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This paper presents a numerical and experimental analysis study of the temperature distribution in a cylindrical specimen heat treated by laser and quenched in ambient temperature. The cylinder studied is made of AISI-4340 steel and has a diameter of 14.5-mm and a length of 50-mm. The temperature distribution is discretized by using a three-dimensional numerical finite difference method. The temperature gradient of the transformation of the microstructure is generated by a laser source Nd-YAG 3.0-kW manipulated using a robotic arm programmed to control the movements of the laser source in space and in time. The experimental measurement of surface temperature and air temperature in the vicinity of the specimen allows us to determine the values of the absorption coefficient and the coefficient of heat transfer by convection, which are essential data for a precise numerical prediction of the case depth . Despite an unsteady dynamic regime at the level of convective and radiation heat losses, the analys i s of the averaged results of the temperature sensors show s a consistency with the results of microhardness measurements. The feasibility and effectiveness of the proposed approach lead to an accurate and reliable mathematical model able to predict the temperature distribution in a cylindrical workpiece heat treated by laser.

Laser surface heat treatment, which aims to improve the properties of materials such as steel and some aluminum alloys, offers many advantages over conventional hardening processes [

T. Miokovic et al. presented in 2006 a study on the superficial hardening of AISI-4140, study based on the effect of inhomogeneous formation of austenite due to conditions of austenization and quenching locally different, and at a dilatometry at high speeds of heating and cooling during the process. Data from this study has been implemented in the ABAQUS finite element program as laws of user-defined materials for a calculation that couples temperature distribution and phase development during heating and cooling process. The results obtained in this study faithfully reproduce the evolution of the temperature as well as the value of the resulting hardness [

Sen-Yung Lee et al. presented in 2014 a new method to solve temperature distribution equations in a workpiece heat treated by laser. The proposed resolution method is based on minimizing the mean squared error between the experimental data obtained and the estimated data of the analytical solution, with time-dependent boundary conditions. The distribution of temperature and heat flux over the entire space-time domain were predicted for real cases to illustrate the simplicity, the effectiveness and precision of the proposed method [

In the field of laser heat treatment an interesting but little studied prediction using the finite difference method to predict the temperature distribution inside of a cylindrical workpiece in rotation was developed for a stationary point by the team of R. Fakir et al. in 2018 [

In this study a mathematical model of laser heat treatment of cylindrical workpiece was developed to predict the temperature distribution and the case depth of the transformation of the microstructure.

The model is based on the calculation of heat flux with an adjustment, by experimental data, of the absorption coefficient of the metal and the coefficient of heat transfer by convection, for a good consideration of the variations of the thermal properties of the material according to the laser heat treatment parameters (including laser beam scanning speed). By means of temperature sensors we measured the evolution of the surface temperature of the cylinder and the surrounding air of the workpiece, according to the laser heat treatment parameters. These temperature measurements were used to estimate the values of the absorption coefficient and the heat transfer coefficient, which are essential data for a good numerical prediction of the case depth related to the eutectoid transformation mechanism. Equations (1) and (2) present the prediction polynomials of the variation of thermal conductivity and specific heat according to the temperature.

λ ( T ) = { 45.06 T + 0.000786 T − 0.000025 T 2 , T ≤ 775 ˚ C 30.8 , 775 ˚ C ≤ T (1)

C p ( T ) = { 400.30 + 0.5317 T , T ≤ 800 ˚ C – 6270 + 8.869 T , 800 ˚ C ≤ T ≤ 887.5 ˚ C 11754 − 11.44 T , 887.5 ˚ C ≤ T ≤ 975 ˚ C 600 , 975 ˚ C ≤ T (2)

For the evaluation of the value of the coefficient of heat transfer by convection, we used an empirical correlation adapted to scientific calculations and which groups the dimensionless number Reynolds ( R e D ), Nusselt ( N u D ), and

Prandtl ( P r ). Equation (3) presents the expression of the empirical correlation for rotating cylinders with a constant rotation speed Ω (rad/s) [

N u D ¯ = ( h D / κ ) = 0.133 R e D 2 / 3 P r 1 / 3 (3)

h = 0.133 ( κ / D ) ( Ω D 2 / β ) 2 / 3 ( β / γ ) 1 / 3 (4)

For the estimation of the value of the heat transfer coefficient through the empirical correlation of Equation (4) it is necessary to know the exact values of the thermo-physical properties of the air (κ, β and γ) in the vicinity of the cylindrical workpiece. Properties that were determined by experimental trials and that are dependent of laser heat treatment parameters (laser power, scanning speed and rotational speed of the cylindrical part).

