# Estimation of a Moving Heat Source due to a Micromilling Process Using the Modified TFBGF Technique

Authors: Sidney Ribeiro; Ana Paula Fernandes; Daniel Fernandes da Cunha; Marcio Bacci da Silva; Jerry Shan; Gilmar Guimaraes
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Moving heat sources are present in numerous engineering problems as welding and machining processes, heat treatment, or biological heating. In all these cases, the heat input identification represents an important factor in the optimization of the process. The aim of this study is to investigate the heat flux delivered to a workpiece during a micromilling process. The temperature measurements were obtained using a thermocouple at an accessible region of the workpiece surface while micromilling a small channel. The analytical solution is calculated from a 3D transient heat conduction model with a moving heat source, called direct problem. The estimation of the moving heat source uses the Transfer Function Based on Green’s Function Method. This method is based on Green’s function and the equivalence between thermal and dynamic systems. The technique is simple without iterative processes and extremely fast. From the temperature on accessible regions it is possible to estimate the heat flux by an inverse procedure of the Fast Fourier Transform. A test of micromilling of 6365 aluminium alloy was made and the heat delivered to the workpiece was estimated. The estimation of the heat without use of optimization technique is the great advantage of the technique proposed.


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## Body

## 1. Introduction

Mechanical manufacturing technology has advanced rapidly in the last few years, having a considerable impact on the development of new materials, processes, and products. The process of mechanical micromachining is becoming an important manufacturing technology due to the increasing demand for miniaturized products [1]. This development had the effect of considerable changes in many areas, such as the mobile phone industry, aerospace, electronics, optics, and biomedicine among others. The development in these fields has attracted the attention of many researchers for the mechanisms and the micromachining processes.Thermal fields and temperatures during a convectional machining process are some of the major factors that affect the surface quality of the material, tool life, and the precision of the manufactured parts. These effects are also present in the micromachining processes since the high temperatures resulting from the process are concentrated in small areas near the cutting interface. Thus, the estimation of heat generated at this interface is extremely important when high accuracy is required. However its direct estimation is very difficult since the only part of the generated heat goes to the workpiece, while the rest is dissipated to the tool, chip, and environment. Moreover, measurements of temperatures at the tool-chip-workpiece interfaces are very difficult due to the cutting movement and the small contact areas involved.Literature presents various works that deal with the solution of the thermal problem that appear in conventional cutting process. For direct measurement of temperature, tool-work thermocouple technique is a good alternative to measure the average temperature of the cut interface. Kaminise et al. [2] have investigated experimentally the influence of tool-holder material on tool-chip interface temperature and on the surface temperatures of the cutting tool and tool-holder. The study was conducted in dry machining of grey iron with uncoated cemented carbide inserts, using identical cutting parameters. Five tool-holders were made with materials having different thermal conductivity: copper, brass, aluminium, stainless steel, and titanium alloy. Although it is a good method, it can be observed that the use of the tool-workpiece thermocouple is limited to tools that can conduct electricity. In addition the thermocouple does not measure the temperature at a specific point, but an average temperature at the heat affected zone between the tool and the workpiece.Due to these experimental difficulties many analytical and numerical methods solutions have been employed to predict cutting temperature. For example, thermal problem during drilling [3–5] or milling can be cited [6, 7].Sorrentino et al. [3] have analyzed the influence of cutting parameters measuring the temperature during drilling on tool and in the laminate for both CFRP and GFRP. Temperature trends as a function of cutting speed and feed rate were obtained and dangerous values of cutting parameters were identified. A numerical model was developed to simulate the temperature rising in the material during drilling. The great problem is the fact that the heat flux generated at interface is unknown.Another work that deals with thermal model in drilling is presented by Arrizubieta et al. [4]. This work [4] discusses the mechanisms of formation of the holes in the laser percussion drilling process of an AISI 304 plate, evaluating the removed material volume in each laser pulse and obtaining the evolution of the hole geometry for the complete pulse sequence. In addition, the experimental analysis has been applied also for the development of a numerical model that can simulate the resulting hole geometry for different pulse sequences. The numeric model is based on a 3D thermal model, developed and particularized specifically for laser material processing. The thermal model does not take into account the phase transformation from solid-liquid and liquid-vapour but in order to incorporate this phenomena a variable specific heat is considered for the material.Zhang et al. [6] have presented a three-dimensional (3D) finite element (FE) simulation model to investigate the complex nonlinear process in hard milling of hot work tool steel. First, the geometric model of workpiece was established considering the previous machined surface profile in actual milling process. Secondly, the prediction validation of the simulation model about hard milling process was verified by comparing the simulated cutting forces with the experiment results. The problem of not knowing the heat flux generated at interface is again avoided by considering converting of mechanical work done in machining into heat energy and by using cutting forces to obtain the heat flux.Karaguzel et al. [7] proposed a new experimental technique to measure the face milling temperatures which are used to verify the analytical model developed. Results indicate a good agreement between the model predictions and dry cutting experiments at different cutting speeds. In this thermal model, heat generation is calculated by assuming that all the mechanical work done in machining operation is converted into heat energy.One thing that the models have in common is that the generated heat source must be known or defined