Exploration of graph-theoretic algorithms for numerical simulation of hydrogen fuel supersonic combustion characteristics

As the power unit of hypersonic vehicle, the supersonic combustion ramjet engine is mainly composed of inlet, isolation section, combustion chamber and tail nozzle [1]. With fewer moving parts and relatively simpler structure than turbojet and turbofan engines, the required oxidizer can be taken directly from the atmosphere, and the subsonic or supersonic combustion can be realized in the combustion chamber, which can have a better specific impulse in high-speed flight, and the cost is lower than that of other propulsive modes, which has a broad application prospect [2–4]. Therefore, mastering the key technology of the super-combustion ramjet engine is a necessary prerequisite for realizing hypersonic flight.

For the study of the combustion flow field of the scramjet engine, there are mainly three means: ground test, real flight test and numerical simulation. Generally speaking, the ground test is more convincing, but the test cost is expensive, the implementation cycle is long, and the combustion process in the super-combustion ramjet is extremely complex, and the flow field information obtained by the observation equipment is limited, which affects the effectiveness of the ground test [5–7]. Secondly, the external conditions of real flight tests are closest to the real situation, but the tests consume too much money, labor and time costs, and are extremely difficult to implement [8–9]. Comparatively, numerical simulation methods are an economical and effective means for the design and flow analysis of scramjet engines [10–13]. Numerical simulation has low cost, short implementation period, and can provide more details of the flow field to make up for the shortcomings of experimental measurements, etc. However, numerical simulation is limited by the development of related theoretical models, and it still can not completely replace the test [14–16]. In recent years, with the development of computer hardware level, high-performance parallel clusters provide a reliable hardware basis for numerical simulation, while the field of computational flow dynamics (CFD) used in numerical simulation has produced a series of new methods, which are becoming more and more prominent in super-combustion research.

Literature [17] analyzed the advantages of hydrogen fuel in super-combustion ramjet engine, on the one hand, the small ignition delay of hydrogen fuel allows rapid combustion and large amount of thrust, on the other hand, hydrogen fuel is a clean energy source with abundant production sources, however, its lower volumetric energy density and wider range of flammability also bring challenges for super-combustion ramjet engine design. Literature [18] evaluated the effect of different positions of strut-type flame stabilizers on the performance of hydrogen fuel supersonic combustion and solved the Navier-stokes equations by the SST k-ω turbulence model, and found that reasonable strut positions will improve the overall fuel efficiency and make the whole system energy efficient. Literature [19] conducted a numerical simulation study on the combustion chamber of a hydrogen-fueled scramjet engine under flight conditions and found that a two-stage combustion chamber combining a center strut injector and a wall-mounted ramp injector improves the mixing effect and possesses a better combustion performance compared to a single-stage combustion chamber with a center strut injector. Literature [20] investigated the effect of inlet air temperature and pressure on the combustion performance of a hydrogen-fueled supersonic combustion ram chamber at constant Mach number using a two-dimensional compressible Navier-Stokes (RANS) turbulence model and a finite rate/vortex dissipative reaction model. Literature [21] introduced specific shapes of multibranch plates in the combustion chamber of a cavity-based super-combustion ramjet engine and analyzed them in detailed numerical simulations, and the experimental results showed that the multibranch plates formed a distinct separation layer, which contributed to the efficient mixing of the fuel and oxidant by generating a strong vortex volume, and improved the combustion efficiency. Literature [22] showed that two types of backstepped branched plates are a better flame keeper within a supercombustion ramjet engine, so a set of in-house computational programs were developed to calculate the effect of the horizontal staggering distance of the two branched plates, and it was found that the asymmetric branched plates favored the combustion of the fuel, while the symmetric branched plates minimized the total pressure loss. A numerical simulation study of a fuel-air mixing scheme based on parallel stub plates has been carried out in the literature [23] to verify the accuracy and applicability of the proposed computational grid and numerical format for solving the hot and cold flow problems by comparing the numerical results with the experimental results.

