EFFECTS OF HEATER DIMENSIONS ON NANOFLUID NATURAL CONVECTION IN A HEATED INCINERATOR SHAPED CAVITY CONTAINING A HEATED BLOCK

The present work reports a numerical study of natural convection in an incinerator shaped enclosure with a localized heated source situated at the bottom. Lattice Boltzmann Method (LBM) is used to simulate nanofluid (water-Al2O3) flow and heat transfer. Simulations have been carried out for the pertinent parameters: Rayleigh number (Ra=10−10), solid volume fraction 0 6%    relative heat source high ( 0 0.5    ), relative heat source width ( 0.02 0.8    ), and inclination angle of the incinerator ( 0 360     ). The comparison of the obtained results is in excellent agreement with results from literature. It may be noted that the Rayleigh number, the solid volume fraction, the heat source tallness enhances the heat transfer and influences the flow pattern and the thermal structures. However for the relative heat source width plays opposite role for values superior to 0.4.


INTRODUCTION
Augmentation of heat transfer is an important goal in generating energy systems.Different mechanisms of heat transfer enhancement have been highlighted in the literature focusing mainly on modifying the geometry, swirl flow devices, use of porous media, partitioned cavities and other techniques, which are well documented [1].Recently, an innovative way of heat transfer enhancement is the use of nano-suspensions (particles, fibers) of relatively higher thermal conductivities suspended in the base fluids.Bergles [2] classified the techniques of heat transfer enhancement as active or passive.Active methods require external power to maintain the enhancement mechanism, whereas passive enhancement techniques don't require external power.The addition of nano-sized particles to a base fluid in order to enhance its thermal conductivity is classified among the passive enhancement techniques.The obtained solution is called nanofluid which is introduced first by Choi [3].Nanofluids are referred to the fluids with suspensions of nanoparticles having a diameter below 100nm.According to Mahbubul et al. [4], nanofluids is a solid-liquid mixture which consists of nanoparticles and a base liquid.Nanoparticles are basically metal (Cu, Ni, Al, etc), oxides (Al2O3, TiO2, CuO, SiO2, Fe2O3, Fe3O4, BaTiO3,etc) and some other compounds (AIN, SiC, CaCO3, graphene, etc) and base fluid usually include water, ethylene glycol, propylene glycol, engine oil, etc.Compared to other techniques used for enhancing heat transfer in practical applications, nanofluids have the advantage of behaving like pure fluids, because of the small size of nanoparticles.This technique can be used for the thermal machines which will be constructed in the future or even for the ones which are already functioning with little modifications.A review of recent research made with nanofluids can be found in [5].Natural convection heat transfer due to a localized heat source in complex geometries is ubiquitous in the majority of industrial systems.
It has been studied extensively because of its interest in many practical systems, including cooling of electronic equipments, material processing, chimneys, furnaces, heat exchangers, solar energy collectors, nuclear engineering, fire research, building engineering and geophysical flows.Ben Cheikh et al. [6] investigated numerically two-dimensional natural convection flow of a dielectric fluid in a square inclined enclosure with a fin placed on the hot wall.They found that for small enclosure inclination angles and moderate values of Rayleigh number (Ra), the flow is steady and mainly constituted by two counterclockwise eddies.For higher values of Ra, secondary eddies were observed between the fin sides and the lower passive walls.They also demonstrated that the inclination angle controls the transition to the unsteady state.Abbassi et al. [7] made a numerical studied of nanoparticle Brownian motion and magnetic field effects by natural convection in a nanofluid-filled open cavity with non uniform boundary condition.The Lattice Boltzmann Method is used as a simulation tool.The numerical results show a decrease in heat transfer with an increase in particle volume fraction.In addition, it is observed that the Brownian motion greatly influences the heat transfer rate depending on the Hartmann number, Rayleigh number and nanoparticle volume fraction.Öztop et al. [8] reviewed the effects of curvilinear boundaries on heat and nanofluid flow by natural convection.They clarified the energetic value of achieving highly efficient energy transport at reduced cost via different approaches, including change in the structural configuration of thermal systems and using nanofluids.They observed that curvilinear surface is the main effective parameter on heat and fluid flow in a nanofluid filled enclosure due to natural convection.Moumni et al. [9] investigated numerically two-dimensional mixed convection fluid flow and heat transfer of water-nanofluids in a two-sided facing liddriven cavity.Two discrete heat sources are located at the bottom wall of the enclosure.The governing equations are solved using a second order accurate finite volume approach.It was found that significant heat transfer enhancement can be obtained.The addition of nanoparticles decreases the fluid motion and increases the heat transfer rate considerably.The location of the two heat sources also affects the average Nusselt number.The latter increases as the locations of the two heat sources move respectively toward the left and the right walls.Mahfoud and Bendjaghloli [10] studied natural convection in a truncated cone filled with