Thermodynamic Solution for Combustion of PETN/TNT Products with Air

Transcription

1 Thermodynamic Solution for Combustion of ETN/TNT with Allen L. Kuhl & John B. Bell University of California Lawrence Livermore National Laboratory, Livermore, CA USA University of California Lawrence Bereley National Laboratory, Bereley, CA USA Corresponding author, A. L. Kuhl: Introit We consider the explosion of a.5-g spherical ETN/TNT charge in air in a.-l calorimeter []. The hot detonation products are rich in fuel ( C (s), CO and H ), and when they mix with air, they release,5 cal/g (in addition to the, cal/g released by the detonation) by a nonpremixed turbulent combustion process []. Here we trace the evolution of combustion in Thermodynamic State Space to elucidate the mechanisms (constraints) that control this process. The analysis is formulated in the Le Chatelier plane of Oppenhiem [], i.e., specific-internalenergy versus temperature diagram, shown in Fig.. The thermodynamic properties of the components (-F, -A, eactants-, and -) were computed by the Cheetah code [], and fit with quadratics to establish the Equations of State (EOS) of the components: u (T ) = a T + b T + c ( = F, A,,) () These quadratic EOS functions (curves in Fig. ) well approximate the thermodynamic values (circles in Fig. ) and fully characterize the thermodynamic behavior of the system []. Combining this with a combustion model, we find a combustion locus in thermodynamic space; we show that this locus is a strong attractor to numerical simulations of the combustion field. Thermodynamics We divide the process into two steps: (i) mixing: of F and A to form ; (ii) combustion: material transformation from to in a thermodynamically-isolated chamber. Mixing is governed by: Mass: Y F + Y A = () Volume: Y F v F + Y A v A = v c () Energy: Y F u F + Y A u A = u c ) () where the specific volume of the chamber is given by v c = /(m F + m A ), and c = Y A /Y F = 5.7 represents the chamber air/fuel ratio. Given the initial conditions ( T A = K & T F =,7K ), the constraint () specifies the chamber energy: u c ) = 5 cal /g and reactants temperature T =, K. and air combine in c proportions to form reactant ). Combustion at constant energy and volume (UV) is expressed by: Mass: Y + Y = (5) Volume: Y v + Y v = v c () Energy: u = u = u c ) (7) Constraint (7) provides the combustion locus: u = u = u c ) depicted in Fig. as the - line.

2 Gas Dynamics We model evolution of the combustion field in the limit of large eynolds and eclet numbers. Then the flow field is governed by the multi-component gas-dynamics for the mixture (m): Mass: t m + ( m u) = () Momentum: t m u + ( m uu) = p m (9) Energy: t m (u m + u u/)+ m (u m + u u/)u = ( p m u) () Components: t Y + u Y = Y s with Y = () where = F,A, and = {, s,(+ s )} denotes the stoichiometric source/sin coefficients based on the stoichiometric air/fuel ratio: s =.77. The term Y s embodies inetic effects. Here it is modeled by the fast-chemistry, large-damöhler-number limit which is consistent with the gas dynamic model (-). This system is closed by Equations of State (EOS) for components- (A, A, A5 & A7) of Fig., while mixture EOS is based on the mixing laws for perfect gases: EOS: T m = [ b m + b m a m (c m u m )]/a m & p m = m m T m (a,b) roperties: a m = Y a, b m = Y b, c m = Y c, m = Y (a,b,c,d) The above model equations (-) were integrated using a high-order Godunov scheme, and Adaptive Mesh efinement (AM) was used to follow the thin reaction zones on the grid. We simulated the explosion and combustion of.5-g ETN/TNT charge in a.-l cylinder; details of this numerical simulation are described in a companion paper of this Colloquium [5]. rojection The numerical solution was projected from physical space to Thermodynamic space via the following volume-averaging operators: Mass: M m (t) = M (t) where M (t) = (t,x)dv (a,b) Amagat s Law: V m (t) = V (t) where V (t) = N (t,x)dv /N m (5a,b) Dalton s Law: p m (t) = p (t) where p (t) = p (t,x)dv / (a,b) Specific Energy: u m (t) = u (t) where u (t) = (t,x) u (t,x) dv / M (t) (7a,b) Specific Entropy: s m (t) = s (t) where s (t) = (t,x) s (t,x) dv / M (t) (a,b) Temperature: T m(t) = T (t) where T (t) = (t,x) T (t,x) dv / M (t) (9a,b) where denotes the volume of the chamber.

