ISSN 00125016, Doklady Physical Chemistry, 2009, Vol. 428, Part 1, pp. 178182. Pleiades Publishing, Ltd., 2009.Original Russian Text S.A. Rashkovskii, Yu.M. Milekhin, A.V. Fedorychev, I.G. Assovskii, 2009, published in Doklady Akademii Nauk, 2009, Vol. 428, No. 2, pp. 206210.

178

For motion control in rocketry, adjustable multimode solidpropellant rocket motors (SPRM) arewidely used [1]. Analysis of various designs of adjustable multimode SPRM showed that the most efficientpropulsion systems (PS) are variablethroat PS usingpropellants with high sensitivity of the combustion

rate to a change in pressure: = 1. Combus

tion of propellants with such in a semiclosed space atconstant nozzle throat area is fundamentally unstable[2]. The stable operation of adjustable PS with 1 isensured by using a pressure stabilization system (PSS)[3], which varies the nozzle throat area, depending onthe programmed pressure, thus compensating for thegrowth of perturbations because of the combustioninstability at 1.

PSS widely uses a combination of proportional andintegral laws of change in the nozzle throat area [3]:

(1)

where p0 is the programmed pressure in the combustion chamber, 0 is the nozzle throat area corresponding to this pressure, t

is the delay time of PSS actua

tors with respect to the moment of measuring the pressure p, and a and b are constant parameters of PSS.

The proportional component in control law (1)controls the pressure for its deviation in quasistationary and transition modes. The integral component in

uln pln

t( ) 0 a p t t( ) p0[ ] b p t( ) p0[ ] t,d

0

t t

+ +=

control law (1) compensates for the statistical error ofcontrol.

Experiments showed that operation of such SPRMcan be accompanied by lowfrequency instability ofcombustion, which leads either to oscillations or tostoppage of PS [3].

The theory of lowfrequency instability in constantthroat SPRM using propellants with < 1 [47]is inapplicable to the SPRM under consideration for atleast two reasons. The first is that the combustion ofpropellants with 1 at constant nozzle throat area isfundamentally unstable even without unsteady combustion effects, and the second is that the processinstability is determined by a combination of dynamiccharacteristics of the combustion chamber, propellant, and PSS. The latter means that criteria of lowfrequency instability of combustion in adjustableSPRM at 1 should significantly differ from thosefor constantthroat SPRM using conventional propellants ( < 1); in particular, these criteria should dependon the PSS parameters a and b in law (1).

The purpose of this work is to study the lowfrequency instability of combustion in SPRM with PSSfor propellants with 1.

MATHEMATICAL MODEL

Unsteady combustion of propellants is characterized in terms of a phenomenological model [4, 5] withconsideration for the Mache effect. The Mache effectis described using a general zerodimensional model[6, 7] as applied to endburning charges under pressure stabilization law (1).

The parameters corresponding to the programmedpressure p0 satisfy the equation

(2)

where u0 = u(p0) is the steady combustion rate at thepressure p0, is the propellant density, S is the chargecombustion surface area, and A0 is the exhaust coefficient [6, 7].

u0S A0p00,=

PHYSICALCHEMISTRY

Stability of Combustion in SolidPropellant Rocket Motorswith Pressure Stabilization System

S. A. Rashkovskiia, Corresponding Member of the RAS Yu. M. Milekhinb,A. V. Fedorychevb, and I. G. Assovskiic

Received April 1, 2009

DOI: 10.1134/S001250160909005X

a Ishlinskii Institute of Problems of Mechanics,Russian Academy of Sciences,pr. Vernadskogo 101/1, Moscow, 119526 Russia

b Soyuz Federal Center for Dual Technologies Federal StateUnitary Enterprise, ul. Akademika Zhukova 42,Dzerzhinskii, Moscow oblast, 140090 Russia

c Semenov Institute of Chemical Physics,Russian Academy of Sciences,ul. Kosygina 4, Moscow, 119991 Russia

DOKLADY PHYSICAL CHEMISTRY Vol. 428 Part 1 2009

STABILITY OF COMBUSTION IN SOLIDPROPELLANT ROCKET MOTORS 179

In terms of small deviations from unperturbedparameters (2) for the PS under consideration by analogy with published results [6, 7] under law (1), weobtain the equation

(3)

where

(4)

are dimensionless constants characterizing PSS; theother notation is as previously [6, 7]:

where W is the combustion chamber volume; is thethermal diffusivity of the propellant; C is the residencetime of the gas portion in the combustion chamber (inthe unsteady process, C is a function of time); is thetemperature gradient in the propellant near the combustion surface; R and n are the gas constant and theadiabatic index of the combustion products, respectively; TB is the propellant combustion temperature;the parameters with the subscripts 0 describe thesteady process; and the parameters without the subscript characterize the unsteady process.

