CC BY
24
YflK (519.9+518,5):532.54
S.S. Bondarchuk
MATHEMATICAL SIMULATION OF A LARGE SIZE ROCKET MOTORS
Tomsk State Pedagogical University
1. introduction
The methods of numerical simulation of processes in solid rocket motors (SRM) are now widely used at all stages of problem solving of their designing and improvement. Special importance of these problems gains in cases, when the full-scale experiment is expensive or gives the rather limited information. Designs of the newest propulsion systems, geometry of their charges and the engineering solutions frequently have non-traditional character. The charges of SRM are characterized by complexity of structure used propellants, containing a significant portions of the metal additives and explosives; have complex geometry of a combustion surface [1,2]. The areas of motion of combustion products in a number of cases require the 3-D description of multiphase (including burning metal particles) flow combustion products. A number of computational problems are connected to a design of charges and particular mission of engines. So, for example, one from effective applications large-sized SRM with a sectional charge is their use in quality start-accelerating engines [3]. The features of similar engines result in necessity to provide a number of the new requirements to pressurization (ignition time and dynamics of pressure raise in combustion chamber). The research activity was focused on large size engines with complex shape of charge bonded with the composite casing [1]. A wide set calculations was conducted on filling the elastic stagnant zones in under-cuff cavities of the rocket motors and evaluation of stress-strain deformed state of the propellant charge, on ignition of charge and transition to developed combustion, on erosion and burnout processes [1]. For the last process the account was made of decomposition of the liner material that may give undesirable gain in thrust at the moment of stages separation.
When launching a rocket from underwater in conditions of rather high pressure of an environment, the studying of processes, happening at it, is also actual and complex problem. With the purpose of reliability increase (equalization of a pressure differential between the combustion chamber and water environment) the practical organization of SRM start-up is designed with use of initially pressurizing the combustion chamber with inert gases from separate pressurized gas cylinders. There is another more effective method of pre-ignition or pre-launch super-
charges, using a low-temperature gas generator. One more way of reducing the gas charge and preventing excessive pressures is to fill the rocket motor with water [4-6]. In this activity such capability is analyzed also.
The computer codes developed are based on contemporary physical description of the governing processes and involve 2-D and 3-D mathematical models. A set of available codes provides opportunity to calculate gas dynamic and thermal fields in combustion chamber.
The software was tested and used in design and development of rocket motors with metal and composite casings, for selection of the igniter and refinement of the required characteristics of a charge, as well as for analysis of emergencies arising owing to a separation or burnout of inhibiting (thermal protection) layers at the end faces of a charge. The examples of the results of calculations and fundamental research are presented. They show typical design of the geometry of complex shape of propellant charge, which were used in different calculations of the motor efficiency and evaluation of operating parameters. In particular, the outcomes:
- of research of SRM stagnation zones filling;
- of calculations of 3-D flows in combustion chamber of SRM;
- of simulation of large engine ignition and pressurization;
- of modelling gasdynamics in packages of SRM;
- of analysis of the ignition characteristics of starting solid propellant motor initially filled with water are obtained and submitted.
2. Mathematical Model
The mathematical model of the processes described above is based on the integral equations of the gas dynamics derived from the laws of mass, momentum, and energy conservation. Additional special equations define the thermal and physical properties of the mul-ti- component combustion products from the igniter charge and the air which initially fills the chamber. This system of equations is applied to a reference volume, V, bounded by a closed surface comprised of both gas-permeable (A) and gas-impermeable (5) materials. Heat and mass are exchanged between gas flow and rocket motor elements. The change in the mass, momentum and energy in the control volume is
connected through volume interactions, the influence of the external surroundings on every of the mentioned quantities, and their transfer through A. This system of conservation equations is written below in the integral form with a generalized coordinate system [2, 7].
