Transient evaporative laser cutting with finite volume method.

Begic, Derzija ; Bijelonja, Izet ; Kulenovic, Malik 等

1. INTRODUCTION

Laser cutting is a very important application of laser technologies in the manufacturing industry. There have been various investigations about the material removal processes by laser beam from the early days (Schuocker, 1986). Thermal models for investigation of laser cutting may be divided into two main categories: models with detailed treatment of thermal conduction and models where details of phase transition (melting, vaporization) are considered. Further, the methods of investigations can also be divided into three categories: analytical, numerical and experimental methods. Major numerical methods used so far are finite element method (FEM) and boundary element methods (BEM). (Junke & Wang, 2008) have used FEM software Ansys to analyse the temperature and thermal stress distributions in laser cutting of glass. The glass is machined by a single CO2 laser beam and by dual C[O.sub.2] laser beams. (Kim, 2004) developed an unsteady convective heat transfer model using BEM considering moving continuous Gaussian laser beam for the prediction of groove shape, groove depth, temperature and flux distribution. A FEM based unsteady heat transfer model for the prediction of amount of material removal and groove smoothness during evaporative cutting with a Gaussian wave pulsed laser has also developed (Kim, 2001).

The objective of this paper is to develop a numerical model based on the finite volume method in order to predict the groove shapes in evaporative laser cutting. Considering the heat input from laser beam as a fixed heat source, an unsteady heat transfer equation is considered that deals with the material cutting process using a continuous laser beam. The convergence analyses are performed for the groove shapes with various meshes and time steps.

2. MATHEMATICAL FORMULATION

A diagram of the laser processing is illustrated in Fig.1. Dimension of aluminium sample is 10 mm x 1 mm and the sample is subjected by a continuous CO2 laser beam.

In order to build a mathematical model, some assumptions should be made: (1) material is isotropic and physical parameters of aluminium are temperature-independent, (2) material removal is a surface phenomenon and phase change from solid to vapour occurs in one step, (3) evaporated material is transparent and does not interfere with incident laser beam, and (4) heat loss by radiation is neglected.

[FIGURE 1 OMITTED]

Based on the above assumptions, the mathematical heat transfer model can be written as follows:

[partial derivative]/[partial derivative]t [[integral].sub.v] [rho]cTdV = [[integral].sub.S] q x n dS + [[integral].sub.V] [h.sub.s] dV (1)

T(t) = [T.sub.o] at t = 0, (2)

where V is volume of sample bounded by the surface S, q is the heat flux vector, n is the unit outward surface normal, [h.sub.s] is the sink of heat; c and [rho] are the specific heat and the density, respectively. [T.sub.o], denotes the initial temperature of sample.

The relation between the heat flux and temperature gradient is given by Fourier's law:

q = k grad T, (3)

where k is the thermal conductivity.

The boundary condition on the surface which is subjected to the laser beam is obtained from the balance of energy on the surface as:

-k [partial derivative]T/[partial derivative]z + h([T.sub.s] - [T.sub.[infinity]]) = [alpha]I, (4)

and boundary conditions on the other boundaries are:

-k [partial derivative]T/[partial derivative]n = h([T.sub.n] - [T.sub.[infinity]]) (5)

where, h is the convection heat transfer coefficient, [T.sub.s] denotes the temperature of the heated zone surface and [T.sub.n] denotes the temperature of the sample area without laser heating, [alpha] is the laser absorptivity, I is the intensity of the laser beam, and [T.sub.[infinity]] is the environment temperature.

3. NUMERICAL METHOD

The governing equation, together with non-linear boundary conditions, applied to the two-dimensional cutting geometry is solved using finite volume method. In order to obtain the discrete counterparts of equations (1), (4) and (5), the time is discretized into an arbitrary number of time steps of the size [delta]t and the solution domain is subdivided into a finite number of contiguous quadrilateral controls volumes (CV) or cells of volume V bounded by the surface 5. The computational points (nodes) are placed at the centre of each CV, while boundary nodes, needed for the specification of boundary conditions, reside at the centre of boundary cell-faces. The inertial term is approximated using the backward differencing, and volume integrals are estimated using midpoint rule. A detailed description of the finite volume method applied in this work is given in (Demirdzic & Muzaferija, 1994). The set of linearized equations for each variable is solved at each time step by iteration. A fully implicit scheme is used for the time dependent terms.

