2. College of Chemical Engineering, Northwest University, Xi′an 710069, China;
3. Department of Mathematics/Institute of data analysis and computation chemistry, Northwest University, Xi′an, 710069, China;
4. College of Communication Science and Engineering, Northwest University, Xi′an 710069, China
2. 西北大学化工学院，陕西 西安 710069;
3. 西北大学数学系/数据分析和计算化学研究所，陕西 西安 710069;
4. 西北大学信息科学与工程学院，陕西 西安 710069
In 1964, Guo Yuxian proposed a nitrogen equivalent (NE) equation for predicting the detonation velocity (D) of CHNO explosives. In the early 1980′s, Guo Yuxian and Zhang Housheng[1-2] proposed two NE equations for predicting the D and detonation pressure (p) of CHNO explosives. Here we plan to propose two empirical NE equations for predicting the values of D and p with more approaching the values of D and p in Kamlet-Jacobs equations than Guo Yuxian-Zhang Housheng′s NE equations.
By substituting the 1631 sets of original data (Table S2), Di, Mi, ρi and xi, i=1, 2, …, 1631, for 324 CHNO single-compound explosives (Table S1) into eqns. (1), (3), (5) and (7), eqns. (2), (4), (6) and (8) are obtained via solution of eqns. (1), (3), (5) and (7) using the trust region approach.
| $ \begin{array}{l} \mathop {\min }\limits_{a,b,{N_{{{\rm{N}}_2}}},{{N}_{{{\rm{H}}_2}{\rm{O}}}},{{N}_{{\rm{C}}{{\rm{O}}_2}}},{N_{{{\rm{H}}_2}}},{{N}_{{{\rm{O}}_2}}},{N_{\rm{C}}}} \;\sum\limits_{i = 2}^n {\left[ {{D_i} - \frac{{100}}{M}({a + b}\rho )} \right.} \\ ({{x}_{{{\rm{N}}_2}}}{{N}_{{{\rm{N}}_2}}} + {x_{{{\rm{H}}_2}{\rm{O}}}}{{N}_{{{\rm{H}}_2}{\rm{O}}}} + {x_{{\rm{C}}{{\rm{O}}_2}}}{N_{{\rm{C}}{{\rm{O}}_2}}} + {x_{{\rm{CO}}}}{N_{{\rm{CO}}}} + \\ {\left. {{x_{{{\rm{H}}_2}}}{N_{{{\rm{H}}_2}}} + {x_{{{\rm{O}}_2}}}{N_{{{\rm{O}}_2}}} + {x_{\rm{C}}}{N_{\rm{C}}})} \right]^2} \end{array} $ |
| $ \begin{array}{l} {\rm{s}}{\rm{.t}}{\rm{. }}690.00 \le a \le 690.01,{\rm{ }}1160 \le b \le 1160.01,{\rm{ }}\\ 1.000 \le {N_{{{\rm{N}}_2}}} \le 1.001,{\rm{ }}0.5400 \le {N_{{{\rm{H}}_2}{\rm{O}}}} \le 0.5401,{\rm{ }}\\ 1.3500 \le {N_{{\rm{C}}{{\rm{O}}_2}}} \le 1.3501,{\rm{ }}\\ 0.7800 \le {N_{{\rm{CO}}}} \le 0.7801,{\rm{ }}0.2900 \le {N_{{{\rm{H}}_2}}}\\ \le 0.2901,{\rm{ }}0.5000 \le {N_{{{\rm{O}}_2}}} \le 0.5001,{\rm{ }}\\ 0.1500 \le {N_{\rm{C}}} \le 0.1501 \end{array} $ | (1) |
| $ \begin{gathered} D = \frac{{100}}{M}(690 + 1160\rho ){\rm{ (}}1.00{x_{{{\rm{N}}_2}}} + 0.54{x_{{{\rm{H}}_2}{\rm{O}}}} \hfill \\ + 1.35{x_{{\rm{C}}{{\rm{O}}_2}}} + 0.78{x_{{\rm{CO}}}} + 0.29{x_{{{\rm{H}}_2}}} + 0.50{x_{{{\rm{O}}_2}}} + 0.15{x_{\rm{C}}}) \hfill \\ \end{gathered} $ | (2) |
where D is the detonation velocity, m·s-1; M is the mole weight of explosive, g·mol-1; 690 and 1160 are constants; ρ is the initial densities of explosives, g·cm-3; 1.00, 0.54, 1.35, 0.78, 0.29, 0.50, 0.15 are the nitrogen equivalent coefficient of gaseous detonation products N2, H2O, CO2, CO, H2, O2, C of explosive; xi (i=N2, H2O, CO2, CO, H2, O2, C) is the numbers of moles of gaseous detonation products.
