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One-Dimensional Burning Wave in a Bed ofMonopropellant ParticlesDONALD A. DREW aa Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York,12180-3590Published online: 27 Apr 2007.
To cite this article: DONALD A. DREW (1986) One-Dimensional Burning Wave in a Bed of Monopropellant Particles, CombustionScience and Technology, 47:3-4, 139-164, DOI: 10.1080/00102208608923870
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Cornburr. Sci. and Tech., 1986. Vol. 47. PP. 139-164 0010-220218614704-0139E25.0010
0 1986 Cordon and Breach Science Publishers. Inc. Printed in Grcat Britain
One-Dimensional Burning Wave in a Bed of Monopropellant Particles
DONALD A. DREW Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, New York 12 180-3590
(Received February 28. 1984; in fital form August 23, 1985)
Abstract-A one-dimensional burning wave in a monopropellant particle bed is studied. Approxi- mate analytical solutions are obtained assuming low Mach number flow, rapid burning and efficient heat transfer between the particles and the gas. A preheat zone, where compression heats the gas, precedes the burning zone. A conducting layer occurs in the vicinity of the leading edge of the burning zone. In the remainder of the burning zone, the temperature remains constant and the particles burn up.
NOMENCLATURE
a wave speed b co-volume B dimensionless co-volume c relative specific heat ck specific heats C+ speed of sound in cold gas Cc parameter in drag correlatign e, unit vector in propagation direction fk body force density G dimensionless temperature gradient in conduction layer h effective interfacial heat transfer parameter ho Heaviside (unit step) function ,hch chemical enthalpy hr enthalpy hk+ dimensionless interfacial heat transfer parameter .Hk interfacial heat transfer parameter Icy conductivity K "elasticity" of particles L+ length scale in unburned region L- length scale in burning region LC length scale in conducting region tn exponent in drag M Mach number Ff lk interfacial momentum transfer density n scaled number density ii: number density N exponent in burning rate p pressure pe contact pressure P'r Prandtl number
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D. A. DREW
qk energy flux r scaled particle radius t: particle radius R gas constant Re, particle Reynolds number S + scaled drag in unburned region S - scaled drag in burning region S drag t time tr parameter in drag Tk temperature - T k stress tensor U relative particle speed vk velocity vr interface velocity V relative gas speed x coordinate in propagation direction x spatial coordinate y scaled spatial variable in burning region y* position of fluidization z scaled spatial variable in preheat region - Z scaled spatial variable in conducting region Zl parameter in drag
Greek Symbols
a volume fraction of particles a* volume fraction of material k a, volume fraction of compacted bed a+ volume fraction in cold mixture y ratio of specific heats
effective ratio of specific heats pk interfacial mass transfer rate Sk scaled density E dimensionless chemical enthalpy 81, internal energy density
interfacial energy density ech chemical energy
translating spatial variable 6'1, scaled temperatures 01 scaled interfacial temperature Of scaled flame temperature 8 scaled matching temperature K* scaled gas conductivity h burning rate parameter A scaled burning rate parameter A* scaled burning rate p viscosity [ r interfacial energy transfer a scaled pressure
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MONOPROPELLANT PARTICLE BURNING WAVE
a, scaled contact pressure prc densities Z scaled drag parameter + burn fraction
Subscripts
g gas i interface k P o r g p particle
Superscripts
+ preheat region - burning region
INTRODUCTION
Conibbstion of particulates is a wide-spread energy conversion process which utilizes the large area-to-volume ratio for efficiency. Burning of fuel droplets and coal particles, and dust explosions are examples where an oxidant is supplied to the fuel by a flow of gas relative to the interface.
Explosives are based on reactions where the reactants are simultaneously present. These reactions rely on reaching and maintaining some ignition requirement, often some threshold temperature.
Our purpose here is to examine the steady one-dimensional propagation of a burning wave into a monopropellant particle bed (see Figure 1). We start the analysis with the full two-phase flow equations with reactions (Drew, 1983). We discuss the many assumptions leading to the one-dimensional burning wave (sce also Kuo and Summerfield, 1974; Kuo et a/ . , 1976; Krier et a/. , 1978).