3 Solution Temporal Space The evolution of the mean thermodynamic variables from the numerical solution (averaged over ) is presented in temporal space in Figs. and. While component variables in general change over time due to combustion, conserved variables such as the mixture mass, volume and energy remain constant thereby illustrating conservation principles ()-(7) for the system. In addition, Figs. b and d illustrate Amagat s Law of partial volumes and Dalton s Law of partial pressures, respectively, for the mixture. Thermodynamic Space The mean solution is presented in Thermodynamic Space in Figs. 5-. In this depiction, time appears only implicitly as the evolution parameter that lins corresponding thermodynamic values, depicted as data points (circles) on those figures. Figure 5 presents the evolution of numerical solution in the u T plane. Computed points for fuel, air and products lie on the F, A and lines corresponding to the equation of state functions (). Computed points for the mixture rapidly (by. ms) approach the combustion locus: u c = 5 Cal / g denoted as the - curve. Figure presents the numerical solution in the p v plane. Computed points for the mixture lie along the combustion isochor: v c = 9 cc / g (red curve -). Figure 7 presents the numerical solution in the p T plane. Computed points for the mixture lie between the reactants isochor and products isochor. Figure presents the numerical solution in the s T plane. While the reactants and products entropies obey the usual logarithmic dependence on temperature, the mixture entropy exhibits a linear dependence on temperature in accordance with the linear mixing laws ()-(9) of perfect gases. Model The above comparisons suggest the following model for the thermodynamic solution of combustion in a chamber (denoted by subscript c): Temperature: T c (t) = T + Y (t)[t T ] () ressure: p c (t) = p + Y (t)[p p ] () Entropy: s c (t) = s + Y (t)[s s ] () Here the model parameters (T =,K & T =,9K ; p =.bars & p =.9 bars; s =.5 cal /g K & s =.7 cal /g K ) come from the thermodynamic solution, while the products function Y (t) comes from the gas-dynamic solution. In the present case, the products function is well approximated by the exponential: : Y (t) = e.t () This model is depicted in Fig. 9. One can see that the numerical solution (circles) lies close to the model functions ()-(), thereby validating the model of this process.

4 Assessment To assess the validity of the model, we have compared the theoretical model predictions with experimental data; results are shown in Table. For the.5-g ETN booster charge, the model predicts.75 bars versus the measured value of. bars; their ratio suggests that 97% of the charge detonated. For the composite charge (.5-g ETN +.-g TNT) in nitrogen, the model predicts. bars versus the measured value of. bars; their ratio suggests that % of the charge detonated. For the composite charge in air, the model predicts.9 bars versus the measured value of.5 bars; their ratio suggests that 9% of the charge was consumed by combustion during the course of the experiment. Table. Comparison of experimental data with theoretical predictions. CASE p c (bars) p (%) T (K) ETN booster charge: experiment (in N ).±.7 97% Thermodynamic Model (in N ).75 ETN/TNT Composite charge: experiment (in N ).±. % Thermodynamic Model (in N )., K experiment (in air).5±. 9% Thermodynamic Model (in air).9,9 K ésumé Mixing and combustion in a thermodynamically-isolated chamber is controlled by u c : the initial energy in the system. It establishes not only the initial state (temperature) of the reactants and the final state (temperature) of the products, but also the locus of combustion states in thermodynamic space. Implementing the quadratic EOS model for stoichiometric combustion in the multi-component gas dynamic conservation laws allows one to simulate the evolution of combustion in physical space. Note that combustion in physical space always occurs under locally stoichiometric conditions ( s =.77), while the combustion locus represents global combustion effects for the chamber (i.e., at the chamber stoichiometry: c = 5.7). The most important contribution of this paper is that we show that the gas dynamic solution, when projected from physical space to thermodynamic space, rapidly approaches the combustion locus predicted by the Thermodynamic model thereby demonstrating that the combustion locus is a strong attractor for this reactive system. This model is especially useful in cases where the inetics of the system is not nown. Acnowledgements This wor was performed under the auspices of the U. S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-75- Eng-. It was sponsored by the Defense Threat eduction Agency under IACO # 5-7.