At each time step, the energy equation is solved to obtain the temperature in each computational point of control volumes. When the temperature at any cell centre achieves the evaporation temperature, the last term in equation (1) is activated. Via this last term, the sink of heat is activated so it simulates the latent heat. For a time step, a size of sink of heat is determined so that the temperature of cell remains at a constant value, in other words, the same as the evaporation temperature. After a convergent solution is obtained for the given time step, it is checked whether the sink of heat is less than the latent heat. If the sink of heat is less than the latent heat the next time step will start. This procedure is repeated until the remains latent heat is spent. In the case that in a time step the sink of heat is greater than the latent heat, which is a physical non realistic solution, the size of time step decreases so that the sink is equal the latent heat. In this case after convergent solution is obtained, the cells by this status are considered as the evaporated cells and the new boundaries of the solution domain must be generated.

4. RESULTS AND DISCUSSION

For numerical analyses an aluminimum sample of size 10 x 1 mm is subjected by a continuous laser beam. The top side of the sample is subjected to the laser beam as shown in Fig.1. The initial temperature of sample is the same as the environment temperature. Due to the symmetry only a half of the aluminium plate is analysed.

The physical parameters of aluminium, air and laser are shown in Table 1. The computations are performed using various meshes and time steps in order to get a converged solution. The computations are performed for three different the time step 2 x [10.sup.-4], 1 x [10.sup.-4] and 0.5 x [10.sup.-4] s. There were not considerable differences among the results with different time steps. Also, the computations are carried out for four various meshes. For all cases uniform meshes are used. Mesh refinement is stopped, when a difference between results obtained by two consecutive mesh refinement were less than 1%.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

In Figs. 2, 3 and 4, the groove shapes are shown for the finest mesh 200 x 10 CVs at the times 6.1096 x [10.sup.-5] s, 3.0679 x [10.sup.-4] s and 5 x [10.sup.-4] s, respectively. The obtained numerical results yield the qualitative good results. A quantitative estimate of results including the shape and the depth of the groove and temperature fields will be performed by the experiment.

5. CONCLUSION

A finite volume method to study the time-dependent material removal of a sample subjected to a continuous laser beam of constant power is developed. For this analysis, the sample and the laser are fixed at a position to simulate the transient drilling process.

Studies show that the method yields convergent results with both mesh refinement and time step. These results showed a qualitative analysis in respect of depth and shape of a groove. A quantitative estimate of numerical results will be performed by experiments.

Also, in the future work, numerical analysis of transient evaporative laser cutting with a moving laser beam by finite volume method will be considered.

6. REFERENCES

Demirdzic, I. & Muzaferija, S. (1994). Finite volume method for stress analysis in complex domains. International Journal for Numerical Methods in Engineering, 37, 21, (January 1994) page numbers (3751-3766), ISSN: 0029-5981

Junke, J. & Wang, X. (2008). A numerical simulation of machining glass by dual CO2-laser beams. Optics & Laser Technology, 40, 2, (March 2008) page numbers (297-301), ISSN 0030-3992

Kim, M.J. (2004). Transient evaporative laser cutting with moving laser by boundary element method. Applied Mathematical Modelling, 28, 10, (October 2004) page numbers (891-910), ISSN 0307-904X

Kim, M.J. & Zhang, J. (2001). Finite element analysis of evaporative cutting with a moving high energy pulsed laser. Applied Mathematical Modelling, 25, 3, (January-February 2001) page numbers (203-220), ISSN 0307-904

Schuocker, D. (1986). Dynamic Phenomena in Laser Cutting and Cut Quality. Applied Physics B: Lasers and Optics, 40, 1, (May 1986) page numbers (9-14), ISSN 1432-0649

Table 1. Physical parameters of material (Al), air and laser

 Properties Values

Aluminium [T.sub.evap] 933 K
 [H.sub.L] 3.88 x [10.sub.5] J [kg.sub.-1]
 c 921 J [kg.sup.-1] [K.sup.-1]
 [alpha] 0.15
 [rho] 2700 kg [m.sup.-3]
 k 238 W [m.sup.-1] [K.sup.-1]
Air h 30 W [m.sup.-2] [K.sup.-1]
Laser P 1000 W
 r 0.1 mm