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The effect of beam oscillation rate on Al evaporation behavior in the electron beam melting process Nakamura, Hideo 1989

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THE EFFECT OF BEAM OSCILLATION RATE ON AL EVAPORATION BEHAVIOR IN THE ELECTRON BEAM MELTING PROCESS by HIDEO NAKAMURA M.Eng., Tokyo I n s t i t u t e of Technology, 1979 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Metals and Materials Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1989 © Hideo Nakamura, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Metals and""Materials Department of Engineering The University of British Columbia Vancouver, Canada Date March 18, 1989  DE-6 (2/88) Abstract Electron beam hearth melting process i s widely used i n producing superalloys, T i and i t s a l l o y s because of i t s excellent m e t a l l u r g i c a l c h a r a c t e r i s t i c s . However, one of the disadvantages of the process i s that a l l o y i n g elements with high vapor pressure evaporate under the highly reduced operating pressure. This makes i t very d i f f i c u l t to carry out an accurate chemical composition c o n t r o l . In order to prevent the excess evaporative los s by reducing the superheating of the molten pool, the beam scanning technique i s employed i n normal operation. Although the e f f e c t of t h i s technique i s well known empirically, few fundamental studies' have been made to date. The purpose of t h i s study i s , therefore, to c l a r i f y the quantitative e f f e c t of the beam o s c i l l a t i o n rate on the evaporation behavior. Small amounts of Ti-6A1-4.V a l l o y were melted i n an EB melting furnace. The temperature on the melt surface was measured i n s i t u by an o p t i c a l pyrometer during the melting period. The evaporation los s of both T i and A l was also investigated. On the basis of the experimental r e s u l t s , a two dimensional unsteady heat and mass transf e r model was developed. (A one dimensional model was developed i n the case of a stationary beam.) The model was used to investigate the e f f e c t of the beam i i o s c i l l a t i o n r a t e on the evaporation behavior and a l s o to di s c u s s the optimum beam scanning r a t e . I t was c l e a r l y shown that the evaporative l o s s of both T i and A l could be suppressed by the increase of the beam o s c i l l a t i o n r a t e . With the beam o s c i l l a t i o n r a t e a t more than 1.0 Hz, however, t h i s e f f e c t could not be observed c l e a r l y anymore. I t was a l s o found t h a t the beam scanning technique i s u s e f u l not only i n c o n t r o l l i n g A l con c e n t r a t i o n , w t % A l , but a l s o i n suppressing the t o t a l evaporative l o s s of both T i and A l . i i i Table of Contents Abstract . . . i i L i s t of Figures v i i L i s t of Tables i x L i s t of Photos x L i s t of P r i n c i p a l Symbols x i Acknowledegment x i i i Section 1 INTRODUCTION 1-1 General 1 1-2 EB melting process 1 1-2-1 Drip melting process 1 1-2-2 Hearth melting process 3 1-3 L i t e r a t u r e review 6 1-3-1 Evaporation phenomena i n vacuum metallurgy 6 1-3-2 Process modeling of EB hearth melting process 15 1-3-3 Moving heat source problem 16 1- 4- Objective of t h i s study 20 Section 2 EXPERIMENTAL PROCEDURE AND RESULTS 2- 1 Experimental apparatus 22 2-2 Experimental procedure 30 2-3 Results -41 i v S e c t i o n 3 MATHEMATICAL MODEL 3-1 Formulation 64 3-1-1 Basic equation 64 3-1-2 E f f e c t of metal flow 65 3-1-3 Heat input and heat l o s s 68 3-1-4- Boundary c o n d i t i o n 71 3-2 S o l u t i o n 71 3-3 Results and d i s c u s s i o n 73 3-3-1 E f f e c t of the number of nodes 73 3-3-2 E f f e c t of FF and beam o s c i l l a t i o n r a t e 75 3-3-3 Comparison w i t h experimental r e s u l t s 85 3-3-4 Temperature contour on metal surface 89 3-3-5 T o t a l weight l o s s 91 3-3-6 Decrease i n wt%Al during melting 93 3- 4 I m p l i c a t i o n s of t h i s study f o r i n d u s t r i a l process • 99 S e c t i o n 4 SUMMARY AND RECOMMENDATIONS FOR THE FUTURE WORK 4- 1 Summary 101 4-2 Recommendations f o r the f u t u r e work 102 L i s t of References , 104 Appendix 1 v ESTIMATION OF RATE CONTROLLING STEP A-1-1 Complete d i f f u s i o n c o n t r o l case 107 A-1-2 E f f e c t of the temperature and the metal flow on the ra t e c o n t r o l l i n g step ...110 Appendix 2 PROGRAM LIST AND FLOW CHART A-2-1 Program l i s t of 1-D model 115 A-2-2 Program l i s t of 2-D model 121 A-2-3 Flow chart of 2-D model 129 v i L i s t of Figures Figure Page 1.1 Schematic diagram of EB d r i p melting process 2 1.2 Schematic diagram of EB hearth melting process 4 1.3 Dual coordinate system f o r moving heat sources. ( E=x-vt ) 18 2.1 Schematic diagram of experimental apparatus 24. 2.2 Wiring diagram of EH 30/20 25 2.3 Water-cooled copper mold 28 2.4 Schematic diagram of i n - s i t u temperature measurement system 29 2.5 C a l i b r a t i o n curve of the pyrometer, IRCON 1100 31 2.6 Schematic drawing of arrangement of charged material 34-2.7 Standard heating-up pattern 35 2.8 Sketch of specimen during experiment 36 2.9 Ty p i c a l r e s u l t of SEM/EDX analysis 38 2.10 Relation between wt%AL and I ^ l ^ T i 39 2.11 Cooling curve of CP-Ti 4-2 2.12 Temperature changes with time 43 2.13 Changes i n mV output and calculated temperature during experiment 48 2.14 Variations of AT (T ,-T. ) of cooling water with ,. out i n b time 50 2.15 D i s t r i b u t i o n of wt%Al i n the pool 56 2.16 Changes i n t o t a l weight loss with time 60 2.17 Changes i n wt%Al with time 61 2.18 E f f e c t of beam o s c i l l a t i o n rate on wt%Al at 5 min and 10 min 62 v i i 3.1 Heat balance taken i n control volume 66 3-2 Coordinate system used for two dimensional model 67 3-3 Meshes used i n the calculation 74-3.4 Comparison of the number of nodes taken i n the calculation 76 3.5 Calculated temperature changes with time i n the case of stationary beam 79 3.6 Temperature d i s t r i b u t i o n at 600 sec with various FF values 80 3-7 Calculated temperature changes with time (0.1Hz) 82 3.8 Calculated temperature changes with time (1.0Hz) 83 3.9 Calculated temperature changes with time (10.0Hz) 84 3.10 Comparison with experimental results 86 3.11 Relation between calculated and observed temperature changes with time 88 3-12 Calculated temperature contours on the melt surface at 600 sec 90 3.13 Calculated change i n the t o t a l weight loss with time 92 3.14 Calculated t o t a l weight loss at 5 min and at 10 min 94 3.15 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 6% 95 3.16 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 4-5% 97 3.17 Calculated wt%Al at 5 min 98 A-1.1 Changes i n wt%Al with time calculated from Machlin's model 109 A-1 .2 Geometry used i n the calculation 111 A-1 .3 Evaporation control map 114 v i i i L i s t of Tables Table Page 2.1 Experimental conditions 32 2.2 Wt%Al analyzed by emission spectroscopy and I. T/I„. obtained by SEM/EDX ....^ 40 2.3 A l l r e s u l t s obtained i n t h i s experiment 46 3.1 Constants used i n the c a l c u l a t i o n 72 3.2 Examples of computer inputs and outputs 77 i x L i s t of Photos Photo Page 2.1 Appearance of 30 kW EB furnace at UBC 23 2.2 Cross section of samples after experiment 51 x L i s t of P r i n c i p a l Symbols A m or cm surface area C mol/cm concentration C J/kgK heat capacity P 2 D cm /sec d i f f u s i v i t y 2 D' cm /sec modified d i f f u s i v i t y FF multiplying factor of heat conductivity. considering metal flow f f multiplying factor of d i f f u s i v i t y considering metal flow AH kJ/kg heat of evaporation h cm depth of molten metal Erp W/m K heat transfer c o e f f i c i e n t between mold and metal j mol/sec.cm mass f l u x k W/mK heat conductivity k' W/mK modified heat conductivity k"Me 1 c m / / s e c mass transfer c o e f f i c i e n t i n l i q u i d phase M kg/mol molecular weight m kg/sec evaporation rate P° Pa vapor pressure of pure element 2 q' W/m sum of heat input and output B 2 q W/m heat loss through mold EB 2 q W/m t o t a l power input of EB EV 2 q W/m evaporation heat loss R 2 q W/m radiation heat loss x i R J/mol.K gas constant, 8.315 T K temperature Ta K ambient temperature, 298 v m/sec v e l o c i t y of heat source x mole f r a c t i o n a m /sec heat d i f f u s i v i t y y Raoultian a c t i v i t y c o e f f i c i e n t • e emi s s i v i t y 0 sec l i f e time of a small element at the surface £ m moving coordinate p kg/m density 2 L. — 8 a W/m .K Stefan-Boltzmann constant, 5-67x10 x i i ACKNOWLEDEGMENT I would l i k e to express gratitude to Professor Alec M i t c h e l l f o r his kind guidance and courteous advice throughout the course of t h i s study. I also thank Rudy Cardeno f o r h i s s i g n i f i c a n t contribution to the experimental work. I am very g r a t e f u l to fellow graduate students and f a c u l t y members at Department of Metals and Materials Engineering. I s h a l l never forget my family's part i n accomplishing t h i s study. My wife, Hitomi, has supported me with love and patience, p a r t i c u l a r l y i n typing out the d r a f t of t h i s t h e s i s . Also, my l i t t l e daughters, Emi and Risa, have given inestimable comfort to me during t h i s study. The f i n a n c i a l support of NKK Corporation i s deeply appreciated. x i i i S e c t i o n 1. INTRODUCTION 1-1 General In recent years, the EB process has been widely adopted f o r 1 ) the production of s o - c a l l e d advanced metals . The energy d e n s i t y 2 2 of the e l e c t r o n beam i s so high ( i n the order of 10 kW/cm ) th a t elevated ' temperature can be obtained e a s i l y . Another c h a r a c t e r i s t i c s of t h i s process i s the h i g h l y reduced pressure i n the melting chamber(in the order of 10 - 10 T o r r ) , which i s r e q u i r e d f o r emission of the e l e c t r o n beam. These two m e t a l l u r g i c a l f e a t u r e s a l l o w even r e f r a c t o r y and r e a c t i v e metals to be melted r e l a t i v e l y e a s i l y . Among s e v e r a l v a r i a t i o n s of the EB furnace the d r i p - m e l t i n g mode and the hearth-melting mode are now the two main methods i n commercial use. 1-2 EB melting process 1-2-1 Drip melting process Fig.1-1 shows a schematic diagram of the d r i p m e l t i n g process. In t h i s process a piece of s o l i d feedstock supported from the top i s slowly lowered i n t o the path of one or more 1 EB gun Feed stock Water cooled copper mold Ingot Fig.1-1 Schematic diagram of EB d r i p melting process. 2 e l e c t r o n beams. The molten metal d r i p s o f f the bottom of the feedstock and f a l l s i n t o the water-cooled mold to s o l i d i f y . A t y p i c a l use of the d r i p melting process i s f o r the melting and r e f i n i n g of r e f r a c t o r y metals such as Nb, Ta, Hf,and Zr, whose melting p o i n t s exceed 1900 degC. Previous workers have claimed t h a t because of the high temperature and the high vacuum environment which are created i n t h i s process, a number of m e t a l l u r g i c a l r e f i n i n g r e a c t i o n s can take place. Degassing r e a c t i o n s (CO, H^, N^) and evaporation r e a c t i o n s ( A l , NbO) from 2) the Nb i n g o t are some of these examples . Successive m u l t i p l e melting operations are of t e n c a r r i e d out i n order to produce h i g h - p u r i t y metals. 1-2-2 Hearth melting process The hearth m e l t i n g process i s used f o r producing s u p e r a l l o y s , T i and i t s a l l o y s . Fig.1-2 shows a schematic diagram of t h i s process. S o l i d or p a r t i c u l a t e feedstock i s charged i n t o one end of a hearth beneath one or more e l e c t r o n beams. The molten metal flows i n the hearth by g r a v i t y and pours i n t o a mold to s o l i d i f y . M e t a l l u r g i c a l advantages of t h i s process are as f o l l o w s : 1) Vacuum r e f i n i n g 2) Removal of non-metallic i n c l u s i o n s 3) F l e x i b i l i t y of forms of feedstock 3 E B gun Fig.1-2 Schematic diagram of EB hearth melting process. The vacuum r e f i n i n g effect i s b a s i c a l l y the same as that of the drip melting process as described previously. However, since the residence time and the reaction area are much larger, t h i s effect i s more enhanced i n the hearth melting process. Non-metallic inclusions can be removed very e f f e c t i v e l y when the metal i s traversing the hearth. High density inclusions 3) (HDI) such as TaC and WC sink to the bottom while low density inclusions (LDI) such as A^O^ i n superalloys f l o a t up to the su r f a c e ^ . In addition some kinds of inclusions such as TiN i n T i can dissolve because the melt i n the hearth has r e l a t i v e l y high superheat. As a resul t ultra-clean metal and alloys can be produced i n th i s process. Another advantage of t h i s process i s that return scrap can be d i r e c t l y fed into the hearth, unlike i n conventional processes such as VAR. Since the cost of scrap i s much lower than that of raw materials, the production cost can be reduced d r a s t i c a l l y i n th i s process. Although the hearth melting process has both the metallurgical and economic advantages described above, i t also has disadvantages. Alloying elements with high vapor pressure evaporate vigorously under highly reduced pressure. For example, the removal of Al from T i - 6 A 1 - 4 V a l l o y and the removal of Cr and Mn from Ni-base superalloys take place. I t i s not unusual to observe a 1-2% y i e l d loss after the evaporation of the primary element. In the actual process, A l i s added after EB processing, 5 while Cr i s i n t e n t i o n a l l y enriched i n the s t a r t i n g feedstock Another technique used for preventing the evaporation los s i s the beam scanning technique. In t h i s technique the electron beam i s scanned on the melt surface at a very high frequency. Since the residence time of the beam at one spot i s very short i n t h i s method, overheat of the melt can be minimized. T y p i c a l beam o s c i l l a t i o n rate used i n the actual process ranges from 0 to 1000 Hz. The d i f f i c u l t y i n the precise composition adjustment described above i s one of the p r i n c i p a l problems i n the hearth melting process. 