AIAA 99-2498 Modeling Pressure and Heat Flux Responses of Nitramine Monopropellants with Detailed Chemistry W. W. Erikson and M. W. Beckstead Brigham Young University Provo, UT

35th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit

20-23 June 1999 Los Angeles, California

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AIAA-99-2498

MODELING PRESSURE AND HEAT FLUX RESPONSES OF NITRAMINE MONOPROPELLANTS WITI3 DETATLED CHEMISTRY*

William W. Erikson+ and Merrill W. Beckstead$ Brigham Young University, Provo, Utah

ABSTRACT

A numerical model of unsteady solid monopropellant combustion has been developed and applied to RDX (cyclotrimethylenetrinitramine) and HMX (cyclotetramethylenetetranitramme). This model allows three distinct modes of simulations to be performed within the framework of a single code: (1) steady-state, (2) quasi-steady gas phase, and (3) fully unsteady gas phase propellant combustion. Detailed chemistry is included. This paper presents a description of the model and results from all three modes of simulations under a variety of conditions. Steady-state results agree reasonably well with experimental data for many combustion parameters with the notable exception of the HMX temperatore sensitivity. The model was also used to simulate the burning rate responses to oscillating pressure and laser heat flux inputs. Oscillatory pressure simulations yielded results which match experimental T-burner trends for both ingredients. Experimental laser-recoil trends were also matched by simulations-particularly for RDX. The validity of quasi-steady assumption was evaluated by performing simulations in both quasi-steady and fully unsteady modes at identical conditions. Evidence is given for significant discrepancies at relatively low frequencies-particularly for high-pressure conditions.

INTRODUCTION AND REVIEW

Combustion instability, particularly with regard to pressure-combustion interactions, has been a topic of prime interest to the solid propellant community for many years. Also in recent years, heat flm coupled responses have become more of interest. These responses are generally given in terms of the dimensionless response functions, % and &, as given by equations (1) and (2). The goal of this work is to obtain pressure and heat flux coupled response functions by means of a combustion model which includes detailed chemistry. In the next few sections we will briefly review previous work relating to models involving either detailed chemistry or unsteady combustion.

( r) R, =f$$ (1)

(2)

Combustion Models with Detailed Chemistrv Recently, several researchers have examined the

detailed chemistry involved in solid propellant combustion, in hopes of gaining forther understanding to this complex process. Jncreasingly difficult steps have been taken toward an ultimate goal of modeling composite propellants under unsteady conditions. First, several research groups have examined steady-state combustion of single monopropellant ingredients such as RDX,1,2*3 HMX,4 or GAP.’ Subsequently, models evolved to include homogeneous mixtures of more than one ingredient, such as ultrafme AP-HTPB mixtures6 or RDX-GAP pseudo-propellants.’ Others have modeled counterflow diffusion flames of AP burning with a fuel gas.* Still others have included unsteady combustion effects, such as the unsteady ignition process of RDX.’ Unsteady RDX combustion under oscillatory pressure conditions has also been modeled,l’ as will be explained in the next section.

Pressure-Couoled Combustion Resuonse Models Over the years, numerous combustion models have

been developed and applied to the problem of pressure coupled combustion with varied degrees of success. To compare combustion models, they are often separated into different classes. Historically, two main paradigms have developed: the Zeldovich-Novozhilov”.’ 2,13,1 4,1 5,1 6 (ZN) approach in Russia; and the Flame Modeling (FM) method in the West (see papers by Hart and McClure,” Denison and Bamn,‘* Krier, et al.” and Culick,2°,21 for instance). Though there are others, one fundamental difference between the two approaches lies in their treatments of the gas phase flame. The flame modeling method generally assumes some sort of kinetic scheme for the gas flame. On the other hand the ZN method treats the flame phenomenologically and, in general, bypasses the need for a rigorous gas

* This work was sponsored partly by Brigham Young University and partly by the Caltech Multidisciplinary University Research Initiative under ONR Grant No. NOOO14-95-1-1338, Program Manager Dr. Judah Goldwasser. + Research Associate t Professor, Department of Chemical Engineering; Associate Fellow AIAA Copyright 0 1999 by the American Institute of Aeronautics and Astronautics

model in favor of a more direct use of steady-state experimental data. In spite of their differences, both the ZN and FM approaches have been shown to be fundamentally equivalent,22’23a24~25 and yield similar results. The traditional ZN (left) and FM (right) form of the pressure-coupled response function is given in equation (3).

