j. m. powers, d. s. stewart and h. krier- two-phase steady detonation analysis

8/3/2019 J. M. Powers, D. S. Stewart and H. Krier- Two-Phase Steady Detonation Analysis

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'.

Two-Phase Steady Detonation AnalysisJ. M. Powers, D. S. Stewart, and H. Krier

Reprinted from Dynamics of Explosions, edited by A. L. Kuhl, J. R.Bowen, J.-C. Leyer, and A. Borisov, Vol. 114 of Progress in Astronauticsand Aeronautics, AlAA, Washington, DC, ISBN 0-930403-47-9, 1988.


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Two-Phase Steady Detonation AnalysisJ. M. Powers,'" D. S. Stewart,t and H. Krier:!: _University of Illinois at Urbana-Champaign, Urbana, Illinois

AbstractSteady solutions to a set of two-phase reactive flow model equationsare studied to test the hypothesis that observed deviations from Chapman-Jouguet (C-J) detonation states in porous solid propellants aremanifestations of the two-phase nature of the flow. Shock jump relationsare presented. A simple expression for a minimum detonation wave speedanalogous to a C-J detonation for a single phase is given. The analogousC-J point is a sonic point. In the appropriate Emit, the minimumdetonation velocity varies linearly with initial bulk density and thecorresponding detonation pressure varies with the square of initial bulkdensity. Nonideal gas effects play an important role in determining C-Jconditions. The effect of reaction zone structure is s.tudied to determine itseffect on detonation end states.

I. IntroductionDetonation, a shock-induced chemica!. reaction, has been described bythe theory of Chapman ana Jouguet The simple C-J theory analyzesa detonation as a jump discontinuity and solves algebraic equations thatrepresent the conservation of mass, momentum, and energy for thedetonation end states (pressure, density, etc.) as functions of thedetonation velocity. C-J theory identifies a unique wave speed and sonicend state for an unsupported detonation. In addition to end state analysis,the structure of the detonation (i.e., the distribution of pressure, density,etc., in the reaction zone behind the wave head) must be considered whenclassifying possible steady soiutions. In some cases, certain end states canbe ruled out because the end state cannot be attained from the initial state atthe wave head. (The discussion of detonation structure is the topic of ZNDtheory.)

Cqpyright 1988 by J. M. Powers. Published by the American Institute ofAeronautics and Astronautics, Inc. with permission.. * Visiting Assistant Professor, Dept of Mech. and Ind. Eng.t Associate Professor, Dept. of Theo. and Appl. Mech.* rofessor, Dept. of Mech. and Ind. Eng.341


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342 J. M. POWERS ET AL.States and wave speeds other than those predicted by C-J theory may

be realized. Under appropriate circumstances, it can be shown for one-phase models that a so-called weak detonation (which travelssupersonically relative to the detonation products) is possible. Asdiscussed by Fickett and Davis (1979), weak detonations are observed inexperiments. Also, the numerical work of Butler, et al. (1982) suggeststhat both C-J and weak detonations may be admitted by equations of two-phase reactive flow used to model solid propellants and granulatedexplosives.The evaluation of numerical solutions to the two-phase reactive flowequations [the work of Butler, et al. (1982) is an example] requires asystematic study of the steady solutions. Knowledge of the one-dimensional steady solutions is necessary in order to develop the stabilitytheory for two-phase detonation models. Steady solutions are alsonecessary to verify that numerical predictions of apparently steadysolutions by unsteady model equations are not numerical artifices.Our work analyzes the steady form of a two-phase, unsteadydetonation model similar to Butler and Krier's (1986) and Baer andNunziato's (1986). The first step in a complete structure analysis is theidentification of detonation shock states and detonation end states. We firstdiscuss shock jumps admitted by the equations. We then consider endstates admitted by the governing equations and, in particular, identify theminimum detonation wave speed admitted by our two-phase model. Inthis process, we identify a C-J wave speed analogous to the C-J wavespeed of the simple one-phase theory. These steps require the analysis ofalgebraic conditions. We next pose the steady equations in a reduced formin which the structure can be studied. The additional constraints that aphysically admissible detonation structure places on this problem and theidentification of other admissible detonation states is given by Powers(1988).

