Abstract
The performance of commercial explosives is an important subject in rock blast ing modeling and simulation. As a result of its nonideal behavior, these explosives usu ally react below their ideal detonation velocity. In these cases, the multidimensional effects, heterogeneities and confinement conditions become important for properly quantifying the detonation state. In this sense, an engineering approach to model twodimensional steady nonideal detonations for cylindrical stick explosives is used to quantify the expected detonation velocity for given reaction rate parameters and con finement conditions. Founded on an ellipsoidal shock shape approach (ESSA), the pro posed model combines the quasionedimensional theory for the axial solution with the unconfined sonic postflow conditions at the edge of the explosive. A mechanistic confinement approach is coupled with the ESSA model to estimate the effect of the inert confiner on the detonation flow. Finally, the proposed model is used to estimate the expected detonation velocity of two typical commercial explosives in a number of different confinement conditions.
keywords:
nonideal detonation; confined detonation; velocity of detonation; rock blasting
1. Introduction
Since the ability of modeling is inherent to any optimization strategy, a reliable explosive energy release quantifi cation is required for a more realistic rock blasting simulation. In this regard, the interest of studying nonideal detonations is considered to be a fundamental step for further downstream mining modeling. The explosive performance must be car ried out using realistic approaches, such as those based on the Euler reactive flow analysis, and including Direct Numerical Simulations (DNS), slightly divergent flow theory, quasiunidimensional analysis or streamline approximations, and oth ers, rather than those based on ideal thermodynamic codes, like CHEETAH, WDETCOM, ATLASDet, and others.
The detonation process in highly nonideal explosives and its interaction with the inert confiner material is still a matter of research and discussion (Sharpe & Braithwaite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
; Sharpe & Bdzil, 2006; Sellers, 2007SELLERS, E. J. Velocity of detonation of nonideal explosives: investigating the influence of confinement. In: ANNU AL CONFERENCE ON EXPLOSIVES AND BLASTING TECHNIQUE, 33., 2007, Nashville, Tennessee, USA. Proceedings […]. Nashville: ISEE, 2007. p. 111. ; Esen, 2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
; Sharpe et al., 2009; Sellers et al., 2012; Braith waite & Sharpe, 2013). Their interaction is complex, especially in cases where the acoustic velocity of the confiner is higher than the velocity of detonation. However, when combined with a mechanistic model for the confiner material, these nonideal detonation models can predict, within the experimental scatter levels, impor tant properties of the detonation, such as detonation velocity, pressure, specific volume, and others.
Mining explosives are strongly dependent of blasthole diameter, densi ties, reaction rates, confinement and others. These characteristics require the use of some classes of nonideal detona tion models to properly quantify the explosive’s performance. These methods must be able to describe the reactive flow solution of the problem, including pressure profiles, densities and others. In this sense, one practical indicator of the explosive characteristics is the detonation velocity. This election is a consequence of the relatively simple method of recording the confined inhole detonation velocity in real scale shots. On the other hand, it is a good indicator of the explosive’s nonideality (Bilgin & Esen, 1999BILGIN, H. A.; ESEN, S. Assessment of ideality of some commercial explosives. In: ANNUAL CONFERENCE ON EXPLOSIVES & BLASTING TECHNIQUE, 25., 1999, Nashville, Tennessee, USA. Proceedings […]. Nashville: ISEE, 1999. p. 3544. ) and can be normally associated with a large set of factors, such as explosive type, density, temperatures, reaction rates, blasthole diameter, confinement, primer size and many others (Sanchidrián & Muñiz, 2000SANCHIDRIÁN, J. A.; MUÑIZ, E. Curso de tecnología de explosivos. Madrid: Fundación Gómez Pardo, 2000. ).
Thus, the prediction of the inhole confined detonation velocity becomes particularly important to quantify the performance of a given nonideal explosive and confinement conditions. Herein, a nonideal detonation model based on the Ellipsoidal Shock Shape Approach (ESSA) is used to properly describe the inhole confined detona tion velocities of two different typical mining explosives in a large set of ex perimental and simulated data.