The aim of this study is to explore the super-combustion characteristics of hydrogen fuel through numerical simulations and the application of graph theory algorithms in this field. In this paper, the combustor required for the study of hydrogen supersonic combustion characteristics is developed and the N-S system of equations is selected as the combustion control equations. Based on the combustion phases in the combustion chamber, a gas phase model and a particle phase model were designed. The application of graph theory in dynamics is modeled to analyze the variation rules of combustion chamber temperature, CO emission, and combustion flame stability when hydrogen is used as fuel.

2

Numerical simulation of combustion characteristics based on graph theory algorithms

2.1

Numerical simulation approach

The CFD solution process is shown in Fig. 1. Numerical simulation [24–25] and experiments compared to simulation experiments usually have the advantages of time-consuming, low cost, and can be more convenient to get the desired data. At the same time, the experiment can also not take into account the influence of the instrumentation and measurement process, simulation research, theoretical calculations is the main way adopted in this paper used to study the combustor. 1)

Pre-processing: First of all, the actual measurement of the specific dimensions of the burner parameters with tools, and then through the SOLIDWORKS software to draw the threedimensional model of the burner. And then, the three-dimensional stereo model was imported to support the vast majority of models, while being able to compensate for the model and precalculation processing. The most powerful meshing software is ICEM CFD software, which can not only generate tetrahedral unstructured mesh according to the characteristics of the model, but also generate hexahedral structured mesh and the boundary of the prismatic mesh and so on.

2)

Solving process: the process of calculating the computational area of the exploration model through the solver. In the experimental simulation of fluid flow, chemical coupling, and heat transfer, FLUENT simulation solver is a good application. Its main work is to have completed the mesh division of the fluid model to accept, according to the characteristics of the model and the data to be obtained to select the appropriate mathematical model for calculation.

3)

Post-processing: The results obtained from the solver are analyzed and summarized, and presented to the reader in the form of images and data, which use Origin, Tecplot, and CFD-POST and other plotting software.

2.2

Control equations

In this paper, the compressible N-S system of equations is used as the controlling equations [26], which is based on the laws of conservation of mass, conservation of momentum energy and conservation of components, and the transient, compressible, multicomponent N-S in a threedimensional rectangular coordinate system is shown in Eq. (1): 1 $\frac{\partial U}{\partial t} + \frac{\partial F}{\partial x} + \frac{\partial G}{\partial y} + \frac{\partial H}{\partial z} = \frac{\partial F_{v}}{\partial x} + \frac{\partial G_{v}}{\partial y} + \frac{\partial H_{v}}{\partial z} + S$ where the subscript “v” means that the term is viscous, and the expressions of the vectors in Eq. are shown below: 2 $\begin{array}{l} U & = [\begin{matrix} ρ \\ ρ u \\ ρ v \\ ρ w \\ ρ E \\ ρ Y_{i} \end{matrix}], F = [\begin{matrix} ρ u \\ ρ u^{2} + p \\ ρ u v \\ ρ u w \\ (ρ E + p) \\ ρ u Y_{i} \end{matrix}], G = [\begin{matrix} ρ v \\ ρ u v \\ ρ v^{2} + p \\ ρ v w \\ (ρ E + p) v \\ ρ v Y_{i} \end{matrix}] \\ H & = [\begin{matrix} ρ w \\ ρ u w \\ ρ v w \\ ρ w^{2} + p \\ (ρ E + p) w \\ ρ w Y_{i} \end{matrix}], S = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ ω_{i} \end{matrix}] \end{array}$ 3 $F_{v} = [\begin{matrix} 0 \\ τ_{x x} \\ τ_{x y} \\ τ_{x z} \\ u τ_{x x} + v τ_{x y} + w τ_{x z} + q_{x} + \sum_{i = 1}^{N} ρ D_{i m} h_{i} \frac{\partial Y_{i}}{\partial x} \\ ρ D_{i m} \frac{\partial Y_{i}}{\partial x} \end{matrix}]$ 4 $G_{y} = [\begin{matrix} 0 \\ τ_{x y} \\ τ_{y y} \\ τ_{y z} \\ u τ_{x y} + v τ_{y y} + w τ_{y z} + q_{y} + \sum_{i = 1}^{N} ρ D_{i m} h_{i} \frac{\partial Y_{i}}{\partial y} \\ ρ D_{i m} \frac{\partial Y_{i}}{\partial y} \end{matrix}]$ 5 $H_{v} = [\begin{matrix} 0 \\ τ_{x z} \\ τ_{y z} \\ τ_{z z} \\ u τ_{x z} + v τ_{y z} + w τ_{z z} + q_{z} + \sum_{i = 1}^{N} ρ D_{i m} h_{i} \frac{\partial Y_{i}}{\partial z} \\ ρ D_{i m n} \frac{\partial Y_{i}}{\partial z} \end{matrix}]$ where u, v, and w are the velocities in the three directions in the right-angle coordinate system, p is the pressure, ρ is the density, q is the heat conduction term, Y_i is the mass fraction of the gas, ω_i is the mass production rate of the i th component, D_im is the diffusion coefficient of the i th component, and h_i is the corresponding enthalpy, and the equation for calculating the enthalpy is: 6 $h = \int_{T_{0}}^{T} c_{p} d T + h_{0}$ where h₀ is the enthalpy produced by the component at the T₀ reference temperature, and the constant pressure specific heat of the component as well as the enthalpy can be fit by a polynomial: 7 $\frac{c_{p}}{R} = a_{1} + a_{2} T + a_{3} T^{2} + a_{4} T^{3} + a_{5} T^{4}$ 8 $\frac{h}{R T} = a_{1} + \frac{a_{2}}{2} T + \frac{a_{3}}{3} T^{2} + \frac{a_{4}}{4} T^{3} + \frac{a_{5}}{5} T^{4} + a_{6} T^{- 1}$

