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International Journal of Mechanical Engineering (IJMET), ISSN 0976 – 6340(Print), International Journal of Mechanical Engineering and Technology and Technology (IJMET), ISSN 0976 – 6340(Print), © IAEME ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), IJMET ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), pp. 109-123 ©IAEME © IAEME, http://www.iaeme.com/ijmet.html CONICAL GAUSSIAN HEAT DISTRIBUTION FOR SUBMERGED ARC WELDING PROCESS Aniruddha Ghosh Dept. of Mechanical Engineering Govt. College of Engg. & Textile Technology Berhampore, WB, India Somnath Chattopadhyaya Dept of ME&MME ISM, Dhanbad, India ABSTRACT: Studies on temperature distribution during welding are very important because this may pave the way for application of microstructure modeling, thermal stress analysis, residual stress/distribution and welding process simulation. An attempt has been made in this paper to predict of temperature variation of entire plates during welding and after welding through an analytical solution is derived from the transient three dimensional heat conduction of an infinite plate to finite thickness. The heat input that is applied on the plates is exactly same amount of heat lost for electric arc is assumed to be a moving heat source with Gaussian distribution. The prediction was compared with experimental results with good agreement. List of Symbols: α =Thermal diffusivity(m2/s) c =Specific heat(j/kg/K) Q =Heat distribution Q0 =Net heat input per unit time(W) T =Temperature of body(K) T0 =Initial Temperature of body(K) v =Travel Speed (m/s) w =Distance in x direction in a moving =coordinate of speed v (w=x-vt)(m) y =Distance in y direction (m) z =Distance in z direction (m) ρ =Density(kg/m3) dQ =Change of heat flux dT =Change of temperature 109 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME dt =Change of time INTRODUCTION: More that forty years ago solution of a travelling point heat source for the case of welding was first attempted. Then moving and stationary heat sources are frequently employed in many manufacturing processes and contact surfaces. In recent years application of localized heat source have been related to development of laser and electron beams in material processing, such as welding, cutting, heat treatment of metals and manufacturing of electronic components[1-2].Analytical and numerical models for the prediction of the thermal fields induced by the stationary or moving heat sources useful tools for studying the afore mentioned problems[2].In some laser beam applications, such as surface heat treatment, the contribution of convective heat transfer must also taken into account[3].Quasi-steady state thermal fields induced by moving localized heart sources have been widely investigated[3,4], whereas further attention seems to be devoted to the analysis of temperature distribution in transient heat conduction because temperature distribution has a significant influence on the residual stresses, distortion and hence the fatigue behavior of welded structures. From literature, presently many related work found. Classical solutions for the transient temperature field such as Rosenthal’s solutions [5] dealt with the semi infinite body subjected to an instant point heat source, line heat source or surface heat source. These solutions can be used to predict the temperature field at a distance far from the heat source. Rykalin [6] pointed out a heat flow model need to take the factors-non constant thermal properties, heat of phase transformation, heat magnitude and distribution, plate geometry, convection and surface depression in weld pool [7].Although Christensen’s experimental results indicate that Rosenthal’s solution gives good agreement with the actual weld bead size over several orders of magnitude, the scatter can be as much as a factor of three. The work of Grosh [8] showed that that the effect of non-constant thermal properties can only make a 10-15 percent difference in the weld pool geometry, and Malmuth [9] has shown that the effect of latent heat can only make a 5-10 percent difference. Obviously the heat input magnitude solution of Rosenthal modified for non constant thermal properties and latent heat can not explain the difference between theoretically predicted and experimentally measured importance of the heat distribution on the plate surface. Fortunately several 110 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME investigators have measured actual heat distributions in arcs on a water cooled copper anode[10].Using these results it is possible to determine whether the presence of a distributed rather than a point source of heat can explain the range of weld shape variation measured by Chirstensen and others. It should be emphasized that this solution retains all but one of the simplifying assumptions used by Rosenthal including absence of convection in weld pool, constant thermal property, and no latent heat of phase transformation. The only change in the use of a distributed rather than a point source of heat. Inspite of these simplifications, it will be shown that the results not only agree with the Rosenthal solution in the limit, but that they are capable