JOURNAL OF PROPULSION AND POWER Vol. 4, No. 5, September October 8 Analytical Solution or Pressure-Coupled Combustion Response Functions o Composite Solid Propellants Michael Shusser and Fred E. C. Culick Caliornia Institute o Technology, Pasadena, Caliornia 9115 and Norman S. Cohen Cohen Proessional Services, Los Angeles, Caliornia 935 DOI: 1.514/1.15 This paper etends the classical analytical solution or small perturbation analysis o the pressure-coupled response o a homogeneous propellant to any two-component composite propellant. The solution obtained is general and can be used with any particular model or propellant combustion. As an eample, the Cohen and Strand ammonium perchlorate propellant model or a single ammonium perchlorate particle size was used in this work. The results and their mechanistic signiicance are presented and discussed. It is shown that, or a two-component composite propellant, two orms o pressure eponents arise rom the analysis. The signiicance o the second eponent is that it enables the composite propellant to be viewed as a homogeneous propellant with a requencydependent eponent via the coupling coeicients. It is ound that the ammonium perchlorate is the main source o instability because o its condensed phase eothermicity and monopropellant lame kinetics. This will be a problem with energetic materials in general. The inert binder provides a stabilizing inluence because o its endothermicity and the diusion lame ormed with the ammonium perchlorate. Eects o ammonium perchlorate particle size and pressure stem rom the changing lame structure and its eect on burning rate. Nomenclature A = dimensionless parameter characteristic o surace decomposition B = dimensionless parameter characteristic o heat eedback law C = coupling parameter D = deined by Eqs. (3) and (4) F = classical response unction, Eq. (17) = requency o oscillations n = burn rate pressure eponent n = deined by Eq. (49) p = pressure R = response unction r = burning rate S = raction o propellant surace area occupied by a component T = temperature T = bulk temperature t = time = distance (negative into the solid) z = dimensionless distance = raction o propellant mass low rate rom ammonium perchlorate = = 1 = dimensionless temperature = thermal diusivity = ammonium perchlorate comple requency Presented as Paper 346 at the 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conerence and Ehibit, Salt Lake City, UT, 8 11 July 1; received 1 December 5; revision received 18 May 8; accepted or publication 6 May 8. Copyright 8 by the American Institute o Aeronautics and Astronautics, Inc. All rights reserved. Copies o this paper may be made or personal or internal use, on condition that the copier pay the $1. per-copy ee to the Copyright Clearance Center, Inc., Rosewood Drive, Danvers, MA 193; include the code 748-4658/8 $1. in correspondence with the CCC. Postdoctoral Scholar; currently Faculty o Mechanical Engineering, Technion, 3 Haia, Israel. Member AIAA. Proessor o Mechanical Engineering and Jet Propulsion. Fellow AIAA. Subcontractor. Associate Fellow AIAA. 158 = binder comple requency = density = dimensionless time = dimensionless requency Subscripts = binder Im = imaginary part prop = propellant Re = real part s = surace value = oidizer 1 = oidizer = binder Superscripts (Im) = imaginary part (Re) = real part = perturbations I. Introduction UNDERSTANDING o unsteady combustion o solid propellants is a basis or analyzing, predicting, and suppressing o combustion instabilities in solid rocket motors. The main cause o these instabilities is the response o the combustion process to pressure oscillations in the combustion chamber. Hence, the most important physical quantity characterizing unsteady combustion is the response o mass burning rate to pressure oscillations. The simplest models o the unsteady combustion o solid propellants [quasi-steady homogeneous one-dimensional (QSHOD) models [1]] assume that the propellant is homogeneous, the gas phase is quasi steady, and the solid phase is one-dimensional. This approimation results in the well-known classical analytical solution or small perturbation analysis o a homogeneous propellant []. I one deines the pressure-coupled response unction R p R p r r p (1) p
SHUSSER, CULICK, AND COHEN 159 where r, p are steady-state values (overbars denote steady-state values throughout the paper) o burning rate and pressure, and r, p are their perturbations, respectively, then nab R p A= 1 A1 B Here, is a nondimensional comple requency o oscillations, whereas each o the basic parameters n, A, and B characterize the dependence o the steady-state burning rate r on a certain physical quantity. Thus, the pressure eponent n characterizes the dependence o r on pressure, A (characteristic o the surace decomposition) is related to the dependence on the surace temperature, and B (characteristic o the heat eedback law) shows the dependence on the surace temperature gradient, that is, heat lu. The classical solution has been used etensively in combustion instability studies. It is, however, limited to homogeneous propellants, whereas most modern solid propellants are composite propellants. To model composite propellant combustion, one must account or the heterogeneity and two-dimensionality o the problem. Following the classical work o Nachbar [3], a widely used approach has been to describe these eects approimately by using a sandwich model. This