II. The Unsteady Two-Phase ModelHere a two-phase model is proposed. The model is similar to modelsused by Butler and Krier (1986) and Baer and Nunziato (1986). Theproposed model utilizes simpler drag and heat transfer relations and asimpler solid equation of state in order to facilitate the analysis. As in theButler-Krier model, but not in the Baer-Nunziato model, the proposedmodel allows momentum changes in response to a gradient in the porosityof the gas phase. Also, as Butler-Krier did, but Baer-Nunziato did not, wedo not consider compaction work in our energy equations. Details aboutwhy these choices concerning the momentum equation and compaction

work are given by Powers (1988). Intraphase diffusive thermal andmomentum transport is ignored. Details about the rationale for the modelin the context of continuum mixture theory can be found in Wallis (1969),Baer and Nunziato (1983), and Krier and Gokhale (1978). For theproposed model, it is assumed that each phase is a continuum;consequently, partial differential equations resembling one-phase equationsare written to describe the evolution of mass, momentum, and energy in


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TWO-PHASE STEADY DETONATION ANALYSIS 343

each constituent. In addition, each phase is described by a thermal staterelation and a corresponding caloric state relation. One phase is assumedto be a solid, the other, gas. In order to close; the system, a compactionequation similar to that of Baer and Nunziato (1986) is adopted. Asdiscussed there, the compaction law is suggested by an assumed form ofthe second law of thermodynamics. The compaction law states that thesolid volume fraction changes in response to pressure ,differences betweenphases and to combustion. Choosing such a compaction equation alsoinsures that the characteristics are"real; thus, the initial value problem iswell-posed. We emphasize that the choices we have adopted for theclosure problem place a premium on simplicity so that explicit analyticcalculations can be made whenever possible. At the same time, the modeladopted here is thought to be representative of a wider class of two-phasedetonation models currently in use.In order to describe the transfer ofmass, momentum, and energy fromone phase to another, phase interaction terms must be specified. The solidphase is assumed to be composed of spherical particles so that certainempirical correlations can be invoked [Butler and Krier (1986), Kuo andNydegger (1978), Denton (1951)]. These correlations describe theburning of spherical particles, particle/gas drag, and particle/gas heattransfer. In contrast to Butler and Krier (1986) and Baer and Nunziato(1986), we explicitly enforce the conservation of number of particles; thatis, we assume the particles do not agglomerate or disintegrate.The unsteady equations are

if 24>2u] + ;x[ p24>2+ p2 >2 ]= + u1-u2 (4)


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344 J. M. POWERS ET AL.

i[PI2( e2 u; 12)] +iJP2P2U( 2 12 +P2/pJ]= - ( e2 /2)( +/3\2ui UU2+h ::!2(T(T2) (6)

ar I> I?] + ar 2 Ir3] = 0ad. 2 axt 2

.p2

P2 + P s20 +q( 'Y.- 1 )p2 2

+ = 11 2

(7)

(9)

(10)

(11)

(12)

(13)


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where the subscript "0" denotes the undisturbed condition, 1 the gasphase, 2 the solid phase, p the density, the volume fraction, u thevelocity, r the solid particle radius, P the pressure, m the burn index, a theburn constant, the drag parameter, e the internal energy, h the heat-transfer coefficient, R the gas constant, b the covolume correction, c theconstant volume specific s the nonideal solid sound parameter, thecompaction viscosity, y" the Tait parameter, and q the heat of reaction.Undisturbed condloons are specified as

P - P P - P "'1 = "'10' u1 = 0- 10 ' 2 - 20 ''I' 'I '