Substitution of perturbations in the form () =1exp(), v() = v1exp(), () = 1exp(), and() = 1exp() into Eq. (3) gives

(5)

dd n 1+

2n ( ) 1

2 n 1

n

( ) =

12 ( ) ( )

+ Q ( ) B/( ) *( ) *d

0

+ ,

Qap00, B

bp0p0

= =

pu0

2

,

TBln pln

, TBln ln

p

,= = =

pW

A0RTB0,

u02

t, v u

u0 1,= = =

0 1, p

p0 1,

Cp 1,= = =

1n n 1+

2n n 1

n

1 12 ( )exp

+

+ Q B+

( )exp 1

v1 112 ( )exp

1 0.=

Equation (5) is completed with equations ofunsteady combustion theory [4, 5]: it is easy to show[6] that the characteristic equation takes the form

(6)

Here, the parameter y is defined by the expression

(7)

where k, r, , and are standard parameters ofunsteady combustion theory [4, 5].

Parameter (7) differs from a similar parameter intro

duced before [6] by the term Q + exp().

Characteristic equation (6) determines a range ofPS parameters at which the process is stable at any parameter values, including 1. In particular, atfixed parameters of the propellant and PS in Eq. (6),one can find a range of PSS parameters B and Q atwhich the PS operation is stable.

For PS with PSS, there are two types of lowfrequency instability. The instability of the first type mayoccur in largevolume combustion chambers ( 1),in which unsteady combustion effects are insufficient.Criteria obtained at 1 determine a parameterrange at which such PS can, in principle, operate in astable mode. The instability of the second type isrelated to unsteady propellant combustion and occursin smallvolume combustion chambers, in which thecharacteristic time of the combustion chamber, p, iscomparable to the characteristic time of relaxation of

a heated propellant layer, , i.e., ~1.

PROCESS STABILITY WITHOUT CONSIDERATION FOR UNSTEADY

COMBUSTION EFFECTS

Let us first consider the stability of the process inPS without consideration for unsteady combustion butwith consideration for the Mache effect and the delayin PSS. In this case, p is conveniently used as a timescale and the corresponding dimensionless frequencyis denoted by '.

Since unsteady combustion is not taken intoaccount, the process is described only by Eq. (5). Thiscase formally corresponds to the limit of Eq. (5) at

(8)

1 4+ 12

ky ( ) 1 12 ( )exp

+

k r 1+( )y +

1 4+ 1+2

ry +( ) 1 12 ( )exp

+ = 0.

y 1n n 1+

2n n 1

n

1 12 ( )exp

+=

+ Q B+

( ),exp

B

u02

1, 0, v1 1.=

180

DOKLADY PHYSICAL CHEMISTRY Vol. 428 Part 1 2009

RASHKOVSKII et al.

Thus, we obtain

(9)

Let us analyze the case where = 0 (no delay).

For imaginary values ' = i0, Eq. (9) is decomposedinto two:

(10)

(11)

The particular case where B = 0 is of interest if control law (1) involves only a proportional component.Then Eq. (11) takes the form

and has the solution 0 = 0, which corresponds to acritical value of the parameter Q:

(12)

The PS operation is stable if the following inequality is valid:

(13)

If B 0, the set of Eqs. (10) and (11) determines thestability boundary in the (B, Q) plane. In the particularcase where n = 1 and = 0 (isothermal process in thecombustion chamber and constant temperature of thecombustion product), Eqs. (10) and (11) have thesolution

(14)

(15)

As previously, the stability range is determined bycondition (13).

