## 2 Materials and Methods  
### Analytical Studies  
Existing analytical studies are predominantly based on the assumption of an ideal nozzle in which losses are not considered due to a significant pressure difference. Under such conditions, the Saint-Venant-Wanzel formula [16] can be applied:  
\[m=F_{2}\sqrt{\frac{2k}{k-1}\rho_{1}p_{1}\left[\left(\frac{p_{2}}{p_{1}}\right) ^{\frac{2}{k}}-\left(\frac{p_{2}}{p_{1}}\right)^{\frac{k+1}{k}}\right]}, \tag{1}\]  
where \(m\) is mass flow rate, kg/s; \(p_{1}\), \(p_{2}\) are inlet and outlet pressures, respectively, Pa; \(\rho_{1}\) is inlet gas density, kg/m\({}^{3}\); \(F_{2}\) is outlet cross-sectional area of the nozzle, m\({}^{2}\); \(k>1\) is thermodynamic constant.  
After introducing the pressure ratio \(\beta=p_{1}/p_{2}\) and considering the parabolic feature of its impact on the mass flow rate (Figure 1), the following approximation of Equation (1) can be proposed:  
\[m(\beta)=a_{0}+a_{1}\beta+a_{2}\beta^{2}, \tag{2}\]  
where \(a_{0}\), \(a_{1}\), and \(a_{2}\) are unknown parameters.  
The parameters \(a_{1}\), \(a_{2}\), and \(a_{3}\) can be determined analytically using the following physical conditions:  
\[m(0)=0;m(\beta_{cr})=m_{cr};m(1)=0, \tag{3}\]  
where \(m_{cr}\) is the critical value of the mass flow rate, kg/s, that corresponds to the critical pressure ratio \(\beta_{cr}\) (in a range of 0.5-1.0) when the supersonic mode occurs.  
These conditions allow for determining the unknown parameters:  
\[a_{0}=0;a_{1}=-a_{2}=\frac{m_{cr}}{\beta_{cr}(1-\beta_{cr})}. \tag{4}\]  
Thus, Formula (2) takes the form:  
\[\frac{m}{m_{cr}}=C.\frac{p_{1}}{p_{2}}\left(1-\frac{p_{1}}{p_{2}}\right), \tag{5}\]  
where \(C>4\) is the following dimensionless parameter determined by the critical pressure ratio:  
\[C=\frac{1}{\beta_{cr}(1-\beta_{cr})}. \tag{6}\]  
Figure 1: The design scheme (**a**) and the dependence between the mass flow rate and pressure ratio (**b**): 1-1— inlet section; 2-2—outlet section; \(C\)—critical point.  
Since the critical pressure ratio for an air-abrasive mixture significantly differs from the ideal gas, the parameter \(C\) should be evaluated using the best approximation of experimental data by Formula (5). In this case, after using the least square approach [17], the following error function should be minimized:  
\[R(C)=\sum_{i=1}^{n}\biggl{[}C\beta_{i}(1-\beta_{i})-\frac{m_{i}}{m_{cr}}\biggr{]} ^{2}\to min, \tag{7}\]  
where \(n\) is the total number of experimental points; \(i\) is experiment no. (\(i=1,2,\ldots,n\)); \(\beta_{i}\) is the pressure ratio for the \(i\)-th experiment; \(m_{i}\) is mass flow rate at the \(i\)-th experiment, kg/m\({}^{3}\).  
The minimization condition  
\[\frac{dR}{dC}=2\sum_{i=1}^{n}\bigg{[}C\beta_{i}(1-\beta_{i})-\frac{m_{i}}{m_{ cr}}\biggr{]}\beta_{i}(1-\beta_{i})=0 \tag{8}\]  
allows for evaluating the actual value of the unknown parameter:  
\[C=\frac{\sum_{i=1}^{n}\frac{m_{i}}{m_{cr}}\beta_{i}(1-\beta_{i})}{\sum_{i=1}^ {n}\beta_{i}^{2}(1-\beta_{i})^{2}}. \tag{9}\]  
Moreover, Formula (6) allows for further determining the actual value of the critical pressure ratio for the air-abrasive mixture:  
\[\beta_{cr}=\frac{1}{2}\bigg{(}1+\sqrt{1-\frac{1}{C}}\bigg{)}. \tag{10}\]