Fatimah Bey

M Jarray

K Chetehouna
email: khaled.chetehouna@insa-cvl.fr

N Gascoin

Dr Khaled

Ceramic panel heating under impinging methane-air premixed flame jets

Keywords: Fireproof burner, Numerical modeling, Impinging flame jet, Heattransfer, Fluid flow, CFD

come

Ceramic panel heating under impinging methane-air premixed flame jets

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Introduction

Heat transfer due to flame jet impingement is a very important process in industry and it is used for many applications like melting scrap metal, shaping glass, brazing, welding, etc. A lot of research has been carried out in this area, both experimentally and numerically. Most of existing studies on impingement flame jets were concentrated on circular jet utilizing methane or natural gas [START_REF] Milson | Studies of methane and methane-air flames impinging on a cold plate[END_REF][START_REF] Hargrave | Forced convective heat transfer from premixed flames -Part 2: Impingement heat transfer[END_REF][START_REF] Fairweather | Stagnation point heat transfer from laminar, high temperature methane flames[END_REF][START_REF] Hargrave | Forced convective heat transfer from premixed flames -Part 1: Flame structure[END_REF][START_REF] Baukal | Heat transfer from oxygen-enhanced/natural gas flames impinging normal to a plane surface[END_REF][START_REF] Viskanta | Heat transfer to impinging isothermal gas and flame jets[END_REF][START_REF] Sumrerng | High efficiency heat-recirculating domestic gas burners[END_REF]. Examples of experiments quantifying the heat-transfer characteristics of impinging flame jets can be found in references [START_REF] Dong | Heat transfer of a row of three butane/air flame jets impinging on a flat plate[END_REF][START_REF] Dong | Heat transfer and wall pressure characteristics of a twin premixed butane/air flame jets[END_REF][START_REF] Kwok | Heat transfer characteristics of an array of impinging pre-mixed slot flame jets[END_REF][START_REF] Zhao | Impinging premixed butane/air circular laminar flame jet -influence of impingement plate on heat transfer characteristics[END_REF][START_REF] Chander | Heat transfer characteristics of three interacting flame jets impinging on flat surface[END_REF]. One of the applications that encounter emerging interest and that could take a strong benefit from numerical simulation is related to fireproof certification tests which are imposed by Federal Aviation Agency (FAA) and its European equivalent (EASA).

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Indeed, the composite panels that form the current planes must be fireproofed and experimental tests show rapid cost increase. Thus, numerical simulation could be a way to investigate panel behavior under fire conditions in order to facilitate related engineering work. To do so, many numerical parameters must be carefully managed in order to ensure the reliability of the calculations.

According to the studies of Dong et al. [START_REF] Dong | Heat transfer from an impinging premixed butane/air slot flame jet[END_REF] the premixed butane-air laminar flame jet impingement heat transfer was dependent on Reynolds number, on equivalence ratio of the air/fuel jet, and on configurations of the air-fuel nozzle and of the impingement plate. Chander and Ray [START_REF] Chander | Flame impingement heat transfer: A review[END_REF] presented a very comprehensive and informative review on impingement heat transfer, where flame shape, stabilization and burner geometries are considered. Kwok et al. [START_REF] Kwok | Heat transfer characteristics of an array of impinging premixed slot flame jets[END_REF] suggested the importance in matching the flame length with the nozzle-to-plate distance to achieve the best heat transfer performance.

Hsieh and Lin [START_REF] Hsieh | Methane flame stability in a jet impinging onto a wall[END_REF] have explored the stability of a methane flame jet impinging normally to a wall. They stated that the flame is relatively stable at lower inlet velocity or lower burner to plate distance.

Hindasageri et al. [START_REF] Hindasageri | Heat transfer distribution for impinging methane air premixed flame jets[END_REF] studied heat flux distribution for premixed methane-air flame jet for Reynolds number varying from 600 to 1400 at an equivalence ratio of 1 for nozzle tip-target plate distance varying from 2 to 4. The temperature distribution of the quartz plate is recorded using a technique based on infrared thermography where the back side and flame side heat flux are compared. These authors stated that the back side measurement would enable capturing thermal images of the impingement plate without loss of aspect ratio, which is the case for flame side measurement, due to obstruction of image by the burner tube. It should be noted that this obstruction can be avoided using numerical approaches.