I
pdV + J pNciA = \YjMjdS,
d_ dt
dt
S j=l
I pRdV + J pRNdA = lY.RjMjdS,
s m
J PCpdV + J PCpNdA = \Y,CPjMjdS>
s M
dt dt
J pudV + j HdA + jndS+\pfdS = 0.
V ASS
JEdV + J(£ + p)NdA + J qdS =
J Z^jMjdS,
s M
N = (u,n),
II = pn + puN,
E
7-1
p • и
R
ck dt
■ F(c, T)
F(c,T)
Zc
Po
■exp
E„
where y - axis normal to propellant surface; T, rt/ c - propellant temperature, density and mass concentration; K - thermal diffusivity of solid propellant; Q - ratio of propellant explosion heat to propellant specific heat; Ea - ratio of propellant activation energy to universal gas constant; Z - pre-expo-nential factor.
Here, the rate of the chemical reactions is characterized by the Arrhenius relation. The effect of hot gaseous combustion products on the charge is simulated by establishing the boundary conditions of the third kind at the solid - propellant surface and by setting the heat flux at the infinity to be zero.
y = 0:a[Tg-T(0,t)] = -X
ô T
ây'
y = oo:
dT ду
О,
where t, p, p, u- time, pressure, density and gas velocity vector; R, Cp - gas «constant» and specific heat of gas components or mixture; y - adiabatic index (ratio of specific heats); n - unit vector of external normal to surface; q, f - heat flux density and friction stress; L - number of gaseous components of mixture; M,H- mass supply density and enthalpy of combustion product; j - subscripts individual features of the components.
One of the difficult problems to be solved in parallel with predicting the gas dynamic flow field is the surface heating and subsequent ignition of a solid - propellant surface element. It is assumed that the ignition of the surface element occurs instantaneously when the surface temperature reaches a certain value, at which point the heat flux to the propellant surface generated by the propellant combustion products becomes larger than the heat flux from the ignition system. In other words, the ignition time is based on the condensed -phase ignition model developed by Vilyunov, Merzh-anov, Freizer, Hicks and other authors [8]. Mathematically, this model of ignition of a condensed substance (solid propellant) involves the transient equation for thermal conductivity which also takes into account heat release term and chemical kinetics:
dT d2T —= K 2 + Q-F(c,T)
dt d y
where a - effective heat emission coefficient, which includes radiant and convective components; X, Tg -temperature of gas and thermal conductivity of solid propellant.
The thermodynamics of the gas mixture in the igniter (indestructible perforated casing) is described by averaged parameters. In this case, the mathematical model describing the variation of gas dynamic parameters in the volume of the igniter casing is a system of lumped or non-dimensional differential equations for interior free volume, W. It is also assumed that the entire surface, D, of the igniter grains is instantaneously ignited and that the perforated casing prevents the escape of these grains. The system of equations for the conservation of mass and energy and for variation in the free interior volume in the igniter then becomes:
d_ dt d_ dt
d_ dt
d_ dt
dW
i=l
(pRW)=fj(l-zl)RiM-RMt,
i=l
(pcpw)=±(i - z,.)cpiM-cpM.,
Г P
r-1 J Ti r-1 p
M„,
_ •yidco
dt ,=/ dt
where z - portion of condensed phase in combustion products.
The gas flow, M„ through the perforation holes, SI is calculated by the familiar quasi - stationary equations:
M*=Sl
y+1
ymr"' pp,
y-l
2npp^2/y 2
y + 1'
-¥
l/n
y/ <mn, m" <y/ <1,
m ■
¥ =
Pk
where pk is external (in the chamber) pressure.
The mass flow rate of gas is related to the change in the volume, w, of burning igniter material as follows:
da>i dt
i = 1,...,D.
The instantaneous volume of the pyrotechnic grains depends on the flame depth (mo/po > (o > 0, mo, pa - mass and density) and is the function of the grain shape. For a wide spectrum of shapes, the relationship between the instantaneous volume of these grains, o), and their sizes is very simple:
CO: ■
rife—<■)
FK
J=i
where e.. (j=l, 2, 3) is the initial geometric characteristic features of the /-grain.