1-3 L i t e r a t u r e review 1-3-1 Evaporation i n vacuum metallurgy In vacuum metallurgy the evaporation phenomena i s highly important. V o l a t i l e elements evaporate e a s i l y from molten metals under the very low operating pressure. This phenomena has been u t i l i z e d i n a number of m e t a l l u r g i c a l processes such as r e f i n i n g , 5) d i s t i l l a t i o n , and coating processes . Among these m e t a l l u r g i c a l 5) _17) processes r e l a t i v e l y many studies have been done about the evaporation i n the vacuum induction melting (VIM) processes which are mechanistically quite s i m i l a r to EB. In t h i s section studies on evaporation phenomena i n the induction melting process was reviewed. 1-3-1-1 Theory 6 The removal by vaporization of an element from a melt involves three basic steps. These basic steps are: Step (A): Transport of atoms from the bulk of metal to the metal/vacuum i n t e r f a c e . Step (B): Evaporation at the metal/vacuum i n t e r f a c e . Step (C): Transport of atoms or molecules i n the gas phase away from the i n t e r f a c e . Usually the condensation step can . be neglected i n VIM processes. Transport from the bulk of metal to the in t e r f a c e 18) Machlin gives the mass-transfer c o e f f i c i e n t f o r transport to the in t e r f a c e i n i n d u c t i v e l y s t i r r e d melt as: J = K L (Co - Cs) (1-1 ) where J = Flux of solute, = Liquid mass transf e r c o e f f i c i e n t given by Eq.1-2, Cs = Concentration of solute at the in t e r f a c e , and Co = Concentration of solute i n bulk metal. K L = ( 4D / TTt' )* (1-2) where D = D i f f u s i v i t y of solute and t' = L i f e time of the element moving along the int e r f a c e given by Eq.1-3-7 t 1 = r / V (1-3) where r = Radius of cr u c i b l e and V = Mean surface v e l o c i t y of solute. The basic idea of t h i s model i s exactly the same as Higbie's 19) "penetration theory" . This model has been used i n a number of studies and proved to be very reasonable. Evaporation reaction The maximum rate of evaporation from the free surface i s 19) given by the Hertz-Knudsen-Langmuir equation which, f o r an element i n d i l u t e s o l u t i o n , take the form: J = K E Cs (1-4) where = Evaporation mass tr a n s f e r c o e f f i c i e n t given by Eq.1-5. K-, = a P°y / P m (2TTRTM)^ (1-5) where a = Condensation f a c t o r , P° = Vapor pressure of pure solute, Y = Raoultian a c t i v i t y constant, p m = Molar density, M = Molecular weight of solute, R = Gas constant, T = Temperature. This equation holds true when the p a r t i a l pressure of the d i l u t e 8 elements i n the gas phase i s n e g l i g i b l y small. Transport i n the gas phase away from the in t e r f a c e The rate of mass transport i n the gas phase away from the melt surface through t h i s gas phase boundary layer can be represented by: J = K Q / RT (Ps - Pb) (1-6) where = Gas phase mass transf e r c o e f f i c i e n t , Ps = P a r t i a l pressure at the in t e r f a c e , and Pb = Pressure of gas phase. w i l l be proportional to a power of the d i f f u s i v i t y ranging from u n i t y to f r a c t i o n a l values depending on the convection i n the tank. I f the thermal convection i s unimportant, can be expressed as Eq.1-7. K Q = K Q ° / Pb (1-7) where K^ 0 = Gas phase mass transf e r c o e f f i c i e n t c h a r a c t e r i s t i c s of the system. 14.) Another form of i s represented by : K Q = D / 1 K H (1-8) where 1 = D i f f u s i o n distance from i n t e r f a c e to condensation place 9 and = Henry's constant. This step can be u s u a l l y neglected when operating pressure i s very low as i n the EB process. O v e r - a l l mass t r a n s p o r t c o e f f i c i e n t Combining Eqs .1-1 , 1-4-» and 1-6 and rearranging, we can obtain o v e r - a l l f l u x of molecules J = Co / ( 1/KL + 1/KE + 1/KQ ) (1-9) 3 O v e r - a l l mass t r a n s f e r c o e f f i c i e n t , K , i s defined as 1/KS = 1/KL + 1/K£ + 1/KQ (1-10) The v a p o r i z a t i o n of most of elements i s known to be a f i r s t -order r e a c t i o n when mono atomic v a p o r i z a t i o n i s considered. Then the equation r e l a t i n g the conc e n t r a t i o n of a v a p o r i z a t i o n element i s In (C -C f) / (C ± - C f) = K S(A/V)t (1-11) where C^ = F i n a l l y a t t a i n a b l e concentration of s o l u t e , C^ = I n i t i a l c o ncentration of s o l u t e , A = Melt surface area, and V = Volume of melt. 1-3-1-2 Studies on evaporation k i n e t i c s i n VIM processes 10 The pioneering work on the evaporation reaction i n the VIM 6) 7) was c a r r i e d out by Ward ' . He studied the pressure and the temperature dependence of the evaporation c o e f f i c i e n t i n i r o n -based melt. His r e s u l t s are summarized as follows: 3 (1) K i s independent of the pressure of the tank below 70 um; that i s , the rate c o n t r o l l i n g step then, i s both step (A) and step (B). (2) At lower temperature the predominant control step i s step (B), whereas at higher temperature i t i s step (A) under the low pressure. His approach f o r determining the rate c o n t r o l l i n g step has been used i n other researcher's l i t e r a t u r e s . n. , , 8),9),11 ),12) . . . Ohno et a l . investigated the evaporation behavior of Mn, Cu, Sn, S, Cr from the iron-base a l l o y i n the -5 -6 VIM (10 - 10 atm). They showed that the evaporation rate of each a l l o y i n g element follows f i r s t order k i n e t i c s and the s p e c i f i c evaporation constant decreases i n the following order of element: Mn>Cu>Sn>S>Cr. In the Fe-Si-S a l l o y s , the constant increased with the increase of s i l i c o n content and there was a l i n e a r r e l a t i o n s h i p between logK and % S i . S 7) Metal s t i r r i n g e f f e c t on K was also studied by Ward ( Fe 8) 9) 11) 12) system) and Ohno ' ' ' (Fe and Cu system). In t h e i r 11 experiments the s t i r r i n g i n the melt was reduced by inserting a graphite crucible between the refractory and the c o i l . I t was 5 cle a r l y shown that K i n shielded charges i s smaller than that i n unshielded one. However no quantitative discussion was carried out. "15) "16) 2 2 ) Harris ' ' studied the evaporation of B i , As, and Sb from copper melt. In order to calculate he derived an empirical equation on the basis of Eq.1-2 considering the temperature dependence on D and V. K L = (8DV/TTr)* = C (A /V) * r * T exp(-C2/T) f* (1-12) where' C^  , C^ = Constant and f = Characteristic frequency of induction furnace. This equation was derived using Eq.1-13 and Eq.1-U. = DQ exp(- E D / RT) (1-13) V = V Q [T^ (A/V) f]* L 2 (1-U) where = Temperature independent factor r e l a t i n g D to T, E^ = Activation energy for d i f f u s i o n , V^ = Temperature independent factor of V to T and (A/V). This i s an interesting r e s u l t since Eq.1-12 indicates that the melt mass transfer c o e f f i c i e n t increases as the melts radius 12 increases and that melt temperature i s a s i g n i f i c a n t f a c t o r i n induction s t i r r e d melt phase mass transport. He calculated by-using Ks, Kg and K^, to propose an experimental expression of K Q = a + b (OPR) (1-15) where a, b = Constant and OPR = ( t o t a l i n i t i a l melt vapor pressure)/(chamber pressure). 17) Ozberk studied the evaporation of B i , Pb, As, and Sb from copper melt. K^ was calculated from Eqs.1-2 and 1-3> i n which V was determined to have f i x e d value of 10 cm/s. K^ was calculated g from K , Kg, and K^, and concluded that the evaporation rates were l a r g e l y c o n t r o l l e d by step (C). Barnhurst investigated the evaporation of Mn from cast i r o n . K^, Kg, K^ value were calculated from Eqs.1-2, 1-5» 1-7 r e s p e c t i v e l y . These r e s u l t s were much smaller then experimentally obtained K . He assumed t h i s was because condensation f a c t o r , _2 was l e s s than 1, that i s 10 order. This can be true i n a large furnace used i n his experiments. 10)13) Irons ' showed a comparison between experimentally measured vaporization rates of Mg from an i n d u c t i v e l y s t i r r e d melt, and those predicted on the basis of a numerical so l u t i o n of the turbulent (Navier-Stokes equation) electromagnetically driven f l u i d flow equation (Maxwell's equation). He assumed that step 13 0 (A) was a rate c o n t r o l l i n g step i n the p a r t i c u l a r system. V i n 20) Eq.1-3 was calculated by the r e s u l t s of Szekely's report . obtained from the experiment was 8.81 x 10 cm/s, which agreed well with t h e o r e t i c a l l y calculated r e s u l t , 7.1 x 10 cm/s. Oxygen and s u l f u r are known as surface active elements to i n t e r f e r e the rate of the evaporation of the elements. For example, the e f f e c t of the dissolved oxygen on rates of evaporation of l i q u i d i r o n and copper was studied by Hayakawa et 21 ) a l . They showed that the rates of evaporation of l i q u i d i r o n and copper decreased remarkably with increasing oxygen at the l i q u i d i n t e r f a c e . The evaporation rate of Fe was expressed as a function of [%0] as follows: J F e = P (M/2TrRT)*{l-0.57(K a d[%0])/(l+K a d[%0])} (1-16) where K ^ = Adsorption c o e f f i c i e n t represented as a function of the temperature. Since the a c t i v i t y c o e f f i c i e n t of these elements i n T i i s much smaller than i n i r o n , the e f f e c t of these elements on the rates of evaporation i n the T i system i s considered to be correspondingly smaller. As described above the evaporation model developed i n various induction melting processes i s r e l a t i v e l y well established and the phenomena can be explained reasonably. Since these theories deal with the evaporation phenomena i n the molten U pool system , b a s i c a l l y they could be applied to the evaporation phenomena i n the EB melting process. 1-3-2 Process modeling of EB hearth melting process Like other m e t a l l u r g i c a l processes, i t i s beyond dispute that the development of an excellent process model enables the i n t e l l i g e n t control of the chemical compositions i n the EB hearth melting process. However, because of the complexity of t h i s process, only a few works on t h i s matter have been reported to H ; 23)-30) date 27) Herbertson examined evaporation reactions i n the EB hearth melting process by r e l a t i n g the evaporation rate constant, kg, calculated by Langmuir equation to the o v e r a l l rate constant, kg, obtained from the actual operation. A well-known r e l a t i o n s h i p between kg and k^ was found c l e a r l y i n his study, that i s , with small k-g (as i n the example case of Cr i n s t a i n l e s s steels) kg becomes nearly equal to k^. As a r e s u l t k^ can be predicted j u s t by c a l c u l a t i n g kg only. Although t h i s analysis can be applied to a rough estimation of the solute evaporation l o s s , the precise chemical c o n t r o l cannot be made because only a macroscopic mass balance was taken i n t h i s treatment. Since the hearth i n the actual process i s r e l a t i v e l y stable from a convective viewpoint, a v a l i d d e s c r i p t i o n of the hearth thermal regime allows the estimation of the mass-flow 28) variables associated with the process. Kheshgi and Gresho 15 t r i e d to make an unsteady-state heat t r a n s f e r model, but t h e i r 29) c a l c u l a t e d r e s u l t s were not converged. Tripp s u c c e s s f u l l y developed a 3-D steady s t a t e heat t r a n s f e r model i n the EB hearth by assuming the hearth has a f i x e d time-averaged temperature. The maximum evaporation l o s s of T i and A l were a l s o estimated i n h i s study. However, si n c e an e l e c t r o n beam i s scanned on the melt surface r a p i d l y i n a c t u a l processes, the problem of an unsteady-s t a t e heat t r a n s f e r , i n other words, the e f f e c t of the beam scanning on the l o c a l surface temperature, must be taken i n t o account i n i n t e r p r e t a t i o n of the evaporation r e a c t i o n i n t h i s process. 1-3-3 Moving heat source problem The .subject d e a l i n g w i t h the process i n which heat i s su p p l i e d from one or more moving heat sources as i n the present case i s c a l l e d "Moving Heat Source Problem". Many s t u d i e s have been made to date p a r t i c u l a r l y i n the f i e l d of welding metallurgy, where various kinds of heat sources such as gas a r c , e l e c t r o n beam, plasma gas, and l a s e r are used f o r j o i n i n g m a t e r i a l s . Both a n a l y t i c a l and numerical treatments were a p p l i e d to the p r e d i c t i o n of m e t a l l u r g i c a l r e a c t i o n s and m i c r o s t r u c t u a l t r a n s f o r m a t i o n i n the weld puddle and the heat a f f e c t e d zone. 16 The exact a n a l y t i c a l theory was developed by Rosenthal i n 31 ) the early 194-0's . The major assumptions made i n h i s theory were i ) quasi-stationary heat flow, i i ) a point heat source to represent the weld arc, i i i ) neglect of convection and r a d i a t i o n e f f e c t s , i v ) two dimensional heat flow, v) neglect of heat of transformation, v i ) temperature independent material properties. Using a coordinate system shown i n Fig.1-3, the basic d i f f e r e n t i a l equation was represented by Eq.1-17. 