet temperatures, the normalized flame height decreased with Re due to the higher axial momentum of the air. Additionally, it was found that for all air inlet temperatures studied, a constant normalized flame height was approached with the increase in Re possibly due to the compression limit of the hydrogen fuel.

At the temperature regime of 300 to 500 K, there is a large drop in conductive heat flux at the fuel inlet edge due to a detaching flame. For temperatures of 600 to 1000 K, the conductive heat flux was near constant and sustained longer at higher Re, which strongly indicated an attached flame. Overall, it was identified that convective heat flux was the dominant mode of heat transfer. The convective heat flux at temperatures between 600 and 1000 K sharply increased as a result of the increased turbulent energy mixing.

In examining the influence of temperature on the gas-phase chemistry, specifically changes with reaction pathways, it was determined that higher air inlet temperatures activated an endothermic chain branching reaction. The additional contribution of the endothermic reaction allowed for increased radical production and pool, leading to smaller reaction timescales more comparable to transport scales at high Re. Endothermic reaction's contributions to radical pool development were also seen in a recent study by Rabbani et al., where explosion modes were activated from these reactions and potentially led to a superadiabatic temperature phenomenon.²⁴ Similarly, it was also noted in their work that the endothermic reaction contributions enhanced radical production and led to auto-ignition and combustion sustainment with decreased explosive timescales. The enhanced radical production is specifically the cause for sustained conductive heat flux values, sharper increase in convective heat flux, and flame attachment for higher air inlet temperatures at high Re.

A.G. and E.M.K. would like to acknowledge the support of the CSULB Foundation. J.K. would like to acknowledge the support of the Office of Naval Research Advanced Energetic Materials program under Award No. N00014-20-1-2711.

The authors have no conflicts to disclose.

Andrew Gonzalez: Conceptualization (equal); Formal analysis (equal); Validation (equal); Visualization (equal); Writing - original draft (equal). Joseph Kalman: Conceptualization (equal); Writing - original draft (equal); Writing - review and editing (equal). Ehsan Madadi-Kandjani: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Software (equal); Supervision (equal); Writing - original draft (equal); Writing - review and editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Da for different Re numbers of each reaction plotted at each air inlet temperature (Fig. 12).

Reaction mechanism for the hydrogen combustion (Table IV).