The variation of typical sizes of a grain as a result of burning is described through the burn depth, e. is
de-
defined by the equations —- = , where r - burning rate. dt
These equations of motion were integrated a the second order modification to method developed by S. Godunov [7]. The mathematical problem in the ignition of a solid propellant was solved using an adaptable calculation grid that varied depending on the heating condition [5].
3. Calculation Results
3.1. Optimization of Large Rocket Motors Start
up. As the character of process on an initial period of the motor activity in many respects is determined in parameters of the igniter - it's design, thrust characteristics, nature of igniter compounds and arrangement, the set forth above factors were main object of researches [3]. Besides it is well known, that the level of the bursting pressure of the nozzle membrane Pd renders noticeable influence to dynamics of wave processes in the combustion chamber of the rocket motor and, therefore, on ignition of a charge of solid propellant. Therefore in calculations the pressure Pd also was varied parameter. A number of calcula-
tions results obtained during the analysis of optimum ignition of a charge, rate of rising of a combustion chamber pressure and motor thrust is shown below.
On Fig. 1 the relations p(t) in region of the front bottom are adduced. The curve 1 characterizes increase a combustion chamber pressure with two igniters. One from them (weight of the igniter charge is 25 kg) is in a head part of the motor; second was between fifth and sixth sections with igniter charge= = 10 kg. The purpose of use of two igniters was maintenance of faster ignition remote sections of a charge at active replacement and intermixing of an air, which was filling in the motor in pre-launch times. The bursting pressure of the nozzle membrane Pd for this version made 0.3 MPa. Curve 2,3 on this figure concerns to version one igniter, equipped double - based propellant and located in a head part of the combustion chamber. The device provides the program of linear
P, MPa
■4 4/ <v / 'if <£
V /
)4 w M
0.1 0.2 0.3 0.4
Fig. 1. Combustion chamber pressure
ts
increase of the consumption on the first half of an motor work with the consequent constants with value 50 kg/s. Pressure Pj= 1.5 MPa for version curve 2 and P= 1 MPa for a curve 3. The nonuniformity of character of submitted relations is caused both response to bursting of the nozzle membranes, and sharp, in some versions, connection to combustion of large elements of a charge surface. The latter is illustrated by curves (in the same denotations) time history of the propellant combustion area submitted on Fig. 2. More fluent character has the curve 4, appropriate to version with one igniter ensuring the constant gas supply during 50 kg/s during 0.2 s of all operating time. The nozzle membrane in this case is calculated on cutting at bursting pressure 0.3 MPa.
For this last version on Fig. 3 the distributions of gas dynamics parameters on length of a channel of the combustion chamber for various moments of time
0.16
0.08
t,s
1 / 4/
G, kg/s
1/3 2/3
Fig. 2. Combustion surface
x/L
7.2
5.4
3.6
1.8
P,MPa
U,m/s
• pressurt - velocity i
t=0.45s h \ r
jbO.OSs tf0.11s XV.
8 oo
m
400
2 00
3000
2250
1500
750
r ÀP ф у У .....-
У л / /1 G
Í 1/ V
Г/
I 1 1
P.kPa
450
300
150
0.1
0.2
0.3 t, s
■150
0 0.2 0.4 0.6 0.8 x/L
Fig. 3. Pressure and velocity distributions
are given. Large initial volume and, therefore, fair quantity of a cold air in the chamber, causes peak of the consumption (Fig. 4); i.e. the thrust variation has essentially nonmonotone character and is accompanied by sharp splash called replacement of a cold air and immediately by formed wave structure of flow. On the same Fig. 4 shown the averaged dependency of pressure drop between head and nozzle parts of the SRM AP~P(x=0)-P(X=L)- From the analysis of this dependency is seen that in initial moment of time in consequence of both a functioning an igniter and beginning of charge ignition axial pressure drop quickly increase to it maximal value (/»0.08 s). Here-
Fig. 4. Consumption and pressure drop
inafter, in consequence of sufficiently intensive wave processes, caused by the nozzle plug destruction and displacement of initial cool gas, with passing of rarefying wave from nozzle to head part of the charge pressure drop takes inverse character.