Equations (1) and (2) describe the evolution of each phase's mass,Eqs. (3) and (4) the momentum evolution, and Eqs. (5) and (6) the energyevolution. Here, a principle of two-phase modeling has been employed;the motion of an isolated constituent is determined by the action of otherconstituents on it. Homogeneous mixture equations are formed by addingEqs. (1) and (2), Eqs. (3) and (4), and Eqs. (5) and (6), respectively.Thus, for the mixture, conservation of mass, momentum, and energy ismaintained.Inhomogeneities in Eqs. (1-6) model interphase momentum, energy,and mass transfer. The model incorporates momentum transfer in the formof Stokes drag and energy transfer as constant-coefficient convective heattransfer. For mass transfer, we utilize an empirical relation for theregression of particle radius. It is observed that the rate of change ofparticle radius is proportional to the surrounding pressure raised to somepower. As we consider a constant number of compressible particles, ourmodel must also allow the particle radius to change in response to a changein density of the particle phase. By combining the particle mass evolutionequation (2) with Eq. (7) for number conservation, it is seen that theproposed model has the desired features, as

Or ar , m r ( ap2 ap2).,....,+u - = -aP -- at 2 ax 1 3p at (IX2 (14)that is, as the motion of a particle is followed, its radius may change inresponse to both burning and compression.The compaction law is expressed in Eq. (8). Constituent 1 is a gasdescribed by a virial equation of state (9). Likewise, constituent 2 is asolid described by a Tait equation of state (11). Assumption of a constantspecific heat at constant volume for each phase allows caloric equations(10) and (12) consistent with the assumptions of classical thermodynamicsto be written for each phase. The variable 4> is defmed as a voJume fraction,


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346 J. M. POWERS ET AL.c> == constituent volume/total volume. All volume is occupied byconstituent 1 or 2; no voids are permitted. This is enforced by Eq. (13).By using standard techniques [Whitham (1974)], it can be shown that thecharacteristic wave speeds are u1' u2 u1 C1 and u2 c2 where cirepresents the sound speed of each phase. the uncouplIng of thecharacteristic wave speeds is a consequence of the assumed form of thecompaction law. Other closure relations will, in general, couple thecharacteristic wave speeds.

III. Steady State AnalysisSteady state analysis of two-phase equations has been performed byKuo and Summerfield (1974a, 1974b), Krier and Mozaffarian (1978),Taylor (1984), and Drew (1986). The studies of both Kuo andSummerfield and Drew considered only the deflagration of incompressibleparticles. Krier and Mozaffarian numerically studied two-phase detonationreaction zones with the assumption of incompressible particles. Asdiscussed by Macek (1959), for typical condensed phase detonationpressures, this assumption is unrealistic. Others such as Taylor (1984)studied shock jumps in two-phase reactive flow equations. Here, weconsider a steady two-phase detonation model with compressible particles.The discussion we present of two-phase detonation end states is new to thebest of our knowledge.Equations (1-13) can be recast in a more tractable form. First, wewrite the equations in dimensionless form. For a right-running steadywave, we make the Galilean transformation = x - Dt, v = u - D under

which Eqs. (1-8) collapse to eight ordinary differential equations. Here, Dis a constant defined as the steady wave speed. Next, Eqs. (1), (3), and(5) may be eliminated in favor of homogeneous mixture equations formedby the addition of the steady form of Eqs. (1) and (2), (3) and (4), and (5)and (6). The resultant mixture equations and the steady form of Eq. (7)may be integrated to form algebraic equations. Thus, the steady two-phasemodel is described by four ordinary differential equations and ninealgebraic equations.Dimensionless Model Equations

Define dimensionless variables where "*" indicate a dimensionlessquantity, as follows:

= p. = p. / po1 I 1

v . =v. /DI 12e . = e./D1 1

2p . = P/(poD )1 1 12T . = c . T./D1 VI 1 r. = r /L

i = 1,2


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where D is the wave speed, L an undefined length scale to be associatedwith the reaction zone length, p.o the initial density of phase i, and c . thespecific heat at constant volome of phase i. Define the independent dimensionless parameters as3 mD2m-l1t l = ap 10

1t = m4

1t3 = h L 2/3 / c DP20 vI

1t =c 1 /c 2v v

1t9 = P DL/Il20 c21t = q/D10 1t -11 - 10

1t13 = bp 10 21t = c 2TO/D14 v1t - 'V17 '2

and the following dependent dimensionless parameters as:1t 18

I-1t 11= 1t + ----=-=-11 1t5= [1t - 1] 1t 1t- [I + 1t ]7 6 14 13

= I-1t111t = [1t - 1] 1t - 1t21 17 14 8


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348 J. M. POWERS ET AL..