We performed parametric studies of Eqs. (10) and(11) at different values of n, , and . The boundary ofthe stability range was sought as the function Qcr(B).As follows from expression (12), if the integral component in control law (1) is absent (B = 0), then Qcrdepends only on and . Introduction of an integralcomponent into control law (1) leads to a change inQcr: the function Qcr(B) behaves differently at differentvalues of and n. At = 0 and n = 1.24 (the Macheeffect is taken into account and the temperature of thecombustion products is independent of pressure), anincrease in the parameter B causes a decrease in thecritical value Qcr; i.e., introduction of an integral component into control law (1) stabilizes the process. At

1n' n 1+

2n n 1

n

1 12 '( )exp

+

+ Q B'+

'( )exp .=

n 1+2n

n 1n

1 1

2 0cos

Q+ ,=

1n0

12 n 1

n

0B

0sin 0.=

1n0

12 n 1

n

0sin=

Qcr 1( )2.+=

Q Qcr.>

Qcr 1,=

0 B.=

= 0 and n = 1, the critical value Qcr is determined byformula (14) and independent of the parameter B.A similar result (with another Qcr value) was observedat = 0.25 and n = 1.24: the dependence Qcr(B) is veryweak and introduction of an integral component intocontrol law (1) neither increase, nor decrease the stability of PS operation in the steady mode.

Calculations demonstrated that monotonicallyincreasing dependence of propellant combustion temperature on pressure has a destabilizing effect. Thisalready follows from relation (12). At large , e.g., =0.25, the PS operation is more stable than in the casewhere control law (1) involves only a proportionalcomponent.

Thus, at = 0, an increase in and a decrease in

n lead to destabilization of PS process at > 1.

The dimensionless frequency of pressure oscillations in the combustion chamber of PS at the stabilityboundary is dependent on the parameter B and, as follows from Eq. (11), independent of . Calculationsshowed that all the solutions of Eq. (11) are close toparabolic dependence (15) and are weakly dependenton the parameters and n.

Let us consider the effect of delay in PSS on the stability of PS operation:

0. In this case, Eq. (9) is

conveniently rewritten as

(16)

At ' = i0, Eq. (16) is a parametric representationof the boundary of the stability range of PS operationin the (B, Q) plane.

Equation (16) was solved at various combinationsof the parameters , , n, and

. Some characteristic

calculation results are presented in Fig. 1.

Note that the functions 0(B) in the case considered are also close to parabolic dependence (15) andweakly dependent on , , n, and

.

Analysis showed that an increase in the parameter leads to noticeable destabilization of the process in

PS; moreover, it is seen that, with an increase in theparameter

, the integral component in control law

(1) has a destabilizing effect. In this case, the Qcr valuescalculated without consideration for the Mache effectmay be several times smaller than those calculatedwith consideration for the Mache effect.

As in the case where = 0, an increase in the

parameter and a decrease in the parameter n, otherconditions being the same, destabilizes the process,with the change in affecting more strongly than thechange in n.

Q B'+ 1

n' n 1+

2n n 1

n

+=

1 12 '( )exp

'( ).exp

DOKLADY PHYSICAL CHEMISTRY Vol. 428 Part 1 2009

STABILITY OF COMBUSTION IN SOLIDPROPELLANT ROCKET MOTORS 181

PROCESS STABILITY WITH CONSIDERATION FOR UNSTEADY COMBUSTION EFFECTS

In this case, the stability boundary in the (B, Q)plane is found by solving the complete equation (6).

As shown above, in the models taking into accountthe Mache effect, the integral component in controllaw (1) plays a destabilizing or, at least, neutral role.Therefore, of the most interest is the case where B = 0.The results described below present this case. Calcula

tions were performed at r = , = 0, and = 0; the

results at other values of r, , and are basically thesame. The calculation results are given in Fig. 2. Thestability boundary is shown in coordinates (, k),which are characteristic of unsteady combustion theory [4, 5]. Above and below the boundaries are stabilityand instability regions, respectively.

Figures 2a and 2b describe an ideal control systemin which

= 0. Other conditions being the same, an

increase in the parameter Q leads to expansion of thestability region and consideration of the Mache effectcauses narrowing of the stability region in the coordinates (, k). Calculations showed that, at = 1.0, thedependence of the frequency 0 on the parameter k isnot monotonic but has a minimum.

At B = 0, the minimal critical value of the parameter Q with consideration for unsteady combustioncoincides with value (12) obtained without consideration for unsteady combustion.

