Numerical Simulation Study on Combustion Flame Performances of a Diffusion Burner

Abstract

ANSYS-Fluent was applied to simulate diffusion combustion flame in a two-dimensional (2D) industrial burner to determine the contours of the mass fraction of gas emissions, velocity, and combustion temperature. The effects of the boundary conditions, including momentum, thermal, and species (inlet air, inlet fuel, and outlet pressure) on combustion temperature and mass fraction (gas emissions) were analyzed in the designed burner. The present study focused on using and analyzing the volumetric reaction and the turbulence-chemistry interaction of the eddy dissipation model for the diffusion flame model. The simulation used the discrete ordinate model and p1 for radiation and the k-ε model for turbulence with enhanced wall treatment. Based on the results, the magnitude velocities of air and fuel, inlet temperature, and mass fractions of oxygen and inert gas can influence the parameters of flame temperature and gas emissions in the industrial burner. The flame shape for all the cases of inlet velocity was predominantly symmetric about the x = 0 mm for all the axial distances towards the outlet. The radial velocity contour at 0.01 m/s (300 K) gave better results with an area of 1.31 m/s to 4.08 m/s, which was wider than that of the case at 0.01 m/s (700 K). By varying the inlet temperature and oxygen mass fraction, the flame configurations on temperature, CO₂, and H₂O formed a symmetric flame structure. The temperature distribution resulted in the centerline being hotter than other radial positions for all of the inlet temperatures. The emissions of CO₂ and H₂O generally increased with the addition of the oxygen mass fraction.

1. Introduction

A challenge of combustion is numerical simulation. Combustion research investigated the gas emissions of NOx and CO using 2D axisymmetric burners for methane–air diffusion flame combustion mode [1] and the numerical analysis of the partially premixed burner for industrial gas turbine combustion application [2,3]. The research used the Eddy Dissipation Model (EDM), a non-premixed and partially premixed flame model, and the simulation of syngas combustion by a non-premixed flame model coupled with the DO approach, which showed better agreement than experimental data [4].

In experimental and numerical investigations on the effect of different air–fuel mixing on the performance of a lean liquid-fueled swirl combustor, the numerical results show different temperatures and species fields predicted for the non-premixed and partially premixed models. This study used the combustion chamber (internal diameter of 0.14 m and length of 0.29 m) with the size of the central injector of 0.5 mm [5]. A numerical study applied the flamelet progress variable approach to the local flame structure for the partially premixed Dimethyl Ether (DME)/air flame. A straight tube with an inner diameter of 0.8 cm and a length of 40 cm was used, and it obtained excellent results for the temperature and species mass fraction [6]. A large eddy simulation (LES) of kerosene spray injection in an aeronautic combustion chamber (a rectangular shape of 130 mm × 130 mm) to assess the capability of an energy deposition model and to reduce the chemical kinetic and global spray injection was conducted by [7].

A type of burner, burner moderate and intense low oxygen dilution (MILD), was tested by the eddy dissipation concept [8], a numerical study in a non-premixed micro combustor with different fuel inlet velocities [9], and direct numerical simulation (DNS) in non-premixed MILD combustion. The research investigated auto-ignition, flame propagation, and aspects of the combustion with a lean mixture. Concepts applied preheating and dilution of the reactant mixture to obtain a higher temperature than the auto-ignition temperature of the used fuel. It utilizes exhaust gas recirculation (EGR) to improve gas emissions and increase the combustion efficiency and stability [10]. However, this method also raised the peak flame temperature and NOx. Inert gases from internal gas recirculation (IGR) have been one strategy to reduce NOx emissions. This method can also minimize oxygen use, known as moderate or intense low oxygen dilution (MILD) and flameless combustion. Both ways can enhance thermal efficiency and decrease gas emissions [11].

A combustion strategy was applied again, using two spray flames with diameters of 10.5 mm and 25 mm to outline the effect of mixing models. The CO₂ mass fraction for the three models had similar trends, with the result value ranging from 0.01 to 0.125 [12]. The mixing model has determined the flame shape, flow, turbulence, temperature, and species compositions. The predictions of the product H₂O by different models were similar to the prediction of temperature [13]. The reduction of nozzle diameter is one of the other ways to obtain an optimum operating condition, as carried out experimentally and numerically by [11] in premixed cyclone combustion. The nozzle diameter of 30 mm and the equivalence ratio of 0.89 can yield better gas emissions and flame temperature. The research observed that the sizes of the fuel orifice, air orifice, combustor diameter, and tailpipe diameter were 1 mm, 2 mm, 5.9 cm, and 2.7 cm, respectively, with pipes of three different lengths, 25.4 cm, 40.7 cm, and 56 cm [14].

Making suitable mixtures between the reactant flow and surroundings is a technique widely applied by modifying the nozzle geometry. It determines combustion products. Three nozzle types, namely circular, square, and rectangular, were investigated. The mean velocity contours using non-circular nozzles can reduce the potential core length by approximately 33%, compared to the circular nozzle [15]. The rectangular nozzle can produce a centerline velocity decay rate higher than that of the circular nozzle. However, the slot nozzle produces a centerline velocity decay rate higher than both the rectangular and circular nozzles. The research observed eight different nozzle geometries, such as a round, square, cross, eight-corner start, six-lobe daisy, equilateral triangle, ellipse, and rectangle. The effect of nozzle geometry was significant in the mean velocities and turbulent quantities [12], which can determine the diameter size of droplets produced. Applying a nozzle size of 20 mm for the gas turbine application, which was generally carried out. The small droplet sizes can evaporate much faster than bigger droplets [16].

The heat transfer process inside the burner was analyzed by coupling spray combustion, forced convection on the wall surface, and conduction in the solid with air heated over the wall. The results showed that above and below the cone possessed different behavior. Above the cone, the temperature gradient stimulated a hot flame, so the temperature of the fresh air elevated. However, below the torch, the negative temperature gradient and stable stratification were achieved [17]. The prediction of wall temperature can use multiphysics simulation with large eddy simulation (LES), conjugate, and radiative heat transfers. The gas temperatures downstream of the burner chamber were cold. It took place when considering the radiation phenomena, and it was close to the adiabatic temperature when close to the combustion centerline. The maximum temperature remained the same because the sudden temperature increase through the flame front was influenced by radiation heat transfer [18].

A comparative study tested non-premixed and partially premixed combustion models in a realistic Tay model combustor to identify temperature and species concentration [19]. To reduce gas emissions, a catalyst can also be used to obtain complete combustion [20]. The flame stability can be achieved by raising the temperature of the reactant species, which can also change other combustion characteristics of the flame temperature and the quality of gas emissions. As noted, the smaller geometry size of the combustion chamber can shorten the residence time of the combustion reaction of reactants. For better combustion, the residence time should be longer than the chemical reaction time. The rise of the mixture fraction temperature can shorten the combustion chemical reaction time [21,22]. As conducted by a researcher in [23] to determine a better chamber design, different chamber geometries were used to simulate species transport and a non-premixed combustion model. A combustion technique stabilized the combustion flame using a stoichiometric ratio. The reactant mixtures were acetylene, H₂, and air [24]. The other ways accelerated combustion chemistry and reduced CO on flame characteristics and chemical kinetics in a swirl gas turbine combustor using large eddy simulation (LES) [25], used a non-premixed combustion model, and distributed combustion conditions with the methane as a fuel [26]. A study performed the combustion research of the H₂/air mixture with CFD modeling for a 3D and 2D computational domain of a micro-cylindrical combustor [27]. That was using the eddy dissipation concept (EDC) and the combustion characteristics of the wall and fluid temperature, hydrogen mass fraction, velocity, and also pressure.

This numerical simulation uses a 2D burner for the diffusion combustion flame. Preheating the mixture fraction of the reactants can influence and determine the characteristics of the contours, flame temperature, gas emissions, and flame stability. This work aims to test the combustion performance of the designed geometry burner. Three parameters investigated are fuel and air inlet velocities, inlet temperatures, and oxygen concentrations.