In this paper we find an approximate analytical solution of the one-dimensional, two-phase reactive flow equations of conservation of mass, momentum and energy. We assume a steady travelling wave. The steady equations will be non-dimensionalized using three scales corresponding to three regions: the preheat, conduction, and burn- ing regions. The conduction region will be divided further into a region "before ignition" and a region "after ignition". The burning region is also divided into two subregions, one where particles remain in contact, and one where particles lose contact.
Flow in the preheat region is isentropic due to assumptions made for that region. Conditions at the preheat-conduction zone interface are temporarily assumed known and are determined Inter. In the conduction region, the energy equation is solved using the assumed temperature at the preheat-conduction interface as one boundary condition. Two solutions must be obtained in the conducting region, one for each subregion. These solutions are related using the conditions of continuity of tempera- ture and heat flux at the subregion interface. They are then matched to conditions in the other two regions. The low Mach number approximation of a constant down- stream pressure fixes the matching conditions at the conduction-preheat interface. This allows the burn rate and therefore the wave speed to be determined based on
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burned
D. A. DREW
i g n i t i o n conduction- I
l a y e r ! "tr
I I I I preheat
burnuo bu rn i n
c o n t a c t FIGURE 1 Flow geometry.
upstream conditions and the downstream pressure. Solutions of the equations in the burning region allows the point where particles lose contact to be determined.
We then discuss many approximations to the mechanics and thermodynamics, and their results. We hope that the balances derived and the resulting wave structure represent those attained in practice. We feel that the actual physical situation is almost hopelessly complicated, and that by seeking approximate balances, we are able to identify the most important mechanisms acting i n a flame in a monopropellant bed. This parallels the large volume of recent work in laminar flames; see the text by Buckmaster and Ludford (1982).
THE MODEL
The averaged equations of motion for two reacting materials are:
1) Conservation of mass
2) Conservation of momentum
3) Conservation of energy
where ar is the volumetric concentration (volume fraction) of phase k , pk is the density, v k is the (mass averaged) velocity, r k is the rate of production of mass of phase k at thc interface, ?=k is the stress, vl is the interfacial velocity, Mn. is the interfacial
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MONOPROPELLANT PARTICLE BURNING WAVE 143
force, f k is the body force, is the internal energy density, q k is the heat flux, and f k is the rate of production of energy of phase k at the interface.
The jump conditions are
We shall use the notation a l=a for the particle phase, with a2= 1 -al= 1 -a for the gas phase.
In order to arrive at a working model, a large number of assumptions are needed.
1) The particle density is assumed to be constant.
2) The particles are spheres of radius i. Thus,
where ii is the number density. 3) The particles are assumed to remain whole during the process. That is, no
agglomeration or fracture is allowed. This leads to the equation
aii - + V . n v p = O at
for conservation of particles. This equation, along with Assumptions 1 and 2, give
This relates the particle surface regression rate to rp. 4) The particle surface regression rate is assumed to be zero before ignition, and
proportional to the pressure raised to a power after ignition. The ignition criterion is assumed to be based on the interfacial temperature, Tt. Specifically, the particles are assumed to be ignited if the interfacial temperature is greater than a prescribed ignition temperature. Thus (Kuo and Summerfield, 1974),
where ho is the Heaviside (unit step) function.
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144 D. A. DREW
5) The effects of turbulence and viscous stresses are assumed negligible. Thus, -. -. Tg = -PI. (10)
6 ) The stress in the solid phase is assumed to consist of two parts: one due to the fluid pressure, and one due to the particle-particle contacts. I t is noted that the stress inside each particle is related to its (microscopic) deformation; however, the model assumes that the externally applied stresses in each particle are transmitted instantaneously to the particle interior. Thus,
We shall further assume that p,=pc(a), with pc=O for a < a c and p, increasing rapidly for a > a,. Kuo and Summerfield use
for a a,, where K = 4 . 8 8 x lo6 g/cm2.