Cu-water nanofluid.They used the finite volume approach to solve the governing equations coupled with the SIMPLE algorithm.Their results showed an enhancement in cooling system when using a nanofluid.Also they demonstrated that the inclination angle of inclined sidewalls and heat source length affect the heat transfer rate and the maximum temperature.Hassan [11] investigated by the FVM heat transfer inside horizontal and vertical enclosures filled with Cu-water nanofluid.A heat sink with rectangular fins represents the base plate of the enclosure where it is heated by a discrete heat source at the bottom of the heat sink.The effect of volume fraction and diameter of the Cu nanoparticles on the average Nusselt number of the heat source is studied.He demonstrated that the average Nusselt number on the heat source increases with the increase of Rayleigh number and nanoparticle volume fraction and with decrease nanoparticle diameter.Using nanofluid increases the number of circulating cells and decreases their size compared with enclosure without nanofluid.The temperature of the heat source decreases when increasing Rayleigh number, increasing of nanoparticle volume fraction and with decrease of nanoparticle diameter.Öztop et al. [12] presented an exhaustive review of natural convection heat transfer and fluid flow in different enclosures with localized heat sources.The common objective of their review work is to investigate the effects of type and location of local heat sources as well as the effects of the different configurations and boundary conditions on the enclosed fluid/nanofluid flow and heat transfer.Kaluri et al. [13] studied laminar natural convection in a square cavity with distributed heat sources by the help of heatline and streamline approach.They considered four different cases, depending on the location of the heat sources on the walls of the cavity.They showed that the used method is found to be adequate to visualize and understand heat energy distribution and thermal mixing occurring inside the considered cavity.Multiple heat sources are also studied by Aminossadati and Ghasemi [14], they considered the problem of natural convection in a 2D cavity filled with nanofluid.Two pairs of heat source-sink are situated on the bottom wall of the cavity while the other walls are thermally insulated.The governing differential equations are solved by the control volume approach.The main goal of their study was to compare the heat transfer performance at different Rayleigh numbers and solid volume fraction of the nanofluid.The results show that regardless of the position of the pairs of source-sink, the heat transfer rate increases with an increase of the Rayleigh number and the solid volume fraction.Mussa et al. [15] studied numerically natural convection in an enclosure using Cubic-Interpolated-Pseudo-Particle (CIP) lattice-Boltzmann method.A D2Q9 lattice model was coupled with the D2Q4 lattice model to represent density and internal energy distribution functions, respectively.
The effect of the Rayleigh number on the flow pattern is studied.The enclosure is filled with air heated by a small localized heat source at two different positions on the bottom wall.Their results showed that natural convection increases with the Rayleigh number and heat source locations.Corvaro and Paroncini [16] analyzed experimentally convective heat transfer generated by a source with three different heights.The source is located in the middle of a square enclosure.The experimental analysis was carried out using holographic interferometry for the study of heat transfer and a 2D-PIV system to analyze the dynamic behavior of the phenomenon.The comparison is based both on the evaluation of the Nusselt number at different Rayleigh numbers and the study of the velocity fields at the same Rayleigh numbers in different configurations.Their experiments showed that the natural convective heat transfer worsens with the source height increase.In the last decades, the Lattice Boltzmann Method (LBM) is emerged as a powerful numerical technique, based on kinetic theory, to simulate fluid flow, heat and mass transfer.It has become a novel alternative to conventional Computational Fluid Dynamics (CFD) solvers like Finite Difference Method (FDM), Finite Element Method (FEM) and Finite Volume Methods (FVM) for solving the Navier-Stokes Equations (NSE) [17].The advantages of LBM include simple calculation procedures and ease in implementation of boundary conditions.It is well suitable for parallel computation, ease and robust in handling of multiphase flow and can be applied for complex geometries [18,19].Moreover, when using LBM, the coupling between pressure and velocity field is avoided.In conventional classical methods this linkage is handled by algorithms such as SIMPLE, SIMPLER [20] and others which are CPU time consumers.Another advantage it can capture turbulence without any turbulence models [19].The main aim of the present paper is to investigate numerically the heat transfer rate and the fluid flow in nanofluid filled incinerator shaped geometry.A heated block is situated on the bottom wall of the incinerator, whereas the vertical walls are considered cold and the top wall is supposed to be adiabatic.The LB method is applied to solve the coupled momentum and energy equations.The grid independency test and the validation of the present model are conducted on simple cavity cases.The effects of the main parameters (Rayleigh number, solid volume fraction, inclination angle, heat source width, heat source height, and heat source position) on flow, thermal fields and Nusselt number are presented and discussed.