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The aim of this work is to explore the possibility of predicting, numerically, the thermal behavior of a test panel, impinged by a methane-air premixed flame jet. The results of the calculations have been compared to the experimental data of Hindasageri et al. [START_REF] Hindasageri | Heat transfer distribution for impinging methane air premixed flame jets[END_REF].

Material and Methods

Experimental data of Hindasageri et al.

Hindasageri [17] for circular flame jet issuing from the burners of diameter 10 mm and 8.7 mm. For the burner of diameter 10 mm at Re=1000, the heat flux decreases with the increase in z/d from 2 to 4. For further increase in z/d from 4 to 6, the change in heat flux is negligible. These outcomes are in agreement with the observations reported by

Grinstein and De Vore [START_REF] Grinstein | Dynamics of coherent structures and transition to turbulence in free square jets[END_REF] and Miller et al. [START_REF] Miller | Numerical simulation of non-circular jets[END_REF].

Physical and numerical modeling

The ANSYS Fluent 14 code [20] was used to solve the compressible Navier-Stokes equations in transient conditions. The pressure-velocity coupling was done using the SIMPLE scheme.

Meshing

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The quality of a CFD solution is highly dependent on the quality of the mesh.

Therefore, it is necessary that the grid is of high quality before proceeding to the next step. The grid structure must be fine enough where strong gradients of the variables are expected. In our study a non-uniform meshes is used with hexahedral and quadrilateral elements, more nodes accumulated around the reaction zone. A grid independence analysis was conducted using five meshes of varying cell number. Each mesh was processed using the enhanced wall treatment, with same boundary condition and a convergence of residual error set to 10 . The mesh of 124248 nodes and 654742 elements was generated and adopted for circular burner and a mesh of 122295 nodes 644527 elements for square burner (Figure 1). By examination of different cell sizes of this mesh, no further significant change was found for finer cells; this suggests that the grid independence has been achieved. An example of test configuration is provided for illustrative purpose (Figure 2).

Computational domain and conditions

The boundary conditions are as follows. At the inlet, a fixed velocity profile is used.

Different numerical models were tested and a comparative analysis was carried out. The objective was to comply as much as possible with the experimental test set-up.

Once this process had been finalized, it was possible to present some of the results and compare it to the experimental data. In addition the mixture of air-methane is prescribed in FLUENT. For the turbulent flame speed, the Zimont model is used [START_REF] Zimont | An Efficient Computational Model for Premixed Turbulent Combustion at High Reynolds Numbers Based on a Turbulent Flame Speed Closure[END_REF]. It The effectiveness for the flame impingement process, and the Nusselt number are given by Eq.( 4) and ( 5), as defined in [START_REF] Kuntikana | Isothermal air jet and premixed flame jet impingement: Heat transfer characterisation and comparison[END_REF].

(

) 4
(5)

Governing equations

The flow field is determined by the celebrated Navier-Stokes equations:

Mass equation = 0 (6)
For an incompressible flow 0 0

Momentum equation

2 ′ ′ ) (8)
Energy equation + = ( ). The Reynolds decomposition separates a velocity into its temporal mean and fluctuating components

′ , ̅ ′ (10)
where is a varying mean fields and ′ a rapidly varying turbulent components ,

Reynolds operator can be thought of as the integration operator, thus we can apply it as follows:

= ̅ so ′ 0 & ( 12
)
Where f denotes a scalar such as pressure or a vector such as velocity. For unsteady flow may be a slow function of time.

Substituting ′ , ̅ ′ into equation ( 8), we obtain

0 ( 13
) + = - + (14) So 0 ( 15
)
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+ = - ̅ + ( ) (16)
Where are the Reynolds stress tensor.

