Steady Deflagration Structure in Two-Phase Granular Propellants - PDF document

Steady Deflagration Structure in Two-Phase Granular Propellants Joseph M. Powers 1 , Mark E. Miller2, D. Scott Stewart3, and Herman Krier4 presented at 12th ICDERS, Ann Arbor, Michigan July 23-28, 1989 1 Assistant Professor, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana 2 Member of Technical Staff, The Aerospace Corporation, Los Angeles, California 3 Associate Professor, Department of Theoretical and Applied Mechanics, University of illinois at Urbana-Champaign, Urbana, illinois 4 Professor, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois This work was performed with the support of the U.S. ONR, Contract N00014-86-K-0434; Dr. Richard S. Miller, Program Manager

~ :.:i':\~-·A • Envisioned Two-Phase Deflagration granular propellant • ........... ~:,~I/~ ... ~"'· ... reacted gas --------- reaction I zone deflagration 1 wave front D = deflagration wave speed Review of Two-Phase Deflagration 1973--Kuo, Vichnevetsky, and Summerfield, AIAA Journal 1974--Kuo and Summerfield, AIAA Journal 1975--Kuo and Summerfield, 15th Combustion S_ym_Q_osium 1986--Drew, Combustion, Science and Technology Related Work 1986--Baer and Nunziato, International Journal of Multiphase Flow 1988--Powers, Stewart, and Krier, Dynamics of Explosions, AIAA Progress 1989--Powers, Stewart, and Krier, Journal of Applied Mechanics 1989--Powers, Stewart, and Krier, Combustion and Flame

Model Features -Representative of a larger class of two-phase models -Each phase obeys a mass, momentum, and energy evolution equation -Mixture mass, momentum, and energy conserved <j> = phase volume I total volume) utilized -Volume fraction ( -PDE's are hyperbolic -Characteristic wave speeds: u1, u2, u1 + c1, u2 + c2 -Dynamic compaction equation employed for closure -Number of particles conserved -Compressible spherical reactive particles -Simplified drag and convective heat transfer relations -Virial gas equation of state for inert gas -Tait equation of state for reactive particles -Viscosity or heat conduction in gas not considered -Viscosity or heat conduction in solid not considered -Radiation not considered

Two-Phase Model Equations -ordinary differential equations in steady wave frame -~ = distance in steady wave frame ~ = ~ - Dt -D = steady wave speed -with additional algebraic equations, the model can be represented by four differential equations in four unknowns particle mass <1>1 [T -T J de2 dv2 p v -+P - = -h- 2 1, particle energy 2 2 dl; 2 dl; r 1/3 dynamic pore collapse

Conservation Relations -obtained by integrating conservative differential equations -initial conditions specify integration constants 1) Mixture mass, momentum, and energy: 2 /2+P /P 1] + p 2 q, 2 vJ /P 2] = p 1 <j> 1 v e 2 +v}2+P -Pa1::{ ea+D /Pal 1[ e 1 +v~/2+P mixture energy - "a" denotes apparent or bulk initial property apparent initial density Pa = P10<l>10 + P20<l>20' apparent initial pressure pa = P10<l>10 + P20<l>20· apparent initial energy 2) Particle number equation:

Gas and Particle State Equations 1) Gas: = p RT ( 1 + bp ) p gas thermal 1 1 1 1 ' gas caloric el = cvlTl" 2) Particle: particle thermal particle caloric Saturation condition: cj> + cj> = 1. I 2

Initial Conditions -Eight independent initial conditions specified for original eight differential equations -Temperature and density for each phase -Velocity for each phase -Initial particle radius -Initial volume fraction -Specified so that initial state is an equilibrium state -Remaining initial conditions fixed by state equations and saturation condition:

~ Dimensional Input Parameters [m/ (s Pa)] 2.90 x 10-9 a [kg /m 3] 1.00 x 100 Pio 1.00 x 100 m [kg I (s m2 )] 1.00 x 104 [kg/m3] 1.90 x 103 P20 1.00 x 10 7 [JI (s K m 8 f3)] h [JI (kg K)] 2.40 x 103 Cvl 1.50 x 103 [JI (kg K)] Cv2 8.50 x 102 R [JI (kg K)] [(m I s)2] 7.20 x 106 O' 5.84 x 106 q [JI kg] [m] 1.00 x 10-4 To [m3 I kg] 1.10 x lQ-3 b 5.00 x 10° Y2 [kg I (ms)] 1.25 x 102 µc [K] 3.00 x 102 To [K] 3.00 x 102+ Tig