The quantity of power density of the Gaussian beam I(x, y) transmitted by conduction in the workpiece, is modeled based on the heat conduction Equation (5) for a revolution system with an axial temperature gradient depending on the time.

λ ( T ) ( ∂ 2 T ∂ r 2 + ∂ T r ∂ r + ∂ 2 T ∂ x 2 ) = ρ C p ( T ) ∂ T ∂ t (5)

Equation (6) presents the transversal profile in the space of the Gaussian laser beam that propagates in the direction of the z-axis [

I ( x , y ) = P exp ( − r ( x , y ) 2 r 0 ) (6)

With r ( x , y ) = ( x − x 0 ) 2 + ( y − y 0 ) 2 and x 0 , y 0 the coordinates defining the center of the laser beam. P is the maximum value of I(x, y) measured at the center defined by x 0 and y 0 .

The initial temperature of the cylinder is supposed to be uniform T_{in} = 27˚C. This initial condition makes it possible to determine the coefficients which are solutions of the equations of the system at the beginning of the laser heat treatment (at t = 0 s). The laser beam moves from the start-laser position (reference position x = 10-mm) to stop-laser position (end position of the course x = 40-mm) with a constant scanning speed SS. The sample is continuously rotating on a test bench with a fixed rotation speed Ω. The diameter of the focal spot of the laser beam is set at 0.50-mm. Equation (7) shows the expression of the course distance of the focal spot at time t and according to the average scanning speed of the course.

l ( t ) = S S ⋅ t (7)

Boundary conditions are expressed according to equation 8, for a good consideration of the scanning speed of the laser beam and starting and stopping positions at the outer surface of the cylinder (r = R).

{ x < l ( t ) or x > l ( t ) + ϕ s , Condition 1 l ( t ) ≤ x ≤ l ( t ) + ϕ s , Condition 2 (8)

Equation (9) presents the expression of the boundary conditions outside and inside the zone irradiated by the laser beam. The first term of condition 1 represents convective heat losses and the second term represents the heat losses by radiation. The first term of condition 2 represents the amount of energy transmitted by the laser beam to the workpiece and the second term represents thermal losses by convection and radiation.

λ ∂ T ∂ r | r = R = { h ( T − T ∞ ) + ε σ ( T 4 − T ∞ 4 ) , Condition 1 P Λ e − r ( x , y ) 2 r 0 2 π R ϕ s − ( h ( T − T ∞ ) + ε σ ( T 4 − T ∞ 4 ) ) , Condition 2 (9)

Finite difference method [

∂ T ∂ r = ( T ( i + 1 , j , t ) − T ( i , j , t ) ) ζ r (10)

∂ T ∂ t = ( T ( i , j , t + 1 ) − T ( i , j , t ) ) ζ t (11)

∂ 2 T ∂ r 2 = ( T ( i + 1 , j , t ) − 2 T ( i , j , t ) + T ( i − 1 , j , t ) ) ζ r 2 (12)

∂ 2 T ∂ x 2 = ( T ( i , j + 1 , t ) − 2 T ( i , j , t ) + T ( i , j − 1 , t ) ) ζ x 2 (13)

Equation (14) presents the expression of the temperature, at the moment t + dt according to the value of the temperature at time t, by means of an explicit discretization of the Fourier-Kirchkoff equation.

λ ( T ( i , j , t ) ) ( ( T ( i + 1 , j , t ) − 2 T ( i , j , t ) + T ( i − 1 , j , t ) ) ζ r 2 + ( T ( i + 1 , j , t ) − T ( i , j , t ) ) ( R ζ r − ( i − 1 ) ζ r 2 ) + ( T ( i , j + 1 , t ) − 2 T ( i , j , t ) + T ( i , j − 1 , t ) ) ζ x 2 ) = ρ C p ( T ( i , j , t ) ) ( ( T ( i , j , t + 1 ) − T ( i , j , t ) ) ζ t ) (14)

The initial temperature of the cylinder is assumed to be constant T(i, j, 0) = 27˚C. Equations (15) and (16) present a discretization with an explicit finite difference scheme of conditions 1 and 2 of Equation (9). This represents the boundary conditions outside and inside the area irradiated by the laser beam, during its course at a scanning speed SS.