2. Governing equations
Modeling nonideal detonations are often based on the reactive Euler equations for the conservation of mass, momentum and energy where u is the velocity; ρ is the density; P is the pressure; E is the internal energy; λ is the reaction progress (λ = 0, for an unreacted product and λ = 1 for a complete reaction process); and W the is reaction rate, and the operator D / Dt = ∂ / ∂t + u ∙∇.
The governing equations are closed by defining the equation of state E(P, ρ, λ) and the reaction rate W(P, ρ, λ). In the Detonation Shock Dynamics (DSD) theory, a change of variables is necessary to describe the behavior of the detonation process along the normal direction n, at any point of the shock front. Thus, the Euler re active flow equations must be transformed from the cartesian coordinate system (x,y,t) into a shockattached coordinate system (n, ξ, t). A robust mathematical founda tion of this transformation is discussed elsewhere (Stewart, 1993STEWART, S. Lectures on detonation physics: introduction to the theory of detonation shock dynamics. Illinois: The University of Illinois, 1993. (Report Number WLTR947089). ; Yao & Stewart, 1996YAO, J.; STEWART, D. S. On the dynamics of multidimensional detonation. Journal of Fluid Mechanics, v. 309, p. 225215, 1996. DOI: https://doi.org/10.1017/S0022112096001620.
https://doi.org/10.1017/S002211209600162...
; Sharpe & Braithwaite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
).A typical unconfined nonideal detonation structure is presented in Figure 1. When particularized to the axis of the charge, these equations can describe the complete axial flow solution through a unique rela tionship between the normal detonation ve locity and axial shock curvature, constitut ing the basis of the quasionedimensional Q1D theory (Sharpe & Braithwaite, 2005).
Schematic representation of a typical unconfined nonideal detonation structure. The detonation is propagating upward with a constant velocity of detonation, D_{o}.
2.1 Axial flow solution
The ESSA model requires a complete description of the axial flow solution to construct the ellipsoidal extension of the detonation shock front to the twodimensions. Although any axial solution could be used, such as those based on the slightly divergent flow theory (Kirby & Leiper, 1985KIRBY, I. J.; LEIPER, G. A. A small divergent detonation theory for intermolecular explosives. In: SYMPOSIUM (INTERNATIONAL) ON DETONATION, 8., 1985, Albuquerque, New Mexico, USA. Proceedings […]. Albu querque: Office of Naval Research, 1985. p. 176186. ), the Q1D model (Sharpe & Braithwaite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
) is adopted in this study due to its solid improvements over other axial models. One of the most important improvements is the substitution of the divergent term  which is unknown  by the axial curvature. Since its development, Q1D has been demonstrating excellent results regarding the axial flow solutions, even in highly nonideal detonations (Sharpe et al., 2009).
The Q1D theory (Sharpe & Braith waite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
) assumes that the axial solution depends only parametrically on the detonation velocity or the shock front curvature, suggesting that it is governed by a simple Dnκ law. When combined with the quadratic pseudopolytropic equation of state (Sharpe & Braithwaite, 2005)
where ρ is the density; Q is the heat of reaction and the quadratic gamma γ* is given by:
where ρo is the initial density; γo, γ1 and γ2 are coefficients obtained from an ideal thermo dynamic detonation code, the following set of ordinary differential equations:
can be obtained. Where un is the normal par ticle velocity; ρ is the density; Dn is the normal velocity of detonation; c is the sound speed; Q is the heat of explosion; γ is the adiabatic gamma; κ*=κ(1+nκ)1; and W is the reaction rate, given by a pressure dependent equation
where n, m and τ are fitting parameters; Pref is a reference pressure. This type of pressurebased reaction rate law presents a maximum at the shock (Cartwright, 2016CARTWRIGHT, M. Modelling of nonideal steady detonations. 2016. 188 f. Thesis (PhD)  Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom, 2016. ). Therefore, once defined the equation of state, the sound speed c can be obtained as where ρ=1/v.
Depending of the “form” of the reaction rate equation, the number of differential equations needed to solve the problem can be reduced. In this article, since W is dependent on the pres sure, and consequently on the specific density or volume, the pressure can be expressed as by assuming the strong shock approxi mation.