model represents a composite structure as side-by-side slabs o uel and oidizer. Steady burning o such sandwiches has been the object o numerous studies [4]. Cohen and Strand perturbed their steady-state model [5] to describe the nonsteady combustion o a composite propellant [6,7]. They were able to calculate the linearized combustion response and to eplain eects o ammonium perchlorate (AP) particle size, pressure, and crosslow on combustion response properties. Shusser et al. [8] used a sandwich model to numerically compute nonlinear combustion response properties o a composite propellant by solving the heat conduction equation in each component, with the Cohen and Strand model [5] as a boundary condition. A similar approach was used by Rasmussen and Frederick [9] and Surzhikov et al. [1]. Recently, a combustion model based on calculating the burning o a three-dimensional array o randomly packed spheres was proposed [11,1]. This model is irst to enable detailed consideration o the microstructure o the propellant. However, as with simpler sandwich models, the predictive value o the model is limited by the approimate constitutive relations it uses. Moreover, the need or computationally ast models remains or uture coupling o the propellant combustion with the gas dynamics o the combustion chamber in comprehensive motor analyses. It is well recognized that closed-orm solutions are very useul, as they provide insights not otherwise available. A general analytic solution or the response unction o composite propellants has not yet been obtained. Previous attempts to use the classical solution () or AP composite propellants were unsuccessul. For eample, Brown and Muzzy [13], who tried to use solution () to match their eperiments, obtained physically unrealistic values o the parameters. The reason or this ailure is obvious. Although a total propellant burning rate eists, surace temperature and heat lu are dierent or the AP and the binder components o the propellant. There are no average or total surace temperatures or surace temperature gradients and, thereore, no A and B parameters or the propellant as a whole can be deined. In the present work, we obtain the irst closed-orm analytical solution or a heterogeneous unsteady model o composite propellants. Such solution can serve several purposes. First, it can give more insight into the physics o the process. It also provides a better way to see trends with changes in physical conditions, such as pressure or temperature. It can be very helpul or adjusting parameters to achieve better modeling. Finally, it serves as a benchmark or numerical computations. Though only pressure-coupled response is considered in this analytical paper, the model can be etended to include velocity coupling by including an appropriate dynamic erosive burning model, assuming that dynamic erosive burning is the essential mechanism. This was done in the numerical work [8]. () In the net section, we will demonstrate how to obtain such an analytical solution or a composite propellant or a single (monomodal) particle size. For such a coupled two-component system, each component has its own set o basic parameters analogous to A, B, n, whereas two additional coeicients are due to the coupling between the oidizer and the binder. This solution is general in the sense that it is not limited to any particular ingredient or model or propellant combustion. A choice o a speciic model inluences only the values o the parameters, as they are obtained by linearizing the model. In Sec. III, we calculate the coeicients or the Cohen and Strand AP propellant model [5 7], show how they change with pressure and particle size, and compare the analytical solution with numerical computations (the numerical scheme has been described elsewhere [8]). We then proceed with a parametric study o the general solution in Sec. IV. Intrinsic instability is considered in Sec. V. The pressure eponent o a composite propellant is studied in Sec. VI. II. Analytical Solution The physical model used or the analysis is the classical AP/binder sideways sandwich. The approach yields a system o two unsteady one-dimensional heat conduction equations coupled through their boundary conditions at the solid surace. Their linearized orm, which is appropriate or small perturbation analysis, is as ollows: @ 1 @ 1 @ 1 @ @ @ @z 1 @ 1 @z 1 @z @ @z r 1 r 1 e z 1 (3) r r e z (4) Here, the subscripts 1, relate to the AP and the binder, respectively, the overbar denotes steady-state values, and the primed quantities are their perturbations. The dimensionless temporal and spatial coordinates 1,, z 1, z, and temperature perturbations 1, are deined by z 1 r 1 1 ; 1 r 1 t 1 ; z r (5) r t (6) 1 T 1 T 1 T 1s T ; T T T s T (7) Here, is the distance rom the solid surace (negative into the solid), t is the time, 1 and are AP and binder thermal diusivities, T 1s and T s are AP and binder surace temperatures, and T is the temperature in the bulk o the solid. The classical theory [] uses the ollowing boundary condition at the solid surace o a homogeneous propellant: @ p z nb @z p 1 B 1 r (8) A r Following the classical ormulation [], we write the boundary conditions at the solid surace and the relationships between burning rate and temperature perturbations as z 1 @ 1 @z 1 n B p p 1 B 1 A r 1 r 1 C r r (9) We use AP to represent the oidizer, but the generality o the approach is applicable to any two-component system.