2.2.4.Turbulence modelling

The closure of the Navier-Stokes equation is performed with the help of a turbulence model. Many models are available in this context (e.g. Speziale, 1987) [START_REF] Speziale | On nonlinear K-l and K-ε models of turbulence[END_REF]. However, the classic model is that due to Launder et al. (1975) [START_REF] Launder | Progress in the development of a Reynoldsstress turbulence closure[END_REF] who wrote a two-equation model for the transport of turbulent kinetic energy and turbulence dissipation from which the Reynolds stresses could be derived. The Reynolds-decomposition approach is generally adopted for practical engineering calculations, and models such as the kmodel and its variants, k-and the DES are typically used. The interested reader may find the details of these models in the appendix, or in [START_REF] Celik | Introductory Turbulence Modeling[END_REF][START_REF] Launder | Progress in the development of a Reynoldsstress turbulence closure[END_REF][START_REF] Wilcox | Turbulence Modeling for CFD[END_REF][START_REF] Yakhot | Renormalization group analysis of turbulence: I. basic theory[END_REF][START_REF] Bardina | Turbulence Modeling Validation[END_REF][START_REF] Cen | Challenges of Power Engineering and Environment[END_REF][START_REF][END_REF][START_REF] Sarkar | Application of a Reynolds-Stress Turbulence Model to the Compressible Shear Layer[END_REF][START_REF][END_REF]. The suitability of a turbulence model to provide accurate predictions for a given case varies greatly;

depending on the geometry, mesh and on the value of the local Reynolds number. It is therefore not possible to determine beforehand which turbulence model is best suited for a given study. The simulations have therefore been carried out with several turbulence models, ie. the k-standard, k-realizable , k-RNG, k-, and DES models. The obtained results have then been compared to the experimental reference data of of Hindasageri et al.

Results and discussions

The accuracy of available turbulence models was evaluated on both circular and square burners for all ranges of Reynolds number and of z/d ratio (see Figure 3). From For the case of the square burner z/d=6; Fig. 3 does not show a clear superiority of the realizable k-turbulence model, but in the main test cases, it can be seen that there is a better agreement between the realizable kmodel results and experimental data, Furthermore the RNG k-and DES models tend to over-predict the effectiveness η at the stagnation point. It is also noticeable that the maximum Nusselt number and effectiveness at the stagnation zone for the square burner are around 5% higher than those of the circular burner for K for the DES, RNG and realizable k-model which is in agreement with the experimental results. From this analysis, it can be stated that the realizable k-model gives very satisfactory results for the complex phenomena presented, even being a RANS two-equation model.

Considering the k- realizable model, in complementarity to Figure 3, the heat flux distribution of the quartz plate of thickness 3 mm impinged by a flame jet for circular burners after 15s is presented (Figure 4). In both figures, it can be seen that η decreases almost linearly as a function of r/d starting from the stagnation point for all different cases. For all z/d, η is at a maximum at the stagnation point and then decreases along the plate. It was noticed that a small increase of effectiveness value is perceptible for z/d=4, where the impingement distance is too high and where the Reynolds number is too small to produce a clear effect of heat exchange. Furthermore, the heat flux intensity in the plate decreases when the z/d ratio is increased from 2 to 6 (Figure 4). This is in agreement with the experimental results. Particularly, it was found that for a 13/34 small z/d ratio the heat flux in the center jet area was relatively low. The heat transfer performance was enhanced with Re due to an increase in the premixed cone height.

According to Kwok et al. [START_REF] Kwok | Heat transfer characteristics of an array of impinging pre-mixed slot flame jets[END_REF], the heat transfer characteristics of the impinging flame jet system were enhanced with the increasing Reynolds number. So, Figure 4 is of great importance for further numerical fireproof tests because the clear impact of geometrical distances (z, r and d) on the effectiveness of the transfer will impose a sensitivity analysis when testing complex geometries. Indeed, the problem of repeatability which is experimentally observed could be due to a problem of accurate positioning.

According to the wall heat flux, the circular shape of the temperature distribution is strongly dependent on the burner-plate distance (Figure 5). Further away from the hot spot, an exponential decrease can be observed (that is more or less the same for all cases), since the contribution of the heat flux outside the hot spot of the plate is less than the contribution of the heat flux inside. This highlights the importance of the flame speed on the material panel and it is in good agreement with regulation and norms that impose a strict control of mass flow rate and equivalence ratio during the fireproof tests.

The variation of the Nusselt number and the effectiveness along the radial direction of the impingement plate starting from the stagnation point for different Re and jet exit-plate distances is shown in Figure 6 for circular burner. It can be seen that η decreases almost linearly as function of r/d starting from the stagnation point for all different cases.