~ Two-Phase Deflagration End States -arbitrarily assume complete reaction -mixture equations define two-phase Rayleigh line and Hugoniot equations Rayleigh line Hugoniot -In general, two physical deflagration solutions for a given wave speed D 1) Low pressure, supersonic, strong solution 2) High pressure, weak, subsonic solution -Maximum deflagration wave speed at CJ condition, sonic solution 1WO-PHASE HUGONIOT /WEAK SOLUTION (SUBSONIC) STRONG SOLUTION #"' (SUPERSONIC) CJ SOLUTION_.. (SONIC) 1/pl RAYLEIGH LINE J' D<DcJDEF RAYLEIGH LINE D=DcJDEF

~ ~ ~ ~ Complete Reaction End State -assume exhaust pressure can be controlled -deflagration wave speed and all gas phase properties then known as functions of exhaust pressure -<l>io = 0. 70 CJ Deflagration Pressure = / k:' 201.2 MPa 140 - 120 ..._., 13 100 8. Strong Deflagration Cl) 80 Branch (Supersonic) Weak Deflagration 60 c:: Branch (Subsonic) 0 ·a 40 ro i;::: 20 Cl.) Cl 0 400 0 100 200 300 500 Final Gas Pressure (MPa) 0 - -1000 ~ ..._., 0 ...... -2000 - 8 Weak Deflagration > Branch (Subsonic) -3000 CJ Deflagration Vl ro 0 ca .s -4000 Strong Deflagration Branch (Supersonic) -5000 0 100 300 400 200 500 Final Gas Pressure (MPa)

CJ Deflagration State -CJ state can be determined numerically -Simple analytic expression in two limits 1) ideal gas b = 0 2) PJ(Pa ea)~ 0 2 e - a T CJ= -, 'Y1 ("(1 + 1) cvl 2 = e e 'Y 1("(1 + 1) a' CJ - 2 ('Y1 - 1) e ) r + 1 ( a 1

Two-Phase Deflagration Structure -Ordinary differential equations integrated for D = 100 m/s -Arbitrarily assumed that no shocks exist in structure or no sonic points -With this assumption the end state is weak subsonic end state -exhaust pressure ,.., 400 MPa -Extreme deflagration exhaust conditions because some parameters arbitrarily chosen so that a numerically resolved structure could be presented -No two-phase steady detlagration structure going to complete reaction was found

~ ~ 0.70 N -e- 0.69 0.68 ____ ...__..__,___...__._.,__..__.... ___ -A.- ________ ...._. -12 -10 -8 -6 -2 0 (mm) Figure 7 Solid Volume Fraction Structure, <1>20 = 0.70, D = 100 m/s 100 80 60 - a (ll 40 ........, ::s 20 0 -20 -12 -10 -8 -6 4 -2 0 (mm) Figure 8 Gas and Solid Lab Velocity Structure, <1>20 = 0.70, D = 100 m/s

103 solid 102 '2 Q.. 101 ::E '-" Q.. 100 gas 10-l -12 -10 -8 -4 -2 0 -6 ; (mm) Figure 9 Gas and Solid Pressure Structure, 4>20 = 0.70, D = 100 m/s 103 - ~ '-" solid E-< 102 101 -10 -8 -12 -6 -4 -2 0 ; (mm) Figure 10 Gas and Solid Temperature Structure, 4>20 = 0.70, D = 100 m/s

~ 10-l gas 10-2 10-3 solid 10-4 <'I 10-5 :;E 10-6 10-7 10-8 10-9 -12 -10 -6 -4 -2 0 -8 (mm) Figure 11 Gas and Solid Mach Number Squared, <1>20 = 0.70, D = 100 m/s

, Conclusions -Possible to predict gas phase deflagration end state and wave speed as function of exhaust pressure and initial conditions -For the region of parameter space studied, no steady two-phase deflagration structure exists -Processes that support detonation are not sufficient to support a two-phase deflagration -It may be necessary to include heat conduction and radiation to model two- phase deflagrations -Combustion of granulated propellants could possible accelerate into steady, self-propagating two-phase detonation