a priori considering the basic cutting forces. Unfortunately in real machining processes the heat flux generated at the chip-tool interface is unknown and strongly dependent on the cutting condition and on the types of workpiece and cutting tool used. Based on these mentioned aspects, the use of inverse heat conduction techniques represents a good alternative since this technique takes into account temperatures measured from accessible positions.De Sousa et al. (2012) [5] based on [8] have developed a technique that can estimate the heat flux at interface and subsequently obtain the field temperature in the material during drilling process. A 3D transient model with moving interface and an inverse technique based on temperature measurement from thermocouple attached to the surface were developed. This work developed an inverse technique based on Green’s function and a dynamic observer with global heat transfer function applied to solve the drilling problem represented by a three-dimensional transient model.A complete analysis of the thermal problem arising from the micromachining process involves thermal modeling which is clearly a three-dimensional transient heat conduction problem with a moving heat source applied to the surface to be machined.As mentioned the solution of the direct problem, can be obtained knowing the boundary conditions and the moving heat source (heat generation) generated due to the friction produced by the contact between the microtool and the workpiece during machining.Unlike conventional machining, few works are found in literature that address the thermal problem arising from micromachining processes. Among them, most involve numerical solutions based on finite element method [9–11]. Usually the heat generated is assumed to be known or estimated from hypotheses based on experimental measurements of shear velocity and shear force.For example, Mamedov and Lazoglu [9] made an analysis for the micromilling of a Ti-6Al-4V titanium alloy using a finite element model and experiments. The developed model uses a semianalytical approach for calculating the heat generated in the primary and secondary deformation zones. In fact, the heat generated is found by shearing power.In fact, the heat generated is estimated using the shearing power predicted from the product of shear velocity and shear force on shear plane (primary shear zone) and by the frictional power predicted from chip velocity and friction force. Heat losses to the environment are not considered.Wissmiller and Pfefferkorn [10] used an infrared camera during micromilling of (6061-T6) aluminium and (1018) steel with 300 microns two-flute tungsten carbide end mills. The measured temperature compares favorably with temperature distributions predicted by a two-dimensional, transient, heat transfer model of the tool. The heat input is estimated by applying Loewen and Shaw’s heat partitioning analysis.Chen et al. [11] carried out microcutting experiments to measure temperature of the microcutter and workpiece using a fast response and self-renewing thermocouple which was installed on a special cylindrical workpiece. In addition, an energy density-based ductile failure material model was proposed and applied to a finite element model to predict the microcutting temperature distribution. The measured temperatures were compared with simulations results.Some experimental works are also reported. Samuel et al. [12] used graphene as an additive to improve the lubrication and cooling performance of semisynthetic metal-working fluids used in micromachining operations. Microturning experiments were conducted in the presence of metal-working fluids containing different concentrations of graphene plates. Temperature was measured by type K thermocouples attached to the rake face of the tool at a distance of approximately 0.8 mm from the cutting edge. According to the author, incorporation of graphene plates in the metal cutting fluid suppresses significantly the peak temperature of the tool during cutting.Numerical solutions applied to micromachining usually exhibit the limitation of the use of very refined mesh.The numerical treatment of the very small size of the microtool is the major difficulty in the use of numerical methods. This problem is due to the transition necessary for the construction of the numerical mesh. Typically the microtool has dimensions of the order of microns while the workpiece dimensions are of the order of centimeters. As the refining of the mesh in the workpiece region should be smaller than the diameter of the microtool (micrometers), an appropriate mesh results in millions of nodes, thus making the numerical technique very costly. An alternative is the use of analytical solutions. The great strength is that the solutions are valid for any point in the domain and the heat flux generated in the interface can be obtained with the real dimensions of the microtool.Another point concerns available inverse techniques. Most of them use optimization techniques which require the direct problem solution to be calculated several times, increasing considerably the computational cost.This work proposes the estimation of the moving heat source produced during the micromilling process using a modification of the TFBGF technique [13] which is based on analytical solution and transfer function.The estimation of the moving heat source without the use of least square minimization or optimization technique is the great advantage of the technique proposed.The moving heat source can be obtained directly from the temperature measured in different positions of the contact area and the transfer function of the associated problem.In a previous work [14], the potency of TFBGF method was analyzed considering only numerical simulations in synthetic data. Hypothetical heat was applied to a 3D thermal system and then recovered considering numerically simulated temperatures on the surface of model. Random errors were added to the simulated temperatures and the technique proved to be suitable for use in actual machining tests such as the drilling process which is presented here.The temperatures, here, were measured using thermocouples at accessible regions of the workpiece surface while the transfer function is calculated analytically from a 3D transient heat conduction model with a moving heat source.In the inverse problem, the estimation of the moving heat source uses the Transfer Function Based on Green’s Function (TFBGF) method adapted here for heat source estimation. This method is based on Green’s function and in the equivalence between thermal and dynamic systems. The technique is simple without iterative processes and therefore extremely fast. With the knowledge of the temperature profile/evolution (experimental temperature far from the heat source) and of the transfer function it is possible to estimate the heat flux by an inverse procedure of the Fast Fourier Transform (IFFT).


