Analysis of the Exothermic Reaction of Flame Ignition in the Combustion Chamber of a Gas Turbine Unit

: This article analyses the exothermic reaction of ﬂame ignition in the combustion chamber of a gas turbine unit, which is characteristic of combustion chambers operating on traditional hydrocarbon fuels. The combustion of gases as an explosive process in conﬁned and semi-enclosed areas remains a poorly understood section of thermal physics. Without a detailed review of the physical and chemical processes taking place in the combustion chamber, it cannot be said whether the gas turbine unit will run sustainably. It is also important to know what combustion modes are in principle possible after a loss of stability in the combustion chamber in order to take action against this in advance. To describe ﬂame ignition and quenching in the ﬂow of the fuel–air mixture through a combustion chamber, a system of differential conservation equations of energy and reactive species supplemented with the equation of state is used. Nonstationary combustion processes in gas-turbine engines were studied, and ﬂame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes. Calculation methodologies for various operating modes of the combustion chamber of a gas turbine unit are developed and realized. The results of the calculations that were carried out are presented with graphical interpretation and with their analysis provided in sufﬁcient detail. Based on this analysis, recommendations are then provided. From the graphs, it can be observed that the combustion chamber of a gas turbine unit reaches its maximum limit of stable operation at the optimum value of the reduced ﬂow velocity in the openings of the air supply to the combustion and the mixing zones of the ﬂame tube ( λ OC ) opt = 0.22 when the fuel–air mixture is at maximum depletion, ensuring that combustion does not stop and ﬂame failure does not occur. The topic of this article relates to the intensiﬁcation of hydrocarbon fuel combustion and the technological improvement of combustion chambers in gas turbine units. This topic is of exceptional importance and relevance, emphasizing its signiﬁcance. The purpose of this work is to develop and implement a methodology for calculating various modes of operation of the combustion chamber of a gas turbine unit.


Introduction
The purpose of this scientific research is to develop and implement a methodology for calculating various modes of operation of the combustion chamber of a gas turbine unit, as well as to create a systematic approach for analyzing and understanding the exothermic reaction of flame ignition in the combustion chamber.This paper contributes to the existing body of knowledge by proposing a systematic approach to understanding and optimizing the combustion process in gas turbines.
The exothermic reaction of flame ignition in the combustion chamber of a gas turbine unit is a critical process that significantly impacts the operation and sustainability of gas turbine units.A thorough investigation of these processes is necessary to ensure the stable performance of gas turbine units.Furthermore, anticipating and effectively addressing potential instability issues that arise from the loss of stability in the combustion chamber requires a comprehensive understanding of the possible combustion modes.To model and describe flame ignition and quenching in the flow of the fuel-air mixture through the combustion chamber, a system of differential conservation equations is employed, which includes energy, reactive species, and equation of state.In the context of gas turbine engines, nonstationary combustion processes, flame ignition, and blow-off are studied by considering heat balance and the continuity of chemical processes.
Nonstationary processes include the ignition of the combustion chamber and the flame failure resulting from sudden changes in the gas sector's position.It is especially important to solve the startup problem for gas turbine installations operating in highaltitude conditions at negative air and fuel temperatures.Currently, the refinement of the combustion chamber is being carried out using an expensive trial and error method, which involves the utilization of special equipment and testing a large number of options.The process of ignition of the fuel-air mixture in the combustion chamber is characterized by a sharp increase in the temperature of the gas in a short period of time and a decrease in the concentration of combustible gas due to its transformation into the final products of the reaction in the same period of time.
One of the main tasks in the design of combustion chambers of gas turbine plants is to maintain a stable combustion process during transient modes of operation with sharp depletion or enrichment of the mixture.As practice shows, measures to reduce nitrogen oxide emissions in most cases lead to a sharp narrowing of the range of stable operation of the combustion chamber.The absence of workable analytical dependencies that take into account the influence of the mode and design parameters of the combustion chamber on its stall characteristics necessitates a large number of expensive tests on stands that simulate the real operating modes of a gas turbine unit's combustion chamber.Therefore, developing a generalized dependence that links the geometry of the flowing portion of the combustion chamber and its operational modes to the value of the excess air ratio at combustion stall becomes an urgent task.
The cessation of combustion in the combustion chamber due to the depletion of the mixture, unlike the start-up process, is accompanied by a sharp decrease in the gas temperature and an increase in the fuel concentration due to the cessation of its chemical reaction with the oxidizer.
In the case of combustion chamber extinguishment due to an enrichment of the fuelair mixture, the process initially proceeds with a sharp increase in gas temperature and a decrease in fuel concentration.