3 2T/85 2 + 8 2T/3y 2 + 3 2T/3z 2 = - (v/ct) 3T/3? (1-17) where a = heat d i f f u s i v i t y , v = v e l o c i t y of heat source ^ = x - vt moving coordinate Many researchers applied h i s theory to the temperature analysis i n various welding processes and confirmed that t h i s was an excellent mathematical model as a f i r s t approximation even 32) though many assumptions were included Since Rosenthal's theory assumes a point or l i n e heat source, i t cannot provide any information concerning the shape of the weld pool. To improve t h i s , recent studies presented a more general s o l u t i o n of a t r a v e l i n g d i s t r i b u t e d heat source, assuming 33) the heat source has a Gaussian d i s t r i b u t i o n . Eagar examined a t r a v e l i n g d i s t r i b u t e d heat source problem by solving a modified 17 z 18 Rosenthal's equation a n a l y t i c a l l y and showed t h a t the t h e o r e t i c a l p r e d i c t i o n s were i n good agreement w i t h experimental r e s u l t s on carbon s t e e l s , T i , and A l . The recent progress of the numerical approaches f o r s o l v i n g a governing equation makes i t p o s s i b l e to d e a l w i t h a more complex m e t a l l u r g i c a l process. The e f f e c t of the convection and the r a d i a t i o n heat l o s s i n the weld pool can be taken i n t o c o n s i d e r a t i o n simultaneously by using a numerical method and more accurate p r e d i c t i o n becomes 34) p o s s i b l e . For example, Kou developed a three-dimensional heat t r a n s f e r model i n c l u d i n g the e f f e c t of the c o n v e c t i o n a l f l o w i n the case of both the moving l a s e r and the moving arc welded metals to f i n d out the good agreement of experimental r e s u l t s w i t h observed ones. Although the b a s i c idea can be a p p l i e d to the heat a n a l y s i s i n the EB melting processes, there are some major d i f f e r e n c e s between two processes. F i r s t , the v e l o c i t y of the heat source i n the EB melting process i s much higher than that employed i n the welding processes. Second, since the whole metal i s molten s t a t e i n the EB melting processes, the r a d i a t i o n and evaporation heat l o s s cannot be neglected and must be considered not only i n the beam impinged area but a l s o i n the r e s t of sample surface. Because of the above reasons, n e i t h e r the a n a l y t i c a l or the numerical treatment developed i n the f i e l d of the welding 19 processes can be a p p l i e d d i r e c t l y to the present case. 1-4- Objective of t h i s study From an o p e r a t i o n a l p o i n t of view, the i n f o r m a t i o n concerning the optimum beam o s c i l l a t i o n r a t e i s very important. However, due to the l a c k of fundamental s t u d i e s as mentioned above, very l i t t l e i s known on the optimum beam o s c i l l a t i o n r a t e . Consequently melters tend to adopt higher beam o s c i l l a t i o n r a t e s ( t y p i c a l l y around 100 Hz) which r e q u i r e an a d d i t i o n a l expensive beam c i r c u l a t i n g and c o n t r o l l i n g system equipment. The o b j e c t i v e of t h i s study, t h e r e f o r e , i s to i n v e s t i g a t e the e f f e c t of the beam o s c i l l a t i o n r a t e on the evaporation behavior of a l l o y i n g elements by developing an unsteady s t a t e heat t r a n s f e r model, which w i l l a s s i s t i n c o n s t r u c t i n g a more general model of the EB hearth melting process i n the f u t u r e . In t h i s study, A l evaporation r e a c t i o n s from Ti-6A1-4V a l l o y , which i s used as an aerospace engine component, were i n v e s t i g a t e d . One of the reasons why t h i s a l l o y was chosen i s tha t the a c t i v i t y c o e f f i c i e n t of A l i n T i - A l - V , Y ^ l ' w n i c ^ a very important constant i n the a n a l y s i s of evaporation r e a c t i o n s , 30) was already determined by Takagi's study . In s e c t i o n 2, experimental procedure and r e s u l t s of the small EB melting t e s t s w i l l be described. In s e c t i o n 3» a d i s c u s s i o n of the optimum beam o s c i l l a t i o n r a t e i n the EB melting process w i l l be made by 20 constructing a mathematical model on the basis of the fundamental results obtained i n section 2. 21 S e c t i o n 2. EXPERIMENTAL PROCEDURE AND RESULTS 2-1 Experimental apparatus The UBC's 30kW EB furnace shown i n Photo.2-1 was used i n t h i s study. Fig.2-1 shows a schematic diagram of the experimental apparatus. This apparatus mainly c o n s i s t s of the f o l l o w i n g p a r t s : Power supply system EB gun Beam r o t a t i n g device High vacuum system Water-cooled mold I n - s i t u temperature measurement system The power supply system was sup p l i e d by NHE (NORTH HILL ENTERPRISES INC.). The maximum power output i s 37.5kw. The voltage (and the current) of the f i l a m e n t , the e x c i t e r and the beam can be c o n t r o l l e d by a manual adjustment of d i a l s on the c o n t r o l l i n g panel. The EB gun made by Von Ardenne (EH-30/20) was used i n t h i s study. The nominal power i s 30kW when the a c c e l e r a t i n g voltage i s 35) 20kV. A w i r i n g diagram i s given i n Fig.2-2 . The heated 22 Photo 2-1 Appearance of 30 kW EB furnace at UBC. 23 Mechanical Diffusion Pump Pump Power Supply [Waveform Genera to r ] Chamber ' n F lowmete r To Mold The rmome te r Fig.2-1 Schematic diagram of experimental apparatus. 24 -cr°v DC Anode - "\ J 0 - -20KV, - 1 . 5 A Solid Cathode I o< DC J 0 - - 1 KV, - 0 . 5 A Filament I I o< A C J 0 - - 5 V , 0 - -25 A Filament III Fig.2-2 Wiring diagram EH 30/20. 25 tungsten f i l a m e n t gives o f f e l e c t r o n s , which bombard the tungsten s o l i d cathode. This bombardment causes the s o l i d cathode to be heated so t h a t high d e n s i t y e l e c t r o n beams can be generated. Next, the e l e c t r o n beams emitted are a c c e l e r a t e d i n an e l e c t r o s t a t i c f i e l d to a high l e v e l of k i n e t i c energy and d e f l e c t e d by magnetic l e n s e s . By c o n t r o l l i n g the strength of the magnetic f i e l d f o r both X & Y d i r e c t i o n s , v a r i ous beam t r a j e c t o r i e s , or beam patterns can be obtained. S i g n a l s given to magnetic lenses were produced by a wave form s y n t h e s i z e r board (WSB-10 made by QUATECH INC.), which were i n s t a l l e d i n the IBM-PC. This device can give output s i g n a l s w i t h d e s i r e d frequencies according to the user's own program. Two boards were employed i n t h i s study; one r e f e r r e d to as a 'Master' board f o r X d i r e c t i o n i n a magnetic l e n s , another r e f e r r e d to as a 'Slave' board f o r Y d i r e c t i o n . A program generating a sine wave f o r a 'Master' board and a cosine wave f o r a 'Slave' board was made f o r each beam o s c i l l a t i o n r a t e (0.1, 1.0, 10.0 and 50.0 Hz). The s i g n a l s generated were magnified by the a m p l i f i e r and sent to magnetic lenses. The amplitude of these s i g n a l s can be c o n t r o l l e d manually. Two independent vacuum systems were employed i n t h i s furnace; one f o r the gun chamber and the other f o r the melt chamber. In the gun chamber a vacuum environment i s r e q u i r e d f o r beam generation and c o n t r o l . The maintenance of a vacuum environment i n the melt chamber i s a l s o necessary to prevent excessive s c a t t e r i n g of the beam. I t i s usu a l to use two 26 separate vacuum systems to guarantee the high vacuum i n the gun chamber without the sudden pressure r i s e caused by the gas evolution from the melt chamber. Both of the vacuum systems consist of an o i l d i f f u s i o n pump and a mechanical pump. The vacuum achieved by t h i s system i s <1x10 Torr f o r the gun _5 chamber and <5x10 Torr f o r the melt chamber. A water-cooled copper mold was used as a c r u c i b l e . Fig.2-3 shows a drawing of the mold. Water flow rate i n t h i s system was approximately 30 l/min. Two thermometers monitoring the i n l e t and o u t l e t water temperature were used i n order to estimate the heat loss through the mold during melting. Fig.Z - U shows a schematic of the temperature measurement set-up. An IRCON 1100 i n f r a r e d o p t i c a l pyrometer was used f o r the i n - s i t u temperature measurement of the molten metal. A sp e c i a l set-up was i n s t a l l e d to protect the window glass from being coated during evaporation of the metal. A c y l i n d r i c a l s h i e l d (20 cm diameter and 28 cm high) was placed on the mold to condense the metal vapor on an inner w a l l . A small hole (6 mm diameter) was d r i l l e d i n order to allow observation of the molten metal surface from the pyrometer set outside the chamber. Furthermore, a copper tube (2 cm diameter) was i n s t a l l e d to avoid glass coating. The observation angle and s i t e was adjusted to be approximately 60 degree and 3 cm from the center of the melt surface. The mV output from the pyrometer was recorded using a recorder f o r temperature a n a l y s i s . 27 Fig.2-3 Water-cooled copper mold. ( u n i t : mm ) 28 29 Temperature can be read o f f the c a l i b r a t i o n curve shown i n Fig.2-5. The (Output% of f u l l scale) i s calculated using Eq.2-36) 1 , i f the emi s s i v i t y of the molten metal of i n t e r e s t i s known, (measured mV) x 100 = (0utput% of f u l l scale) (2-1) 50 x (emissivity) 2-2 Experimental procedure The e m i s s i v i t y i n t h i s p a r t i c u l a r furnace was estimated i n the following way. The emi s s i v i t y mentioned here means the t o t a l e m issivity correction which includes any factors such as the observation angle and the window glass absorption. The rearrangement of Eq.2-1 gives Eq.2-2. 2 x (measured mV) (emissivity) = (2-2) (Output% of f u l l scale) To obtain the em i s s i v i t y of molten T i the l i q u i d u s temperature (1667 degC) was used as a 'known' standard temperature. The' (0utput% of f u l l scale) at l i q u i d u s temperature can be obtained from Fig.2-5. The measurement of the mv output of pyrometer allows the c a l c u l a t i o n of the em i s s i v i t y of the molten T i . The cooling curve was obtained by melting pure Ti(CP-Ti) blocks using the temperature measurement set-up described before. Table 2-1 shows the experimental conditions used i n t h i s 30 Ill _J < o (/) _J _l D Li. LL O ^ D 0. r-D O 100 90 80 70 60 50 40 30 20 10 9 8 7 6 5 4 1 TE • M P =:RA T U R F C A L U 3RA1 H O N / 1100 S E R I E S / ) 1 Ul V I M 1 RAI I O i» M G E 210 0°C _ r"l r K u r / / / IRC ON, INC 00 12 00 13 00 14 00 1500 16 00 1700 18 00 1900 20 00 2 00 2200 23 00 2400 25 Temp. (°C) Fig.2-5 C a l i b r a t i o n curve of the pyrometer, IRCON 1100, 31 Table 2-1 Experimental conditions. EB power Beam travelling radius Frequency Time 15.5Kw (18KV-0.86A) 1. 5cm 0, 0.1, 1, 10Hz 0', 1', 5', 10' 32 study. Four pieces of Ti-6A1-4.V a l l o y (50 mm x 55 mm x 25 mm (1), 60 mm x 60 mm x 15 mm (1), 10 mm x 15 mm x 4-5 mm (2) ) were cut from a large round ingot and placed i n the mold. Fig.2 - 6 shows the arrangement of the charged materials. Both the melt and the gun chamber were evacuated to a determined vacuum pressure. Since rapid power input causes the vigorous evolution of the gas from charged materials which deteriorates beam generating conditions, a preheating and premelting operation was necessary. The standardized heating-up pattern i s shown i n Fig.2-7. The beam spot was moved around on the charged material during an 11kV period. The molten pool produced was then observed to have a diameter of 1.5-2cm and the other portion remained unmelted. Since the amount of gas i n the charged material varied from one to another, i t was d i f f i c u l t to keep the melting pattern constant. After t h i s premelting period the beam was set back to the center p o s i t i o n and the power was raised to 15-5 kW as quickly as possible. Beam o s c i l l a t i o n rates chosen were 0, 0.1, 1.0, 10.0 and 50.0 Hz. In the case of the 0 Hz experiment the beam was set to the center of the charged material, whereas i n the case of moving beam experiments the beam t r a v e l i n g radius was adjusted to be 1.5 cm. Although beam spot sizes could not be measured p r e c i s e l y , they appeared to be 1-2mm i n diameter. The molten surface was observed by seeing an image r e f l e c t e d i n a glass set i n s i d e the vacuum chamber. Fig.2-8 shows a sketch of specimen during t e s t s . 33 Fig.2-6 Schematic drawing of arrangement of charged material. 34 2 5 2 0 1 5 1 0 5 0 0 . 8 0 . 6 0 . 4 0 . 2 0 2 0 1 5 1 0 5 0 0 . 1 A 0.15A 7 K V 9 K V 15.5KW 2 . 2 K W 0 . 2 0 A 0 . 8 6 A ' Beam Moved Power Raised Experiment 11KV 1 8 K V 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 Time (sec) After a determined holding time ( 1 , 5 and 10 min.) the power was turned o f f . Using a cooling curve obtained from each experiment , the emissivity was calculated f o r a l l charges i n the same manner as was described before. A l l samples were polished using a 6um diamond polis h e r and analyzed by SEM/EDX, where X-radiation was c o l l e c t e d f o r 200 seconds. Fig.2 -9 shows a t y p i c a l SEM/EDX r e s u l t . Gaussian deconvolution was car r i e d out to integrate each element's Ka count. Analysis was done three times at the same point (6mm below the surface along the center l i n e ) and the averaged value was calculated. To investigate A l d i s t r i b u t i o n i n the pool, the whole pool area was analyzed i n several samples. Six standard samples were analyzed by emission spectroscopy (ca r r i e d out by OREMET Co.). F i g . 2-10 shows the r e l a t i o n between %A1 by emission spectroscopy and the r a t i o , I ^ l ^ T i ^ a count of A l to Ka count of Ti) obtained by SEM/EDX. As can be seen i n t h i s f i g u r e , good c o r r e l a t i o n was found. An experimental equation was obtained by regression analysis as follows; wt%Al = 19590.6R3 -1034-2.1R2 + 351 .74-R + 0.01879 (2-4) R = I A 1 / I T i ( 0 < A l < 6.5 % ) Table 2-2 shows d e t a i l e d r e s u l t s . Metal which deposited on the inner wall of the s h i e l d was 37 CO CD © Ii") © U J CO cn trj I © O.J OJ 00 o 111 w © © O.J II : Ii o -H H : 1L1 W : (•'j Q. • U 'ti : i _ •—i : a.. U : m +-•• i ~ o o © © © in -p 111 CO z> • 1 1 Fig.2-9 T y p i c a l r e s u l t of SEM/EDX an a l y s i s . 38 R = U l / l T i {% AO = 195903.6R3-10342.1R2 + 351.74R + 0.01879 7 . 6 5 < 4 S 3 2 1 / / / / / / / / s / / / i — « H /£m—i / / / / / / / / / / / / / / / / / / / J I L 0 1 2 3 X10 l A l / l T i - 2 Fig.2-10 Relation between wt%Al and 1^/1^. 39 Table 2 - 2 Wt%Al analyzed by emission spectroscopy and obtained by SEM/EDX. Sample # w t % U l / l T i <DATA> Al V 27 0.83 5.30 0.00235 0 . 0 0 1 7 0 2 8 4 2 2 5 0 . 0 0 3 1 9 2 3 5 1 7 8 23-2 2.87 4.80 0.0118 0 . 0 1 1 2 1 2 2 0 . 0 0 1 2 0 23-1 3.97 4.54 0.0181 0 . 0 2 0 3 1 7 3 0 . 0 0 1 6 8 21 4.00 4.51 0.0152 0 . 0 1 6 0 1 3 5 0 . 0 0 1 6 3 0-2 6.26 4.38 0.0291 0 . 0 2 9 5 2 8 4 0 . 0 0 2 9 4 30 6.28 4.40 0.0286 0 . 0 2 9 1 2 8 8 0 . 0 0 2 7 9 4-0 also analyzed to check the composition of the evaporant. In t h i s case "Standardless EDX analysis technique" method (ZAF corrections v i a MAGIC V), one of the software routines incorporated i n the EDX was used f o r c a l c u l a t i n g each wt%, since A l concentration was lar g e r than the upper l i m i t of the above experimental equation. 2-3 Results Fig.2-11 shows a t y p i c a l r e s u l t of a cooling curve of pure T i . A d i s t i n c t h o r i z o n t a l l i n e corresponding to T i li q u i d u s temperature was obtained. From t h i s r e s u l t e was calculated to be 0.116. Since s i m i l a r cooling curves were obtained a f t e r a l l experiments, each £ could be calcu l a t e d . A t y p i c a l c value was i n the range of 0.11-0.13-Fig.2-12 shows temperature changes with time. In the case of a 0 Hz experiment, the temperature fluctuated within approximately ±20 degC. Temperature cycles found i n 0.1 and 1.0 Hz experiments are 10 sec. and 1 sec. res p e c t i v e l y , which are i n good agreement with expected time cycles. No temperature cycles were detected i n the 10.0 Hz and 50.0 Hz experiments, since time cycles are very short (< 0.1 sec). Since there appears to be no large dif f e r e n c e i n the temperature behavior of the 10.0 Hz and 50.0 Hz t e s t s , 50.0 Hz experiments were not ca r r i e d out. A l l r e s u l t s obtained i n t h i s study are shown i n Table 2-3, which includes chemical compositions of raw materials and 41 0 Time S t a r t 1.78mV 44mV Time 0 Hz Stop S t a r t 1.46mV 1.42mV Stop Time 0.1 Hz - .. Fig.2-12-a) Temperature changes with time 0 Hz and 0.1 Hz. 43 4 Start 2.44mV 1.46mV Time 1 .0 Hz Stop Start Stop Time 10.0 Hz Fig.2-12-b) Temperature changes with time 1 .