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A two-dimensional transient model for convection in laser melted pool

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Abstract

A two-dimensional transient model for convective heat transfer and surface tension driven fluid flow is developed. The model describes the transient behavior of the heat transfer process of a stationary band source. Semi-quantitative understanding of scanning is obtained by a coordinate transformation. The non-dimensional forms of the equations are derived and four dimensionless parameters are identified, namely, Peclet number (Pe), Prandtl number (Pr), surface tension number(S), and dimensionless melting temperature(@#@ Tm * @#@). Their governing characteristics and their effects on pool shape, cooling rate, velocity field, and solute redistribution are discussed. A numerical solution is obtained and presented. Quantitative effects of Prandtl number and surface tension number on surface velocity, surface temperature, pool shape, and cooling rate are presented graphically.

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Abbreviations

C :

constant which defines the interface

d :

width of laser beam

D :

thickness of workpiece

D eff :

effective diffusion coefficient

k :

thermal conductivity

I :

length of laser beam

L :

length of workpiece

n:

normal vector along the interface

p :

pressure

Pe:

Peclet numberu o d/k

Pr:

Prandtl numberv / k

q :

net heat flux from laser

Re:

Reynolds numberu o d/v

r o :

radius of laser beam

S :

surface tension number

T :

temperature

T m :

melting temperature

T* m :

dimensionless melting temperature °K

T∝ :

temperature of metal when it is not heated

u:

velocity vector

u:

x-component of u

u o :

scanning speed of the laser beam

u n :

normal velocity of the interface

v:

y-component of u

w :

z-component of u

W :

width of workpiecex,y,z Cartesian coordinate

p :

density

k :

thermal diffusivity

λ:

latent heat of fusion

v :

kinematic viscosity

μ :

viscosity

σ :

surface tension

l :

liquid

s :

solid

i, j :

indices of node for the discretization of thex,y plane

m :

melting

∝:

ambient

*:

dimensionless quantities

':

derivative with respect to its independent variable

n :

n th time step in the finite difference equation

∇:

del operator i ∂/∂x/+j ∂/∂y

(u ⋅ ∇ ):

convective operator u ∂/∂x/+v∂/∂y

∇ 2 :

Laplacian operator ∂2/∂x2+ ∂2/∂y2

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Authors and Affiliations

  1. The University of Illinois, 1206 West Green Street, 61801, Urbana, IL

    C. Chan (Research Assistant), J. Mazumder (Associate Professor) & M. M. Chen (Professor)

Authors
  1. C. Chan
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  2. J. Mazumder
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  3. M. M. Chen
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Additional information

This paper is based on a presentation made at the symposium “Fluid Flow at Solid-Liquid Interfaces” held at the fall meeting of the TMS-AIME in Philadelphia, PA on October 5, 1983 under the TMS-AIME Solidification Committee.

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Chan, C., Mazumder, J. & Chen, M.M. A two-dimensional transient model for convection in laser melted pool. Metall Trans A 15, 2175–2184 (1984). https://doi.org/10.1007/BF02647100

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