16 SHUSSER, CULICK, AND COHEN z @ p n @z B p C r1 1 B r 1 r (1) 1 A r r 1 r 1 A 1 j z 1 ; r r A j z (11) One sees that now both AP and binder have their own set o basic parameters n, A, B, describing the relations between corresponding surace temperatures, heat lues, and burning rates. The n and n arise rom the burning o the composite propellant: n is not the value or AP by itsel as a monopropellant, and or the inert binder by itsel, n would be meaningless. Their meaning will be discussed in Sec. VI. The coupling coeicients C, C characterize how each heat lu is inluenced by the other component s burning rate. Physically, they are caused mainly by the coupling through the multiple lame structure. It should be noted that, due to the coupling, each component (AP or binder) behaves dierently when in a propellant and when by itsel. Thus, the values o the AP parameters A, B will also be dierent rom those or pure AP [14], as we shall see in the net section. We emphasize that the values o the parameters cannot be obtained by considering each component separately. They are determined by the physics o unsteady combustion o a composite propellant and can be obtained rom its suitable model. To complete the statement o the problem, we add the boundary conditions in the bulk o the solid z 1! ; z!1 (1) We now impose on the system an oscillatory pressure perturbation o requency. Deining two dimensionless oscillatory requencies 1 1 r ; 1 r (13) and AP and binder response unctions R r 1 p r 1 p ; R r p r p (14) one obtains (the details can be ound elsewhere [15]) n B C F R (15) =A 1= 1B 1=A C C =n B F n B C F R (16) =A 1= 1B 1=A C C =n B F Here, F is shorthand or the classical response unction () n F A B A = 1 A 1 B (17) with a corresponding ormula or F. The comple requencies, are obtained rom i 1 ; i (18) One sees rom Eqs. (15) and (16) that, when one o the coupling coeicients vanishes, the corresponding response unction reduces to its homogeneous orm [Eq. ()]: R F or C ; R F or C (19) The real and imaginary parts o R and R are written out in the Appendi. III. Cohen and Strand Model or Constitutive Combustion Relations The solution obtained in the previous section depends on several parameters. The values o the parameters can be determined by a particular model or propellant combustion. We have used the Cohen and Strand model [5] or this purpose, as was veriied by incorporating it as the constituent combustion model in our response unction work and comparing numerical solutions o the present model with a large amily o response unction data in [8]. Thus, to estimate the physically reasonable values o the combustion response parameters, we are justiied in using the Cohen and Strand model [5] to irst calculate the coeicients. The way to derive the eight coeicients A, A, B, B, n, n, C, C is to linearize the nonlinear surace boundary condition as obtained rom the original model (see [5 8]) and to compare its linearized orm with Eqs. (9) and (1). The resulting detailed epressions are given elsewhere [15]. Figure 1 compares the analytical solution with numerical computations or the individual component AP and binder response unctions as they behave in the composite propellant or a particular case. Both real and imaginary parts are shown in the igure. One sees that the agreement is very good. We studied the inluence o absolute pressure and particle size on the coeicients. AP concentration was kept at 87% throughout the calculations. Propellant bulk temperature was 5 C. We show here only the main results. The details can be ound in [15]. The B parameter is shown in Fig.. We see immediately that B in a propellant is much higher than in the classical ormulation. Although the B parameter usually lies between.6 and 1, B can be almost 3. On the other hand, or high pressures and large particle sizes, B becomes less than one; the reason or this is that, under these conditions, C, as we shall see later, and hence the less stable AP is practically decoupled rom the binder. B decreases with increasing pressure mainly because o decreasing lame heights. The jump at about 8 atm or the m case is caused by a change in the competing lame structure, as relected in the F model parameter (see [5,16] or more details). The binder parameters A, B, n may not be compared with the classical homogeneous A, B, n because an inert binder does not burn by itsel. Here, we show only B in Fig. 3. One sees that B is even larger than B and always stays above.4. Thus, an inert binder is highly stabilizing to the AP. The calculations also demonstrate [15] that n decreases much aster than n with increasing pressure because the binder is heated only by the diusion lame. AP is also heated by its monopropellant lame, which is the destabilizing inluence, as can be inerred rom our results in [14]. Figures 4 and 5 are plots o the coupling coeicients C and C. C is always negative and increases with pressure and particle size. This relects the stabilizing inluence o the binder on the AP by the coupling. The binder decomposition is endothermic and urnishes a diusion lame. C vanishes or high pressures and large sizes, where the AP becomes decoupled rom the binder and behaves more like pure AP. The dependence o C on pressure and particle size is more complicated, as seen rom Fig. 5. It is positive ecept or m AP at low pressures. This means that the AP generally has a destabilizing inluence on the binder when the binder is in act burning. The AP monopropellant lame and eothermic condensed phase tend to be destabilizing. The analytical solution has served to clariy this or a composite propellant. IV. Parametric Study A parametric study o the analytical solution [Eqs. (15) and (16)] is not easy because o the large number o parameters. The solution depends on eight coeicients A, A, B, B, n, n, C, C, and also It is possible that B will decrease with decreasing pressure at very low pressures, atm, due to the importance o condensed phase eothermicity o the AP at low pressures [8,14]. However, the presence o the binder has a stabilizing inluence on the AP.