then the dimensionless model equations (for compact notation the stars aredropped) can be written economically as

r = - v 2


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TWOPHASE STEADY DETONATION ANAl.YSIS

P 5 12

e s 1'2 =1= n17 ft, =1= n2 (f t = 1 ) P 1017 2

cp. =1= CP2 s 1

Inm Shoek Jump Ct)ftdirionll

348

(23)

(24)

(25)

(26)

(27)

B'lUltiOH8 (15=27) IN villd tftmushout the (Ueld), fiowfield. I f Idetonltion 801ution with I ItidinS IIhoek i6 186umM, It il l neee681f)' toknow 8hoek jump eondition8. In order to form two=phll6e 8hoek jumpeonditiollll, it i8 1l88umed thlt I thin 8hoek wive exilu8 in whlen noIlJlPfeeilble fe&etion, drlS, htlt tfln8fef, Of eomp&etion oeeUf8: Thu8,1I'l8. (15=18) mlY be repl&eed by the 8hoekjump eondition8,


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(31)where "s" denotes the shocked state and "0" the undisturbed state.Equations (28-31) and state relations [Eqs: (25) and (26)] are sufficient tocalculate the shock jumps for phase 2. The shock jumps for phase 1 areimplied by the mixture Eqs. (19-21) and state relations [Eqs. (23) and(24)]. Since porosity does not change through the shock, the shock jumpsfor the two phases are calculated independently.Three solutions to the gas shock jump relations exist: the inertsolution, a nonphysical solution, and a shock solution. In the limit as 1t13(or b) approaches zero, the nonphysical prediction of gas densityapproaches -l/1t13 and is therefore rejected. In the same limit, the shocksolution can be thought of as a perturbation to the ideal-gas shock solution.The asymptotic gas shock solution is

For 1t13 > 0, the nonideal shock relations predict lower shockpressures than the ideal relation. Figure 1 shows a plot of the dimensionalgas phase shock pressure vs the shock wave speed. Unless otherwisenoted, dimensional input parameters for this and all other cases are listed inTable 1. These input are representative of the granular solid

'2t3' - "'"l..,-

1.0

0.8

0.6

0.4

0.20.0 + __ __ "r '__

3000 4000 5000 6000 7000 8000 9000 10000D (mls)

Fig. 1 Gas-phase shock pressUIe vs shock wave speed.


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a, ml (s Pa)PIO, kgl m3mp, kg 1 s m2)P20, kg 1m3 .lJh, J I(s K m2)cvl' J 1(kg K)lJcv2, J 1(kg K),lJ

Table 1 Dimensional input parameters

2.90 x 10-9 R, J 1 kg K)1.00 x 101 s, (m l s)2#1.00 x 100 q, J 1kg1.00 x 1()4 ro, mlJ1.90 x 103 b, m3 /kg1.00 x 107 12#2.40 x 103 Jl, kg l (m s)c1.50 x 103 To,K

8.50 x 1028.98 x 1065.84 x 1061.00 x 10-41.10 x 10-35.00 x 1001.00 x 1063.00 x 102

propellant HMX. Table 1 notes references for some of these parameters.Those parameters without reference have been estimated to match data orare able to be arbitrarily specified. We allow the initial gas density andtemperature to be arbitrarily specified. The heat transfer and dragparameters, not important in this work, have been estimated to roughlymatch the trends predicted in experiments. The Tait equation parametershave been estimated to match experimental shock and compaction data[Powers, et al. (1988)]. The gas state equation parameters have beenchosen to match detonation state predictions of the thermochemistry codeTIGER [Coperthwaite and Zwisler (1974)] as reported by Butler and Krier(1986).For the solid phase, two solutions are admitted: the inert solution andthe shock solution. The exact expression for solid phase shock pressure is605040

302010

3000 4000 5000 6000 7000 8000 9000 10000D (mls)

Fig. 2 Solid-phase shock pressure vs shock wave speed.