5 eferences [] Neuwald,., eichenbach, H. & Kuhl, A.L. () Shoc-Dispersed Charges Combustion in Chambers and Tunnels, Energetic Materials, th ICT Conference,.-. [] Kuhl, A. L., Ferguson,.E & Oppenheim, A.K. (999) Gasdynamics of Combustion of TNT in, Archivum Combustionius 9:-, 7-9. [] Oppenheim, A. K. and Maxon, J. A. (99) Thermodynamics of Combustion in an Enclosure, rogress in Aeronautics and Astronautics, American Institute of Aeronautics and Astronautics, New Yor, pp. 5-. [] Kuhl, A.L, Howard, M and Fried, L. (999) Thermodynamic Model of Afterburning in Explosions Energetic Materials, th ICT Conference, [5] Bell, J. B., Day, M., Kuhl, A.L., Becner, V., & Neuwald,. (5) Numerical Simulation of Combustion of TNT/ETN in a Calorimeter, th ICDES, Montreal, Canada. Appendix: Equations of State for the Composite ETN/TNT Charge FUEL: u F = T +.79 T. (cal/g F ) (A) AI: u A =. 5 T +.7 T 7. (cal/g A ) (A) The fuel and air are assumed to mix at frozen composition to form reactants at an air/fuel ratio of Y A /Y F ; eactants properties may be calculated from: u ( ) (u F + u A )/(+ ) (A) EACTANTS: u ) =.95 5 T T. (cal/g ) (A) EACTANTS: u ( s ) = T +.9 T. (cal/g ) (A5) where c = 5.7 and s =.77. The products are assumed to be in thermodynamic equilibrium; at a pressure of bar, a fit to Cheetah calculations gives ODUCTS: u, b) = T +.7 T 5.5 (cal/ g ) (A) ODUCTS: u ( s, b) =.5 5 T +. T.5 (cal/ g ) (A7) JWL arameters: Charge =. g/cc, MW=5.5 g/mol A = 5.79 Mbars, B =. Mbars, C =.97 = 7., =., = =.7, =.59 H d = 95 cal/g F H c = 5. cal/g =,.75 cal/g F Chapman-Jouguet jump conditions Shoc velocity = 5.9 m/s article velocity=. m/s Sound speed =.99 m/s E = -7.9 cal/g CJ =79,. bars V CJ =.75 cc/g T CJ =,. K S CJ =.7 cal/g-k H CJ =59.5 cal/g E CJ =.9 cal/g 5

6 Figure. Locus of states in the Le Chatelier diagram for the combustion of the detonation products gases generated by the explosion of a composite (ETN/TNT) charge in the.-liter calorimeter. Chamber stoichiometry is c = A CJ u,c ( s ) u (Cal/g ) -5 - H c = -5 cal/g u c ( s ) F Combustion at: T = K T =,K T =,5K u d,575k,k,7k -5 T (K) Figure. Locus of states in the Le Chatelier diagram for stoichiometric combustion ( s =.77) of the explosion products from the composite (ETN/TNT) charge in the.-liter calorimeter. The green lines illustrate combustion loci starting at K, K and 5 K.

7 (a) masses (d) pressures x 9 7 M (g) p (bars) 5 (b) volumes (e) moles V (cm ) N.. (c) densities. (f) entropies. -. (g/cc) - s (cal/g-k) mixing -. Figure. Evolution of the mean thermodynamic variables of the components in the chamber. 7

8 (a) specific internal energies 5 u (cal/g ) -5 - (b) temperatures 5 T (K) 5 5 Figure. Evolution of the mean specific internal energies and temperatures of the components in the chamber.

9 A u (cal/g) - - u,d. ms F (t) (t) (t) (t) Combustion Locus T (K) Figure 5. Evolution of the mean thermodynamic solution in the u T plane. Numerical solution for the mixture is attracted to the Combustion Locus: u c = 5 Cal / g (-). x 9 7 ms ms ms eactants isotherm (T=K) isotherm (T=9K) (t) (t) Combustion Locus p (bars) 5.ms v (cm /g) Figure. Evolution of the mean thermodynamic solution in the p v plane. Numerical solution for the mixture rapidly approaches the combustion isochor: v c = 9 cc / g (-). 9

10 x ms ms ms p (bars).ms eactants isochor (v=9cc/g) isochor (v=9cc/g) (t) Combustion Locus T (K) Figure 7. Evolution of the mean thermodynamic solution in the p T plane. Numerical solution for the mixture rapidly approaches the combustion locus -... ms ms. s (cal/g-k). ms.ms. eactants isochor (v=9 cc/g) isochor (v=9 cc/g) (t) Combustion Locus T (K) Figure. Evolution of the mean thermodynamic solution in the s T plane. Numerical solution for the mixture rapidly approaches the combustion locus -.

11 (a). (b). 5. Y (t) T(K) 5.. D AM fit=-e -.t 5 T c (t) (c). x (d) p (bars) 5 s (cal/g-k). p c (t) (t). s c (t) (t). Figure 9. Comparison of computed mixture environment histories with the analytical chamber model: (a) mass fraction of products produced, (b) temperature, (c) pressure, (d) entropy.