In heat transfer, moving heat sources is an engineering problem, particularly in welding. In the early 20th century, welding engineers began studying moving heat sources in thin plates, both empirically and theoretically. Depending on welding parameters, plate geometry and material properties, the solution takes three different forms: semi-infinite, intermediate, or thin plate. The temperature distribution and cooling rates can be determined from theoretical solutions to the problem, allowing engineers to better understand the consequences of heat sources on weldability and end item performance.
Historical solutions
Empirical In the 1930s metallurgists Albert Portevin and D. Seferian attempted to experimentally determine heat transfer characteristics in welding. They correlated the effects of several factors—material properties, welding process, and part dimensions—on temperature distribution, by performing oxyacetylene (gas) and covered electrode (arc) welds on plates and bars of various profiles, and multiple materials, including steel, copper, and aluminum. Their work showed that arc welding temperature gradients were steeper and cooling rates were faster than those of gas welding, which were more sensitive to material thickness than those of arc welding. In addition to process, material properties, and dimensions, the authors noted that preheat played a role in temperature distribution. G.E. Claussen and W. Sparagen did not detail other attempts to determine temperature distribution in welding, because the variety of approaches employed by the investigators resulted in data that were not comparable. They did note that the data generally revealed the effect of weld process on heat affected zone (HAZ) width, with gas welding having the widest HAZ, bare electrode arc processes the narrowest, and covered electrode falling in the middle.
Theoretical Until the mid-1930s the study of the theory of heat transfer from a moving source was neglected, and temperature distribution due to moving heat sources could only be calculated approximately. In 1935, Daniel Rosenthal published the first literature applying the exact theory of heat flow from a moving source to arc welding. Rosenthal's theoretical model included several assumptions:
Material properties are constant The heat source is a point source The surface of the work piece does not lose heat to the atmosphere Heat created by the Joule effect is neglected Rosenthal's solution has been shown to agree well with measured results over a wide range of parameters, although with some scattering of data. The assumption of a point, line, or plane heat source leads to inaccuracy in the vicinity of the fusion zone (where temperature is within about 20% of the melting temperature) and prohibits predicting the shape of the weld pool. Following Rosenthal, researchers were able to approximate weld pool shape by assuming a Gaussian heat source defined by the equation:
Q ( x , y ) = q 2 π σ 2 e − ( x 2 + y 2 ) 2 σ 2 {\displaystyle Q(x,y)={q \over 2\pi \sigma ^{2}}e^{-{(x^{2}+y^{2}) \over 2\sigma ^{2}}}}
where:
Q : heat source, q : net power input, σ : distribution parameter. and later, other heat source distributions, such as semi-ellipsoidal and double ellipsoidal.
Equations The governing equation for 3D transient heat transfer in a solid of semi-infinite dimensions, with no heat generation or surface losses, is:
λ ∂ 2 θ ∂ x 2 + λ ∂ 2 θ ∂ y 2 + λ ∂ 2 θ ∂ z 2 = ρ C ∂ θ ∂ t {\displaystyle \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}}
where:
θ : temperature, x : direction parallel to weld travel, y : direction in plane and perpendicular to weld travel, z : through-thickness direction, λ : thermal conductivity, ρ : density, t : time, C : specific heat. In the case of a moving heat source applied to a plate that is so thin that temperature does not vary in the through-thickness dimension, the third term becomes zero, and the problem is two-dimensional conduction. The factors that determine whether temperature varies through the thickness include:
… excerpt ends here. Continue reading the full article.