SHUSSER, CULICK, AND COHEN 161 Fig. B parameter or the Cohen and Strand model [5]. Fig. 3 B parameter or the Cohen and Strand model [5]. Fig. 1 Comparison o analytical and numerical solution or composite propellant response unction; 87% AP ( m)/hydroyl-terminated polybutadiene, 68 atm, 98 K: a) AP response unction, b) binder response unction. on the ratio o dimensionless requencies that we will denote as : = 1 () We thereore adopt the ollowing approach. We will choose reasonable basic values o each parameter rom Sec. III results and vary only one o them at a time. Only the real part o the AP response unction as it behaves in the propellant will be considered due to its being the most important driver or practical applications. Results o this eercise can provide guidance or more study in uture work. The basic parameters chosen are given as ollows: A 1, B 1:5, n 1, A 8, B 3, n :8, C :3, C :5, :. Fig. 4 Coupling coeicient C or the Cohen and Strand model [5].
16 SHUSSER, CULICK, AND COHEN Fig. 5 Coupling coeicient C or the Cohen and Strand model [5]. All o the parameters are naturally divided into our groups: 1) AP coeicients A, B, n ; ) binder coeicients A, B, n ; 3) coupling coeicients C, C ; and 4) requency ratio. The inluence o the AP coeicients turned out to be analogous to A, B, n in the classical solution. The increase in A increases the response and moves its maimum to higher requencies. Decreasing B makes the response unction curve more peaked, whereas larger n increases the value o the response but not its requency dependence. The inluence o the binder coeicients is shown in Figs. 6 8. We see that or A it is rather limited. One has to vary it considerably to achieve a small change. This is analogous to the small eect o binder decomposition kinetics on steady-state burning rate. Increasing B makes the binder heat eedback smaller and thereore the AP response larger. A certain shit o the peak requency to the right is also seen in Fig. 7. On the other hand, increasing n decreases the AP response as more pressure-sensitive heating goes to the binder. Figures 9 and 1 are plots o the response unction or several C and C. As epected, the AP response is larger or less negative C and smaller C (less coupling rom or less coupling to the binder). No change in the orms o the curves can be observed in the igures. Again, we see the stabilizing inluence o the binder. Finally, Fig. 11 shows the inluence o the requency ratio. What is interesting is that it is requency dependent. Although or low requencies ( 1 < 5), the response is larger or lower, or high requencies ( 1 > ), it is larger or higher. There is an intermediate region in between. To understand the reason or this, one needs to observe that increasing or given 1 makes the binder requency higher. I is small, it is shited toward the peak area, and so the binder response becomes larger and the AP response correspondingly smaller. On the other hand, i is higher than the peak requency, then the binder response decreases and the AP response increases. Frequency ratio relects dierences in mean regression rates and thermal diusivities between AP and binder. Again, we see the binder being more or less a stabilizing inluence on the system. Fig. 6 Real part o AP response unction or several A. Fig. 7 Real part o AP response unction or several B. V. Intrinsic Instability It is well known that QSHOD-type models o homogeneous propellants have intrinsic stability limits, that is, there are regions o its parameters in which no steady burning is possible because the combustion is inherently unstable []. The goal o this section is to consider the possibility o intrinsic instability or the present model. We shall derive suicient conditions to prevent intrinsic instability and use the Cohen and Strand model [5] to see i they are satisied in practice. Fig. 8 Real part o AP response unction or several n.