-0 Hz and 10.0 Hz. 44 Start Time 50.0 Hz Fig.2-12-c) Temperature changes with time 50.0 45 Table 2-3 A l l r e s u l t s obtained i n t h i s experiment. NOTE after 3' 15.5kw * eccentric SEMI QUANTITATVE ANALYSIS /—\ <t 4.77% 5.77% a * s -ci-C D — C O C O C O C M 5.26% 3.82% 0.97% a a a C D C M C D C O C O C O C M « — ^ ^ ^ C D eo C O L O C O ^J- C O C M C O C O Al W/o Ti W/o V W/o 28 68 4 DEPTH CENTER EDGE (0)-8 C D O — oo — 1 1 1 co co r— OQ C O C O 1 1 1 co r— C D * — C D 1 1 1 — iO C M C D C D C D 1 1 1 < D cr> to 1 Al W/o Ti W/o V W/o 28 68 4 WATER < oco L O iO C O C O — •^ r L O co C M C D C O C O C O cn — C D ^J. C O C O WATER TOUT Q C D C D C O C M C O cz> — L O r-~ r-» C D O O co co r-— O C O C O L O co r«-WATER TIN J C M L O o r-^  o C M O C D C M — C O C O C D C M « — «— r-» oo C D C M — WATER m t rt Lu 29.4 30.2 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.4 29.0 29.4 WEIGHT (g) < 32.4 18 46.4 C O co — C O C O 15.7 30 57.0 14.1 26 41.5 WEIGHT (g) Wf 762.5 661.0 i AAA OUH 653.2 591 585.4 697.4 541.4 732.7 fi 1 0 0 1 z 547.5 WEIGHT (g) Wo 624.4 / 00 . L 693.4 699.6 605 640.2 713.1 598.4 746.8 000 589 PREHEAT & MELT TIME o C D C D C D co t>-C D r-- C D C O 1"*— co co r^ . 650" 645" 640" C D C D C D C D C O ^ J " C O C O C O C M O C D C O C D L O C O C O f-* 1 POWER KW L O C D C D L O L O L O m m r-- r-- co L O L O L O C M t*- co L O L O L O co r-— r— L O L O L O POWER < 0.86 i n 1 . u 0.80 0.85 0.88 n Q7 U. 0/ 0.90 0.80 U. 0/ 0.90 0.85 U. of 0.88 POWER KV co ir^  cn co C O O C D r- oo C D C D C D C D oo r*-C D C D O O C O O O f-— TIME C D C D io co — UTJ C D «—• L O C D — L O O HZ o C D - C D Raw Material Deposition Mark C O — C M - W < N CsJ N CNI C O « — L O cn C M eo C D co r*- Raw Material Deposition 46 deposited metal on an inner wall of the s h i e l d . Fig.2-13 shows the mV output obtained from the pyrometer and calculated temperature during melting. In order to estimate temperature, an emissivity, which was also shown i n the same f i g u r e , was calculated using a r e s u l t of a cooling curve obtained a f t e r each experiment. While the mV output sc a t t e r i n g i s shown i n the case of 0 Hz and 10.0 Hz experiments, the maximum and minimum mV output are shown i n the case of 0.1 and 1.0 Hz experiments corresponding to time cycles observed c l e a r l y i n F i g . 2-12. The output mV decreases with time; the "decreasing r a t i o " i s around 14.-20% at 5 min, around 20-26% at 10 min. This indicates that coating of the glass window could not be avoided completely. Fig.2-14 indicates changes i n AT (=difference between the temperature at the out l e t and at the i n l e t ) of cooling water. After 2 min a constant AT (= 6 degree C) was obtained regardless of the difference i n beam o s c i l l a t i o n rates. The steady state heat trans f e r condition appeared to be attained a f t e r 2 min i n th i s experiment. Photo 2-2 shows pool p r o f i l e s of samples. As can be seen c l e a r l y , 1 min samples have r e l a t i v e l y shallow pool depth, that i s , the pool was not f u l l y developed. This agrees well with the data of AT mentioned above. The pool depth of the other samples was more than 15 mm. In some samples, however, the p o s i t i o n of the deepest pool depth was not centered because the 47 0 Hz 0 . 1 H z 3.00 -^ 2.52 > £ 2.00 8> 1.50 CD ° 1.00 H-18 2.48 A2 * 1% • 3 . 0 1.44 £ = 0.128 2 . 0 -1.0 H-11 J 1 = 14% 2.1 J« = 18% 1.46 1 80 E = ^*°^—Lj~\ 0.133 i i i i 1.20 3.0 2.0 -1.0 H -15 J' = 27% J ! = 10% 1.76 0 60" 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 1900 H-18 1800 E 1700 v 1— 1600 1823 -1900 1800 1700 H-11 1600 1730 1645 -19001-1800 -1700 1600 -H -15 1740 60' T i m e (min) 1 2 3 4 5 T i m e (min) 0 1 2 3 4 5 6 7 8 9 10 T i m e (min) 0.1 Hz .2-13-a) Changes i n mV output and calculated temperature during experiment 0 Hz and 0.1 Hz. 48 1Hz > 3.0 2.0 1.0 H-19 -t = 0.133 0 60" 0 ' 2 3 4 5 6 0 1 2 3 4 5 6 1900 ^ 1800 o I 1700 l-1600 H-19 1900 1795 TTio 1705 1800 1700 1600 0 60" Time (min) H-22 1900 1800 .1700 1600 6 7 8 9 10 H-16 1710 1660 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 Time (min) T i m e (min) 8 9 10 1 .0 Hz 10Hz 3.Oh > J. 2.0 1.0 -H-20 , £ = 0.128 3.0k 2.0 1.0 H-13 3.0 2.0 .1.0 H-17 :1.23 £ = 0.094 0 65" 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 9'20" 1900 _ 1800 o % 1700 I-1600r H-20 1900 18001 1745 I 1700: 1600 H-13 1900 H-17 - 1800 — • —^ 1750 - 1705. 1720 i690 ' 7 0 0 : 1728 1705 - 1600 i_ i i 1 1 1 1 1 1 1— 0 65" 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 9'20" Time (sec) Time (min) Time (min) 10.0 Hz Fig.2-13-b) Changes i n mV output and calculated temperature during experiment 1.0 Hz and 10.0 Hz. 49 OHz O O I-8 6 4 2 0 2* 4. 2 4 6 8 10 Time (min) -ft X K X -I 1 1 [_ 2 4 6 8 10 Time (min) 1Hz • 19 o|2 x 16 p I--A p - 9 * -X X J I ' ' ' 2 4 6 8 10 Time (min) o * — x -I 1 1 I I L. 2 3 4 6 8 10 Time (min) 0.1Hz 18 o i l x 15 10Hz • 20 o!3 x17 Fig.2-U Variations of AT (T . - T. ) of cooling water with b , . out i n time. 50 ( ZH L'O ) sa-rdures j o uox^oes ggojrj (q-Z"Z °10l{d uxui OL (° nospitai Supply Canada Inc. hospitalier American du Canada 111 , i|2 uxra c, (q Canlab American Hospital Supply Canada Inc Materiel hospitalier American du Canada inc III , Ii2 , I|3 Photo 2-2-c) Cross section of samples ( 1.0 Hz ). 53 Canlab® Hospital Supply Maienei' * hospitaller beam could not be centered accurately. Therefore care was taken i n the i n t e r p r e t a t i o n of these r e s u l t s . Fig.2-15 indicates the d i s t r i b u t i o n of %wtAl i n the pool obtained a f t e r 5 min experiment. In a l l samples %wtAl was almost constant throughout i n the pool. In Fig.2-15-a) %wtAl on a metal surface i s also shown. There i s no large difference between A l concentration on the surface (3.23%) and that i n the pool (3.96%). These r e s u l t s a l l imply that the evaporation reaction was mainly c o n t r o l l e d by the in t e r f a c e reaction under the experimental conditions. Fig.2-16 shows the change i n the weight loss with time. The weight l o s s increases almost l i n e a r l y with time; 4-0-60g of the samples was l o s t i n 10 min. The e f f e c t of beam o s c i l l a t i o n rate i s also observed i n the present experiment. Fig.2-17 shows wt%Al changes with time. Fig.2-18 shows wt%Al at 5 min and 10 min. After 10 min experiments, A l was reduced to 1-2.5% i n a l l experimental conditions. There appears to be no large dif f e r e n c e between the beam o s c i l l a t i o n rate changes as expected. After each experiment a r e l a t i v e l y large amount of evaporant was deposited on the inner wall of the s h i e l d . A deposit sampled a f t e r a 0 Hz experiment was analyzed by the method described previously. As seen i n Table 2-3 the composition of the deposit 55 45mm 10,|J0 .|X1 J Pool C e n t e r 6 +-> 5 3.96% 3.23% (#) 3 < O N +-> 5 0 3 6 9 12 16 Dis tance F rom Sur face (mm) 20mm from C e n t e r 4.11% - A A -< 4-> 5 0 3 6 9 12 14 Dis tance F rom Su r face (mm) 5 -0 6 -10mm from C e n t e r 3.96% O- -O 3 6 9 12 16 Dis tance F rom Sur face (mm) 30mm from C e n t e r 4.00% -x 0 3 6 910 Dis tance F rom Su r face (mm) F ig.2-15 - a ) D i s t r i b u t i o n o f wt%Al i n t h e p o o l ( 0 H z ) , 56 45mm Poo l C 5 Centerline 3.55% j i i_ '0 3 6 9 12 15 17 Distance From Surface (mm) 5 5 H-11 (0.1Hz-20mm from Center 3.29% — O — 0 3 6 9 12 14 Distance From Surface (mm) 5 3, -40mm from -Centerline " 3.37% A i i 3 6 Distance From Surface (mm) F ig.2-15 -b ) D i s t r i b u t i o n o f wt%Al i n t h e p o o l . ( 0.1 Hz) 57 45mm Pool Centerline 2.38% • r _i i_ < 6 9 12 Distance From Surface (mm) 20mm from Centerline 2.54% -A. A £ •0 3 6 910 Distance From Surface (mm) < 5 t; 4 H-12 (1Hz~5) 10mm from Centerline 2.41% 6 9 13 Distance From Surface (mm) 2.38% X 30mm from Centerline Distance From Surface (mm) Fig.2-15-c) D i s t r i b u t i o n of wt%Al i n the pool. ( 1.0 Hz) 58 20mm , 20mm Pool 20mm H-13 (10Hz~5) < 5 5 Cente r 3.79% _] I L. J0 3 6 9 12 15 18 Distance From Surface (mm) 6h < 5 5 20mm left from Cente r 3.81% 0 3 6 9 12 Distance From Surface (mm) 40mm left from Center 3.82% 0 3 5 Distance From Surface (mm) < O N 3.91% J I L 0 3 6 9 Distance From Surface (mm) F ig.2-15 -d ) D i s t r i b u t i o n o f wt%Al i n t h e p o o l . ( 10.0 Hz) 59 to o lap '(U T i m e ( m i n ) Fig.2-16 Changes i n t o t a l weight loss with time. 60 7 3 0 •pre-melt A ® • OHz A 0.1Hz ® 1Hz x 10Hz J 1 I L 2 4 6 Time (min) A J L_ 8 10 Fig.2-17 Changes i n wt%Al during experiment. 61 0.1 1.0 10.0 Hz Fig.2-18 E f f e c t of beam o s c i l l a t i o n rate on wt%Al at 5 min and 10 min. 62 was Al=28%, Ti=68%, and V=4%. This indicates T i and V also evaporate heavily along with A l i n t h i s experiment. The r a t i o of % (Ti + V) to %A1 i s around 2.6. A l was highly concentrated i n the deposit because of i t s high vapor pressure. 63 Section 3 MATHEMATICAL MODEL 3-1 Formulation 3-1-1 Basic equation As described i n Section 2, the shape of the molten pool observed i n t h i s study was almost i d e n t i c a l when the beam was positioned at a center, of the sample, and hence i t can be regarded as a cylinder with approximately 9 cm diameter and 1.5 cm depth. Since the molten pool was r e l a t i v e l y shallow i n the experiment (the r a t i o of diameter to depth i s around 6.7) the moving beam problem can be described as a two dimensional heat flow model and the stationary beam problem as a one dimensional heat flow model. Assuming material properties such as heat conductivity, heat capacity, and density are constant, a governing equation i n a c y l i n d r i c a l coordinate system can be expressed as follows: 1 3 3T 1 3 2T q ' 1 3T _ _ _ ( r _ _ ) + _ _ + _ = _ _ _ ( 3 _ U r 3r 3r r 2 38 s k' a 3t 64 Where k' = Heat conductivity of T i incl u d i n g the e f f e c t of molten flow (W/mK), 2 q' = Sum of heat input and output (W/m ), a = Heat d i f f u s i v i t y of T i (m 2/sec), (=k'/C pp), Cp = Heat capacity of T i (J/kgK), and p = Density of T i (kg/m 3). Fig.3-1 shows a schematic diagram of the heat balance taken i n t o account i n t h i s c a l c u l a t i o n . A coordinate system used f o r the 2-D model i s i l l u s t r a t e d i n Fig.3-2. 3-1-2 E f f e c t of metal flow In the present experiment, a s l i g h t metal flow was observed during melting period. The d r i v i n g force of the flow observed i n t h i s study can be considered to be a temperature gradient i n the metal, which causes two flow types: i ) Natural convection 37) flow and i i ) Surface tension driven flow . These two flow types originate from the change i n phys i c a l properties, that i s , gravity and surface tension r e s p e c t i v e l y according to the existence of the temperature gradient on a metal surface, as can be seen i n t h i s case. The molten metal tends to expand with temperature, so that the metal flow takes place on a metal surface r a d i a l l y from an EB "hot spot". This phenomenon has been observed i n various casting 65 Heat Input Heat Output EB power, q EB 0 Radiation, q r ; Evaporation, <=£> Conduction, q Cond Fig.3-1 Heat balance taken i n control volume. 66 9Omm0 g.3-2 Coordinate system used f o r two dimensional model. processes and many studies have been reported to date. On the other hand, the f l u i d w i l l be drawn along the surface from the region of the lower surface tension to the region of the higher surface tension. Since the surface tension tends to decrease with temperature, f l u i d flow outward from an EB 'hot spot' i s anticipated again i n the case of surface tension driven flow. This flow has been observed mainly i n the welding process. In order to take account of the e f f e c t of the metal flow as mentioned above, a multiplying f a c t o r FF was considered i n the mathematical model as follows. k' = FF k (3-2) where k represents an 'ordinary' heat conductivity of T i (W/mK). FF i s reported to be around 7 i n the case of a natural QQ \ convection . However no reports have been given yet when both natural convection flow and surface tension driven flow take place. 3-1-3 Heat input and heat loss q' i n Eq.3-2 i s represented by Eq.3-3-68 EB R EV , . qi = q _ q _ q (3-3) EB 2 Where q = Total power input of EB (W/m ), R 2 q = Radiation heat los s (W/m ), and EV 2 q = Evaporation heat los s (W/m ). The energy e f f i c i e n c y of the EB power input f o r T i i s reported to be around 80%, when beam impinges normal to the 39) metal . The r e s t of the input energy i s l o s t by the backscattering of electron beams. In the present case, where the beam impinged almost normal, 83 degrees, to the molten T i a l l o y s , the e f f e c t i v e power input can be calculated to be 12 kW (l5-5kW x 0.8). R 2 Radiation heat l o s s , q (W/m ) was represented by the following equation. q R = e a ( T^ - ) ( 3_4) a where £ = Em i s s i v i t y of T i ^ ° \ 0.4-, O = Stefan-Boltzmann constant, 5.67x10"8 (W/in¥), T^ = Ambient temperature, 298K. EV 2 Evaporation heat l o s s , q (W/m ) was given as follows. FV q = mT. AH T. + m A 1 AH A 1 (3-5) 69 2 where m^ ,^ ,m^  = Evaporation rate of T i and A l (kg/m sec) and AH T 1, AH A 1 = Heat of evaporation of T i and A l (kJ/kg). In t h i s study the evaporation behavior of Vanadium was included i n that of T i f o r s i m p l i c i t y because the vapor pressure of both elements i s almost the same. I t i s important to know the reaction control step f o r the estimation of the evaporation rate of the element of i n t e r e s t . Since, as described i n Appendix 1, the evaporation r e a c t i o n at the surface appeared to play a main ro l e i n the study, the evaporation rate of each element can be expressed using 19) Langmuir equation as follows m. = [y. P.° M. / (2 M. 7T R T)*] X. (3-6) l 1 1 1 1 l ( i = T i and Al) where y = A c t i v i t y c o e f f i c i e n t (-), M = Molecular weight (kg/mol), R = Gas constant (=8.315J/Kmol), X = Mole f r a c t i o n (-), and P° = Vapor pressure of pure element (Pa). follows The vapor pressure of pure T i and -Al i s expressed as 41). 70 P ° = 1 . 3 3 x 1 o ( - 2 3 2 0 0 / T - 0 ' 6 6 l ° g T + 1 3 - 7 4 )  p "o = 1 > 3 3 x 1 0 ( - l 6 3 8 0 / T - l o g T + 1 2 . 