Thus, the set of ordinary differ ential Equations (4), (5) and (6) form a boundaryvalue problem, in other words, an eigenvalue problem in κ (Dn) for a given Dn (κ). The most common resolution method is the shooting method, subjected to the jump shock conditions at the shock front and generalized CJ conditions at the sonic locus (Stewart & Yao, 1998STEWART, D. S.; YAO, J. The normal detonation shock velocitycurvature relationship for materials with nonideal equation of state and multiple turning points. Combustion and Flame, v. 113, n. 1/2, p. 224235, 1998. ; Sharpe, 2000SHARPE, G. The structure of planar and curved detonation waves with reversible reactions. Physics of Fluids, v. 12, n. 11, p. 30073020, 2000. DOI: https://doi.org/10.1063/1.1313389.
https://doi.org/10.1063/1.1313389....
; Sharpe & Braithwaite, 2005).
3. Ellipsoidal shock shape approach
In the case of a highly nonideal deto nation, experimental shock measurements indicate that the detonation shock front can be well represented by an elliptic arc (Kennedy, 1998KENNEDY, D. L. Multivalued normal shock velocity versus curvature relationships for highly nonideal explosives. In: SYMPOSIUM (INTERNATIONAL) ON DETONATION, 11., 1998, Snowmass, Colorado. Proceedings […]. Snowmass: Office of Naval Research, 1998. p. 181192. ; Sharpe & Braithwaite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
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). This observation is also supported by Direct Numerical Simulations (DNS). Additionally, Watt et al., (2009WATT, W.; BRAITHWAITE, M.; BYERS BROWN, W.; FALLE, S.; SHARPE, G. Maximum entropy of effective reaction theory of steady nonideal detonation. In: APS TOPICAL CONFERENCE ON SHOCK COMPRES SION OF CONDENSED MATTER, 16., 2009, Nashville, Tennessee, USA. Proceedings […]. Nashville: American Physical Society, 2009. p. 213216. DOI: http://dx.doi.org/10.1063/1.3295106.
http://dx.doi.org/10.1063/1.3295106...
), using the Maximum Effective Entropy Theory (Byers Brown, 2002) in the construction of the detonation driving zone (DDZ), obtained better results when an ellipsoidal shock shape function was used to represent the detonation shock front. These evi dences suggest that an ellipse of the form can be assumed to well represent the shock shape. Here, z and r are the axial and ra dial directions, respectively; and α and β are the semiminor and semimajor axis of the ellipse. In this study, the semiaxis of the ellipse is called shock shape param eters. The origin of the adopted coordinate system is located at the central axis of the shock front, where r =0 and z =0. An as sumed shock shape function allows one to calculate the slope angle and curvature κ at any point along of the shock zf.
3.1 Twodimension model expansion
There is a unique relationship between the axial flow solution, shock shape parameters and radial dimension of the charge. Since the axisymmetric curvature κ is the solution of the set of differential Equations (4), (5) and (6) for a given velocity of detonation Dn and reaction rate parameters n, m and τ, a further relationship between κ, α and β can be addressed in order to ensure a proper dependence between the shock shape parameters and axial solution. This is a necessary condition for the problem.
The shock front shape function zf can be differentiated twice in order to express the curvature at any point along of the shock. Thus, at r =0, zf '' can be related to the axisymmetric cylinder shock front curvature κaxis by the fol lowing expression which relates the axial solution with the shock shape parameters.
Once the interdependence be tween the shock shape parameters and axial solution is established, the first boundary condition at the edge of the unconfined charge can be defined in terms of the shock slope. This condi tion establishes that the postshock flow must be exactly sonic at the charge edge, r = R (Cartwright, 2016CARTWRIGHT, M. Modelling of nonideal steady detonations. 2016. 188 f. Thesis (PhD)  Department of Applied Mathematics, University of Leeds, Leeds, United Kingdom, 2016. ). That is, given by the shock polar analysis as:
where R is the charge radius, z'f=dzf/dr and γ is the polytropic gamma. Thus, the first derivate of Equation (10) must be equal to (12) in order to comply with the first boundary condition.