SHUSSER, CULICK, AND COHEN 163 The solutions (15) and (16) can be written as R n B D C n B D D C C (1) Here, R n B D C n B D D C C () D =A 1= 1 B 1=A (3) D =A 1= 1 B 1=A (4) For intrinsic instability, the denominator in Eqs. (1) and () must vanish. This is equivalent to Fig. 9 Real part o AP response unction or several C. D Re D Re D Im D Im C C (5) where D Re D, D., D Re D Re D Im D Im D Re (6), D Im, D Im are the real and imaginary parts o We consider the case where both D Re Then, to satisy Eq. (6), we would need, D Re are positive: D Re > ; D Re > (7) D Im D Im < (8) Comparing Eqs. (5) and (8) one sees that i Eq. (7) is satisied then no intrinsic stability is possible or C C <.IC C > then, to prevent intrinsic instability, one needs to impose the additional condition D Re D Re >C C (9) Fig. 1 Real part o AP response unction or several C. In Sec. III, we obtained that, or the Cohen and Strand model [5], C C is almost always negative (see Figs. 4 and 5). This is consistent with the physics o the problem, as one would epect a destabilizing inluence o the AP on the binder (C > ) and a stabilizing inluence o the inert binder on the AP (C < ). However, or ine AP and low pressure, C C > was obtained. Whether this is an artiact o the model or relects the real physical situation is at present unclear. Nevertheless, we will consider both C C < and C C > in our analysis. To obtain conditions or the parameters, we write the real parts eplicitly: r Re B (3) D Re A Fig. 11 Real part o AP response unction or several. r Re B (31) D Re A Here, Re, Re are the real parts o the comple requencies, (see the Appendi). Equation (18) was used in obtaining Eqs. (3) and (31). Calculating the minimum values o D Re 1 Re < 1, one sees that Eq. (7) is equivalent to A, D Re or 1 Re < 1, s B 1 > 1 A (3)
164 SHUSSER, CULICK, AND COHEN n 1 @ ln r 1 @ ln p ; n @ ln r @ ln p (36) However, the situation or composite propellants turns out to be more complicated. In this section, we will demonstrate the eistence o two types o pressure eponents or a composite propellant and discuss the physical reasons or this phenomenon. A. Two Types o Pressure Eponent The classical pressure eponent or a homogeneous propellant n can be deined in two ways. Besides the usual deinition [Eq. (35)], one can use the surace boundary condition (8) to deine it as n p @ @ r B @p @z (37) z Fig. 1 s B 1 > 1 A (33) A Equations (3) and (33) are the suicient conditions to prevent intrinsic instability or C C <. IC C >, another condition must be added: B s A Suicient condition to prevent intrinsic instability. 1 1 B A s A 1 1 A >C C (34) To see i Eqs. (3 34) are satisied in practice, we use the values o the coeicients or the Cohen and Strand model [5] calculated in Sec. IV (see Figs. 5). One sees rom these igures that Eq. (33) is always satisied and Eq. (34) is satisied or C C >. However, this is not the case or condition (3). The let-hand side o condition (3) is plotted vs pressure or our particle sizes in Fig. 1. The igure shows that condition (3) is violated or high pressure and coarse AP. For m AP, the limit value o pressure is slightly above 16 atm. In numerical computations o composite propellant combustion [8], intrinsic instability was ound at 17 atm. Hence, the criterion o condition (3) is not only a suicient condition but probably close to the necessary condition as well. It should be stressed that, or high pressure and coarse AP, C and AP is practically decoupled rom the binder and behaves like pure AP. Only under these conditions, the Cohen and Strand model [5] allows intrinsic instability. One can thereore reach a tentative conclusion that the stabilizing inluence o an inert binder prevents an intrinsic instability under most conditions o interest. We show again the importance o avoiding coarse AP at high pressures, this time eplained with mathematics. VI. Pressure Eponents in Composite Propellants Pressure eponents represent the pressure dependence o the steady-state burning rate. For a homogeneous propellant, it is deined as @ ln r n @ ln p (35) Similarly, or a composite propellant, one can deine AP and binder pressure eponents that represent the pressure dependence o the steady-state burning rates r 1 and r : The deinitions o Eqs. (35) and (37) describe two aspects o the same phenomena. The ormer shows the pressure dependence o the steady-state burning rate, whereas the latter relates heat eedback and pressure oscillations. Burning rate is determined by surace temperature, which