2. Burner and Simulation Setups

The mathematical model considered several assumptions about the chemical and physical processes to solve and yield results in the combustion study cases. Even though simplifications were performed, perfect setups of mathematical forms have to have similar cases in turbulent flames. The fundamental study of current research is to attend to mathematical tools. This study investigated the turbulence–chemistry interaction of the eddy dissipation model by solving the general transport equations of mass, momentum, energy, and species concentrations.

The combustion chamber for the current simulation is shown in Figure 1. The sizes of which are 100 cm in length, 50 cm in diameter, 2 cm in fuel injection, and 3 cm in air injection. The computational grid used triangle methods together with three-edge refinement around fuel and air inlets. The max face size, minimum size, and size function are 2 × 10⁻³ m, 1.333 × 10⁻⁴ m, and curvature, respectively, which obtained the numbers of 146,973 nodes and 291,830 elements. To simulate the governing equations of mass, momentum, energy, and heat transfer for diffusion combustion flame models that use FLUENT-ANSYS 17.1 software. Pressure–velocity coupling uses a SIMPLE algorithm with a second-order upwind scheme to discretize the governing equations. The other solver setting with the gradient evaluation used the least-squares cell method. The momentum and energy use second-order upwind. The turbulent kinetic energy and turbulent dissipation rate are calculated using first-order upwind. The scale residuals for convergence are 1 × 10⁻⁶ for continuity, energy, and species mass fractions.

The k-ε model is commonly used in diffusion models for combustion analysis by modeling the turbulent kinetic energy (k) and its dissipation rate (ε). This can improve the mixing of fuel and oxidizer in diffusive flames, leading to faster combustion in complex flows. Diffusion flames are often turbulent, and these turbulent eddies are essential for mixing fuel and oxidizer, which drives the combustion process [28]. The boundary conditions were C₁₂H₂₃ as fuel and air as an oxidizer, consisting of 23% oxygen and 77% nitrogen. The inlet magnitude velocities of the fuel tested were 0.01, 0.03, 0.05, 0.075, 0.1, 0.3, 1, 5, and 10 m/s. The constant of the used air was 0.5 m/s. At the exit combustion chamber, a fixed pressure of 0 Pa was specified, and by choosing stainless steel as the combustion chamber material walls. The boundary conditions of the inlet fuel, inlet air, walls, and pressure in the current study are described in more detail in

Table 1. Boundary conditions of inlet fuel, inlet air, wall, and pressure.

Boundary Conditions	Parameters	Values
Inlet air
Momentum	Velocity (m/s)	0.5
	Hydraulic diameter (m)	0.03
	Turbulent intensity (%)	10
Thermal	Temperature (K)	300
Species	Oxygen (mass fraction)	0.23
Inlet fuel
Momentum	Velocities (m/s)	0.01, 0.03, 0.05, 0.075, 0.1, 0.3, 1, 5 and 10
	Hydraulic diameter (m)	0.02
	Turbulent intensity (%)	10
Thermal	Temperature (K)	300
Species	Kerosene (mass fraction)	1
Walls	Wall slip	0
	Material	Steel
	Thermal condition	Mixed
	Heat transfer convection (W/m2·K)	0
Outlet pressure	Gauge pressure	0
	Hydraulic diameter (m)	0.5
	Turbulent intensity (%)	10

. The setup of the simulation is shown in

Table 2. Setup of simulation.

Models	Parameters
Viscous model	K-ε Standard
Radiation model	P1
Combustion model	Diffusion combustion flame
Mixture properties	Kerosene (C12H23)-air
Turbulence chemistry interaction	Eddy dissipation model (EDM)
Reaction	Volumetric
NOx	Thermal and prompt NOx

. The simulation used an Intel (R) Core (TM) i5-7400 CPU @ 3.00 GHz and an installed RAM of 8 GB.

3. Numerical Combustion Model

Most combustors in industrial applications operate using the concepts of spray and reactive flows within the turbulent regime characterized by high Reynolds numbers. The simulation of multi-component fuel spray requires better treatment concepts to yield good turbulence within both the liquid and gaseous phases [28]. The oxy/air-fuel flow in many combustion systems is turbulent. The other types are laminar and transition. The transition flow exists between those two regimes. The turbulent flow is more complex mathematically than both of the flows [29].

3.1. Transport Equation and Chemical Model

The mathematical concept of fluid inside the combustion chamber is described by a set of governing Equations (1)–(4) for mass, momentum, species, and energy in Cartesian coordinates, respectively:

$${\displaystyle \frac{\partial \mathsf{\varrho }}{\partial \mathrm{t}}}+\nabla .(\mathsf{\varrho }\mathbf{u})=0$$

(1)

$${\displaystyle \frac{\partial \mathsf{\varrho }\mathbf{u}}{\partial \mathrm{t}}}+\nabla . (\mathsf{\varrho } \mathbf{u} \mathbf{u})=-\nabla \mathrm{p}+\nabla \mathsf{\tau }$$

(2)

$${\mathrm{c}}_{\mathrm{p}}\left[{\displaystyle \frac{\partial \mathsf{\varrho }\mathrm{T}}{\partial \mathrm{t}}}+\nabla . (\mathsf{\varrho }\mathbf{u}\mathrm{T})\right]=\nabla .\left(\mathsf{\varrho }{\mathrm{c}}_{\mathrm{p}}\nabla \mathrm{T}\right)+\sum _{\mathrm{i}}{\mathrm{c}}_{\mathrm{p},\mathrm{i}} \mathsf{\varrho }\left({\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{L}}_{\mathrm{e},\mathrm{i}}}}\nabla {\mathrm{Y}}_{\mathrm{i}}+{\mathrm{Y}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{c},\mathrm{i}}\right)\nabla \mathrm{T}+{\mathrm{r}}_{\mathrm{T}}$$

(3)

$${\displaystyle \frac{\partial \mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{t}}}+\nabla \mathsf{\varrho }\mathbf{u}{\mathrm{Y}}_{\mathrm{i}}=\nabla (\mathsf{\varrho }{\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{L}}_{\mathrm{e},\mathrm{i}}}}\nabla {\mathrm{Y}}_{\mathrm{i}}+\nabla (\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\mathrm{V}}_{\mathrm{c},\mathrm{i}})+\nabla {\mathrm{r}}_{\mathrm{i}}$$

(4)

where $\mathsf{\varrho }$ is the air–fuel mixture density, u is the velocity vector, and T is the mixture temperature. ${\mathrm{Y}}_{\mathrm{i}}$ is the mass fraction of species $\mathrm{i}$, $\mathsf{\alpha }$ is the thermal diffusivity, ${\mathrm{c}}_{\mathrm{p},\mathrm{i}}$ is the specific heat at a constant pressure of species $\mathrm{i}$, and ${\mathrm{r}}_{\mathrm{T}}$ is the heat source terms due to chemical reaction. ${\mathrm{V}}_{\mathrm{c},\mathrm{i}}$ is the correction velocity of species i. The shear stress takes place due to a velocity gradient. The stress tensor Equation (5) is

$$\mathsf{\tau }=\mathsf{\mu }{\displaystyle \frac{\mathrm{du}}{\mathrm{dy}}}=\mathsf{\mu } \left(\nabla \mathbf{u}+{(\nabla \mathbf{u})}^{\mathrm{T}}-{\displaystyle \frac{2}{3}}\left(\nabla \mathbf{u}\right)\mathrm{I}\right)$$

(5)

where $\mathsf{\mu }$ is the air-fuel dynamic viscosity, $\mathrm{I}$ is the identity matrix, and ^T is the transpose operation. The Lewis number (Equation (6)), the specific heat at a constant pressure of the air-fuel mixture (Equation (7)), and the correction velocity (Equation (8)), respectively, are calculated with the following equations.

$${\mathrm{L}}_{\mathrm{e},\mathrm{i}}={\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{D}}_{\mathrm{i}}}}$$

(6)

$${\mathrm{c}}_{\mathrm{p}}=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{Y}}_{\mathrm{i}} {\mathrm{c}}_{\mathrm{p},\mathrm{i}}$$

