DOI 10.1007/s10891-015-1171-0
Journal of Engineering Physics and Thermophysics, Vol. 88, No. 1, January, 2015
HEAT AND MASS TRANSFER IN COMBUSTION PROCESSES SIMULATING THE COMBUSTION OF N POWDER WITH ADDED FINELY DIVIDED ALUMINUM V. A. Poryazov, A. Yu. Krainov, and D. A. Krainov
UDC 536.46+536.24
A mathematical model for combustion of N powder with added aluminum particles is presented. It takes account of the exothermal chemical reaction in the gas phase, convection and diffusion, heating, and combustion of aluminum particles in the gas flow, the motion of combustion products, and the lag of the particle velocity behind that of the gas. The results of calculation of the burning velocity of powder correspond to the experimental data on the dependence of this velocity on pressure and aluminum particle size. It has been established computationally that for aluminum particles of diameter less than 20 μm the burning velocity of N powder depends substantially on the size of these particles. Keywords: N powder, gas-dispersive medium, aluminum particles, ignition, combustion. Introduction. The ballistic efficiency of rockets is essentially defined by the efficiency of the specific thermodynamic momentum of their propulsion engines. To raise the temperature and energy of powder combustion products, a finely divided metal is added to the powder [1–4]. The results of experimental investigations of the combustion of N powder with additions of finely divided aluminum are presented in [3, 4]. According to these results the aluminum powder present in the composition of a high-energy condensed substance exerts its influence on the linear rate of substance combustion, which is not always a positive factor. The experimentally revealed influence of finely divided aluminum with different degrees of dispersion on the energy characteristics of N powder combustion has not been given a theoretical description. Therefore of practical interest is to derive the dependence of the linear rate of combustion on the mass concentration and sizes of aluminum particles that enter into the high-energy composition. The linear rate of combustion of a high-energy condensed substance (ballistic powder, composite solid propellant) depends on pressure, the initial temperature of the powder, its composition, and other parameters. Works [5–8] laid the foundation of the powder combustion theory, works [9–12] presented models of combustion of a powder and of composite solid propellants, and [13, 14] described the present situation with modeling combustion of high-energy substances. Work [15] presented a model of erosional combustion of solid rocket propellants and showed that the turbulent transfer of heat in the boundary layer leads to the rearrangement of the temperature field above the combustion surface, increases the heat flux to the condensed phase, and causes an increase in the rate of combustion. The results obtained agree well with the results of experimental investigations in the presence of blowing [16] and without it [6]. In the present work, to determine the dependence of the burning velocity of N powder on the temperature of its surface, we used the model of [15] developed for the case of the absence of the tangential velocity component of gas above the propellant surface. To model the burning out of aluminum particles in a flow of powder combustion products, we used the experimental results of [17] showing that at pressures above 20 atm the rate of combustion of aluminum particles in an oxidant flow is independent of pressure. Construction of the Mathematical Model. The following assumptions were made to formulate the mathematical model. According to the model of Belyaev–Zel′dovich [5, 6, 18–20], the linear rate of combustion of a powder is determined by the surface temperature of the condensed phase. An exothermal reaction of the first order proceeds by the Arrhenius law Tomsk National Research State University, 36 Lenin Ave., Tomsk, 634050, Russia; email:
[email protected]. ru. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 88, No. 1, pp. 93–101, January–February, 2015. Original article submitted April 4, 2014. 94
0062-0125/15/8801-0094 ©2015 Springer Science+Business Media New York
in the gas phase. Convection and diffusion of the reagent as well as the powder combustion occur under isobaric conditions in which the pressure is independent of the distance to the combustion surface and the gas expands on heating. Aluminum particles have a spherical shape, the same size in the powder, and they are distributed uniformly within the latter. The combustion of aluminum particles is described on the basis of the experimental data of [16, 17], and the ignition of an aluminum particle occurs after it reaches a definite temperature. The particles and gas exchange heat by the Newton law. The particles move under the action of the gas-induced friction force described by the Stokes law. Because of the small volume concentration of particles in the gas, the motion of particles does not exert its influence on the motion of the gas, no carbon skeletons (agglomeration sites) are formed on the propellant surface, and the surface is not blown by air. Agglomeration of particles on the condensed phase surface [21] and the interaction of particles with themselves in the gas phase are absent. With account for the assumptions made, the mathematical model is constructed on the basis of the energy conservation equations for the gas and aluminum particles, mass conservation equations of the gas and particles and of the number of aluminum particles, the equation of the burning-out of a reagent in the gas phase, the equation of particle motion in a gas flow, and the equation of gas state. The system of equations written in the coordinate system connected with the origin on the combustion surface has the form
⎛ − E2 ⎞ ∂T ⎞ ∂ 2T2 ⎛ ∂T 2 c2 ρ 2 ⎜ 2 + u 2 ⎟ = λ 2 + Y ρ 2 k0Q2 exp ⎜ ⎟ + 4παrk n (T3 − T2 ) , 2 ∂x ⎠ ⎝ ∂t ∂x ⎝ Run T2 ⎠
(1)
⎛ − E2 ⎞ ∂Y ⎞ ∂ 2Y ⎛ ∂Y u D + = − Yk0 exp ⎜ 2 ⎟, ⎜ ⎟ 2 ∂x ⎠ ⎝ ∂t ∂x ⎝ RunT2 ⎠
(2)
∂ρ 2 ∂ (ρ 2u ) + = −G , ∂t ∂x
(3)
∂T ⎞ 2μ Al ⎛ ∂T c3ρ 3 ⎜ 3 + w 3 ⎟ = −4παrk2 n (T3 − T2 ) + GQAl , t x 3μ O ∂ ∂ ⎝ ⎠
(4)
∂ρ 3 ∂ (ρ 3w) + = G, ∂t ∂x
(5)
∂w ∂w +w = −τ fr , ∂t ∂x
(6)
∂n ∂ ( nw) + = 0, ∂t ∂x
(7)
P = ρ 2 RT2 = const .
(8)
This system comprises the energy equations for the gas phase and particles (1) and (4), the equation for the burn-up fraction (2), the mass conservation equation for the gas phase (3), the mass conservation equation for particles (5), the equation of particle motion (6), the equation for the number of particles (7), and the equation for the state of an ideal gas (8). The coordinate x = 0 corresponds to the combustion surface. According to the Belyaev–Zel′dovich model [5, 14], where the decomposition of a powder to a gas phase is considered as a gross reaction with the thermal effect Q1 and activation energy E1, the linear rate of combustion of the powder is defined by the relation
⎛ − E1 ⎞ Vk = K υ exp ⎜ ⎟, ⎝ 2 RTs ⎠
(9)
where Kυ is an empirical constant. The conditions on the boundary x = 0 express the mass and energy conservation laws [14]:
ρ1Vk = ρ 2 u x = 0 ,
ρ1Vk Y = D2
∂Y ∂x
x =0
+ ρ 2 uY
x =0
,
95
λ2
∂T ∂x
x =0
(
)
= ρ1Vk c2T2 x = 0 − Q1 − c1T1, 0 ,
ρ 3, kVk = ρ 3w x = 0 ,
ρ2 x =0 =
n x =0 =
T3 x = 0 = T2 x = 0 ,
P , RT2 x = 0
ρ3 x = 0
(10)
P = const ,
.
3 ρk (4 3) πrAl,0
Here rAl,0 is the radius of an aluminum particle emerging from the combustion surface of the N powder into the gas flow and T1,0 is the initial temperature of the powder, T2 x = 0 ≡ Ts. The following boundary conditions are specified on the boundary x = ∞:
∂T ∂x
∂Y ∂x
= 0, x=∞
= 0.
(11)
x=∞
The initial conditions have the form
T2 ( x, 0) = Tign , ρ 2 ( x, 0) =
P , RT2 ( x, 0)
T3 ( x, 0) = Tign , ρ 3 ( x, 0) = 0,
Y ( x, 0) = 0 ,
w ( x, 0) = 0,
n ( x, 0) = 0 .