Equation (2) is known as Guo Yuxian-Zhang Housheng′s NE equation for predicting the value of D of CHNO explosives.
The relative error (Δδ) of eqn. (2) is:
| $ \begin{array}{l} \Delta \delta = \sqrt {\sum\limits_{i = 1}^n {{{\left[ {{D_{{\rm{cald,}}{i}}} - {D_{{\rm{K - J,}}{i}}}} \right]}^2}/\sum\limits_{i = 1}^n {p_{{\rm{K - J,}}{i}}^2} } } = 0.0499\\ \mathop {\min }\limits_{c,{N_{{{\rm{N}}_2}}},{N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}},{N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}},{N_{{\rm{CO}}}},{N_{{{\rm{H}}_{\rm{2}}}}},{N_{{{\rm{O}}_{\rm{2}}}}},{N_{\rm{C}}},d} \sum\limits_{i = 2}^n {\left\{ {{p_i} - c\left[ {\rho \frac{{100}}{M}(} \right.} \right.} \\ {x_{{{\rm{N}}_{\rm{2}}}}}{N_{{{\rm{N}}_{\rm{2}}}}} + {x_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}}{N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + {x_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}}{N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} + {x_{{\rm{CO}}}}{N_{{\rm{CO}}}} + \\ {\left. {{{\left. {{x_{{{\rm{H}}_{\rm{2}}}}}{N_{{{\rm{H}}_{\rm{2}}}}} + {x_{{{\rm{O}}_{\rm{2}}}}}{N_{{{\rm{O}}_{\rm{2}}}}} + {x_{\rm{C}}}{{\rm{N}}_{\rm{C}}}} \right]}^2} + d} \right\}^2} \end{array} $ |
| $ \begin{array}{l} {\rm{s}}{\rm{.t}}{\rm{. }}1.09200 \le {c} \le 1.09201,{\rm{ }}1.000 \le {N_{{{\rm{N}}_{\rm{2}}}}} \le 1.001,0.5400 \le \\ {N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} \le 0.5401,{\rm{ }}1.3500 \le {N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} \le 1.3501,0.7800 \le {N_{{\rm{CO}}}} \le 0.7801,{\rm{ }}\\ 0.2900 \le {N_{{{\rm{H}}_{\rm{2}}}}} \le 0.2901,0.5000 \le {N_{{{\rm{O}}_{\rm{2}}}}} \le 0.5001,{\rm{ }}0.1500 \le {N_{\rm{C}}}\\ \le 0.1501,{\rm{ }}0.57400 \le d \le 0.57401{\rm{ }} \end{array} $ | (3) |
| $ \begin{array}{l} p = 1.092\left[ {\rho \frac{{100}}{M}} \right.(1.00{x_{{{\rm{N}}_{\rm{2}}}}} + 0.54{x_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + 1.35{x_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} + 0.78{x_{{\rm{CO}}}}\\ {\left. { + 0.29{x_{{{\rm{H}}_{\rm{2}}}}} + 0.50{x_{{{\rm{O}}_{\rm{2}}}}} + 0.15{x_{\rm{C}}})} \right]^2} - 0.574 \end{array} $ | (4) |
where p is the detonation pressure, GPa; 1.092 and 0.574 are constants.
Equation (4) is known as Guo Yuxian-Zhang Housheng′s NE equation for predicting the value of p of CHNO explosives.
The relative error (Δδ) of eqn. (4) is:
| $ \begin{array}{*{35}{l}} \Delta \delta =\sqrt{\sum\limits_{i=1}^{n}{{{\left[ {{p}_{\text{cald},i}}-{{p}_{\text{K-J},i}} \right]}^{2}}/\sum\limits_{i=1}^{n}{p_{\text{K-J},i}^{2}}}}=0.0885 \\ \underset{a,b,{{N}_{{{\text{N}}_{2}}}},{{N}_{{{\text{H}}_{\text{2}}}\text{O}}},{{N}_{\text{C}{{\text{O}}_{\text{2}}}}},{{N}_{\text{CO}}},{{N}_{{{\text{H}}_{\text{2}}}}},{{N}_{{{\text{O}}_{\text{2}}}}},{{N}_{\text{C}}}}{\mathop{\min }}\,\sum\limits_{i=2}^{n}{\left[ {{D}_{i}}-\frac{100}{M}(a+b\rho )( \right.} \\ {{x}_{{{\text{N}}_{\text{2}}}}}{{N}_{{{\text{N}}_{\text{2}}}}}+{{x}_{{{\text{H}}_{\text{2}}}\text{O}}}{{N}_{{{\text{H}}_{\text{2}}}\text{O}}}+{{x}_{\text{C}{{\text{O}}_{\text{2}}}}}{{N}_{\text{C}{{\text{O}}_{\text{2}}}}}+{{x}_{\text{CO}}}{{N}_{\text{CO}}}+{{x}_{{{\text{H}}_{\text{2}}}}}{{N}_{{{\text{H}}_{\text{2}}}}}+ \\ {{\left. {{x}_{{{\text{O}}_{\text{2}}}}}{{N}_{{{\text{O}}_{\text{2}}}}}+{{x}_{\text{C}}}{{N}_{\text{C}}}) \right]}^{2}} \\ \end{array} $ |