7) The interfacial velocity is taken to be the particle velocity:
8 ) The interfacial force is assumed to consist of a buoyancy term and a drag term. Thus,
If S = 9 ~ / 2 i ~ , where p is the viscosity of the gas, then the drag force is Stokes drag. We shall use Ergun's (1952) expression for the packed bed, so that
S = So[l +0.023 Re,(l -a)/a], where
and Anderson's (1961) correlation for the fluidized region
where
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MONOPROPELLANT PARTICLE BURNING WAVE 145
These expressions are those used by Kuo and Summerfeld (1974). 9) Body forces are neglected.
10) (a) The internal energy of the particles consists of the chemical energy to be released in the reaction, thermal energy, and a stored energy due to the microscopic deformation. The deformation energy will be neglected, leaving
where cp is the specific heat of the particles. 10) (b) The internal energy of the gas consists of heat and energy of an cbm-
pression. Thus, % = hg-plpg = ~ J g - p l ~ g , 0 7)
where c, is the specific heat at constant pressure for the gas. Both c, and c, are taken to be constant.
11) The gas is assumed to satisfy Clausius' equation
where R is the gas constant and b is the co-volume. The ratio of specific heats is Y = cgl(c9 - R).
12) Molecular heat conduction (thermal diffusion) is a very inefficient mechanism; however, temperature gradients may be large in small layers. Thus, we shall retain thermal conductivity in the gas. We assume
Heat conduction in the particles is retarded by the necessity of the heat to conduct from particle to particle across relatively small contact areas. We assume that this mechanism is inefficient, so that
qp = 0. (1 9b) 13) Viscous heating is neglected. 14) The rate of heat flow to each phase is assumed to be linear in the difference
between the temperature of that phase and the temperature at the interface. Thus,
where H k depends on the concentration and surface area of the particles. Denton's (1951) convective heat transfer coefficient for the packed bed is
where Pr=pc,/k, is the Prandtl number. For the fluidized region, the Rowe-Claxton (1965) correlation is
0.305 (2 1 b)
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146 D. A. DREW
where m = (0.6667 + 1 .55Rep-0,28)/(4.65Rep-0.zs+ 1). (2 1 c)
Strictly speaking, no constant value of Hp gives the exact value of the heat flux per unit particle volunle. An adequate value may be obtained by assuming spherical symmetry, and retaining one eigenmode. This results in
ONE-DIMENSIONAL BURNING
Let us seek solutions to this model which correspond to one-dimensional traveling waves. Thus, we assume that for any functionf,
where a is the wave speed. Furthermore, we assume that
As boundary conditions, let us assume that
U + - a V + - a P + P f a + a * as 5 - t a, i + r +
Tg + T + Tp -+ T +
and P - + p - 0 as C + - m . (25b) a + O
The ordinary differential equations for the steady propagation of a plane burning wave are
d Xa pp- [aUl = = - -pNho(Tt - Ttgn),
d l r (26)
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MONOPROPELLANT PARTICLE BURNING WAVE 147
The jump conditions for the conservation of mass and momentum have been incor- porated in Eqs. (26)-(33). The jump condition for the conservation of energy is
This equation can be solved for Ti, giving
where hch= E C ~ + P / ~ .
NONDIMENSIONALIZATION
Let us nondimensionalize the problem as follows. The velocity scale is a, so let
The density scale is taken to be p,+=p+/RT+. Then define
The pressure scale is pgC(C+)2, where CC is the speed of sound at l= co. The dimensionless pressures T I and nc are defined by
nc = P ~ I ~ , + ( C + ) ~ . The Mach number is
M = a/C+.
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148 D. A. DREW
The temperature scale is taken to be T+. Thus
The particle radius is nondimensionalized by
and we nondimensionalize the number density by
where
The natural length scale is determined by the equations of motion, and will be different for Tr < Ti,, and for Ti> Ti,,. For the preheat region, the scale will be determined by drag and heat transfer. We write
Then
We define
s+ = sjso. (47)
The dimensionless thermal conductivity is
K + = k,/(L+ P,+ a c,). We shall assume that K+ < 1.