Materials and Methods
The operating process in the combustion chamber of a gas turbine unit is determined by the spatial irreversible transfer of energy, matter, and quantity of motion in the reaction volume.These effects are described by a system of differential equations that involve scalar and vector quantities, which are transferred by a turbulent gas flow in the presence of sources and sinks of heat and mass resulting from a chemical reaction.The special feature of differential conservation and transfer equations is that they are written for a specific volume of gas, which is bounded by a control surface.These equations enable us to assess the processes occurring within this volume by analyzing the gas parameters at the limits of the surface.
The integral math model, resulting from solving the differential transfer equations in the criterion-parametric form, serves as the theoretical basis for the equations derived in this work to calculate the combustion chamber.The progression of chemical reactions is described by changes in temperature and pressure of the medium and the composition of the fuel-air mixture, which includes the concentration of the combustible and oxidizer and depends on the type of fuel.The efficiency of heat and mass transfer in turbulent gas flow is determined by geometric and hydrodynamic criteria and flow parameters.The numerical values of the coefficients in the calculation equations are determined based on experimental data obtained from refining combustion chambers in various types, sizes, and designs of gas turbine engines.These engines operate on both liquid and gaseous fuels.The optimization task of a combustion chamber is to determine the geometric parameters of its flow part and characteristics that ensure the maximum (minimum) of some target function (quality criterion) when the given conditions and constraints are satisfied.In order for optimization to be possible, the mathematical model, which is a system of equations, must be closed by introducing a target function into it.Then out of the set of solutions, one that satisfies the quality criterion will be found.The target function of a particular system should not be formed from the parameters of the system itself, but the parameters of the wider system should be used for this purpose.
To describe flame ignition and quenching in the flow of the fuel-air mixture through a combustion chamber, a system of differential conservation equations of energy and reactive species supplemented with the equation of state is used.Because the transient processes of flame ignition and quenching are represented by nonzero time derivatives, one can write the following inequalities [1]: where E = C p T; T is the gas temperature; C p is the specific heat at constant pressure; C r is the fuel concentration in the flow; ρ and P are the gas density and pressure, respectively; W represents the gas velocity vector (in turbulent flow, Reynolds-averaged velocity); µ I is the molecular viscosity; µ I T is the turbulent viscosity; Pr and Sc are the molecular Prandtl and Schmidt numbers, respectively; Pr T and Sc T are the turbulent Prandtl and Schmidt numbers, respectively; S E and S r are the energy and species concentration source terms, respectively, as functions of fuel type, P, T and α; and α signifies the excess air coefficient.
After integrating inequalities (1) and (2) over the volume and applying proper transformations [1][2][3] while considering the turbulence model, the following inequalities can be derived [4][5][6][7]: where δT and δC r are the differences of temperature and fuel concentration at combustor outlet and inlet, respectively; G is the flow rate through volume V p where flame ignition or quenching occurs; Īp is the normalized length of the combustor; W is the parameter characterizing the swirl intensity in the flow [1,[7][8][9][10][11]; N s is the number of mass and heat sources; and index m denotes the mean value [12][13][14][15].Equations are considered to describe the boundaries of nonstationary processes in the gas turbine unit combustion chamber, including the start and stop of the flame: formula of current lines in differential form [1,[16][17][18][19][20]: continuity formula for a quasi-stationary process: gas states: where u, υ, w-gas flow velocity vector projections spread out along Cartesian axes of coordinates; dG-mass gas flow through the elementary site dF, normal to the gas flow velocity vector; W p -mass speed of chemical reaction [21][22][23]: where k 0 ~Pm -molecule impact constant; m-value characterizing the effect of pressure on combustion rate; E-activation energy; C r -concentration of fuel in the gas flow by mass; C 0 -concentration of an oxidizer in the gas flow by mass; R-universal gas constant; n 1 = const, n 2 = const-procedure for the oxidation reaction of hydrocarbon fuels [1,7]; H M -the amount of heat generated by the combustion of the fuel-air mixture: where η-fuel completeness factor; H u -the lowest specific amount of heat generated during fuel combustion [1,24].