When the total surface of charge is connected to the combustion, second peak of axial pressure drop values is observed, after which pressure drop gradually and after small amplitude fluctuations, approaches to its quasistationary value.
3.2. Gasdynamics of Filling and Ignition of Rocket Motors' Elastic Stagnation Zones. Large-sized SRM in most cases have narrow unloading butt end cavities (stagnation zones) between the case of the motor and inhibited protection of the charge. Solution of the problem on the estimation of static strength of constriction on certain characteristics of the pro-pellant, frame, sizes, loads effecting the charge presents a considerable practical interest [9-11]. Indicated places with the absence of any defects present by themselves a danger for a charge from the point of view of strength because almost all of them experience a considerable tension by all kinds of loading. In real construction of SRM the ends of the clamp of such charges are made in a form of so called cuff functions (pliable elastic wedges), unloading special contact zones of propellant. Initial quantity of the width of cavity and its opening while filling of the internal volume can be estimated only on the basic of quite simple experiments (as a rule static loading) only partly reflecting the real process. Modelling of filling of stagnation zones in starting up period is interesting also from the point of view of investigation of temperature regimes of construction elements, which form the cavity, estimation of the quantity of changes of pressure between stagnation
Fig. 5. Calculating and standard experimental pressure -time diagrams
T,K
i/L=0.0
' l/L=0.33
l/L=0M
l/L=1.0
0 0.05 0.10 t,S
Fig. 6. Temperature of cuff surface
zones and motor chamber including the analysis of possibly abnormal development of the process because of the exfoliation of the cuff, appearance of longitudinal charge vibrations, etc.
The process of filling of stagnation zone of large-sized SRM has been examined, the basic parameters of which are the following: diameter 2.3 m; free volume of combustion chamber 0.9 m3; volume of stagnation zone 0.04 m3; butt end of the charge and bottom of the case are covered by rubber-like heat-protecting material.
Calculating and standard experimental pressure -time diagrams are demonstrated on Fig. 5. It is seen from the conduct of curves that after reaching of the
ignition device the working time i«0.05 s, the pressure in chamber increases gradually and then decreases a little with opening of a nozzle plug. From the moment /-0.045 s the ignition of the working surface of the charge starts, leading to the sharp increase of pressure and practically reaching the regime quantity to i«0.13 s. The same Fig. demonstrates the dependence of the velocity of combustion products' flowing into the place of stagnation zone. On time the behaviour of the curve follows the change of pressure in motor chamber.
Thus till the moment /«0.02 s velocity is increasing approximately to 120 m/s. Then, because of compression of cold gas, initially filling the cavity, velocity decreases gradually but from the time moment f«0.05 s it decreases faster because of depres-surization in combustion chamber on account of opening of nozzle. Further sharp increase of pressure («60 MPa/s) leads to no less sharp increase of velocity of flowing and considerable gas compression in the deadlock end of stagnation zone, that causes starting from the moment t~0.1 s the decrease of flowing velocity. By the time of motor's going out to the stationary regime, the flowing velocity is about 35 m/s and falls down non - symptomatical-ly due to the cooling of the gas in cavity because of the heat exchange with surfaces of bottom and cuff.
A considerable part of a calculating period of time of distribution velocity along the surface of stagnation zone has a diminution character towards the deadlock end of the cavity. This fact and also the accumulation of initial cold gas at the end of stagnation zone condition the behaviour of dependencies of temperature of cuff surface on time, demonstrated on Fig. 6. The curves are represented for four approximately equidistant elements of the cuff surface, starting from the entrance part of the cavity (l/L=Q) to a deadlock one (l/L-l).
