1. Introduction
The impinging flame jet (IFJ) is applied in many domestic and industrial heating systems because of its high heat and mass transfer characteristics and importance in many practical applications [
1], such as domestic cooking and heating, gas geysers, boiler wall heating, glass processing, and melting scrap metal parts [
2]. IFJ can provide local heating, has short start–up and cool–down times, and can obtain greater heat flux than high–cost radiant heating technology [
3]. Therefore, its primary fuel demand is less, its overall efficiency is improved, and it can even reduce pollutant emissions [
4]. Finally, IFJ has become a very attractive alternative. As a result, many researchers have conducted experimental and numerical research on flame jet heat transfer (FJHT) to help comprehensively understand its thermal characteristics and provide valuable insights [
5,
6,
7,
8].
In general, the structure of IFJ can be divided into a free jet region, a stagnation region, and a wall jet region. Each region has different flow, combustion, and heat transfer characteristics. The mechanism of FJHT mainly consists of several complex physical and chemical processes, including thermal conduction, convection, radiation, condensation and boiling, and thermochemical heat release (TCHR) [
9]. Therefore, its heat transfer process is much more complex than traditional isothermal impact jets. According to previous research reports, IFJ mainly uses forced convection as the heat transfer mechanism [
10]. Previous studies have confirmed that in impinging flame jets that use air as the oxidant, forced convection heat transfer accounts for 70% to 90% of the total heat transfer in the impinging region [
11,
12]. The thermal radiation from the flame can be ignored since the emissivity of the hot gas layer is very low [
13]. Cremer et al. [
14] pointed out that the TCHR mechanism is significant in hydrocarbon fuel flames under pure oxygen or oxygen–rich conditions and can reach up to 60% of the total heat transfer. However, the effect on heat transfer in premixed methane–air flames is minimal and, therefore, can be ignored. In addition, the interaction between the flame and the wall produces flame structures with different characteristics depending on the position of the collision flame [
15,
16], and it plays a critical role in the near–wall combustion process of various combustion equipment and power engines [
17,
18].
Many parameters affect the heat transfer performance of IFJs. These include the geometry and design of the burner, the flame type, the operating conditions (fuel composition, Reynolds number of the jet, and the equivalence ratio of the mixture), and the distance of nozzle–to–plate. Chander and Ray [
19] studied the influence of different burner geometries (tube, nozzle, and orifice) on the heat transfer characteristics of IFJs. It was found that the heat flux in the stagnation and wall jet regions is different, resulting in different flame lengths and shapes due to different exit velocity profiles. Compared to the heat flux distribution of the nozzle and orifice burner, the tube burner is non–uniform. Milson and Chigier [
20] studied the heat transfer characteristics of diffusion and premixed methane flame impingement on cold steel plates. Compared with diffusion flame, the maximum heat flux of premixed flame is higher because premixed flames can increase the combustion rate and promote faster combustion completion. Hargrave et al. [
21] indicated that as the Reynolds number increases, the reaction region of the premixed methane flame will extend further downstream and become wider. For the premixed methane impinging flame jet (PMIFJ), the peak heat flux is located near the reaction region, and both the velocity and temperature are found to reach the highest values in this region. This is due to the diffusion and exothermic recombination of dissociated substances in the boundary layer around the heated object, raising the convective heat transfer rate and heat flux [
22]. Kwok [
23] studied the effects of Reynolds number and equivalence ratio on premixed butane IFJs at a fixed nozzle–to–plate distance. The results indicate that the heat transfer to the impingement plate increases with the Reynolds number due to the turbulence enhancing the mixing of fuel and air, increasing the combustion rate. Furthermore, it is found that maximum heat transfer occurs under slightly fuel–rich conditions. Finally, it is proposed to match the flame length to the distance from the nozzle to the plate, which is crucial for achieving optimal heat transfer. Kuntikana and Prabhu [
24] experimentally studied the heat transfer performance of PMIFJs. The thermal efficiency was found to increase with increasing Reynolds number and equivalence ratio. The thermal efficiency decreases as the distance of nozzle–to–plate increases. They also established an empirical correlation between the stagnation point Nusselt number and three parameters (Reynolds number, equivalence ratio, and nozzle–to–plate distance). The maximum deviation is 12% compared to the experimental stagnation point Nusselt number. Dong et al. [
25] conducted experiments using premixed butane IFJs to establish empirical correlations between the average Nusselt number and Reynolds number, equivalence ratio, and nozzle–to–plate distance. Compared with the experimental average Nusselt number, the maximum deviation is 13.3%.
In addition to using experimental methods for research, numerical simulation methods can provide an augmented understanding of the correlation between the flow field and heat transfer of IFJs. Chander and Ray [
26] used the computational fluid dynamics (CFD) method and simplified methane chemical reaction mechanisms to study the flow field and heat transfer characteristics of PMIFJs. The results show that the local peak heat flux is away from the stagnation region, which is caused by the change of the peak axial velocity close to the plate. Hidasageri et al. [
27] studied PMIFJs produced by square and rectangular burners through CFD and a simplified methane chemical reaction mechanism. The simulated results show that the vortices formed near the corner of the tube produce an axis-switching phenomenon, causing the heat flux distribution on the impingement plate to be rotated. Singh et al. [
28,
29] adopted the CFD and a methane–air two–step global reaction mechanism to simulate the behavior of the flow field and heat transfer of several double-swirling flames impinging a flat plate without simplifying complex geometries and gain insights into their correlations. Laguillo et al. [
30] used the CFD and the detailed chemical reaction mechanism of methane (GRI–Mech 3.0) to accurately capture the relevant phenomena that occur in the partially premixed flame of methane, such as the jet’s velocity decay, the structure of inner premixed cone and diffusion zone, the temperature distribution on the wall, and pollutants emission. It further provides the factual standard for the design of natural gas burners. Numerical simulation has become an indispensable part of today’s engineering. It provides a powerful tool that enables engineers to evaluate and analyze various engineering systems and processes more quickly and effectively [
31,
32]. CFD is commonly utilized to optimize the design and operating parameters of practical engineering problems [
33,
34,
35]. However, the difficulties in CFD optimization are high computational costs and time constraints due to the need to consider the complexity of the geometry, the number of meshes, the applicability of the model, and the complexity of the chemical reaction mechanism.