Calculation Methodology for Unsteady Combustion in the Combustion Chamber of a Gas Turbine Unit
Inequality (3) with «<» sign for the ignition process represents the condition when the heat release rate due to chemical reactions exceeds the rate of heat removal from the volume by the flow.
On the contrary, flame quenching may take place, which corresponds to inequality (3) with «>» sign.
Inequality (4) with «>» sign represents the continuity condition for chemical reactions after mixture ignition in the combustion chamber, i.e., the rate of the fuel delivery to the volume exceeds the rate of its burning-out in the volume.If the fuel-air mixture burns out faster than it is delivered to the volume with the incoming flow, the flame will be quenched [3].
As a result of processing the experimental data on flame ignition and quenching in the combustion chambers of gas-turbine engines of different designs and sizes, the inequalities (3) and ( 4) take the form: for flame ignition: for flame quenching: exp(e 11 W) where P k and T k are the air pressure and temperature at the combustor inlet; E CB is the electric power of the ignition discharge, d M is the median diameter of liquid fuel drops; V C is the volume of the combustor flame tube; F C is the cross-section area of the flame tube; N i , is the number of fuel injectors; and ∑µF OC is the total effective area of all orifices in the flame tube; where µF 3 is the effective area of the swirler; W is the intensity of air swirling at the outlet of the transition section swirler, which depends on the swirler geometrical characteristics; λ OC is the velocity coefficient; L 0 is the theoretical amount of oxidant required for burning 1 kg of fuel; and alpha α max is the maximum value of the excess air coefficient at which the flame blows off.Equations ( 5) to (7) describe how the boundaries of combustion processes in the combustion chamber are solved in parametric form: where λ 0 -coefficient of the velocity of the gas flow over the combustion site (function of the ratio of mass gas flow to the volume of the combustion site); E S -spark plug discharge energy; and index Z-out of volume [7].
The groups included in expression (15) of criteria and parameters describe the following: the rate of chemical reaction; the rate of heat emission as a result of chemical reaction; the intensity of phase transformation processes in the combustion chamber-p k ; T K ; T Z ; E S ; α; type of fuel; convective energy transfer rate of the substance; and the amount of movement through the reaction volumes of the combustion chamber and the hearth supporting its combustion at the boundary of its failure, G V p , λ 0 [1,3,[25][26][27].At maximum α, the optimum value of the relative swirl capacity is determined with µ − F 3 .