3 2 ) Constants used i n the c a l c u l a t i o n are summarized i n Table 3-1 3-1-4- Boundary condition Heat l o s s through a mold, which gives a boundary condition of Eq.3-1, i s given by the following equation q B = h T A AT (3-7) where - Heat t r a n s f e r c o e f f i c i e n t between mold and molten metal (W/m^ ) A = Interface area (m ) AT = Temperature diffe r e n c e between mold and molten metal (K), 1690 K (= 1693-283 )• ~ 2 h T was estimated to be 1140 W/m K by considering the increase i n the temperature difference of cooling water ( AT"=8 degC, flow rate = 29-4 1/min) measured during experiment. 3-2 Solution Eq.3-1 was solved by a numerical method. The f i n i t e d i f f e r e n c e method ( e x p l i c i t method) was used f o r c a l c u l a t i o n . In 71 Table 3-1 Constants used i n the c a l c u l a t i o n . Molten Pool Size 9cm<j>X 1. 5cm h (425g) a 5 .67X10" 8 (w/m 2 -k 4 ) e 0.4 J H T , 8 . 8 8 X 1 0 6 ( J / k g ) J H A I 1.08X 107 ( J / kg ) k-n 28.2 (wm/k) P 4540 (kg/m 3) C p T i 690 ( J / kg -k ) 7 T i 1.0 7AI 0.028 72 order to cope with the non-linear term, q', an extrapolation L2) method was used i n order to calculate an adjusted temperature Denoting the previous step by t , then, T ( t n + l / 2 ) = 3/2 T ( t n ) - 1/2 T ( t n - 1 ) (3-8) Meshes used are shown i n Fig.3-3• In t h i s f i g u r e 'Beam' indicates nodes where the beam i s impinged at a c e r t a i n period. The basic scheme used i n the c a l c u l a t i o n was as follows. F i r s t , the beam was placed at one of the 'Beam' po s i t i o n s . Using an adjusted temperature calculated by Eq.3-8, r a d i a t i o n and heat loss were calculated. Second, mass balance was calculated by taking account of the evaporation weight loss of both T i and A l . Third, the f i n i t e d i f f e r e n c e c a l c u l a t i o n f o r heat trans f e r equations was c a r r i e d out. The above c a l c u l a t i o n was repeated u n t i l c a l c u l a t i o n time reaches beam residence time at one node. F i n a l l y , the beam was moved to the next node. A l l the above ca l c u l a t i o n s were repeated u n t i l a determined time step was achieved. Both programs (1D and 2D) and the flow chart of 2D model are shown i n Appendix 2. 3-3 Results and discussion  3-3-1 E f f e c t of the number of nodes 73 2D ID O ' B e a m ' • r = 2.7cm — A r = 3. 6cm Fig.3-3 Meshes used i n the c a l c u l a t i o n . 74 Fig.3-4- shows the e f f e c t of the number of nodes. Although the difference i n the temperature was found within a 1 cm region, the temperature d i s t r i b u t i o n was almost the same. From an economical point of view i n computer c a l c u l a t i o n , NR=5 was used. NS, the number of nodes f o r 6 d i r e c t i o n was determined to be 10 i n order to meet the s t a b i l i t y condition i n the c a l c u l a t i o n . Consequently the t o t a l number of nodes used was 51 i n the case of the 2D model and 6 f o r the 1D model. Examples of computer outputs are shown i n Table 3-2 together with the input data. 3-3-2 E f f e c t of FF and beam o s c i l l a t i o n rate Fig.3-5 shows a calculated r e s u l t of the stationary beam case at FF=10.0. A steady state can be achieved i n a very short time. While the temperature at the beam hot spot reaches around 2750 degC, the temperature at 2.7 cm i s as low as 1670 degC. A steep temperature gradient i s observed. Fig.3-6 shows the e f f e c t of FF on the temperature d i s t r i b u t i o n i n an ingot at 600 sec. Obviously the higher FF gives a f l a t t e r temperature gradient. The difference between the temperature at r=0 cm (EB hot spot) and that at r= 4«5 cm (surface of the ingot) i s around 1200 degC i n the case of FF=10, while i t i s around 150 degC i n the case of FF=100. The e f f e c t of the metal flow appears to be very large. 75 I Distance from center (cm) Fig.3-4 Comparison of the number of nodes taken i n the c a l c u l a t i o n . 76 Table 3-2-a) Examples of computer inputs and outputs 0 Hz and 0.1 Hz. •DIMENSION OF MOLTEN METAL" RA0IUS-O.O45M DEPTH-0.0I5M AREA-0.6362E-02M2 V0LUME-O.9543E-O4M3 WEIGHT- 0.423KG •EB CONDITIONS** BEAM POWER- 12.000KW BEAM AREA-0.6362E-O4M2 POWER DENSITY- O.189E*06KW/M2 FREQUENCY- 0.0 HZ •CALCULATING CONOITIOS"* RNODE : NUMBER- 5 SIZE- 0.0090M TIME STEP- 0.050 SEC END TIME • 600.000 SEC FF- 20.00 HEAT CONDUCTIVITY OF TI- S64.00WM/K HEAT TRANSFER COEFFICIENT- 1140.0O0W/M2/K INITIAL AL WEIGHT PERCENT- G.00 •CONTROLLER*• TOTAL STEPS-12000 PRINT FREQUENCY- 400 ••RESULTS-* •INITIAL TEMPERATURE-1700.OODEGC THROUGHOUT IN MELT •TEMPERATURE DISTRIBUTION IDEGCI AT TIME- 600.OOO SEC 2471.60, 2041.21, 1903.82, 1825.95. 1774.29. 1737.71, •TOTAL TI LOSS-0.4 175E-OMKG) •TOTAL AL LOSS-0. 1353E-OMKG) I •FINAL AL CONCENTRATION- 3.22WTV. i ) 0 Hz •DIMENSION OF MOLTEN METAL** RA0IUS-0.045M 0EPTH-0.015M AREA-O.G362E-02M2 VOLUME-0.9543E-04M3 WEIGHT- 0.423KG •EB CONDITIONS** EB TRAVELLING RADIUS-0.OISM BEAM POWER- 13. COOKW BEAM AREA-0.101BE-03M2 POWER DENSITY - 0.118E+06KW/M2 FREQUENCY- 0.10HZ •CALCULATING CONOITIOS** RNODE : NUMBER- 5 SIZE- 0.0090M THETA NODE: NUMBER- 10 SIZE-0.628RAD TIME STEP- 0.050 SEC END TIME - 600.000 SEC FF- 20.00 HEAT CONDUCTIVITY OF T I - 564.O0WM/K HEAT TRANSFER COEFFICIENT- 1140.OOOW/M2/K INITIAL AL WEIGHT PERCENT- 6.00 •CONTROLLER*• TOTAL STEPS-120OO STEPS/NODE- 20 PRINT FREQUENCY- 20 ••RESULTS** •INITIAL TEMPERATURE-1700.OOOEGC THROUGHOUT IN MELT •TEMPERATURE DISTRIBUTION (DEGC) AT TIME-1931 . 10 1973 .51 1906 74 1874 . 15 1866 .31 1873 .97 1967 .89 1849 97 1818 .49 1820 .28 1836 .93 1853 .81 1784 .76 1775 .72 1784 . 35 1806 .37 1774 .61 1735 .59 1735 .92 1752 . 81 1777 . 15 1729 .31 1699 85 1703 .97 1722. 19 1746 .86 600.000 SEC 1B91 .85 1918 .39 1955 40 2006 .43 2059.23 1861 .98 1894 .32 1939 99 2034 26 2348.28 1834 . 12 1865 .87 1901 . 40 1946 .54 2019.36 1805 .52 1835 .82 1864 . 38 1882 .36 1875.01 1774 .98 1604 .36 1830. S3 1840 80 1813.09 •TOTAL TI LOSS-0.2850E-OKKG) •TOTAL AL LOSS-0. 1299E-0KKG) •FINAL AL CONCENTRATION- 3.25WTX i i ) 0.1 Hz 77 Table 3-2-b) Examples of computer inputs and outputs 1.0 Hz and 10.0 Hz. ••DIMENSION OF MOLTEN M E T A L * ' RA0IUS-O.O45M DEPTH=O.015M AREA-0.63S2E-02M2 VOLUME"O.954 3E-04M3 WEIGHT- 0 .423KG • • E 8 CONDIT IONS" EB TRAVELLING RADIUS'O.015M BEAM POWER- I2.0O0KW BEAM AREA-0.1018E-03M2 POWER DENSITY" O .118E»06KW/M2 FREQUENCY. t O O H Z ••CALCULATING CONDITIOS** RNOOE : NUMBER- 5 S I Z E - 0.0090M THETA NODE: NUMBER- 10 SIZE=0.628RA0 TIME STEP- .0.050 SEC END TIME « 600 .000 SEC F F • 20 .00 HEAT CONDUCTIVITY OF T I - 564.00WM/K HEAT TRANSFER COEFFICIENT- 1140.0O0W/M2/K INITIAL AL WEIGHT PERCENT- 6 .00 • •CONTROLLER** TOTAL STEPS-12000 STEPS/NODE- 2 PRINT FREQUENCY = 20 • R E S U L T S * * • IN IT IAL TEMPERATURE - 1700. OOOEGC THROUGHOUT IN MELT •TEMPERATURE DISTRIBUTION (OEGC) AT T IME- S00 .000 SEC 1950.23 1924.40 1903.79 1857. 14 1818.61 1926 51 1933 63 194 1 87 1950 90 19E0 03 1968 62 1977 30 1974 1 1 196 1 39 1899 78 1907 71 1916 87 1927 61 1940 74 1957 64 1981 54 2013 88 21 19 27 1861 62 1866 49 187 1 63 1876 99 1882 42 1887 44 1890 72 1889 73 1868 67 1820 63 1822 35 1823 67 1824 49 1824 62 1823 82 1821 87 1818 73 1816 56 1785 03 1785 61 1785 67 1785 18 1784 16 1782 82 1781 59 1781 28 1782 59 •TOTAL TI L O S S - 0 . 1 8 6 2 E - 0 K K G ) •TOTAL AL L O S S - 0 . 1220E-OHKG) •FINAL AL CONCENTRATION- 3 36WT% i ) 1.0 Hz -•DIMENSION OF MOLTEN M E T A L " RA0IUS-0 .045M DEPTH-0 015M AREA-0.6362E-02M2 VOLUME-0.9S43E-04M3 WEIGHT- 0.423KG • • E B CONDIT IO NS " ! EB TRAVELLING RAOIUS-0 015M BEAM POWER- 12.000KW BEAM AREA-O.10I8E-03M2 POWER DENSITY- O.118E*06KW/M2 1 FREQUENCY* 10.00HZ " C A L C U L A T I N G C O N D I T I O S " ANODE : NUMBER- 5 S I Z E - 0.0090M THETA NODE: NUMBER- 10 SIZE-O.628RAD | TIME S T E P - 0 .010 SEC ENO TIME • 60O.000 SEC I F F - 20.OO HEAT CONDUCTIVITY OF T I - 564.00WM/K HEAT TRANSFER COEFFIC IENT- 1140.0O0W/M2/K INITIAL AL WEIGHT PERCENT- 6 .00 ; ' ! " C O N T R O L L E R " TOTAL STEPS-60000 STEPS/NOOE- 1 PRINT FREQUENCY- 2COO " R E S U L T S " • IN IT IAL TEMPERATURE-1700.OOOEGC THROUGHOUT IN MELT •TEMPERATURE DISTRIBUTION (DEGC) AT TIME- 600 .000 SEC 1952.77 1954 05 1954 34 1954 57 1954 73 1954 81 1954 80 1954 71 1954 51 1954 21 1953 71 1948 64 1950 76 1952 97 1955 28 1957 70 I960 23 1962 88 1965 66 1968 58 1972 06 1877 45 1877 61 1877 74 1877 82 1877 87 1877 86 1B77 80 1877 69 1877 52 1877 26 1823 80 1823 80 1823 80 1823 80 1823 79 1823 79 1823 78 1823 78 1823 79 1823 79 1785 99 1785 99 1785 99 1785 99 1785 99 1785 99 I78S 99 1785 99 1785 99 1785 99 •TOTAL TI L O S S - 0 . I 6 9 6 E - 0 K K G ) •TOTAL AL L O S S - 0 . 1 1 9 6 E - 0 K K G ) •F INAL AL CONCENTRATION* 3.40WTX i i ) 10.0 Hz 78 Po-2900 2700 2500 O 2300 o ri. E K 2100 1900 1700 k . - » — • -1500 -*-X X-12kW,FF=10.0 OHZ Center r = 0 cm r = 0.9cm • r — 2. 7cm X X 0 100 200 300 400 500 600 Time (sec) Fig.3-5 Calculated temperature changes with time i n the case of stationary beam. 79 80 Fig.3-7 shows the temperature changes at three points, a beam impinged p o s i t i o n , r=2.7 cm and r=3.6 cm res p e c t i v e l y , when the beam o s c i l l a t i o n rate i s 0.1 Hz. As can be seen i n t h i s f i g u r e , a p e r i o d i c a l temperature change corresponding to the beam o s c i l l a t i o n rate (time cycle = 10 second) was obtained. This r e s u l t also c l e a r l y shows that the beam movement unlike the stationary beam case gives a l o c a l unsteady-state condition to each p o s i t i o n . Similar to the 0 Hz case the temperature gradient becomes f l a t t e r with the increase of FF. The temperature di f f e r e n c e among three points becomes smaller with the increase of FF. E s p e c i a l l y i n the case of FF=100, there i s only a s l i g h t temperature gradient i n the melt. Fig.3-8 shows calculated r e s u l t s when the beam o s c i l l a t i o n rate i s 1.0 Hz. A shaded area i n t h i s f i g u r e indicates the temperature range; an upper l i n e connects a l l maximum temperature points obtained i n one time cycle and a lower l i n e connects a l l minimum temperature points. More d e t a i l e d temperature changes from 50 sec to 51 sec are also shown i n the same f i g u r e , i n which maximum and minimum points are depicted f o r each p o s i t i o n . S i m i l a r to the r e s u l t s on 0 Hz and 1.0 Hz c a l c u l a t i o n s , the e f f e c t of FF i s very large. Fig.3-9 shows calculated r e s u l t s when the beam o s c i l l a t i o n rate i s 10.0 Hz. More d e t a i l e d temperature changes from 9-0 sec •to 9-1 sec are shown i n the same f i g u r e . The differ e n c e between the maximum and minimum temperature i s very small even at FF=10. Similar to three other cases, the e f f e c t of FF i s again 81 2900 2700 2500 p 2300 1500, 12Kw 0.1Hz F F = 10.0 ° 'Beam' • r = 2.7cm «r = 3.6cm] V 0 10 20 30 40 50 60 Time (sec) 2700 2500 2300 12kW 0.1Hz FF=20 .0 o'Beam' • r = 2 . 7 c m * r = 3.6cm 0 '0 20 30 40 50 60 Time (sec) FF = 10 FF = 20 2700 2500 2300 O £ 2100 ID H 1900 1700 1500 12kW 0.1Hz FF=100.0 ° Beam » r = 2 . 7 c m *r=3 .6cm 0 10 20 30 40 50 60 Time (sec) FF = 100 Fig.3-7 Calculated temperature changes with time(0.1 Hz) 82 2700 12Kw 1Hz F F = 10.0 o 'Beam' « r = 2.7cm x r = 3.6cm Time (sec) Time (sec) FF = 10 2700 2500 2300 1500 2700 2500 2300 12kW 1 Hz FF=20.0 o 'Beam' • r = 2.7cm x r = 3.6cm Max Min _i i_ 10 20 30 40 50 60 Time (sec) 15001—L Min Max 50 51 Time (sec) FF = 20 d. 2100 E 1900 1700 1500 12Kw 1Hz F F = 100.0 o 'Beam' * r = 2.7cm * r = 3 . 6 c m —1 L. 2300 2100 1900 1700 0 10 20 30 40 50 60 Time (sec) 1500'—^ Max(O) f Min(») \ Min(X) Max(X) 50 51 Time (sec) FF = 100 i Fig.3-8 Calculated temperature changes with time.(l.OHz) 83 2700 2300 a E : 4) • ! 2 K w 10Hz F F = 10.0 o ' B e a m ' « r = 2 .7cm * r = 3 .6cm 0 10 20 30 40 50 60 T i m e ( s e c ) FF = 10 2700 2500 2300 p E <D H 12kW 10Hz F F = 2 0 . 0 o ' B e a m ' «>r = 2 . 7 c m x r = 3 . 6 c m 20 30 40 T i m e (sec) 9.0 9.1 T i m e (sec) FF = 20 a E <D r-2700 2500 2300 2100 1500 12Kw 10Hz F F = 100.0 o ' B e a m ' « r = 2 .7cm X r = 3 .6cm J L. 2300 2100 1900 1700 1500 M ^ ) ^ * X ( 0 ) Min(») Mj"'°) 9.0 9.1 10 20 30 40 50 60 T i m e ( s e c ) T i m e ( s e c ) Fig.3-9 Calculated temperature changes with time(10.0Hz) FF = 100 8U rather large i n t h i s case. When FF=100, the difference among three points i s reduced to be within as small as around 60 degC. 3-3-3 Comparison with experimental r e s u l t s In order to determine the FF i n t h i s experiment, calculated values were compared with experimental r e s u l t s . From an experimental point of view that i ) t h e beam may not have been centered p r e c i s e l y and i i ) t h e beam t r a v e l i n g radius may not have been adjusted p r e c i s e l y , both calculated temperatures at r=2.7 cm and at r=3«6 cm were used as an observation s i t e temperature i n determining FF, although i n r e a l i t y the observation s i t e was set to be r=3cm. Fig.3-10-a) indicates the case of 0 Hz (Stationary beam). A shaded region shows the temperature range obtained i n the experiment whereas VV shows that obtained i n the preliminary te s t , i n which CP-Ti was melted. FF=10-20 seems to agree well with the experimental r e s u l t s . Fig.3-10-b) indicates the case of 0.1 Hz. Although the discrepancy found was more than i n the case of 0 Hz, FF=10-20 can account f o r experimental r e s u l t s e s p e c i a l l y i n the CP-Ti t e s t . Fig.3-10-c) indicates the case of 1.0 Hz and 10.0 Hz. The difference between the experimental and calculated values i s 85 Temp. ("C) oo TO -I O o 3 03 H-to O 0 s; H-c+ ts" CD X t3 CD H-3 CD c+ "n \ \ \ \ \ \ \ \ \ \ 1> Temp. (*C) FT Temp. ('C) Temp. ('C) Temp. (*C) 11 X • II II CO to cn O O 3 3 CD o a1 03 —' H —^  o c+ —^  • W o o —i • • o tn N tsi tn N 8° O tn r e l a t i v e l y large f o r Ti-6A1-4.V. .However, i n the case of CP-Ti any FF value seems to agree with experimental r e s u l t s r e l a t i v e l y w e l l . From the point of view of accuracy i n the temperature measurement, data obtained from CP-Ti experiment are considered to be more r e l i a b l e compared with those obtained by Ti-6A1-4.V t e s t s , since the degree of glass coating on the viewing window was much l e s s i n the case of CP-Ti. By examining Fig.3-10 and taking the above discussion i n t o consideration, FF=20 seems to be reasonable at t h i s stage. This i s supported by the experimental r e s u l t s on the t o t a l weight l o s s from the specimen. When FF i s small, f o r example FF=10, the beam spot temperature becomes higher and the t o t a l weight los s predicted by the model i s much larger than that obtained i n the experiment. On the other hand, when FF i s large, f o r example FF=100, the weight los s predicted by t h i s model i s much smaller than that obtained i n the experiment. The value of FF=20 i s twice as much as that estimated i n the case of natural convection. Considering the a d d i t i o n a l e f f e c t of the surface tension driven flow i n the EB process, t h i s might be a gopd estimation. Fig.3-11 shows a comparison of the temperature changes measured i n the experiment (CP-Ti) and those calculated by the 87 9 Fig.3-11 Relation between calculated and observed temperature changes with time. 88 model at FF=20. As seen i n t h i s f i g u r e , the model i s i n very good agreement with the experimental r e s u l t s . 