1 Journal of Marine Science and echnology, Vol. 17, No., pp. 1-17 (9) EVAPORAION EFFEC IN NONLINEAR PENERAION OF HIGH ENERGY BEAM DRILLING Je-Ee Ho* and Chen-Lung Yen** Key words: enthalpy ethod. ABSRAC Due to the inappropriate assuption with the neglect of evaporation effect in previous research, a linear penetration behavior with a significant deviation fro the experiental result was usually proposed in the lower energy density region. o reedy the defect, the evaporation ass deterined fro the difference between the elting rate and flow expulsion by pressure difference is reconsider in this study. With a -D quasic-steady odel based on the enthalpy theory, the unifor penetration velocity estiated fro the Stefan boundary condition provides a special advantage in calculating efficiency. Meantie, the divergent iteration has been effectively avoided by setting up a non-unifor distribution of grids in the nuerical schee; which also enables a successful prediction of nonlinear penetration behaviors, such as the aterial reoval rate and penetration velocity versus incident energy density. Copared with the experiental data of Allen [1], present odel shows a good agreeent for copper drilling in higher energy density region (> 7 1 1 w/ ), where the relative errors between the calculated and experiental data are no ore than 1%. Even the linear drilling result in lower energy density region has been further iproved in this study. I. INRODUCION Nonlinear penetration behavior is an iportant phenoenon in high energy bea drilling, including the laser bea (L.B) and electron bea (E.B) ethods. However, describing the rapid reaction between the evaporating atos at the cavity base sees so coplex that a full understanding of energy Paper subitted 11/8/7; accepted 5//8. Author for correspondence: Je-Ee Ho (e-ail: jeho@niu.edu.tw). *Departent of Mechanical and Electrical Engineering, National I-Lan University, I-Lan County, aiwan, R.O.C. **Departent of Mechanical Engineering, National aiwan University, aipei, aiwan, R.O.C. transport inside the work piece is still lacking. Previous studies had discussed sipler odels on penetrating velocity with energy density, such as the pure elting odel [6] and single evaporating theory [5]. Both proposed echaniss, a linear relation between the penetration velocity and energy density, seeed to only describe drilling behavior in lower energy densities. Allen [1] predicted the penetrating efficiency by easuring the aterial reoval rate, and those experiental results showed that a nonlinear relationship was observed during the energy intensity range of 5 Mw/c ~ Mw/c. Chiou and Wei [7] developed an axial syetrical quasi-steady odel to calculate the fluid flow of the liquid layer by considering the surface tension as the driving force; a surprising result showed that the calculated evaporation rate was only 1/1 of the elting ass. Ho and Young [3] proposed a 1-D odel to describe the nonlinear behavior in a high energy bea. heir analytical solution was expressed as an exponential function in theral property. his approach presented an excellent agreeent in higher input energy density, but the predicted value was still overestiated during the lower input energy density. In view of above unreasonable linear results, a ore relevant relation between the energy absorption in evaporation and nonlinear behavior should be reconsidered. his is the objective of the present work. II. ANALYSIS o siplify the siulation odel without losing penetration behavior, several reasonable assuptions should be ade as follows: 1. Convective ters, due to a sall Pelect nuber estiated near the cavity base, can be ignored without causing significant error.. Hydraulic pressure gradient whose order is uch greater than that in surface tension force will be taken as the driving echanis in the flow otion. 3. A EM distribution of incident energy density [3] is assued to irradiate on the cavity base with radius.1 σ bounded.