(12)
We will write expressions for the quantities τfr and G on the right-hand sides of the equations of system (1)–(8). The force with which the aluminum particles interact with the gas is calculated by the formula
τ fr =
Ffr
4 3 πrk ρ k 3
Ffr = Cr S m
,
ρ 2 ( w − u) u − w . 2
Here the drag coefficient is determined from the empirical formula of [22]:
Cr =
(
)
24 1 + 0.15 Re 0.682 , Re
Re =
2rk ρ k u − w . η
The heat transfer coefficient is equal in this case to
α =
Nuλ 2 , 2rk
Nu = 2 +
Nu l2 + Nu t2 ,
(13)
where Nul = 0.664Re0.5 and Nut = 0.037Re0.8. To determine the rate of change in the mass of the burning particles G we assume that the product of aluminum combustion is the oxide Al2O3 that remains on the particle, and the density of the particle does not change in the process of combustion. We will denote the initial mass of the aluminum particle as mAl,0, the current value of the particle mass in the process of its combustion as mk, and the mass of the unburnt part of aluminum in the particle as mAl. On partial burning-out of aluminum in the particle, the mass of the latter is defined by the relation
mk = mAl +
μ Al + (3/2)μ O (mAl,0 − mAl ) . μ Al
At a constant density of the particle ρk the following relation is valid: 96
mk =
4 ρ πρ k rk3 = 3 . 3 n
(14)
The mass of the particles in a unit volume is equal to Mk = nmk and the time derivative of this mass determines the value of G
G = −
3μ O 2 drAl nρ k 4πrAl . 2μ Al dt
(15)
It was established from the experimental data in [16] that at P > 20 atm the time of the burning out of the aluminum particle is independent of pressure and it is determined by the initial diameter of the particle dAl,0 and by the relative concentration of 1.5 d Al, 0 the oxidant a: τc = 1.062·104 0.9 . Differentiating the right- and left-hand sides of this expression and making elementary a dr a 0.9 transformations, we obtain that Al = –2.22·10–5 . Substituting the relation obtained into (15), we obtain an expresdt rAl sion for the rate of change in the mass of particles during their combustion:
G =
3μ O a 0.9 2 nρ k 4πrAl k Al , 2μ Al rAl
k Al = 2.22 ⋅ 10 −5 m1.5 /s .
(16)
The quantity of aluminum that remained in the particle rAl is determined from the number of particles in a unit volume n and from their reduced density ρ3 with the use of the equality
⎛ ⎛ 3⎞ μ Al + ⎜ ⎟ μ O ⎜ 4 3 3 3 ⎝2⎠ + − rAl mk = πρ k ⎜ rAl rAl,0 μ 3 ⎜ Al ⎜ ⎝
(
)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠
and of the following formula derived from Eq. (14): 1
rAl
⎡⎛ ⎤3 ⎞ 3 ⎢⎜ μ Al + μ O ⎟ 2μ Al ⎥ ρ3 3 2 ⎥ . ⎟ rAl,0 = ⎢⎜ − 3μ O ⎥ μ Al ⎛4⎞ ⎢⎜ ⎟ ⎜ ⎟ πnρ k ⎟ ⎢⎜⎝ ⎥ ⎝ 3⎠ ⎠ ⎣ ⎦
(17)
In deriving Eq. (17) it was assumed for simplification that the oxide Al2O3 that remained in the particle has the shape of a spherical layer [22, 23]. This assumption is used for determining the current radius of an aluminum sphere in the process of its burning and in no way influences the rate of combustion of the aluminum particle, since this rate was selected with account for experimental data. Procedure of Calculations and Results. The system of equations (1)–(8) with boundary and initial conditions (10)–(12) and expressions for the right-hand sides of Eqs. (13), (16), and (17) was solved numerically by the methods outlined in [24, 25]. Equations (1) and (2) were solved by an implicit difference scheme with the aid of the sweeping method, and Eqs. (3)–(7), by the explicit difference scheme with the use of the approximation of convective components by upstream differences. For the stability of the numerical solution, use was made of the stability condition of the form Δt < Δx/max[ui], where Δt is the time step, Δx is the space step, and ui is the velocity at the points of the computational mesh, The system of equations (1)–(10) was solved according to the following algorithm of computation of unknowns on the n + 1th time layer (counting of time step). The velocity Vk was calculated by Eq. (9), the burning out of the gas-phase combustible on the n + 1th time layer, by Eq. (2) written in a difference form, the temperature of the gas-phase combustible on the n+1th time layer, by Eq. (1), the gas velocity, by Eq. (3), the particle temperature, by Eq. (6), the number of particles in a unit volume, by Eq. (7), and the density of particles, by Eq. (5). The temperature of the powder surface on the new time layer and the linear rate of powder combustion are determined with the use of the difference approximation of boundary 97