(7)

$${\mathrm{V}}_{\mathrm{c},\mathrm{i}}={\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{L}}_{\mathrm{e},\mathrm{i}}}} {\displaystyle \frac{\nabla \mathrm{W}}{\mathrm{W}}}-{\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{L}}_{\mathrm{e},\mathrm{j}}}} \left(\sum _{\mathrm{j}=1}^{\mathrm{N}}\nabla {\mathrm{Y}}_{\mathrm{j}}\right)-{\displaystyle \frac{\mathsf{\alpha }}{{\mathrm{L}}_{\mathrm{e},\mathrm{j}}}} {\displaystyle \frac{\nabla \mathrm{W}}{\mathrm{W}}}\left(\sum _{\mathrm{j}=1}^{\mathrm{N}}\nabla {\mathrm{Y}}_{\mathrm{j}}\right)$$

(8)

where $\mathrm{W}={\left({\sum }_{\mathrm{j}=1}^{\mathrm{N}}{\displaystyle \frac{{\mathrm{Y}}_{\mathrm{j}}}{{\mathrm{W}}_{\mathrm{j}}}}\right)}^{-1}$ is the local mean molecular weight of the air–fuel mixture, ${\mathrm{D}}_{\mathrm{i}}$ is the mass diffusivity of species $\mathrm{i}$. The Schmidt number is the ratio between viscosity and mass diffusion, ${\mathrm{S}}_{\mathrm{c}}=\mathrm{v}/\mathrm{D}$. The Prandtl number is the ratio between viscosity and heat transfer, ${\mathrm{P}}_{\mathrm{r}}=\mathrm{v}/\mathsf{\alpha }$. The Lewis number is the ratio between heat transfer and mass diffusion, ${\mathrm{L}}_{\mathrm{e}}=\mathsf{\alpha }/\mathrm{D}$. If ${\mathrm{L}}_{\mathrm{e}}=1$, the Schmidt and Prandtl numbers are ${\mathrm{P}}_{\mathrm{r}}={\mathrm{S}}_{\mathrm{c}}$.

The combustion process in the combustor can influence the flow field through the density parameter. The equation of state for ideal gases expresses the density, $\mathrm{p}=\mathsf{\varrho } \mathrm{R} \mathrm{T} {\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\displaystyle \frac{{\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{W}}_{\mathrm{i}}}}$. The local heat release rate (Equation (9)) and the relation between the enthalpy and the temperature (Equation (10)) are

$${\mathrm{r}}_{\mathrm{T}}=\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{h}}_{\mathrm{i}}{\mathrm{r}}_{\mathrm{i}}$$

(9)

$${\mathrm{h}}_{\mathrm{i}}={\mathrm{h}}_{\mathrm{i},\mathrm{o}}+{\int }_{{\mathrm{T}}_{\mathrm{o}}}^{\mathrm{T}}{\mathrm{c}}_{\mathrm{p},\mathrm{i}} \left(\mathrm{T}\right)\mathrm{dT}$$

(10)

The specific enthalpy of species $\mathrm{i}, {\mathrm{h}}_{\mathrm{i}}$ can be expressed as the sum of formation enthalpy at a reference temperature ${\mathrm{T}}_{\mathrm{o}}$, and the specific heat capacity at constant pressure for species i due to the temperature change from the reference temperature (${\mathrm{T}}_{\mathrm{o}}$) to the final temperature (T).

For the combustion reaction model, the field of chemistry concerning the reaction rate is called chemical kinetics. The chemical reaction rate, $\mathrm{Rate}={\displaystyle \frac{∆{\mathrm{C}}_{\mathrm{i}}}{∆\mathrm{t}}}$, is the decrement change in concentration of the air–fuel reactant ($∆{\mathrm{C}}_{\mathrm{i}}$) to the gas emissions as combustion products at the time per time ($∆\mathrm{t}$).

For the combustion process, the concentration of the fuel and air as the reactants depends on time. That always decreases with it, and the concentration of the combustion products in the reaction always increases with time. With this, the concentration of reactants always has a negative sign. For complete combustion reactions without involving N₂ and some inert gases, the main chemical reaction (Equation (11)) inside the combustion chamber and the fuel consumption rate (Equation (12)) can be determined as

$$\mathrm{p} \mathrm{Fuel}+\mathrm{q}{ \mathrm{O}}_2\hspace{1em}\begin{array}{c}{\mathrm{r}}_{\mathrm{f}}\\ \rightleftarrows \\ {\mathrm{r}}_{\mathrm{r}}\end{array}\hspace{1em}\mathrm{r} {\mathrm{H}}_2\mathrm{O}+\mathrm{s} \mathrm{C}{\mathrm{O}}_2$$

(11)

$${\mathrm{r}}_{\mathrm{f}}=\mathrm{k} {\left[{\mathrm{C}}_{\mathrm{Fuel}}\right]}^{\mathrm{p}} {\left[{\mathrm{C}}_{{\mathrm{O}}_2}\right]}^{\mathrm{q}}$$

(12)

The calculated p and q depend on the initial consumption rate and initial concentrations of fuel and oxygen. The sum of p and q is as an overall reaction order. Computing the combustion rate constant (Equation (12)) that can use the following Arrhenius equation (13)

$$\mathrm{k}=\mathrm{zp} {\mathrm{e}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}=\mathrm{A} {\mathrm{e}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$$

(13)

where A, z, and p are the frequency factor, the collision frequency, and the steric factor (p < 1), respectively, and ${\mathrm{e}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$ is the collision fraction at sufficient energy to produce a chemical reaction. Equation (12) can be rewritten as (Equation (14))

$${\mathrm{r}}_{\mathrm{f}}=\mathrm{A} {\left[{\mathrm{C}}_{\mathrm{Fuel}}\right]}^{\mathrm{p}} {\left[{\mathrm{C}}_{{\mathrm{O}}_2}\right]}^{\mathrm{q}} {\mathrm{e}}^{-{\mathrm{E}}_{\mathrm{a}}/\mathrm{RT}}$$

(14)

Equation (13) can also be expressed in the logarithmic form (Equation (15)) to find each of the variables

$$\ln \left(\mathrm{k}\right)=-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}}\left({\displaystyle \frac{1}{\mathrm{T}}}\right)+\ln \mathrm{A}$$

(15)

which is identically a linear equation, $\mathrm{y}=\mathrm{mx}+\mathrm{b}$, where $\mathrm{y}=\ln \left(\mathrm{k}\right), \mathrm{x}={\displaystyle \frac{1}{\mathrm{T}}},$ slope $\mathrm{m}=-{\displaystyle \frac{{\mathrm{E}}_{\mathrm{a}}}{\mathrm{R}}}$ and intercept $=$ $\mathrm{b}=\ln \left(\mathrm{A}\right)$.

Assuming the chemical equilibrium of gaseous emissions with a complete combustion reaction is occurring without involving N₂ and some inert gases, the equilibrium constant inside the combustion chamber (Equation (16)) is

$$\mathrm{K}={\displaystyle \frac{{\left[{\mathrm{H}}_2\mathrm{O}\right]}^{\mathrm{r}}{\left[\mathrm{C}{\mathrm{O}}_2\right]}^{\mathrm{s}}}{{\left[\mathrm{Fuel}\right]}^{\mathrm{p}}{\left[{\mathrm{O}}_2\right]}^{\mathrm{q}}}}={\displaystyle \frac{{{(\mathrm{C}}_{{\mathrm{H}}_2\mathrm{O}})}^{\mathrm{r}}{{(\mathrm{C}}_{{\mathrm{CO}}_2})}^{\mathrm{s}}}{{{(\mathrm{C}}_{\mathrm{Fuel}})}^{\mathrm{p}}{{(\mathrm{C}}_{{\mathrm{O}}_2})}^{\mathrm{q}}}}$$

(16)

For the complete reversible reaction, the equilibrium rate constant on the left side, as described in Equation (17), is

$${\mathrm{K}}_{\mathrm{r}}={\displaystyle \frac{{\left[\mathrm{Fuel}\right]}^{\mathrm{p}}{\left[{\mathrm{O}}_2\right]}^{\mathrm{q}}}{{\left[{\mathrm{H}}_2\mathrm{O}\right]}^{\mathrm{r}}{\left[\mathrm{C}{\mathrm{O}}_2\right]}^{\mathrm{s}}}}={\displaystyle \frac{1}{\mathrm{K}}}$$