R = ‘+‘bl) nAB+n,(h-1) P

Jr+:-(r+k)+l = A++-(l+A)+AB (3)

Combustion models may also be categorized according to the assumptions made in their derivation. One example is the quasi-steady assumption, which, in essence, means that because gas phase processes (e.g. reactions, diffusion, etc.) occur much faster than condensed phase processes, the gas phase may reasonably be treated as steady-state. The quasi-steady assumption begins to break down at high frequencies, high pressures, and with extended flame zones.26J27 Most early work using either the ZN or FM approach imposed the quasi-steady constraint, with the notable exception of Hart and McClure’s.17 One advantage to applying the quasi-steady assumption is the closed form analytic solution (e.g. the ‘A-B’ response equations in western literatore18~21 or the corresponding ZN form13) which is obtained. The simple forms of the equations allow response functions to be easily computed, based either on direct experimental data or on modeling results. Examples of this technique have appeared in recent literature,28~2g even though the basic quasi-steady theories were developed in the 1960’s.

As time went on, researchers began to relax the quasi-steady assumption. In 1972, T’ien26 developed a model which included a folly unsteady gas phase with a single step gas phase reaction. Margolis and Williams3o developed a model in which they included not only an unsteady gas phase, but multidimensional effects as well. This model used a simplified global gas phase reaction scheme and added the additional constraint of constant gas phase density-which is valid only for limited conditions. Huang and Micci31 extended the work of T’ien26 by including three gas phase reaction steps. Clavin and Lazimi27 used an approach in which two separate models (one for high frequency and one for low) are combined to get the overall response- once again with simplified kinetics. Novozhilo$* extended the ZN approach to include the effects of gas phase time lags, but with limited usefulness as it is dif6cnlt to determine the correct time lag a priori.

Erikson and Beckstead” developed a numerical model for unsteady propellant combustion. Unlike

those mentioned above, this model included detailed gas phase chemistry. It was used to examine the combustion response of RDX to oscillating pressure. However, due to the solution methods employed, the model tended to be unwieldy; required excessive computational time; and had signiiicant inaccuracy. In the current work, all of these problems were addressed and significant improvement was attained in ease of use, efficiency, and accuracy.

Heat Flux-Couuled Combustion Response Models Because of the difficulty and expense in obtaining

experimental pressure coupled response data, in recent years, it has been more and more common for investigators to measure the combustion response to oscillatory heat flm. Probably the first to do so were Mihlfeith, et al. in 1970.33 Since that time, obtaining combustion response by laser recoil has proliferated considerably. Some experiments have been performed on real propellauts34~35~36~3737.38 and others have involved single ingredients such as nitramines.38~3g~40~4’~42~43

Models have been developed in order to interpret experimental data, and to relate the heat flux response, %, to the more useful pressure response, %. Ibiricu and Williams44 investigated the effect of radiant heat fhtx on propellant combustion, though not necessarily unsteady combustion such as with an oscillatory radiant source. Son, Brewster, and coworkers25,34,35,41’45 and DeLuca46 have done considerable work on developing theoretical formulations for the propellant combustion responses under the influence of radiation. Son and Brewster have developed analytical forms for Rg and a transfer function, relating Rp to &, by following an approach similar to the original ZN method derivation.

PHYSICAL DESCRIPTION OF MODEL

To maintain tractability and focus on including detailed chemical kinetics, tbis work has been restricted to a one-dimensional analysis of a single homogeneous ingredient It is assumed to burn with a laminar flame. The physical domain consists of three regions: the solid, liquid, and gas pbases. In this paper, we will group the solid and liquid regions together as a single condensed phase. The interface between the liquid layer and the gas phase is the surface. Figure 1 shows a simple diagram of monopropellant combustion as considered in the current work.

In the solid phase, the propellant is largely inert and is primarily influenced by heat conduction-though under laser assisted combustion, in-depth radiant energy deposition may also contribute. As the

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Solid Phase Liquid Gas

-CO Melt

Fig. 1: Physical Propellant Combustion Model

temperature rises due to heat transfer, phase changes may occur. Included in these are both crystal phase changes (for HMX) and melting (for both RDX and HMX). The propellant may undergo slight decomposition in the solid phase, but once it is liquefied through melting, decomposition accelerates.

In the liquid layer, propellant decomposition may occur-conceptually forming gaseous decomposition products, which would gather in bubbles. There is experimental evidence4’ of bubbles in a liquid layer during propellant combustion, and others have theoretically considered their behavior.48 However, as a practical matter, there are many parameters associated with bubbles such as shape, surface tension, number density, relative velocity, etc. which are still unknown. To rigorously account for these effects would require

additional assumptions for some of these parameters and possibly introduce multi-dimensional effects. Others’~z44g have attempted to account for bubbles in terms of a void fraction, however they have not accounted for all the complex processes occurring and were forced to make additional assumptions about bubble behavior. In the current model, we have not included bubbles per se, but since distributed condensed phase energy release has been shown to significantly impact unsteady combustion,2o~5o we have retained distributed decomposition effects. Similar to the work of Prasad, et al.,3 decomposition products are assumed to be ‘dissolved’ in the liquid, and to have no effect on the local transport properties, though thermodynamic effects (i.e. enthalpy and specific heat) are retained and calculated on a mass basis, and condensed phase density is considered a constant. We also consider the effect of propellant evaporation at the surface (liquid-gas interface), since both decomposition and evaporation have been observed experimentally.51~52 Material leaving the liquid region enters the gas phase where final combustion occurs.

