User:Benatar.1/Article 8

= Moving Heat Source in Thin Plates = Moving Heat Sources is a topic in heat transfer that applies to determination of temperature distribution during welding. In 1935, Daniel Rosenthal published the first literature applying the exact theory of heat flow from a moving source to arc welding. Prior to Rosenthal's publication, the problem of heat flow from a moving point source had only been solved approximately. Although Rosenthal's work can be applied to various welding scenarios, it is particularly evident in welding a butt joint between two uniform plates. Depending on welding parameters, plate geometry and material properties, the solution takes three different forms: semi-infinite, intermediate, or thin plate.

If a moving heat source is applied to a thin plate, the problem becomes a 2 dimensional heat transfer problem, with constant temperature through the thickness of the plate at any point. The plate is considered thin if a dimensionless thickness (ζ) that is a function of heat source velocity (V), thickness (h), and thermal diffusivity (κ), is much less than unity.

$$\zeta=\frac{Vh}{2\kappa}$$

Governing Heat Transfer Problem
[INSERT PICTURE OF ARC ON A SURFACE WITH COORDINATE SYSTEM]

The governing equation for three dimensional transient heat transfer in a plate of infinite dimensions, no heat generation, and no surface convection or radiation, with temperature (θ), through thickness direction (z), direction parallel to heat source movement (x), direction perpendicular to heat source movement (y), time (t), thermal conductivity (λ), density (ρ), and specific heat (C) is:

$$\lambda{\partial^2\theta\over\partial x^2}+\lambda{\partial^2\theta\over\partial y^2}+\lambda{\partial^2\theta\over\partial z^2} = {\rho C}{\partial\theta\over\partial t}$$

Because the plate is thin relative to the travel speed and thermal diffusivity, there is insignificant thermal gradient in the through thickness direction, and the third term is removed. For convenience, the problem is solved in a new coordinate system that moves in the same direction and with same speed as the heat source. When observed from the new moving origin, the temperature profile is constant in time, and the term on the right is removed. Rosenthal refers to this condition as "quasi-stationary state." The through thickness direction (z) and direction perpendicular to the direction of travel (y) are unchanged, but the direction parallel to travel (w) is:

$$w=x-Vt$$

[INSERT PICTURE OF MOVING COORDINATE SYSTEM SUPERIMPOSED ON FIXED COORDINATE SYSTEM]

Because the heat source does not appreciably raise the temperature of the plate at distances far from the source, two boundary conditions apply--temperature gradient perpendicular to travel is zero at large values of y; and temperature gradient parallel to travel is zero at large values of w. Another boundary condition that is attributed to Rosenthal is considering the rate of energy, or power (P) transferred from the arc to the plate is equal to the heat transferred outward from a cylinder with height equal to the thickness of the plate and infinitely small radius :

$$P=\lim_{r \to 0}-{\partial\theta \over \partial r}2\pi r\lambda h $$

The assumed form of the solution to the differential equation is the sum of the initial temperature and the product of an unknown function (φ) of location and an exponential function of location in the travel direction. Taking advantage of his treatment of arc power, Rosenthal converts the "quasi-stationary" differential equation from Cartesian coordinates to radial coordinates, with the dimension r equal to distance from the source in the plane of the plate (for any value of z) :

$${\partial^2\varphi\over\partial r^2}+{1\over r}{\partial\varphi\over\partial r} - {\Bigl({V\over 2\kappa}\Bigr)}^2\varphi=0$$

Subject to the following:

$${\partial\theta\over\partial w}=0$$ for $$w\rightarrow\infty$$

$${\partial\theta\over\partial y}=0$$ for $$y\rightarrow\infty$$

Solution
The solution of the radial "quasi-stationary" equation is the modified Bessel function of the second kind and zeroth order:

$$\varphi=K_0\Bigl({Vr\over 2\kappa}\Bigl)$$

Substituting φ into the equation Rosenthal assumed for the solution of the original differential equation:

$$\theta=\theta_0+\Biggl(\frac{P}{2\pi\lambda h}\Biggr)e^{\Bigl(-\frac{Vw}{2\kappa}\Bigr)}K_0\Biggl(\frac{Vr}{2\kappa}\Biggr)$$

[INSERT MY VIDEO FROM ABAQUS OF HEAT TRANSFER IN A THIN PLATE]

Applications
Rosenthal's solution of temperature distribution due to a moving heat source has several practical uses in welding engineering, including calculating peak temperature and cooling rate, which have implications in metallurgical results of welding, and as-welded properties of joints.