Where R is the universal gas constant R = 8.3145J/(mol·k). E is the total internal energy per unit mass: 9 $E = e + \frac{1}{2} (u^{2} + v^{2} + w^{2})$ where e is the specific internal energy per unit mass.

The shear stress tensor τ_ij is expressed as follows: 10 $τ_{x x} = μ_{e f f} (- \frac{2}{3} \nabla \cdot \vec{V} + 2 \frac{\partial u}{\partial x})$ 11 $τ_{y y} = μ_{e f f} (- \frac{2}{3} \nabla \cdot \vec{V} + 2 \frac{\partial v}{\partial y})$ 12 $τ_{z z} = μ_{e f f} (- \frac{2}{3} \nabla \cdot \vec{V} + 2 \frac{\partial w}{\partial z})$ 13 $τ_{x y} = τ_{y x} = μ_{e f f} (\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y})$ 14 $τ_{y z} = τ_{z y} = μ_{e f f} (\frac{\partial w}{\partial y} + \frac{\partial v}{\partial z})$ 15 $τ_{x z} = τ_{z x} = μ_{e f f} (\frac{\partial w}{\partial x} + \frac{\partial u}{\partial z})$ 16 $\nabla \cdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}$ where μ_eff is the effective coefficient of viscosity of the molecule. It is calculated as follows: 17 $μ_{e f f} = μ + μ_{t}$ where μ is the laminar viscosity coefficient, μ_t is the turbulent viscosity coefficient, μ_t can be derived from a turbulence model, and μ is derived from Sutherland’s formula: 18 $μ = μ_{0} {(\frac{T}{288})}^{15} \cdot \frac{288.15 + C}{T + C}$ where μ₀ = 1.789×10⁻⁵ N · s/m², C = 110.4K.

The heat transfer term q_i is expressed as: 19 $q_{x} = K_{e f f} \frac{\partial T}{\partial x}, q_{y} = K_{o f f} \frac{\partial T}{\partial y}, q_{z} = K_{e f f} \frac{\partial T}{\partial z}$ where K_eff is the effective conductivity factor. Its calculation formula is as follows: 20 $K_{e f f} = K + K_{t}$ where K is the laminar heat transfer coefficient and K_t is the turbulent heat transfer coefficient.

