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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 [1]
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)
[2], 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;
aml35@yandex.ru
Fig. 1. Calculated dependences upr (p): curve 1 is borrowed from [2]; curves 2 and 5 are constructed on the
basis of the data [5]; curve 3 is similar to that in [5];
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 [3]
in processing experimental data obtained, e.g., in a
constant-pressure vessel [4].
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 [1]:
ϕ2 B(k)Fth p
.
RTp
ν=
where
6
δXi
whence it follow that
Spr ρpr upr (p)f (Tch ) =
Following [1], 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 [5]. 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 [5]. 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 [6] 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)].
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