Three-dimensional flow was simulated for a similar large SRM with a slotted-tube charge during motor transition to a steady-state operation and filling of nozzle stagnation zone. Isobars in Fig. 7 show the pressure distribution in the combustion chamber. The pressure magnitudes are related to a reference value determined from mass and energy balances in the SRM chamber under critical conditions of isoentropic flow of gases through the nozzle. Fig. 7 shows that the nozzle inlet recessed into the combustion chamber is approached, the relatively equal distances between isobars are reduced and their profile is distorted due to a complex interaction of counter flows. A high-pressure region is formed in the zone of the encountering flows. Then, when the motor aft end is approached, the distances between isobars increase near the nozzle; the pressure difference approaches zero in the stagnation zone at the aft end within the SRM casing. The same figure shows the velocity distribution corresponding to the pressure distribution in the zone of the recessed noz-
Fig. 7. Distribution of velocity and pressure in combustion chamber of a large-size motor
Fig. 8. Velocity distribution in cross section of a slotted-tube charge
Fig. 9. Flow over end (nozzle) closure
zle. As the inlet of the recessed nozzle is approached, the profile (practically uniform in the channel and uniformly inclined in the slot) is deformed, particularly in the regions in which the gas efflux from the slot to the channel.
Fig. 8 shows the distribution of the transverse velocity of gas in the cross section of A-A. The flow pattern (velocity vector field) that occurs during filling of the stagnation zone in the vicinity of the aft closure due to pressure rise in the combustion cham-
ber is shown in Fig. 9. Solid lines indicate the disposition of slot axes in the charge. The flow pattern is determined by the more intense compression of gas flow in those regions near the slots.
3.3. Ignition Pressure Transient in Solid Rockets Initially Filled with Water. When launching a rocket from under water, or from the surface, conventional ignition methods can easily generate excessive pressures in the combustion chamber and in any surrounding structures. This problem is currently avoided by
initially pressurizing the combustion chamber with inert gases from separate pressurized gas cylinders. There is another more effective method of pre-igni-tion or pre-launch supercharges, using a low-temper-ature gas generator. However, if the combustion products have a temperature above 100 °C, they may produce an unplanned ignition of the rocket motor pro-pellant. Preliminary experiments are needed to prevent this accidental ignition. One more way of reducing the gas charge and prevent excessive pressures is to fill the rocket motor with water [1,4, 5]. This method requires proper studying as the contact between the water, gas charge and propellant surface may change the propellant's ignition characteristics as a result of propellant ingredients being leached [5]. Direct contact of the seawater with the propellant surface can be excluded by using a thin protective layer.
The motion of the boundary between gas and water inside the motor is coupled to the motion of the water through the nozzle area, Sr The rate water flow through the nozzle is determined by the difference between instantaneous gas pressure and exterior water pressure p through the following formula:
P, MPa
7.5
d ~dt
jdV = S. -(p-p.).
The numerical comparative analysis of local and integral parameters of the physical processes were made for a motor with the following general characteristics: 3 m in length, 16 m2 propellant surface area; 2.2 m motor diameter; 0.16 m nozzle throat diameter. The propellant's combustion temperature was assumed to be 3340 K. The igniter combustion products weighed 2.5 kg, contained up to 50 % of condensed phase material and had a temperature 3320 K.
The analysis of igniting motors with their chamber initially filled with water is illustrated in Figs. 10—13. Fig. 10 shows the pressure transient while Fig. 11 shows the corresponding thrust transient for different external pressures. The fundamental difference produced by the water is rather high level of the motor thrust at the very initial moments of igniter operation caused by water displacement since it's density is several orders higher than the gas density. For all the conditions studied, the following pattern characterizes the evolution of the ignition process. The igniter starts working at the high initial pressure over a small free volume. As a result, the gas pressure and temperature increase sharply. Simultaneously with a gradual displacement of the fluid, the charge surface is exposed to intensive heating and propellant ignition starts practically immediately (in the case of an external pressure 2.0 MPa, it is even a little earlier) after displacing water (¿«0.0023-S-0.0025 s). Accordingly, the initial pressure in the combustion chamber can somewhat exceed the steady - state value which is achieved after a complete combustion of an igniter.