2.3

Gas phase modeling

2.3.1

Gas-phase turbulence modeling

Turbulence phenomenon is a chaotic and strongly varying, multi-scale coupled and dissipative threedimensional fluid motion state in space and time controlled by the N-S equations, and there is a great deal of uncertainty in the turbulence phenomenon compared to laminar flow [27]. DNS (direct solution), LES (large eddy simulation), and RANS (Reynolds’ time-averaged equations) are the three classical methods for solving turbulence.

In this paper, we choose to use RANS to solve the turbulence inside the combustion chamber [28]. RANS, on the other hand, utilizes various types of assumptions and statistical theories to average the N-S equations in time to solve the time-averaged quantities needed in engineering. It is characterized by low computational effort and a wide range of applicable Reynolds number has a good application in engineering.

Due to the strong decompressibility of the supersonic flow, a two-equation SSTk – ω model is chosen for the computation, which has good adaptability by using equation k – ω for the near-wall surface and equation k – ε for more distant locations, in the following form: 21 $\frac{D ρ k}{D t} = \frac{\partial}{\partial x_{j}} [(μ + σ_{k} μ_{i}) \frac{\partial k}{\partial x_{j}}] - ρ \bar{u_{i}' u'} \frac{\partial u_{i}}{\partial x_{j}} - β^{*} ρ ω k$ 22 $\frac{D ρ ω}{D t} = \frac{\partial}{\partial x_{j}} [(μ + σ_{ω} μ_{t}) \frac{\partial k}{\partial x_{j}}] - ρ \bar{u_{i}' u_{j}'} \frac{γ}{v_{t}} \frac{\partial u_{i}}{\partial x_{j}} - β ρ ω^{2} + 2 (1 - F_{1}) ρ σ_{ω 2} \frac{1}{ω} \frac{\partial k}{\partial x_{j}} \frac{\partial ω}{\partial x_{j}}$ 23 $μ_{t} = \frac{a_{1} k}{\max (a_{1} ω; Ω F_{2})}$ Where: Ω is the vorticity, $F_{2} = \tanh (\arg_{2}^{2})$ , $\arg_{2} = \max (2 \frac{\sqrt{k}}{0.09 ω y}; \frac{500 v}{y^{2} ω})$ y are the distances from the wall.

2.3.2

Gas-phase combustion modeling

In this paper, the main combustible components in the rich combustion gas produced by the gas generator include hydrogen, carbon monoxide, carbon particles and boron particles, etc., of which hydrogen and carbon monoxide are in the gas phase. The carbon particles and boron particles are the granular phase or condensed phase.

In order to simplify the calculation, this paper chooses to simulate the combustion of gas-phase components through the eddy dissipation model (EDM). The gas-phase reaction equations are as follows: 24 $2 C O + O_{2} \to 2 C O_{2}$ 25 $2 H_{2} + O_{2} \to 2 H_{2} O$

The EDM model equation is as follows: 26 $W_{s} = \min (W_{s, E B U}, W_{s, A r})$ 27 $W_{s, E B U} = C_{R} ρ \frac{k}{ε} \min (Y_{F}, \frac{Y_{o x}}{β})$ 28 $W_{s, A r r} = B_{s} ρ^{2} Y_{F} Y_{o x} \exp (- \frac{E_{s}}{R T})$ where W represents the reaction rate (mol/L · s), E represents the activation energy (kJ/mol), Y represents the mass fraction of phase components (%), β represents the equivalence ratio (dimensionless).

2.4

Particle phase modeling

2.4.1

Particle term motion model

Since the model also solves the N-S equations, there are also three methods: RANS, LES, and DNS. The granular phase introduces some second-order matrices when using the RANS method, which can only be dealt with by empirical formulas, leading to limitations in its scope of application and relatively low accuracy. The particle orbital model is based on the idea of Lagrangian method, the particle mass point as the object of study is to track the trajectory of a specific mass point through the continuous flow, and the position and velocity of each particle is obtained by integrating the ordinary differential equations.

