## 3 Thermal Deformation Model for Steel Joint
### Establishing Temperature Function  
The temperature \(T\) is a function of \(r\). The function relationship between \(T\) and radius \(r\) is studied as follows.  
Because the joints temperatures were changing uniformly, there is a linear relationship between the temperature function and \(r\). The least square linear method is used to fit the temperature function \(T\left(r\right)\).  
The linear equation is as follows:  
\[T=c+d\cdot r+\varepsilon \tag{21}\]  
where \(\varepsilon\) is the deviation; and \(c\), \(d\) are regression coefficients.  
There would be  
\[\sum\limits_{i=1}^{n}\varepsilon_{i}^{2}=\sum\limits_{i=1}^{n}\left(T_{i}-c- dr_{i}\right)^{2} \tag{22}\]  
The least-squares constraint would be:  
\[Q=\sum\limits_{i=1}^{n}\varepsilon_{i}^{2}=\sum\limits_{i=1}^{n}\left(T_{i}-c -dr_{i}\right)^{2}=\min \tag{23}\]  
where \(Q\) is the sum of the squared deviations.  
We take partial derivatives of c and d, respectively, and set them equal to 0.  
\[\left\{\begin{array}{l}\frac{\partial Q}{\partial c}=2\sum\limits_{i=1}^{n} \left(T_{i}-c-dr_{i}\right)=0\\ \frac{\partial Q}{\partial d}=2\sum\limits_{i=1}^{n}\left(T_{i}-c-dr_{i}\right) r_{i}=0\end{array}\right. \tag{24}\]  
Further, there would be  
\[\left\{\begin{array}{l}\sum\limits_{i=1}^{n}T_{i}-nc-d\sum\limits_{i=1}^{n} r_{i}=0\\ \sum\limits_{i=1}^{n}T_{i}r_{i}-c\sum\limits_{i=1}^{n}r_{i}-d\sum\limits_{i=1}^{n }r_{i}^{2}=0\end{array}\right. \tag{25}\]  
The estimated parameter values of \(c\) and \(d\) would be  
\[\left\{\begin{array}{l}\dot{d}=\frac{\sum\limits_{i=1}^{n}r_{i}\ T_{i}-n \overline{r}\ \overline{r}}{\sum\limits_{i=1}^{n}r_{i}-n\overline{r}\ \overline{r}}\\ \dot{c}=\ \overline{T}-d\ \overline{r}\end{array}\right. \tag{26}\]  
where \(\overline{T}\) is the average value of \(T_{i}\), \(\overline{T}=\frac{1}{n}\sum\limits_{i=1}^{n}T_{i}\); and \(\overline{r}\) is the average value of \(r_{i}\), \(\overline{r}=\frac{1}{n}\sum\limits_{i=1}^{n}r_{i}\).  
Because \(a\leq r_{i}\leq b\), \(r_{1}=a\) and \(r_{5}=b\), \(r_{2}\) to \(r_{4}\) are equally spaced. For point B there are \(r_{1}=0.069\), \(r_{2}=0.074\), \(r_{3}=0.079\), \(r_{4}=0.084\) and \(r_{5}=0.089\). According to Equation (26), we get the linear equation coefficients \(c\) and \(d\) of T in cross-sections of points B under 0.016 m/s, 0.038 m/s and 0.061 m/s at 300 \({}^{\circ}\)C, 400 \({}^{\circ}\)C, 500 \({}^{\circ}\)C, as shown in Table 2.  
According to Formula (20), the sectional radial deformation is as shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{\(v\)/m/s} & \multicolumn{2}{c}{**300 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**400 \({}^{\circ}\)C**} & \multicolumn{2}{c}{**500 \({}^{\circ}\)C**} \\ \cline{2-7}  & \(d\) & \(c\) & \(d\) & \(c\) & \(d\) & \(c\) \\ \hline
0.016 & 113.75 & 282.35 & 264.00 & 362 58 & 474.25 & 443.28 \\
0.038 & 164.75 & 276.93 & 295.00 & 353.75 & 327.75 & 422.93 \\
0.061 & 229.75 & 270.25 & 365.00 & 339.32 & 345.00 & 377.50 \\ \hline \end{tabular}
\end{table}
Table 2: Linear equation coefficients of point B.  
As can be seen from Figure 7, the increase rate of thermal deformation is increased with the increase of \(r\). The deformations of the three cooling water flow rates are very similar when the temperature is 300 \({}^{\circ}\)C. With increasing initial heating temperatures, the deformation difference under three cooling water flow rates increases rapidly. In a word, with increasing initial heating temperatures, the change of the same cooling water flow has a greater influence on the thermal deformation.