It was also observed under all the conditions, for the circular burner, that the radial distribution of Nu has the same bell-shaped trend curves which is not the case for the square burner particularly for higher Reynolds number. For the circular burner with a plate distance higher than 2 and a Reynolds number higher than 1400, the maximum 14/34

Nusselt number and effectiveness at the stagnation point are no longer differentiated and the effectiveness for z/d=4 and for z/d=6 are nearly equal.

Conclusions and Perspectives

Within the framework of fireproof tests, numerical simulation is a way of saving time and cost. However, tools must demonstrate to be reliable in the conditions that

  wall, (K) Temperature of the flame (K) u Velocity of the fluid, (m.s -1 )

  is a premixed combustion model based on the prediction the turbulent flame speed. It takes into account methane concentration, temperature, and molecular diffusion properties, as well as the detailed chemical kinetics and the influence of large eddies and small eddies on flame front wrinkling and stretching.The solution is declared converged if the residuals of the continuity, RANS and turbulence equations are below 10of their original values. The adiabatic wall temperature is estimated during the post-processing of the results using a transient thermal solver (Ansys workbench) that has been coupled with the CFD results.For comparison purpose, simulations were conducted with the same dimensions and operating conditions as those of the experimental study of Hindasageri et al.[START_REF] Hindasageri | Heat transfer distribution for impinging methane air premixed flame jets[END_REF]. The heat flux distribution is studied for burner tubes of circular shape (d=10 mm) and square shape (width=10 mm). Distance between burner exit and plate is represented by a nondimensional ratio (z/d); it is varied from 2 to 6. The length to diameter ratio (l/d) of all tube burners are maintained as 50. Three different Reynolds numbers of 600, 1000 and 1400 were considered. Methane-air premixed flame jet of Reynolds number varying from 600 to1400 and an equivalence ratio of Φ 1 is considered. The mixture Reynolds number (Re) is calculated from Eq. (1).

  time; (i=1,2,3) are the velocity components, is the gravitational acceleration vector, is fluid density, p is pressure, T is the Temperature, and the viscous stress tensor, ( + , the rate of strain tensor ′ ′ representing the reynolds stress tensor.

Figure 3 ,

 3 Figure3, the k-RNG and realizable k-models demonstrate a good agreement with

  engineers will use them for predimensionning large structure panels of airplanes. A numerical study of heat transfer in a flat plate impinged by a premixed flame jet with different physical and boundary conditions based on experimental results by Hindasageri et al. has been performed. Circular burner with inlet diameters of 10 mm and square burner of sides 10 mm have been modeled. The efficiency of different RANS turbulence models was studied. It was found that the realizable k-model gives the best overall results compared to the other models. Comparisons between the experimental and numerical results of the heat flux and effectiveness, under a wide different values of Re, burner shape and z/d, were made. The results clearly show that the heat flux varies slowly for large flame top-to-plate distances and increases rapidly for shorter distances, which is in agreement with the experimental results by Hindasageri et al. The sensitivity of the effectiveness parameter in the geometrical distances may explain the problem of experimental repeatability that is observed. Numerical fireproof tests could thus be more efficient than experimental tests. In addition, numerical tests confirm some key points which are found in the fireproof norms of FAA (like the strict control of flame speed and equivalence ratio). The variation of Nusselt number and the effectiveness with Re for different jet exit-plate distances obtained by the present simulation agree well with the experimental results. A 15/34 future work will now focus on a real 3-D aerospace panel to provide understanding of numerical ability under realistic test configuration.

Fig 1 .Fig. 2 .

 12 Fig 1. Close-up view of the numerical meshing near the flame/panel interaction.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Turbulence models effects on effectiveness for circular (a to c) and square (d to f) burners at different z/d ratios (2 and 6) and Reynolds number (600 and 1400)

Fig. 5 .Fig. 6 . 34 Figure 1

 56341 Fig. 5. Temperature distribution for circular burner at 15s

Figure 2 17/ 34

 234 Figure 2

Figure 3 18/ 34

 334 Figure 3

Figure 4 19/ 34

 434 Figure 4

Figure 5 20/ 34

 534 Figure 5

Figure 6

 6 Figure 6

Appendix

A.1.1.Boussinesq Approach & Usual k-ε models

The k-ε model is one of the most widely used turbulence models as it provides robustness, economy and reasonable accuracy for a wide range of turbulent flows.