4.5
3.0
1.5
w/wf
S/2Q
5/ -ree vc iepenc ilume fence -
0.0 1.0 1.5
2.0
t,msj
O 5 10 15 20 25 t,ms
Fig. 10. Pressure and free volume dependence's
0.8
0.6
0.4
0.2
0.0
U, n 1fs/ *=O.C 03
dis 'eioct tribu y ^ Hon ^=2 t=1rr 'Oms' !S
f t=5ms£
t=2m s N
\f=0 5ms
0.2 0.4 0.6 0.8 x/Lj
0 5 10 15 20 25 t,ms
Fig. 11. Thrust dependence and velocity distribution
Fig. 12 shows the time evolution of the charge's burning surface. The time of initialing ignition and the time to ignite the surface completely for these conditions is slightly less (/iteO.OOl s) than in the case of starting up the motor after preliminary gas pres-surization. In Fig. 10, the time dependencies of changing motor free volume during the startup period are given. It is seen that higher levels of external pressure require more time to fill the chamber, though, the differences are generally insignificant. Figures 11 and 13 illustrate the spacial distribution of the gas velocity and pressure at different times for an external pressure 1 MPa. The characteristics of the wave processes during the motor's transition to the steady state regime are slightly «softened» because of the moving surface of the displaced liquid.
1.0
0.8
0.6
0.4
0.2
0.0
S/Sg
15
20 j J
10
3 4 t.ms
Fig. 12. Combustion surface
7.5
6.0
4.5
3.0
1.5
P, MPa
t-10m s t=5ms
t~20m s
t=0.5 m:
i=2ms
t=3ms
0.2
0.4
0.6
0.8 x/L
in free volumes of motor combustion chambers. The main design characteristics of rocket motors were the following: nozzle throat radius 0.0119 m; free volume of the combustion chamber 0.04 m3; L = lm -length of the gas pipe, linking motors; 5^=0.1 m2 combustion surface.
On Fig. 14 the curve of flow velocity distributions in the gas pipe in various moments of time are adduced. In the given version in one from motors weight of the igniter made 0.24 kg, in the friend - 0.1 kg. Various weights of igniters cause both significant
U, m/s
360
270
180
90
t=0.02s
b=0.02s t=0.3s
t=0.5s
0.0
0.2
0.4
0.6
0.8
x/L
Fig. 13. Pressure distribution
3.4. Gas Dynamics Modelling of SRM Packet transitions to the steady-state regime. The units of a various kind of propulsion systems, auxiliary and other systems can be designed in versions with application of gas dynamics communication between motors. The theory and the techniques of calculation of a motor's packet with gas dynamics communication were offered in the supposition or about instantaneous overflow, or at the rather approximate (isothermal) model of gas flow in the gas pipe. The not-con-sideration of combustion products dynamics eliminates from consideration such problems, as determination of the time characteristics of any fluctuations in one from motors on activity of a packet, influence of friction and heat losses in the area of the gas pipe, its thermal operation etc.
The mathematical statement of a problem leans on the above-stated set of equations and is carried out within the framework of calculation of an one -dimensional flow in gas pipe at averaged parameters
Fig. 14. Velocity distributions in the gas pipe (different weight of the igniter)
velocities of combustion products overflow and earlier ignition of propellant in the motor with greater weight of the igniter. After connection to combustion of a working surface of charges of solid propellant owing to gradual equalization of pressure to time ¿«0.44-0.5 s the overflow of gas from the motor in the motor stops.