Figure 2c presents the calculation results at various 0. The stability region narrows with an increase in

the parameter , with the process eigenfrequency

depending weakly on the parameter value. Thus, it

13

is seen that unsteady propellant combustion has only asecondary influence on the stability of the system if

0 and has a decisive effect if = 0.

0.1

0 0.5

0.2

1.0 1.5 2.0B

0.3

0.4

0.5

0.6Q

1

2

3

4

Fig. 1. Stability boundary of the process in PS without consideration for unsteady combustion in coordinates (B, Q)at = 1.1, = 0, n = 1.24 and

= (1) 0, (2) 0.1, (3) 0.2,

and (4) 0.3.

Fig. 2. Stability boundary of the process in PS with consideration for unsteady combustion in coordinates (, k) atthe following parameter values: (a) = 1.0,

= 0, B = 0,

= 0, n = 1, and Q = (1) 0.1, (2) 0.2, (3) 0.5, (4) 1.0, (5)5.0, and (6) 10.0; (b) = 1.1,

= 0, B = 0, = 0.25, n =

1.24, and Q = (1) 0.23, (2) 0.25, (3) 0.3, (4) 0.4, (5) 0.5,and (6) 1.0; and (c) = 1.1, Q = 0.5, B = 0, = 0.25, n =1.24, and

= (1) 0, (2) 0.1, (3) 0.2, (4) 0.3, and (5) 0.4.

0.5

0 0.5

1.0

1.0 1.5 2.0 2.5k

1.5

2.0

2.5

3.0

3.5

4.0

1

2

3

45

0.5

0

1.0

1.5

2.0

2.5

3.0

3.5

1

23

4

5 6

0.5

0

1.0

1.5

2.0

2.5

3.0

3.5

4.0

1

2 3 45 6

()

(b)

(c)

4.0

182

DOKLADY PHYSICAL CHEMISTRY Vol. 428 Part 1 2009

RASHKOVSKII et al.

CONCLUSIONS

Thus, for the first time, we have developed a mathematical model of unsteady processes in the combustion chamber of SPRM with PSS that is controlled bya proportionalintegral control law. The model takesinto account unsteady propellant combustion effectsand the Mache effect. We have performed a parametricstudy of the stability of various modes of SPRM operation with consideration for the PSS parameters,unsteady combustion effects, and the Mache effect.Criteria and limits of stable operation of SPRM withPSS have been determined.

We have shown that the role of unsteady combustion of the filler in the emergence of instability is secondary if there is a delay between the moment of measuring the pressure and the actual moment of creationof a required nozzle throat area.

We have established that, if there is a time delay inPSS, the role of the integral component in the controllaw may be reversed: from stabilizing to destabilizing.The results obtained have shown that an intraballisticmodel of the process in SPRM with PSS must takeinto account the Mache effect; otherwise, the values ofthe critical parameters of the control system at which

instability of the process in PS emerges may be significantly underestimated.

REFERENCES

1. Bobylev, V.M., Raketnyi dvigatel tverdogo topliva kaksredstvo upravleniya dvizheniem raket (SolidPropellantRocket Motors as Means of Motion Control in Rocketry), Moscow: Mashinostroenie, 1992.

2. Orlov, B.V. and Mazing, G.Yu., Termodinamicheskie iballisticheskie osnovy proektirovaniya raketnykh dvigatelei na tverdom toplive (Thermodynamic and Ballistic Foundations of Design of SolidPropellant RocketMotors), Moscow: Mashinostroenie, 1979.

3. Ivanov, S.M. and Tsukanov, N.A., Fiz. Goreniya Vzryva,2000, vol. 36, no. 5, pp. 4556.

4. Novozhilov, B.V., Nestatsionarnoe gorenie tverdykhraketnykh topliv (Unsteady Combustion of Solid Propellants), Moscow: Nauka, 1973.

5. Zeldovich, Ya.B., Leipunskii, O.I., and Librovich, V.B.,Teoriya nestatsionarnogo goreniya poroka (Theory ofUnsteady Combustion of Gunpowder), Moscow:Nauka, 1975.

6. Assovskii, I.G. and Rashkovskii, S.A., Fiz. GoreniyaVzryva, 1998, vol. 34, no. 6, pp. 5258.

7. Assovskii, I.G. and Rashkovskii, S.A., Fiz. GoreniyaVzryva, 2001, vol. 37, no. 3, pp. 8393.

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