## 2. Theoretical Fundamentals

### 2.1. Thermal Model

Figure1 shows the thermal model of the workpiece and microtool assembly described by the transient three-dimensional heat diffusion equation in the x, y, z coordinate system.Figure 1
Thermal model scheme.The governing equation with temperature dependent thermal properties can be written as(1)∂2T∂x2+∂2T∂y2+∂2T∂z2+qgkδx-vtδy-Pyδz-Pz=1α∂T∂t-u∂T∂xsubjected to the first kind boundary conditions(2)T0,y,z,t=TL1,y,z,t=T0,Tx,y,z,t=Tx,L2,z,t=T0and the initial condition(3)Tx,y,z,0=T0.The effect of the moving heat source can be treated in two ways. One hypothesis is to consider the heat flux as released at the surface. Thus the heat flux is considered as a boundary condition. The second hypothesis, used here, is to consider the heat flux as generated by a heat source. This hypothesis gives a better approximation for engineering applications such as machining, grinding, cutting, and sliding of surfaces, which has heat generated as a result of friction heating in an area with penetration depth, that is, a volume heat source. In this work, the moving heat source will be considered a point heat source of constant strengthδ(W), releasing its energy continuously over time while moving in the positive x direction with a constant velocity u, in a stationary medium that is initially at zero temperature.A transformation is proposed [15, 16] in order to remove the last term from (1). It means a new variable W(x,y,z,t) is introduced:(4)Tx,y,z,t=Wx,y,z,texp⁡ux2α-u2t4α.In addition, in the solution of moving heat source problems, it is convenient to let the coordinate system move with the source [14]. This is achieved by introducing a new coordinate ξ defined by(5)ξ=x-ut.Using (4) and (5) the governing equation is obtained as (6)∂2W∂ξ2+∂2W∂y2+∂2W∂z2+qgkδξ-Pξδy-Pyδz-Pz=1α∂W∂t.subjected to the boundary conditions (7)W0,γ,z,t=WL1,y,z,t=0,Wξ,0,z,t=Wξ,L2,z,t=0Wξ,y,0,t=Wξ,y,L3,t=0,k∂W∂zz=L3+WL3,th+ku2α=hW∞e-uL3/2α-u2t/4α,where (8)h+ku2α=heff.The termheff in (8) is the effective heat convection coefficient [14].Equation (6) can then be solved by Green’s function method [15] as(9)W=∫τ=0t∫ξ′=0L1∫y′=0L2∫z′=0L3G∗t-τqgτdz′dy′dξ′dτ,where(10)G∗ξ,y,z,t∣ξ′,y′,z′,t-τ=Gt-τδξ′-Pξδy′-Pyδz′-Pz×euPξ/2α-u2τ/4αand Green’s function for this problem can be obtained by [16](11)Gξ,y,z,t∣ξ′,y′,z′,t-τ=8L1L2L3∑m=1∞∑n=1∞∑p=1∞e-βm2ue-βn2ue-βp2u×sen⁡βmxsen⁡βmx′sen⁡βny×sen⁡βny′sen⁡βpzsen⁡βpz′βp2+B2βp2+B2+B,where βm=mπ/L1, βn=nπ/L2, βpcot(βp)=-B, B=hL3/k, and u=α(t-τ).The original temperature can then be recovered by (4), that is,(12)Tx,y,z,t=Wx,y,z,texp⁡ux2α-u2t4α.If the value of the heat flux,qg(t), is known, (12) represents the direct problem solution related to the inverse problem studied.