Surrogate models are proposed as solutions to solve the above problems of CFD optimization. Therefore, this method is called surrogate–based optimization (SBO). Common surrogate models include the response surface method (RSM) [
36], support vector regression (SVR) [
37], radial basis function (RBF) [
38], and Kriging model (KM) [
39]. In these surrogate models, KM takes values from regionalized variables in finite regions and makes optimal and unbiased estimates from the perspective of the correlation and variability of variables. In addition, KM provides predicted mean and variance and establishes various learning functions to estimate failure probability and uncertainty accurately. Therefore, compared with other models, KM has higher fitting accuracy and can better approximate complex functions [
40,
41,
42]. Surrogate models must cautiously be used due to the possible impact of new errors, although they can quickly obtain analytical results. Hence, the suitability of surrogate models must be carefully evaluated to ensure valid and accurate results, allowing for correct decisions to be made [
43].
According to the previous literature review, empirical correlation was used to predict the heat transfer performance of unknown points for IFJs in the past, and the predicted values were found to deviate significantly from the actual values. In addition, since the optimization method for the heat transfer performance of IFJs has yet to be given, the optimal operating parameters of PMIFJs are worth exploring. This study uses experimental and numerical simulation methods to study PMIFJs. This study used the equivalence ratio, Reynolds number, and nozzle–to–plate distance, which are common in the previous literature, as research parameters. The experimental data serve as a validation of the numerical simulation to ensure its reliability. In terms of numerical simulation, obtaining the global heat transfer behavior and optimal solution using the CFD method and the detailed GRI–Mech 3.0 mechanism is quite challenging because of the complexity of the mechanism and the large number of computational resources required to consider all three parameters. Therefore, the SBO method is used to solve this problem, which is also the main contribution of this paper.
The KM is chosen as the surrogate model. Its Gaussian basis function has independently different variance values in different dimensions, which enables the model to capture changes in the parameter space more accurately, thereby obtaining more accurate and optimized response surfaces. Hence, global information within specific parameters can be gained by fewer simulations, and target values at unknown parameters can be further predicted. Furthermore, the genetic algorithm (GA) is used to determine three optimization problems: maximizing the likelihood function in KM, the new infilled points according to maximizing root mean square error (RMSE), and the operating parameters of optimal heat transfer performance.
The subsequent sections of this paper are structured as follows.
Section 2 introduces the experimental setup and measurement methods.
Section 3 provides the working process of the CFD method, mesh independence testing, and model validation results.
Section 4 introduces the principles and working process of the SBO method and further elaborates on the details of each process.
Section 5 provides optimized KM results and discusses the effect of operating parameters on heat transfer performance and the results of sensitivity analysis and optimization. Finally,
Section 6 provides the conclusions of this study.
2. Experimental Methods
The PMIFJ system used in this study consists of a burner, the flow output and control system of fuel and air, and a water–cooled alumina plate, as shown in
Figure 1. The burner comprises a diverging tube, three equalizing chambers, and a converging nozzle. Methane (purity 99.95%) and dry air are controlled by two mass flow controllers (MFC, DPC–47, Aalborg Instruments & Control Int., New York, NY, USA) with an accuracy of
0.01 SLPM. Methane and air are mixed in a mixing chamber, and the mixture is introduced into the burner. Five layers of stainless steel mesh are installed in the burner to reduce the flow fluctuations of the mixture, ensure uniform flow, and prevent flame flashback. Gaskets are utilized to prevent methane and air leaks. Finally, the mixture flows to the nozzle exit to produce a laminar premixed methane–air flame. The volume flow rate of methane and air is determined based on different equivalence ratios and Reynolds number operating conditions. The equivalence ratio (
) and Reynolds number (
) of the PMIFJ are defined as follows:
where
and
are the volume flow rates of methane and air, respectively.
where
is the density of the mixture,
is the velocity of nozzle exit,
is the nozzle diameter, and
is the dynamic viscosity of the mixture.
where
is the mass fraction of species
, and
is the density of species
.
where
is the dynamic viscosity of species
,
is the molar fraction of species
, and
is the molar mass of species
.
In order to observe the flame shape and characteristics of PMIFJ, luminous images of the flame were taken by a 4K CMOS camera. In addition, the cone’s leading edge defines the height of the premixed cone to provide validation for numerical simulations. To verify the heat transfer performance of the simulated PMIFJ, a water–cooled alumina plate was used as a heat exchanger to measure its total heat transfer rate. Aluminum sticks fixed the heat exchanger and adjusted the distance of nozzle–to–plate (
). A water pump and water flow meter supply the cooling water at a constant room temperature and flow rate to the heat exchanger. Two K–type thermocouples were used to measure the temperature of the water flowing into and out of the heat exchanger. After the mixture is ignited, the inlet and outlet water temperatures are recorded through the temperature acquisition card (NI USB-9162, National Instruments, Austin, TX, USA) once the outlet water temperature reaches a steady state. According to the energy balance equation for cooling water under steady-state conditions [
44], Equation (5) calculates the total heat transfer rate at different conditions.
where
is the total heat transfer rate,
is the mass flow rate of water,
is the specific heat of water, and
and
are the water temperatures at the inlet and outlet of the heat exchanger, respectively.