Physical properties for RDX and HMX used in the model were obtained from experimental data and are shown in Table I. The chemical mechanisms used in the model are given in Table II. Condensed phase chemistry was based on Brill’s work.‘l For RDX, gas phase chemistry was directly from Yetter, et aZ.53354 This was slightly modified by Davidson4 for HMX.

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MATHFlMiTICAL FORMULATION

Governing Eauations The physical situation outlined above can be

described mathematically by an appropriate set of governing equations. In this case we have a one- dimensional time dependent system which is described by the following set of differential equations. The conservative forms of the gas phase mass contimrity, momentum, species contimrity, and energy equations are given by (4) to (7).

&“,+~mu)+~-$~(p~)=o (5)

&~~,)+$~uY~)+~(PU,V,)-C~~W~ = 0 (6)

~(pH)+~&~-u~-~(“~)

+F;(pY.V,hi)-$.t ’ =0 (7)

The equations for condensed phase mass, species, and energy are given by equations (8) to (10). Note that since density has been assumed constant in the condensed phase, equation (8) reduces to the trivial relation that the velocity, u+ is spatially constant, though it can vary in time. This is consistent with the physics of the problem-the condensed phase is not actually moving with a velocity K, but rather it is stationary and the surface is regressing at a rate, rb. However, if the coordinate system is defined such that x = 0 always corresponds to the surface, the flow would appear to be going through the surface to an observer riding on the surface, and mathematically they are equivalent (I& is equal in magnitude to rb, but opposite in direction). Finally note that because u, is constant in space, the momentum equation is not necessary and is redundant to the continuity equation.

~+-$J3.11.)= 0 l3Y. m.

pc at -l+pouoI--~,Wi =o ax p$, ~+p,u,c, s-i h; ( 1

(9)

-qraa,&bs exP(K,,x)+Cbiwihi =O (10) i

Equations (4) to (10) require suitable boundary and matching conditions. These are given below in (11) through (18). The condensed phase equations require given temperature, (11) and species mass fractions, (12) at the inlet (unburned propellant). At the propellant surface (interface between condensed and gas phases), the continuity of mass fhrx, species fluxes, and energy flux must be preserved ((13), (14), and (15) respectively). Zn addition, we require that the temperature of both gas and condensed phases be equivalent at the surface (16). The downstream (burned) boundary conditions for the gas phase include a zero gradient condition for species and temperature, as well as a specified pressure (a constant for steady- state, a fonction of time for unsteady), as given by (17) to (18).

- Lys- = -h,Ei,, + wcYop,s-hoppap T,- = T,,

p.m = pambimt ct)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

With the boundary conditions, the problem is fully specified with one exception-the propellant burning rate. The conservation equations above are true for any arbitrary burning rate. An additional constraint is required to solve for the burning rate eigenvalue. This extra condition may be m many forms. Many steady- state models (including the often-used Pl?li%fX flame code65) impose a specified temperature at a location to solve for the burning rate. However, a fixed temperatore at a given location such as a constant surface temperature has been shown to be au unreasonable condition for unsteady burning.13 Unsteady models typically use an additional relationship between burning rate and surface temperature such as an Arrhenius expression’***’ or power-law.‘g As in a previous work,” we have chosen a relationship involving the chemical kinetics of the condensed phase. That is, the mass burning rate at any

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given time is equal to the overall rate of propellant decomposition and evaporation at that time, as given by equations (20) to (22). Spatially integrating the local decomposition rates yields the aggregate decomposition rate,*O and the net evaporation rate is calculated from the difference of evaporative and condensing fluxes at the surface.‘~”

s Iii deoomposition = I ~,W,~ (21)

Numerical Annroach The requirements of this work mandated that the

solution method be flexible, robust, efficient, and accurate for three types of conditions: (1) steady-state; (2) quasi-steady gas phase, unsteady condensed; and (3) fully unsteady gas, unsteady condensed. Since steady- state solutions are used as initial conditions for subsequent quasi-steady and fully unsteady calculations, it is imperative that the exact same numerical grids and algebraic discretizations (with the exception of the temporal terms) for all three types of calculations be used. Failure to do so would result in a ‘jump’ rather thau a smooth transition from steady to unsteady combustion.

