c Pleiades Publishing, Ltd., 2017. ISSN 0010-5082, Combustion, Explosion, and Shock Waves, 2017, Vol. 53, No. 5, pp. 569–573. c A.M. Lipanov. Original Russian Text Calculation of Pressure in a Solid-Propellant Rocket Motor with the Use of a Real Dependence of the Solid Propellant Burning Rate on Pressure A. M. Lipanova UDC 621.454.3 Published in Fizika Goreniya i Vzryva, Vol. 53, No. 5, pp. 87–92, September–October, 2017. Original article submitted November 30, 2016; revision submitted December 26, 2016. Abstract: Five variants of calculating the burning rate of a solid propellant as a function of the pressure in a solid-propellant rocket motor are considered. Two variants of analytical expressions are proposed for approximating real dependences. In all variants, the pressure in the rocket motor can be presented by simple analytical expressions as a function of solid propellant parameters, charging conditions, and structural factors of the charge and motor. Keywords: solid propellant, burning rate, rocket motor. DOI: 10.1134/S0010508217050100 The burning rate of a solid propellant as a function of pressure in the solid-propellant rocket motor (SRM) is usually considered as a power-law dependence  νst upr = ust , 1p (1) where ust 1 and νst are constants. Concerning the experimental dependences upr (p), they diﬀer signiﬁcantly from power-law curves. Figure 1 shows ﬁve curves upr (p). Curves 1, 2, 3, and 5 are borrowed from various publications, and curve 4 is proposed by the author. As it follows from the analysis of curve 1 (see Fig. 1) , upr = 0 at p = 0. As the pressure increases within the SRM operation range (up to 30–60 MPa; active jetpropelled missiles are also classiﬁed here as rockets), the burning rate increases in a manner similar to a power law. In the vicinity of the inﬂection point, this curve can be approximated by a straight line. With a further increase in pressure, the dependence upr (p) asymptotically transforms to a straight line, i.e., a single-term law of the burning rate u1 p, which is satisﬁed within the artillery range of pressure (more than 60 MPa). It is seen that the curve upr (p) is a more complicated function than a power-law curve. It is demonstrated below that this curve is adequately approximated by the function a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia; email@example.com Fig. 1. Calculated dependences upr (p): curve 1 is borrowed from ; curves 2 and 5 are constructed on the basis of the data ; curve 3 is similar to that in ; curve 4 is proposed by the author of the present work. upr = u1 p + apm e−np , (2) where u1 , a, m, and n are positive coeﬃcients, 0 < m < 1. According to Eq. (2), we obtain upr = 0 at p = 0. At p > 0, the propellant burning rate increases, but this increase occurs with a negative second derivative because m < 1. The exponent e−np strictly decreases. Therefore, the product pm e−np passes through the point of the maximum deviation from the asymptotic curve c 2017 by Pleiades Publishing, Ltd. 0010-5082/17/5305-0569 569 570 Lipanov u1 (p) and then starts to approach the latter. As p → ∞, the dependence upr (p) tends to a single-term law. It follows from here that the coeﬃcient u1 should be determined outside the range of the rocket range of pressure and should be assumed to have a known value within this range. Thus, we have to ﬁnd three coeﬃcients: a, m, and n. Let us rewrite Eq. (2) as upr − u1 p = apm e−np , ln(upr − u1 p) = ln a + m ln p − np. (3) This equation describes the equality of the incoming amount of combustion products [left side of Eq. (3)] to their amount outgoing through the engine nozzle. Here Spr is the burning surface of the solid propellant charge, ρpr is the solid propellant density, Tch is the charge temperature, f (Tch ) is a function characterizing the dependence of the solid propellant burning rate on the charge temperature, Fth is the area of the minimum cross section of the nozzle (throat), Tp is the thermodynamic temperature of combustion products at constant pressure, R is the gas constant of combustion products, ϕ2 is the ﬂow rate coeﬃcient, k