Criterion of Stable Gas Flow in the Diffuser of the Combustion Chamber of a Gas Turbine Unit
Ensuring the stability of the processes occurring in the combustion chamber is one of the main tasks in creating a gas turbine installation.The source of instability can be the combustion chamber diffuser if it is not properly designed.For example, the instability of the gas flow in the diffuser leads to an increase in the unevenness of the gas temperature field at the outlet of the combustion chamber, and the manifestation of increased unevenness is nonstationary.
To analyze the gas flow in the diffuser of the combustion chamber, consider the equation of conservation of the quantity of gas flow and the continuity equation: On the other hand, for the diffuser: where F k and F d -cross-sectional area at the diffuser inlet and at the diffuser outlet; index «m» denotes the average value of the parameter in the range of its change from the input to the output of the object under consideration; index «δ» denotes the change in the parameter value from the output to the volume input; χ-characteristic size; l d -diffuser length; l d -diffuser length divided by the height of the inlet section; V d -diffuser volume; ν k -gas flow velocity at the diffuser inlet.Substituting this expression into the left side of the inequality ( 16), assuming that 2 , l d = χ, and, using the developed model of turbulent phenomena in the flow of a continuous medium, we finally obtain the condition of unstable gas flow in the diffuser: For a planar diffuser, the complexes included in (18) are equal: where α-is the angle between the axis and the diffuser wall.
For gas flow with numbers Re > 10 5 ratio l d Re 0. The condition of transition to unstable gas flow in the diffuser, which we denote by A, we transform to the form: As a parameter characterizing the instability of gas flow in the combustion chamber diffuser, we take the ratio of: where ; T max -maximum value of the local gas temperature in the outlet section of the combustion chamber; T z -the average mass temperature of gas in the outlet section of the combustion chamber; T k -the air temperature at the combustion chamber inlet; ∆θ max -maximum value of the variation of the value of the circumferential nonuniformity coefficient of the gas temperature field for different copies of the combustion chamber of the same design; (θ max ) av -the average value of the coefficient of circumferen- tial unevenness of the gas temperature field at the outlet of the combustion chamber of a certain design for different instances [5].

Results
One of the characteristics of gas turbine units is the limitation of stable operation in the combustion chamber at maximum depletion of the fuel-air mixture.This refers to the stability of the working process in the combustion chamber at the highest possible value of the excess air ratio, where combustion continues without stopping and flame failure does not occur.
On the basis of transforming and solving differential equations related to energy transfer and fuel concentration in the gas flow, as well as generalizing experimental data, the optimum value of the reduced flow velocity in the inlets to the combustion and mixing zones was determined.
Figures 1 and 2 depict the boundaries described by inequalities (10) to ( 13) and specify the limits of flame ignition or quenching in the combustor.Inequalities ( 10) and ( 12) correspond to curve a in the figure, while inequalities ( 11) and ( 13) correspond to the b curve for ignition and extinguishing, respectively.Boundaries of flame quenching (refer to Figure 2).It appears from the graphs that the maximum limit of stable operation of the combustion chamber of a gas turbine unit at maximum depletion of the fuel-air mixture, at which combustion does not stop and flame failure does not occur, is provided at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and mixing zones of the flame tube (λOC)opt = 0.22.Formulas ( 10)-( 13) can be recommended for calculating the characteristics of combustion chamber start-up and combustion failure, commonly known as «flame slippage», which occurs due to fuel impoverishment in an air mixture in order to reduce emissions.Additionally, these formulas can predict the occurrence of flow variations near the boundaries of unsteady combustion in a gas turbine unit.Boundaries of flame quenching (refer to Figure 2).It appears from the graphs that the maximum limit of stable operation of the combustion chamber of a gas turbine unit at maximum depletion of the fuel-air mixture, at which combustion does not stop and flame failure does not occur, is provided at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and mixing zones of the flame tube (λOC)opt = 0.22.Formulas ( 10)-( 13) can be recommended for calculating the characteristics of combustion chamber start-up and combustion failure, commonly known as «flame slippage», which occurs due to fuel impoverishment in an air mixture in order to reduce emissions.Additionally, these formulas can predict the occurrence of flow variations near the boundaries of unsteady combustion in a gas turbine unit.shows that they represent the ratio between the speed of convective and molecular transport of an impulse, substance, and energy [2,[28][29][30].

