# Effects of Valve, Armature, and Armature Pin Guidance on Diesel Injector Performance
## 3 Results and Discussion
### Correlation Matrix  
To understand the interactions among the pertinent parameters of the injector, the actual behavior of the injector was analyzed using a Python program. Subsequently, a correlation matrix was generated based on the obtained results. In Python, correlation matrices are calculated using libraries such as \"pandas,\" which computes the correlation coefficient between pairs of variables in a dataset. The correlation coefficient, often denoted by \(\rho\) (rho) or \(r\), measures the strength and direction of the linear relationship between two variables. There are different methods to compute the correlation coefficient; this study utilized the Pearson correlation coefficient, which measures linear correlation between two variables. It is calculated as the covariance of the two variables divided by the product of their standard deviations. Mathematically, for two variables \(X\) and \(Y\), the Pearson correlation coefficient \(r\) is calculated as follows:  
\[r=\frac{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)\left(Y_{i}-\overline{Y} \right)}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\overline{X}\right)^{2}}\sqrt{\sum_{i =1}^{n}\left(Y_{i}-\overline{Y}\right)^{2}}}, \tag{7}\]  
where \(X_{i}\) and \(Y_{i}\) are the individual data points, \(\overline{X}\) and \(\overline{Y}\) are the means of \(X\) and \(Y\), respectively, and \(n\) is the number of data points. The output of this calculation is a correlation matrix for the variables \(X\) and \(Y\), where each entry represents the correlation coefficient between corresponding variables. The correlation coefficient ranges from \(-1\) to \(1\):  
* \(r=1\) indicates a perfect positive linear relationship;
* \(r=-1\) indicates a perfect positive linear relationship;
* \(r=0\) indicates no linear relationship. This study revealed significant interactions between the functions of the injector and other sub-component parameters. These findings enable providing feedback to independent processes for optimal production without physical prototyping. The interpretation of the correlation matrix is as follows:  
Figure 16: Nozzle hole flow rate at _maxP_. * The terms such as \"valve inner diameter\", \"armature inner diameter\", \"armature outer diameter\", and \"clearance between the armature and armature pin\" in the correlation matrix represent inputs, while the other terms represent results. * A value approaching +1 indicates a strong positive interaction between two variables, suggesting that as one variable increases, the other also increases. * A value approaching \(-1\) indicates a strong negative interaction between two variables, where one variable's increase corresponds with the other's decrease. * A value approaching 0 suggests a weak or no interaction between two variables, indicating they are not significantly affected by each other. The correlation matrix, encompassing all groups, was generated using the Python program, incorporating flow rates obtained from the simulation. This matrix allowed us to identify characteristics that interacted significantly across all groups. Figure 17 reveals a robust positive correlation between the backflow rate from the armature to the armature pin, the total backflow rate, and the spray amount at maximum pressure. Conversely, there is minimal correlation between the backflow rate from the armature to the valve and the total backflow rate. This suggests that the primary determinant of backflow is the interaction between the armature and armature pin. This is attributed to the mechanism by which fuel, upon contact with the valve when the armature is open, is directly expelled out of the valve and into the main tank rather than moving between the armature and valve. Figure 17: Correlation matrix including all groups. Backflow rates increase with injection pressure. The highest correlation is observed at \(maxP\), indicating that backflow amounts are also greatest at this pressure level. In summary, the backflow rate positively correlates with the injector spray value.