is the ratio of the isobaric (cp ) and isochoric (cv ) heats, and (k+1)/[2(k−1)] √ 2 k. B(k) = k+1 Equation (3) yields δu0pr δSpr δρpr p ∂upr δp Tch ∂f δTch + + + + Spr ρpr upr upr ∂p p f ∂Tch Tch δϕ2 δFth 1 Tch ∂RTp δTch δp = + − + . ϕ2 Fth 2 RTp ∂Tch Tch p (4) p ∂upr upr ∂p [see the fourth term on the right in Eq. (4)] and obtain 6 δp 1 δXi = , p 1 − ν i=1 Xi i=1 This is a linear relation in logarithmic coordinates. It can be easily solved by the least squares technique  in processing experimental data obtained, e.g., in a constant-pressure vessel . Let us consider pressure as a dimensionless variable normalized to 1 atm. The propellant burning rate is presented as the ratio to its value, e.g., at p = 500 MPa. Then the burning rate in the rocket range of pressure is always smaller than unity. Curve 1 in Fig. 1 is plotted with the coeﬃcients u1 = 0.0002, a = 0.0004280, m = 0.99753, and n = 0.00663044. Curve 1 shows the data for propellants without catalysts. Let us consider the behavior of the parameter ν for curve 1. This parameter appears due to variations of the factors in the following equation : ϕ2 B(k)Fth p . RTp ν= where 6 δXi whence it follow that Spr ρpr upr (p)f (Tch ) = Following , we denote Xi = δSpr δρpr δu0pr δTch δϕ2 δFth + + +β − − , Spr ρpr upr Tch ϕ2 Fth β= 1 Tch ∂RTp Tch ∂f + . f ∂Tch 2 RTp ∂Tch If upr is calculated with the use of a power law, the resultant value of ν coincides with νst . δXi as inConsidering small normalized deviations Xi dependent random variables with zero mathematical expectations and using known values of dispersion, we can ﬁnd the pressure dispersion: Dp = 6 1 Di . (1 − ν)2 i=1 From here, we ﬁnd the root-mean-square deviation σ: 6 1 σ= Di . 1 − ν i=1 After that, we apply Sorkin’s formula for the limiting deviation 6 Δp → n Di . p i=1 Here n depends on the probability of determining the limiting deviation Δp/p. It is necessary to have n = 3 for the required probability of 0.997, n = 3.3 for 0.999, etc. Δp As ν → 1, we obtain → ∞ for the ﬁnal p 6 Δp → value equal to n Di . As ν → 0, we have p i=1 6 6 Δp n < n Di . If ν < 0, then Di , because p i=1 i=1 6 Δp n 1 = < 1. Di , where p 1 + |ν| i=1 1 + |ν| Let us return to the analysis of the parameter ν. Using Eq. (2), we calculate the value of this parameter as Calculation of Pressure in a Solid-Propellant Rocket Motor 571 Fig. 2. Behavior of the parameter ν for curves 1–3 shown in Fig. 1. ν= u1 p + apm (m − np)e−np . u1 p + apm e−np (5) Using the coeﬃcients a, m, and n corresponding to curve 1 in Fig. 1, we obtain the dependence ν(p) in the form of curve 1 in Fig. 2. It is seen that the parameter ν ﬁrst decreases, reaches a minimum value equal to 0.518 approximately at p = 240, and then increases and tends to unity as p → ∞. As a whole, the values of ν are suﬃciently high. To ﬁnd the pressure in the SRM, we rewrite Eq. (3) in the form ϕ2 B(k)Fth u1 p . (6) Spr ρpr u1 f (Tch ) = RTp upr (p) Let us denote u1 p . upr (p) (7) Spr ρpr u1 f (Tch ) RTp = A. ϕ2 B(k)Fth (8) ϕ= Then, Eq. (6) yields ϕ= Relation (7) with allowance for Eq. (2) can be written as ϕ= 1 1+ ae−np /u 1p 1−m . (9) It follows from Eq. (9) that ϕ → 0 as p → 0 and ϕ → 1 as p → ∞, i.e., the function ϕ changes in the interval from zero to unity. It steeply increases around p = 0 and slowly approaches unity as p → ∞. The shape of the dependence ϕ(p) for curve 1 in Fig. 1 in the pressure interval from 10 to 200 corresponds to curve 1 in Fig. 3a. We can see that this is almost a straight line (especially bearing in mind a limited range of pressure for a particular SRM). Let us denote the minimum pressure for such an engine by pmin and the corresponding value of ϕ by ϕmin . Then, we have ϕ = ϕmin + a1 (p − pmin ). Fig. 3. Dependences ϕ(p) corresponding curves in Fig. 1: (a) to curves 1 and 4; (b) to curves 2, 3, and 5. The maximum pressure pmax corresponds to ϕmax . Now the coeﬃcient a1 can be found by the formula ϕmax − ϕmin . a1 = pmax − pmin Then we obtain p − pmin . (10) ϕ = ϕmin + (ϕmax − ϕmin ) pmax − pmin Let A = 0.42, pmin = 60, pmax = 90, ϕmin = 0.407, and ϕmax = 0.452. Equating ϕ to A, we ﬁnd from Eq. (10) that A − ϕmin . p̄ = pmin + (pmax − pmin ) ϕmax − ϕmin Substituting the values of pmin , pmax , ϕmin , ϕmax , and A, we obtain p̄ = 68.67. 