# A GTA Welding Cooling Rate Analysis on Stainless Steel and Aluminum Using Inverse Problems
## 2 Materials and Methods  
### Thermal Model  
Figure 1 describes the GTA welding problem which consists of a GTA torch that applies a heat flux over an aluminum plate. This GTA torch follows direction \(s\), with velocity \(u\). The heat input will cause a thermal gradient on the plate that could be described by the heat diffusion equation. When the welding starts, a heat loss due to the spontaneous effects of convection and radiation begins.  
The GTA welding process may be expressed by the heat diffusion equation for space:  
\[\frac{\partial}{\partial x}\bigg{(}\lambda(T)\frac{\partial T}{\partial x}\bigg{)} +\frac{\partial}{\partial y}\bigg{(}\lambda(T)\frac{\partial T}{\partial y} \bigg{)}+\frac{\partial}{\partial z}\bigg{(}\lambda(T)\frac{\partial T}{ \partial z}\bigg{)}=\rho\frac{\partial H(T)}{\partial t}, \tag{1}\]  
where \(x\), \(y\) and \(z\) are the Cartesians coordinates, \(T\) the temperature, \(\lambda\) the thermal conductivity, \(\rho\) the density. The enthalpy function \(H\) is defined as:  
\[H=\int CdT+fL, \tag{2}\]  
where \(C\) is the specific heat, \(L\) the latent heat and \(f\) is the Heaviside step function defined as a function of the melting temperature, \(T_{m}\):  
\[\begin{array}{lcl}f(T)=1&&T>T_{m}\\ f(T)=0&&T<T_{m}\end{array} \tag{3}\]  
The problem analyzed is subject to the boundary conditions of convection and radiation:  
\[-\lambda(T)\frac{\partial T}{\partial\eta}=h(T)(T-T_{\infty})+\sigma\epsilon (T)\Big{(}T^{4}-T_{\infty}^{4}\Big{)}, \tag{4}\]  
where \(\eta\) is the normal direction, \(h\) the heat transfer coefficient by convection, \(\sigma\) the Stefan-Boltzmann constant, \(\epsilon\) the emissivity and \(T_{\infty}\) the room temperature.  
The following boundary condition of heat flux, \(q^{\prime\prime}\), is applied to the area \(A_{xy}\):  
\[-\lambda(T)\frac{\partial T}{\partial z}=q^{\prime\prime}(x,y,t), \tag{5}\]  
The initial condition of imposed temperature is used for the entire domain as:  
\[T(x,y,z,0)=T_{\infty}, \tag{6}\]  
The solution of the temperature field is obtained through the numerical approximation of Equation (1) by using finite difference method with the implicit formulation in time. The linear system of algebraic equations is solved by using MSI (modified strongly implicit) procedure [10].  
The heat flux, \(q(x,y)\) (see Figure 1) is applied to a circular region with radius \(r\) and area \(A_{xy}\). It has a Gaussian distribution and releases its energy continuously over the time as it moves with a constant positive velocity \(u\) in the \(x\) direction [2]:  
Figure 1: Scheme representation of the three-dimensional thermal welding process.  
\[q(x,y)=\frac{3Q}{\pi\times r^{2}}e^{-3\frac{(x-y)^{2}}{r^{2}}}e^{-\frac{3r^{2}}{r^ {2}}} \tag{7}\]  
where the \(Q\) is the heat rate function estimated through BFGS inverse technique [11].  
Due to the temperature gradients in the air and the gravitational field, there is an induction of natural convection currents around the sample. The Nusselt number obtained from empirical correlations of the literature [12] was used to determine the convection coefficient \(h\).