of explaining most of the experimental scatter. Rosenthal fail to predict the temperature in the vicinity of the heat source [5]. Eager and Tsai [11] modified Rosenthal’s theory to include a two dimensional (2-D) surface Gaussian distributed heat source with a constant distribution parameter (which can be considered as an effective solution of arc radius) and found an analytical solution for the temperature of a semi-infinite body subjected this moving heat source. Their solution is a significant step for the improvement of temperature prediction in the near heat source regions. Jeong and Cho [12] introduce an analytical solution for the transient temperature field of fillet welded joint based the similar Gaussian heat source but with different parameters (in two directions x and y). Using conformal mapping technique, they have successfully transformed the solution for temperature field in the plate of finite thickness to the fillet welded joint. Even though the available solutions using the Gaussian heat sources could predict the temperature at regions closed to the heat source, they are still limited by the shortcoming of the 2-D heat source itself with no effect of penetration. This shortcoming can only be overcome if more general heat source are implemented. Goldak, et al. (3-D)first introduced the three dimensional double ellipsoidal moving heat source. Finite element modeling (FEM) was used to calculate the temperature field of a bead-on-plate and showed that this 3-D heat source could overcome the shortcoming of the previous 2-D Gaussian model to predict the temperature of the welded joints with much deeper penetration. However, up to now, an analytical solution for this kind of 3-D heat source was not yet available [13], and hence, researchers must rely on FEM for transient temperature calculation or other simulation purposes, which requires the thermal history of the components. Therefore, if any 111 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME analytical solution for a temperature field from a 3-D heat source is available, a lot of CPU time could be saved and the thermal-stress analysis or related simulations could be carried out much more rapidly and conveniently. Nyguyen,et al.[14]are derived analytical solution for the transient temperature field of the semi infinite body subjected to 3-D power density moving heat source (such as semi-ellipsoidal and double ellipsoidal heat source). However results based on are not satisfactory semi-ellipsoidal 3-D heat source with respect to double ellipsoidal heat source. 2. EXPERIMENTAL PROCEDURE 2.1 EXPERIMENTAL SETUP Figure 1 MEMCO Semi Automatic Submerged Arc Welding machine at the workshop of the Indian School of Mines, Dhanbad 2.2 SPECIFICATIONS: Input voltage supply- 380/440 volts Welding speed Trolley-30 to 1200 mm/min 3 Phase,50/60 Hz cycle, Air cooled Wire feed speed-100 to 8000mm/min Output current 600 amps Wire diameter -2 to 5 mm Duty cycle 100% Head movement vertical/horizontal -135 mm Open circuit voltage 56volts, 35 Kva Deposition rate- 4 to 6 kg/hr Flux hopper capacity 12.5 kg Wire flux ratio-1:1 • Flux used: ADOR Auto melt Gr II AWS/SFA 5.17(Granular flux) • Electrode: ADOR 3.15 diameter copper coated wire • Test Piece: 400 x 75 x1 6 mm square butt joint • Weld position flat • Electrode positive and perpendicular to the plate 112 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME 2.3 PROCESS FLOW CHART: Cutting of 20 mm Mild Face Grinding on Automatic V-Groove preparation on Steel Plate by Gas Grinding Machine Shaper Machine Cutting Grinding of the Face where Cutting of Samples by Welding by Submerged further study is to be carried Power Hack Saw Arc Welding Process out Wax Coating of the Ground Removal of wax and Carbon coating on the Surface cleaning of surface surface Temperature reading taken on Study of bead geometry, different points of welded plates by estimation of dimensions of bead laser non contact thermometer geometry SELECTING THE EXPERIMENTAL DESIGN The experiments were conducted as per the design matrix at random to avoid errors due to noise factors. The job 300x300x20 mm (3 pieces) was firmly fixed to a base plate by means of tack welding and then the welding was carried. The welding parameters were noted during actual welding to determine the fluctuations if any. The slag was removed and the job as allowed to cool down. The job was cut at three sections by hacksaw cutter and the average values of the penetration, reinforcement height and width were taken using vernier caliper of least count 0.02mm.Experimental procedure for prediction of Saw process: Same procedure as above, the job was cut at three sections and the average values of the penetration, reinforcement height and width (shown in fig. 2) were taken using vernier calliper of least count 0.02mm. 113 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME Figure 2 Bead geometry, P-Penetration-Reinforcement height-Bead Width Table-1: Thermo physical properties of employed material Thermal Density(m3) Specific Thermal conductivity(W/m/ ) heat(J/kg/ ) diffusivity(m2/s) 54 7861kg/m3 420 1.6×10-5 Steel(C=0.3%) EXPERIMENTAL RESULTS: Table 2 Set of reading obtained are tabulated as below Sl.N Volta Curre Travel