Laser Heat Processing: Advantages Laser radiation very "clean" no containments with other materials Working atmosphere can be controlled as needed

Transcription

1 Laser Heat Processing: Adantages Laser radiation ery "clean" no containments with other materials Woring atmosphere can be controlled as needed Lasers can be focused to small spots ery localized heating at ery high power Careful control of heating power Beam easy to direct into hard to access points Can pass beam through glass to isolated areas Energy generally deposited near the surface

2 General Heat Flow Laser heating is just lie other heat flow problems For the time being assume heat source note that this is not always possible for laser Heat flows by Fourier Law of Heat Conduction Q A A where Q = heat generated: Watts (some boos use H) = hermal conductiity W/m o C or W/cm o C or W/m o K heat flow per unit area per unite distance A = cross sectional area = temperature in o C z = depth into the material his is actually the definition of thermal conductiity Note this assumes steady state conditions laser heating stabilizes in 10-1 to sec hus assume steady state conditions

3 hermal Values of Materials hermal Conductiity changes with temperature Hence may change with position Specific Heat When heat enters a olume temperature change depends Specific Heat C of material, J/g/ o K Specific Heat determine how much heat needed to raise temperature of unit mass one degree C. Density Density of material Kg/m 3 Latent Heats Latent heats gie the energy required for a phase change units of J/g L f Latent Heat of Fusion: energy for melting L Latent Heat of Vaporization: energy to aporize

4 Basic Heat Flow Differential Equations let q. be the energy deposited per unit olume (use H in some boos) W/m 3 Consider heat flowing through a olume hen: energy at left face = change in internal energy + energy out right face hen energy per unit time in left face (Fourier s law) q A hermal Conductiity W/m/ o K If uniform, steady state get Newton s Law of Cooling q A Hence can calculate heat loss if now thermal conductiity

5 hermal Values of Materials hermal Conductiity changes significantly with materials Inerse of thermal conductiity lie resistance in electric circuit Newton heat flow means slope of conductie aries as 1/ q A In construction use R alues, which are 1/ but in British units hr-ft o F/BU Higher the R the less heat flow

6 Basic Heat Flow Differential Equations Now consider heat flowing through a olume In the general case there q deposited and q generated within hen: energy at left face + heat generated = change in internal energy + energy out right face Energy per unit time in left face q A Energy out of right face of +d is q q d q d A d Energy generated within the element per unite olume & time q gen qad Change (loss) in internal energy due to heating of material q int ernal CA t d

7 Basic Heat Flow Differential Equations hus writing the energy balance d ernal int gen q q q q d A d t CA qad A Combining gies heat flow DE t C q If assume thermal conductiity constant with position then t 1 q where hermal Diffusiity in m /s C

8 hermal Diffusiity hermal Diffusiity gies rise caused by an applied heat pulse or how rapidly heat diffuses through the material High thermal diffusiity: low surface temperature rise deep penetration of heat pulse Low thermal diffusiity: high surface temperature rise In three dimensional the heat flow DE becomes t 1 q Use for coordinate system that notes symmetry of problem For Cartesian z y Often useful to use cylindrical coordinates for laser Eample laser spots are circularly symmetric As angle is often uniform reduces from 3D to D problem alues Reduces to a problem in radius and depth But must use proper cylindrical polar equations to get right 1 1 r z r r r

9 emperature Change for Uniform Illumination Assume that the surface is uniformly illuminated by the laser Energy absorbed at the surface in a ery small depth H=I(1-R) where R = reflectiity I = light intensity he heat DE has been soled for depth z and time t (by Carslaw & Jaeger, 1959) H z ( z,t ) t ierfc t he ierfc is the integral of the complementary error function

10 Error Function Related Equations Heat flow equations are related to the Error Function erf erf ( ) e s ds his is the integral of a Gaussian between 0 and he Complementary Error Function erfc 0 erfc 1 erf erfc is the error function but integred from to infinity he ierfc is related to the error function by 1 ierfc erfc s ds ep 1 erf ierfc(1) = 0.05 and is falling rapidly

11 Useful Error Function Approimations Error function erf(), Complementry Error Function erfc() are erf ( ) e s ds s erfc( ) 1 erf ( ) e ds erf() hard to find but easy to approimate with erf ( ) 1 1 t 1 p a 1 t where a 0 t a 3 t 3 e p a 1 = , a = , a 3 = See Abramowitz & Segun (Handboo of Mathematical Functions) Error on this is < for all We are using complementary error function erfc() = 1 - erf() erfc(0) = 1 erfc() = 0 Approimation has <% error for << 5.5 For > 5.5 use asymptotic approimation e 1 erfc( ) 1 as Ecel & Quatropro spreadsheet hae erf() and erfc() built in. Must actiate analysis toolpac & soler first but become inaccurate for >5.4 then use asymptotic For > 5.4 then ierfc() becomes e ierfc( ) as