In a previous work,” to allow for a smooth transition, initial conditions were obtained by calculating the fully unsteady forms an ‘asymptotically’ long time with constant boundary conditions-yielding a ‘steady-state.’ Thus, even steady-state solutions were computationally very expensive. Furthermore, the prior work utilized the ‘time step splitting’ approach66,67 to separate the stifl gas phase chemistry from the fluid mechanics. In this approach chemistry and fluid mechanics are assumed to operate independently; thus the use of very small time steps is inherent to avoid ‘splitting errors.’ In the previous work,*’ steps of 10m6 seconds were used, but comparison with results of another code’ at the same conditions indicated that gas phase chemistry was occurring too quickly (the temperature profile overshot the other results by about 200 K in the near surface region) indicative of a splitting error. Moreover, using the larger time steps as required for a low frequency oscillation would further exacerbate the problem. Hence, in the current work, the time step splitting approach was discarded.

The equations as presented in the previous section represent a series of highly coupled, stiff partial

differential equations. The conservation equations were discretized over a piecewise uniform finite difference grid and solved using a variety of traditional numerical methods.

The condensed phase energy equation was discretized with a spatially centered scheme, while the condensed species equations used the upwind scheme. Upwinding was used to maintain monotonicity-all higher order methods attempted seemed to yield non- physical results such as overshoots, undershoots or oscillatory behavior. For unsteady calculations, the energy equation was treated folly implicitly while the species equations were solved explicitly.

The gas phase was solved using the TWOPiVp5 solver for steady-state and quasi-steady. Although this is the same solver used by the PREM&5 code, the equations used in the current work differ somewhat from PREMIX We have used the conservative rather than nonconservative forms of the gas phase equations and have included the momentum equation, which was not used in PREME For the folly unsteady gas phase cases, the temporal terms in the equations were included and treated folly implicitly. Incidentally, the DASSL68 code was also implemented for the fully unsteady cases with some degree of success-though for most cases TWOPNT was used. DASSL, while very powerful, was somewhat diEcult to apply to this particular problem where gas and condensed phases are coupled together and the burning rate is a function of both phases.

It should also be mentioned that calculating the burning rate in the manner described in the previous section was inherently unstable when treated explicitly. That is, if the burning rate were merely lagged one time step (i.e. determined by results of the prior step), non- physical oscillations in burning rate would quickly arise. This problem was rectified by using an iterative implicit procedure. Outlines of the solution algorithms used are given below.

Steadv-state Algorithm Guess surface temperature and burning rate. (T,,, rb,J Calculate a condensed phase solution based on T,, rb,, to yield species concentrations and a surface heat flux. cYis, qs,J Calculate a gas phase solution based on Ts,a, rb,p, Yis. This yields another surface heat flux, q,,g Calculate burning rate based on gas and condensed phase solution& yielding rb*. Repeat from step 1 until rb* = rb,g and q,,c and q,, satisfy equation (15). A damped Newton-Raphson method was used to determine new guessed values of rb and T,.

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Unsteady (or Quasi-Steadv) Algorithm 1. Update boundary conditions to current time (i.e. P,,

%-ad) 2. Guess burning rate and surface heat flu. (rb,g, qs> 3. Calculate a condensed phase solution based on q,,

rb,g aud the old solution. This yields species concentrations and a surface temperature. (Yis, T,)

4. Calculate a gas phase solution based on T,, rb,p, Yis and the old solution. This yields another surface heat flux, qs,g

5. Calculate burning rate based on gas and condensed phase solutions, yielding rb*.

6. Repeat from Step 2 until rb* = rb,,. A secant or bisection method is used to guess burning rate. (The heat flw, qs,c is determined from eqn. (15) and lagged one iteration. and appeared to converge without trouble).

RESULTS

Steadv-State A suitable initial condition is required for the

subsequent quasi-steady and fully unsteady combustion calculations. The code was run in steady-state mode over a range of pressures and incident heat fluxes both for obtaining the starting conditions and to compare with available steady-state experimental data. The primary goal of the current work is to establish a framework for calculating unsteady combustion results, and to demonstrate results with reasonable inputs. With this in mid, no particular effort was made to ‘fine tune’ input parameters such as kinetic constants or material properties to more closely match steady-state experimental results-this sort of work would be better suited to dedicated steady-state codes (e.g. references 1, 2, or 3). Instead, the approach was to simply use a given set of input data, from experimental or other modeling work and use it-and let the pieces fall as they may.

The propellant burning rate for RDX and I&IX calculated by the steady-state model is shown as a function of pressure in Figures 2 and 3, together with available experimental data,56*6g~70~71,72,73,74 and appears to give reasonable agreement. Note the similarity in burning rate curves between the two ingredients.

