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# 02 Para Gleici - item5-05

(Parte **1** de 2)

The Use of COMSOL and Inverse Problem Technique to Estimate the Heat Flux on a Cutting Tool

R. F. Britoa, S. R. Carvalhob, S. M. M. Lima e Silva*a aInstituto de Engenharia Mecânica, Universidade Federal de Itajubá – UNIFEI, Itajubá, Minas Gerais, Brasil. bFaculdade de Engenharia Mecânica, Universidade Federal de Uberlândia – UFU, Uberlândia, Minas Gerais, Brasil.

e-mail: rogbrito@unifei.edu.br srcarvalho@mecanica.ufu.br metrevel@unifei.edu.br

Keywords: COMSOL, inverse problems, machining process, heat transfer.

Abstract This work proposes the use of inverse problem techniques in connection with COMSOL to estimate the heat flux and the temperature field on a turning cutting tool in transient regime. The main purpose of the present work is to present the improvements performed in relation to the authors’ previous work to develop the complex geometry of a machining process. Specification function, which is an inverse problem technique, was implemented in a program to estimate the heat flux applied on the tool, from the experimental temperature records. Once the heat flux is known, COMSOL is again utilized to obtain the temperature field on the cutting tool. The validation of the methodology is carried out by comparing the numerical and experimental results of temperature.

1 Introduction

Several engineering processes have their performance and quality affected by high temperature values. A typical example is the machining process in which cutting tool temperatures may be higher than 900ºC [1]. High temperatures change the microstructure and physical properties of the tool during machining, thus reducing their capacity to resist mechanical stress [2]. The direct consequence of these alterations is the reduction of their lifespan and performance. This leads to high operation costs and reduction of the end product quality. The right knowledge of the temperature values and applied heat flux in this kind of process, results in advantages like the development of more efficient cooling techniques as well as better specifications of the cutting parameters in machining processes. These temperatures have a controlling influence on the wear rate of the cutting tool as well as on the friction between the chip and the tool. However, the direct measurement of the temperature in a machining process is difficult to accomplish due to the movement of the piece as well as the presence of chips. Thus, the use of inverse heat conduction techniques conveys a good alternative to obtain these temperatures, since these techniques allow the use of experimental data obtained from accessible regions. Inverse problems consist of obtaining the value of a variable through the measurement of another variable measured directly [3].

R. F. Brito, S. R. Carvalho, S. M. M. Lima e Silva

These techniques often use optimization algorithm in order to minimize the error between the calculated and real value of the variable in question. Nowadays, several researchers have proposed the combination of inverse techniques and numerical heat transfer solutions to analyze the thermal fields during machining processes. Conveying a greater availability of computational resources, the use of numerical methods gained terrain, and it did not take long before they started being used, along with experimental methods in the studies of temperature fields on cutting tools. A three-dimensional finite difference-based model to predict temperature in machining processes was presented in [4]. The FDM based model proposed in this paper offered very rapid and reasonably accurate solutions. The simulated results were validated with infrared thermal measurements which were determined from the machining of AISI 1050 and AISI H13 materials under various cutting conditions. In the study of [5] an analytical and numerical model for cutting temperature prediction of 316L stainless steel was developed. The simulation model was set up in commercial FEM software of Abaqus6.8, which is good at nonlinear dynamic calculation. An ALE finite element model, which combines the advantages of both Lagrangian and Eulerian techniques, was used. The Johnson-Cook plasticity model was used to model the workpiece material. The analytical modeling and FEM modeling results match very well. In [6] the temperature distribution of the microcutter in the micro-end-milling process was investigated by numerical simulations and experimental approach. Micro-end-milling processes were modeled by the three-dimensional finite element method coupling thermal–mechanical effects. The micro-cutter cutting temperature distribution, the effect of various tool edge radii on cutting force, and the effective stress during micro-end-milling of aluminum alloy Al2024-T6 using a tungsten-carbide micro-cutter were investigated on. The simulation results showed that with the increase of the tool edge radius the cutting force increases, while the effective stress and mean cutting temperature decreases slightly. Inverse techniques have already been used to study temperature fields on a cutting tool. The solution of a three dimensional inverse heat conduction problem using an Evolutionary Algorithm (EA) was demonstrated in [7]. The heat flux on the tool during the turning process was determined by using evolutionary operations combined with measured temperatures on the tool surface. The threedimensional conduction in the tool and tool holder was simulated using FLUENT. In [8] an inverse method was proposed to estimate the heat sources in the transient two-dimensional heat conduction problem in a rectangular domain with convective bounders. The non homogeneous partial differential equation (PDE) is solved using the Integral Transform Method. The test function for the heat generation term was obtained by the chip geometry and thermomechanical cutting. Then the heat generation term was estimated by the conjugated gradient method (CGM) with adjoint problem. The sequential function specification method was used to estimate the transient heat flux imposed on the rake face of a cutting tool during the cutting operation with two different assumptions [9]. In one of them the thermal conductivity is assumed to be constant, and in the other one it varies with the temperature. The cutting tool was modeled as a three dimensional object. Simulated temperature data was used to recover the heat flux at the cutting tool surface using linear as well as nonlinear solutions. This work proposes the use of inverse problem techniques with the commercial software COMSOL® 4.3 to estimate the heat flux and the temperature field in the contact area under transient regime, in a turning cutting tool. A Matlab program, with Specification Function technique, was developed to estimate the heat flux applied on the cutting tool, using experimental temperature records in a determined point. The validation of the proposed methodology was accomplished in controlled experiments in laboratory.