The equations in the preheat region are
d 8, - (au) = 0,
dz
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MONOPROPELLANT PARTICLE BURNING WAVE 149
where
In the burning region, the length scale is determined by the burning rate r,. We define
L- = pg+ a/[h(pg+(Cf)2)"/r+], and
y = 5/L-. We define
kg- = HgL-If,+ a c,,
E = hCh/cg T f . .Again we assume K - < 1.
Equations (26)-(33) become
dau anN 8,-= --,
dy r
nu d l l dall , M2aSpu- = - a - - -- + M2aS-(v - u),
dv 4 du
n = Y-l 8, O,/(l - BS,) (68)
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D. A. DREW
In addition, Eq. (56) holds throughout the burning region. Thermal conduction is an inefficient process. The result of this is that conduction
is often a limiting mechanism in thermal processes. In the simple laminar flame, it is a balance between convection and conduction which determines flame speed. For thermal conduction to be important, large temperature gradients must occur. The upshot of this is that conduction effects are usually important only in regions which are relatively thin. Such a region occurs in the vicinity of the ignition point, 5=0. In order to describe the conducting layer, we stretch the region around 5=0 so that conductivity is as important as convection. When we scale in this manner, we shall also assume that the heat release due to burning is as i m ~ o r t a n t as conduction. -
The scaling is given by LC = klpg+ a cg,
z = 1;lLC. We define
HaC = HgLC/pg+ a c,,
The equations in the conducting layer are
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MONOPROPELLANT PARTICLE BURNING WAVE I51
APPROXIMATIONS
Let us next discuss several approximations in the mechanics and thermodynamics which will allow us to obtain an analytic, if approximate, solution of the problem.
1 . Particles in Contact
Consider the region where nc>O. The particles remain in contact throughout this region. Recall that n c = nc(a) is assumed to be a steeply increasing function of a for a > a,. This implies that a differs little from ae while the particles are in contact. Thus we shall assumed that
a a, = a+. (86)
Let us now consider the preheat region, where no burning occurs. This implies
and
2. Low Mach Number
We shall restrict our attention to burning waves which are slow, and wherein the gas and particles also move slowly. Thus, we assume that M<1. In the preheat region, the approximate gas momentum equation becomes
The total momentum equation can be integrated to give n+an ,= n++asn,+. In the burning region, we need an assumption about the relative sizes of the drag and the inertia terms. For present purposes, we shall assume that S-=0(1). With this assumption, we have
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152 D. A. DREW
for the region where burning in contact occurs. Finally, when the particles lose contact, we have
3. Eficient Interparticle Heat Trnnsfer in Preheat Region
Let us now make an approximation for the thermodynamics in the preheat region. The heat transfer coefficient H, is usually large. This reflects the ability of the micro- scale convection in the gas to transfer heat to or from the interface. Also, the heat transfer coeficient H, is usually large, but for a different reason, namely, the heat transfer coefficient is proportional to an inverse power of r , modeling the surface area available for heat transfer. For small particles, then, H, is large.
Using (50) and (25) gives S,v = -1. (93)
If we assume k is large, we have
If we add Eqs. (54) and (55), and use (86), (88) and (93), we have
Using the equation of state (53) and integrating gives
where
The temperature is
8 = s,?-l(l - Bf)?(l - BS,)/[l- B?S,]?.
Substituting Eq. (96) in the gas momentum equation (89) yields
The boundary condition at z=O is
S,(O) = g,, which is to be determined. Note that the solution satisfies S,(m) = 1.
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MONOPROPELLANT PARTICLE BURNING WAVE 153
4 . Eficient Burning
Let us now discuss the thermal processes in the burning region. Combustion is (usually) dominated by chemical reaction terms. In this model, these terms make up r,. We anticipate, then, that r, is large. This implies that h,- and 11,- are small compared to 1 .