In this paper, the particle phases present in the combustion chamber are mainly carbon particles, boron particles, and some inert particles. Due to the high temperature of the scramjet engine, the oxide film on the surface of the particles will be melted by the high temperature and become liquid or even some of the small diameter particles will be liquefied directly. This results in the combustion chamber interior actually including three forms of material: gas phase, solid phase, and liquid phase. Since the droplet rupture and deformation process of metal droplets is very complex, and the liquefied particles in this paper only account for a small part of the total number of particles, the effect of the liquefaction of the oxide film on the surface of the metal particles or the liquefaction of the whole particles on the combustion of the particle phase motion is ignored in this paper.

In this paper, we focus on the combustion of the particle phase in the combustion chamber According to the previous comparative analysis of the two models, the random orbit model is chosen to solve the motion of the particle phase in the engine.

Under the Cartesian coordinate system, the particle dynamics equations using the Lagrangian method are: 29 $\frac{d {\vec{X}}_{p}}{d t} = {\vec{V}}_{p}$ 30 $\frac{d {\vec{V}}_{p}}{d t} = {\vec{F}}_{p} + F$ where ${\vec{X}}_{p}$ is the particle position vector, ${\vec{V}}_{p}$ is the velocity vector of the particle phase, ${\vec{F}}_{p}$ is the drag force per unit mass of the particle, and F is the other external forces on the particle phase, including Stephan flow, pressure gradient action, and volume forces.

It is customary to express the drag force per unit mass of particle as: 31 ${\vec{F}}_{p} = \frac{3}{4} \frac{C_{D} ρ}{ρ_{p} d_{p}} (\vec{V} - {\vec{V}}_{p}) | \vec{V} - {\vec{V}}_{p} |$ where $\vec{V}$ is the velocity vector of the gas phase and the trailing coefficient C_D is expressed as: 32 $C_{D} = {\begin{matrix} \frac{24}{{Re}_{p}} (1 + \frac{1}{6} {Re}_{p}^{2 / 3}) & {Re}_{p} < 1000 \\ 0.44 & {Re}_{p} \geq 1000 \end{matrix}$

The particle Reynolds number Re is expressed as: 33 ${Re}_{p} = \frac{ρ d_{p} | \vec{V} - {\vec{V}}_{p} |}{μ}$

From the instantaneous momentum equation of the particle, there is: 34 $\frac{d v_{k i}}{d t} = \frac{v_{i} - v_{k i}}{τ_{r}} = \frac{{\bar{v}}_{i} - v_{i}' - v_{k i}}{τ_{r}}$ where v_i, v_ki are the instantaneous velocities of the fluid and particle, respectively, ${\bar{v}}_{i}$ is the average velocity, v_i is the pulsation velocity, and τ_r is the relaxation time of the particle.

2.4.2

Particulate phase combustion modeling

In this paper, a kinetic/diffusion-limited rate model is adopted to describe the combustion process in the granular phase. The reactions in the particulate phase are mainly: 35 $C + O_{2} \to C O$ 36 $B + O_{2} \to B_{2} O_{3}$

The combustion rate equation is shown in equation (37): 37 ${\begin{matrix} \frac{d m_{p}}{d t} = - π d_{p}^{2} p_{o x} \frac{D_{0} R}{D_{0} + R} \\ D_{0} = C_{1} \frac{{[(T_{p} + T_{\infty}) / 2]}^{0.75}}{d_{p}} \\ R = C_{2} \exp (- \frac{E}{R T_{p}}) \end{matrix}$

Where: D₀ represents the diffusion coefficient (dimensionless), R represents the kinetic reaction rate (mol/L · s); T_p and T_∞ represent the ambient particle and gas temperatures (K) and p_ox represent the partial pressure of oxygen (Pa) and C₁ represent the mass-diffusion limiting rate constant, and C₂ represents the frequency factor.