However, under an ellipsoidal shock shape hypothesis, many pos sible sonic solutions could be found at different charge radii, if Equation (12) is the only boundary condition. Consequently, in order to calculate the charge radius, an additional second and complementary boundary condition must be formulated by considering the interdependence of the shock shape parameters and axial solution.
Exploring the ellipsoidal structure of the problem, for a given set of shock shape parameters, it is expected to find a charge radius R smaller than the semimajor axis of the ellipse zf, so that Rmax=β. Hence, a relationship between the unconfined charge radius and the semimajor axis of the ellipse can be established as fn=R/Rmax, where fn is an expression which relates to the degree of nonideality of the explosive. Com parable to other relationships such as Do / DCJ or λ, the R/Rmax ratio should pres ent a similar behavior when the detona tion approaches its ideal performance. Thus, fn is defined as a dimensionless expression of the following form where λ is the reaction progress at the axis; m is a reaction rate parameter; Do is the velocity of detonation; and DCJ is the ther modynamic ideal velocity of detonation; fa is a function dependent on the pressure and adiabatic coefficient.
Thus, both the charge radius R and velocity of detonation Do can be related to the shock shape parameters by combining the Equations (10), (11), (12) and (13). From these equations, one can observe how R in creases when Do and λ increases. In the limit case, when the detonation approaches to its ideal behavior λ→1 and Do→DCJ, we have that R→Rmax. This behavior is in line with Sharpe & Braithwaite (2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
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) findings about how the shock locus becomes flatter at the axis when the detonation speed increases.
3.2 Confined detonation
The strategy adopted in the present ar ticle is based on the ideas initially proposed by Eyring et al., (1949EYRING, H.; POWELL, R. E.; DUFFEY, G. H.; PARLIN, R. B. The stability of detonation. Chemical Reviews, v. 45, n. 1, p. 69181, 1949. DOI: https://doi.org/10.1021/cr60140a002.
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) and continued by Souers et al., (2004SOUERS, P. C.; VITELLO, P.; ESEN, S.; BILGIN, H. A. The effects of containment on detonation velocity. Propellants, Explosives, Pyrotechnics, v. 29, n. 1, p. 1626, 2004. DOI: https://doi.org/10.1002/prep.200400028.
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). A simple inspection of the diametereffect curves resulting from unconfined and confined detonations reveals that for a given diameter, the detona tion velocity increases as the confinement increases. Similarly, for a given detonation velocity, it shows how the charge radius decreases when the confinement increases. Thus, once the unconfined detonation state is known, the equivalent confined state is found to be dependent on the confinement, which leads to where fc is the confinement factor; Ru and Rc are the unconfined and confined radius, respectively.
Sellers (2007SELLERS, E. J. Velocity of detonation of nonideal explosives: investigating the influence of confinement. In: ANNU AL CONFERENCE ON EXPLOSIVES AND BLASTING TECHNIQUE, 33., 2007, Nashville, Tennessee, USA. Proceedings […]. Nashville: ISEE, 2007. p. 111. ) observed the impor tance of the elastic parameters of the confining material and its effect on the confined detonation velocity. It is expected that fc be proportional to the constitutive properties and thickness of the confining material. Thus, the proposed confinement factor incorporates important features of the problem where ρc and Cc are the density and acoustic velocity of the confining ma terial; ρo and Do are the density and detonation velocity of the explosive; fz is the artificial precompression fac tor due to the subsonic coupling (fz=1 for supersonic; and fz>1 for subsonic). In the case of rock blasting dimen sions, the confining thickness is far bigger than the critical thickness, corresponding to the case of infinite thickness confinement.
