# Optimizing Replaced Nozzle Diameter of Abrasive Blasting Systems Using Experiment Technique Design
## 2 Methodology  
In sand blasting, the cost for cleaning one square meter, \(C_{cl}\), (USD/m\({}^{2}\)) can be calculated as  
\[C_{cl}=\left(C_{m,h}+C_{a,h}+C_{n,h}\right)/\upsilon_{cl} \tag{1}\]  
where \(C_{m,h}\), \(C_{a,h}\) and \(C_{n,h}\) are the cleaning system cost, abrasive cost, and nozzle cost per hour (USD/h), respectively; \(\upsilon_{cl}\) is the cleaning rate (m\({}^{2}\)/h). The abrasive cost is determined by  
\[C_{a,h}=3600\cdot\dot{m}_{a}\cdot C_{a,m} \tag{2}\]  
in which \(C_{a,m}\) is the abrasive cost per kilogram (USD/kg); \(\dot{m}_{a}\) is the abrasive mass flow rate (kg/h). From data in [18], the following regression equation (with the determination coefficient \(R^{2}=0.998\)) was found to determine \(\dot{m}_{a}\):  
\[\dot{m}_{a}=15.14\cdot d_{n}^{-0.0214}\cdot p1.0359 \tag{3}\]  
The nozzle cost per hour, \(C_{n,h}\) can be established by:  
\[C_{n,h}=C_{n,p}/L_{n} \tag{4}\]  
where \(C_{n,p}\) is the nozzle cost per piece (USD/piece); \(L_{n}\) is the nozzle lifetime (h). Regarding the nozzle lifetime, the lifetime of a nozzle made of boron carbide is from 750 to 1500 h [18]. A nozzle can be replaced when the absolute wear reaches 1/16\" (1.59 mm) beyond its original size [19]. As a result, the wear rate, \(\mathbf{WR}\), can be found by  
\[\mathbf{WR}=\mathbf{1.59}/\mathbf{L_{n}}=\mathbf{1.59}/(\mathbf{1500}\div\mathbf{750})\approx\mathbf{0.001} \div\mathbf{0.002}\;(\text{mm}/\text{h}) \tag{5}\]  
Based on the data in [15], the following regression equation (with the determination coefficient \(\mathbf{R^{2}}=\mathbf{0.9984}\)) is found to determine the cleaning rate \(\mathbf{v_{cl}}\):  
\[\mathbf{v_{cl}}=\mathbf{6.6101\cdot 10^{-6}}\mathbf{d_{n}^{2.0121}}\mathbf{p^{1.5988}} \tag{6}\]  
In this case, \(\mathbf{p}\) is the air pressure (kPa); The data analysis in [18] also gives that the following regression equation (with the determination coefficient \(\mathbf{R^{2}}=\mathbf{0.9774}\)) is identified to calculate the air pressure:  
\[\mathbf{p}=\mathbf{3854.7}\cdot\mathbf{d_{n}^{-2.5783}}\cdot\mathbf{p^{1.1801}} \tag{7}\]  
in which \(\mathbf{P}\) is the power (kW); \(\mathbf{d_{n}}\) is the nozzle diameter (cf. Figure 1). The air pressure evolution as a function of the nozzle diameter can be graphically expressed in Figure 2. It is observed that the air pressure strongly depends on the nozzle diameter. When the nozzle diameter increases (or the nozzle lifetime grows), the air pressure decreases. Furthermore, the cleaning rate depends on the nozzle diameter and air pressure (cf. Equation (6)), i.e., an increase in the replaced nozzle diameter leads to a significant decrease in cleaning rate (cf. Figure 3). Figure 1: The shape of a nozzle diameter [19]. Figure 2: Air pressure versus nozzle diameter. From the above analyses, the cleaning cost equals the cost of machine and abrasive per one cleaned square meter (\(C_{m,s}+C_{a,s}\)) and the cost of nozzle per one cleaned square meter \(C_{n,s}\) strongly depends on the nozzle diameter. This performance is displayed in Figure 4 where the cleaning cost (Ccl) can be obtained as the combination of the machine cost and abrasive per one cleaned square meter (\(C_{m,s}+C_{a,s}\)) and the cost of nozzle per one cleaned square meter \(C_{n,s}\). It is noticed that when the nozzle diameter progressively increases, corresponding to the augmentation of nozzle lifetime, the nozzle cost per one cleaned square meter \(C_{n,s}\) descends according to the 2-axis asymptotic rule (cf. Figure 4). The nozzle cost becomes huge if the nozzle lifetime decreases. On the contrary, if the nozzle lifetime grows, the nozzle cost will be minor. In addition, when the nozzle or its lifetime increases, the machine and sand cost linearly grow, as seen in \[C_{a,h}=3600\cdot\dot{m}_{a}\cdot C_{a,m} \tag{2}\]0 Hence, the cleaning cost \(C_{cl}\) will be minimized at the optimum nozzle diameter, d0p.