For various initial combustion surfaces (0.12 m2 and 0.1 m2) on Fig. 15 the curve distributions of gas flow velocities along the gas pipe in various moments of time are given. This version is characterized by that the motor with a smaller working surface of a charge owing to smaller heat losses in this surface is fired faster.
The earlier connection of a charge to combustion causes heavily overflows of gas from one combustion chamber in other 0.2 s) during time between full ignition that and other rocket motor. After the combustion envelops charges of both motors, the mode of overflow of combustion products from rocket motor with the greater working surface of a charge in other (i«0.5 s) with velocity of the order 15+20 m/s is settled. On Fig. 16 for the
U, m/s
P, M Pa
200
20 0
-20
-40
—\ t=0.2s
-
t=0.1s ^OJ02s
t=P.4$
t=0.5s
0.0
0.2
0.4
0.6
0.8
x/L
Fig. 15. Velocity distributions in the gas pipe (different initial combustion surfaces)
given version curve «pressure-time» at the rocket motors packet transition to the steady-state regime with gas connection (curves 1 and 2) and without it (curves 3 and 4).
On Fig. 17 the limits of pressure variation in motors are shown at variations of such data-ins as a surface of combustion, constant of the law of burning rate, consumption coefficient, free volume of the motor and weight of igniter. The range of parameters between curve 1 and 2 belongs to motors without gas connection, shaded (between curve 3 and 4) - if there is the gas pipe between rocket motors. From the analysis of the diagram it is visible, that for the given set of variations of data the maximum difference of pressure between motors owing to gas connection is reduced approximately on 25 %. Besides we shall note, that in difference from models of a rocket motors packet work with instantaneous overflow of gas, for the given version dispersion of pressure in motors makes of the order 40 % from regime value.
4. Conclusion
On the bases of presented universal, developed physics-mathematical model of internal chamber processes a number of numerical researches concerning analysis of particularities of the large-sized rocket motor transition to the steady-state regime was
3.75
2.50
1.25
- with i conn — withe i gas 3/ ection / wt it /^
È - Jf **
H
0.2
0.4
t,s
Fig. 16. The curve «pressure - time» (different initial combustion surfaces)
P, MPa
X1
t I î
; f fy w
m**
O O.I 0.2 0.3 0.4 O.S t,S
Fig. 17. The limits of pressure variation in rocket motors packet
carried out. A number of significant SRM problems was solved concerning practical design, which is also a scientific interest. Analysis of calculation results has shown a high degree of their adequacy and correspondence to real intrachamber SRM processes.
References
1. Velichko I.I. et al. Fresh Approach to Solid Rocket Motor Design // Journal of Propulsion and Power. V. 11. 1995. № 4.
2. Bondarchuk S.S., Vorozhtsov A.B., Kozlov E.A., Feshenko Y.V. Analysis of Multidimensional and Two-Phase Flows in Solid Rocket Motors // Journal of Propulsion and Power. V. 11. 1995. № 4.
3. Bondarchuk S.S., Vorozhtsov A.B. Mathematical Simulation of a Large Size Rocket Motors // Proceedings of the 3-rd International High Energy Materials Conference and Exhibit HEMCE-2000 (6-8 December 2000). Thiruvananthapuram, India.
4. Bondarchuk S.S. et al. Ignition Pressure Transient in Solid Rockets Initially Filled with Water // Journal of Propulsion and Power. V. 15. 1999. №6.
5. Bondarchuk S.S. et al. Solid Propellant Underwater Ignition Modelling // AIM Paper 99-0864 (A99-16700), AIAA, Aerospace Sciences Meeting and Exhibit, 37m, Reno, NV (11-14 Jan. 1999).
6. Bondarchuk S.S. et al. Methods for Recovery of Ingredients from Solid-Propellant Rocket Motors // The First International Symposium on Special Topics in Chemical Propulsion and Combustion of Energetic Materials, 18-22 June, Stresa (Lake Maggiore), Italy.