Penetrati Reinforcem Wel Weigh Weigh MDR o. ge (V) nt (A) Speed on ent d t t (kg/mi (cm/mi (mm) Height(mm Widt (befor (befor n) n) ) h e e (mm weldin weldin ) g) g) 1 19.6 7.36 7.48 25 350 17 6.67 2.74 8 0.07 2 13.9 7.56 7.62 25 350 30 5.26 1.02 5 0.06 3 30.8 7.48 7.66 25 450 17 7.65 3.8 8 0.10 4 15.2 7.38 7.48 25 450 30 6.52 2.1 8 0.10 5 22.7 7.26 7.40 35 350 17 6.02 1.91 6 0.08 6 19.3 35 350 30 5.90 1.80 2 7.33 7.41 0.08 7 25.7 35 450 17 6.25 2.57 7 7.33 7.59 0.15 8 22.5 35 450 30 6.02 2.13 3 7.32 7.49 0.17 Explanation of Table 2-From literature [16] it is found that an increase in travel speed substantially reduces the heat input resulting in lower burn off rate. This reduced burn off rate decreases the metal deposition at the weld joint thereby lowering all the 114 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME three bead parameters i.e. penetration, reinforcement height and the width. It is seen that the penetration increases with increase in the current, the reinforcement height also increases marginally with an increase in the current but the width decreases i.e. it has a negative effect. The increase in current results in decreasing arc length as well as prevents the spreading of the arc cone; which results in higher melting temperature at the tip and parent material, resulting in deeper penetration at the cost of bead width. The decrease in bead width helps in marginally increasing the reinforcement height. There is a small increase in the penetration but the effect on width is higher on the positive side. The reinforcement reduces slightly. So above results are the interaction effects of input factors. From results of metal deposition rate, we may get information about weld bead volume. Table -3: Distribution of temperature on different grid points on the welded surface of the specimen for 7.87Kw heat input at 100th sec (at the end of the welding process, experimental values) i j 1 2 3 4 5 2 300.2 445.9 501.8 540.6 601.8 1 450.7 501.5 545.9 610.5 713.1 0 521.2 530.8 570.9 600.8 798.9 -1 443.7 510.5 559.9 600.5 689.7 -2 390.8 440.9 508.9 530.9 587.7 Here x=ih, y=jk, h=0.075m, k=0.075m, Temperature reading taken in Fig:3 A-Pictorial view of bell shaped Welding arc Fig:3 B-Proposed ellipsoidal heat source in which heat flux is distributed in a Gaussian manner throughout the heat source’s volume[14] 115 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME Fig:3 C-Proposed spherical heat source in which Fig:3 D-Proposed conical heat source in heat flux is distributed in a Gaussian manner which heat flux is distributed in a throughout the heat source’s volume[5] Gaussian manner throughout the heat source’s volume Figure 3 Heat source volume/shape in which heat flux is distributed in a Gaussian manner throughout the heat source’s volume DOUBLE CONICAL HEAT SOURCE: 3-D heat source could overcome the shortcoming of the previous 2-D Gaussian model to predict the temperature of the welded joints with much deeper penetration. However, up to now, an analytical solution for this kind of 3-D heat source was not yet available. Let equation of a double cone 3-D heat source Gaussian model is Q(x,y,z)=A[exp{-(x2+y2-(z+a)2×tan2β1)}+exp{-(x2+y2-(z-b)2×tan2β2)}] (1) Where A is amplititude of Gaussian distribution,(0,0,-a) position of electrode bottom point,(0,0,b) penetration end point along z direction, β1, β2 are semi vertical angles of cones . For the case semi infinite Gaussian i.e. infinite length, breath but finite thickness then equation will be Qo=A (2) , taking tan2β1= tan2β2=1 The solution for the temperature field of semi-infinite body is based on the solution for an instant point source that satisfies the following differential equation of heat conduction in fixed coordinates dTt’= ×exp (- ) (3) 116 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME Where temperature rise dTt’ for a point heat source, dQ(the amount of heat located at position (x’, y’, z’) at time t’) in a very small interval of time dt’. =thermal diffusivity; c =specific heat; ρ=mass density. Let us consider the solution of an instant semi-ellipsoidal heat source as a result of superposition of a series of instant point heat sources over the volume of the described Gaussian heat flux at a point source into Equation 3 and integration over the volume of the heat source double conical heat source . Let us consider the solution of an instant semi-ellipsoidal heat source as a result of superposition of a series of instant point heat sources over the volume of the described Gaussian heat flux at a point source into Equation 3 and integration over the volume of the heat source double conical heat source . dTt’= ×exp (- ) For the case semi infinite Gaussian i.e. infinite length, breath but finite thickness then equation will be T-T0 = × (2z+2k+ab)(0.8862×k0.5× +1) dk [Where, k=4×α× (t-t’), so k=f (t’), w=(x-vt)] = × (2z+2k+a-b)(0.8862×k0.5× +1)dk (4) Solution of above Equation (4) has derived with the help of trapezoidal method for . From this equation (4) temperature has calculated in different grid points at 100th sec for travel speed 17cm/min and heat input 7.87Kw which are tabulated below are good agreement with experimental results. 