2 Theoretical Formulation

2.1 Temperature model

The problem dealt with in this work is represented by Fig. 1a, which represents a set consisting of a cutting tool, a hard metal, a wedge positioned under the cutting tool between the tool and the tool holder. There is also a staple and a bolt to fix the set. In Figure 1a the schematic model for the thermal problem of machining is presented. The heat generation during the machining process is indicated by a distribution of unknown heat flux q”(x,y,t), over the arbitrary area by x and y. A blown up view of the set is shown in Fig. 1b.

a) b)

Figure 1: a) Thermal problem scheme and b) Detail of the contact interface between the tool and the workpiece.

The heat diffusion equation ruling this problem may be given as:

tzyx t

T tzyx

T tzyx

T tzyx subject to the following boundary conditions

on the contact interface with the workpiece (Fig. 1b) (2)

and in the remaining regions of the set (3)

and having the following as the initial condition

Contact area x y

R. F. Brito, S. R. Carvalho, S. M. M. Lima e Silva

The direct problem consists in solving the heat diffusion equation according to the boundary conditions (Eqs. 1 to 4). The COMSOL® 4.3 program, which solves thermal problems by using the finite element method, is used for this purpose. The use of COMSOL for the numerical resolutions of differential equations that rule the physical phenomenon investigated should be highlighted. Also, COMSOL allows adjusting any boundary conditions, as well as modeling the geometry so as to faithfully represent the system investigated as presented in Fig. 1a.

2.2 The Inverse Problem

The inverse technique adopted in this work is the Specification Function [3]. This technique requires the calculation of the sensitivity coefficient which is done numerically from Duhamel Theorem [10]. The sensitivity coefficient is then obtained with the use of a numerical probe which follows the temperature changes in the point equivalent to where the thermocouples were placed in the experiments. Once the sensitivity coefficient is at hand, the heat flux is estimated with the use of a Matlab language program. Another important parameter is the value of future time steps r. In the Specification Function technique, a determined value of future time steps r is used to estimate the heat flux at present instant. In the resolution of the inverse problem, the Specification Function searches for a heat flux value that minimizes the objective function given in Eq. 5, for each time step rp ns j pMjpMjTYF (5)