Fig. 1. Distribution of aluminum particle and gas temperatures (a), gas and aluminum particle velocity (b), aluminum particle and unburnt substance particle radii (c), and of the gas temperature and rate of chemical reaction in the gas (d) over the powder combustion surface at rAl,0 = 15 μm, P = 10 MPa, and Tign = 1300 K: a) 1, aluminum particles; 2, gas; b) 1, gas; 2, aluminum particles; c) 1, unburnt substance particles; 2, aluminum particles; d) 1, gas; 2, Φ. conditions (10). The counting of time step is repeated the needed number of times. The computations are continued until the stationary distribution of the gas-dispersive medium parameters is achieved above the powder combustion surface. Based on the algorithm described, a computer program has been written, using which we analyzed the results of convergence calculations on decrease in the space step. From the convergence of the results obtained on decrease in the space step, we selected the value of the latter that ensured the convergence of results with an error not higher than 0.3%. We carried out calculations with different values of the initial temperature above the powder surface Tign. The stationary distributions of the gas-dispersive medium parameters obtained above the combustion surface for different values of the temperature Tign in the range from 1500 to 2500 K were the same. The satisfaction of the mass and energy conservation laws was controlled in the process of calculations that were fulfilled with an accuracy of up to 98%. The rate of combustion of N powder was calculated by the model, presented in [15], for the case of the absence of the tangential flow of combustion products above the powder surface. At the constants given in [15] for the N powder, our calculations gave the value 0.01047 m/s for the linear rate of combustion. The calculation by model (1)–(8) with the same constants for the N powder in the absence of aluminum particles yielded the value 0.01119 m/s for the linear rate of combustion. The difference amounted to 7%. For the powder surface temperature we obtained the value 724 K and the value 2334 K for the temperature of the combustion particles, which corresponds to the experimental data of [6] for the N powder 98
and to the calculations in [15]. The subsequent testing confirmed the adequacy of the results of calculations by the developed computer program. The combustion of N powder with addition of finely divided aluminum was investigated numerically for the mass concentrations of the latter that were used in the experiments in [3, 4]. The calculations of the combustion of N powder were carried out at the following values of the thermophysical and kinetic quantities: c1 = 1465 J/(kg·K), c2 = 1466.5 J/(kg·K), c3 = 760 J/(kg·K), λ2 = 1 W/(m·K), Q1 = 556,800 J/kg, Q2 = 2,435,300 J/kg, QAl = 36,510,000 J/kg, ρ1 = 1600 kg/m3, ρk = 2600 kg/m3, a = 1, η = 0.00005 Pa·s, E1 = 79,733 J/mole, E2 = 186,107 J/mole, Run = 8.31 J/(mole·K), R = 264.36 J/(kg·K), Kυ = 8.46 m/s, k0 = 0.98·1010 1/s, kAl = 2.22·10–5 m1.5/s, T1,0 = 293 K, μAl = 0.027 kg/mole, μO = 0.016 kg/mole, and D2 = Le λ2/(c2ρ2). In most calculations the aluminum particle ignition temperature was assigned equal to 1300 K. We carried out additional calculations in a wider ignition temperature range. In these calculations we varied the values of gas pressure above the combustion surface within the range 4.0 ≤ P ≤ 20.0 MPa, of the radius of aluminum particles emerging from the N powder combustion surface into the gas flow within the range 1 ≤ rAl,0 ≤ 300 μm, and of the mass concentration of aluminum in the powder within 0–9%. The results of calculations presented in Fig. 1 in the form of steady-state distributions of the powder combustion products were obtained for the case where the content of aluminum particles of radius rAl,0 = 15 μm in the powder was 9 wt.