4. Results and discussion
In order to explore some fea tures of the ESSA model and its potential application in rock blast ing simulations, several detonation cases were modeled to observe the influence of different explosives, rock confinements and blasthole diameters upon the degree of the detonation velocity. The evaluation is centered on the performance of two different explosives, an ANFO (Kirby et al., 2014KIRBY, I.; CHAN, J.; MINCHINTON, A. Advances in predicting the effects of nonideal detonation on blasting. In: ANNUAL CONFERENCE ON EXPLOSIVES AND BLASTING TECHNIQUE, 40., 2014, Denver, Colorado, USA. Proceedings […]. Denver: ISEE, 2014. p. 114. ) and blended Emulsion 70/30 (Sujansky & Noy, 2000), covering blasthole diameters from 165mm to 311mm. The properties of the dif ferent rock masses, where several inhole detonation velocities were measured, are presented in Table 1. These data were presented by Esen (2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
) as part of the DeNE code validation study.
Rock properties (Esen, 2008ESEN, S. A statistical approach to predict the effect of confinement on the detonation velocity of commercial explosives. Rock Mechanics and Rock Engineering, Austria, v. 37, n. 4, p. 317330, 2004. DOI: https://doi. org/10.1007/s0060300400263.
https://doi. org/10.1007/s006030040026... ).
4.1 Explosive characterization
Before proceeding with the con fined detonation cases, it is necessary to characterize the explosives involved in the study. In the ESSA model, and in most of nonideal detonation approaches, the characterization is normally carried out in two parts.
The first step is the thermody namic characterization of such explo sive, which includes its ideal detonation velocity, heat of explosion and the quadratic coefficients of the isentropic gamma. These parameters are calcu lated with ideal thermodynamic codes, such as IDEX and ATLASDet, and are presented in Table 2. The ANFO ideal detonation velocity is taken from Sharpe & Braithwaite (2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
), together with other parameters, such as the heat of explosion and the coefficients of the quadratic gamma function. The thermodynamic description of the EM 70/30 was calculated with the ATLASDet code (Couceiro, 2019).
The second step is quantifica tion of their nonideality behavior. Since these ammonium nitratebased materials are nonideal explosives, multidimensional effects and mixing heterogeneities become extremely im portant to properly quantify their det onation performance, which requires some knowledge about their reaction kinetics. The chemical reaction rate is not known analytically and requires the calibration of the reaction rate parameters against a set of unconfined detonation velocities, distributed in a set of different diameters.
The ESSA model presents an attractive fitting capability for uncon fined detonation data. This is possible, since it is coupled with the Q1D model (Sharpe & Braithwaite, 2005SHARPE, G. J.; BRAITHWAITE, M. Steady nonideal detonations in cylindrical sticks of explosives. Journal of Engineering Mathematics, v. 53, n. 1, p. 2958, 2005. DOI: https://doi.org/10.1007/s1066500555707.
https://doi.org/10.1007/s106650055570...
) for the axial flow solution. The fitting process can be performed when experimental data  diameters and unconfined deto nation velocities  are available. In this study, the ANFO data set is obtained from Kirby et al. (2014KIRBY, I.; CHAN, J.; MINCHINTON, A. Advances in predicting the effects of nonideal detonation on blasting. In: ANNUAL CONFERENCE ON EXPLOSIVES AND BLASTING TECHNIQUE, 40., 2014, Denver, Colorado, USA. Proceedings […]. Denver: ISEE, 2014. p. 114. ) while the EM 70/30 is coming from Sujansky & Noy (2000). The fitting strategy is straight forward. It consists in minimizing the sum of residues formed by the square of the difference between the experimental and calculated inverseradius of the charges by varying the reaction rate parameters n, m and τ. Thus, once both sets of information are available, the ESSA model can reliably reproduce the structure of any unconfined detonation within the set of experimental diameters, including the full mapping of the diametereffect curve of the explosive.
The values of n, m and τ result ing from the minimization process are presented in Table 3. In addi tion, it is noted that a weaker state dependency was required to experi mentally adjust both explosives, since n=1.78 and n=5.77 were obtained. Cowperthwaite (1994) was the first to observe the existence of an inflec tion point in diametereffect curves when n≥1.5 (Watt, et al., 2012WATT, S. D.; SHARPE, G. J.; FALLE, S. A.; BRAITHWAITE, M. A streamline approach to twodimensional steady nonideal detonation: the straight streamline approximation. Journal of Engineering Mathematics, v. 75, n. 1, pp. 114, 2012. DOI: https://doi.org/10.1007/s1066501195060.