Figures 4 and 5 show the two propellants burning rates for simulated laser-augmented burning as a function of radiant heat flux. Experimental data38,3g,40*41 for laser assisted combustion are shown as well. The model for laser assisted combustion of RDX appears to reasonably match the burning rate data however the HMX modeling results appear to be somewhat low. This can in part be attributed to the density of HMX solid used in modeling. We have used the crystalline density of HMX, 1.9 g/cm3, compared to the 1.7 g/cm3 reported by Tang, et al. 40 and 1.8 g/cm3 by Loner and Brewster41, which are representative of pressed powder

0.01

l Zimmer-Galler

1 10 10 Vressure [aim]

Fig. 2: RDX Burning Rate vs. Pressure at 298 K

10 l

l Fogelmng et al. m Boas

, - Hh4X model

1 10 IPressure [Am]

10

Fig. 3: HMX Burning Rate vs. Pressure at 298 K

B $ 0.1 a” 0

‘E 005 z . ~

A l

A.0 0 l

0

0 50 100 150 2ct t-j-ad [w/cm21

Fig. 4: RDX Burning Rate vs. Laser Flux.

samples. If Figure 5 showed mass instead of linear burning rate, the values would more closely coincide, though the modeling results are still a little low. It should also be noted that Loner and Brewster estimated a 50% reflectivity for HMX. Their data as shown in

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0.2 A

A

e -

l Tangetal. A Lpqer & Brewster 0 fim$~od5tlal.

0 ““I”“:““;“”

0 50 100 150 20 C&ad [w/cm21

Fig. 5: HMX Burning Pate vs. Laser Flux.

Figure 5 are therefore corrected values (the actual power output from the laser was hence twice what is shown in the figure).

We can also examine mauy other combustion

1 10 Pressure [atm]

10

Fig 6: RDX Surface Temperature

850

750 700 650 603

l l

- HMX model

Fig. 7: HMX Surface Temperature

parameters with the steady-state model, some of which have corresponding experimental data for validation. Surface temperature for RDX and HMX are shown in Figures 6 and 7 as a function of pressure. Modeled surface temperature for both ingredients are compared with data from Zeniq71 and appear to lie within the experimental scatter. Figures 8 and 9 show condensed phase melt layer thickness as a function of pressure, for both the models and experimental data of Zeniq71 and Brewster and Schroeder.3g The trend of decreasing melt layer thickness is verified by the experimental data-though modeling results appear to show thinner melt layers thau experiment, particularly at high pressures. Figures 10 and 11 show gas phase temperature profiles for RDX and HMX at various pressures and are compared with data from Zenin71 and the Parrs7’. The 1 and 5 atm model data appear to match experimental data reasonably well, especially in the near surface region. The higher pressure data do not match as well-model results appear to be significantly steeper in the near surface region. This

10 Pressure [atm]

1

Fig. 8: RDX melt layer thiclmess.

10 Pressure Iatml

Fig. 9: HMX melt layer thickness.

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3500 - 3cQO -: 2500 +

Y

I 6 Zenin 1 atml Parr 1 atm Zenin 5 atm

n /I

-0.05 0 0.05 0.1 0.15 0.; xb-4

!

0 Zenin 90 atm -model 20 atm -model 90 atm

-0.01 0 0.01 0.02 0.03 x km1

Fig. 10: RDX Temperature Profiles at 1,5,20, and 90 atm. Model and Experimental Data

3500 3ooo 2500

pm bl1500

1ocQ 500

0 -0.05 0 0.05 0.1 0.15 0.2

x km1 -

3500 I. 1

-0.01 0 0.01 0.02 0.02 xlcml

Fig. 11: BMX Temperature Profiles at 1,5,20, and 70 atm. Model and Experimental Data.

may be indicative of inadequacies in the chemical reaction mechanism in the model, though one would expect that lower pressure experimental results to be more accurate than high pressure data. Figures 12 and 13 show major aud minor chemical species concentrations as given by the steady-state RDX model at 0.5 atm pressure. Experimental data of Ermolin et al.76 are given as a comparison. The large data points at the extreme right hand side are equilibrium calculations from the Edwards thermochemical equilibrium code.

Both the temperature profiles in Figures 10 and 11 and the species profiles in Figures 12 and 13 approach equilibrium conditions at the downstream (burned) location. This is consistent with the adiabaticity assumed in the model formulation. In the experiments from which the experimental data were taken for the same figures, true adiabatic conditions are impossible to maintain which may explain some of the discrepancies.

Unsteadv Combustion Parameters In addition to directly obtained data, other useful

parameters can be derived from the steady-state results. The temperatore sensitivity, or,, is one such variable. Figure 14 shows op for HMX aud RDX along with experimental data derived from Boggs et ~1.~~ Siguificant differences are observed experimentally between the two ingredients. The temperature sensitivity of RDX is nearly constant with pressure at a level of about 0.001 K-l. In contrast, large values of err are observed for HMX at low pressures. These diminish with increases in pressure to roughly the same level as RDX. The current model appears to reasonably predict the op trends for RDX, but fails to predict the low-pressure trends for HMX.