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362 J, M, POWERS ET AL ,

2.1C (1C17 1)2P a 14 = 1C28 1C +1 817 (33)In Fig, 2, the dimensionallolid phale Ihock prellure il plotted VI theshOOK wave speed,Similar relationi are available for Ihock density, temperature, andenergy, For the ease of two ideal gales (1C13 a1Ca 0), these equationsreduce to famUiar Bhock relations in terms of the shock Mach number, Forthe ideal,ases, shook Mach numbers may be exprelled al

(34 = 35)

One eflterion for I solid state equltlon is that the state equation in with the RMkine=Husoniot jump conditions be able to mltchexperimentll shock 10ldins dlta. By solvin. the shook jump equltions forthe finll velooi!y, In e'!lu'ession for shook wave speed Dvs piston velocityu2' cln be developed. The relation in term8 of dimensional pllfameters is

(36)

o 400 fiOO 800 1000 liOOPitmi V@looity (mj)

imtI ho@i WIlW p@@d piUon fof Oiitl IIMX:


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TWOPHASE STEADY DETONATION ANALYSIS 363

As shown in Fig. 3 by choosing Y2= 5 the experImental shock loadingdatil of Marsh (1980) is accurately matched.Bnd State AIlM,vsisAn end state has been reached when, at a given location theinhomogeneities of Bqs. (15-18) are identically zero. Equations (1S-18)suggest that an end state has been reached when cj) .. 0 and TJ= t6T2'corresponding to complete reaction and temperatUre equilibnum. Asdiscussed more fully in Sec. IV, it is not established if other end statesexist or even if the complete reaction/temperature eqUilibrium state is alegitimate end state. By assuming that the complete reaction/temperatureeqUilibrium state is an end state, we may draw conclusions about thesteady wave spetds admitted by the two-phase model and show that inmany ways the two-phase model is analogous to one-phase models.In the complete reaction state, the mixture equations (19-21) allow forthe gas phase properties to be detennined. For = 0 ( = ), Eqs. (19-21) can be combined to form an equivalent Rayleigh line[Eq. (37)] and two-phase Hugoniot [Eq. (38)], as,

PI- 7t23 . . ( 1/7t18 - lipJ (7 )[ e - 7t11

7t67t14] I [p +7t ] [lIP -1/7t ]1 7t + 2 1 23. 1 1818= (1 -7t11Xt14 + 7t1O)

7tS7t18(38)

From the state relations [Eqs. (23) and (24)], we can writeel =el(PttPI), which is substituted into the Hugoniot equation (38).The Rayleigh line equation (37) allows Pl to be eliminated in favor of Pl'Substituting this in Eq. (38) results in a cubic equation for Pl' Three casesare possible: 1) three distinct real solutions, 2) two equal rea solutions anda third real solution, and 3) a real solution and a complex conjugate pair ofsolutions. When three distinct real solutions exist, two are analogous tothe weak and strong solutions predicted by the simple one-phase theory.The third solution has no such counterpart and often is a nonphysicalsolution with PI < O. The third solution has not been thoroughly studied.By imposing the condition that two real roots are degenerate (whichforces the Rayleigh line and Hugoniot to be tangent), a minimumdetonation velocity can be found. We will call this condition the C-Jcondition. Because the detonation velocity D is embedded in thedimensionless parameters, it is convenient to return to dimensionalvariables to express C-J conditions. Define the bulk density, bulkpressure, and bulk internal energy as


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354

ea

J. M. POWERS ET AL.

= P10 10 e10 + P 0 20 e20P10 10 + P 0 20

By applying the tangency condition, it can be shown that in the limit asbPa and Pa I (Paea) approach zero that the Taylor series approximation forthe C-J wave speed and pressure are given by

(40)