% at the pressure P = 10 MPa and at the aluminum particle ignition temperature 1300 K [3]. Figure 1d shows the distribution ⎛ − E2 ⎞ of the rate of chemical reaction of the gas Φ = Yk0 exp ⎜ ⎟ . The reaction in the gas phase starts near the combustion ⎝ Run T2 ⎠ surface and ends at a distance of about 0.1 mm from it. After the emergence into the gas phase the aluminum particles do not burn; they are heated in the gas flow up to the temperature of the start of reaction and then start to burn. After the start of combustion the particle temperature is ahead of the gas temperature (Fig. 1a). Due to the heat transfer from the particles, the gas temperature in this region begins to increase faster than in the case where there are no aluminum particles. With increase in the distance from the surface the aluminum particles burn out, and the temperature of the combustion products increases. The aluminum particles burn out completely at a great distance from the combustion surface (Fig. 1a, c). Thus, particles of radius 15 μm burn out at a distance of about 12 mm from the combustion surface. The combustion of aluminum particles increases the gas temperature near the powder surface, leads to an increase in the conductive heat flux to it, and, as a consequence, to an increase in the powder surface temperature and in the linear rate of combustion determined by Eq. (9). The gases leaving the combustion surface have the velocity 0.4 m/s. Due to the heating of the gas by the reaction occurring in the gas phase and by the burning aluminum particles, the gas velocity increases and attains its final value after the burning out of aluminum particles. For the variant presented in Fig. 1 this velocity is equal to 2.5 m/s. The aluminum particles depart from the combustion surface with zero velocity (in the laboratory coordinate system). On coming into interaction with the gas, they are accelerated and gradually acquire the gas velocity (Fig. 1b). With increase in the initial size of aluminum particles emerging into the gas flow from the powder surface, the temperature of the start of their combustion is attained at a greater distance from the combustion surface (Fig. 2). In view of this, the combustion of these particles does not lead to a substantial increase in the powder surface temperature and to an increase in the rate of powder combustion. Larger aluminum particles burn out at a greater distance from the combustion surface, and the final temperature of the combustion products does not change as it depends only on the mass concentration of aluminum in the powder. Fine particles are heated rapidly and start to burn closer to the surface and heat the gas, which leads to an increase in the combustion surface temperature and in the rate of combustion. We investigated the influence of the size of aluminum particles in the composition of a powder on the rate of its combustion. The percentage of aluminum in the calculations was taken equal to 9%. Figure 3 presents the dependences of the rate of power combustion on the initial size of aluminum particles at different pressures. Their form corresponds qualitatively to the dependence predicted by A. F. Belyaev in [1]. At small values of the initial radius of aluminum particles the rate of powder combustion exceeds its rate of combustion without aluminum particles. With increase in the initial radius of particles from small values (1–2) μm, the rate of combustion decreases rapidly. When the initial radius of particles is equal to about 30–40 μm, the rate of N powder combustion becomes equal to its rate of combustion without aluminum particles. With further increase in the initial size of aluminum particles, the rate of powder combustion depends slightly on the size of these particles and is smaller than its rate of combustion without addition of aluminum particles, since the temperature of the powder surface with aluminum particles is lower (the aluminum particles play the role of an inert material whose heating requires a certain quantity of heat). It is also seen from Fig. 3 that the rate of powder combustion starts to depend substantially on the initial size of aluminum particles when their radius is less than 5 μm.
99
Fig. 2. Distribution of aluminum particle temperature near the powder combustion surface at Tign = 1300 K, P = 10 MPa, and rAl,0 = 5 (1), 15 (2), and 30 μm (3). Fig. 3. The rate of N powder combustion vs. the aluminum particle radius at aluminum mass concentration 9% and various pressures: 1) P = 100 atm; 2) 80; 3) 60; 4) 40; 5) 100; 6) 80; 7) 60, 8) 40. The horizontal lines, N powder without aluminum particles.