3-3-4- Temperature contour on metal surface Fig.3-12 shows temperature contours under various beam o s c i l l a t i o n rates at 600 sec, when the steady-state heat trans f e r condition ( i n terms of "time averaged") appears to be achieved. In t h i s ' c a l c u l a t i o n , FF=20 was used i n a l l cases. As c l e a r l y seen i n t h i s f i g u r e , there i s a large difference i n each temperature contour. The EB hot spot temperatures reached at 600 sec are as follows; Beam O s c i l l a t i o n Rate Hot Spot Temperature 0 Hz 2470 degC 0.1 2249 1.0 2121 10.0 1974 The temperature gradient on the metal surface becomes les s steep with the increase of beam o s c i l l a t i o n rate. P a r t i c u l a r l y i n the case of 10.0 Hz, calculated temperatures i n the molten' metal were r e l a t i v e l y low ( i n the range of 1788 - 1974 degC) and a nearly concentric temperature contour was obtained. This r e s u l t implies that with a higher beam o s c i l l a t i o n rate of more than 10.0 Hz, the same kind of temperature contour would be obtained and the e f f e c t of the rapid beam scanning would not be 89 0 Hz 0.1 Hz anticipated anymore. From t h i s c a l c u l a t i o n , i t i s well confirmed that overheating of molten metals can be avoided to a considerable extent by beam scanning. 3-3-5 Total weight loss Fig.3-13 shows the calculated t o t a l weight loss of T i and A l with time. Again here, FF=20 was used i n t h i s c a l c u l a t i o n . The t o t a l weight l o s s increases almost l i n e a r l y i n any beam o s c i l l a t i o n rate. From t h i s model i t i s clear that the lower o s c i l l a t i o n rate gives the l a r g e r weight l o s s . However, the difference between the weight loss i n 1.0 Hz and that i n 10.0 Hz i s very small. This r e s u l t suggests that there i s a l i m i t a t i o n i n increasing the beam o s c i l l a t i o n rate to more than 10.0 Hz. In the same f i g u r e , experimental r e s u l t s obtained are also shown. The t o t a l weight loss at each time was calculated by subtracting 5.7g, which corresponds to a weight loss at 0 min, from the weight loss o r i g i n a l l y measured a f t e r each experiment. By comparing the calculated r e s u l t s with the experimental ones as indicated i n Fig.3-13, i t was confirmed that the tendency of the change i n the t o t a l weight loss obtained i n the experiment can be explained reasonably well ' by the model. Accordingly, t h i s mathematical model can be regarded as a reasonable f i r s t - s t e p model. 91 100 Experimental too in CO o - J +-> _c S1 50 CD "co +-> o • 0 Hz A 0.1Hz © 1.0Hz X 10Hz Calculated line Time (Min) Fig.3-13 Calculated change i n the t o t a l weight los s with time. 92 Fig.3-14- shows a r e l a t i o n between the calculated t o t a l weight l o s s and the beam o s c i l l a t i o n rate at 5 min and 10 min re s p e c t i v e l y . In t h i s f i g u r e , the weight loss of A l and that of T i are i l l u s t r a t e d i n d i v i d u a l l y together with the sum of the weight l o s s of these two elements. As c l e a r l y seen, T i evaporates s u b s t a n t i a l l y as well as A l . This i s a t t r i b u t e d to the f a c t that T i has large X and y i n Eq.3-6 i n spite of the lower vapor pressure. The r a t i o of the weight los s of T i and that of A l i s i n the range of 1 - /+• This agrees well with the experimental r e s u l t , (wt%Ti+wt%V) /(wt%Al) = 2.6, which was obtained by the analysis of the metal deposited on an inner wall of the s h i e l d a f t e r 0 Hz t e s t s . Another i n t e r e s t i n g . point i s that the e f f e c t of the beam o s c i l l a t i o n rate becomes smaller at 1.0 - 10.0 Hz. This r e s u l t can be supported by the r e s u l t s of the calculated temperature contours as previously shown i n Fig.3-12, i n which a r e l a t i v e l y homogenized temperature d i s t r i b u t i o n was obtained i n the case of 1.0 Hz and 10.0 Hz as compared with i n other cases. 3-3-6 Decrease i n wt%Al during melting Fig.3-15 shows calculated r e s u l t s of changes i n wt%Al with time, where the i n i t i a l wt%Al i s taken as 6.0%. Experimental r e s u l t s are also shown i n t h i s f i g u r e . The calculated r e s u l t s reveal that wt%Al decreases almost l i n e a r l y with time. Although 93 .55 10 _o -p top 0 0 . 1 1 1 0 Frequency (Hz) 0 0 . 1 1 1 0 Frequency (Hz) Fig.3-14- Calculated t o t a l weight loss at 5 min and 10 min. 94 [Al]0 =6% ® Experiment • 0 Hz A 0.1Hz © 1 Hz X 10Hz 0 • 1 1 I 1 L_ 0 2 4 6 8 10 Time (sec) Fig.3-15 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 6%. 95 the decrease i n the wt%Al can be suppressed s l i g h t l y with the increase of the beam o s c i l l a t i o n rate, the difference among beam o s c i l l a t i o n rates i s very small. This i s because the r e l a t i v e l y large evaporative los s of T i , as shown previously, makes the apparent change i n the A l concentration expressed by wt% smaller. This appears to be a main reason that the difference among beam o s c i l l a t i o n rates could not be found c l e a r l y i n the experiment: presumably the differ e n c e i s small enough to be beyond the accuracy of the experiment. The behavior of the decrease i n wt%Al could not be explained w e l l when the i n i t i a l wt%Al i s 6.0% as i l l u s t r a t e d i n Fig.3-15. At "time"=0.0, wt%Al must have decreased from the i n i t i a l concentration, 6.1%, due to the evaporative l o s s during a premelting period p r i o r to the actual experiment. Although the true i n i t i a l wt%Al cannot be known exactly, calculated r e s u l t s are shown i n Fig.3-16 when the i n i t i a l wt%Al was a r b i t r a r i l y taken as 4-.5%. As c l e a r l y seen i n t h i s f i g u r e , the model can explain the experimental r e s u l t s r e l a t i v e l y w e ll. Fig.3-17 shows a comparison of the experimental and the calculated wt%Al at 5 min. Wt%Al at 5 min was calculated to be almost constant, around 3«4-%, by t h i s model regardless of the differen c e i n the beam o s c i l l a t i o n rate. As shown i n t h i s f i g u r e , the model i s i n good agreement with the experimental r e s u l t s . 96 [AI]o=4.5% 9^ 1 " - Exper iment • OHz A 0.1Hz ® 1 Hz 10Hz 0 4 6 T ime (min) 8 ® A 1 0 Fig.3-16 Calculated changes i n wt%Al with time when i n i t i a l wt%Al i s taken as 4--5%. 97 [AI]o=4.5% _ 3 < -p * 2 0 • • • • - • Experiment — Calculated i i . 0 0.1 1.0 10.0 Frequency (Hz) Fig.3-17 Calculated wt%Al at 5 min. 98 3-4 Implications of t h i s work f o r i n d u s t r i a l process As described previously, the mathematical model developed i n t h i s study was proven to be reasonable f o r the i n t e r p r e t a t i o n of evaporation phenomena i n a small molten metal (9 cm diameter and 1.5 cm depth), although many assumptions were included. Since a small sample which was considered i n developing the mathematical model can be regarded as a part of a large hearth, the r e s u l t s obtained can be applied to an actual process semi-q u a n t i t a t i v e l y . A r e l a t i o n between the beam scanning v e l o c i t y used i n t h i s study and that used t y p i c a l l y i n the actual process, i n which a beam scanning length i s around 4-0 cm, i s shown as follows; Frequency Beam V e l o c i t y (cm/sec) (Hz) Experiment . Actual process 0.0 0.0 0 0.1 0.94 4 1.0 9-4 40 10.0 94-0 400 The beam v e l o c i t i e s used i n t h i s study are approximately a fourth of those i n the t y p i c a l actual process at each beam o s c i l l a t i o n rate. Although a precise d e s c r i p t i o n i s necessary f o r a complete understanding of the the EB hearth melting process, the. tendency 99 of the temperature p r o f i l e and the evaporative l o s s can be deduced from the r e s u l t s of t h i s model. Namely, i n terms of a composition c o n t r o l , the e f f e c t of the beam o s c i l l a t i o n rate i s not very large. Instead, the t o t a l weight los s during melting process i s suppressed by the increase of the beam o s c i l l a t i o n rate. From an operational point of view, t h i s i s very important to prevent the deposition of the evaporant on a inner wall of the chamber, which makes the y i e l d low, and the decrease i n the beam energy e f f i c i e n c y . However, at beam o s c i l l a t i o n rates over 10.0 Hz the e f f e c t cannot be expected. In conclusion the optimum beam o s c i l l a t i o n rate appears to be 1.0 - 10.0 Hz. 100 Section 4-SUMMARY AND RECOMMENDATIONS FOR THE FUTURE WORK 4.-1 Summary In order to avoid overheating the metal, which sometimes causes a substantial evaporative l o s s o f valuable elements during the EB hearth melting process, the beam scanning technique i s employed i n the actual process. In t h i s study, i n v e s t i g a t i o n on the optimum beam o s c i l l a t i o n rate was conducted by making a mathematical model on the basis of a small scale EB melting experiment. Small amounts of Ti-6A1-4V a l l o y were melted i n the small s i z e EB melting furnace. The influence of the beam o s c i l l a t i o n rate on the evaporation behavior of Al was investigated. The beam o s c i l l a t i o n rates used i n t h i s study were 0, 0.1, 1.0 and 10.0 Hz. The temperature cycle observed i n t h i s study was found to be i n good agreement with the given beam o s c i l l a t i o n r a t e. In a l l samples, a s i g n i f i c a n t amount of the wt%Al decrease and the t o t a l weight loss (Al + Ti) was observed. The e f f e c t of the beam o s c i l l a t i o n rate was observed i n the t o t a l weight loss change, but could not be found c l e a r l y i n the wt%Al decrease i n the present experiment. On the basis of the r e s u l t s from the experiment, i n p a r t i c u l a r the r e s u l t s of the temperature measurement, 2-D and 1-101 D unsteady heat trans f e r models were made: a 2-D model f o r the moving beam problem and a 1-D model f o r the stationary beam problem. In these models, the mass balance with respect to both A l and T i was also taken into account by assuming that the evaporation reaction at the i n t e r f a c e plays a major r o l e . That i s , the evaporation reaction i s a rate-determining step. From a comparison of the calculated and experimental r e s u l t s , FF=20.0, a fa c t o r containing the e f f e c t of the molten metal, appeared to explain the experimental r e s u l t s reasonably well under the present experimental conditions. According to t h i s model, i t was c l e a r l y shown that the evaporative los s of both A l and T i could be suppressed by increasing the beam o s c i l l a t i o n rate to around 1.0 Hz. This r e s u l t was obtained because the overheating of the molten metal was prevented s u b s t a n t i a l l y . With the beam o s c i l l a t i o n rate at more than 1.0 Hz, however, t h i s e f f e c t could not be found c l e a r l y anymore. On the other hand, i n terms of the composition control of A l , the e f f e c t of the beam o s c i l l a t i o n rate was not s i g n i f i c a n t , since T i also evaporated simultaneously during a melting period under the low operating pressure. In conclusion, the optimum beam o s c i l l a t i o n rate can be considered to be i n the range of 1.0 - 10.0 Hz e s p e c i a l l y from an operational point of view. 4-2 Recommendations f o r future work 102 Unfortunately the e f f e c t of the beam o s c i l l a t i o n rate could not be detected c l e a r l y i n the present experiment. In the future, therefore, experimental accuracy should be improved further. For example, we should improve the beam r o t a t i n g equipment and minimize the e f f e c t of the premelting period. In addition, various parameters such as EB power input, vacuum degree, and solute elements having d i f f e r e n t a c t i v i t y c o e f f i c i e n t s should be taken into account i n order to allow the model to be more precise and r e l i a b l e . Furthermore, a three dimensional mathematical model should be developed by considering the e f f e c t of the s o l i d i f i c a t i o n phenomena, which w i l l give a p r e d i c t i o n of the metal pool, and the metal flow driven by both surface tension and grav i t y . This w i l l u l t i m a t e l y lead to the development of a complete process modeling of the EB hearth melting processes commercially used. 103 L i s t of References 1. D.Apelian, C.H.Entrekin : International Metals Reviews, 31, (1986), pp.77-90. 2. 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Chang, R.I.L. Guthrie and J . Szekely : Met. Trans. 9B, (1978), pp.151-154-14- R.J.- Barnhurst and J.E. Gruzleski : Ironmaking & Steelmaking, 5, (1979), pp.294-300. 15. R. Harris and W.G. Davenport : Met. Trans. 13B, (1982), pp. 581-588. 16. i b i d . , pp.589-591 -17. E. Ozberk and R.I.L. Guthrie : Met. Trans. 17B, (1986), pp. 19-29. 104 18. E.S.Machlin : Trans, of Met. Soc. of AIME, 218, (i960), pp. 3U-326. 19- G.H.Geiger and D.R.Poirier : "Transport phenomena i n metallurgy" Readings, Mass, Addison-Wesley, (1973)-20. J . Szekely, C.W. Chang, and W.E. Johnson : Met. Trans. 8B, (1977), p517. 21. S. Hayakawa, T. Choh, and M. Inoue : I.S.I.J. 22, (1982), pp. 637-645. 22. R. Harris : Met. Trans. 15B, (1984), pp.251-257. 23- T.Santala and C.A.Adams : J . Vac. S c i . Technol., 17, (1970), ss.22-29-24. Y.Nakamura and M.Kuwabara : I.S.I.J., 15, (1975), pp.103-108. 25. S . S c h i l l e r and H.FSrster : "Electron Beam Remelting and Refining State of the Art 1984" Proc.Conf., (1984), Bakish Materials Corporation, Englewood, NJ , pp.49-69-26. A.Mitchell and K.Takagi : Proc. Vacuum Metallurgy conf., (1984), pp.89-99. 27. J.Herbertson: "Electron Beam Remelting and Refining State of the Art 1986 P t . I I " Proc.Conf., (1986), Bakish Materials Corporation, Englewood, NJ, pp.19-29. 28. H.S.Kheshgi and P.M.Gresho : "Electron Beam Remelting and Refining State of the Art 1986 Pt. I I " Proc.Conf., (1986), Bakish Materials Corporation, Englewood, NJ, pp.68-79-29. D.Tripp : Master's Thesis, U n i v e r s i t y of B r i t i s h Columbia, (1987). 