572 Fig. 4. Behavior of ν as a function of pressure for curve 4 in Fig. 1. Let us consider curve 2 in Fig. 1 . The coeﬃcients corresponding to this curve have the values a = 0.0017779, m = 0.95, and n = 0.0060655. It is seen that curve 2 ﬁrst increases intensely (up to p ∼ = 100) and then slowly, asymptotically approaching the straight line u1 p. This behavior can be explained by the nature of the propellant being used or by the action of a catalyst. The behavior of ν for this curve corresponds to curve 2 in Fig. 2. At p = 150, we have ν = 0.3. After that, this parameter decreases to ν ≈ 0.02 at p = 300 and then starts to grow. The value of 0.3 is again reached at p = 520. Thus, we have 0.02 ν 0.3 in the pressure range from 150 to 520. The function ϕ for this curve is shown in Fig. 3b (curve 2). It is also very close to a straight line, especially in view of a limited range of pressure for a particular SRM. Let us consider curve 3 in Fig. 1. It is obtained for the coeﬃcients a = 0.0019085, m = 0.99, and n = 0.0078707. The dependence upr (p) of a similar shape was reported in . According to this curve, the burning rate reaches the maximum value approximately at p = 180. The corresponding dependence ν(p) is shown in Fig. 2 (curve 3). It is seen that the value of ν ﬁrst decreases; this parameter changes its sign at p = 180 and becomes negative. However, as the maximum on the curve upr (p) is suﬃciently smooth, the greatest negative value of the derivative of upr with respect to p is not very large. Correspondingly, the absolute value of ν in the negative domain is also fairly small (not greater than −0.1). Nevertheless, if ν < 0, then 6 6 n Δp Δp = < n Di , i.e., Di , p 1 + |ν| i=1 p i=1 and this is an interesting result. Though the curve for ϕ in this case (see Fig. 3b) resembles an S-shaped curve, it is still close to a straight line. Let us consider curve 4 in Fig. 1. It corresponds to the coeﬃcients a = 0.022325, m = 0.8, and n = 0.02. Lipanov This curve diﬀers from other curves because it has a clearly expressed maximum. The maximum value upr = 0.2 is reached approximately at p = 42, after which the burning rate decreases and passes through a minimum at p = 260: upr = 0.061. Thus, the burning rate decreases by more than three-fold in the pressure interval from 42 to 260. We observe that ν < 0 in the entire pressure interval (Fig. 4), and the minimum value ν = −1.18 is reached at p ∼ = 142. At p = 253, the dependence crosses the abscissa axis, and the parameter ν becomes positive. Then it tends to unity with a further increase in pressure. In the entire range of negΔp is smaller ative values of ν, the limiting deviation p 6 than n Di ; moreover, there is more than a two-fold i=1 diﬀerence at the point (−1.18; 142). If a propellant with such a dependence of the burning rate on pressure were created, it would become possible to fabricate a very light-weight SRM casing for a required maximum pressure corresponding to the maximum charge temperature. The ratio ϕ for this variant of calculations is shown in Fig. 3a (curve 4). With an increase in pressure, the function ϕ progressively increases. At pressures typical for each particular SRM, this function can be approximated either by the exponent ϕ = ϕmin ek(p−pmin ) , or by the second-order curve ϕ = ϕmin + a1 (p − pmin ) + a2 (p − pmin )2 . Let us consider the calculation variant with A = 0.3. The sought root is located between the pressure values of 140 and 150. Let us assume that pmin = 140, pmax = 150, ϕmin = 0.283, and ϕmax = 0.328. Then we obtain k = 0.0148. Let us write the equation ϕmin ek(p̄−pmin ) = A. Taking a logarithm, we obtain p̄ = pmin + A 1 ln . k pmin Thus, p̄ = 143.9, which corresponds to curve 4 in Fig. 1. To use a second-order curve, we need one more point in addition to two points with pmin and pmax . For this purpose, we use p̃ = 145 corresponding to ϕ̃ = 0.305. Let us denote B1 = ϕmax − ϕmin ϕ̃ − ϕmin , B̃ = . pmax − pmin p̃ − pmin Calculation of Pressure in a Solid-Propellant Rocket Motor 573 Finally, we can recommend two expressions for approximating the dependence upr (p): upr = u1 p + apm e−np and upr = u1 p + a(1 − e−mp )e−np . Fig. 5. Behavior of ν as a function of pressure for curve 5 in Fig. 1. Then, we have B1 − B̃ , a1 = B1 − a2 (pmax − pmin ). pmax − p̃ The pressure p̄ is found from the quadratic equation a2 = a2 y 2 + a1 y − (A − ϕmin ) = 0, where y = p̄ − pmin . Choosing the plus sign at the square root, we obtain p̄ = 143.9, which coincides with the solution obtained by using an exponent. Thus, there is no need to apply an approximate power-law function for determining the pressure in the SRM chamber. The pressure can be easily calculated by using the function ϕ, which is approximated by a straight line, by a second-order parabola, or by an exponent. Let us return to the analysis of Fig. 1. In addition to the four above-considered curved calculated by the formula u = u p + apm e−np , pr The diﬀerence between these expressions is the use of the function y1 = pm in the ﬁrst case and the function y2 = 1 − e−mp in the second case. Using the ﬁrst formula, one can describe all curves upr (p) except for curve 5. Using the second formula, one can describe all curves upr (p) except for curve 4. The solution of the entire problem can be obtained by using a system of two functions y1 (p) and y2 (p). Let us formulate the main conclusions of this work. 1. The real behavior the solid propellant burning rate as a function of pressure is more complicated than a power-law dependence. 2. Using the dependences of the burning rate on pressure, it is possible to obtain a function ϕ that monotonically increases with increasing pressure in the rocket pressure range and that can be approximated by an exponent, by a straight line, or by a second-order curve. 3. In view of conclusion No. 2, the pressure in the SRM as a function of the solid propellant parameters, charging conditions (charge temperature), and structural factors of the charge and engine can be calculated by using the expression for the real dependence of the burning rate on pressure. 1 it contains one more curve (curve 5). It is calculated by the formula upr = u1 p + a(1 − e−mp )e−np . Curve 5 diﬀers from curve 1 by using the function y2 = 1 − e−mp  instead of the function y1 = pm . The function y2 , similar to the function y1 , is equal to zero at p = 0, but it tends to unity as p → ∞. It is impossible to obtain a dependence with a clearly expressed maximum by using this function. Instead, it is possible to obtain curve 5 very slowly approaching u1 p. The following values of the coeﬃcients were used to construct curve 5: a = 0.092393, m = 0.034616, and n = 0.0011214. The parameter ν corresponding to this curve displays a rapid decrease at the beginning (Fig. 5) and then changes very slowly. In the pressure range within p = 250, it does not increase above 0.3 and does not decrease below 0.18. Concerning the function ϕ (curve 5 in Fig. 3b), its behavior is similar to the behavior of curve 4 in Fig. 1 in the region of decreasing pressure. This means that the simplest option of approximating this curve is to use an exponent. REFERENCES 1. R. E. Sorkin, Theory of Intrachamber Processes in SolidPropellant Rocket Systems (Nauka, Moscow, 1983) [in Russian]. 2. D. I. Abugov and V. M. Bobylev, Theory and Calculation of Solid-Propellant Rocket Motors (Mashinostroenie, Moscow, 1987) [in Russian]. 3. E. S. Ventsel, Probability Theory (Nauka, Fizmatgiz, Moscow, 1969) [in Russian]. 4. Internal Ballistics of Solid-Propellant Rocket Motors, Ed. by A. M. Lipanov and Yu. M. Milekhin (Mashinostroenie, Moscow, 2007) [in Russian]. 5. E. F. Zhegrov, Yu. M. Milekhin, and E. V. Berkovskaya, Chemistry and Technology of Double-Base Powders for Solid Propellants (Moscow State University of Printing Arts, Moscow, 2011) [in Russian]. 6. A. M. Lipanov, “Solid-Propellant Burning Rate As a Function of Pressure,” Fiz. Goreniya Vzryva 49 (3), 34–38 (2013) [Combust., Expl., Shock Waves 49 (3), 283–287 (2013)].