## 2.1. Thermal Model

Figure1 shows the thermal model of the workpiece and microtool assembly described by the transient three-dimensional heat diffusion equation in the x, y, z coordinate system.Figure 1
Thermal model scheme.The governing equation with temperature dependent thermal properties can be written as(1)∂2T∂x2+∂2T∂y2+∂2T∂z2+qgkδx-vtδy-Pyδz-Pz=1α∂T∂t-u∂T∂xsubjected to the first kind boundary conditions(2)T0,y,z,t=TL1,y,z,t=T0,Tx,y,z,t=Tx,L2,z,t=T0and the initial condition(3)Tx,y,z,0=T0.The effect of the moving heat source can be treated in two ways. One hypothesis is to consider the heat flux as released at the surface. Thus the heat flux is considered as a boundary condition. The second hypothesis, used here, is to consider the heat flux as generated by a heat source. This hypothesis gives a better approximation for engineering applications such as machining, grinding, cutting, and sliding of surfaces, which has heat generated as a result of friction heating in an area with penetration depth, that is, a volume heat source. In this work, the moving heat source will be considered a point heat source of constant strengthδ(W), releasing its energy continuously over time while moving in the positive x direction with a constant velocity u, in a stationary medium that is initially at zero temperature.A transformation is proposed [15, 16] in order to remove the last term from (1). It means a new variable W(x,y,z,t) is introduced:(4)Tx,y,z,t=Wx,y,z,texp⁡ux2α-u2t4α.In addition, in the solution of moving heat source problems, it is convenient to let the coordinate system move with the source [14]. This is achieved by introducing a new coordinate ξ defined by(5)ξ=x-ut.Using (4) and (5) the governing equation is obtained as (6)∂2W∂ξ2+∂2W∂y2+∂2W∂z2+qgkδξ-Pξδy-Pyδz-Pz=1α∂W∂t.subjected to the boundary conditions (7)W0,γ,z,t=WL1,y,z,t=0,Wξ,0,z,t=Wξ,L2,z,t=0Wξ,y,0,t=Wξ,y,L3,t=0,k∂W∂zz=L3+WL3,th+ku2α=hW∞e-uL3/2α-u2t/4α,where (8)h+ku2α=heff.The termheff in (8) is the effective heat convection coefficient [14].Equation (6) can then be solved by Green’s function method [15] as(9)W=∫τ=0t∫ξ′=0L1∫y′=0L2∫z′=0L3G∗t-τqgτdz′dy′dξ′dτ,where(10)G∗ξ,y,z,t∣ξ′,y′,z′,t-τ=Gt-τδξ′-Pξδy′-Pyδz′-Pz×euPξ/2α-u2τ/4αand Green’s function for this problem can be obtained by [16](11)Gξ,y,z,t∣ξ′,y′,z′,t-τ=8L1L2L3∑m=1∞∑n=1∞∑p=1∞e-βm2ue-βn2ue-βp2u×sen⁡βmxsen⁡βmx′sen⁡βny×sen⁡βny′sen⁡βpzsen⁡βpz′βp2+B2βp2+B2+B,where βm=mπ/L1, βn=nπ/L2, βpcot(βp)=-B, B=hL3/k, and u=α(t-τ).The original temperature can then be recovered by (4), that is,(12)Tx,y,z,t=Wx,y,z,texp⁡ux2α-u2t4α.If the value of the heat flux,qg(t), is known, (12) represents the direct problem solution related to the inverse problem studied.