2.5

Algorithmic Modeling for Graph Theory Analysis

2.5.1

Graph-theoretic topology analysis algorithms

The proximity centrality of a node i is defined as the reciprocal of the average unit shortest path length between it and all other nodes, and is often used to assess the ease with which it can establish connections with other nodes, expressed as follows: 38 $C l o (i) = \frac{N - 1}{\sum_{j} d_{i, j}}$

Where N is the number of nodes, d_i,j is the unit shortest path length from node i to node j, i.e., the minimum number of directed edges contained in the path. The value of proximity centrality ranges from 0 to 1. The higher the proximity centrality of a node is, the easier it is to transfer information with other nodes.

The process of establishing connections between nodes may not be direct, but requires the participation of intermediary transition nodes. The intermediary centrality can be measured by the role played by a node i in the process of connecting other nodes, which is expressed by the following equation: 39 $B e t (i) = \sum_{j = 1, l = 1}^{N} \frac{S_{j, l} (i)}{S_{j, l} (N - 1) (N - 2) / 2}$ where S_j,l denotes the kind of unit shortest paths that node j has to connect with node l, and S_j,l (i) denotes the number of unit shortest paths among them that can go through node i.

In graph theory, nodes that can be connected by only one directed edge are mutually called neighbor nodes. Taking a node i as the central node, if the network formed by it together with its surrounding neighbor nodes is taken out, the network can be called the subnet work of node i. In order to study how much the complexity of each subnetwork is affected by the connectivity among the nodes in the subnetwork, a clustering factor is introduced as follows: 40 $C l u (i) = \frac{b_{i}}{g_{i} (g_{i} - 1) / 2}$

Where g_i denotes the number of neighboring nodes of node i, and b_i denotes the number of these neighboring nodes that can be connected to each other by only one directed edge.

The clustering coefficient is a measure of the closeness of the network, and subnetworks with high clustering coefficients always have high cohesion, in which case the connectivity among neighbor nodes often does not require the introduction of a central node as a transition node, implying to some extent that the central node may have a lower intermediary centrality.

2.5.2

Graph-theoretic shortest path algorithms

The transfer of information on the shortest path is always the most rapid without considering the effects of perturbations. For plasma topological networks, the reaction represented by the shortest path tends to obtain the maximum production of the product based on the minimum loss of reactants, which provides a new idea to carry out industrial practice to reduce the cost. In the study of this paper, Dijkstra’s algorithm will be used to analyze and simplify the reaction kinetics of hydrogen fuel superfuel characterization in order to obtain a reasonable optimal solution. In the process of determining the path weights, appropriate treatment will be carried out to avoid the uncertainty caused by the appearance of negative weights.

Dijkstra’s algorithm is often used to solve the shortest path problem in single-source directed networks, which adopts a “greedy strategy”. First, a node a is selected as the source, the distance between the source and its neighbors is recorded in the set K, the edge with the smallest distance is kept as the shortest path, and the end node b of the shortest path is recorded, then the above operation is repeated from node b, the distance between the beginning node a and the neighbors of b is recorded, and if there is a shorter path, the length of that path is updated to the set K. The relaxation operation is continued outward, and the length of that path is updated to the set K. The shortest path is then recorded in the set K. The shortest path is the one with the smallest distance. The relaxation process is extended outward until it reaches the last node. By doing so, the shortest path between the originating node and all the remaining nodes will be determined.

3

Experimental results of numerical simulation of hydrogen fuels

3.1

Arithmetic validation based on a super-combustion ramjet combustion chamber

The supersonic combustion ramjet combustion chamber configuration referenced in this paper is the DLR combustion chamber. Where the total length of the combustion chamber is 340 mm and the inlet width is 50 mm. the combustion chamber is separated at a separation angle of 3° at a distance of 100 mm from the upper wall surface. The distance between the hydrogen strut injector and the inlet was 77mm and 25mm, respectively. The strut had a total length of 32mm and a height of 6mm. the angle of the top of the triangle was 12° and the nozzle diameter was 1mm.