Obviously, the model is lacking in some area- possibly related to uncertainties in reaction chemistry, particularly in the condensed phase and near surface gas phase. It may be that the same problem that yields incorrect temperature profiles also contributes to temperature sensitivity problems. Ward et a1.77 have developed a simple model for HMX combustion which uses an experimental temperature profile (such as

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1 0 0.1 0.2 0.3 x [cm1 0. 4 Fig. 12: Major Species (RDX at 0.5 atm) Fig. 14: RDX/HMX oP: Model and Experiment

0 0.1 0.2 0.3 xbl

Fig. 13: Minor Species (RDX at 0.5 atrn)

Zenin’s7’) as an input to fit certain model parameters. Their model is then used to predict combustion characteristics at other conditions, and appears to match experimental HMX temperature sensitivity rather well. Based on their work it seems likely that a chemical reaction mechanism (condensed and gas) which yields correct temperature profiles will likely achieve better fits to other parameters such as or, as well.

0 0.006 -in BoggsHMx 4 Boggs RDX

0.005 -! -Model HMX ek 0.004 -I 0 -Model FtDX

p b 0.003 -: 0 0.002 0

cl l-l 0.001 404

0 1~~~‘1~“‘1”‘~1~‘~‘1”‘1 0 20 40 60 80 10(

Pressure [aim]

0.007 L

Ill

1

Various theories have been derived to utilize steady state combustion parameters to predict unsteady burning characteristics. Both the traditional ZN (Zel’dovich-Novozhilov)i3 and FM (Flame Modeling)21 theories can be applied (directly with ZN, with some conversions for FM) with steady state data to obtain unsteady responses and stability criteria. The ZN theory parameters are k, u, r, v, and 6, while the FM parameters are A, B, n, and n,. It can be shown that the FM theory is equivalent if A = k/r, B = l/k, n = v, and n, = 6/r. In a similar manner, laser assisted combustion parameters can be derived in a manner similar to the ZN theory.25,45 The required parameters are: k, u, r, v, and 6, evaluated at a constaut radiant flux level; and three new parameters: Pi, vq and 6,

Tables III aud IV show ZN parameters for self- deflagration and laser-augmented combustion, as cal- culated by the current-model. These parameters were obtained by computing baseline cases, then independently perturbing initial temperature, pressure, and radiant flux level. These are used to construct theoretical response function curves (i.e. eqn. (3)) as shown in following sections. Note that the Jacobian parameters, 6 and 6,, are nonzero, but very small.

Table IIk ZN-FM Parameters at Various Pressures.

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Table IV: Laser Assisted ZN Parameters at 1 atm. %-ad

[w/cm*] k r V CL 6 Vq h 6, 48 0.2273 0.0179 0.6597 0.0541 -0.0005 0.2920 0.0342 -0.0173

Pressure Counled Combustion Modeling The combustion model was run with an oscillating

pressure boundary condition for many different frequencies at several pressure levels. Both quasi- steady and fully unsteady modes were run Typically 3 to 5 sinusoidal cycles were run to allow the model to reach a steady oscillation pattern. Pressure oscillation magnitudes were kept small to maintain linearity (0.01% for 5 atm or less and 0.001% above 5 atm). During oscillations, various combustion parameters including burning rate, local temperature, species concentrations, gas velocities and pressure were sampled and stored. The pressure coupled responses were calculated by relating time traces of pressure and burning rate, yielding the magnitude aud phase components.

Figures 15 aud 16 show the quasi-steady % magnitudes for RDX and HMX respectively at 200, 500 and 1000 psi, as calculated by the model. In addition, experimental data from Finlinson7* are shown. There appears to be a large amount of scatter in the data, especially for RDX. However both sets of data appear to show an increase in response magnitude with pressure-which is consistent at least in trend with the

modeling results. Figures 17 and 18 show the quasi-steady Rr

magnitude and phase for RDX at six different pressures as calculated from the model (Figure 17) and by the ZN theory (Figure 18). The ZN theory curves were obtained from equation (3) with the ZN parameters given in Table III (i.e. the parameters were obtained from steady-state modeling results n& experimental data). Note that the location of the % peaks is consistent between the model and ZN theory curves, though the ZN curves have somewhat lower peak amplitudes. Also note that the phase curves seem to match very well. In spite of the relatively good match between the model and ZN curves, it should not be construed that the current model represents reality. Rather, it is a test of internal consistency between the steady-state and quasi-steady models, since the ZN parameters were derived from steady-state modeling results. On the other hand, we should not dismiss the results as unsatisfactory because they do not exactly match the ZN curves. This is to be expected since we have included aspects of combustion which are beyond the assumptions of the ZN derivation such as non- constant properties, distributed condensed phase

4 200 psi exp - 200 psi model

3 w 500 8 si exp - _ A 100 psiexp - :&QYi%?ed

Figure 15: RDX Rp Magnitude

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___ latm -5 atm -10 atm . .._........... 200 psi _ - - . . _ _ 500 psi - - - - 1000 ps

0.01 0.1 1 10 100 1000 10000 IRio5

fiequew W I

~ latm -5 atm -10 atm . . . . . . . . . . . . . . . 200 psi _______ 500 psi - - - - 1OOOp :