7. Godunov S.K. (ed) Numerical Solution of Multidimensional Problems of Gas Dynamics. M., 1982.
8. Vilyunov V.N., Zarko V.E. Ignition of Solids. Elsevier, Amsterdam, 1989.
9. Bondarchuk S.S., Vorozhtsov A.B. Gasdynamics of filling of elastic stagnation zones of large-sized solid rocket motors // 30*1 International Annual Conference of ICT (Energetic Materials) (29 June - 2 July 1999). Fraunhafer - Institute Chemische Technologie. Karlsruhe, Fed. Rep. of Germany.
10. Bondarchuk S., Vorozhtsov A., Salko A., Kondratova O. Mathematical Simulation of Airbag Inflation by Low Temperature Gas Generator Products // An International Journal Propellants. Explosives, Pyrotechnics. V. 25, Issue 5, 2000. (Published Online: 3 Nov. 2000 WILEY-VCH Verlag GmbH, Weinheim, Fed. Rep. of Germany).
11. Vorozhtsov A.B., Bondarchuk S.S., Salko A.E., Zima V.P. Inflation of Elastic Airbag under Wather - Experiment and Calculations // Energetic Materials: Ignition, Combustion and Detonation. 32nd International Annual Conference of ICT. (3 July - 6 July 2001). Karlsruhe, Fed. Rep. of Germany.
УДК (519.9+518.5):532.54
B.B. Жолобов*, Е.И. Тарновскии
МОДЕЛИРОВАНИЕ НЕУСТАНОВИВШИХСЯ ТЕЧЕНИЙ УГЛЕВОДОРОДНЫХ СМЕСЕЙ
В ТРУБОПРОВОДАХ
"Управление магистральных нефтепроводов Центральной Сибири, Томский государственный педагогический университет "Томский политехнический университет
Трубопроводный транспорт, являясь доминирующим видом перемещения энергоносителей, играет особую роль для нашей страны. Современная наука и техника не нашли альтернативы трубопроводам, поэтому их значение в ближайшие десятилетия будет лишь возрастать. Эксплуатация трубопроводных систем порождает большое число вопросов, правильно ответить на которые можно, опираясь только на научные проработки. Для теоретического исследования течения в трубах, как правило, используются модели механики сплошных сред. Физические законы сохранения, примененные к элементарному объему транспортируемого флюида, приводят к определяющей системе дифференциальных уравнений, гибкость которой обеспечивается введением ряда параметров, идентифицируемых впоследствии по диспетчерской информации. Неустановившиеся напорные и безнапорные течения в гидравлическом приближении описываются системами дифференциальных уравнений в частных производных гиперболического типа [1].
Уравнения в частных производных этого типа достаточно хорошо изучены и разработан ряд эффективных численных методов для их решения. Привлекательной стороной такого подхода является универсальность и возможность адаптации к реальным условиям. Традиционно напорные и без-
напорные течения в трубопроводном транспорте рассматривались изолированно. Это, по-видимому, связано с тем, что сопряженное решение задачи о возникновении и схлопывании паровых или парогазовых полостей слабо поддается аналитическому анализу. Имеются лишь отдельные разрозненные публикации, в которых изучается поведение изолированных газопаровых полостей в ограниченных объемах жидкости. В случае существенного изменения давления при движении углеводородных смесей и при неустановившихся процессах возникают фазовые переходы, приводящие к образованию паровых и парогазовых объемов различного масштаба (от изолированных пузырьков в объеме жидкости до протяженных участков с неполным заполнением поперечного сечения трубы). Полости, возникающие при смене режимов перекачки, при восстановлении исходного режима могут исчезать (схлопываться) в том случае, если они заполнены только паровыми компонентами углеводородной жидкости. Различие в механизмах выделения и растворения газовых компонентов приводит к тому, что для растворения газовых компонентов необходимо уже настолько больше времени, что реализуются режимы течения с неполным заполнением трубы при давлениях, существенно превышающих давление насыщенных паров жидкости.