Analysis of criteria
Figure 3 shows a comparison of the boundaries of the measured and calculated ignition zones.Boundaries of the ignition domain (refer to Figure 1).Boundaries of flame quenching (refer to Figure 2).It appears from the graphs that the maximum limit of stable operation of the combustion chamber of a gas turbine unit at maximum depletion of the fuel-air mixture, at which combustion does not stop and flame failure does not occur, is provided at the optimum value of the reduced flow velocity in the openings of the air supply to the combustion and mixing zones of the flame tube (λ OC ) opt = 0.22.Formulas ( 10)-( 13) can be recommended for calculating the characteristics of combustion chamber start-up and combustion failure, commonly known as «flame slippage», which occurs due to fuel impoverishment in an air mixture in order to reduce emissions.Additionally, these formulas can predict the occurrence of flow variations near the boundaries of unsteady combustion in a gas turbine unit.

RePr , l p
ReSc shows that they represent the ratio between the speed of convective and molecular transport of an impulse, substance, and energy [2,[28][29][30].
Figure 3 shows a comparison of the boundaries of the measured and calculated ignition zones.The graphs in Figure 3 illustrate a good match between the calculation and experiment, which gives grounds for the application of the calculation methodology in design practice.
The graph in Figure 4 illustrates the dependence of the value of ( ) Δ for straight-flow, counterflow annular and individual combustion chambers and experimental engines on the parameter A (refer to Figure 4).Experimental data from the planar diffuser models presented in the study were used [31].Experimental data of combustion chambers of various types of engines were used [1].The graphs in Figure 3 illustrate a good match between the calculation and experiment, which gives grounds for the application of the calculation methodology in design practice.
The graph in Figure 4 illustrates the dependence of the value of ∆θ max (θ max ) av for straightflow, counterflow annular and individual combustion chambers and experimental engines on the parameter A (refer to Figure 4).
Experimental data from the planar diffuser models presented in the study were used [31].
From the analysis of the graph in Figure 4, it follows that when the value of parameter A is less than 0.374, i.e., when the gas flow in the diffuser is stable (see criterion condition (21) obtained above), the variation of the circumferential non-uniformity coefficient of the gas temperature field at the combustion chamber outlet from instance to instance is minimal and is caused only by the variation of the geometric dimensions of the flowing part of the combustion chamber within the tolerance field during its manufacture.
The instability of the gas temperature field at the outlet of the combustion chamber also increases as the value of A increases above 0.374.Such instability of θ max in the outlet section of the combustion chamber of a high-temperature engine can be the cause of defects on the turbine structural elements during operation [5].From the analysis of the graph in Figure 4, it follows that when the value of parameter A is less than 0.374, i.e., when the gas flow in the diffuser is stable (see criterion condition (21) obtained above), the variation of the circumferential non-uniformity coefficient of the gas temperature field at the combustion chamber outlet from instance to instance is minimal and is caused only by the variation of the geometric dimensions of the flowing part of the combustion chamber within the tolerance field during its manufacture.
The instability of the gas temperature field at the outlet of the combustion chamber also increases as the value of A increases above 0.374.Such instability of θmax in the outlet section of the combustion chamber of a high-temperature engine can be the cause of defects on the turbine structural elements during operation [5].
Therefore, the design of combustion chamber diffusers must take into account the criterion condition (21) and the dependence shown in Figure 4.

Discussion
Nonstationary combustion processes in gas-turbine engines were studied, and flame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes.Based on the encouraging agreement between predictions and measurements, the method under consideration is commendable for power-plant design purposes.
Geometric criteria and parameters in function (15) determine the intensity of the turbulent leveling and mixing processes in the gas flow.
The utilization of a combustion chamber equipped with air inlet openings in the combustion and mixing zones results in an optimal reduced velocity value, significantly enhancing the flame failure limit even at maximum fuel mixture depletion.This approach expands the gas turbine unit's range of stable operation and improves reliability.
The criterion developed for the stability of gas flow in diffusers is one of the components in the automated design system for the combustion chamber of a gas turbine unit.
The results of this calculation study can be used in engineering projects and for the development of strategies to intensify fuel combustion in the combustion chamber of a gas turbine unit.
The final decision on the choice of the preferred organization of the combustion chamber operating mode of a gas turbine unit is made on the basis of a technical and economic analysis, taking into account also the type of fuel-burning system of the turbine unit.Therefore, the design of combustion chamber diffusers must take into account the criterion condition (21) and the dependence shown in Figure 4.

