A Convective Heat-Transfer Model for Partial and Full Penetration Keyhole Mode Laser Welding of a Structural Steel

Abstract

In the keyhole mode laser welding of many important engineering alloys such as structural steels, convective heat transport in the weld pool significantly affects temperature fields, cooling rates, and solidification characteristics of welds. Here we present a comprehensive model for understanding these important weld parameters by combining an efficient keyhole model with convective three-dimensional (3-D) heat-transfer calculations in the weld pool for both partial and full penetration laser welds. A modified turbulence model based on Prandtl’s mixing length hypotheses is included to account for the enhanced heat and mass transfer due to turbulence in the weld pool by calculating spatially variable effective values of viscosity and thermal conductivity. The model has been applied to understand experimental results of both partial and full penetration welds of A131 structural steel for a wide range of welding speeds and input laser powers. The experimentally determined shapes of the partial and full penetration keyhole mode laser welds, the temperature profiles, and the solidification profiles are examined using computed results from the model. Convective heat transfer was the main mechanism of heat transfer in the weld pool and affected the weld pool geometry for A131 steel. Calculation of solidification parameters at the trailing edge of the weld pool showed nonplanar solidification with a tendency to become more dendritic with increase in laser power. Free surface calculation showed formation of a hump at the bottom surface of the full penetration weld. The weld microstructure becomes coarser as the heat input per unit length is increased, by either increasing laser power or decreasing welding velocity.

Appendices

Appendix I

Because the orientation of keyhole is almost vertical, the heat transfer at the keyhole wall takes place mainly along the horizontal plane. A heat balance on the keyhole wall gives the following relation for local keyhole wall angle θ:[7,8]

$$ \tan {\left( \theta \right)} = \frac{{I_{c} }} {{I_{a} - I_{v} }} $$

(Ia)

where I _c is the radial heat flux conducted into the keyhole wall, I _a is the locally absorbed beam energy, and I _v is the evaporative heat flux on the keyhole wall. The value of I _c is obtained from a 2-D temperature field in an infinite plate with reference to a linear heat source. The term I _c is defined as

$$ I_{c} {\left( {r,\varphi } \right)} = - \lambda \frac{{\partial T{\left( {r,\varphi } \right)}}} {{\partial r}} $$

(Ib)

where (r,φ) designates the location in the plate with the line source as the origin, T is the temperature, and λ is the thermal conductivity. The 2-D temperature field can be calculated considering the conduction of heat from the keyhole wall into the infinite plate as[41]

$$ T{\left( {r,\varphi } \right)} = T_{a} + \frac{{P^{\prime } }} {{2\pi \lambda }}K_{0} {\left( {\Omega r} \right)}e^{{ - \Omega r\cos \varphi }} $$

(Ic)

where T _a is the ambient temperature, $P^{'}$ is the power per unit depth, K ₀() is the solution of the second kind and zero-order modified Bessel function, and Ω = v/(2κ), where v is the welding speed and κ is the thermal diffusivity.

The locally absorbed beam energy flux, I _a , on the keyhole wall that accounts for the absorption during multiple reflections and the plasma absorption is calculated as[8]

$$ I_{a} = e^{{ - \beta l}} {\left( {1 - {\left( {1 - \alpha } \right)}^{{1 + \pi /4\theta }} } \right)}I_{0} $$

(Id)

where β is the inverse Bremsstrahlung absorption coefficient of plasma, l is the average path of the laser beam in plasma before it reaches the keyhole wall, α is the absorption coefficient of the work piece, θ is the average angle between the keyhole wall and the initial incident beam axis, and I ₀ is the local incident beam intensity that varies with depth from the surface and radial distance from the beam axis.[8] Constant laser beam absorption coefficient, independent of location, is assumed for the plasma in the keyhole and for the laser beam absorption at the keyhole wall. The keyhole profile is first calculated without considering multiple reflections. With the approximate keyhole dimensions, the average angle between the keyhole wall and incident beam axis is then calculated.[8]

The factor $1 + \frac{\pi } {{4\theta }}$ represents the average number of reflections that a laser beam undergoes before leaving the keyhole.[8] When a laser beam of intensity I ₀ traverses a length l in the plasma before reaching the material surface, (1−e ^−βl)I ₀ is absorbed by the plasma. Of the remaining −e ^−βl I ₀ that falls on the material, (1−α)−e ^−βl I ₀ is reflected. After $1 + \frac{\pi } {{4\theta }}$ reflections, (1−α)^1+π/4θ e ^−βl I ₀ of the intensity is reflected and the remaining (1− (1−α)^1+π/4θ)e ^−βl I ₀ is absorbed. For α = 0.16 and $1 + \frac{\pi } {{4\theta }}$ = 6, l = 0.5 mm, about 5 pct of the local beam intensity is absorbed by the plasma,[44,45] about 65 pct is absorbed by the material, and the remaining 30 pct leaves the keyhole.

The evaporative heat flux, I _v , on the keyhole wall is given as

$$ I_{v} = {\sum\limits_{i = 1}^n {J_{i} \Delta H_{i} } } $$

(Ie)

where n is the total number of alloying elements in the alloy, ΔH _i is the heat of evaporation of element i, and J _i is the evaporation flux of element i given by the modified Langmuir equation:[46–48]

$$ J_{i} = \frac{{a_{i} P^{0}_{i} }} {{{\text{7}}{\text{.5}}}}{\sqrt {\frac{{M_{i} }} {{2\pi {\text{R}}T_{b} }}} } $$

(If)

where a _i is the activity of element i, $P^{0}_{i}$ is the equilibrium vapor pressure of element i over pure liquid at the boiling point T _b , and M _i is the molecular weight of element i. The factor 7.5 is used to account for the diminished evaporation rate at one atmosphere pressure compared to the vaporization rate in vacuum and is based on previous experimental results.[46,47]

Appendix II

Recoil pressure exerted by the metal vapors can be given by the difference between the momentum of the vapors leaving the surface and the z-direction momentum of the liquid near the liquid-vapor interface.

$$ P_{{{\text{rec}}}} = \rho _{g} v^{2}_{g} - \rho _{l} v^{2}_{l} $$

(IIa)

The subscripts g and l stand for gas and liquid, respectively, and the velocities normal to the liquid-vapor interface. Because, by mass conservation,

$$ \rho _{g} v_{g} = \rho _{l} v_{l}\ {\text{and }}\rho _{g} \ll \rho _{l} $$

(IIb)

$$ v_{g} \gg v_{l} $$

(IIc)

Therefore,

$$ P_{{{\text{rec}}}} = {\left( {\rho _{g} v_{g} } \right)}v_{g} - {\left( {\rho _{l} v_{l} } \right)}v_{l} = {\left( {\rho _{g} v_{g} } \right)}{\left( {v_{g} - v_{l} } \right)} \approx \rho _{g} v^{2}_{g} $$

(IId)

$$ P_{{{\text{rec}}}} \approx J_{g} v_{g} $$

(IIe)

where J _g is the vaporization flux, which can be calculated from the Langmuir equation given in Eq. [IIf]:

$$ v_{g} = J_{g} /c $$

(IIf)

where c is the concentration of the metal vapors and is given by

$$ c = \frac{{Mp_{v} }} {{{\text{R}}T}} $$

(IIg)

where M is the molecular weight of iron, and p _v is the vapor pressure of iron at temperature T, calculated from an empirical relation.[49]