30. K.Takagi : Master's Thesis, University of B r i t i s h Columbia, (1984). 31. D. Rosenthal : Trans, of the ASME, 43, (1946), pp.849-866. 32. For example, N. Christensen, V. Davies, K.Gjermundsen : B r i t i s h Welding Journal, 12, (1965), pp.54-75-33- T.W.Eager and N.S.Tsai : Welding Journal, 62, (1983),346s-355s. 34- S.Kou and Y.H.Wang : M e t a l l . Trans., 17A, (1986), pp.2265-2270. 35- Operations Manual of EH-30/20, Von Ardenne, East Germany. 105 36. Operations Manual of Automatic O p t i c a l pyrometer Series 1100, IRCON Inc., I l l i n o i s . 37. C.R.Heiple, J.R.Roper : Welding Journal, 61, (1982), 97s-102s. 38. A.S.Ballantyne : Ph.D. Thesis, University of B r i t i s h Columbia, (1978). 39- S . S c h i l l e r , U.Heisig, and S.Panzer : "Electron Beam Technology", (1982), John Wiley & Sons, Inc., New York, NY. 4.0. "Thermal Properties of Titanium A l l o y s " , Defense Materials Information Center, B a t t e l l e Memorial I n s t i t u t e , Columbus, Ohio. 4.1. O.Kubaschewski, C.B.Alcock : "M e t a l l u r g i c a l Thermochemistry", 5th ed., Pergamon Press, Oxford, (1979). 42. P. Cacciatore : "Modeling of Casting and Welding Processes", (1980), AIME. 43- A.Mitchell, H.Nakamura and D.Tripp : "Electron Beam Remelting and Refining State of the Art 1987" Proc.Conf., (1987), Bakish Materials Corporation, Englewood, NJ , pp. 23-32. 106 Appendix 1 ESTIMATION OF RATE CONTROLLING STEP A-1-1 Complete d i f f u s i o n control case The p o s s i b i l i t y of a "complete" d i f f u s i o n control case was 18) examined by using Machlin's model Basic equation Assuming that Machlin's model, which was developed f o r reaction system i n the induction furnace, can be also applied to t h i s system, the mass transf e r c o e f f i c i e n t , ^ (cm/sec), i s given as follows. Me,l 2 ( D / 7T e )* (A-1-1 ) where D D i f f u s i v i t y of solute element (cm /sec), 10 ^ cm^/sec 6 " L i f e time" of a small element (sec) On the other hand, o v e r a l l mass f l u x of A l , j Al (mole/sec), can be expressed by the following equation. JA1 = A k, :Me,l CA1 (A-1-2) 107 where A = Reaction area (cm ), 3 = A l concentration (mole/cm ) Substituting Eq.A-1-1 into Eq.A-1-2, the following equation can be obtained. (wt%Al) = (wt%Al) 0 exp(-(l/h) k M g j l t) (A-1-3) where h = Depth of molten pool (cm), 1.5 cm and (wt%Al) Q = I n i t i a l wt%Al, 4..5% F i n a l l y , the following equation can be obtained. (wt%Al) - 4.5 exp(-7.52 x 10" 3 0 ~* t) (A-1-4.) By taking 8 as a parameter, the change i n wt%Al with time was calculated. Results Fig.A-1.1 shows calculated curves with various 0. In t h i s f i g u r e , the experimental r e s u l t s are also shown. I t i s c l e a r l y shown that the experimental r e s u l t s can be explained well at 8 = 10 - 4.0 sec. However, from the observation of the metal flow on the surface during experiment, the l i f e time i n t h i s p a r t i c u l a r case was i n the order of 0.1 - 1.0 sec. This i s s u b s t a n t i a l l y l a r g e r than the calculated one. Therefore, the whole reaction 108 5 U < 3 o N 5 2 1 \ \ " A " A N e= (sec) Experiment^, • 0 Hz A 0.1 Hz ® 1 Hz x 1 0 Hz J O ® "A 0 2 4 6 ^ 8 Time (min) 10 Fig.A-1.1 Changes i n wt%Al with time calculated from Machlin's model. 109 would not have been c o n t r o l l e d by a d i f f u s i o n c o n t r o l l i n g step only. A-1-2 E f f e c t of the temperature and the metal flow on the rate  c o n t r o l l i n g step A simple mathematical model was developed i n order to estimate the e f f e c t of the temperature and the s t i r r i n g condition on the rate c o n t r o l l i n g step as follows. Basic equation and s o l u t i o n Fig.A-1.2 i l l u s t r a t e s a geometry used i n t h i s c a l c u l a t i o n . The solute element d i f f u s e s from the bulk metal to the reaction i n t e r f a c e and removed by the evaporation reaction. The influence of the metal flow was taken into account by introducing a f a c t o r , f f , as described i n Section 2. A basic equation can be given by taking mass balance i n t h i s system as follows. 9*C Al 9C A l D (A-1-5) 9t where D' f f D and D 'Ordinary' d i f f u s i v i t y 2 of solute element (cm /sec). The boundary conditions are given as follows. 110 [metal surface] E L O 9Omm0 g.A-1.2 Geometry used i n the c a l c u l a t i o n . 111 B.C.1 C A 1 = A.5 % B.C.2 3C A 1/3x = 0.0 B.C.3 -D»3C A 1/3x = K„ C at t = 0.0 sec at the bottom, x = 15mm at the surface, x=0.0mm The i n i t i a l condition i s expressed by B.C.1. B.C.2 gives a condition that there i s no mass f l u x at the bottom of the pool. B.C.3 gives a heterogeneous reaction rate on the melt surface. Kg, Langmuir rate constant, i s expressed by Eq.A-1-6. (Same as Eq.1-5) K E = Y P A 1 ° / P m (27TMRT)* (A-1-6) Thermochemical data used i n the c a l c u l a t i o n are tabulated below. Temperature(degC) 1600 1700 1800 1900 2000 2100 D (cm 2/sec) 0.70 0.80 0.90 1.0 1.1 1.2 A ± x 10" 4 K_ (cm/sec) 0.15 0.4.0 0.93 2.00 4--01 7.56 E x 10" 3 * D = D Q exp(-E D/RT); D Q=constant, E^ = A c t i v a t i o n 22) energy, 4-OkJ . at 1900 degC was assumed to be 1.0 x10 4 cm^/sec. Eq.A-1-2 was solved by taking f f (1 - 100)and T (1700 -3100 degC) as parameters. 112 Results Results are summarized i n F i g . A-1.3, where numbers r e f e r to U3) the r a t i o of surface concentration to bulk concentration . From t h i s f i g u r e , i t i s seen that there i s a substantial e f f e c t of both f f and temperature on the mode of rea c t i o n c o n t r o l i n evaporation. A low temperature, high v e l o c i t y regime w i l l lead to evaporation co n t r o l ; the converse to d i f f u s i o n c o n t r o l . Since the "average" molten temperature was r e l a t i v e l y low (<1900 degC), namely overheat was suppressed, and also the r e l a t i v e l y strong s t i r r i n g condition, which a r i s e s from the natural convection and the surface-tension driven flow (corresponding to ff=10 - 30), was observed i n the present experiment, the contribution of the evaporation step would have been very large. Therefore, as a f i r s t approximation, the assumption that the rate c o n t r o l l i n g step was the evaporation reaction at the molten surface should be reasonable i n the present case. In addition, t h i s assumption was also supported by the experimental r e s u l t s described i n Section 2; namely, the d i s t r i b u t i o n of %A1 i n the molten pool was v i r t u a l l y f l a t . ( Of course, there i s no question about that the simultaneous solution of the momentum-, heat- and mass- transf e r equations i s necessary f o r the precise i n t e r p r e t a t i o n of the evaporation phenomena i n the system t h i s i s a future work as described i n Section 4«) 113 95% Evaporation control 5-95% •Mixed -control <5% Diffusion control 1700 2100 2500 2900 Temp. ( °C) 3300 Fig.A-1.3 Evaporation control map. 1U Appendix 2 PROGRAM LIST AND FLOW CHART A-2-1 Program l i s t of 1-D model L i s t i n g of STSPOT at 11 :29:27 on JAN 25 , 198B f o r CC1d=NHER on G 1 C PROGRAM * S T S P O T * ((STATIONARY BEAM)) 2 C 3 C CYLINDRICAL COORDINATE SYSTEM (1D MODEL) 4 C 5 DIMENSION T ( 1 0 0 ) , T C ( 1 0 0 ) , T N ( 1 0 0 ) , T A ( 100) , G 1 T 0 ( 1 O O ) , 0 R ( 1 O O ) . 0 E B ( 1 O 0 ) . S T I J ( 1 O O ) . 7 2 S A L J ( 1 0 0 ) , N T Y P E ( 1 0 ) , A 1 ( 1 0 0 ) , A 2 ( 1 0 0 ) , 8 3 A3 (10O) .AREA(100 ) 9 C 10 C DEFINE STATEMENT FUNCTIONS 1 1 C 12 C *RADIATION HEAT LOSS* 13 C 14 TTA(TT)=TT*TT+TAMBT*TAMBT 15 TTB(TT)=TT+TAMBT 16 TTC(TT)=TT-TAMBT 17 T T T ( T T ) = T T A ( T T ) * T T B ( T T ) * T T C ( T T ) 18 ORA(TT)=EMISS*SIGMA*TTT(TT) 19 C 20 C *EVAPORATION HEAT LOSS* 21 C 22 P T I ( T T ) = 1 3 3 . 2 8 * 1 0 * * ( - 2 3 2 0 0 . O / T T - 0 . 6 6 * A L O G 1 0 ( T T ) + 1 1 .74) 23 P A L ( T T ) = 1 3 3 . 2 8 * 1 0 * * ( - 1 6 3 8 0 . 0 / T T - A L 0 G 1 0 ( T T ) + 1 2 . 3 2 ) 24 T I J ( T T ) = G A M T I * P T I ( T T ) * T I M 0 L / S Q R T ( 2 . O * P I * T I M 0 L * R * T T ) * X T I 25 A L J ( T T ) = G A M A L * P A L ( T T ) * A L M O L / S O R T ( 2 . 0 * P I * A L M O L * R * T T ) * X A L 26 Q T I ( T T ) = T I d ( T T ) * H T I . 27 O A L ( T T ) = A L J ( T T ) * H A L 28 QEV(TT)=QTI (TT)+QAL(TT) 29 C 30 C *SUM OF HEAT LOSS TERM (W/M2) 31 C 32 QL(TT)=QRA(TT)+QEV(TT) 33 C 34 C READ INPUT DATA 35 C 36 READ ( 5 , 4 6 0 ) RAD.DEPTH 37 READ ( 5 . 4 7 0 ) NR. IFREQ 38 RNR = FLOAT(NR) 39 READ ( 5 . 4 6 0 ) POWERK, HZ 40 POWERW = POWERK * 1000.0 41 READ ( 5 , 4 6 0 ) WPCTA, TEMPIC 42 TEMPIK = TEMPIC + 273 .15 43 READ ( 5 , 4 6 0 ) D E L T , TIMEN 44 READ ( 5 , 4 6 0 ) F F , HT 45 IMAX = TIMEN / DELT + 0 . 5 46 C 47 C AREA.VOLUME,AND WEIGHT OF LIQUID TI 48 C 49 PI = 3 .141592654 50 RHO = 4 5 4 0 . 0 51 RHOAL = 2700 .0 52 WPCTT = 100.0 - WPCTA 53 RHOM = (RHO*WPCTT + RHOAL*WPCTA) / 100.0 54 C 55 AREAH •= RAD * RAD * PI 56 VOLUME = AREAH * DEPTH 57 WEIGHT = VOLUME * RHOM 58 C 115 L i s t i n g of STSPOT at 11:29:27 on JAN 25. 1988 f o r CC1d=NHER on G 59 C NODE AREA 60 C 61 DELR = RAO / RNR 62 AREAO = (DELR*DELR) * PI / 4.0 63 DO 20 1 = 1, NR 64 RI = FLOAT(I) 65 IF(I.EO.NR) GO TO 10 66 C * INNER* 67 AREA(I)=2.0*PI*RI*DELR*DELR 68 GO TO 20 69 C *OUTER* 70 10 AREA(I)=PI*(RNR-O.25)*0ELR*DELR 71 C 72 20 CONTINUE 73 C 74 C PHYSICAL PROPERTIES (SI-UNIT) 75 C *HEAT CONDUCTION* 76 C 77 TIK = 28.2 78 CPTI = 690.0 79 FTIK = TIK * FF 80 ALPHA = FTIK / (CPTI*RHO) 81 FO = ALPHA * DELT / (DELR*DELR) 82 BI = HT * DELR / FTIK 83 C 84 C *RADTATION* 85 C 86 EMISS = 0.4 87 SIGMA = 5.67E-8 88 TAMBT = 298 . 15 89 C 90 C *EVAPORATION* 91 C 92 R = 8.3144 93 TIMOL = 0.0479 94 ALMOL = 0.027 95 GAMTI = 1.0 96 GAMAL = 0.028 97 HTI = 8.88E6 98 HAL = 1.08E7 99 C 100 C EB POWER 101 C 102 OIK = POWERK/AREAO 103 01 = OIK * 1000.0 104 C 105 C INITIAL CONCENTRATION OF ALLOY MASS BALANCE 106 C 107 WTI0=WEIGHT*WPCTT/100.0 108 WALO=WEIGHT-WTIO 109 XPNAL = WPCTA / ALMOL 110 XPNTI = WPCTT / TIMOL 111 XAL = XPNAL / (XPNAL + XPNTI) 112 XTI = 1.0 - XAL 113 C 114 C WRITE LIQUID DIMENSIONS AND ETC, 115 C 116 WRITE (6,480) RAD, DEPTH 116 L i s t i n g of STSPOT at 11:29:27 on JAN 25. 1988 f o r CC1d=NHER on G 1 17 WRITE (6.490) AREAH, VOLUME, WEIGHT 1 18 WRITE (6,500) POWERK, AREAO, OIK , HZ 1 19 WRITE (6,510) NR, DELR 120 WRITE (6,520) DELT, TIMEN 121 WRITE (6,530) FF, FTIK, HT, WPCTA 122 WRITE (6,540) IMAX.IFREO 123 C 124 C SORT NODE TYPE 125 C 126 DO 70 I = 1, NR 127 NTYPE(I) = 2 128 IF (I .EQ. 1) NTYPE(I) = 1 129 IF (I .EQ. NR) NTYPE(I) = 3 130 70 CONTINUE 131 C 132 C TERMS 133 C 134 DO 80 I = 1, NR 135 RI = F LOAT(I) 13G A1 ( I ) = (RI - 0.5) / RI 137 A2(I ) = (RI + 0.5) / RI 138 80 CONTINUE 139 B = DELR * DELR / (DEPTH*FTIK) 140 D1 = 2.0 * (RNR - 0.5) / (RNR - 0. 25; 141 D2 = 2.0 * RNR / (RNR - 0.25) 142 C 143 C SET INITIAL CONDITIONS 144 C 145 TO = TEMPIK 146 TOO = TEMPIK 147 TOC = TEMPIC 148 STIOJ = 0.0 149 SALOJ = 0.0 150 QEB0=QI 151 DD 100 I = 1, NR 152 T ( I ) = TEMPIK 153 TO(I) = TEMPIK 154 TC ( I ) = TEMPIC 155 S T I J ( I ) = 0.0 156 S A L J ( I ) = 0.0 157 QEB(I)=0.0 158 100 CONTINUE 159 TIME = 0.0 160 ICOUNT = 0 161 C 162 C WRITE INITIAL CONDITIONS 163 C 164 WRITE (6.550) TEMPIC 165 WRITE (7,600) HZ 166 WRITE (7.610) FF, FTIK,POWERK 167 WRITE (7,620) 168 WRITE (7,630) TIME, TOC 169 C 170 C START CALCULATION 171 C 172 1 10 ICOUNT = ICOUNT + 1 173 IF (I COUNT .GT. IMAX) GO TO 420 174 TIME = FLOAT(ICOUNT)*DELT 117 L i s t i n g of STSPOT at 11:29:27 on JAN 25. 1988 f o r CC1d=NHER on G 175 C 176 C TEMPERATURE ADJUSTMENT FOR CALCULATION OF 177 C RADIATION 6 EVAPORATION HEAT LOSS 178 C 179 TOA = 1.5 * TO - 0.5 * TOO 180 DO 140 I = 1, NR 181 T A ( I ) = 1.5 * T ( I ) - 0.5 * TO(I) 182 140 CONTINUE 183 C 184 C RADIATION + EVAPORATION HEAT LOSS 185 C 186 QRO = OL(TOA) 187 DO 160 I = 1 . NR 188 TA1 = T A ( I ) 189 OR(I) = 0L(TA1) •190 160 CONTINUE 191 c 192 c SUM OF TI & AL LOSS AT EACH NODE(KG/M2) 193 c 194 STIOJ = STIOJ + TIJ(TOA) * DELT 195 SALOJ = SALOJ + ALJ(TOA) * DELT 196 DO 180 I = 1, NR 197 TA1 = T A ( I ) 198 T1J1 = T I J ( T A 1 ) 199 ALJ1 = ALJ(TA 1) 200 S T I J ( I ) = STIJ.(I) + TIJ1 * DELT 201 S A L J ( I ) = S A L J ( I ) +.ALJ1 * DELT 202 180 CONTINUE 203 c 204 c TOTAL TI & AL LOSS (KG) 205 c 206 ATIJ = STIOJ*AREAO 207 AALJ = SALOJ*AREAO 208 DO 220 I = 1, NR 209 IF (I .EO. NR) GO TO 190 210 c * INNER* 211 AATIJ = S T I J ( I ) * AREA(I) 212 AAALJ = S A L J ( I ) * AREA(I ) 213 GO TO 200 214 c •OUTER* 215 190 AATIJ = S T I J ( I ) * AREA(I) 216 AAALJ = S A L J ( I ) * AREA(I) 217 c *SUM* 218 200 ATIJ = AT I J + AATIJ 219 AALJ = AALJ + AAALJ 220 220 CONTINUE 221 c 222 c MASS BALANCE : ADJUST TI & AL QUANTITY IN MELT 223 c 224 WTI = WTIO - ATIJ 225 WAL = WALO - AALJ 226 WT = WTI + WAL 227 WPCTA •= WAL / WT* 100 .0 228 XNTI = WTI / TIMOL 229 XNAL = WAL / ALMOL 230 XNT = XNTI + XNAL 231 XAL = XNAL / XNT 232 XTI = 1 . 0 - XAL 118 L l s t l n g ' o f STSPOT at 11:29:27 on OAN 25, 198B for CC1d=NHER on G 233 C 234 C FINITE DIFFERENCE EQUATIONS 235 C 236 C •CENTER PART* 237 C 238 T0N=F0*(4.0*T(1)+B*(QEBO-QRO))+(1-4.0*F0)*T0 239 C 240 C SORT 241 C 242 DO 360 I = 1, NR 243 NTY = NTYPE(I) 244 GO TO (260, 270. 280), NTY 245 c 24G c *TYPE1* 247 c 248 260 TN(I)=F0*(A1(I)*T0+A2(I)*T(1+1)+ 249 1 B*(QEB(I)-QR(I)))+(1.0-2,0*FO)*T(I) 250 GO TO 360 251 c 252 c •TYPE2* 253 c 254 270 TN(I)=F0*(A1 (I )*T(1-1 )+A2(I)*T(1 + 1) + 255 1 B*(QEB(I)-QR(I)))+(1.0-2.0*F0)*T(I) 256 GO TO 360 257 c 258 c *TYPE3* 259 c 260 280 TN(I) = F0*(D1*T(I-1 )+BI*D2*TAMBT+B*(QEB(I)-QR(I 261 1 +(1.0-D1*F0-D2*F0*BI)*T(I) 262 263 360 CONTINUE 264 c 265 c TEMPERATURE MUST BE LESS THAN 3285.0 DEGC (3558.15 K) 266 c 267 IF(T0N.GT.3558.15) T0N=3558.15 268 DO 364 I=1,NR 269 T1=TN(I) 270 IF(T1.GT.3558.15) TN(I)= 3558.15 271 364 CONTINUE 272 c 273 c SUBSTITUTE NEW TEMPERATURE IN OLD ONE 274 c 275 TOO = TO 276 TO = TON 277 DO 380 I = 1, NR 278 TO(I-) = T(I) 279 T(I) = TN(I) 280 380 CONTINUE 281 c PRINT OUT CONTROL 282 IF ( (ICOUNT/1FREQ)*IFREQ .NE. ICOUNT) GO TO 410 283 c 284 c CHANGE TEMPERATURE UNIT: K TO C 285 c 286 TOC = TO - 273.15 287 DO 400 I = 1, NR 288 TC( I ) •= T( I ) - 273 . 15 289 400 CONTINUE 290 c 119 L i s t i n g of STSPOT at 11:29:27 on JAN 25. 1988 f o r CC1d=NHER on G 291 C WRITE TEMPERATURE & TI AND AL LOSS 292 C 293 WRITE (7,630) TIME, TOC,TC(1).TC(3),TC ( 5) , ATIJ, AALJ. WPCTA 294 C 295 C NEXT STEP 296 C 297 410 GO TO 110 298 C 299 C WRITE RESULTS AT END TIME 300 C 301 420 WRITE (6,560) TIME 302 WRITE (6,570) TOC,(TC(I),I=1,NR) 303 WRITE (6,580) ATIJ, AALJ, WPCTA 304 GO TO 1000 305 C 306 C READ FORMAT 307 460 FORMAT (5F10.5) 308 470 FORMAT (5110) 309 C 310 C WRITE FORMAT 311 C 312 480 FORMAT (5X,'**DIMENSION OF MOLTEN METAL**'/8X,'RADIUS=',F5.3, 313 1 'M',3X,'DEPTH='.F5.3,'M') 314 490 FORMAT (8X,'AREA=',E10.4,'M2',3X,'VOLUME=',E10.4,'M3',3X, 315 1 'WEIGHT=',F7.3,'KG'//) 316 500 FORMAT (5X.'