## 3. The TFBGF Method

As mentioned before, the inverse procedure used here is based on the Transfer Function Based on Green’s Function (TFBGF) method [13] adapted for heat source estimation. This procedure will be described in this section.Transfer functions are used to characterize the relationship between input and output of a dynamic system, represented byx(t) and y(t), respectively, in Figure 2.(13)Ys=Hs·Xs,(14)Yt=ht∗Xt=∫-∞+∞ht-τXτdτ.Figure 2
Dynamic system of an input and an output.There is an equivalence between dynamic systems and problems of heat conduction, since both are described by a set of differential equations. Thus, the heat flux is treated as input of the system, and the temperature field as the response.The analysis of dynamic systems is facilitated by the use of the Laplace transform, because it provides the mathematical relationship between input and output of the dynamic system. It is known that, for a linear dynamic system (Figure2), the relation between input and output in the complex variable s domain is given by the multiplication shown in (13), that is,(15)Yx,s=Hx,s·Xs.The inputX in the domain s can be calculated by(16)Xs=1Hx,s·Yx,sor in the time domain by the deconvolution(17)Xt=L-11Hx,s∗Yx,t.Comparing (9) and (15) an equivalent thermal system can be identified, where the output of the system is the auxiliary temperature W, the transfer function is the modified Green’s function G, and the moving heat source qg is the input of the system. That is,(18)G∗ξ,y,z,t∣ξ′,y′,z′,t-τ=ht-τ,qgt=Xt.This Equation can be solved if the transfer function given by the modified Green’s functionG∗ is known.The proposed methodology for identification of the analytical impulse response is based on the theory of dynamical systems of one input and one output.For a given inputX(t) and output Y(t) pair, the transfer function, H(t), is invariable. If the input X(τ)=δ(τ) is applied, the transfer function can then be obtained from the auxiliary temperature distribution, W(t). That is, Y(t)=W(t)=h(t)=G∗(t).Using (9), (14), and (4) and  qg(τ)=δ(τ) and after integration, the transfer function is given by(19)hξ,y,z,t=αk8L1L2L3∑m=1∞∑n=1∞∑p=1∞e-βm2+βn2+βp2αt×sen⁡βmξsen⁡βmPξsen⁡βnysen⁡βnPy×sen⁡βpzsen⁡βpPzeuPξ/2αβp2+B2βp2+B2+B,(20)Ts=qgs·Hs⟹qgs=TsHs.Thus, the solution of the inverse problem (in variableT) is obtained in the Laplace domain using (19) or in the time domain using (4) [13]. The input, the heat source qg(s) was estimated from the impulse response H(s) and from the temperature measured at any position of the system. Heat source estimation in the Laplace domain or in time domain used the software Scilab® with Fast Fourier Transform and inverse Fourier transform (FFT) functions in the following expression:(21)qgs=1Hx,s·Wx,s.