PHYSICAL MODEL AND MATHEMATICAL FORMULATION
Figure 1 shows the considered two-dimensional problem with boundary conditions and coordinate system.It is an incinerator shaped enclosure of length L and height H (ABCDEFGH).The heat source (A1B1C1D1) at high temperature Th is located at the bottom surface of the cavity and has the width a and height b.The other parts of the bottom wall and the ceiling are considered adiabatic.The remaining walls of the cavity are cold.To make a straightforward study, the following assumptions are retained.The flow is steady, laminar and incompressible; all the thermophysical properties of the fluid are constant, except for the density in the buoyancy term which follows the Boussinesq approximation.The base fluid and the nanoparticles are assumed to be in thermal equilibrium and the nanofluid is Newtonian.The nanoparticles are supposed to be of spherical shape and the radiation effects are neglected.The non-dimensional form of the problem variables are implemented in the following: 00 0 2 00 ,, ; , ; , ; ; and Based on the above assumptions, the dimensionless governing equations can be expressed in the incompressible unsteady form as follows: 0

LATTICE BOLTZMANN METHOD
The main variable in LBM algorithm is the density distribution function fi(x,t) of the fluid pseudomolecules that has lattice velocity ci at location x at a time t.The lattice Boltzmann equation with Bhatnagar-Gross-Krook (BGK) collision mode, for incompressible problems, uses two distribution functions, f and g, for the flow and temperature fields respectively.For the flow field, the discretized LBM equations can be written as [7]: For the temperature field where Δt denotes the lattice time step which is set to unity and   ,   are the relaxation times for the velocity and temperature fields, respectively.eq i f , eq i g are the local equilibrium distribution functions (Eq.( 9)) that have an appropriately prescribed functional dependence on the local hydrodynamic properties and Fi is an external force term.The left-hand side of Eq. ( 7) is often called streaming or propagation and accounts for the migration of the fluid molecules from one grid point to its neighboring nodes.The right hand side is the collision part which models the relaxation process of the molecules towards the local equilibrium distribution eq i f , namely the truncated Boltzmann distribution when using the BGK collision model.( . ) where u and  are the macroscopic velocity and density, respectively, c= / x t  is the lattice speed, x  is lattice space and similar to lattice time step is equal to unity, i  is the weighting factor for flow, and temperature.
The double populations D2Q9-D2Q9 thermal model is used in this study.Thus, the weighting factors and the discrete particle velocity vectors are given as follows: The SRTs of the standard LB equations are linked to the diffusion parameters as:

NON DIMENSIONAL PARAMETERS
In the LBM simulations the Rayleigh number, Prandtl number and Mach number are assigned constants values therefore the viscosity and thermal diffusivity are calculated from the definition of these non dimensional parameters.

Pr R a
where n is the number of lattice nodes in y-direction.
To insure an incompressible flow the Mach number was chosen as 0.1 Ma  .transfer rate on the heat source is described by the Nusselt number Nu which is given by: where the heat transfer coefficient is computed from: The thermal conductivity of the nanofluid is expressed as: and using the dimensionless quantities, the local Nusselt number along the left wall can be written as: The average Nusselt number is obtained by integrating the local Nusselt number along the heat source:

GRID INDEPENDENCE STUDY AND VALIDATION
To ensure the grid independence we examine the sensitivity of our results for different uniform grids: 50×50, 80×80, 100×100, 120×120, 150×150, 200×200 and 250×250 and for all the studied Rayleigh numbers.For each Ra value, if the tested quantity converges, the grid size is adopted.This idea avoids time consuming for lower monitoring parameters (Ra) which may be pointless.The results of simulations for the tested grids are presented in Table 1.Yang [22] who used the Lattice Boltzmann Method, in this case also excellent results are found.