Improvements have been made to the standard model which improves its performance.

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Two variants are available in Fluent; the RNG (renormalization group) model and the realizable model.

A common method employs the Boussinesq relationship [START_REF] Celik | Introductory Turbulence Modeling[END_REF] to relate the Reynolds stresses to the mean velocity gradients, so the Reynolds shear stresses is redefined with The Kronecker symbol with j=3 indicating the vertical direction,

The dissipation rate of k, Is a model parameter which is constant for k-standard and RNG and equal to 0.09, but variable for the k-realzeable.

The eddy viscosity is not a fluid property but depends strongly on the state of turbulence; may vary significantly from one point in the flow to another and also from flow to flow.

Equations for k and ε, together with the eddy-viscosity stress-strain relationship constitute the k-ε turbulence model where ε is the dissipation rate of k.

FLUENT gives the user the choice of three models from k-family k-ԑ standard : The standard k-model has been the most widely used model since it was introduced by Jones and Launder [START_REF] Launder | Progress in the development of a Reynoldsstress turbulence closure[END_REF] As a result, its strengths and weaknesses 22/34 are well known. According to Wilcox [START_REF] Wilcox | Turbulence Modeling for CFD[END_REF] it is generally inaccurate for flows with adverse pressure gradient (and therefore also for separated flows) which would limit its applicability for river flows.

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The standard k-model, can guarantee niether the positivity of normal Reynolds stress 0 (nonrealizable) when the strain rate is large. nor the Schwartz inequality for shear stresses in large strain rates ′ ′ ′ ′ [START_REF] Cen | Challenges of Power Engineering and Environment[END_REF]. In fact the normal stress in xdirection can be calculated from

so for the standard k-ԑ model, the normal stress will be negative in flow with high strain rate 0.09 3.7

which is of course unrealstic.

To overcome these problems, the realizable k-model makes the eddy-viscosity coefficient, , dependent on the mean flow and turbulence parameters.

Since the model still relatively new, it is not clear in exactly which instances the realizable k-model consistently outperforms the RNG model. However, initial studies have shown that the realizable model provides the best performance of all the k-ɛ model versions for several validations of separated flows and flows with complex secondary flow features [START_REF][END_REF].

A.1.2 Transport Equations

The turbulence kinetic energy, k, and its rate of dissipation, ԑ, are obtained from the following transport equations: Represents the generation of turbulence kinetic energy due to the mean velocity gradients this term may be defined as

is generation of turbulent kinetic energy that arises due to buoyancy. It's given by (A9)

where is the turbulent Prandtl number for energy, and thermal expansion, where (A10)

Buoyancy occurs due to a difference in indoor-to-outdoor air density resulting from temperature and moisture differences. The greater the thermal difference and the height of the structure, the greater the buoyancy force, so in Fluent when a non-zero gravity field and temperature gradient are present simultaneously, the k-models account for the generation of k due to buoyancy.

represents the fluctuating dilation in compressible turbulence that contributes to the overall dissipation rate, For high-Mach-number flows, compressibility affects turbulence through so-called "dilatation\dissipation", which is normally neglected in the modeling of incompressible flows [START_REF][END_REF] the dilatation dissipation term, , is included in the k equation. This term is modeled according to a proposal by Sarkar where a is the speed of sound (A13) the heat capacity ratio, the ratio of the heat capacity at constant pressure ( ) to heat capacity at constant volume ( ).

The eddy viscosity (A14)

The RNG Transport Equations has a similar form to the standard k-ɛ model

However the turbulent viscosity is modeled by the differential equation [START_REF][END_REF] Where is the value of turbulent viscosity calculated without the swirl modification, is a swirl constant that assumes different values depending on whether the flow is swirl-dominated or only mildly swirling.

The modeled transport equations for k and in the realizable k-ɛ model are

where

The difference between the realizable k-model and the standard and RNG k-models is that no longer constant. It is computed from [START_REF][END_REF] * (A25)

where * ≡ Ω Ω (A26)

and

Where Ω is the mean rate of rotation tensor viewed in a rotating reference frame with the angular velocity . The model constants and are given by

Modeling the Turbulence Production k

The k production may be defined as

Modeling the dissipation of k

The dissipation of k is given by * *

Where is given by:

1 tanh 20