With this assumption, we have
The equations for the gas and particle temperatures are
5 . Conduction Boundary Layer
In this layer, the gas temperature adjusts from the limiting temperature in the preheat region to a matching value from the "outer" solution valid in the burning region. We assume that the particles remain in contact throughout this region, thus a=at . We also assume M < I. This implies F l = n - throughout. Furthermore, in analogy with the assumption of dominant heat release, we shall assume a balance occurs involving the heat release and thermal conduction. The equation for the gas tem- perature in the conduction layer is different for 5 > 0 arid 5 < 0.
For 5 > 0, we have
The solution satisfying 0,(0) = Oi,, and O,(co) = 8 is
e, = 8+(&,, - 8) exp(-Z) For 5 < 0, we have
The variation of r is governed by Eqs. (77) and (85), with r(O)= I . The solution 113
If we further assume 8, $1, we have
1 iza+(n-yT r = l + - Z+ . . . I . .
12 8,
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154 D. A. DREW
It is then consistent to assume that very little change in radius occurs in the conducting layer; that is, the conducting layer is an adjustment in temperature without a change in geometry.
The equation of conservation of mass for the gas is
Thus,
where
The equation for the gas temperature is
d2 0, - ( 1 -a+) ---
do, + [ - ( I -a+)+ i\a+(n-)"'Z]- = Aa+(n-).y[6',- O,].
d Z 2 dZ
The solution satisfying 6',+6j as Z+- w and 8=01,, a t Z=O is given by
0, = of +(B~,,- of) exp * ) [l +G fexp(.- G Z Z ) ~ ~ ] ,
0
where
Continuity of dO,/dZ at Z = 0 gives
This equation gives a relation between the dimensionless burning rates A* and the dimensionless preheat temperature 8, through the dimensionless constant G, which measures the temperature gradient at the ignition point. If for the moment we assume 6 is known, then A * is determined by Eq. (1 13), and this, in turn, determines the burning specd, n. A graph of G versus A* is shown in Figure 2.
BURNING REGION
After making the many approximations mentioned above, we have
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MONOPROPELLANT PARTICLE BURNING WAVE
FIGURE 2 Scaled temperature versus A*.
If the particles are in contact, we have
a = a+,
and if the particles are not in contact, we have
1' .. The solution of (1 18) which matches to (1 17) as y+O is
a, z? Of. (120)
]Finally, the particle temperature profile which is bounded at the point +=a,S, is
o , = o , . . (121)
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156 D. A. DREW
This requires a jump in 0, at y=0. This jump could be resolved by including a particle conductivity. Conductivity across the "points" of particle-particle contact is difficult to model, and is presumably quite small. This being the case, the infor- mation gained by performing the analysis of a particle conduction sublayer is mini- mal, in contrast to that obtained in the gas conduction layer, which determines the wave propagation speed.
Equations (I l4), (1 15) and (I 19) can be attacked by defining
With this, we have Spau = +- Spa+,
In the two regions, namely the particles-in-contact and the fluidized subregions, the motion can be found by solving (I 19), (123) and (124), using the relations between a, r , u and n, together with 1-1 = 1.1- and 8,= O f . For the particles-in-contact region, O > y > y * , or O < + < + * , we have
In the fluidized subregion, Eqs. (123) and (124) yield
1 u = -(+-a+ S,),
asp
Since 8,- is constant, Eq. (1 19b) gives
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MONOPROPELLANT PARTICLE BURNING WAVE 157
The conditions which go with Eq. (133) are
Examination of the singular point at (a+&,, 0) shows that solutions entering a neighborhood of (a'sp, 0) will approach (a+Sp, 0). Thus, (134a) is automatically satisfied.