3 Validation of the Methodology Proposed

A great difficulty in the solution of inverse heat conduction problems is the validation of the technique used. This difficulty is inherent to the problem, once the validation of the estimated heat flux requires the previous knowledge of the experimental heat flux. It is observed that in real inverse problems, as in machining process, the experimental heat flux is not known. Thus, an alternative for the validation of the inverse technique is to carry out a controlled experiment, in which the heat flux and the temperature are measured at the cut tool. In this sense, before the analysis of the real machining process, a cemented carbide tool with dimensions of 0.0127 x 0.0127 x 0.0047 m was used. A heat flux transducer and two thermocouples previously calibrated and a kapton electric heater were used on this tool. This heater was connected to a digital power supply (MCE). The heat flux transducer was located between the heater and the tool, in order to measure the heat supplied to the tool. The temperatures at the tool were measured with two thermocouples. The heat flux and temperatures signals were acquired by a HP Series 75000 data acquisition system, controlled by a PC. Temperatures were measured using type K thermocouples (30AWG) welded by capacitive discharge and calibrated by using a bath temperature calibrator ERTCO with a stability of 0.01 ºC. The solution of the three-dimensional heat diffusion equation is obtained with the use of the finite element method, through the commercial software COMSOL® 4.3. For this, a computational thermal model was used to faithfully represent the experimental model of the sample. This model was discretized in a computational tetrahedral mesh. The validation results are presented in Figs. 2a, 2b and 2c. Figure 2a presents a comparison between the experimental and estimated flux, whereas Fig. 2b compares the experimental and numerical temperatures. Figure 2c presents the deviation between the experimental and numerical temperatures. The Specification Function method for r equal to 10 future time steps was used in Fig. 2a.

a) b) c)

Figure 2. a) Experimental and estimated heat flux, b) experimental and calculated temperatures c) temperature residuals.

4 Experimental Assembly in a Real Machining Process

The machining test was carried out in a conventional lathe IMOR MAXI–I–520–6CV without coolant. The material used in the experimental test was a cylindrical gray cast iron bar FC 20 EB 126 ABNT of 7 m in external diameter. The insert and tool holder used were cemented ISO SNUN12040408 K20/Brassinter and ISO CSBNR 20K12/SANDVIK COROMAT, respectively. The temperatures were measured on accessible locations of the insert, the shim and the tool holder by using K type thermocouples and a data acquisition system HP 75000 Series B controlled by a PC (Fig. 3a). Table 1 presents the location of the thermocouples shown in Fig. 3b.

a) b)

Figure 3. a) Experimental apparatus used to acquire the temperature signals in the tool during machining and b) detail of the position of the thermocouple welded to the tool.

Table 1: Locations of the thermocouples shown in Fig. 3b.

Time (s) Experimental Estimated015304560753040506070 Temperature (ºC)

Temperature residuals (ºC)

Time (s) R1=T1exp-T1num R2=T2exp-T2num tool holder cutting tool thermocouples unknown heat flux

R. F. Brito, S. R. Carvalho, S. M. M. Lima e Silva

The chip-tool contact area determination represents one of most important and delicate aspects among the main sources of errors in the solution of the thermal model problem. Some methods to identify this area can be found in the literature as, for example, the use of image analyzer software [1] or the application of coatings [12]. In both processes, the area is measured after cutting. This procedure is also used here. However, in this work, the interface contact areas were obtained from the three tests carried out with the same cutting condition. In order to measure the contact area an image system program with video camera Hitachi CCD, KP-110 model, an AMD PC- K6 450 MHz and the GLOBAL LAB image software were used. A typical contact area is presented in Figs. 4a and 4b. The contact area value was 1.41mm2, obtained for feed rate of 0.138mm/rot, cutting speed of 135.47m/min and depth of cut of 5.0mm.

a) b) Figure 4. a) Image treatment of the contact area and b) contact area on the computational model.

Many tests were carried out to observe the influence of cutting speed, feed rate and depth of cut in the temperature distribution. However, due to the limitation of pages in this paper the results are presented for only two tests. The test identifications with the cutting conditions are presented in Tab. 2. Each cutting condition was repeated three times to observe the repeatability. In each experiment the total number of measurement of each thermocouple was nt = 180 with a time sample of 0.5s. The thermal conductivity and diffusivity of the tool are respectively, = 43.1 Wm/K and α = 14.8 x 10-06m2/s [13].

Table 2: Cutting conditions.

Cutting parameters Test 1 Test 2 Feed rate 0.138 m/rev 0.138 m/rev

Cutting speed 135.47 m/min 135.47 m/min

Depth of cut 5.0 m 1.0 m Final diameter 72 m 76 m

(Parte **1** de 2)