\(x\) represents the position, \(T\) represents the temperature and \(t\) represents the time.
The area being considered is defined as \(S\) and its surroundings as \(\Gamma\). Assuming that Dirichlet or Neumann-type boundary conditions are given throughout \(\Gamma\), the boundary conditions become as follows:
\[\begin{align}
T = T_1(x, 5)& \qquad X \in \Gamma_1 \label{eq:2.4.2} \\\
k\frac{\partial T}{\partial n} = q(x, T, t)& \qquad X \in \Gamma_2 \label{eq:2.4.3}
\end{align}\]
The function form of \(T_1\) and \(q\) is known. \(q\) is the outflow heat flux from the boundaries.
With this program, it is possible to consider three types of heat flux.
where \(q_s\) is the distributed heat flux, \(q_c\) is the heat flux by the convective heat transfer, and \(q_r\) is the heat flux by the radiant heat transfer.
Furthermore,
\(Tc=Tc(x,t)\)
Convective heat transfer coefficient atmospheric temperature
\(hc=hc(x,t)\)
Convective heat transfer coefficient
\(Tr=Tr(x,t)\)
Radiation heat transfer coefficient atmospheric temperature
Eq.\(\eqref{eq:2.4.8}\) is a formula of non-linear and non-steady-state. The objective now is to discretize it in time by the backward Euler method and calculate the temperature at time \(t = t_0\) (when the temperature at \(t = t_0+\Delta t\) is known) with the following equation:
\[\begin{equation}
[K]_{t=t_0+\Delta t} \lbrace T \rbrace_{t=t_0+\Delta t} + [M]_{t=t_0+\Delta t}
\frac{\lbrace T \rbrace_{t=t_0+\Delta t} - \lbrace T\rbrace_{t=t_0}} {\Delta t}
= \lbrace F \rbrace_{t=t_0+\Delta t}
\label{eq:2.4.13}
\end{equation}\]
The next step is to improve the temperature vector \(\lbrace T \rbrace_{t=t_0+\Delta t}^{(i)}\) which approximately satisfies Eq.\(\eqref{eq:2.4.13}\) to determine the solution \(\lbrace T \rbrace_{t=t_0+\Delta t}^{(i)+1}\) with a good precision.
Therefore, the temperature vector must be expressed as follows:
\[\begin{equation}
\lbrace T \rbrace_{t=t_0+\Delta t}=
\lbrace T \rbrace_{t=t_0+\Delta t}^{(i)} + \Delta \lbrace T \rbrace_{t=t_0+\Delta t}^{(i)}
\label{eq:2.4.14}
\end{equation}\]
The product of the heat transfer matrix and temperature vector, as well as the mass matrix, are expressed approximately by the following equations:
\[\begin{align}
[K]_{t=t_0+\Delta t} \lbrace T \rbrace_{t=t_0+\Delta t} \nonumber
& \cong [K]_{t=t_0+\Delta t}^{(i)} \lbrace T \rbrace_{t=t_0+\Delta t}^{(i)} \\\
& + \frac{\partial [K]_{t=t_0+\Delta t}^{(i)}{\lbrace T \rbrace_{t=t_0+\Delta t}^{(i)}}}{\partial \lbrace T \rbrace_{t=t_0+\Delta t}^{(i)}}
\lbrace\Delta T \rbrace_{t=t_0+\Delta t}^{(i)}
\label{eq:2.4.15}
\end{align}\]
By substituting Eq.\(\eqref{eq:2.4.14}\), Eq.\(\eqref{eq:2.4.15}\) and Eq.\(\eqref{eq:2.4.16}\) into Eq.\(\eqref{eq:2.4.13}\), and omitting the terms of second or higher order, the following equation is obtained:
In non-steady-state analysis, the discretization in time is done through the implicit method; thus, the analysis is normally not affected by the restriction of the size of the time increment \(\Delta t\). However, if the time increment \(\Delta t\) is to large, the number of convergences in the iterative calculation increases. Therefore, this program is equipped with an automatic increment function, which constantly monitors the dimension of the residual vector in the iterative calculation process. If the convergence of the iterative calculation is too slow, it decreases the time increment \(\Delta t\). Moreover, when the number of iterative calculations is too small, it increases the time increment \(\Delta t\).