Condition 1: x < t i ⋅ S S ⋅ ζ t où x > t i ⋅ S S ⋅ ζ t + ϕ s

λ ( T ( R − ζ r , j , t ) − T ( R , j , t ) ζ r ) = ( h ( T ( R , j , t ) − T ∞ ) + ε σ ( T ( R , j , t ) 4 − T ∞ 4 ) ) (15)

Condition 2: t i ⋅ S S ⋅ ζ t ≤ x ≤ t i ⋅ S S ⋅ ζ t + ϕ s

− λ ( T ( R − ζ r , j , t ) − T ( R , j , t ) ζ r ) = I 2 π R ϕ s − ( h ( T ( R , j , t ) − T ∞ ) + ε σ ( T ( R , j , t ) 4 − T ∞ 4 ) ) (16)

Equation (17) shows, a simplified writing of the 4th-order polynomials for solving the equations of conditions 1 and 2. The prediction of the value of the temperature T(R, j, t) on the surface of the cylindrical workpiece can be done by applying the numerical method of Regula Falsi, which consists of repeating the interval division to find the value of T(R, j, t) solution of conditions 1 and 2.

T ( R , j , t ) ← { Ψ T ( R , j , t ) 4 + Π T ( R , j , t ) − Γ = 0 , Condition 1 Ψ T ( R , j , t ) 4 + Π T ( R , j , t ) − Γ = 0 , Condition 2 (17)

Ψ, Π, and Γ, are the respective coefficients of the monomials of degree 4, 1 and 0.The laser energy is zero (I = 0) in the term Γ corresponding to the condition 1.

Ψ = ε σ ζ r λ ( T ) (18)

Π = ( λ ( T ) + h ζ r ) λ ( T ) (19)

Γ = T ( R − ζ r , j , t ) + ζ r λ ( T ) ( I 2 π R ϕ s + ε σ T ∞ 4 + h T ∞ ) (20)

The numerical solution of the partial differential equations of the thermal model was done by the finite difference method. The method requires a convergence study of the mesh to ensure a stability of the numerical scheme, a consistency between the exact solution and the discretized solution and finally a convergence of the results of the thermal model [

provides more accurate results with a considerable calculation time (important and not insignificant),it is necessary to carry out a manual convergence study of the mesh to optimize the parameters: computing time, data storage memory and accuracy of output results. A study that consists of creating a mesh using a reasonable number of elements, followed by an analysis of the output results, then a slight increase in the mesh density and an analysis of the results by comparison with the results of the previous iteration. This study of manual convergence of the mesh allows us to obtain a precise and exact solution with a sufficiently dense mesh and without requiring too many computing resources. Equation (21) presents, for an explicit numerical scheme in finite differences (Equation (14)), the condition of convergence that connects the pitch of the temporal mesh ζ t with the pitch of the spatial mesh ζ r and ζ t .

ζ t < ρ C p λ ( 2 ζ r 2 + 2 ζ x 2 + 1 r ζ r ) (21)

The experimental tests were carried out on specimens of cylindrical geometry in steel AISI-4340, with a diameter of 14.5-mm and a length of 50-mm.

is ensured by a rotating test rig which can reach 10,000-RPM and which is equipped with a central support with three jaws and a counter-head moving along the x-axis. To measure the surface temperature of the workpiece according to the laser heat treatment parameters, a MI-IGAR-12-LO pyrometer and a FLIR thermal infrared camera with a sensitivity of less than 5˚C, and which can cover a temperature range up to 2000˚C, have been exploited. Four J-type precision thermocouples with a sensitivity of 0.05-s were placed in the vicinity of the cylindrical part to evaluate the ambient air temperature according to the laser heat treatment parameters, by using a NI-USB-6009 which offers features data acquisition (DAQ).Before laser heat treatment, the samples were heat-treated in the oven to homogenize the hardness value. The oven heat treatment consisted of placing the samples in an oven at a 900˚C austenitization temperature for a period of 20-min, followed by cooling in the oil and tempering in the oven at a temperature of 650˚C for a period of 100-min to obtain a uniform hardness of 25-HRC (Rockwell C) [

To perform experimental modeling, it is essential to have relevant tests to adequately represent the desired results. The acquisition of validation results with a reasonable number of tests requires the definition of an ordered sequence of experimental tests using an experimental design. Experiment planning based on the Taguchi method optimizes the number of tests to be performed while maintaining the robustness, performance and statistical consistency of a factor design [

_{9} Taguchi, with the results of measurement of the case depth, values of the dimensionless numbers (Reynolds and Prandtl), the thermal conductivity of the air and the absorption coefficient of the material. L_{9} matrix output responses were determined by examining the hardness profile by measuring the microhardness, the surface temperature and the air temperature around the workpiece by using an infrared camera and air temperature