Fig. 4. The rate of N powder combustion vs. the aluminum particle radius at aluminum mass concentration 9%, P = 10 MPa, and Tign = 990 (1), 1100 (2), 1300 (3), 1500 (4), and 1700 K (5): 6, powder without aluminum particles. Fig. 5. Dependences of the rate of N powder combustion on pressure at aluminum mass concentration of 9% calculated for aluminum particles of different sizes: 1) rAl,0 = 1 μm, 2) 2; 3) 3; 4) 5; 5) 7; 6) 10; 7) 15; 8) 30, and 9) 45. It is indicated in [27] that the development of the chemical reaction of oxidizing ASD-4 aluminum particles proceeds in the temperature range 960–1300 K. The development of oxidation of an ultradispersed aluminum powder of "Alex" grade occurs in two stages and starts at a lower temperature. In the model presented, the combustion of aluminum particles begins on attainment of the specified ignition temperature. Therefore we carried out calculations of the rate of combustion of the N powder on addition of aluminum particles at various values of aluminum particle ignition temperature. This temperature 100
TABLE 1. Comparison of Calculations with Experimental Data of [4] Vk, m/s
P, atm 40
60
80
100
N powder Experiment Calculation by (1)–(17)
0.052 0.0066
0.0071 0.0081
0.0085 0.0094
0.0102 0.0112
N powder + 9% Al Experiment (dAl,0 = 40–70 μm) Calculation by (1)–(17) (dAl,0 = 60 μm)
0.006 0.0067
0.008 0.0083
0.0096 0.0097
0.0105 0.0108
varied within the range 990–1700 K; the pressure above the combustion surface was equal to P = 10 MPa. The results of calculations in the form of the dependence of the rate of N powder combustion on the aluminum particle radius are presented in Fig. 4. It is seen from this figure that with increase in the ignition temperature of aluminum particles the rate of powder combustion varied insignificantly when the aluminum particles are of size rAl,0 > 40 μm. When the size of aluminum particles rAl,0 < 5 μm, there is also a strong dependence of the linear rate of combustion of powder on the size of these particles. The higher ignition temperature of aluminum particles in the N powder when their size rAl,0 < 40 μm leads to a decrease in rate of powder combustion. Figure 5 presents the dependences of the rate of combustion of N powder with an admixture of aluminum particles on pressure; they were calculated for aluminum particles of various sizes. It is seen that the rate of powder combustion increases with pressure. This is due to the fact that the zone of intense chemical reactions approaches the combustion surface and, as a consequence, the surface temperature increases. As the initial size of aluminum particles decreases, the zone of their combustion also approaches the powder surface; they heat the gas and increase the powder surface temperature, which leads to an increase in the rate of its combustion. To approximate the experimental dependence of the rate of powder combustion on pressure, the relation Vk = Vk,0Pv was used. According to the Belyaev–Zel′dovich theory [5, 6] v = 0.5 for the reaction of the first order in the gas phase. Calculations of the value of v from the curves presented in Fig. 5 yield the values v = 0.5 ± 0.02 at different values of rAl,0 in the entire pressure range investigated. Table 1 presents the results of calculations of the linear rate of combustion of N powder by the model (1)–(8) and the experimental data presented in [4] as graphs of the rate of combustion of N powder and of N powder with a 9% addition of aluminum particles of diameter 40–70 μm. Satisfactory agreement between the results of calculations and experimental data is seen. Conclusions. A mathematical model has been developed for calculating the dependence of the rate of combustion of N powder on pressure and on the size of aluminum particles introduced into it. The results of calculations by this model are compared with the corresponding experimental data, and satisfactory agreement was obtained. It has been established computationally that the rate of combustion of N powder with aluminum particles of size less than 20 μm depends substantially on the size of these particles, which may complicate the attainment of stable powder combustion in technical facilities. The present paper was carried out with financial support from the Ministry of Education and Science of Russia within the framework of state assignment No. 10.1329.2014.
NOTATION c, specific heat at a constant pressure; Cr, drag coefficient; D, diffusion coefficient; E2, energy of reaction activation in the gas phase; Efr, drag force of a sphere immersed in a flow; G, rate of change of the mass of particles in the course of their combustion; k0, pre-exponential factor in the Arrhenius law; Le, Lewis number (Le = 1 for the gas phase); n, number of particles in a unit volume; Nu, Nusselt number; P, pressure; Q2, thermal effect of reaction in the gas phase; QAl, heat of aluminum combustion; rk, size of the aluminum particle; R, gas constant; Run, universal gas constant; Sm, midsection area; T, temperature; Ts, temperature of combustion surface; Tign, ignition temperature of aluminum particles; u, gas velocity; w, velocity of particles; x, coordinate; Y, concentration of a combustible in the gas phase; α, heat transfer coefficient; η, dynamic 101
viscosity coefficient; λ, thermal conductivity; μAl, μO, molar masses of aluminum and oxygen particles; ρ2, gas density; ρ3, reduced density of particles (mass of particles in a unit volume); ρk, density of a particle in the flow of combustion products; τfr, force of interaction of particles with the gas; Φ, rate of chemical reaction of the gas. Indices: c, combustion; un, universal; ign, ignition; r, resistance (drag); m, midsection; fr, friction; s, surface; 1, 2, and 3, parameters of powder, gas phase, and of the condensed phase of combustion products.
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