# Theoretical Study of Large Uncoupling Coefficient Loading for Surface Blasting
## 3 Critical Uncoupling Coefficient of Large Diameter Drilled Holes  
### Blasting Critical Uncoupling Coefficient  
In the condition of high blasting pressure, the results of Equations (1)-(6) for the blast products can meet the assumption that \(PV^{3}\) = constant. However, as the gas expands, the pressure decreases, and the equation of state of blasting gas follows the law of \(PV^{K}\) = constant (\(K\) = 1.2-1.3). There is a critical pressure between blast products for \(PV^{3}\) = constant and \(PV^{K}\) = constant (\(K\)=1.2-1.3). The critical pressure \(P_{K}\) is  
\[P_{K}=0.154\sqrt{(E-\frac{1}{2}\frac{P_{c}}{\rho_{0}})^{3}\frac{\rho_{0}{}^{3} }{P_{c}}} \tag{7}\]where \(E\) is the amount of energy contained in a unit mass explosive; \(P_{c}\) is blast pressure. The value of \(P_{K}\) is approximately calculated to be 200 MPa.  
When the gas pressure is greater than the critical pressure, the results obtained from Equation (6) are correct. When the expansion reaches the critical pressure, the volume of gas \(V_{K}\) should be  
\[V_{K}=(\frac{P_{c}}{P_{K}})^{\frac{1}{3}}V_{c} \tag{8}\]  
where \(V_{c}\) is volume of charge. Three conditions are discussed as below.  
(1) For radial uncoupled continuous charge, \(l_{b}=l_{c}\); when the blasting gas expands to fill the entire hole, the volume of gas \(V_{K}\) follows:  
\[\frac{\pi}{4}{d_{\rm{bmax}}}^{2}l_{b}=(\frac{P_{c}}{P_{K}})^{\frac{1}{3}}\frac {\pi}{4}{d_{c}}^{2}l_{c} \tag{9}\]  
For this condition, the critical radial uncoupling coefficient \(K_{\rm{dmax}}\) is  
\[K_{\rm{dmax}}=\frac{{\rm{d}}_{\rm{bmax}}}{d_{c}}=(\frac{P_{c}}{P_{K}})^{\frac {1}{6}} \tag{10}\]  
(2) When radially coupled axially uncoupled charges are used, the gas expands to fill the entire blast hole, and the volume of gas \(V_{K}\) follows:  
\[\frac{\pi}{4}{d_{c}}^{2}l_{\rm{bmax}}=(\frac{P_{c}}{P_{K}})^{\frac{1}{3}}\frac {\pi}{4}{d_{c}}^{2}l_{c} \tag{11}\]  
Herein, the critical axial uncoupling coefficient \(K_{\rm{lmax}}\) is  
\[K_{\rm{lmax}}=\frac{{\rm{l}}_{\rm{bmax}}}{l_{c}}=(\frac{P_{c}}{P_{K}})^{\frac {1}{3}} \tag{12}\]  
(3) For a radially and axially uncoupled charge, the gas expands to fill the entire blast hole, and the volume of gas \(V_{K}\) follows:  
\[\frac{\pi}{4}{d_{x}}^{2}l_{x}=(\frac{P_{c}}{P_{K}})^{\frac{1}{3}}\frac{\pi}{4} {d_{c}}^{2}l_{c} \tag{13}\]  
Herein, the uncoupling coefficient \(K_{\rm{dmax}}\) is found to be  
\[K_{\rm{d1 max}}=(\frac{d_{x}}{d_{c}}\sqrt{\frac{l_{x}}{l_{c}}})=(\frac{P_{c}}{ P_{K}})^{\frac{1}{6}} \tag{14}\]  
\[(d_{c}<d_{x}<d_{\rm{bmax}},l_{c}<l_{x}<l_{\rm{bmax}})\]  
The rock emulsion explosive is taken as a second example. The density of this explosive is 1.24 g/cm\({}^{3}\) with a blasting velocity \(D_{1}\) of 4200 m/s. According to Equations (10), (12) and (14), the values of \(K_{\rm{d1 max}}\) and \(K_{\rm{dmax}}\) for this explosive are smaller than 1.54, and values of \(K_{\rm{lmax}}\) are smaller than 2.4. It can be seen that when the gas pressure is greater than the critical pressure and the gas expands to fill the entire blast hole, the radial uncoupling coefficient and the radial axial uncoupling coefficient becomes equal, although they are smaller than that of the radial coupling axial uncoupling charge. When the axial uncoupling coefficient is greater than the critical uncoupling coefficient, the blast hole void interval becomes large, and the blasting gas expands following the ideal gas equation of state. For the large uncoupling coefficient charge, the dynamic impact of the explosion products will continue to weaken, while the quasi-static pressure of the blasting gas enhances this phenomenon further. As a result, when the uncoupling factor exceeds the critical uncoupling factor, the classical theory for impact pressure generated by uncoupled charge in blast hole will no longer apply, and the smooth blasting parameters need to be revised.