(17)

From the ideal gas, $\mathrm{PV}=\mathrm{nRT} \mathrm{or} \mathrm{P}=\left({\displaystyle \frac{\mathrm{n}}{\mathrm{V}}}\right)\mathrm{RT}=\mathrm{CRT}$, the relation between pressure and concentration is

$${\mathrm{K}}_{\mathrm{p}}={\displaystyle \frac{{{(\mathrm{P}}_{{\mathrm{H}}_2\mathrm{O}})}^{\mathrm{r}}{{(\mathrm{P}}_{{\mathrm{CO}}_2})}^{\mathrm{s}}}{{{(\mathrm{P}}_{\mathrm{Fuel}})}^{\mathrm{p}}{{(\mathrm{P}}_{{\mathrm{O}}_2})}^{\mathrm{q}}}}$$

(18)

The equilibrium constant Equations (19) and (20) are

$$\mathrm{K}={\displaystyle \frac{{\left(\frac{{\mathrm{P}}_{{\mathrm{H}}_2\mathrm{O}}}{\mathrm{RT}}\right)}^{\mathrm{r}}{\left(\frac{{\mathrm{P}}_{{\mathrm{CO}}_2}}{\mathrm{RT}}\right)}^{\mathrm{s}}}{{\left(\frac{{\mathrm{P}}_{\mathrm{Fuel}}}{\mathrm{RT}}\right)}^{\mathrm{p}}{\left(\frac{{\mathrm{P}}_{{\mathrm{O}}_2}}{\mathrm{RT}}\right)}^{\mathrm{q}}}}={\displaystyle \frac{{{(\mathrm{P}}_{{\mathrm{H}}_2\mathrm{O}})}^{\mathrm{r}}{{(\mathrm{P}}_{{\mathrm{CO}}_2})}^{\mathrm{s}}}{{{(\mathrm{P}}_{\mathrm{Fuel}})}^{\mathrm{p}}{{(\mathrm{P}}_{{\mathrm{O}}_2})}^{\mathrm{q}}}}{\displaystyle \frac{{\left(\mathrm{RT}\right)}^{\mathrm{p}}{\left(\mathrm{RT}\right)}^{\mathrm{q}}}{{\left(\mathrm{RT}\right)}^{\mathrm{r}}{\left(\mathrm{RT}\right)}^{\mathrm{s}}}}$$

(19)

$$\mathrm{K}={\mathrm{K}}_{\mathrm{p}} {\left(\mathrm{RT}\right)}^{\left(\mathrm{p}+\mathrm{q}\right)-(\mathrm{r}+\mathrm{s})}={\mathrm{K}}_{\mathrm{p}} {\left(\mathrm{RT}\right)}^{∆\mathrm{n}}$$

(20)

where $\mathrm{K}$ and ${\mathrm{K}}_{\mathrm{p}}$ are equilibrium constants in terms of concentrations and partial pressures, respectively. $\left(\mathrm{p}+\mathrm{q}\right)$ is the sum of the stoichiometric coefficients of reactants, and $\left(\mathrm{r}+\mathrm{s}\right)$ is the sum of the stoichiometric coefficients of products (gas emissions coefficients).

To describe species equations in the governing equations, which can use either mole fraction (X) or mass fraction (Y).

$${\mathrm{X}}_1={\displaystyle \frac{{\mathrm{n}}_1}{{\mathrm{n}}_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}}={\displaystyle \frac{{\mathrm{n}}_1}{{\mathrm{n}}_1+{\mathrm{n}}_2+{\mathrm{n}}_3+\dots +{\mathrm{n}}_{\mathrm{i}}}}$$

(21)

From the gas ideal, as described in Equations (21) and (22), the equation becomes

$${\mathrm{X}}_1={\displaystyle \frac{{\mathrm{n}}_1}{{\mathrm{n}}_{\mathrm{Total}}}}={\displaystyle \frac{{\mathrm{P}}_1}{{\mathrm{P}}_{\mathrm{Total}}}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{X}}_2={\displaystyle \frac{{\mathrm{n}}_2}{{\mathrm{n}}_{\mathrm{Total}}}}={\displaystyle \frac{{\mathrm{P}}_2}{{\mathrm{P}}_{\mathrm{Total}}}}$$

(22)

The combustion reactants can also use the mass fraction (Y) in Equations (23) and (24) to symbolize the concentration of species

$${\mathrm{Y}}_1={\displaystyle \frac{{\mathrm{m}}_1}{{\mathrm{m}}_{\mathrm{Total}}}}={\displaystyle \frac{{\mathrm{m}}_1}{{\mathrm{m}}_1+{\mathrm{m}}_2+{\mathrm{m}}_3+\dots +{\mathrm{m}}_{\mathrm{i}}}}$$

(23)

$${\mathrm{Y}}_1={\displaystyle \frac{{\mathrm{m}}_1}{{\mathrm{m}}_{\mathrm{Total}}}}={\displaystyle \frac{{\mathrm{P}}_1}{{\mathrm{P}}_{\mathrm{Total}}}}, \mathrm{a}\mathrm{n}\mathrm{d} {\mathrm{Y}}_2={\displaystyle \frac{{\mathrm{m}}_2}{{\mathrm{m}}_{\mathrm{Total}}}}={\displaystyle \frac{{\mathrm{P}}_2}{{\mathrm{P}}_{\mathrm{Total}}}}$$

(24)

3.2. Radiation Model

ANSYS Fluent offers five radiation models: the discrete transfer radiation model (DTRM), the P1-radiation model, the Rosseland model, the surface-to-surface (S2S), and the discrete ordinate model (DOM). It allows the inclusion of radiation with or without a participating medium in heat transfer simulations. Heating or cooling the surfaces due to radiation or heat sources within the fluid phase is included in one of the above models. Considering the P-1 radiation model, the simplest problem of the more general P-N model, the influence of geometry configuration on the radiative heat transfer, is used as a numerical model in the current study.

Radiative heat transfer should be included in a numerical simulation when the radiant heat flux is large compared to the heat transfer rate due to convection or conduction [30]. Analysis should consider the radiation model if the combustion temperature is more than 2000 K [27]. The volumetric radiative heat transfer involving walls and the fluid medium is more prominent in the combustion chamber, where the gas emissions participate as a medium. Gas emissions can absorb and emit radiation [31]. The radiative transfer Equation (25) for an absorbing, emitting, and scattering medium at position $\overrightarrow{r}$ in the direction $\overrightarrow{s}$ is

$${\displaystyle \frac{\mathrm{dI}(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{s}})}{\mathrm{ds}}}+\left(\mathrm{a}+{\mathsf{\sigma }}_{\mathrm{s}}\right)\mathrm{I}\left(\overrightarrow{\mathrm{r}},\overrightarrow{\mathrm{s}}\right)=\mathrm{a}{\mathrm{n}}^2{\displaystyle \frac{\mathsf{\sigma }{\mathrm{T}}^4}{\mathsf{\pi }}}+{\displaystyle \frac{{\mathsf{\sigma }}_{\mathrm{s}}}{4\mathsf{\pi }}} {\int }_0^{4\mathsf{\pi }}\mathrm{I}\left(\overrightarrow{\mathrm{r}},{\overrightarrow{\mathrm{s}}}^{\prime }\right)\varnothing \left(\overrightarrow{\mathrm{s}},{\overrightarrow{\mathrm{s}}}^{\prime } \right)\mathrm{d}{\mathrm{\Omega }}^{\prime }$$

(25)

where ${\overrightarrow{s}}^{\prime }$ is the scattering direction vector, s is the path length, a is the absorption coefficient. n is the refractive index, ${\sigma }_s$ is the scattering coefficient, $\sigma$ is the Stefan–Boltzmann constant (5.669 × 10⁻⁸ W/m². K⁴), I is the radiation intensity, T is the local temperature, $\varnothing$ is the phase function, and ${\mathrm{\Omega }}^{\prime }$ is the solid angle.