**EB CONDITIONS**'/8X, 317 1 'BEAM POWER=',F7.3,'KW',3X,'BEAM AREA='.E10.4, 318 2 'M2',3X,'POWER DENSITY'',E15.3,'KW/M2'/8X, 319 3 'FREOUENCY=',F7.2,'HZ'/) 320 510 FORMAT (5X,'**CALCULATING CONDITIOS**'/8X,'RNODE : NUMBER 321 1 I5.3X,'SIZE=',F7.4,'M'//) 322 520 FORMAT (8X,'TIME STEP 3',F10.3,' SEC'/8X,'END TIME =',F10.3, 323 1 ' SEC'/) 324 530 FORMAT (8X,'FF=',F7.2,3X,'HEAT CONDUCTIVITY OF TI=',F7.2, 325 1 'WM/K'/8X,'HEAT TRANSFER COEFFICIENT=',F10.3, 326 2 'W/M2/K'/8X,'INITIAL AL WEIGHT PERCENT 3',F5.2/) 327 540 FORMAT (5X, '*'CONTROLLER**'/8X, 'TOTAL STEPS=', 15, 3X. 328 1 'PRINT FREQUENCY 3', 15///) 329 550 FORMAT (5X, '**RESULTS**'//8X, ''INITIAL TEMPERATURE 3', F7.2, 330 1 'DEGC, 3X, 'THROUGHOUT IN MELT'//) 331 560 FORMAT (8X,''TEMPERATURE DISTRIBUTION (DEGC) AT TIME=',F10.3, 332 1 ' SEC'/) 333 570 FORMAT (8X, 15( 1X.F7.2, ', ' ,3X)) 334 580 FORMAT (//8X, ' *TOTAL TI LOSS 3', E10.4, '(KG)'/8X, 335 1 '*TOTAL AL LOSS 3', E10.4, '(KG)'//8X, 336 2 '"FINAL AL CONCENTRATION 3', F5.2,'WT%') 337 600 FORMAT (5X,'FREQUENCY=' . F7.2, 'HZ') 338 610 FORMAT (5X,'F F 3 ' , F7.2,4X,'HEAT CONDUCTIVITY OF MOLTEN T I 3 ' . 339 1 F7.2, 'WM/K'/5X, 'INPUT POWER3' ,F7 . 2, 'KW'/) 340 620 FORMAT (/5X.'DATA AT A CENTER POINT' 341 1 //6X,'TIME(S)',5X,'TEMP.(C)',4X, 342 2 '*TIN(C)*',3X , '*TMID(C)*',3X, '*TOBS(C)*' .4X, 343 3 'TI LOSS(KG)',4X,'AL LOSS(KG)',5X,'WT%AL'/) 344 630 FORMAT (5X,F8.2,4(2X,F10.2),2(5X.E10.4),4X,F6.2) 345 C 346 1000 STOP 347 END 120 A-2-2 Program l i s t of 2-D model L i s t i n g o f MVSPOT at 11 :29:43 on JAN 25. 1988 f o r CC1d=NHER on G 1 C PROGRAM *MVSPOT* 2 C 3 C CYLINDRICAL COORDINATE SYSTEM (2 -D MODEL) 4 C 5 DIMENSION T ( 1 0 0 , 1 0 0 ) , T C ( 1 0 0 , 1 0 0 ) , T N ( 1 0 0 , 1 0 0 ) . T A ( 1 0 0 , 1 0 0 ) , 6 1 T 0 ( 1 0 0 , 1 0 0 ) , 0 R ( 1 0 0 . 1 0 0 ) , O E B ( 1 0 0 , 1 0 0 ) , S T I J ( 1 0 0 . 1 0 0 ) . 7 2 S A L J ( 1 0 0 , 1 O 0 ) , N T Y P E ( 1 0 0 , 1 0 0 ) , A 1 ( 1 0 0 ) , A 2 ( 1 0 0 ) . 8 3 A 3 ( 1 0 0 ) , A R E A ( 1 0 0 ) 9 C 10 C DEFINE STATEMENT FUNCTIONS 1 1 C 12 C 'RADIATION HEAT LOSS* 13 C 14 TTA(TT)=TT*TT+TAMBT*TAMBT 15 TTB(TT)=TT+TAMBT 1G TTC(TT)=TT-TAMBT 17 T T T ( T T ) = T T A ( T T ) * T T B ( T T ) * T T C ( T T ) 18 ORA(TT)=EMISS*SIGMA*TTT(TT) 19 C 20 C *EVAPORATION HEAT LOSS* 21 C 22 P T I ( T T ) = 1 3 3 . 2 8 * 1 0 * * ( - 2 3 2 0 0 . 0 / T T - 0 . 6 6 * A L O G 1 0 ( T T ) + 1 1 . 7 4 ) 23 PAL(TT) = 1 3 3 . 2 8 * 1 0 * * ( - 16380 .O /TT -ALOG10(TT )+12 .32 ) 24 T I J ( T T ) = G A M T I * P T I ( T T ) * T I M 0 L / S Q R T ( 2 . O * P I * T I M O L * R * T T ) * X T I 25 A L J ( T T ) = G A M A L * P A L ( T T ) * ALMOL/SORT(2 .0 *P I *ALMOL*R* T T ) * X A L 26 O T I ( T T ) = T I J ( T T ) * H T I 27 O A L ( T T ) = A L J ( T T ) * H A L 28 QEV(TT)=QTI (TT)+QAL(TT) 29 C 30 C *SUM OF HEAT LOSS TERM (W/M2) 31 C 32 QL(TT)=QRA(TT)+QEV(TT) 33 C 34 C READ INPUT DATA 35 C 36 READ ( 5 , 4 6 0 ) RAD, DEPTH, EBRAD 37 READ ( 5 , 4 7 0 ) NR, NS. IFREO 38 RNR = FLOAT(NR ) 39 RNS = FLOAT(NS) 40 READ ( 5 . 4 6 0 ) POWERK, HZ 41 POWERW = POWERK * 1000.0 42 READ ( 5 . 4 6 0 ) WPCTA, TEMPIC 43 TEMPIK = TEMPIC + 273.15 44 READ ( 5 , 4 6 0 ) D E L T , TIMEN 45 READ ( 5 . 4 6 0 ) F F , HT 46 IMAX = TIMEN / DELT + 0 . 5 47 C 48 C AREA,VOLUME.AND WEIGHT OF LIQUID TI 49 C 50 PI = 3 . 141592654 51 RHO = 4540 .0 52 RHOAL = 2700 .0 53 WPCTT = 100.0 - WPCTA 54 RHOM = (RHO*WPCTT + RHOAL *WPCTA) / 100.0 55 C 56 AREAH = RAD * RAD * PI 57 VOLUME = AREAH * DEPTH 58 WEIGHT * VOLUME * RHOM 121 L i s t i n g of MVSPOT at 11:29:43 on JAN 25, 1988 f o r CC1d=NHER on G 59 c 60 c NODE AREA 61 C 62 DELR = RAD / RNR 63 DELS " 2.0 * PI / RNS 64 AREAO = (DELR*DELR) * PI / 4.0 65 DO 20 I = 1, NR 66 RI = FLOAT(I) 67 IF(I.EO.NR) GO TO 10 68 c •INNER* 69 AREA(I)=RI*DELR*DELR*DELS 70 GO TO 20 71 c •OUTER* 72 10 AREA(I)=0.5*(RNR-0.25)*DELR*DELR*DELS 73 c 74 20 CONTINUE 75 c 76 c NODE ARRANGEMENTS 77 C 78 NNEI = EBRAD / DELR + 0.5 79 CYCLE = 1.0 / HZ 80 TIM1 = CYCLE / RNS 81 ISMAX = TIM1 / DELT + 0.5 82 IF (ISMAX .LT. 1) GO TO 440 83 C 84 C PHYSICAL PROPERTIES (SI-UNIT) 85 C •HEAT CONDUCTION* 86 C 87 TIK = 28.2 88 CPTI = 690.0 89 FTIK = TIK * FF 90 ALPHA = FTIK / (CPTI*RHO) 91 FO = ALPHA * DELT / (DELR*DELR) 92 BI = HT * DELR / FTIK 93 C 94 C •RADIATION* 95 C 96 EMISS = 0.4 97 SIGMA = 5.67E-8 98 TAMBT = 298. 15 99 C 100 C •EVAPORATION* 101 C 102 R = 8.3144 103 TIMOL = 0.0479 104 ALMOL =0.027 105 GAMTI = 1.0 106 GAMAL = 0.028 107 HTI = 8.88E6 108 HAL = 1.08E7 109 C 1 10 c EB POWER 11 1 C 1 12 AEB = FLOAT(NNEI) * DELS * DELR * DELR 1 13 OIK •= POWERK / AEB 1 14 01 « OIK * 1000.0 1 15 c 1 16 C INITIAL CONCENTRATION OF ALLOY MASS BALANCE 122 L i s t i n g of MVSPOT at 1 1:29:43 on JAN 25, 1988 f o r CC i d = NHER on G 1 17 C 118 WTIO = WEIGHT * WPCTT / 100.0 119 WALO = WEIGHT - WTIO 120 XPNAL = WPCTA / ALMOL 121 XPNTI = WPCTT / TIMOL 122 XAL = XPNAL / (XPNAL + XPNTI) 123 XTI = 1 . 0 - XAL 124 C 125 C WRITE LIQUID DIMENSIONS AND ETC, 126 C 127 WRITE (6,480) RAD, DEPTH 128 WRITE (6,490) AREAH, VOLUME. WEIGHT 129 WRITE (6,500) EBRAD, POWERK, AEB, QIK, HZ 130 WRITE (6,510) NR, DELR, NS, DELS 131 WRITE (6,520) DELT, TIMEN 132 WRITE (6,530) FF, FTIK, HT, WPCTA 133 WRITE (6,540) IMAX. ISMAX, IFREQ 134 C 135 C SORT NODE TYPE 136 C 137 DO 70 1 = 1 , NR 138 DO 60 J = 1 . NS 139 IF ( J .EQ. 1) GO TO 40 140 - IF ( J .EQ. NS) GO TO 50 141 C *MIDDLE* 142 NTYPE(I.J) = 5 143 IF (I .EQ. 1) NTYPE(I.J) = 4 144 IF (I .EQ. NR) NTYPE(I.J) = 6 145 GO TO 60 146 C *INITIAL* 147 40 NTYPE(I.J) = 2 148 IF (I .EQ. 1) NTYPE(I.J) = 1 149 IF (I .EQ. NR) NTYPE(I.J) = 3 150 GO TO 60 151 C "LAST* 152 50 NTYPE(I.J) = 8 153 IF (I .EQ. 1) NTYPE(I.J) = 7 154 IF (I .EQ. NR) NTYPE(I.J) = 9 155 60 CONTINUE 156 70 CONTINUE 157 C 158 C TERMS 159 C 160 DO 80 I = 1, NR 161 RI = FLOAT(I) 162 A1(I) = (RI - 0.5) / RI 163 A2( I) = (RI + 0.5) / RI 164 A3(I) = 1.0 / (RI*RI*DELS*DELS) 165 80 CONTINUE 166 B = DELR * DELR / (DEPTH*FTIK) 167 C = 2.0 * DELS / PI 168 01 = 2.0 * (RNR - 0.5) / (RNR - 0.25) 169 D2 = 2.0 * RNR / (RNR - 0.25) 170 D3 = 1.0 / (RNR *(RNR - 0.25 ) *DELS*DELS) 171 C 172 C SET INITIAL CONDITIONS 173 C 174 TO = TEMPIK 123 L i s t i n g of MVSPOT at 11:29:43 on JAN 25, 1988 f o r CC1d=NHER on G 175 TOO « TEMPIK 176 TOC = TEMPIC 177 STIOJ = 0.0 178 SALOJ = 0.0 179 DO 100 I = 1, NR 180 DO 90 J = 1, NS 181 T ( I , J ) = TEMPIK 182 T O ( I . J ) = TEMPIK 183 T C ( I , J ) = TEMPIC 184 S T I J ( I , J ) = 0.0 185 S A L J ( I , J ) = 0.0 186 90 CONTINUE 187 100 CONTINUE 188 TIME = 0.0 189 I COUNT = 0 190 JNODE = 0 191 C 192 C WRITE INITIAL CONDITIONS 193 C 194 WRITE (6,550) TEMPIC 195 WRITE (7,600) HZ 196 WRITE (7,610) FF, FTIK, POWERK 197 WRITE (.7,620) EBRAD, NNEI 198 WRITE (7,630) TIME. TC(NNEI,1) 199 C 200 C START CALCULATION 201 C 202 1 10 NEB = 1 203 120 ICOUNT = ICOUNT + 1 204 IF (ICOUNT .GT. I MAX) GO TO 420 205 JNODE = JNODE + 1 206 TIME = FLOAT(ICOUNT)*DELT 207 C 208 C TEMPERATURE ADJUSTMENT FOR CALCULATION OF 209 C RADIATION & EVAPORATION HEAT LOSS 210 C 21 1 TOA = 1 . 5 * TO - 0 . 5 * TOO 212 DO 140 I = 1, NR 213 DO 130 J = 1, NS 214 T A ( I , J ) = 1.5 * T ( I , J ) - 0.5 * TO(I,J) 215 130 CONTINUE 216 140 CONTINUE 217 C 218 C RADIATION + EVAPORATION HEAT LOSS 219 C 220 QRO = OL(TOA) 221 DO 160 I = 1, NR 222 DO 150 J = 1, NS 223 TA1 = T A ( I , J ) 224 OR(I,J) = 0L(TA1) 225 150 CONTINUE 226 160 CONTINUE 227 C 228 C SUM OF TI & AL LOSS AT EACH NODE(KG/M2) 229 C 230 STIOJ • STIOJ + TIJ(TOA) * DELT 231 SALOJ = SALOJ + ALJ(TOA) * DELT 232 DO 180 I = 1, NR 124 L i s t i n g of MVSPOT at 11:29:43 on JAN 25. 1988 f o r CC1d=NHER on G 233 DO 170 J = 1. NS 234 TA1 = T A ( I . J ) 235 TIJ1 = TI J ( T A 1 ) 236 ALJ1 = ALJ(TA1 ) 237 S T I J ( I . J ) = S T I J ( I . J ) + TIJ1 * DELT 238 S A L J ( I . J ) = S A L J ( I . J ) + ALJ1 * DELT 239 170 CONTINUE 240 180 CONTINUE 241 C 242 C TOTAL TI 8 AL LOSS (KG) 243 C 244 ATIJ = STIOJ * AREAO 245 AALJ = SALOJ * AREAO 246 DO 220 I = 1, NR 247 DO 210 J = 1, NS 248 AATIJ = S T I J ( I . J ) * AREA(I) 249 AAALJ = S A L J ( I . J ) * AREA(I ) 250 C *SUM* 251 ATIJ = ATI J + AATIJ 252 AALJ = AALJ + AAALJ 253 210 CONTINUE 254 220 CONTINUE 255 C 256 C MASS BALANCE : ADJUST TI & AL QUANTITY IN MELT 257 C 258 WTI = WTIO - ATIJ 259 WAL = WALO - AALJ 260 WT = WTI + WAL 261 WPCTA = WAL / WT * 100.0 262 XNTI = WTI / TIMOL 263 XNAL = WAL / ALMOL 264 XNT = XNTI + XNAL 265 XAL = XNAL / XNT 266 XTI = 1 . 0 - XAL 267 C 268 C SET EB MOVEMENT 269 C 270 QEBO=0.0 271 DO 240 I = 1. NR 272 DO 230 J = 1. NS 273 QEB(I,J) = 0.0 274 IF (I .NE. NNEI) GO TO 230 275 IF ( J .EQ. NEB) QEB(I.J) = QI 276 230 CONTINUE 277 240 CONTINUE 278 C 279 C FINITE DIFFERENCE EQUATIONS 280 C 281 C "CENTER PART* 282 C 283 SUMT = 0.0 284 DO 250 J = 1, NS 285 SUMT = SUMT + T ( 1 , J ) 286 250 CONTINUE 287 TON=C*F0*(SUMT-RNS*TO)+(QEBO-QRO)*B*F0+TO 288 C 289 C SORT 290 C 125 L i s t i n g of MVSPOT at 11:29:43 on JAN 25, 1988 f o r CC1d=NHER on G 291 DO 360 1 = 1 , NR 292 DO 350 J = 1. NS 293 NTY = NTYPE(I.J) 294 GO TO (260, 270. 280, 290, 300. 310. 320. 330. 340). NTY 295 C 296 C *TYPE1* 297 C 298 260 TN(I,J)=FO*(A1(I)*T0+A2(I)*T(1 + 1 ,J) + A3(I )*(T(I,NS) + 299 1 T(I,J+1))+B*(OEB(I,J)-0R(I,J)))+(1.0-2.0*F0-2.0*FO* 300 2 A 3 ( I ) ) * T ( I . J ) 301 GO TO 350 302 C 303 C *TYPE2* 304 C 305 270 TN ( I , J ) = F 0 * ( A 1 ( I ) * T ( 1 - 1 , J ) + A 2 ( I ) * T ( 1 + 1 , J ) + A 3 ( I ) * ( 306 1 T(I,NS)+T(I,J+1))+B*(QEB(I,J)-QR(I,J)))+(1.0-2.0* 307 2 FO-2.0*FO*A3(I))*T(I.J) 308 GO TO 350 309 C 310 C *TYPE3* 311 C 312 280 TN(I,J)=F0*(D1*T(I-1,J)+D2*BI*TAMBT+D3*(T(I,NS)+T( 313 1 I.J+1))+B*(OEB(I,J)-OR(I,J)))+(1.0-D1*F0-D2*BI*F0 314 2 -2.0*D3*FO)*T(I,J) 315 GO TO 350 316 C 317 C *TYPE4* 318 C 319 290 T N ( I , J ) = F O * ( A 1 ( I ) * T 0 + A 2 ( I ) * T ( 1 + 1 , J ) + A 3 ( I ) * ( T ( I , J -320 1 1 )+T(I,J+1))+B*(OEB(I,J)-OR(I,J))) + (1.0-2.0*F0-321 2 2 . 0 * F O * A 3 ( I ) ) * T ( I , J ) 322 GO TO 350 323 C 324 C *TYPE5* 325 C 326 300 TN ( I , J ) = F 0 * ( A 1 ( I ) * T ( I - 1,J)+A2(I)*T(I + 1,J)+A3 (I )* ( 327 1 T ( I , J - 1 )+T(I,J+1 ) )+B*(OEB(I,J)-OR(I,J))) + (1.0-2. 328 2 0*FO-2.0*FO*A3(I))*T(I.J) 329 GO TO 350 330 C 331 C *TYPE6* 332 C 333 310 TN(I,J) = F0*(D1*T(1-1,J)+D2*BI*TAMBT+D3*(T(I , J-1) + 334 1 T(I,J+1))+B*(OEB(I,J)-OR(I.J)))+(1.0-D1*FO-D2*BI* 335 2 FO-2.0*D3*FO)*T(I,J) 336 GO TO 350 337 C 338 C *TYPE7* 339 C 340 320 TN ( I , J ) = FO*(A 1(I)*T0+A2(I)*T(1 + 1,J) + A 3 ( I ) * ( T ( I , J -341 1 1 )+T(I , 1))+B*(QEB(I,J)-QR(I.J)))+( 1 .0-2.0*F0-2.0*F0* 342 2 A3(I ) ) * T ( I , J ) 343 GO TO 350 344 345 C 346 C *TYPE8* 347 C 348 330 TN ( I , J ) = F O * ( A 1 ( I ) * T ( 1 - 1 , J ) + A2(I)*T(1+1 , J) + A3( I )*( 126 L i s t i n g of MVSPOT at 11:29:43 on JAN 25. 1988 for CC1d=NHER on G 349 1 T(I . J - 1 )+T(I,1))+B*(QEB(I,J)-QR<I.J))) + (1.0-2.0* 350 2 FO - 2.0*FO*A3(I))* T(I,J) 351 GO TO 350 352 C 353 C *TYPE9* 354 C 355 340 TN(I,J )=F0*(D1 *T(I-1.J )+D2*BI*TAMBT+D3*(T(I,J-1) + 356' 1 T(I, 1 ) )+B*(OEB(I,J)-OR(I.J))) + (1.0-D1*FO-D2*BI*FO-357 2 2.0*D3*FO)*T(I.J) 358 C 359 350 CONTINUE 360 360 CONTINUE 361 C 362 C TEMPERATURE MUST BE LESS THAN 3285.0 DEGC (3558.15 K) 363 C 364 IF(TON.GT.3558.15) T0N=3558.15 365 DO 364 1=1,NR 366 DO 362 J=1,NS 367 T1=TN(I,J) 368 IF (T1.GT.3558 .15) TN(I.J)= 3558.15 369 362 CONTINUE 370 364 CONTINUE 371 C 372 C SUBSTITUTE NEW TEMPERATURE IN OLD ONE 373 C 374 TOO = TO 375 TO = TON 376 DO 380 I = 1 , NR 377 DO 370 J = 1, NS 378 T0(I.J) = T(I,J) 379 T(I , J ) = TN(I,J) 380 370 CONTINUE 38 1 380 CONTINUE 382 C PRINT OUT CONTROL 383 IF ((ICOUNT/IFREO)*IFREO .NE. ICOUNT) GO TO 410 384 C 385 C CHANGE TEMPERATURE UNIT: K TO C 386 C 387 TOC = TO - 273.15 388 DO 400 I = 1, NR 389 DO 390 J = 1. NS 390 TC(I,J) = T(I,J) - 273.15 391 390 CONTINUE 392 400 CONTINUE 393 C 394 C WRITE TEMPERATURE & TI AND AL LOSS 395 C 396 WRITE(7,630)TIME.TC(NNEI.1),TC(3.2).TC(4.2),OEB( NNEI .1).ATIJ. 397 1 AALJ,WPCTA 398 C 399 C NEXT STEP & CONTROL 400 C 401 410 IF (JNODE .LT. ISMAX) GO TO 120 402 JNODE = 0 403 NEB = NEB + 1 404 IF (NEB .LE. NS) GO TO 120 405 GO TO 110 406 C 127 L i s t i n g of MVSPOT at 11:29:43 on JAN 25. 1988 f o r CC1d=NHER on G 407 C WRITE RESULTS AT END TIME 408 C 409 , 420 WRITE (6,560) TIME 410 ' 1 WRITE (6,565) TOC 411 DO 430 I » 1, NR 412 WRITE (6.570) (TC(I.J).J=1.NS) 4 13 430 CONTINUE 414 WRITE (6,580) ATIJ, AALJ, WPCTA 415 GO TO 1000 416 440 WRITE (6,590) 4 17 GO TO 1000 418 C 419 . C READ FORMAT 420 C 421 460 FORMAT (5F10.5) 422 470 FORMAT (5110) 423 C 424 C WRITE FORMAT 425 C 426 480 FORMAT (5X, ' "DIMENSION OF MOLTEN METAL**'/8X,' RADIUS''.F5.3, 427 1 'M' ,3X, 'DEPTH'' ,F5 . 3, 'M' ) 428 490 FORMAT (8X,'AREA=',E10.4,'M2',3X.'VOLUME=',E10.4,'M3',3X, 429 1 'WEIGHT=',F7.3,'KG'//) 430 500 FORMAT (5X,'**EB CONDITIONS**'/8X,'EB TRAVELLING RADIUS''.F5.3 431 1 'M'/8X,'BEAM POWER=',F7.3,'KW',3X,'BEAM AREA=',E10.4, 432 2 ' M2 ' , 3X, 'POWER DENSITY'' ,E15.3, 'KW/M2'/8X, 433 3 'FREQUENCY'',F7.2.'HZ'/) 434 510 FORMAT (5X, ' ""CALCULATING CONDITIOS**'/8X . 'RNODE : NUMBER' 435 1 I5.3X,'SIZE'',F7.4,'M'/8X.'THETA NODE: NUMBER'',15. 436 2 3X, 'SIZE'',F5.3. 'RAD'/) 437 520 FORMAT (SX.'TIME STEP'' ,F 10.3, ' SEC'/8X,'END TIME =',F10.3, 438 1 ' SEC'/) 439 530 FORMAT (8X, ' FF= ' . F7 . 2.3X, 'HEAT CONDUCTIVITY OF TI = ',F7.2, 440 1 'WM/K'/8X,'HEAT TRANSFER COEFFICIENT=',F10.3, 441 2 'W/M2/K'/8X,'INITIAL AL WEIGHT PERCENT'',F5.2/) 442 540 FORMAT (5X. '*"CONTROLLER**'/8X, 'TOTAL STEPS'', 15, 3X, 443 1 'STEPS/NODE''. 15, 3X, 'PRINT FREQUENCY'', 15///) 444 550 FORMAT (5X, ' ""RESULTS**'//8X, '* INITIAL TEMPERATURE'', F7.2, 445 1 'DEGC, 3X, 'THROUGHOUT IN MELT'//) 446 560 FORMAT (8X, '"TEMPERATURE DISTRIBUTION (DEGC) AT TIME'' .F 10.3, 447 1 ' SEC'/) 448 565 FORMAT (9X.F7.2) 449 570 FORMAT (8X, 10(1X,F7.2,1X)) 450 580 FORMAT (//8X, '"TOTAL TI LOSS'', E10.4, '(KG)"/8X, 451 1 '"TOTAL AL LOSS''. E10.4, '(KG)'//8X, 452 2 '"FINAL AL CONCENTRATION'', F5 . 2 , ' WT"/.' ) 453 590 FORMAT (SX.'TIME STEP IS LARGER THAN TO BE SPENT IN ONE NODE') 454 600 FORMAT (5X,'FREQUENCY=' . F7.2. 'HZ') 455 610 FORMAT (5X.'FF=', F7.2.4X.'HEAT CONDUCTIVITY OF MOLTEN TI=', 456 1 F7.2.'WM/K'/5X.'POWER INPUT'',F7.2.'KW'/) 457 620 FORMAT (/5X,'DATA AT A POINT( R='. F10.3. '(M):'.1X, 458 1 'RNODE NO.»',13, ' )'//7X, 'TIME(S) ' ,3X , 'TEMP.(C)' , 3X , 459 2 '"TMID.(C)*' , 2X , '"TOBS.(C)"' ,4X. 460 3 'EB(KW/M2)',6X . ' TI LOSS(KG)' .4X, 'AL LOSS(KG ) ' ,4X. 461 4 ' WT*/.AL' ) 462 630 FORMAT (3X.F10.3,3(2X.F10.2).3(5X.E10.4).2X,F7 . 2 ) 463 C 464 10O0 STOP 465 ENO 128 A-2-3 Flow chart of 2-D model ( S T A R T j >(p I C O U N T = 0, J N O D E = 0 , T I M E = 0.0 — f ~ Z NEB=1 t I C O U N T = I C O U N T + l J N O D E = J N O D E + l T I M E = T I M E + D E L T » Tempera ture ad jus t . Radiat ion & Evapora t ion Heat L o s s t T i & Al loss M a s s Ba lance F IN ITE D I F F E R E N C E C A L C U L A T I O N ( E N i [ ) 129 

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