Factors | Range |
---|---|

Laser power (W) | 2800, 2900 and 3000 |

Scanning speed ( mm/s) | 8, 9 and 10 |

Rotation speed (RPM) | 4000, 5000 and 6000 |

Tests | Factors | Reponses | ||||||
---|---|---|---|---|---|---|---|---|

P (W) | SS (mm/s) | Ω (RPM) | C_{d} (μm) | k (W/m-K) | Pr | Re | Λ | |

1 | 3000 | 8 | 6000 | 250 | 6.13E−02 | 0.718 | 1316.4 | 0.935 |

2 | 3000 | 9 | 5000 | 200 | 6.01E−02 | 0.716 | 1154.3 | 0.912 |

3 | 3000 | 10 | 4000 | 100 | 5.77E−02 | 0.710 | 1017.3 | 0.861 |

4 | 2900 | 8 | 5000 | 200 | 6.01E−02 | 0.716 | 1154.3 | 0.918 |

5 | 2900 | 9 | 4000 | 150 | 5.89E−02 | 0.713 | 968.5 | 0.905 |

6 | 2900 | 10 | 6000 | 100 | 5.77E−02 | 0.710 | 1525.9 | 0.890 |

7 | 2800 | 8 | 4000 | 200 | 6.01E−02 | 0.716 | 923.5 | 0.953 |

8 | 2800 | 9 | 6000 | 150 | 5.89E−02 | 0.713 | 1452.7 | 0.936 |

9 | 2800 | 10 | 5000 | 100 | 5.77E−02 | 0.710 | 1271.6 | 0.922 |

sensors. We note that the thermal conductivity of the air, the Reynolds and Prandtl numbers are proportional to the laser power and the rotation speed of the cylinder and are inversely proportional to the scanning speed of the laser beam. With a maximum difference of about 0.36E−02 W/m-K for the thermal conductivity of the air, of 347 for the Reynolds number and a small difference of about 0.008 for the Prandtl number. And that the absorption coefficient is proportional to the scanning speed and inversely proportional to rotational speed and laser power, for a difference of about 0.074 between the maximum laser hardening configuration and the minimum configuration.

To estimate the value of the convective heat transfer coefficient and the absorption coefficient according to the laser power, the scanning speed and the rotation speed, an ANOVA analysis of variance with the general step-by-step method, detailed in

For each process parameter, the value of the variance ratio F was compared with the values of the standard F tables for high statistical significance levels. It was concluded that the heat transfer coefficient is mainly linked with the rotation speed Ω, for a model response value of about 98%. And that the absorption

Factor | DOF | Contribution (%) | Sum of squares | F-Value | P-Value |
---|---|---|---|---|---|

P | 1 | 0.2 | 0.678 | 3.00 | 0.144 |

SS | 1 | 0.5 | 1.752 | 7.76 | 0.039 |

Ω | 1 | 98.92 | 324.553 | 1437.11 | 0.001 |

Error | 5 | 0.34 | 1.129 | ? | ? |

Total | 8 | 100 | ? | ? | ? |

Factor | DOF | Contribution (%) | Sum of squares | F-Value | P-Value |
---|---|---|---|---|---|

P | 1 | 29.51 | 0.000394 | 4.74 | 0.0095 |

SS | 1 | 49.20 | 0.000399 | 4.79 | 0.0094 |

P^{2 } | 1 | 8.02 | 0.000481 | 5.77 | 0.0074 |

P.SS | 1 | 7.71 | 0.000462 | 5.55 | 0.0078 |

Error | 4 | 5.56 | 0.000333 | ? | ? |

Total | 8 | 100 | ? | ? | ? |

coefficient depends mainly on the two factors P and SS, for a model response value of about 94%. Equations (22) and (23) present the empirical prediction relationships, of the heat transfer coefficient and the absorption coefficient, on the basis of a linear regression.

h = 19.61 − 0.00262 P + 0.724 S S + 0.007200 Ω (22)

Λ = 11.83 − 0.00819 P + 0.290 S S + 0.000002 P 2 − 0.000108 P ⋅ S S (23)

the comparison values of the surface temperature at the stop-laser position (40-mm) using an infrared camera with an emittance coefficient of 0.123 and a distance between the camera lens and the cylinder surface of about 700-mm. Visualization with the thermal camera allowed us to give identical surface temperature values for a constant emissivity coefficient and this regardless of the laser heat treatment parameters. At the start position of the laser heat treatment (10-mm), the surface temperature rapidly increases from 27˚C to about 300˚C in 50-ms.Then from the 10- to 15-mm position, it rises from 300˚C to about 850˚C, followed by a slight increase every 5-mm of displacement.