3.3. Diffusion Combustion Flame

The combustion research mode consists of diffusion, premixed, and partially premixed flames. The diffusion flame (non-premixed combustion flame) occurs where oxidizer and fuel are not mixed or enter the combustor with distinct streams [32]. Combustion occurs in the mixing layer, which is very small compared to the combustion system, and the mixing step brings reactants into the mixing zone. Controlling combustion and the chemical reaction is very difficult [29]. The time required for turbulent mixing to occur through convection and diffusion processes is significantly larger than that for most of the combustion chemical reactions. The turbulent mixing and chemical reactions are the rate-limiting processes [33].

The combustion process is simplified to a mixing case where transport equations for one or two mixture fractions are solved. In the diffusion model in ANSYS Fluent, the thermochemistry processes of the mixture fraction between fuel and oxidizer are calculated using PDFs [28]. Applying the diffusion combustion model in ANSYS Fluent, which offers the modeling of secondary stream, empirical secondary stream, and empirical fuel stream. When reactant species of fuel and oxidizer mix in the reaction zone, the chemistry can be determined using the other approaches [28].

The mixing of fuel and oxygen, and the chemical reaction occur at the same time. Therefore, it is the rate of mixing that controls the combustion rate. Depending on the state of the fuel, there are gaseous diffusion combustion and liquid spray combustion. Gaseous diffusion consists of laminar and turbulent combustion. Depending on the flow condition of the fuel gas discharge into the combustion chamber. Since the process occurs inside the flame, it is a complicated case. The simplifications of differential equations for diffusion flames are

• On the flame surface, fuel and air should mix in a proper ratio. Chemical reactions occur only on the flame surface, and they are instantaneous.
• The flows of fuel and air are one-dimensional with uniform velocity. The diffusion of the reactants only takes place along the radial directions.
• The mole number does not change in the combustion reactions. The pressure does not change throughout the whole process.
• The diffusion of fuel and oxygen in inert gases is regarded as the diffusion of two components. Their diffusion coefficients are equal.
• The density $(\rho )$ and diffusion coefficient (D) of mixed gases do not correlate with temperature, so that both of them are constant in the radial direction $(\rho D=\mathit{constant})$.

At constant pressure, $\rho \approx {\displaystyle \frac{1}{T}}$, the diffusion coefficient of the two components is $D\approx T^{1.75}$. Therefore, the effects of temperature on $\rho D$ are small. The value of which is approximated as a constant in the radial direction. According to the above assumptions, the mass fraction conservation equation for component k with the chemical reaction is

$${\displaystyle \frac{\mathrm{D}\ln {\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{t}}}}={\displaystyle \frac{{\mathrm{w}}_{\mathrm{i}}}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}$$

(26)

Each component as follows

$${\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}={\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{x}}\right)}{\partial \mathrm{x}}}+{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{k}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{y}}\right)}{\partial \mathrm{y}}}+{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{k}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{z}}\right)}{\partial \mathrm{z}}}$$

(27)

$${\displaystyle \frac{\mathrm{D}\ln {\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{t}}}}={\displaystyle \frac{1}{{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\mathrm{D}{\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{t}}}}={\displaystyle \frac{1}{{\mathrm{Y}}_{\mathrm{i}}}}\left(\mathrm{u}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{x}}}+\mathrm{v}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{y}}}+\mathrm{w}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{z}}}\right)$$

(28)

By referring to assumption (b), the parameters are $u=v=0$ and $w=\mathit{constant}$, and for a steady-state diffusion combustion process where ${\displaystyle \frac{D{Y}_i}{D_i}}=0$, Equation (28) becomes

$${\displaystyle \frac{\mathrm{D}\ln {\mathrm{Y}}_{\mathrm{i}}}{{\mathrm{D}}_{\mathrm{t}}}}={\displaystyle \frac{1}{{\mathrm{Y}}_{\mathrm{i}}}}\left(\mathrm{w}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{z}}}\right)$$

(29)

Note that the chemical reaction only takes place on the flame surface by referring to assumption (a), but when the process on the flame surface is not considered, ${\mathrm{w}}_{\mathrm{i}}=0$, then by using assumption (d), ${\mathrm{D}}_{\mathrm{F}}={\mathrm{D}}_{\mathrm{O}}=\mathrm{D}=\mathrm{constant}$.

The diffusion velocity ${\overrightarrow{v}}_{\mathit{DI}}$ is described in Equation (30)

$${\overrightarrow{\mathrm{v}}}_{\mathrm{DI},\mathrm{y}}=-\mathrm{D}\nabla \ln {\mathrm{Y}}_{\mathrm{i}}$$

(30)

The second term on the right side of Equation (26) can be simplified by referring to the forms of ${\mathrm{w}}_{\mathrm{i}}=0$ and ${\mathrm{D}}_{\mathrm{F}}={\mathrm{D}}_{\mathrm{O}}=\mathrm{D}=\mathrm{constant}$ and assumption (e)

$$-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}=-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{v}}}_{\mathrm{DI},\mathrm{y}}\right)}{\partial \mathrm{y}}}=-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}\nabla \left[\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}\left(-\mathrm{D}\nabla \ln {\mathrm{Y}}_{\mathrm{i}}\right)\right]$$

(31)

$$-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}={\displaystyle \frac{\mathrm{D}}{{\mathrm{Y}}_{\mathrm{i}}}}\nabla \left[{\mathrm{Y}}_{\mathrm{i}}{\displaystyle \frac{1}{{\mathrm{Y}}_{\mathrm{i}}}}\nabla {\mathrm{Y}}_{\mathrm{i}}\right]={\displaystyle \frac{\mathrm{D}}{{\mathrm{Y}}_{\mathrm{i}}}}{\nabla }^2{\mathrm{Y}}_{\mathrm{i}}$$

(32)

For the case of an axisymmetric diffusion flame, the cylindrical coordinate system is used

$${\nabla }^2{\mathrm{Y}}_{\mathrm{i}}={\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial }{\partial \mathrm{r}}}\left(\mathrm{r}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{r}}}\right)+{\displaystyle \frac{1}{{\mathrm{r}}^2}}{\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{\phi }}^2}}+{\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{z}}^2}}$$

(33)

$Y_i$ is not related to $\phi$ when it is axisymmetric. By referring to assumption (b) and neglecting diffusion throughout the z-direction

$$\begin{gathered} {\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathsf{\phi }}}=0 {\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathsf{\phi }}^2}}=0 \\ {\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{z}}}=0 {\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{z}}^2}}=0 \end{gathered}$$

(34)

Equation (32) becomes

$${\nabla }^2{\mathrm{Y}}_{\mathrm{i}}={\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial }{\partial \mathrm{r}}}\left(\mathrm{r}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{r}}}\right)={\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{r}}^2}}+{\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{r}}}$$

(35)

By substituting Equation (35) into Equation (32), the following is found (Equation (36))

$$-{\displaystyle \frac{1}{\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}}}{\displaystyle \frac{\partial \left(\mathsf{\varrho }{\mathrm{Y}}_{\mathrm{i}}{\overrightarrow{\mathrm{u}}}_{\mathrm{DI},\mathrm{j}}\right)}{\partial {\mathrm{x}}_{\mathrm{j}}}}={\displaystyle \frac{\mathrm{D}}{{\mathrm{Y}}_{\mathrm{i}}}}\left[{\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{r}}^2}}+{\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{r}}}\right]$$

(36)

The diffusion flame equation can be simplified by substituting ${\mathrm{w}}_{\mathrm{i}}=0 \mathrm{and} {\mathrm{D}}_{\mathrm{F}}={\mathrm{D}}_{\mathrm{O}}=\mathrm{D}=\mathrm{constant}$ and Equation (36) into Equation (31), as formed in Equation (37)

$${\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{z}}}={\displaystyle \frac{\mathrm{D}}{\mathrm{w}}}\left[{\displaystyle \frac{{\partial }^2{\mathrm{Y}}_{\mathrm{i}}}{\partial {\mathrm{r}}^2}}+{\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial {\mathrm{Y}}_{\mathrm{i}}}{\partial \mathrm{r}}}\right]$$

