# The Predictive Model of Surface Texture Generated by Abrasive Water Jet for Austenitic Steels
## 3 Results and Discussion
### Evaluation of Measurement Results from the Mathematical-Statistical Point of View
As the cutting head feed rate increases, the surface roughness value \(Ra\) also increases, which corresponds to the above results (Figure 4). The interaction of the distance between the nozzle and the material and the cutting head feed rate is also statistically relevant, with a 20.14% effect on the change in surface roughness value \(Ra\). Equally significant is the effect of high-pressure water flow, with an 8.38% impact on the change in the roughness value \(Ra\), abrasive mass flow with an 8.34% effect on the variability of \(Ra\), and their interaction with 8.26% effect on the change in surface roughness value \(Ra\). The thickness of the cut material, as the last statistically significant factor, affects the change in surface roughness value \(Ra\) by 10.68%. In this case, the influence of the diameter of the abrasive nozzle proved to be statistically insignificant, but this may be due to the interval of experimental input parameters used. Based on the above, the regression dependence can be predicted (10):  
\[ \begin{array}{l}Ra=31.733+4.832\cdot 10^{-2}\cdot x_{3}-7.428\cdot x_{1}-1.173 \cdot 10^{-3}\cdot x_{4}-0.775\cdot x_{5}-7.435\cdot 10^{-2}\cdot x_{6}+\\ 0.219\cdot x_{1}\cdot x_{5}+3.224\cdot 10^{-3}\cdot x_{3}\cdot x_{4}-1.780\cdot 10^{-3} \cdot x_{1}\cdot x_{5}^{2}+6.701\cdot 10^{-3}\cdot x_{5}^{2}\end{array}  \tag{11}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Feed Rate [mm\(\cdot\)min\({}^{-1}\)]** & **Valid [N]** & **Sum of Ranks** & **Mean Rank** \\ \hline
50 & 17 & 350.500 & 20.618 \\
75 & 17 & 535.500 & 31.500 \\
100 & 17 & 713.000 & 41.941 \\
125 & 17 & 747.000 & 43.941 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Results of Kruskal-Wallis scattering analysis for cutting head feed rate. Figure 5: Effect of high-pressure water flow (**a**) and abrasive mass flow (**b**) on the change in surface roughness value \(Ra\). On the interpretation and justification of cross-correlations and the model (10): the members (products of parameters) in the concerned Equation (10) express correlations between pairs of significant parameters, namely correlations of physical significance for the used AWJ technology. The product \(x_{1}\cdot x_{5}\) in Equation (10) represents the area velocity \(v_{A}\), dimensionally corresponding to:  
\[x_{1}\cdot x_{5}=[L]\cdot[v_{p}]=\text{mm}\cdot\text{mm}\cdot\text{min}^{-1}= \text{mm}^{2}\cdot\text{min}^{-1}\]  
The product in Equation (10) represents the volume flow rate \(Q_{V}\) for time \(t\) as the volume of liquid for time flowing through one place (the whole cross-section at one point (tubes, nozzles) per a time unit):  
\[x_{1}\cdot(x_{5})^{2}=[L]\cdot[(v_{p})^{2}]=\text{mm}\cdot\text{mm}^{2}\cdot \text{min}^{-2}=\text{mm}^{3}\cdot\text{min}^{-2}\]  
The total effect of \(v_{A}\) and \(Q_{V}\) over time presents not only the isolated effects of the distance of the nozzle outlet from the machined surface, but their combination in the sense that the resulting value should not be extreme (maximum, minimum), but optimal with respect to the desired roughness value. The product \(x_{3}\cdot x_{4}\) in Equation (11) represents the square of the mass flow rate \(Q_{m}\) related to the distance of the nozzle outlet from the surface to be treated, dimensionally corresponding to:  
\[x_{3}\cdot x_{4}=[Q_{V}]\cdot[m_{a}]=\text{1}\cdot\text{min}^{-1}\cdot\text{g }\cdot\text{min}^{-1}=\text{g}^{2}\cdot\text{min}^{-2}\]  
The total effect of the subject product \(x_{3}\cdot x_{4}\) also presents not only the isolated effects of the volume flow rate and the amount of abrasive but their combination in the sense that the resulting value should not be extreme but again optimal in relation to the required roughness. A graphical representation of the variation of the surface roughness value \(Ra\) from the change in the cutting head feed rate and the abrasive mass flow is shown in Figure 7. The graph shows that the minimum values of the studied variable \(Ra\) can be achieved by low cutting head speeds and high abrasive mass flow values.