Rosenthal solution

The Rosenthal solution defines the steady state thermal field produced by a point heat source moving on the surface of a semi-infinite solid. In an Eulerian reference frame with x-axis pointing towards the heat source velocity and originating at its location:

$$T=T_0 + \frac{P}{2\pi d \lambda}\exp{\left[-\frac{U}{2\kappa}\left(d + x\right)\right]}$$

where:

$T_0$ is the initial (or preheating) temperature
$P$ is the thermal power transferred to the solid
$U$ is the velocity magnitude of the heat source relative to the solid
$d$ is the distance from the heat source
$\lambda$ is the thermal conductivity
$\kappa$ is the thermal diffusivity.

This solution is represented below on the domain surface given the process parameters, material properties, and a temperature threshold that — in addition to limiting the contour interval for display purposes — identifies the isotherm whose transverse size (half-width or penetration) and length-to-width aspect ratio are specified in the output fields.

Mattia Moda

Parameters and properties

Velocity (mm/s):

Thermal conductivity (W/m/K):

Threshold penetration:

Power (W):

Thermal diffusivity (mm²/s):

Threshold aspect ratio:

Preheating (K):

Temperature threshold (K):

Fix axes:

Please note that the Rosenthal solution implies — besides those already mentioned — the assumptions of adiabatic boundary and homogeneous-isotropic material with thermophysical properties independent of temperature, thus neglecting advection, radiation, convection, and phase transitions. In addition, the concentrated heat input originates a temperature singularity at the origin. On the flipside, owing to its axisymmetry about the motion trajectory, the above solution applies, more generally, to a point heat source moving on the apex of an infinite wedge of angle $\theta \in \left(0, 2\pi\right]$, provided that $P$ is multiplied by $\frac{\pi}{\theta}$.