J.-E. Ho and C.-L. Yen: Evaporation Effect in Nonlinear Penetration of High Energy Bea Drilling 13 keyhole energy bea. v vapor-liquid interface Solid Liquid σ z liquid-vapor interface elting interface δ e. j = ϖrδv δ f r r liquid-solid interface u Fig. 1. Scheatic sketch of high energy density bea.. = ρuϖr Fig.. Flow distribution. 1. Governing Equations he forulation of the enthalpy equation with an axial syetrical, quasi-steady state in both olten and solid zones can be expressed as: h 1 h h ρ u = kir + ki (1) r r r cs, hs1 h = cs + ( ε c1 + hs1, + ε ), < ε ε + ε < + ε () Knight [4], is required to deterine the surface teperature while the vapor-liquid interface is specified. h1 g 1 1 1 1 β pb exp = γ + (4) Rg b R1 R Where β =.55 was also calculated by Knight, considering the therodynaics non-equilibriu at the evaporating surface. he surface tension γ at the botto of the cavity is assued to be a linear function of teperature along the free surface. As to the energy conservation along the interface, the incident energy density dissipated by heat conduction and evaporation absorption yields. Boundary Conditions r q exp = k σ r df dr + ρuη v (5) kl = hc σ r r k s = h c ( ) at z = 3 = at = ( ) at z = 3σ = at r = Where σ is the distribution radius that defines the region in which 75% of the incident energy is deposited and the enthalpy function h soothes the discontinuous enthalpy at the solid-liquid interface. 1) Vapor-Liquid Interface An oentu conservation in (4), the balance between effective surface pressure and surface tension given by (3) ) Liquid-Solid Interface Penetration velocity u deterined fro the Stefan boundary conditiona at the liquid-solid interface can be expressed as kl z= δ = ks z= δ ρuh 3. Evaporation Ratio As far as the ass flow conservation is concerned as sketched in Fig., the elting rate & at the olten base will be shared by the evaporation rate & and the flow explusion rate, l & due to the pressure difference, at the botto of the cavity. he & l can be approxiately estiated by Bernoulli s v sl (6)
14 Journal of Marine Science and echnology, Vol. 17, No. (9) equation with the absence of friction drag set at r = and the pressure difference developed leads to B p = Ae Ae Evaporation ratio η v in (7) is defined as the evaporation rate for unit elting rate B & v v δ ηv = = 1 ( ) ( ) & u r Where & = ρuπr, & = πrδ vρ, & v = &, and v can be expressed as 1 p ρ l & l 4. Penetration Efficiency he penetration efficiency η is defined as the extracted aterial volue per unit input power at r = and can be expressed as where u η = q (7) q = k 1 + ρuη (8) v Z III. NUMERICAL PROCEDURE he discrete for of (1) and () with boundary conditions (3)-(6) can be obtained by using the central finite differences. A nuerical schee with 4 3 nodal points ensures the independence of the solution on the grid. Non- unifor nodal points are distributed in both the r- and z- directions, but have a greater concentration near the cavity base. o solve this proble, key steps developed are as follows. (i) Specify the shape of the cavity first, and then calculate the teperature distribution along the vapor-liquid interface fro (4). (ii) Given the penetration velocity. (iii) Iterate the enthalpy equations (1)-() with boundary conditions (3)-(4) using successive over-relaxation ethod with a relaxation factor of 1.5 until the solutions converge to a relative error liit of.5%. (iv) Estiate the penetration velocity fro (6). (v) If the relative error of the given penetration velocity and estiated value excesses 3%, steps (iii) and (iv) should be kept running. (vi) Copare the newest shape estiated fro (5) and last shape of the cavity. If the relative error is ore than 5%, update the latest geoetry and repeat steps (i) ~ (iv). (vii) Deterine the evaporation rate v, and penetration efficiency η fro (7) and (8), respectively. (viii) Give the another input power q, repeat steps (i) ~ (vii). IV. EXPERIMENAL PROCEDURES o verify the validation of the evaporation odel in the nonlinear penetrating process, an E.B drilling equient with the working capacity of accelerating voltage 6 kv and working current 6 A was used. During the experiental proceeding, the accelerating voltage was set at 6 kv; and A, 3 A, 4 A, 5 A of working current was regluated by turns. In the eantie, the focal spot was restricted on the surface of the test saple, which provided an equivalent incident energy density of 4 1 1 w/ ~ 1 1 1 w/. he copper sheets with diensions of.3.3.15 were selected as the workpieces and a deagnetis polishing should be involved in the pre-processing, which prevented the influence of residual surface agnetic intensity on the acccelerating electrons. Four target spots were evenly distributed on the surface of the workpiece with a distance of.3 respectively. A survey of E.B with a currant of 5 A on these positions was necessary in advance. o guarante a high working quality during the drilling process, a vacuu