(37)

where ${\mathrm{Y}}_{\mathrm{i}}$ is the mass fraction of component $\mathrm{i}$, D is the diffusion coefficient of two components, w is the flow rate of gaseous fuel or oxidizer, z is the distance of a certain plane of the flame from the coordinate of the discharge opening. The concentration fields of fuel ${\mathrm{Y}}_{\mathrm{F}}$ and oxygen ${\mathrm{Y}}_{\mathrm{O}}$ can be determined by partial differential Equation (38), but the boundary conditions are different (not including the point on the flame surface, because at point ${\mathrm{w}}_{\mathrm{k}}\ne 0)$. Thus, two differential equations have to be solved for the distribution of ${\mathrm{Y}}_{\mathrm{F}}$ and ${\mathrm{Y}}_{\mathrm{O}}$. Adopting the mathematical method by Back and Schumann, a new variable is defined

$$\mathrm{D}=\left\{\begin{array}{c}{\mathrm{Y}}_{\mathrm{F}} 0\le \mathrm{r}<{\mathrm{r}}_{\mathrm{f}}\\ 0 \mathrm{r} <{\mathrm{r}}_{\mathrm{f}}\\ {-\mathrm{R}}_{\mathrm{FO}}{\mathrm{Y}}_{\mathrm{O}} {\mathrm{r}}_{\mathrm{f}}<\mathrm{r}\le {\displaystyle \raisebox{1ex}{${\mathrm{d}}^{\prime }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\end{array}\right.$$

(38)

where ${\mathrm{r}}_{\mathrm{f}}$ is the radius of the flame surface. The fuel–oxygen ratio $\left({\mathrm{R}}_{\mathrm{FO}}\right)$ calculated by Equation (39)

$${\mathrm{R}}_{\mathrm{FO}}={\displaystyle \frac{{\mathrm{Y}}_{\mathrm{F}}}{{\mathrm{Y}}_{\mathrm{O}}}}, {\displaystyle \frac{\partial \mathrm{Y}}{\partial \mathrm{z}}}={\displaystyle \frac{\mathrm{D}}{\mathrm{w}}}\left[{\displaystyle \frac{{\partial }^2\mathrm{Y}}{\partial {\mathrm{r}}^2}}+{\displaystyle \frac{1}{\mathrm{r}}}{\displaystyle \frac{\partial \mathrm{Y}}{\partial \mathrm{r}}}\right], 0<\mathrm{r}\le {\displaystyle \frac{{\mathrm{d}}^{\prime }}{2}}$$

(39)

The concentration distribution function of the fuel inside the flame and oxidizer outside the flame (on each plane of the flame) can be correlated by the following boundary conditions, as shown in Equations (40)–(42), where L is the flame height.

$${\left.{\displaystyle \frac{\partial \mathrm{Y}}{\partial \mathrm{z}}}\right|}_{\mathrm{r}=0}=0, {\left.{\displaystyle \frac{\partial \mathrm{Y}}{\partial \mathrm{z}}}\right|}_{\mathrm{r}={\displaystyle \frac{{\mathrm{d}}^{\prime }}{2}}}=0 0\le \mathrm{z}\le \mathrm{L}$$

(40)

$${\left.\mathrm{Y}\right|}_{\mathrm{z}=0}=\left\{\begin{array}{c}{\mathrm{Y}}_{\mathrm{F},0} 0\le \mathrm{r}\le {\displaystyle \frac{\mathrm{d}}{2}}\\ {-\mathrm{R}}_{\mathrm{FO}}{\mathrm{Y}}_{\mathrm{O},0} {\displaystyle \frac{\mathrm{d}}{2}}<\mathrm{r}\le {\displaystyle \frac{{\mathrm{d}}^{\prime }}{2}}\end{array}\right.$$

(41)

$$\begin{gathered} {\mathrm{Y}}_{\mathrm{F}} \left(\mathrm{z},\mathrm{r}\right)\hspace{1em}\hspace{1em}\hspace{1em} 0\le \mathrm{r}<{\mathrm{r}}_{\mathrm{f}}\hspace{1em}\hspace{1em}\hspace{1em} 0\le \mathrm{z}\le \mathrm{L} \\ {\mathrm{Y}}_{\mathrm{O}} \left(\mathrm{z},\mathrm{r}\right)\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{r}}_{\mathrm{f}}<\mathrm{r}<{\displaystyle \frac{{\mathrm{d}}^{\prime }}{2}}\hspace{1em}\hspace{1em}\hspace{1em} 0\le \mathrm{z}\le \mathrm{L} \\ {\mathrm{Y}}_{\mathrm{O}}={ \mathrm{Y}}_{\mathrm{F}}=0 \hspace{1em}\hspace{1em}{\mathrm{r}}_{\mathrm{f}}=0\hspace{1em}\hspace{1em}\hspace{1em}0 \le \mathrm{z}\le \mathrm{L} \end{gathered}$$

(42)

4. Results and Discussion

As explained by previous research in [34], referring to research conclusions, the spray flame depends on the nozzle, fuel, burner, and operating parameters. The diffusion combustion flame model, coupled with energy, p1 for the radiation model, and k-ε for the viscous model, is used to determine the combustion performance of a burner. The contours of the temperature and gas emissions are used as references to evaluate the combustion characteristics of this current work, as performed by previous work in [3]. For the high combustion temperature, gas emissions react with each other to form emission products. Therefore, the parameters under various conditions are essential parameters to be investigated [3,27,28,35,36,37,38]. Carbon dioxide (CO₂) indicates complete combustion, whereas hydrocarbon (HC) and carbon monoxide (CO) indicate incomplete combustion [38,39]. The temperature of a cone is plotted along the axial axis at the radial axis of the centerline. That confirms the symmetry of the simulation result inside the combustion flame [17].

Each fuel has limits to burn (called flammability limits), so that the magnitude of velocities chosen in this work must cover the combustible ranges of the used fuel. The flammability limit of fuels depends on parameters such as temperature, pressure, fuel type, ignition source, and others, but that is not absolute. The lower and upper flammability limits of the kerosene are 0.65 and 1.45, respectively [16].

4.1. Influence of the Velocity Magnitude of Air and Fuel Inlets

Figure 2 presents the contours of temperature by keeping the inlet temperature constant and the mass fraction of oxygen in the inlet flow of fuel and air at 300 K and 0.23. As shown in Figure 2, the higher contour temperature for the same conditions occurs at a velocity magnitude of the inlet fuel of 0.05 m/s, compared to the others. As is expected from this measurement, the temperature along the centerline is low and does not show a constant value. It also shows that the temperatures at axial distances from 0 cm to 50 cm, on the other hand, are more uniform and higher. This result is a little different from the previous research work in [3], where the temperature along the centerline showed a constant value of 1775 K, indeed, at axial distances > 80 mm. The plots of temperature for the reference case in Figure 2 show that the maximum temperature takes place in the flame zone, namely around the centerline of the combustion chamber, as indicated in [40].

The temperature along the axial distances at various radial distances shows a relatively constant value for the axial distance > 650 mm, indeed. This leads to the flame position forming good stability at that position. The flame shape for all cases is predominantly symmetric about the x = 0 mm plane concerning the centerline of the combustion chamber for all axial distances towards the outlet. The results agree with the experiments in [36,41]. Some research results have obtained an asymmetric flame shape or disagreed with the symmetric flame, as mentioned in [3]. In an asymmetric flame, the temperature distribution along the abscissa is not uniform due to factors, namely air–fuel mixing, heat loss, and burner geometry. This asymmetry can lead to variations in temperature distribution, potentially affecting combustion efficiency and gas emissions. This phenomenon is evident, as indicated by the diameter ranging from 0 to 0.05 m at the axial positions from 0.2 m to 0.4 m. The case of the transition from symmetric to asymmetric flame shape is seen in the present simulation of the NO mass fraction. Increasing fuel velocity enhances the entrainment of flue gas, which can create a lower O₂ concentration [42]. This result differs from the previous research, where the temperature of the centerline in the combustion zone and preheat zone increased slightly by applying a microporous combustor [43].