This slight increase in temperature between the position 15-mm and 40-mm is due to the heating phenomenon by advancement and can be explained by the fact that the thermal properties (thermal conductivity and specific heat)change with the temperature, and that convection and radiation heat losses become more important for high temperatures. This phenomenon induces a slight modification of the case depth along the longitudinal of the cylinder. To remedy this, it’s important to control the laser heat treatment parameters being processed.

The transformation start temperature allows us to estimate the case depth to improve the mechanical performance of the sample. The AISI-4340 steel used has an initial allotropic transformation temperature of 727˚C (Ac1), a complete austenitization temperature of about 927˚C (Ac3), and a melting temperature of about 1417˚C. To validate the developed model, it was assumed that the depth reaching a complete austenitization temperature (Ac3) will have a martensitic microstructure if the rate of the cooling cycle is greater than 8˚C/s.

[

These results confirm that the mathematical model developed is accurate in predicting temperature during heating and cooling. The temperature values predicted by the model fit well with the case depth measured using the Clemex^{®} microhardness machine. The developed model provides comparable results in the parameter range of

In this study, the temperature distribution in a cylindrical workpiece heat treated by laser was simulated using the finite difference method. The major findings are summarized below:

The algorithm developed with the FDM model for AISI-4340 steel, by adding the displacement parameter of the laser beam, is very effective in simulating the temperature distribution in a workpiece heat treated by laser. The results of theoretical predictions correspond to the experimental tests.

The variation of the thermal properties of the material has been integrated so as not to affect the output results of the thermal model. The coefficient of convective heat transfer has been evaluated by empirical correlations using measurements of the air temperature around the cylinder. Measurements provided by precision thermocouples that have been positioned very close to the outer surface of the cylinder.

The variation of the absorption coefficient that influences the effective penetration of heat flux, according to interaction conditions between the laser energy and the workpiece surface was approximated using experimental tests based on Taguchi experience planning and ANOVA analysis of variance.

Experimental validation of the model by the use of an infrared camera confirmed the values of the predicted temperature on the surface, in the range of defined parameters by the experimental design.

It would be interesting to consider completing this model of prediction of the temperature profile, by integrating a control of the laser heat treatment parameters for a uniform case depth along the longitudinal axis of the cylinder. We believe that this type of approach is the most appropriate way to develop a prediction model of the case depth for workpieces of cylindrical geometry.

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Fakir, R., Barka, N., Brousseau, J. and Caron-Guillemette, G. (2018) Numerical Investigation by the Finite Difference Method of the Laser Hardening Process Applied to AISI-4340. Journal of Applied Mathematics and Physics, 6, 2087-2106. https://doi.org/10.4236/jamp.2018.610176

Λ Absorption coefficient of the material,

Cp Specific heat, J∙kg^{−1}∙K^{−1},

h Thermal transfer coefficient, W∙m^{−2}∙K^{−1},

k Thermal conductivity of the air, W∙m^{−1}∙K^{−1},

λ Thermal conductivity of steel, W∙m^{−1}∙K^{−1},

σ Stefan-Boltzmann constant, W∙m^{−2}∙K^{−4},

ϕ_{s} Diameter of the laser spot, m,

D Diameter of the cylinder, m,

γ Thermal diffusivity, m^{2}∙s^{−1},

ε Emissivity of material surface,

L Cylinder length, m,

nt Number of temporal mesh nodes,

ρ Density, kg∙m^{−3},

nr Number of mesh nodes following R,

nx Number of mesh nodes following X,

Nu Nusselt’s number,

Pr Prandtl’s number,

Re Reynolds number,

ζr Pitch of the spatial mesh following R, m,

ζx Pitch of the spatial mesh following X, m,

ζt The pitch of the temporal mesh, s,

P Laser power, W,

I Efficient laser power, W,

SS Scanning speed, mm/s,

R Radius of the cylinder, m,

C_{d }Case depth, μm,

l(t) Spatiotemporal position of beam, m,

T_{∞} Ambient air temperature, ˚C,

Ac_{1} Heating temperature at point A_{1}, ˚C,

Ac_{3} Heating temperature at point A_{3}, ˚C,

T Temperature of the material, ˚C,

T_{in} Initial material temperature, ˚C,

β Kinematic viscosity, m^{2}∙s^{−1},

Ω Rotation speed of the cylinder, RPM,

r Distance from laser beam centre, m,

r_{0} Laser beam radius, m.