Similar expressions can be obtained for C-J density, temperature, andenergy.For Pa = b = 0, our formulas show that it is appropriate to treat thetwo-phase C-J condition as a one-phase C-J condition using Pa and ea aseffective one phase properties. Fickett and Davis (1979) give equationsfor one-phase C-J properties. In these equations, one can simplysubstitute the bulk density for the initial density and the bulk internalenergy for the chemical energy to obtain the two-phase C-J equations.From Eqs. (39) and (40), the C-J velocity depends linearly on Pa, thebulk density, and the C-J pressure varies with the square of bulk density "inagreement with the trends discussed by Johansson and Persson (1970).This dependence is demonstrated in Figs. 4 and 5. These plots wereobtained by solving the full set of nonlinear equations to determine the C-Jstate. The bulk density was varied by varying the initial porosity. Alsoplotted in Figs. 4 and 5 are predictions of the approximations (39) and (40)and predictions of the thermochemistry code TIGER reported by Butlerand Krier (1986).Equations (39) and (40) indicate that the C-J state is quite sensitive tothe nonideal parameter b. In particular, it suggests that, when thedimensionless group bPa is of order 1, that nonideal effects become quiteimportant. This is demonstrated in Fig. 6, which for constant bulkdensity, plots C-J wave speed vs the nonideal parameter b. The figurewas obtained by solving the full nonlinear equations. In other works(Butler and Krier 1986), the parameter b has been treated as adjustable inorder to match C-J data.

By numerically studying exact two-phase C-J conditions, it can beinferred that the C-J point is a sonic point -- that is the gas velocity is equalto .e local gas phase sonic velocity. In addition, numerical studies


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.....uCl

"2f5.......

....


10000

8000

6000

4000

2000

Estimate

TIGERExact

o 500 1000 1500 20003P a ( kg /m)

Fig. 4 Minimum detonation velocity vs bulk density for a nonideal gas.

40 TIGERExactEstimate30

10

o 500 1000 1500 2000Pa (kg/m3)

Fig. 5 C-J pressure vs bulk density for a nonideal gas.


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356 J. M. POWERS ET AL.16000 Initial Bulk Density = 333 kg/m 31400012000

..-. --10000

.... 8000ut:l 600040002000

0.000 0.001 0.002 0.003 0.004b (m

3/kg)Fig. 6 Minimum detonation wave speed vs nonideal gas parameter b with constantbulk density.

indicate that for D > DC-J the nonideal strong point is subsonic, while thenonideal weak point is supersonic. This corresponds to the results of thesimple one-phase theory.IV. Reaction Zone Structure

It is important to note that the two-phase C-J solution is notnecessarily admitted by our model equations when detonation structure isconsidered. A global phase space analysis to the full set of differential-algebraic equations will detennine which wave speeds correspond tophysically admissible solutions. In this section, we outline the problem oftwo-phase detonation structure and describe solution techniques.It is possible to reduce Eqs. (15-27) to a set of four uncoupleddifferential equations in four unknowns P2' v'l} Pz' and 2' For 1t8 = 0(an ideal solid equation of state), Eqs. (25) and can be used to writee as a function of P2 and P2' The expression for e2 can be substitutediJto Eq. (17). The resulting set of ordinary differentlal equations can beuncouyled using the techniques of linear algebra to solve explicitly for dv dP and These equations are written below.

dp...:...2. =

(41)


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dV2 = -AP2 + BP2V2 - CP2( 1t 17- 1)

dP2-=p 2

r( -1t17P/P2)AP2v2 - B1t17P2 +CP2Vi t17 - 1)r( - 1t17P /P2)

(42)

(43)

p = 6. - e (46)1 nJn2-A

- v 2 2I -1 tU

1 PIT= -1 1t - 1 P7 1

1 PIe = -1 1t - 1 P7 I1 P2T = -2 1t - lp17 2

(47)

(48)

(49)

(50)

(51)


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(52)

= 1-1 2 (53)

(54)

(56)

(57)

1x + 1 x 18 + x-P22v2

A = X> {X22 1 u to + : ; ) (58)

(59)


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(60)

In these equations, A, B, C, D, .0, A, fl, and e are introduced asintennediate variables. From these equations, it is straightforward to seehow all variables may be written as functions of the variables P:r v2, P2,and


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360 J. M. POWERS ET AL.differential equations, the nature of the end state will be revealed as either asource, sink, or saddle. End states that are saddle points exist in someone-phase detonation models and are associated with eigenvaluedetonations. The eigenvalue detonation wave speed is the only steadywave speed admitted by the model equations. Such eigenvalue detonationsmay also be associated with a two-phase detonation model. I f so, thewave speed associated with the eigenvalue detonation could be used in anumerical integration of the full nonlinear set of differential equations todescribe the detonation structure. A numerical integration at any otherwave speed would necessarily diverge from the end state, should that endstate be a saddle. It is for thi:; reason that great care must be exercised inundertaking a numerical integration of Eqs. (41 -60).