1

-100 -; -120 <

0.01 0.1 1 10 100 1000 10000 lBio5

f iequew P W

Figure 17: RDX Rr, Magnitude and Phase-Quasi-Steady Modeling Results. I

~ 1 atm -5 atm - 10 atm . . . . . . . . . . 200 psi _ _ _ _ _ _. 500 psi - - - - 1000 ps

t I 2.5 -I

0 “I “1”“: -“: “““: “““: ’ ““‘.E ( “‘Y ” ,,-1 0.01 0.1 1 10 100 1000 10000 lEio5

frequency W I -

~ 1 atm -5 atm -10 atm . . 200 psi _ _ _ _ _ _ _ 500 psi - - - - 1000 psi

Tl 0 & -20

s-40 ho # -80

-100 -120

0.01 0.1 1 10 100 1000 10000 lEia5

fiequew W I Figure 18: RDX & Magnitude and Phase-ZN theory (parameters from SS model)

decomposition, etc. Similar ZN theory comparisons were made for HMX and yielded very similar results, hence they are not shown.

Figures 19 and 20 compare fully unsteady Rp calculations with corresponding quasi-steady results for RDX and HMX at three different mean pressure levels. In these figures, the solid lines represent quasi-steady results while the dashed lines are for folly unsteady at the same conditions. A few important conclusions can be drawn. First, the fully unsteady results coincide with quasi-steady at low frequencies for each pressure, while as frequency increases, small deviations occur and become larger. Second, the existence of a high frequency peak is evident similar to that predicted in works by T’ien26 aud Clavin and Lazimi.* Third, the frequency at which the quasi-steady assumption becomes invalid (where deviations begin) appears to be a function of pressure. For both RDX and HMX at 1 atm, the quasi-steady assumption appears to be valid below about 200 Hz; below about 500 Hz at 200 psi; and below 1000 Hz at 1000 psi. At 1 atm, deviations

- 1 atm QS ~ 200 psi QS

- - - - - - - 1 atm unsteady

- 1000 psi QS _._________.__.. 200 si unsteady - - - - 1008psi unstea@ Y

: Circles indicate frequencies vhere

begin way down the ‘tail’ of the curve, at 200 psi it is near the peak, and at 1000 psi it occurs at a frequency below the peak. Interestingly, the location Clavin and Lazimi” indicated for deviations from quasi-steady for

0.1 1 10 100 loo0 10000 lE-to5 1 frequency PI

Fig. 19: RDX % QS and Fully Unsteady Models 1 11

American Institute of Aeronautics and Astronautics

------- 1 atmunsteady _____........... 200 si m&e&y - - - - lOOxpsiunstead!

Circles indicate frequencies tiere :

-: to break dew.

OL 0.1 1 10 100 1030 loo00 lE+OS

frequency FW Fig. 20: HMX % QS and Fully Unsteady Models 1

50 atm (see figure 3 of ref. 27) were at virtually the same location on the quasi-steady curve as we observe for the 1000 psi case here.

Heat Flux Coupled Combustion Modeling Cases were run for oscillatory laser assisted

combustion of HMX and RDX at several mean flux level. The amplitude was 1% of the mean for RDX and 0.1% of the mean for HMX, though this did not appear to have any significant influence-both are still in the linear regime.

Results for RDX modeling at 1 atm and incident laser flux levels of 48, 60, and 90 W/cm* are shown in Figure 20, along with experimental data for 48 W/cm* from Brewster and Schroeder.3g Both quasi- steady and fully unsteady results are shown. Note the existence of the normal low frequency peak at about 10 Hz and au additional high frequency peak (several hundred Hz) in the experimental data. The modeling

OS Model Unsteadv Model Laser Recoil -48 QS - - - --48 mu. x 48 single sine

_____.m (3)~s. l 48 log sueep ..~.~~.~..~~~-.~ 90 uns. (Brew&et t Schroede

1.2 Legend numbers are radiant flux level [W/cm21

0.1 1 10 100 loo0 1OKx frequency FW

- Fig. 20: RDX % : Q uasi-steady Model, Fully Unsteady Model, and Experimental Laser Recoil

results reasonably match the low frequency peak. Quasi-steady modeling results monotonically decrease from that point Fully unsteady results, however, show a slight second peak at 48 W/cm*. At higher flux levels the high tiequency peak increases in amplitude- approaching the size of the experimental peak, though at a fhrx level a factor of two higher.

Similar HMX results are shown in Figure 21 for three incident flux levels. Quasi-steady and unsteady modeling results are compared with experimental data from Loner and Brewster:* and Tang, ef ~1.~’ (note: Tang, ef al. had data for phase but not magnitude at 35 W/cm* and magnitude but not phase at 90 W/cm*) The modeling result show poor agreement with laser recoil data for all flm levels. The peak responses predicted by the model are much lower amplitude thau experiment. Once again, two peaks were observed in the unsteady modeling results. No corresponding second peak existed in the experimental data but the highest frequency data were only about 200 Hz, which is about the point where the model shows a secondary peak beginning to form.