| $ \begin{array}{l} {\rm{s}}{\rm{.t}}{\rm{. }}650 \le a \le 695,{\rm{ }}1150 \le b \le 1165,{\rm{ }}0.800 \le {N_{{{\rm{N}}_{\rm{2}}}}} \le \\ 1.001,{\rm{ }}0.340 \le {N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} \le 0.640,{\rm{ }}1.150 \le {N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} \le 1.350,{\rm{ }}\\ 0.250 \le {N_{{\rm{CO}}}} \le 0.780,{\rm{ }}0.110 \le {N_{{{\rm{H}}_{\rm{2}}}}} \le 0.290,{\rm{ }}\\ 0.010 \le {N_{{{\rm{O}}_{\rm{2}}}}} \le 0.500,{\rm{ }}0.110 \le {N_{\rm{C}}} \le 0.150 \end{array} $ | (5) |
| $ \begin{array}{l} D = \frac{{100}}{M}(695 + 1150\rho )(1.00{x_{{{\rm{N}}_{\rm{2}}}}} + 0.64{x_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + 1.34{x_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}}\\ + 0.72{x_{{\rm{CO}}}} + 0.18{x_{{{\rm{H}}_{\rm{2}}}}} + 0.50{x_{{{\rm{O}}_{\rm{2}}}}} + 0.12{x_{\rm{C}}}) \end{array} $ | (6) |
where 695 and 1150 are constants; 1.00, 0.64, 1.34, 0.72, 0.18, 0.50, 0.12 are the nitrogen equivalent coefficient of gaseous detonation products N2, H2O, CO2, CO, H2, O2, C of explosive.
Equation (6) is known as the empirical NE equation for predicting the value of D of CHNO explosives
The relative error (Δδ) of eqn. (6) is:
| $ \begin{array}{l} \Delta \delta = \sqrt {\sum\limits_{i = 1}^n {{{\left[ {{D_{{\rm{cald,i}}}} - {D_{{\rm{K - J,}}{i}}}} \right]}^2}/\sum\limits_{i = 1}^n {D_{{\rm{K - J,}}{i}}^2 = 0.0373} } } \\ \mathop {\min }\limits_{c,{N_{{{\rm{N}}_2}}},{N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}},{N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}},{N_{{\rm{CO}}}},{N_{{{\rm{H}}_{\rm{2}}}}},{N_{{{\rm{O}}_{\rm{2}}}}},{N_{\rm{C}}},d} \sum\limits_{i = 2}^n {\left\{ {{p_i} - c\left[ {\rho \frac{{100}}{M}(} \right.} \right.} \\ {x_{{{\rm{N}}_{\rm{2}}}}}{N_{{{\rm{N}}_{\rm{2}}}}} + {x_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}}{N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + {x_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}}{N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} + {x_{{\rm{CO}}}}{N_{{\rm{CO}}}} + \\ {\left. {{{\left. {{x_{{{\rm{H}}_{\rm{2}}}}}{N_{{{\rm{H}}_{\rm{2}}}}} + {x_{{{\rm{O}}_{\rm{2}}}}}{N_{{{\rm{O}}_{\rm{2}}}}} + {x_{\rm{C}}}{N_{\rm{C}}})} \right]}^2} + d} \right\}^2} \end{array} $ |
| $ \begin{array}{l} {\rm{s}}{\rm{.t}}.{\rm{ }}1.060 \le a \le 1.500,{\rm{ }}1.000 \le {N_{{{\rm{N}}_{\rm{2}}}}} \le 1.001,{\rm{ }}0.6400 \le {N_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}}\\ \le 0.6401,{\rm{ }}1.3400 \le {N_{{\rm{C}}{{\rm{O}}_{\rm{2}}}}} \le 1.3401,{\rm{ }}0.7200 \le {N_{{\rm{CO}}}} \le 0.7201,{\rm{ }}\\ 0.1800 \le {N_{{{\rm{H}}_{\rm{2}}}}} \le 0.1801,{\rm{ }}0.0.500 \le {N_{{{\rm{O}}_{\rm{2}}}}} \le 0.501,{\rm{ }}0.1200 \le \\ {N_{\rm{C}}} \le 0.1201,{\rm{ }}0.001 \le d \le 0.874 \end{array} $ | (7) |
| $ \begin{array}{l} p = 1.060\left[ {\rho \frac{{100}}{M}} \right.(1.000{x_{{{\rm{N}}_{\rm{2}}}}} + 0.64{x_{{{\rm{H}}_{\rm{2}}}{\rm{O}}}} + 1.34x{\rm{C}}{{\rm{O}}_{\rm{2}}} + 0.72{x_{{\rm{CO}}}}\\ {\left. { + 0.18{x_{{{\rm{H}}_{\rm{2}}}}} + 0.50{x_{{{\rm{O}}_{\rm{2}}}}} + 0.12{x_{\rm{C}}})} \right]^2} - 0.619 \end{array} $ | (8) |
where 1.060 and 0.619 are constants.