117 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME Table -4: Distribution of temperature on different grid points on the welded surface of the specimen for 7.87Kw heat input at 100th sec (just at the end of the welding process) i j 1 2 3 4 5 2 390 450 478 540 580 1 460 511 523 601 690 0 590 550 555 630 780 -1 460 511 523 601 690 -2 390 450 487 540 580 Here x=ih, y=jk, h=0.075m, k=0.075m, Temperature reading taken in . Both numerical and experimental results from this study have showed that the present analytical solution could offer a very good prediction for transient temperatures near the weld pool, as well as simulate the complicated welding path. Figure 5 Graphical representation for temperature distribution [data taken from table-4]. 118 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME Figure 6 Temperature values according to distance from welding nugget [data taken from table-4] Figure 8 Effect of weld current on temperature values [data taken from table-4] 119 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME TEMPERATURE CALCULATION AFTER WELDING: Figure 9 Mesh grid for numerical solution two dimensional heat equation Figure 10 Mesh grid for numerical solution two dimensional heat equation with nth plate (for temperature calculation after time t) Two dimensional heat equation is + = × (5) 120 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME If h is step- size then mesh point (x,y,t) =( ih, hj, nl) or may be denoted as simply(i,j,n). By applying Numerical method, Equation No. (5) Can be solved. Solution: + = × Or Ti,j,n+1 = Ti,j,n +(l/α×h2)[Ti-1,j,n + Ti+1,j,n +Ti,j+1,n + Ti,j-1,n - 4Ti,j,n] (6) From Eqn. No. 4, 5, 6 and taking l=60 sec, h=0.075m, α=thermal diffusivity=9×10-6 m2/sec.Temperature of different grid points are calculated which are tabulated below; these have good agreement with experimental values. Prediction of temperature on different points on welding pates after welding has been made through this numerical method .It is also newly found solution which will be very helpful to predict rate of cooling of metal so hardness, heat affected zone etc. Table -5: Distribution of temperature on different grid points on the welded surface of the specimen for 7.87Kw heat input at 100th sec (at the end of the welding process) i j 1 2 3 4 5 2 301 345 423 445 480 1 423 401 466 475 560 0 520 429 475 610 680 -1 423 401 466 475 560 -2 301 345 423 423 480 Here x=ih, y=jk, h=0.075m, k=0.075m, l=60sec, Temperature reading taken in . Here also both numerical and experimental results from this study have showed that the present analytical solution could offer a very good prediction for transient temperatures near the weld pool, as well as simulate the complicated welding path. CONCLUSION: This type of numerical investigation is made to estimate two and three dimensional transient heat conduction field in infinite and semi infinite metallic solid because surface heat transfer strongly affected the temperature distribution in the welded pates. In this study, analytical solutions for the transient temperature field of a semi 121 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 1, Number 1, July - Aug (2010), © IAEME infinite body subjected to 3-D power density moving heat source (such as double conical heat sources) were found and experimentally validated. Also, it was shown the analytical solution obtained for double conical heat source was a general one that can be reduced to 2-D Gaussian distributed heat source and classical instant point heat source. The analytical solution for double conical heat source was used to calculate transient temperatures at selected points on a mild steel plates which are welded by taking x- axis along welding line, origin is starting point of welding, y-axis is perpendicular to welding line and z-axis towards plate thickness. Both numerical and experimental results from this study have showed that the present analytical solution could offer a very good prediction for transient temperatures near the weld pool, as well as simulate the complicated welding path. Furthermore, very good agreement between the calculated and measured temperature data indeed shows the creditability of the newly found solution and potential application for various simulation purposes, such as thermal stress, residual stress calculations and microstructure modeling. At last prediction of temperature on different points on welding pates after welding has been made through numerical method .It is also newly found solution which will be very helpful to predict rate of cooling of metal so hardness, heat affected zone. REFERENCES: [1] Tanasawa, I. and Lior, N., 1992, Heat and Mass Transfer in Material Processing, Hemisphere, Washington, D.C. [2] Viskanta, R. and Bergman, T. L., 1998, Heat Transfer in Material Processing, in Handbook of Heat Transfer, Chap. 18, McGraw-Hill, New York. [3] Shuja, S. Z., Yilbas, B. S., and Budair, M. O., 1998, Modeling of Laser Heating of Solid Substance Including Assisting Gas Impingement, Numer. Heat Transfer A, 33, pp. 315-339. [4] Bianco, N., Manca, O. and Nardini, S., 2001, Comparison between Thermal Conductive Models for Moving Heat Sources in Material Processing, ASME HTD, 369-6, pp. 11-22. 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