in turn is obtained rom the heat lu boundary condition at the solid surace. For a composite propellant, we have introduced two parameters n and n in boundary conditions (9) and (1). They can be deined by etending Eq. (37) to composite propellants n p @ @ 1 r B @p @z (38) 1 z1 1 r n p @ @ r B @p @z (39) z 1 r However, there are other ways, besides Eqs. (38) and (39), to etend Eq. (37) to composite propellants. The deinitions o Eqs. (38) and (39) assume that one calculates the p derivative while keeping both r1 and r at zero. However, it is not necessary to keep both o them at zero. We can etend Eq. (37) to a composite propellant while keeping each component s corresponding steady-state burning rate, that is, p @ @ n 1 lim 1! B @p @z (4) 1 z1 p n lim! B @ @ @p @z z r 1 r (41) The limits in Eqs. (4) and (41) are needed in accordance with the steady-state limit. To calculate n 1 and n, we use the surace boundary conditions (9) and (1) and the response unction deinition Eq. (14), to obtain rom Eqs. (4) and (41) n 1 n C B R j ; n n C B R j (4) As will be proven later [see Eq. (5)], it ollows rom Eq. (4) that n 1 n C B n ; n n C B n 1 (43) Substituting into Eq. (4) the zero-requency limits o both response unctions that can be retrieved rom Eqs. (15) and (16), one obtains n 1 B n B C n (44) B B C C
SHUSSER, CULICK, AND COHEN 165 n B n B C n B B C C (45) B. Method o Homogenization To understand the meaning o n and n, we will use what one might call a homogenization o the propellant. Consider the AP surace boundary condition (9) @1 p n @z B 1 p 1 B 1 r 1 r C A r (46) 1 r By deinition o the binder response unction R, r r R p p (47) Hence, Eq. (46) can be written as @1 n p @z B 1 p 1 B 1 r 1 (48) A r 1 Fig. 13 AP pressure eponent in a propellant. where n n 1 C R n B (49) Equation (48) looks eactly like the surace boundary condition or a homogeneous propellant, Eq. (8). O course, it is an unusual homogeneous propellant because its pressure eponent n depends on the oscillatory requency through R. Nevertheless, the classical relationships remain valid or this propellant. For eample, its response unction R is given by n R A B A = 1 A 1 B (5) Because o n being a unction o R, Eq. (5) yields a relation between AP and binder response unctions R and R. One can use Eqs. (15) and (16) to veriy Eq. (5). The same method can also be applied to the binder component, or which n can be similarly deined. Thus, we can make an important conclusion that each component in the composite propellant behaves like a homogeneous propellant with a requency-dependent pressure eponent. The n and n, thereore, are pressure eponents characteristic o the nonsteady coupling. For C, n n n 1. For C, n n n ; the binder can still burn because o the steady-state heat eedback but there is no nonsteady interaction. The steady-state pressure eponents are obtained by taking the zero-requency limit o n and n, which yields Eq. (4) or n 1 and or n. One sees rom Eq. (43) that, without coupling, n 1 n i C ; n n i C (51) This is consistent with the result that, when one o the unsteady coupling coeicients vanishes, the corresponding response unction reduces to its homogeneous orm. It can also be veriied that, as or homogeneous propellant, the response unction value or zero requency is equal to the corresponding pressure eponent: R n 1 or ; R n or (5) The method o homogenization can be used to calculate the response unction o a multicomponent propellant (see [15]). Fig. 14 Binder pressure eponent in a propellant. C. Practical Values To estimate the physically reasonable values o the pressure eponents, we calculated them or the Cohen and Strand model [5]. As in Sec. III, AP concentration was kept at 87% throughout the calculations and propellant bulk temperature was 5 C. Both pressure eponents n 1 and n are plotted in Figs. 13 and 14. One can see that the dependencies o n 1 and n are more or less like n and n. However, n 1 <n because C < and n >n because C > [see Figs. 4 and 5 and Eq. (43)]. One can also deine the pressure eponent or the whole propellant: n prop @ ln r prop @ ln p The total propellant burning rate is given by (53) r prop S r 1 S r prop (54)