RESULTS AND DISCUSSIONS
In this paper natural convection inside an incinerator shaped enclosure filled with Al2O3-water nanofluid and having a hot rectangular source situated in the bottom face have been investigated.
Results are presented as streamlines, isotherms, and average Nusselt numbers.

EFFECT OF RAYLEIGH NUMBER (Ra)
Figure 5 shows the effect of Rayleigh number on streamlines and isotherms inside an incinerator including a hot rectangular heat source (ε=0.2, δ=0.5) and filled with pure fluid (water) and nanofluid with =4%.x=0.5 represents a symmetric plane for both isotherms and streamlines.For low Rayleigh numbers (Ra=10 3 -10 4 ), isotherms are parallel to the three faces of the hot source and are perpendicular to the bottom surface of the incinerator and horizontal thermal like stratification is formed around the heater.Streamlines form two counter rotating big cells from either part of the heater.Increasing Rayleigh number to Ra=10 5 , convection currents are more expressed and the isotherms are slightly tightened near hot walls to form a thermal boundary layer and the stratification is distorted.The streamlines still form a symmetric cells more elongated to the top due to the ascendant fluid motion and are occupying the whole medium below y=0.7.For Ra=10 6 the symmetry of the flow and thermal behavior is broken.One can remark the formation of two new counter rotating cells of different sizes.These two cells are formed due to the like Rayleigh-Bénard convection that hold at the upper wall of the heat source where the upper corner of the two big cells serve as moving (lid) boundary conditions to the formations of the small cells.
It is also worth to mention that the big rotation cells take the geometric forms of the surrounding space due to the intensified pressure.For the isotherms, boundary layers are clearly formed near the isotherm (hot and cold) walls and a like vertical stratification is formed between vertical walls of the heater and those of the incinerator.The former two small cells intensify the heat transfer rate from the upper heater wall.The overall heat transfer from the heated block is 4.034, 4.182, 6.909 and 14.222 for pure water and 4.680, 4.772, 7.1952 and 15.266 for nanofluid at Ra=10 3 , 10 4 , 10 5 , and 10 6 respectively.Thus, increasing the Rayleigh number from 10 3 to 10 6 and the solid volume fraction from 0 to 6% enhances about three times the heat transfer.

EFFECT OF THE NANOPARTICLE VOLUME FRACTION
For all Rayleigh numbers considered in this study heat transfer is obviously increased as the solid volume fraction increases.At Rayleigh numbers (Ra=10 3 -10 4 ), the addition of nanoparticles augments the heat transfer.However, if the Rayleigh number increases (Ra=10 5 -10 6 ), the addition of nanoparticles leads to a heat transfer rise on the heater vertical side walls whereas, a heat transfer reduction on its top wall is remarked.This pattern can be explained by the formation of two eddies placed at the heater top wall which decrease fluid flow motion to the incinerator top and consequently reduces the heat transfer.Besides, it can be seen from Figure 5 that the addition of nanoparticles does not affect the global dynamic and thermal structures.Figure 6 depicts the wall average Nusselt number at the left (or the right) of the hot block as a function of the Rayleigh number variation for 10 3 ≤Ra≤10 6 .First, one can see easily the rapid variation of the average Nusselt number with the Rayleigh number rise at constant solid volume fraction  ; and second the average Nusselt number increases by increasing the volume fraction  of nanoparticles.However, for the top facet average Nusselt number a reduction is observed by increasing Ra to 10 5 .This is explained by the fact that the ascendant flow near vertical walls creates a convective boundary layer parallel to hot walls which augments the heat transfer.These currents drive with them the fluid from either side of the heater top wall and its boundary layers become more spaced and the cells from either part of the hot block become more elongated to strangle the flow above the top wall.These two effects induce the decrease of heat transfer due to the heater top wall.Besides, for Ra=10 6 , the flow is characterized by the appearance of two new cells of different dimensions above the heater top wall.The corresponding Nusselt number is significantly increased compared to Ra=10 5 because the counter rotating two small cells drive with them the cold fluid from the incinerator neck.The tendency is same for all nonzero solid volume fractions.structure has changed widely at the right of the hot block.Besides, one can say that the effect of adding nanosuspensions does not affect the pure dynamic and thermal structure for γ=60° tilting case.For a quarter turn rotation ( 90   ), the streamlines for the pure fluid are formed by two eddies, whereas for the nanofluid these two eddies are merged on one big eddy occupying the whole medium.Increasing continuously the tilt angle, we obtain again two eddies for the pure fluid and the nanofluid and the head of the top cell begins to detach to fuse with the right cell and form a large cell extended from right to the top of the heater.The large cell dimension diminishes as the tilt angle approaches 180° where we obtain symmetric acting two equal cells.