AcknowledgmentsThis work was supported by the U. S. Air Force Office of Scientific

Research under Grant AFOSR-85-0311 and the U. S. Department ofEnergy, Los Alamos National Laboratories under Contract DOE LANL9XR6-5128Cl.References

Baer, M. R., and Nunziato, J. W. (1983) A theory for deflagration-to-detonationtransition (DOn in granular explosives. SAND82-0293, Sandia NationalLaboratories, Albuquerque.Baer, M. R., and Nunziato, J. W. (1986) A two-phase mixture theory for thedeflagration-to-detonation transition (DOn in reactive granular materials. Int.J. Muitiphase Flow 12, 861-889.Butler, P. B., and Krier, H. (1986) Analysis of deflagration to detonation transitionin high-energy solid propellants. Combust. Flame 63,31-48.Butler, P. B., Lembeck, M. F., and Krier, H. (1982) Modeling of shockdevelopment and transition to detonation initiated by burning in porouspropellant beds. Combust. Flame 46, 75-93.Coperthwaite, M., and Zwisler, W. H. (1974) "TIGER" Computer codedocumentation. Report PYV-1281, Stanford Research Institute.Denton, W. H. (1951) The heat transfer and flow resistance for fluid flow throughrandomly packed spheres. Proc. General Discussion on Heat Transfer, !MEand AS ME, London.Drew, D. A. (1986) One-dimensional burning wave in a bed ofmonopropellantparticles. Combust. Sci. Technoi. 47, 139-164.Fickett, W., and Davis, W. C. (1979) Detonation, University of California Press,Berkeley.Johansson, C. H., and Persson, P. A. (1970) Detonics of High Explosives.Academic, London.


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Krier, H., and Gokhale, S. S. (1978) Model ing of convective mode combustionthrough granulated propellant to predict detonation transition. AIM J. 16,177-183.Krier, H., and Mozaffarian, A. (1978) Two-phase reactive particle flow throughnormal shock waves. Int. J. 0/Multiphase Flow 4, 65-79.Kuo, K.K., and Nydegger, C.C. (1978) Flow resistance measurements andcorrelation in a packed bed ofWC 870 spherical propellants. J. Ballistics 2,1-25.Kuo, K. K., and Summerfield, M. (1974a) Theory of steady-state burning of gas-permeable particles. AIM J. 12, 49-56.Kuo, K. K., and Summerfield, M. (1974b) High speed combustion of mobilegranular solid propellants: wave structure and the equivalent Rankine-Hugoniot relation. Fifteenth (International) Symposium on Combustion, TheCombustion Institute, Pittsburgh, PA.Macek, A. (1959) Transition from deflagration to detonation in cast explosives. J.Chem. Phys. 31, 162-167.Marsh, S. P., ed. (1980) LASL Shock Hugonoit Data, University of CaliforniaPress, Berkeley.Powers, J. M. (1988) Ph.D. Thesis, Theory 0/detonation structure/or two-phasematerials. Department of Mechanical and Industrial Engineering, University oflllinois at Urbana-Champaign.Powers, J. M., Stewart, D. S., and Krier, H. (1988) Analysis of steady compactionwaves in porous materials. To appear in J. Appl. Mech.Taylor, P. A. (1984) Growth.and decay of one-dimensional shock waves in

multiphase mixtures. Acta Mech. 52, 239-267.Wallis, G. B. (1969) One-Dimensional Two-Phase Flow. McGraw-Hill, NewYork.Whitham,G. B. (1974) Unear and Nonlinear Waves. Wiley, New York.

j. m. powers, d. s. stewart and h. krier- two-phase steady detonation analysis

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