pup was operated continuously to keep a pressure of 1-6 Pa inside the working chaber. With the post processing on the drilling cavity, the saples were subsequently cut, polished and etched to reveal the patterns and the outlines of the fusion zone as shown in Figs. 3 ~ 5 were captured by a DINO digital icroscope which was connected with a coputer through a tranducer wire. V. RESULS AND DISCUSSION he work piece was chosen to be copper and all figures provided were diensional coordinates for coparison with experiental results and the data by Allen. he photographs in Figs. 3, 4, and 5 showing the drilling cavity were obtained for copper drilling under an incident energy density of 4 1 1 w/, 6 1 1 w/, and 8 1 1 w/, respectively. A regular drilling hole without residual solidification left in Fig. 3 predicts that ost of the olten etal ight have evaporated into the keyhole to produce a continuous foration of the cavity and penetration echanis is believed to be deterined by evaporation odel. In Fig. 4, the observed outline with a regular cavity shape and a sooth solidification with a flat level above the cavity base deonstrate that the olten etal in the cavity was nearly in static state, instead of a flowing otion. Another view fro the cavity covered with a black liquid fil leads to the prediction that the static liquid inside the cavity ust have been overheated. According to above description, a strong evaporation, subjected to be a prior paraeter, is predicted to be
J.-E. Ho and C.-L. Yen: Evaporation Effect in Nonlinear Penetration of High Energy Bea Drilling 15 Fig. 3. Section view of copper cavity under the incident energy density 4 1 1 w/. Fig. 5. Section view of the copper cavity under the incident energy density 8 1 1 w/. Penetration velocity (/s) 18 16 14 1 1 8 6 4 Nuerical value Allen s data Young and Ho 4 5 6 7 8 9 1 11 1 13 Energy density (*1 1 w/ ) Fig. 6. Penetration velocity vs. energy density in copper drilling. Fig. 4. Section view of copper cavity under the incident energy density 6 1 1 w/. responsible for the foration of the cavity and should not be disregarded as the energy density was below 6 1 1 w/. In contrast to the characteristics shown in Figs. 3 and 4, a jagged shape with a rough surface in Fig. 5 was caused by the expulsion of the olten flow under the action of the pressure difference; which not only had a deeper penetration, but also stirred an unsteady disturbance inside the cavity, where a non-unifor solidification of the olten part was left. In this odel, the flow otion, due to the pressure difference, starts working and which will be taken as the driving echanis to for the drilling cavity if the energy density is ore than 8 1 1 w/. Figure 6 illustrates the penetration velocity versus energy density. At the lower energy density (< 7*1 1 w/ ), the penetration velocity increases slightly fro.3 /s ~ 4 /s, and the slower penetration echanis is priarily caused by the evaporation effect,which takes too uch tie to absorb the evaporation latent heat to penetrate into cavity as illustrated in
16 Journal of Marine Science and echnology, Vol. 17, No. (9) Penetration efficiency (*1 11 3 /J) 16 Nuerical value 14 Allen s data 1 1 8 6 4 4 5 6 7 8 9 1 11 1 Energy density (*1 1 w/ ) Fig. 7. Penetration efficiency vs. energy density in copper drilling. Evaporation ratio (%) 9 8 7 6 5 4 3 1 4 5 6 7 8 9 1 11 1 Energy density (*1 1 w/ ) Fig. 8. Evaporating ratio vs. energy density in copper drilling. Figs. 3 and 4. But for the higher energy density (> 8*1 1 w/ ), the penetration velocity exhibits a nonlinear increase fro 4 /s ~ 15 /s which corresponds to a olten flow driven quickly by pressure difference, instead of evaporation, to fro a new cavity as shown in Fig. 5. Coparing with Allen s experients, the results are in close agreeent with each other and the axiu relative error is less than 1% when the energy density excesses 7*1 1 w/. Another coparison fro a linear penetration odel, 6 /s ~ 14 /s, proposed by Young and Ho [3] shows that the drilling result will be apparently over estiated in the energy density below 7*1 1 w/ if the evaporation effect is not involved. Concerning the variation in penetration efficiency with different energy densities, Fig. 7 shows that a poor drilling efficiency of about 6*1-11 3 /J abruptly clibs up to 1.4*1-1 3 /J as the energy density varies fro 4*1 1 w/ to 1*1 1 w/ where the penetration velocity in Fig. 7 quickly increases fro /s ~ 15 /s with a faster growth than the energy density (< 1*1 1 w/ ) dose. he opposite result, i.e., a slow decrease of efficiency fro 1.5*1-1 3 /J to 1.3*1-1 3 /J, occurs when the energy density is ore than 1*1 1 w/, while the stable value 15 /s in drilling velocity grows slowly than the increent of input energy density. Both tendencies, in Figs. 6 and 7, exhibit a siilar distribution of the penetration velocity and efficiency with input power density applied. Moreover; the axiu drilling efficiency 