Equation (8) is known as the empirical NE equation for predicting the value of p of CHNO explosives
The relative error (Δδ) of eqn. (8) is:
| $ \Delta \delta = \sqrt {\sum\limits_{i = 1}^n {{{\left[{{D_{{\rm{cald, i}}}}-{p_{{\rm{K-J, }}\mathit{i}}}} \right]}^2}/\sum\limits_{i = 1}^n {p_{{\rm{K -J, }}\mathit{i}}^2 } } }=0.0682 $ |
Compared with the values of Δδ of eqns. (2) and (4), the ones of Δδ of eqns. (6) and (8) decrease by 25.2% and 23.0%, respectively, indicating that eqns. (6) and (8) can be used to predict the values of D and p of CHNO explosives with more approaching the values of D and p in Kamlet-Jacobs equations than common used nitrogen equivalent equations.
Associated Content: Supporting information
The supporting information of the structure formula (Table S1) and original data (Table S2) is available free of charge on the website of Chinese Journal of Energetic Materials.
| [1] |
The explosive theory writing group. Theory of Explosive[M]. Beijing: National Defence Industry Press, 1982.
|
| [2] |
Guo Yu-xian, Zhang Hou-sheng. Nitrogen equivalent (NE) and modified nitrogen equivalent (MNE) equations for predicting detonation parameters of explosives-prediction of detonation velocity of explosives[J]. Explosive and Shock Waves, 1987, 7(4), 348 (in Chinese).
|
| [3] |
Keshavarz M H. Prediction of the condensed phase heat of formation of energetic compounds[J]. Journal of Hazardous Materials, 2011, 190(1-3): 330-344. DOI:10.1016/j.jhazmat.2011.03.043 |
| [4] |
Ovchinnikov V. Thermochemistry of heteroatomic compounds: interdependence between of some thermochemical parameters of the different classes organic Nitro compounds and a number of valence electrons in their molecules[J]. American Chemical Science Journal, 2013, 3(1): 11-23. DOI:10.9734/ACSJ |
| [5] |
Tian De-yu, Zhao Feng-qi, Liu Jian-hong. Handbook of energetic materials and the related compounds[M]. Beijing: National Defence Industry Press, 2011.
|
| [6] |
Mohammad H K. Simple correlation for predicting detonation velocity of ideal and non-ideal explosives[J]. Journal of Hazardous Materials, 2009, 166(2-3): 762-769. DOI:10.1016/j.jhazmat.2008.11.117 |
| [7] |
Mohammad H K. Theoretical prediction of condensed phase heat of formation of nitramines, nitrate esters, nitroaliphatics and related energetic compounds[J]. Journal of Hazardous Materials, 2006, 136(2): 145-150. DOI:10.1016/j.jhazmat.2005.12.006 |
| [8] |
Rothstein L R, Petersen R. Predicting high explosive detonation velocities from their composition and structure[J]. Propellants, Explosives, Pyrotechnics, 1979, 4(3): 56-60. DOI:10.1002/(ISSN)1521-4087 |
| [9] |
Mohammad H K. Novel method for predicting densities of polynitro arene and polynitro heteroarene explosives in order to evaluate their detonation performance[J]. Journal of Hazardous Materials, 2009, 165(1-3): 579-588. DOI:10.1016/j.jhazmat.2008.10.019 |
| [10] |
Kamlet M J, Jacobs S J. Chemistry of detonations. I. A simple method for calculating detonation properties of CHNO explosives[J]. The Journal of Chemical physics, 1996, 48(1): 23-45. |
| [11] |
Hu Rong-zu, Zhao Feng-qi, Gao Hong-xu, et al. Estimation of detonation performances of explosives using M、ρ、ΔfHmθ、ΔdHmθ、Cp、Tig or b of CHON explosives and ΔfHmθ of detonation products[J]. Chinese Journal of Explosives and Propellants, 2013, 36(2): 20-23. |
Two empirical nitrogen equivalent equations for predicting the detonation velocity (D) and detonation pressure (p) of CHNO explosives with more approaching the values of D and p in Kamlet-Jacobs equations than common used nitrogen equivalent equations were presented.