Discussion
Nonstationary combustion processes in gas-turbine engines were studied, and flame ignition and blow-off were determined by the heat balance and by the continuity of chemical processes.Based on the encouraging agreement between predictions and measurements, the method under consideration is commendable for power-plant design purposes.
Geometric criteria and parameters in function (15) determine the intensity of the turbulent leveling and mixing processes in the gas flow.
The utilization of a combustion chamber equipped with air inlet openings in the combustion and mixing zones results in an optimal reduced velocity value, significantly enhancing the flame failure limit even at maximum fuel mixture depletion.This approach expands the gas turbine unit's range of stable operation and improves reliability.
The criterion developed for the stability of gas flow in diffusers is one of the components in the automated design system for the combustion chamber of a gas turbine unit.
The results of this calculation study can be used in engineering projects and for the development of strategies to intensify fuel combustion in the combustion chamber of a gas turbine unit.
The final decision on the choice of the preferred organization of the combustion chamber operating mode of a gas turbine unit is made on the basis of a technical and economic analysis, taking into account also the type of fuel-burning system of the turbine unit.
The stated theory allows for solving the problem of ensuring the start-up and stability of a gas turbine unit under given conditions at the design stage.
represent the ratio between the speed of convective and molecular transport of an impulse, substance, and energy[2,[28][29][30].

Energies 2023 ,
16, x FOR PEER REVIEW 9 of 12Experimental data of combustion chambers of various types of engines were used[1].

Figure 3 .
Figure 3.Comparison of the boundaries of the measured and calculated ignition zones.Symbols 1-6 correspond to experimental data obtained during tests of combustion chambers of various types of engines and various parameters [1]; curves are constructed according to formulas (10) and (11): 1-turbojet twin-circuit engine, jet fuel, centrifugal nozzle, candle discharge energy E = 6 J, height at ground level H = 0 km, air and fuel temperature ta = tf = 15 °C; 2-5-turboprop, candle discharge energy E = 6 J, with auxiliary air injection at start-up into the second circuit of the centrifugal nozzle: 2-jet fuel, height H = 4.5 km, air and fuel temperature ta = tf = −40 °C; 3-jet fuel, height H = 10 km, air and fuel temperature ta = tf = −40 °C; 4-diesel fuel, height H = 6.5 km, air and fuel temperature ta = tf = −40 °C; 5-jet fuel, aerating front device, height H = 4.5 km, air and fuel temperature ta = tf = −40 °C; 6-turboshaft engine, jet fuel, centrifugal nozzle, candle discharge energy E = 2 J, height H = 4 km, air and fuel temperature ta = tf = 15 °C.

Figure 3 .
Figure 3.Comparison of the boundaries of the measured and calculated ignition zones.Symbols 1-6 correspond to experimental data obtained during tests of combustion chambers of various types of engines and various parameters [1]; curves are constructed according to formulas (10) and (11): 1-turbojet twin-circuit engine, jet fuel, centrifugal nozzle, candle discharge energy E = 6 J, height at ground level H = 0 km, air and fuel temperature t a = t f = 15 • C; 2-5-turboprop, candle discharge energy E = 6 J, with auxiliary air injection at start-up into the second circuit of the centrifugal nozzle: 2-jet fuel, height H = 4.5 km, air and fuel temperature t a = t f = −40 • C; 3-jet fuel, height H = 10 km, air and fuel temperature t a = t f = −40 • C; 4-diesel fuel, height H = 6.5 km, air and fuel temperature t a = t f = −40 • C; 5-jet fuel, aerating front device, height H = 4.5 km, air and fuel temperature t a = t f = −40 • C; 6-turboshaft engine, jet fuel, centrifugal nozzle, candle discharge energy E = 2 J, height H = 4 km, air and fuel temperature t a = t f = 15 • C.