In addition, free flow air enters the combustion chamber from the isolation section at a speed of Mach 2, while hydrogen fuel is injected from the stub at a speed of Mach 1. The static pressures of both the free-flow air inlet and the hydrogen inlet were 100 kPa, and the static temperatures were 340 K and 250 K. In addition, the air fractions of oxygen, nitrogen, and water were set to be 0.25, 0.71, and 0.04. Dirichlet boundary conditions were employed to define the free-flow air inlet and hydrogen inlet, while Neumann boundary conditions were employed to define the outlet. All walls were set to a fixed no-slip boundary and a constant temperature of 330K.

In this paper, to verify the accuracy of the numerical results obtained, the solver is set up using the same simulation conditions as in the DLR scramjet engine combustion chamber experiments. Fig. 2 Results of air and hydrogen mixing efficiency and total pressure loss compared with Kummitha model. The results indicate that the mixing efficiency and total pressure loss calculated from the present numerical results are extremely similar to those obtained by Kummitha. The absolute values of the difference between the calculated results for mixing efficiency and total pressure loss range from 0.001 to 0.04. Overall, this experiment provides strong evidence for the accuracy of the numerical results.

3.2

Numerical simulation analysis results

3.2.1

Combustion temperature field analysis

This section explores the temperature variation in the combustion chamber for different hydrogen doping ratios. The combustion chamber is operated at a thermal load of 180 KW, with a boiler thermal efficiency of 01 and an air excess coefficient of α of 1. The fuel gas composition is a mixture of natural gas mixed with hydrogen at ratios ranging from 0 to 0.3. The fuel gas composition is a mixture of natural gas and hydrogen at ratios ranging from 0 to 0.3.

Due to the small difference between the hydrogen doping ratio of each condition, in order to compare clearly, the cloud selected the minimum, maximum and intermediate groups of hydrogen doping ratio of a total of three groups of conditions, respectively, Rf = 0, Rf = 0.15, Rf = 0.3, compared with its high temperature zone temperature cloud.

Figure 3 shows the results of the comparison of the temperature maps in the high temperature region of the combustion chamber. It can be seen from Fig. 3 that the combustion temperature distributions of the conditions in the interval of hydrogen doping ratio from 0 to 0.3 are roughly the same, and they are all in the range of 400 K to 2000 K. The high temperature region is in the axial direction, and it is also in the axial direction. The range of the high temperature region in the axial direction increases as the hydrogen doping ratio increases.

In order to further compare the axial temperature distributions for each condition, this section also observes the temperature variations at different axial distances for gases with different hydrogen doping ratios. The results of different axial temperature distributions for each working condition are shown in Fig. 4. The data in the figure shows that the temperature change at different axial distances has a trend of increasing, then decreasing, and stabilizing. And the highest temperature point of different working conditions all appeared between 2m and 2.5m. In addition, the axial maximum temperature increases with the increase of hydrogen doping ratio, as shown in the figure, the maximum temperature increases by 3.65% with the hydrogen doping ratio from 0 to 0.3. This indicates that hydrogen, when used as fuel, has a supersonic combustion characteristic that increases the temperature of the combustion chamber.

3.2.2

CO emission patterns for different particle combustion

This section analyzes the CO emission pattern at different temperatures for carbon particles in the combustion chamber.

Keeping the oxygen concentration in the environment as 25%, the wind speed as 0.5m/s, and the ambient temperature between 1200K and 1500K, the effect of temperature on the CO emission law of the carbon particles combustion process is investigated, and the results of the CO emission law at different temperatures are shown in Fig. 5.