The quantity $* is determined by the condition that n , = 0 there. The contact pressure at the ignition point is
and, consistent with the approxinlations made herein, is also equal to
This relates +* to a+n,+, the contact pressure far upstream of the burning wave. For Stokes drag, when m=2, we have
M2 S- n + - n - + , + n , + = - --- ] I.(I - *)(-a+8p)
a+ Sp
- [a+S,+(l -a+)&-
(1 -a+)& -] +*). (137)
Finally, the burning speed a is related to 6 through Eq. (1 13). The assumption of small Mach number impl~es that 1-1 =constant throughout the conduction layer and the burning regions. Thus, f l = n- , which is determined by conditions in the burned gas. Then 4 can be determ~ned from Eqs. (96) and (98). It is then straightforward to plot the burning speed versus the pressure behind the wave for given values of all other parameters. The parameter values used for this plot are (Kuo and Summerfield, 1974) T+=293.91 OK, Tt,,=615"K, Tf=286O0K, A=36.3 cm/sec (k bar)"', N=0.8867, pp = 1.6 g/cm3, c,= 1.7 x 104 cm2/secoK, b = 1.26 cm3/g, p,+=O.OO1 g/cm3, a+ =0.57, r+=0.015 cm, y = 1.4, p,+c,= ppcP, p+ = 1 atmosphere. The thermal conductivity k, is taken to be 1.1 1 x I04 gm cm/sec? OK. A plot of the wave speed vs. pressure is shown in Figure 3. The profiles of the gas temperature in the conduction region and in the preheat region, and a family of particle volume fractions for different y*s are shown in Figures 4a, b, c.
SUMMARY A N D DISCUSSION
The non-dimensional burn rate, in terms of dimensional numbers can be written as
A(P-)N kg a+ A* = --
(ps+ a)? r + c,(l -a+)
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D. A. DREW
FIGURE 3 Speed of propagation (cm/sec) versus pressure (atm) assuming laminar heat con- duction in the gas.
This equation relates the non-dimensional burn rate A* to the wave speed, o. T o determine the wave speed, u, a series of equations must be solved. It can be seen that the wave speed is dependent on the system parameters as follows:
The downstream pressure (TI - or P-) is allowed to be specified. This allows 8, the preheat-conduction intellicial temperature, to be determined as follows:
Consider the preheat region equations (96) and (98) when n = I-[-. This gives 8 as a function of n-, a+, S,, C+, and 7. Once 8 is known, the burn rate A* may be determined by solving Eqs. (1 12b) and (1 13). Once A* is known it is a simple matter to use Eq. (138) to determine the burn speed o. Given a (therefole M-a/cC), then the point where particles lose contact can be determined as a function of particle contact stress, initial porosity and other parameters by solving Eq. (137) for +*, a transformed distance given by
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MONOPROPELLANT PARTICLE BURNING WAVE
I M) -3500 -25.00 -15.W ' 5 . 00 5111
z FIGURE 4a Gas temperature vs. Z in the conduction region.
where a l lN / r is the burn rate and y* is the distance in which the particles lose contact.
With the many approximations, the burning wave has the structure shown in Figure 5. The approximate equations are shown in the regions where they are valid.
A preheat region precedes the region where burning occurs. At the leading edge of the burning region sits a conductive subregion, where the released heat diffuses ahead against the gas convection. This heat maintains the ignition temperature against the heat lost to convection. This region controls the speed at which the total dynamical interactions occur. Specifically, the wave speed is selected so that the convective-conductive balance in this layer is maintained. The remainder of the burning region is maintained at the flame temperature.
The gas pressure rises in the preheat region as the ignition point is reached. This compression heats the gas and drives hot gas into the preheat region. The pressure remains constant throughout the burning region. The value of the elevated pressure far downstream of the wave is determined by the pressure needed to heat the gas and particles to the ignition point.
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Y
FIGURE 4b Gas temperature vs. y in the preheat zone.
The particles do not move while they are in contact and unignited. After ignition, up to the point where they are fluidized, they move together to maintain contact as the average radius decreases. After the particles lose contact, they move rearward and decrease in radius until they disappear. The particles lose contact at the point where the contact stress becomes zero. This point depends on the level of contact stress upstream of the wave.
The description of the burning wave given above is qualitative, but is expected to be approximately valid for more complicated models for the interfacial drag and heat transfer, the gas equation of state, and particle surface regression rate. This approxi- mate analysis suggests that the exact form of the interfacial heat transfer and the particle surface regression rate are unimportant in determining the temperature profiles. Tile interfacial drag and gas equation of state are important in the preheat region. Ano:her qualitative conclusion concerns the equation of state. The ideal gas law yields pressures and propagation speeds which are too low. This suggests
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MONOPROPELLANT PARTICLE BURNING WAVE
FIGURE 4c Particle volume fraction vs. 7 in the burning region, for y'=O. - 5 , -10, -15, - 20.
that it is important to have an equation of state capable of handling the high densities, pressures and temperatures encountered in the process.