EFFECT OF THE INCINERATOR TILT ANGLE
Figure 7b) depicts the effect of the incinerator inclination angle on the absolute stream function maximum for pure fluid and nanofluid at 4%   and for Ra=10 5 .It can be seen easily that for the studied inclination angles, the maximum of the stream function in the case of pure fluid is almost higher than in the case of the nanofluid.This behavior can be explained by the augmentation of friction when nanoparticles are added ( nf f   ). the Nusselt number varies sinusoidally as the inclination angle increases for the left or the right side of the heater source.Besides, it is observed that the inclination angle is a key parameter for the calculation of heat transfer in the three sides of the source heater.Figures 6.In this section the heater dimensionless height is varied from 0.1 to 0.5 and its width is fixed at 0.2   in the middle of the incinerator for all simulations.For all considered Rayleigh numbers (Ra=10 3 -10 6 ) the isotherms and flow structures exhibits a symmetrical patterns from either sides of the heater as shown in Figures 6 (a, b).For Rayleigh numbers 10 3 and 10 4 (Figure 8) isotherms are parallel to the heater vertical walls.This behavior is more pronounced with the augmentation of the heater dimensionless height.At the heater to facet, isotherms are also parallel to the horizontal surface.The distance between isotherms decreases by increasing heater height δ.Two symmetric counter rotating cells are formed.The right cell is rotating clockwise and the left one is rotating anti-clockwise.
For Rayleigh numbers 10 5 and 10 6 (Figure 9), isotherms are parallel to the heater vertical walls and are very smoothed near the vertical walls.However for the top face of the heater, isotherms have a parabolic shape.This parabolic shape is well shown with increasing the Rayleigh number.For 0,3   and Ra=10 6 , isotherms θ=0.05 are perpendicular to the top face of the incinerator which is adiabatic.Increasing  from 0.3 to 0.4 at Ra=10 6 , the isotherms θ=0.05 is at a position equal to 0.82; this can be explained by the stream function where we can see the formation of two cells on the top face of the heater source which are counter-rotating.The isotherms shape is reversed from the top of the incinerator to top side of the heater source.The thermal boundary layer thickness at the bottom side of heater source is decreased.This can be explained by the formation of two new contrarotative cells at the top face of the heater.These two cells contribute to the redistribution of energy to the left and right inclined walls and minimize the heat transfer to the top of the incinerator.Therefore on the two inclined sides of the incinerator the thermal boundary layer thickness decreases also.
Figure 10 shows that the average Nusselt number from the heater left side increases as the relative heat source height increases.This is evident because if the height  is increased, more energy is introduced to the medium.

EFFECT OF THE RELATIVE HEATER DIMENSIONLESS WIDTH
Figures 7(a, b and c) show respectively the effect of relative heat source width on isotherms, streamlines and average Nusselt number for Ra=10 3 -10 6 and 0, 02 0,8   in the case of pure water.The height of the heat source δ is maintained constant at 0.2.
Figures 7(a and b) demonstrate that the heater dimensionless width increase (with Ra=10 3 -10 5 ) leads to formation of two symmetric cells which occupy the whole of the free space of the incinerator.Isotherms are normal to the adiabatic bottom incinerator wall and are tightened near the heater vertical sides.
For Ra=10 6 and 0.4   the symmetry of isotherms is broken, it is well pronounced for 0.6

 
where we can see that the two symmetric cells are broken to give four cells; one big cell situated at the top side of that heater and three other little eddies two on the right side and the third at the left side of the incinerator.The symmetry of isotherms is also broken and we can see the formation of thin boundary layer on the faces of the heater block and on the cold inclined walls of the incinerator.For Ra=10 6 and 0.8   , Figure 12 shows the formation of six eddies occupying the whole area of the incinerator, results are quasi-symmetric and energy is transferred to the total area of the studied medium which gives also quasi-symmetric isotherms.The regime is unsteady due to the Rayleigh-Bénard like situation.
Figure 13 shows the effect of relative heat source width on average Nusselt number on the left side of the hot source.One can see easily that if the relative height ε is inferior to 0.6 and Ra=10 3 -10 4 , the average Nusselt number increases slightly with increasing relative heat source.).