The temperature contours inside the combustion chamber are not uniform, so the evaluation of temperature contour uniformity is usually needed by determining the pattern factor (PF). The parameter, which is defined by Equation (43), is determined by measuring the homogeneity of the combustion chamber temperature and the maximum temperature contour. The value of PF is closer to 1, which indicates that the outlet temperature contour distribution of the combustion chamber is more uniform [40].

$$\mathrm{PF}={\displaystyle \frac{{\mathrm{T}}_{\max }-{\mathrm{T}}_{\mathrm{in}}}{{\mathrm{T}}_{\mathrm{avg}}-{\mathrm{T}}_{\mathrm{in}}}}$$

(43)

where T_in is the average inlet temperature of both air and fuel, ${\mathrm{T}}_{\max }$ is the maximum temperature at the combustor outlet, and ${\mathrm{T}}_{\mathrm{avg}}$ is the average temperature contour at the outlet combustion chamber of the burner.

The H₂O contours, as shown in Figure 3, under various air magnitude velocities, with the fuel velocity magnitude kept constant at 0.5 m/s. The inlet temperature was kept constant at 300 K by using an oxygen mass fraction of 0.23. The maximum mass fraction of H₂O takes place almost near the inlet air at all of those magnitudes of velocity. Complete combustion requires enough oxygen, which is provided near the air inlet. The location close to the air inlet is in good agreement with the other mass fraction results. A more visible discrepancy is noticed at the far-field location, as explained by the higher predicted temperature. The characteristics of the H₂O flame contour are different at each of the various magnitude velocities and are very lovely at 0.0005 m/s.

The flame contour area of radial velocity, as displayed in Figure 4 at a magnitude velocity of 0.1 m/s (inlet temperature = 300 K), is better than the other contour. For the boundary condition at the inlet fuel of 0.01 m/s and temperature of 700 K, the radial velocity of 0.25–0.75 m/s dominates the flame contour with an area of more than 47%, and the maximum radial velocity is 1.93 m/s with a flame contour area of around 2.5%. For the boundary condition at the inlet fuel of 0.01 m/s and temperature of 300 K, the velocity of 0.5–1.4 m/s dominates the contour, with an area of about 45%, and the radial velocities of 3–4 m/s are about 5% of the flame contour area. The velocity of 0 m/s to 1 m/s dominates the flame contour with an area of more than 70% of the boundary condition at the inlet fuel of 0.1 m/s and a temperature of 300 K. The radial velocities of 4 m/s to 5 m/s are around less than 5% of the flame contour area for this condition. For the same fuel inlet, as shown in Figure 4a,b, both of them have contour sharp differences. The radial velocity contour in Figure 4b gives better results with an area of 1.31 m/s to 4.08 m/s, which is wider than that of the case in Figure 4a. This wider area of radial velocity is often desirable in combustion applications, where a uniform distribution of fuel is crucial for efficiency and performance.

Figure 5a–d displays the mass fraction of CO. An asymmetric curve for all the fuel inlets and four axial distances is almost formed. The lowest CO is obtained at a fuel inlet of 1.5 m/s. The concentration of CO is lower than that of the other cases of fuel inlets. At the centerline of the radial position, CO gets a significant decrease for all cases except for the case in Figure 5b. The lower CO indicates a better combustion process for the operating condition, and it is expected to take place to achieve a higher temperature. CO is an intermediate product of gas emission formed and further oxidized into CO₂. A chemical combustion reaction between air and fuel results in the following products: $\mathrm{a} \mathrm{fuel}+\mathrm{b} {\mathrm{O}}_2 \to \mathrm{c} {\mathrm{CO}}_2+\mathrm{d} {\mathrm{H}}_2\mathrm{O}+\mathrm{CO}+\mathrm{other} \mathrm{products}$ and $2\mathrm{CO}+{\mathrm{O}}_2 \to {2\mathrm{CO}}_2$. CO can be detected in flue gases. The occurrence is due to the following reasons: if not enough residence time for both the air and fuel to react and burn in a combustor. The burner chamber is too cold. The air–fuel ratio is out of stoichiometry (the ideal mixing ratio). The fuel and air in a burner chamber are insufficiently mixed [44,45].

Figure 5e–h is the CO₂ mass fraction versus radial and axial distances for various boundary conditions of the fuel inlet at a constant air inlet of 0.5 m/s. Both the temperature fuel and temperature air inlets are kept at 300 K. The curves for all the fuel inlets and four axial distances form a symmetric curve with a similar trend when the radial distance of 0 m is used as a centerline. The curves obtained almost resemble the CO mass fraction. The highest CO₂ mass fraction is obtained at a fuel inlet of 0.05 m/s by the highest concentration of around 0.52. With the centerline of radial position, CO₂ shows a significant decrease in all cases. The higher CO₂ indicates a complete combustion process for the operating condition. The characteristic of the CO₂ contour is close to all of those. CO₂ forms near the oxygen inlet because at this place, where the oxygen provided is more than at the wall and the center of the combustion chamber. This current research agrees with previous work in [46]. Figure 5i–l shows HCO gas emission. This pollution formed with very small concentrations, as identified in Figure 5i–l. In the context of combustion and air pollution, HCO is called Hydrogen Monoxide, which refers to the unburned fuel and a gas emission of Hydrocarbon oxygen caused by incomplete combustion due to problems like fuel, injector, combustion chamber, air fuel ratio, etc. This unburned hydrocarbon-oxygen (HCO) is a form of air pollution and contributes to smog. High levels of HCO that escape from a combustion process can cause human health and environmental damage.

4.2. Influence of Inlet Temperature

In this section, the mass fraction of oxygen is kept constant at 0.23 with an air velocity of 0.5 m/s. The mass fraction and magnitude velocity of kerosene kept constant are 1 and 0.01 m/s, respectively. The effects of the inlet temperature of fuel and air on temperature distribution inside the combustion chamber at 500 K, 700 K, 900 K, and 1500 K are shown in Figure 6. The increase in temperature can raise the combustion chamber temperature to all selected radial positions. The maximum temperature occurs at the inlet temperature of 1500 K, followed by 900 K, 700 K, and 500 K. The chemical reaction occurs rapidly at a higher temperature, and the mixing of fuel and oxygen is faster [32]. The increment of the maximum temperature from 500 K to 900 K is 280 K, from 1573 K to 1853 K. However, it is 170 K from 1853 K to 2023 (lower at 700 K to 900 K). Therefore, from 900 K to 1500 K, the increment is higher, 375 K from 2023 K to 2398 K. This is due to the higher inlet temperature, which results in a more uniform temperature distribution. A similar result is shown in reference [42]. The configuration of flame forms a symmetry flame, which occurs at the radial positions of 0.1 m and −0.1 m for the temperature inlet. The temperature distributions at the centerline and close to the combustor centerline are hotter for all the inlet temperatures than at other radial positions. However, that is colder when far from the centerline. This yield differs from the research in [18].

The addition of inlet temperature from 500 K to 1500 K raises the CO mass fraction, as shown in Figure 7. The trend of the CO curve is very smooth, as seen in the temperature profiles in Figure 6. CO at the radial position of 0 m is prominent and significantly decreases up to 0.2 m axial distance. However, at the other radial positions, the values increase significantly, occurring for all inlet temperatures. The levels of CO emission are almost constant at the axial distances > 0.4 m. The temperature range from 300 K to 1800 K corresponded to the prediction of CO, whereas the oxidation of CO dominated at temperatures above 1800 K [47].