We shall now compare and/or contrast our results with those obtained by Kuo and Summerfield (1974). First, many of the assumptions we use result in simpler equations than theirs. Thus, we have sacrificed some "reality" for a model which is amenable to some analysis. In contrast, we include the thermal conductivity in the gas, which they ignore (except in gas-particle heat transfer). Their results show the particle and gas temperature approximately equal in the preheat region; we derive this from the assumption of large /I,+ ahd hgf. They show the volume fraction nearly constant in the preheat region; we derive this result from the assumption of steep p,(a). The constancy of v, is an immediate consequence. Their results show a very steep thermal gradient near the ignition point. Indeed, they mention difficulties in the numerical procedure, and that the transition becomes a jump when the positions
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BURNING I I PREHEAT free / contact
I I a = a+
\ f CONDUCTION LAYER , I I
FIGURE 5 Temperature profile, showing equations valid in each subregion.
of ignition and ablation are coincident. We resolve the thermal layer using the con- ductivity scale. This region controls the flame speed. We feel that failure to resolve this region correctly jeopardizes any flame speed calculations. In the burning region, they show that the gas temperature is nearly constant; we derive this from asymptotic considerations for large burning rates. Their variation of porosity is comparable t o ours, and on the same scale. Their pressure varies in the burning region, while ours
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MONOPROPELLANT PARTICLE BURNING WAVE 163
is constant. The difference lies not in any approximations, but in their (incorrect) model (their Eq. 5) without the so-called buoyancy force, which removes the porosity . from the pressure gradient term. The low Mach number approximation on their equation leads to (1 - alp-constant, so that p varies when a varies. Their variation of p and a is not inconsistent with this approximation. Thus, our model, with its many approximations, captures the features of the numerical solution of the more complicated model. I t also allows a qualitative understanding of which parameters and assumptions affect what features.
The design of burning systems may be facilitated by the understanding which this analysis gives of the relation between the microscopic mechanisms and the macro- scopic parameters. For example, mechanisms which increase the effectiveness of the turbulent heat transfer increase k, and hence increase the burning speed a propor- tionally. The roles of the ignition temperature and the flame temperature are subtle; both modify the preheat temperature 6. On the other hand, certain potential design parameters are seen to have small influence on the macroscopic process. For example, the exact forms of the interfacial heat transfer and drag and the particle contact pressure do not appear in the expression for the flame speed. We expect that the actual flame speed should be relatively insensitive to these quantities.
We also note the potential problem with not including a model for conduction into the governing equations. This implies that there is no mechanism for moving the released heat from the reacting ahead to ignite the particles. This implies that without a conduction model, gas convec.:ion and conlpression must preheat the particles to the ignition temperature, 6=eig, . Although the model used here differs in several ways from that used by Kuo and Summerfield (1974), we suspect that their prediction of very high pressures and b= &,, is due to their use of a model without conductivity.
It appears that extension to higher Mach numbers is possible. It involves modify- ing Eq. (89) to include inertial effects, and possibly a shock. The conduction layer dynamics would also be affected. Extension to sufficiently slowly varying conditions is trivial-the results can be used with the external fields given as slowly varying functions of time and position. Extension to marked transients such as ignition from cold conditions appears very difficult. Conditions change so that different assumptions would be violated at different times.
ACKNOWLEDGEMENT
This work was supported by the U.S. Army Research Office. It was begun while the author was visiting the U.S. Army Research Ofice under an IPA agreement. Discussions with A. K. Kapila are gratefully acknowledged. A referee offered several constructive comments which have been incorporated in the text.
REFERENCES
Andersson, K. E. B. (1961). Chem. Eng. Sci. 15, 276. Buckmaster, J., and Ludford, G . (1982). Theory of Latninar Flarnes. Cambridge University,
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