CONCLUSIONS

In conclusion, we have developed a numerical monopropellant combustion model. The model includes detailed chemical kinetics and is applicable to (1) steady-state, (2) quasi-steady gas-phase, and (3) a folly unsteady condensed phase. The code was applied to the nitramines, RDX and HMX and tested under a variety of simulated conditions, including pressure and radiant heat lkx oscillations.

Reasonable agreement with experimental data was obtained for many combustion parameters including burning rate, surface temperature, melt layer thickness,

OS Model Unsteadv Model Laser Recoil -48 QS - - - --48 uns. x 48 smgle sme

----% 8: _____.m 60~~3. l 48 log sweep

401, ---------------. 90 UIIS. (Brewter & Schroeder

I 20 0

52 -20 a -40 aa -60

jj -80 k -100 3 -120

-140 -160 -180

Legend numbers are radiant flux level

12 American Institute of Aeronautics and Astronautics

perRe;;$ Unsteadv Model OS McK&~~ _ _ _ _

-88 8 8 _. . ._. . 60 ::’ 6 go & 0 $i TEg

Legend numbers are radiant flux level [W/cm2]I

B l #a 0.8

8 0.6 % 0.4

0.2 0

0.1 1 10 loo loo0 1OOOt IQ-w-w WI I - L -

OS Model Unsteadv Model Laser Recc -35QS - - - -3Sum.. x 35 Loll -60 QS _______ 6Oms. 4 60 Tim -90 QS . . . . . . . . . . . . . . . go MS.

4ob I

Legend numbers sxe radiant flux level [W/cmz] ’ 1111”‘1 b ‘1’1”‘1 ’ 111”1’1 ““‘1*1 ““~uL

0.1 1 10 100 loo0 looo(

Fig. 2 1: HMX &: Quasi-steady Model, Fully Unsteady Model, and Experimental Laser Recoil 1 and low pressure temperature and species profiles. Modeled temperature profiles at high pressures showed a steeper near-surface region than experimental data This was attributed to inadequacies iu the chemical mechanisms.

Quasi-steady pressure responses matched theoretical curves reasonably well and experimental trends were reproduced, notwithstanding the large degree of scatter in the data. The fully unsteady model showed a second, high-frequency peak in the pressure response curve. The quasi-steady assumption was evaluated by comparing quasi-steady results with folly unsteady. It was determined that the assumption is valid below 200 Hz at 1 atm and below about 1000 Hz at 1000

97 si. This corresponded with another theoretical

analysis. Heat flux responses were computed, and compared

with experimental data. For RDX, reasonable agreement was reached, including a qualitative match of data with a second high frequency peak. HMX results had poor agreement with experimental data for heat flux response, substantially under-predicting the peak amplitude, though the location of the peaks coincided.

NOMENCLATURE

A ‘A-B’ model parameter (see ref. 2 1) B ‘A-B’ model parameter (see ref. 21)

2 specific heat capacity frequency

H mixture enthalpy h species enthalpy

kibs (-1)” Beer’s law absorption coefficient

k ZN parameter: k = (IS-T,) (&r&YI’,,), m mass burning rate

n ‘A-B’ model parameter: pressure index n, ‘A-B’ model parameter (see ref. 21) P pressure qmd radiant heat flux %,c = (A dT/dx),. %,g = (-1 dT/dx), % pressure coupled response function

2 heat flux coupled response function universal gas constant

r ZN parameter: r = (ST$aT,), rb linear burning rate

; surface Temperature

t temporal variable U velocity V diffusion velocity W molecular weight

; spatial coordinate mass fraction

Greek svmbols

F thermal diffusivity ZN parameter: 6 =vr-pk

% ZN parameter: 6, =vqr-hk h = % + %(1+4in)” 3\. thermal conductivity P viscosity P ZN parameter: u = l/(T,-T,) (aT,ldlnp)T, I% ZN parameter: uq = l/(TS-T,) (Z!Js/dmq&o,p V ZN parameter: v = (%t r&Jhtp)T, Vq i??d parameter: vq = (ah rbu,/t?.hqm&,p P density =P temperature sensitivity: op = (i31nr&7T0)p sz dimensionless frequency = 2tic&,,* Q volumetric molar production rate

13 American Institute of Aeronautics and Astronautics

Suuerscriuts & Subscripts C condensed phase evap evaporation i3 gas phase i species index 0 initial oP original propellant ingredient S surface S- condensed side of surface s+ gas side of surface w vapor -CO inlet (cold propellant) +CXJ outlet (burned gases)

REFERENCES

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14 American Institute of Aeronautics and Astronautics

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15 American Institu~ of Aeronautics and Astronautics