Figure 8 displays CO₂ mass fraction curves versus the axial direction at varying inlet temperatures from 500 K to 1500 K. Increasing the inlet temperature can decrease the CO₂ mass fraction for all radial positions. This result differs from the research in [34,37]. The kerosene fuel burns first before entering the combustion chamber because the inlet temperature given exceeds the temperature at which the fuel ignites by itself. The autoignition temperature of kerosene is around 210–220 °C (484–494 K). This is the lowest temperature at which kerosene will ignite spontaneously without an external ignition source, like a spark or flame. The radial distribution of species, including CO₂, is influenced by mixing, reaction rates, and heat transfer within the combustion chamber. In combustion processes, theoretically, an increase in inlet temperature typically leads to higher flame temperatures. Higher flame temperatures generally promote more complete combustion, which results in an increased production of CO₂ and H₂O as end products. The trend of the CO₂ curve is almost similar to that of H₂O. The CO₂ decreases in all the radial directions except at d = 0 m. The emission of CO₂ is more significant than that of CO because the concentration of oxygen is more dominant in forming CO₂ than other emissions. The level of CO₂ can change significantly up to the axial distances of 0.2–0.4 m because oxygen is still more available around the positions. CO₂ is almost constant at axial distances > 0.4 m in all cases. The curves of CO₂ for all inlet temperatures under the radial positions of 0.1 m and −0.1 m show slight differences and form a symmetrical flame. Figure 9 shows the H₂O mass fraction by blowing up air and fuel contours. The H₂O mass fraction almost forms a symmetry flame configuration. The higher H₂O concentration is obtained at the air inlet because forming this gas emission requires enough oxygen. The rise in inlet temperature can increase the CO₂ level, but it is not significant.

4.3. Influence of Oxygen Mass Fraction

The current analysis compares previous research and simulation results. Results display different study fields of various oxygen mass fractions. The velocity magnitude and inlet temperature of air kept constant are 0.5 m/s and 300 K, respectively. For the fuel inlet, the values provided are kept at 0.01 m/s and 300 K, respectively. Figure 10 shows the effect of oxygen on the temperature distribution inside the combustion chamber under various boundary conditions of oxygen from 0.23 to 0.8. Know that increasing the oxygen concentration can raise contour temperatures. The higher oxygen concentration can result in a more uniform temperature distribution by yielding perfect combustion. The simulation results are similar to those in reference [42]. The oxygen reduction can affect the field of temperature as described in [26] by applying various oxygen concentrations of 21%, 18%, and 15%. The results agree with reference [47] for the temperature contour, with the radial position of 0.2 m closer to the wall temperature.

This simulation not only provides the flame temperature distribution contours as a result of the combustion process but also provides the gas emissions, as presented in Figure 10 and Figure 11. The effects of oxygen are that the emissions of H₂O and CO₂ increase, while the emissions of CO decrease. The predicted gas emission results agree well with theory. The addition of oxygen level forms CO decreases, and CO₂ increases because perfect combustion takes place with enough oxygen [44,48]. The rise of temperature and oxygen for stoichiometric burning can increase the amount of CO₂ [39]. The results of NO_x and CO decreased up to nearly zero, but CO₂ increased slightly under the oxygen concentrations of 18% and 15% by volume [26]. The formation of NOx depends upon reaction duration, inlet temperature, and oxygen availability in the combustor [49]. NOx is reduced by dual-stage low premixed (DLF) flame technology because DLF controls combustion temperature by limiting the use of the equivalence ratio of premixed gas [50]. The level of NOx is dependent on the oxygen concentrations, and CO emission is dependent on the air–fuel ratio [51].

Carbon monoxide (CO) is available in combustion gas as a result of the partial oxidation of the carbon content in fuel, the presence of which in gas emission is an indication of low combustion efficiency because it is not completely oxidized to CO₂. The form that can reduce the thermal combustion efficiency, of course, can increase fuel consumption. CO is present more when the combustion process is carried out with too little air than required stoichiometry, and therefore, the oxygen is insufficient for completing carbon oxidation reactions. The combustion process occurs with excess air/oxygen higher than the stoichiometric requirement [37,44].

The geometric size of the combustion chamber influences gas emissions. The emission of CO is higher for a small combustion chamber than for a large combustion chamber [48]. The higher temperature of the flame, which appears when forming H₂O and CO₂, is more dominant because the synthesis of the emissions yields an enthalpy of the combustion reaction higher than that of CO. The oxygen provided must be more than that required to increase the fuel conversion efficiency to gas emissions. The emissions of CO₂ and H₂O, as shown in Figure 11a–d, generally increase with the addition of the oxygen mass fraction at all the selected lines, except in the case of Figure 11d. The highest CO₂ and H₂O emissions occur in the case as shown in Figure 11c and Figure 11l, respectively. Those cases describe the complete combustion for these operating conditions and can be used in the real application, as proved by the flame temperature in Figure 10. The flame configuration of CO₂ and H₂O almost forms a symmetrical shape, especially when the axial position is less than 0.25 m. The values of CO₂ range from more than 0 to 0.6, which are more similar to the results in [12].

For an excellent understanding, Figure 12 displays the flame temperature contours at various oxygen mass fractions. The addition of oxygen mass fraction can increase the flame temperature in the combustion chemical reaction, which can change the contour form, contour area, and flame stability of temperature. For the same oxygen concentration, as shown in Figure 12a,b, it is clear that both have sharp differences. The temperature contour in Figure 12b gives better results with an area of 2061 °C to 2400 °C, which is wider than that of the case in Figure 12a. The condition, as shown in Figure 12b, involves a gas of CO that can react with enough oxygen to form CO₂. Another reason for this case is to predict the effect of the transport direction of O₂ species on the near fuel distribution by an injector.

The perfect combustion is more serious, occurring near the fuel injector area. The explanations, as shown in Figure 12c, form a sharp difference when the presence of CO, in Figure 12d, is neglected in the case. The condition in the case of Figure 12d only involves CO₂ and H₂O, so that the temperature contour shows enough difference compared with the cases in Figure 12b,c, displaying that temperatures near the fuel inlet are more significant than the air inlet. The temperatures near the nozzle are generally higher for both axial and radial temperature profiles; these results agree well with previous research in [3].

5. Conclusions

The simulation results for various operating conditions, including inlet temperatures, velocity magnitudes, and oxygen mass fractions, are compared with those from prior research. The radial velocities of 0.25–0.75 m/s, 0.5–1.4 m/s, 0–1 m/s dominate the flame contour with areas of more than 47%, 45%, and 70%, respectively, for the boundary conditions at the inlet fuels of 0.01 m/s (700 K), 0.01 m/s (300 K), and 0.1 m/s (300 K). For the same fuel inlet, the contour has sharp differences. The radial velocity contour at 0.01 m/s (300 K) gives better results with an area of 1.31 m/s to 4.08 m/s, which is wider than that of the case at 0.01 m/s (700 K).

The given tests to the combustion chamber using kerosene fuel have resulted in better performance when applying inlet velocity magnitudes for fuel = 0.01 m/s and air = 0.5 m/s. The flame shape for all cases of inlet velocity is predominantly symmetric about the x = 0 mm for all axial distances towards the outlet. The maximum mass fraction of H₂O takes place almost near the inlet air at all of those magnitudes of velocity. An asymmetric curve for all fuel inlets and four axial distances forms on the mass fraction of CO, with the lowest CO mass fraction obtained at the fuel inlet of 1.5 m/s. The curves obtained are almost similar to the CO mass fraction. The highest CO₂ mass fraction is obtained at the fuel inlet of 0.05 m/s by the highest concentration of around 0.52%. The curves for all fuel inlets and four axial distances form a symmetric curve with a similar trend when the radial distance of 0 m is the centerline.

The increase of inlet temperature can raise the combustion chamber temperature to all selected radial positions, and the maximum temperature takes place at the inlet temperature of 1500 K, followed by 900 K, 700 K, and 500 K. The CO₂ curve trend is almost similar to H₂O. The increase in the inlet temperature of air and fuel can raise the CO mass fraction but can decrease CO₂. Higher oxygen levels leading to a higher peak temperature in the flame, which can reduce fuel consumption. Adding the mass fraction of oxygen can raise the combustion flame temperature and can change the contour form, contour area, and flame stability with temperature. The addition of N₂ and CO can influence temperatures and the emissions of CO₂, CO, and H₂O in the combustion chemical reaction. The emissions of CO₂ and H₂O form a symmetry flame when the axial position is less than 0.25 m.