166 SHUSSER, CULICK, AND COHEN Here,,, and prop are AP, binder, and propellant densities; S and S are ractions o propellant surace area occupied by AP and binder, respectively. This also yields the propellant response unction rom the component contributions. Dierentiating Eq. (54) with respect to ln p and using Eq. (36), we obtain n prop n n (55) where is the raction o propellant mass low rate rom AP. The calculation shows [15] that n prop is high or ine and coarse particles but can be considerably smaller or intermediate sizes where diusion control is important in the lame. VII. Conclusions We have obtained a general analytical solution or the pressurecoupled response o a monomodal composite propellant, applied to AP and inert binder. The solution depends on the AP coeicients A, B, and n, the binder coeicients A, B, and n, the coupling coeicients C and C, and the requency ratio. The inluence o these parameters on the real part o the AP response unction in the propellant has been investigated. The inluence o the AP coeicients is analogous to the A, B, and n parameters o the classical homogeneous propellant solution. The increase in A increases the response and moves its maimum to higher requencies. Decreasing B makes the response unction curve more peaked, whereas larger n increases the value o the response but not its requency dependence. The inluence o A was ound to be limited due to the small eect o binder decomposition kinetics on burning rate. Increasing B makes the binder heat eedback smaller and thereore the AP response larger, whereas increasing n decreases the AP response as more heat goes to the binder. Less coupling rom the binder (less negative C ) or less coupling to the binder (smaller C ) increases the response. The inluence o the requency ratio was ound to be requency dependent. The parameters A, A, B, B, n, n, C, and C were calculated or a particular case o the Cohen and Strand model [5]. The analytical solution was shown to be in ecellent agreement with numerical computation results, and has the advantage o revealing important mechanisms. Thus, we see the destabilizing inluence o the monopropellant AP and the stabilizing inluence o the inert binder in the system. The inluence o absolute pressure and AP particle size on the parameters has been studied. B in a propellant turned out to be much higher than in the classical ormulation due to the stabilizing inluence o the binder in the coupling. B decreases with increasing pressure mainly because o decreasing lame heights. The B parameter was ound to be even larger than B. This shows that an inert binder is highly stabilizing to the propellant system. Because o the binder being heated only by the diusion lame, n decreases much aster than n with increasing pressure. The AP coupling coeicient C is always negative. This relects the stabilizing inluence o the binder on the AP by the coupling. However, C becomes less negative with increasing pressure and particle size. C vanishes or high pressures and large sizes, where the AP becomes decoupled rom the binder and behaves more like pure AP. The binder coupling coeicient C is generally positive, showing that the AP has a destabilizing inluence on the binder. The AP monopropellant combustion is the destabilizing inluence on the system, whereas the diusion lame with binder is stabilizing. Suicient conditions preventing intrinsic instability in a composite propellant were obtained. Using the Cohen and Strand model [5], we demonstrated that these conditions are satisied in most practical situations. In agreement with previous numerical results, it was ound mathematically that intrinsic instability is possible only at high pressures and with coarse AP. At this condition, AP is practically decoupled rom the binder and behaves like pure AP. Thus, it appears that in most practical cases, the stabilizing inluence o an inert binder prevents intrinsic instability. To show this, it is necessary to treat the two-component system as it really is rather than as an averaged one-component system. We have shown that, or a two-component composite propellant, two orms o pressure eponents arise rom the analysis. Whereas the conventional n 1 and n show the pressure dependence o both AP and binder steady-state burning rates, n and n arise rom the oscillations o the unsteady heat eedback and unsteady coupling. The general epressions or n 1 and n in propellants have been obtained and their dependencies on pressure and AP particle size were studied or the Cohen and Strand model [5]. In addition, the pressure eponent o the aggregate composite propellant was deined. We also demonstrated that both AP and binder in the composite propellant system behave like a homogeneous propellant with a requency-dependent pressure eponent characteristic o the nonsteady coupling. The