12 emperature Rise for Uniform Illumination From DE solution since ierfc is small for >1 hus find that rise is small when z t 1 Hence small rise when z 4 t Heat will diffuse a depth L in time of order L t 4 Change in surface temperature with time substitute z = 0 and note 0 ierfc 1 hus surface temperature change is: H (0,t ) t ierfc 0 hus temp increases with t H t

13 emperature Change with Finite ime Laser Pulse If hae a square pulse of duration t p he for t < t p follow the preious formula For ime greater than the pulse z,t z,t z,t t t t p eg. Consider Cooper with H = W/m for t p = 10-6 sec From table = m /s rises highest at surface (z=0) and changes fastest At pulse end heat has diffused about L p L 4 t 4( )(10 ) At depth pea occurs much later, and lower alues m

14 Laser Focused into a Spot If laser focused into uniform spot radius a then formula changes to (by Carslaw & Jaeger, 1959) ( z,t ) H t ierfc z t ierfc erm on right caused by sideways diffusion At the centre of the spot (z = 0 ) ( z,t ) H t his gies same as uniform heating if 1 ierfc z a t a ierfc 1 t his is true for ierfc(>1) thus a t 4 eg for Copper with a = 1 mm and = m /s, 3 10 t or t a t s

15 Laser Focused into a Spot As t goes to infinity (ery long times) it can be shown H z,t z a z hus for finite spot temperature reaches a limit Highest surface temp 0, Ha Effects of beam Gaussian distribution is not that different In practice as thermal conductiity, reflectance R, thermal diffusiity all ary with temperature hus tend to use numerical simulations for real details

16 Eample Focused Laser Spot Calculation

17 Phase Changes and Energy Balance Energy Balance: Energy in = Energy to raise temp + heat flow Note: a rough rule of thumb if near steady state half the energy goes into heat flow so energy required is twice that to raise temperature As heating increase will get melting of the surface Eentually also get aporization point All requires energy to heat In general the specific heat of the material changes C s = specific heat of solid C l = specific heat of liquid phase L f = Latent Heat of Fusion: energy for melting L = Latent Heat of Vaporization: energy to aporize Energy required to melt a unite olume of material E C L m s m f where m is the melting point, the starting temp. = density of material Note: this does not include energy lost to heat flow

18 Phase Changes and Energy Balance When aporization occurs E C s C m m L f L Generally true that heat capacity does not change much with C m C C Generally Latent heat of apourization >heat of fusion L f L Vapourization temperature is much > base or melting < m < Energy input required is approimately E C L

19 Melting Depths Consider light pulse on surface Will get melting to some depth Eentually also raise surface to aporization point his is the laser welding situation Can estimate the depth of melt front after some time t Recall temperature distribution ( z,t ) H t ierfc z t Ratio of the emperatures changes with depth are t ( z,t ) z ierfc (0,t ) t Eentually surface raises to aporization point It cannot rise higher without apourization thus stays at Hence can calculate the melt pool depth with this.

20 Melt Depth Estimate Estimate the depth of melt front for that after some time t Bottom of melt is at melting point, op at apourization point Assume base temperature is near 0 o C (ie ~room temp) z,t m 0,t Recall that at the surface m ierfc (0,t ) H z t t hus time can be eliminate by soling for Depth of melt is gien by H zmh ierfc t Note: for a gien material Hz m is fied hus large welding depths gien by low heat intensities applied for long time proided that there is sufficient energy in the beam m

21 Eample of Melting Calculations What is the heat flow required for weld depth of 0.1 mm in copper From the table for copper m = 1060 o C = 570 o C K = 400 W/m o C hus zmh ierfc From the graph or calculation hus H ( ) 400 z m m ierfc(=0.44) = ( ) W / m

22 Vaporization of Material When material remoed by aporization Get a melt front and a heated front liquid front moes with elocity s From the heat balance, assume that all power goes into heating hen the melt front should be H s C L where H is power density per square area Note this is the minimum power alue Good rule of thumb is actual power required twice this loss about the same by heat flow to substrate Can calculate depth d of holes by nowing laser pulse duration t p and front elocity d t s p

23 Eample Depth of Hole with Vaporization Heat pulse of H = W/m and t = 500 microsec hits copper. What will be the resulting ma hole depth From the tables = 570 o C = 8960 g/m 3 C = 385 J/g o C L = J/g d t s p Ht p C L 3 m 0.95mm

24 Keyholes and Increased Welding/Cutting Depths When the laser forms hole in material Beam penetrates to much greater depth Creates a large deep melt pool behind moing beam Melt fills in hole behind moing beam If not true welding limited 1 mm in steel

25 Keyholes Formulas Modeled by Swift, Hoo and Gic, 1973 Assume linear heat source power P (W) Note P is total power while H is unreflected power because eyhole absorbs all the power (reflections do no escape) Etends into metal depth a Moing forward with elocity (weld speed) in direction y direction across weld is (centred on the heating point) emperature distribution becomes P y ep K 0 a where K 0 is the Bessel function nd ind order 0 Width w of the weld is gien by the point where =melting P w a m