## 4. Experimental Test

The CNC micromilling machine tool used for the experimental tests is a Mini-Mill/GX model, manufactured by Minitech Machinery. This machine has a positioning resolution of 0.0001 mm. In order to eliminate vibration interferences originated from the external environment while machining, an inertial table was projected to support the machine tool (Figure3). A Nakanishi model E3000 controller, maximum speed of 60,000 rpm, was used to control the speed of the tool. Cemented carbide tools manufactured by Performance MicroTools were used in the experiments. The specifications of the tools are model TR-2-0150-S, two flutes, standard flute length, and 0.015 in. diameter. Temperature was acquired using a type K thermocouple connected to a 34970A model data acquisition data logger switch unit, manufactured by Keysight Technologies. The equipment uses a default 27 Hz acquisition rate for one channel monitoring.Figure 3
Micromilling machine equipment.Table1 shows the parameters of the cutting tests.Table 1
Experimental test parameters.
Tool diameter (mm)
0.381
Spindle speed (kRPM)
30–40
Feed rate (mm/tooth)
0.001–0.0015
Depth of cut (μm)
30.0A thermocouple was welded to the workpiece surface using a capacitive circuit. The thermocouple was positioned in the trajectory of the cutting tool, at 5.0 mm from the first contact tool/workpiece as shown in Figure1. As the distance between the cutting tool and the thermocouple decreases, the temperature acquired by the sensor is expected to increase.The cutting tool was positioned to machine the thermocouple, as in a destructive test. The major source of error is due to the bead of the welded thermocouple, which is approximately the same size as the cutting tool diameter. For all wet conditions the cutting fluid was stationary on top of the workpiece surface.


## 5. Results and Discussion

Figure4 shows the temperature evolution in dry machining with the conditions of Table 1. It was used to estimate the heat flux generated at the interface. Due to the existence of noise, these temperatures were filtered using a moving average filter Figure 5. Data for times greater than 115 seconds were disregarded because they have no representativeness. The higher values of temperature in Figure 4 were when the cutting tool was close to the welded thermocouple.Figure 4
Temperature evolution atx=0.005m, y=0.055m, and z=0.032m position.Figure 5
Filtered temperature evolution atx=0.005m, y=0.055m, and z=0.032m position.The workpiece was of a inconel 718 alloy with thermal propertiesk=11.8Wm-1K-1, ρ=8190Kgm-3, and Cp=435JKg-1K-1 and dimensions x=0.045m, y=0.011m, and z=0.032m as shown in Figure 1. The micromilling was positioning at x=0m, y=0.055m, and z=0.032m and has speed u=0.00015ms-1.Figure6 shows the estimated heat source.Figure 6
Moving heat source estimation.Using the heat source data and the analytical solution the temperature evolution was estimated and compared with experimental measured temperatures. Figure7 shows this comparison. The maximum difference between the profiles of temperature evolution is 4.5%. This comparison shows the success of using the TFBGF method with analytical solutions.Figure 7
Comparison between temperatures.Once the moving heat source has been obtained the direct problem can be solved. In following, the temperature at the interface can now be obtained. Figures8 and 9 show the calculated temperatures at the limits of the contact area between workpiece and tool in xy and yz planes, respectively.Figure 8
Workpiece temperature field: view atxy plane.Figure 9
Workpiece temperature field: view atyz plane.The results presented here are in the range of values found in literature [17]. For example, Wissmiller and Pfefferkorn found a maximum temperature of 323 K (53°C) for micromilling a workpiece of 6061-T6 Aluminium under machining conditions: federate of 300 mm/min, feed of 3.75μ/tooth, and depth of cut of 100μm. Under these conditions, they estimated the heat input into tool, qg of 63.8 mW with a uncertainty between +41.3 mW and -34.9 mW. In this case the amount of heat transferring into the tool was estimated by employing Loewen and Shaw’s heat partitioning analysis for orthogonal machining [17]. Wissmiller and Pfefferkorn affirmed that the large uncertainty in the heat input into the tool, qg is a result of the large uncertainty in the machining force values caused by the variations in force values among tools and machining conditions. The technique proposed presents the great advantage of estimating the heat transferred to the tool using only thermocouple measurements without any dependence of machining parameters.


## 6. Conclusion

A 3D transient analytical solution of a workpiece during a micromilling process was obtained. Both thermal model solutions and transfer function of the system were obtained using Green’s function method. The moving heat source was obtained by using the TFBF method adapted for heat source estimation. Once the moving heat source was obtained the direct problem could then be solved. This work shows the temperature field calculated in the region including the contact zone and a comparison with measurements values in the direction of the micromilling presented an uncertainty of a4.5%.



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