convenience o an analytical solution will acilitate urther studies o the response unction o composite propellants. Other advanced energetic ingredients may be more destabilizing than AP and can be evaluated or linearized analysis by studying the monopropellants themselves in the ramework o this theoretical approach, which began with the study o AP [14]. Appendi: Real and Imaginary Parts o Binder and Ammonium Perchlorate Response Functions The real and imaginary parts o the binder and AP response unctions R and R are as ollows: Here, R e 1e 5 e e 6 e 5 i e e 5 e 1 e 6 e 6 e 5 e 6 R e 3e 5 e 4 e 6 e 5 i e 4e 5 e 3 e 6 e 6 e 5 e 6 (A1) (A) e 1 n B Re A Re 1 B 1A n B C Im (A3) e n B Im A Re Im e 3 n B r A Re Im e 4 n B Im A Re Im (A4) 1 B 1A n B C e 5 Re A Re 1 B 1A Im Re A Re 1 B 1A Im Im Im A Re Im A Re C C Im (A5) (A6) (A7)
SHUSSER, CULICK, AND COHEN 167 e 6 Im A Re Im Re A Re 1 B 1A Im Im A Re Im Re 1 B A Re 1A (A8) Im The real and imaginary parts o the comple requencies and are given by Re 1 p 1 16 1 = 1 (A9) Im 1 p 1 16 1 1 Re 1 1 1= p 1 16 = Im 1 p 1 16 1 1= (A1) (A11) (A1) Acknowledgments This work was sponsored partly by the Caliornia Institute o Technology and partly by the U.S. Oice o Naval Research Multidisciplinary University Research Initiative under Grant No. N14-95-I-1338. Judah Goldwasser was the Navy Program Manager. The supplemental support under Multidisciplinary University Research Initiative Grant No. N14-95-I-1339, through the courtesy o Herman Krier, University o Illinois at Urbana-Champaign, is grateully acknowledged. Reerences [1] Price, E. W., and Flandro, G. A., Status and Prospects or Future Developments, Nonsteady Burning and Combustion Stability o Solid Propellants, edited by L. De Luca, E. W. Price, and M. Summerield, Vol. 143, Progress in Astronautics and Aeronautics, AIAA, Washington, D.C., 199, pp. 849 873. [] Culick, F. E. C., Review o Calculations or Unsteady Burning o a Solid Propellant, AIAA Journal, Vol. 6, No. 1, 1968, pp. 41 55. doi:1.514/3.498 [3] Nachbar, W., Theoretical Study o the Burning o a Solid Propellant Sandwich, Solid Propellant Rocket Research, edited by M. Summerield, Vol. 1, ARS Progress in Astronautics and Rocketry, Academic Press, New York, 196, pp. 7 6. [4] Price, E. W., Sambamurthi, J. K., Sigman, R. K., and Panyam, R. R., Combustion o Ammonium Perchlorate Polymer Sandwiches, Combustion and Flame, Vol. 63, No. 3, 1986, pp. 381 413. doi:1.116/1-18(86)97-6 [5] Cohen, N. S., and Strand, L. D., Improved Model or the Combustion o AP Composite Propellants, AIAA Journal, Vol., No. 1, 198, pp. 1739 1746. doi:1.514/3.813 [6] Cohen, N. S., and Strand, L. D., Combustion Response to Compositional Fluctuations, AIAA Journal, Vol. 3, No. 5, 1985, pp. 76 767. doi:1.514/3.8981 [7] Cohen, N. S., and Strand, L. D., Eect o AP Particle Size on Combustion Response to Crosslow, AIAA Journal, Vol. 3, No. 5, 1985, pp. 776 78. doi:1.514/3.8983 [8] Shusser, M., Culick, F. E. C., and Cohen, N. S., Combustion Response o Ammonium Perchlorate Composite Propellants, Journal o Propulsion and Power, Vol. 18, No. 5,, pp. 193 11. [9] Rasmussen, B., and Frederick, R. A., Jr., Nonlinear Hterogeneous Model o Composite Solid Propellant Combustion, Journal o Propulsion and Power, Vol. 18, No. 5,, pp. 186 19. [1] Surzhikov, S. T., Murphy, J. J., and Krier, H., Two-Dimensional Model or Unsteady Burning Heterogeneous AP/Binder Solid Propellants, AIAA Paper -3573, July. [11] Jackson, T. L., and Buckmaster, J., Heterogeneous Propellant Combustion, AIAA Journal, Vol. 4, No. 6,, pp. 11 113. [1] Massa, L., Jackson, T. L., Buckmaster, J., and Campbell, M., Three- Dimensional Heterogeneous Propellant Combustion, Proceedings o the Combustion Institute, Vol. 9, No.,, pp. 975 983. doi:1.116/s154-7489()8363-7 [13] Brown, R. S., and Muzzy, R. J., Linear and Nonlinear Pressure Coupled Combustion Instability o Solid Propellants, AIAA Journal, Vol. 8, No. 8, 197, pp. 149 15. doi:1.514/3.599 [14] Shusser, M., Culick, F. E. C., and Cohen, N. S., Combustion Response o Ammonium Perchlorate, AIAA Journal, Vol. 4, No. 4,, pp. 7 73. [15] Shusser, M., and Culick, F. E. C., Analytical Solution or Composite Solid Propellant Response Function, Guggenheim Jet Propulsion Center, Caliornia Inst. o Technology, Rept. CI 1-, May 1. [16] Beckstead, M. W., Derr, R. L., and Price, C. F., Model o Composite Solid-Propellant Combustion Based on Multiple Flames, AIAA Journal, Vol. 8, No. 1, 197, pp. 7. doi:1.514/3.687 V. Yang Editor-in-Chie