1.5*1-1 3 /J occurring at 1*1 1 w/ in Fig. 7 is produced when the drilling velocity reaches the stable value in Fig. 6. Coparing the nuerical and experiental results shows that the agreeent is good in ost energy regions, especially in higher energy densities (> 9*1 1 w/ ) where the axiu relative error of 15% eerges at an input energy density of 1*1 1 w/. According to the ass flow conservation, the evaporation ass ratio is defined as the evaporating rate per elting rate. Fro above definition, the distribution of evaporation ratio in Fig. 8 illustrates that about 6% ~ 8% of elting flow will evaporate into the drilling cavity when the energy density varies fro 4 1 1 w/ to 6 1 1 w/. It tells that the doination of the evaporation effect in the foration of cavity can be predicted here. On the other hand, the continuous decrease of the ratio, fro 6% towards 1%, occurs within an energy density of 6 1 1 w/ ~ 8 1 1 w/, where the evaporation effect has fully lost its influence. In such case, alost all the olten flow with a inor evaporation will be carried to a radial direction by pressure difference, which is taken as the driving source in the foration of cavity and is also in concordance with above discussion. VI. CONCLUSION he discussion above leads to the following iportant conclusions: 1. During the energy density below 7 1 1 w/ for copper drilling, a continuous foration of cavity is priarily caused by evaporation effect which should not be ignored.. Due to the extra duration required to absorb the latent heat in the evaporation process, it will slow down the penetration velocity and reduce the drilling efficiency. Conversely, a significant iproveent of penetration behavior will be ade as the flow otion, driven by pressure difference, is doinate in higher energy density region. 3. he regular shape with sooth wall of cavity will be captured during the slower evaporation process. Otherwise, a coarse surface of keyhole pattern appears in faster penetration odel. 4. he distribution of nonlinear penetration can be odified by the consideration of the evaporation effect in the lower energy density and an assuption of neglecting the convective ters without losing the drilling characteristic has been also identified in this study. 5. Detail analysis of various energy distributions in the workpiece will provide an extensive understanding on the nonlinear variation of drilling behavior in the future work. NOMENCLAURE A epirical constant for copper 3.35 1 1 [Nt/ ]
J.-E. Ho and C.-L. Yen: Evaporation Effect in Nonlinear Penetration of High Energy Bea Drilling 17 B epirical constant for copper 464 [K] C pi specific heat in both phases [J/kg K] f the location of liquid-vapor interface [] h enthalpy function [J/kg] h c convection coefficient [W/ K] h lg latent heat of evaporation [J/kg] h sl latent heat of elting [J/kg] k i theral conductivity, k l or k s [W/ K] & elting rate [kg/s] & l ass flow rate carried away by pressure difference [kg/s] & v evaporation rate [kg/s] p vapor pressure [pa] P b saturated vapor pressure at boiling teperature Q incident power [kw] q axiu incident energy flux, Q/πσ [W/ ] R g specific gas constant [J/kg K] teperature [K] elting teperature [K] b boiling teperature [K] botto teperature of the cavity [K] abinet teperature [K] u penetration velocity [/s] z, Z diensional and diensionless vertical coordinate, Z = z/σ, as illustrated in Fig. 1 r, R diensional and diensionless radial coordinate, R = r/σ, as illustrated in Fig. 1 R 1, R principal curvatures of vapor-liquid interface [] α s theral diffusivity in solid phase [/s ] α l theral diffusivity in solid phase [/s ] β.55 was considered for therodynaics nonequilibriu at evaporating surface. δ liquid thickness [] γ surface tension at elting teperature [Nt/] ρ density of working aterial [kg/ 3 ] σ distribution radius, reference length [] η penetration efficiency [ 3 /J] η v evaporation ratio [%] dγ d surface tension gradient [Nt/ K] REFERENCES 1. Allen, M. V., Laser drilling velocity in etals, Journal of Applied Physics, Vol. 47, pp. 546-5463 (1976).. Ho, J. E. and Ching, Y. H., Coputing the absorption of a drilling or welding hole for laser by using onte carlo ethod, Journal of the Chinese Society of Mechanical Engineers, Vol. 7, No. 1, pp. 61-67 (6). 3. Ho, J. E. and Young, H.., he analysis on penetrating efficiency in high-energy bea drilling, Key Engineering Material, Vol. 364, pp. 38-314 (8). 4. Knight, C. J., ransient vaporization fro a surface into vacuu, AIAA Journal, Vol., pp. 95-954 (198). 5. Ol'Shanskii, N. A., Moveent of olten etal during electron bea welding, Svarochnoe Proizvodstvo, Vol. 1, pp. 1-14 (1974). 6. Ready, J. F., Effects due to absorption of laser radiation, Journal of Applied Physics, Vol. 36, pp. 46-468 (1965). 7. Wei, P. S. and Chiou, L. R., Molten etal flow around the base of a cavity during a high-energy bea penetrating process, Journal of Heat ransfer, Vol. 11, pp. 918-93 (1988).