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; Cart wright, 2016). Figure 2 shows the modeled diametereffect curves for the ANFO and EM 70/30 together with the experimental data (Kirby et al., 2014KIRBY, I.; CHAN, J.; MINCHINTON, A. Advances in predicting the effects of nonideal detonation on blasting. In: ANNUAL CONFERENCE ON EXPLOSIVES AND BLASTING TECHNIQUE, 40., 2014, Denver, Colorado, USA. Proceedings […]. Denver: ISEE, 2014. p. 114. ; Sujansky & Noy, 2000). The presence of an inflection point is evident in both curves. In the ANFO case, it indicates a critical diameter of 78.8 mm, which is very close to the critical experimental diameter of 79mm. On the other hand, the EM 70/30 experimental data suggest a critical diameter less than 40mm. In turn, the ESSA model identifies a criti cal diameter of 36mm, in perfect co herence with the experimental data. It is further noted that as the diameters increase, the nonideal detonation velocities approximate to the ideal detonation velocity, as expected.
4.2 ESSA model application in rock blasting
Since the reproduction of realscale detonation experiments in the laboratory are restrictive, the validation of nonideal detonation models is normally carried out against experimental inhole detonation velocity taken in production blasts. In real applications, the detonation velocity can be a good indicator of the degree of nonideality of a given explosive. In this study, a set of published inhole detonation velocities (Esen, 2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
), together with results obtained with DeNE code, were used to compare the prediction capability of the ESSA model. The DeNE code, developed by Esen (2008), is based on the slightly divergent flow theory, combining the pseudopolytropic equation of state, pressurebased reaction rate law and statistical expressions to model the effect of confinement.
Thus, the ESSA model was applied in similar circumstances of application, such as blasthole diameters, rock types and explosive properties as defined by Esen (2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
). The corresponding inhole confined velocities of detonation and their comparison with the experimental and DeNE values were evalu ated. The results are presented in Table 4.
According to Esen (2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
), the mean error in the experimental measurements was 3.5%, presenting a variation between 2.2% and 7.2%. As can be seen in Table 4, the results achieved with the ESSA model were very good, presenting errors consistent with the experimental tests. The total mean error of the ESSA model was 2.4% while the DeNE was 3.8%, as presented in Table 5. Analyzing by groups of explosives, the simulations with DeNE for the ANFO showed an average error of 4.7 %, while ESSA was 2.9%. For the blended emulsion, the errors were 3.3% and 2.0%, respectively.
The ESSA model shows a good response to the sensitivities imposed by different confinement conditions. This sensitivity characteristic is important to obtain a reliable prediction of the inhole detonation velocity, and the fully axial reactive flow solution, in a sort of conditions where commercial explosives are normally applied. Thus, since the observed errors  shown in Table 5  are within the range of experimental errors presented by Esen (2008)ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
, it is believed that the ESSA model is adequate for its practical application in rock blasting simulation and modeling.
5. Conclusions
A twodimensional steady nonideal detonation model based on the Ellipsoidal Shock Shape Approach (ESSA) was successfully used to predict the inhole detonation velocity under several confinement conditions. Once the reaction rate parameters were fitted, the ESSA model could provide an excellent prediction of the unconfined diametereffect curves for both ANFO and EM 70/30 explosives, including their failure diameters. The experimental inhole detonation velocity data was presented alongside with the predictions of the DeNE (Esen, 2008ESEN, S. A Nonideal detonation model for evaluating the performance of explosives in rock blasting. Rock Mechanics and Rock Engineering, Netherlands, v. 41, n. 3, p. 467497, 2008. DOI 10.1007/s0060300601192.
https://doi.org/10.1007/s006030060119...
) and compared with the ESSA model outcomes. The overall results indicate a better response of the proposed model to the different confine ment conditions, explosives types and blasthole diameters, within the set of analyzed experimental data, endowing the ESSA model as a reliable information source for a more realistic rock blasting analysis and simulation.
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Publication Dates

Publication in this collection
22 June 2020 
Date of issue
JulSep 2020
History

Received
17 Aug 2019 
Accepted
19 Feb 2020