As can be seen from Fig. 5, at the beginning of the sintering process, the carbon particles are in the warming stage, and the fuel is not burned, so there is no CO production. As the temperature reaches the ignition point of the coke, the coke begins to burn, at which time the generation of CO increases sharply and reaches its highest value. After the start of the combustion reaction, the oxygen diffused into the carbon particles is rapidly consumed, at which time the burning rate of the coke decreases rapidly, and the amount of CO generated is decreasing at a very rapid rate. As the combustion reaction proceeds, the internal coke of the carbon particles is gradually consumed, the fuel particle size decreases, the reaction rate decreases, and the amount of CO generated decreases. The consumption of CO mainly occurs in the process of diffusion to the outside as well as in the secondary combustion reaction with oxygen on the surface of the carbon particles, and with the decrease in the internal coke size, the internal porosity of the carbon particles increases, and the resistance of the diffusion of oxygen to the inside decreases, and the oxygen concentration inside the particles increases accordingly. Concentration is also increased accordingly, and the higher the degree of secondary reaction of CO generated by fuel combustion inside the carbon particles, the higher the complete combustion rate, so the amount of CO generation is gradually reduced.

As can be seen from Fig. 5, the incomplete combustion rate of the fuel in the sintering process of carbon particles gradually decreases as the temperature rises, and the total CO emission from carbon particles appears to decrease by about 5.32% to 7.51% when the ambient temperature increases from 1200 to 1500K. This is mainly due to the increase in the reaction rate of CO with oxygen within the stagnant layer on the surface of carbon particles when the temperature is increased. The thermodynamic analysis of the carbon reaction during sintering combustion revealed that an increase in temperature favors the complete combustion of fixed carbon.

3.2.3

Variation of Blow-Out Limit and Flame-Out Limit with Hydrogen Doping Ratio

In this section, the trends of blowout limit and flame quenching limit with hydrogen doping ratio are investigated to evaluate the effect of hydrogen as fuel on combustion flame stability.

The flame blowout limit and flame quenching limit for different hydrogen doping ratios are shown in Fig. 6. It can be seen from the figure that the blowout limit increases gradually with the increase of hydrogen doping ratio. When the hydrogen doping ratio is 30%, the blowout limit is increased by 104.84% compared with that of non-hydrogen doped fuel; the flame quenching limit is 0.135 m/s without hydrogen doping; when the hydrogen doping ratio is 15%, the flame quenching limit decreases to 0.084 m/s; when the hydrogen doping ratio is increased to 30%, the flame quenching limit is further reduced to 0.055 m/s. Obviously, with the increase of hydrogen doping ratio, the flame stabilization interval is significantly widened, which means that combustion stability is significantly enhanced. This means that the combustion stability is significantly enhanced. In conclusion, with the increase of hydrogen doping ratio, the combustion chamber is prone to flashback and oscillation, which will change the flame blowout limit speed.

4

Conclusion

In this study, experiments were carried out through numerical simulations to investigate the supersonic combustion characteristics of hydrogen fuels. The combustion characteristics of fuels with different hydrogen doping ratios are explored by controlling equations, gas-phase and particle-phase modeling, and graph theory algorithms.

In this paper, using the DLR combustion chamber simulation conditions, the mixing efficiency and total pressure loss results were measured and the absolute values of the differences from the Kummitha model were distributed between 0.001 and 0.04. The axial maximum temperature of the combustion chamber occurs from 2m to 2.5m and increases by 3.65% when the hydrogen doping ratio is increased from 0 to 0.3. As the temperature of the combustion chamber increased, the CO emission from carbon particles showed a decreasing trend of 5.32% to 7.51%. The more hydrogen doping of the fuel, the blowout limit of the combustion chamber flame increased, while the flameout limit showed a decrease.

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Exploration of graph-theoretic algorithms for numerical simulation of hydrogen fuel supersonic combustion characteristics

Pinghua Yan

Shufen Yang

Pubblicato online: 19 mar 2025

Ricevuto: 03 nov 2024

Accettato: 05 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0394

Parole chiaveHydrogen fuel, Numerical simulation, Kummitha model, Graph theoretic algorithm, DLR combustion chamber

© 2025 Pinghua Yan, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Parole chiave
Hydrogen fuel, Numerical simulation, Kummitha model, Graph theoretic algorithm, DLR combustion chamber