EFFECT OF THE HOT BLOCK POSITION
Figure 14 presents the effect of heat source positions on isotherms and streamlines for pure water and nanofluid for Ra=10 5 .Evidently, one can say that moving the hot block toward the vertical cold side walls leads to symmetry breaking of the thermal and flow structures.At a middle position we note the formation of two cells.The lower is due to the like heated differentially cavity where intensively the heat transfer was verified and proved to increase significantly.The upper cell is due the like Rayleigh-Bénard convection.The thermal and flow structures in the presented three cases are significantly changed and the heat transfer is decreased by approaching the incinerator side wall.This can be explained by the fact that the left wall of the hot block will be inactive and then less heat will be transferred to the fluid.

CONCLUSION
Natural convection flow and heat transfer enhancement by Al2O3 nanosuspensions in an incinerator shaped enclosure containing a rectangular hot block is investigated in the present study.An indepth analysis of the monitoring parameters effects on the flow and thermal structures and the heat transfer enhancement is conducted by help of a SRT thermal lattice Boltzmann model.
From this study, the following conclusions are drawn: 1.Based on the quantitative and qualitative validation, the excellent agreement between ours results and former experimental and numerical data let us to rely on the powerfulness of the LB method to scrutinize such a twophase like problem. 2. The Rayleigh number increase enhances for about three times the heat transfer for both pure fluid and nanofluid.3. The addition of nanoparticles to the base fluid leads to the decrease of the activity of the fluid motion measured by the stream-function magnitude and causes a substantial increase in the heat transfer rate.4. The inclination angle of the incinerator is found to be a key parameter to control the heat transfer for both pure water and nanofluid.The tilting angle affects significantly the heat transfer for the heater three walls.5.The heat source tallness increase enhances the heat transfer for all its tested values and all Rayleigh numbers.However, the relative heat source width plays opposite role for dimensionless values superior to 0.4.

Figure 2 .
Figure 2. Domain boundaries and direction of streaming velocities, D2Q9

Figure 4 ..
Figure 4. Nusselt number distribution along the hot wall for Ra=10 3 -10 6 at a solid volume fraction

Figure 7a )6
Figure 7a) presents the effect of the inclination angle (  ) on the streamlines and isotherms for two volume fractions

Figure 7c )
Figure 7c) presents the effect of inclination angle average Nusselt number on the heater three hot facets for both cases of base fluid and nanofluid with 4%  

Figure 14 .
Figure 14.Effect heat source position on isotherms and streamlines for Ra = 10 5 and δ=0.4,ε=0.2 ( ̶ ̶ Base fluid; ---nanofluid).CONCLUSIONNatural convection flow and heat transfer enhancement by Al2O3 nanosuspensions in an incinerator shaped enclosure containing a rectangular hot block is investigated in the present study.An indepth analysis of the monitoring parameters effects on the flow and thermal structures and the heat transfer enhancement is conducted by help of a SRT thermal lattice Boltzmann model.From this study, the following conclusions are drawn: 1.Based on the quantitative and qualitative validation, the excellent agreement between ours results and former experimental and numerical data let us to rely on the powerfulness of the LB method to scrutinize such a twophase like problem. 2. The Rayleigh number increase enhances for about three times the heat transfer for both pure fluid and nanofluid.3. The addition of nanoparticles to the base fluid leads to the decrease of the activity of the fluid motion measured by the stream-function magnitude and causes a substantial increase in the heat transfer rate.4. The inclination angle of the incinerator is found to be a key parameter to control the heat transfer for both pure water and nanofluid.The tilting angle affects significantly the heat transfer for the heater three walls.5.The heat source tallness increase enhances the heat transfer for all its tested values and all Rayleigh numbers.However, the relative heat source width plays opposite role for dimensionless values superior to 0.4.