MODELING OF CONSTANT VOLUME COMBUSTION OF PROPELLANTS FOR ARTILLERY WEAPONS

Transcription

1 MODELING OF CONSTANT VOLUME COMBUSTION OF PROPELLANTS FOR ARTILLERY WEAPONS Lt.col. lect. dr.ing. Daniel ANTONIE, Col.(r). rof.as. dr.ing. Sorin GHEORGHIAN, Col.(r). C.S.II dr.ing. Nicolae MĂRUNȚELU MILITARY TECHNICAL ACADEMY, BUCHAREST, ROMANIA Abstract: Combustion of artillery roellants is a very comlex hysical and chemical rocess which deends on a large number of arameters. This aer resents a mathematical model consisting of the Nobel- Abel equation of state, the energy conservation equation, the equation of conservation of mass and burning rate law. The system of equations associated with this mathematical model is solved numerically using a comuter rogram. Finally it resents theoretical results obtained with this rogram for M3 trile-base seven erforated granular roellant, comared with exerimentally results measured using closed vessel technique. Keywords: combustion, solid roellant, closed vessel, mathematical model, comuter rogram. 1. INTRODUCTION During combustion the roellant gradually turns into roducts, mostly in the gas hase. The amount of roellant that turns into gases er unit time, called gas formation rate, deends on the nature of the owder, the shae and size of the roellant and the ressure at which combustion occurs, increasing with increasing ressure. The heat released by roellant combustion, heats resulting gases, raising their temerature. The increase of the temerature and the amount of gases released during roellant combustion at constant volume, leads to the increase of the gas ressure, which in turn increases the rate of gas formation. Therefore, the roellant combustion at constant volume is a self-accelerating non-stationary rocess. During combustion, a small art of the reaction heat is lost through the walls of the closed vessel. Due to the comlexity of the combustion rocess, the evaluation of these losses is mainly exerimentally. For the combustion rocess modeling the following assumtions are done, which together are known as geometric combustion law [1]: the roellant elements forming the load are identically in terms of size and hysic and chemical roerties; the roellant elements ignition occurs instantaneously when using a secial igniter; the combustion takes lace at constant seed on all roellant elements surfaces on the normal to these surfaces (combustion takes lace in arallel layers); In addition, when assessing the amount of the heat transferred during combustion to the closed vessel walls, it is considered that it is roortional to the ressure [2]. The variables that describe the roellant combustion rocess versus time on the bases of these assumtions are: roellant mass, ressure and gas temerature, thickness of burned owder, combustion surface area and combustion rate. These variables can be derived analytically for simle shaed elements (shere, cylinder, tube), or evaluated numerically for comlicated shaed elements (e.g. multi-erforated). The analytical derivation of the roellant gas ressure versus time at the constant volume is resented in detail in [1]. In this case, for the variables simlification and searation, the following aroximations are made: burning surface area is constant and equal to its average value; burning rate linearly varies with the gas ressure and heat transfer losses are neglected. Further the numerical evaluation of the variables that describe the combustion rocess is resented which, unlike analytical calculation does not require making any simlifying aroximations and can be alied for any form of the roellant elements. 2. THE MATHEMATICAL MODEL It is considered a closed vessel having interior volume W [m 3 ] in which was introduced a roellant load with an initial mass of ω [kg] and a secial igniter (mass ω a [kg]). All roellant TERMOTEHNICA 1/

2 Daniel ANTONIE, Sorin GHEORGHIAN, Nicolae MĂRUNȚELU elements are instantly ignited and start to burn at time t = s, after the igniter comlete its combustion. During combustion inside the vessel is a two-hase mixture of gases (gas from roellant combustion redominantly lus a small amount of gas roduced by the burning of the igniter and the air resent inside the vessel before combustion starts) and solid unburned roellant. Burning ends at the moment t = t k, when the unburned roellant mass is equal to zero. Let consider ω(t) [kg] the unburned roellant mass at a time t, < t < t k. The gas mixture ressure and temerature, are P [Pa] and resectively T [K], deend on the unburned roellant mass ω [kg]. The functions (ω) and T(ω) describing this deendence is determined by solving the system formed using Nobel-Abel equation of state and the energetic balance equation: W ( t) aa aeraer ( t)/ ( t) R ara aerraer T ( t) c T c T v f ( t) c v c a v, a a v, a f, a c aer v, aer T Q max, c (1) max, c where: α [m 3 /kg], T f [K], c v [J/(kg K)] and R [J/(kg K)] are covolume, adiabatic flame temerature, secific heat and roellant gas constant resectively, ω a [ kg], α a [m 3 /kg], T f [K], c v, a [J/ (kg K)] and R a [J/(kg K)] mass, covolume adiabatic flame temerature, secific heat and igniter gas constant resectively, ω air [kg], α air [m 3 /kg], c v,air [J/(kg K)] and R air [J/(kg K)] mass, covolume secific heat and air constant resectively, Q max, c [J] the maximum amount of heat lost through closed vessel walls and max,c [Pa] maximum gas ressure exerimentally measured in the closed vessel exactly in the conditions of the roblem. The maximum amount of heat lost is calculated solving the system (1) for the end of combustion, when ω(t k ) = and = max,c. The thickness of the burned roellant layer e[m] deends on the unburned roellant mass ω(t) and is determined by solving the equation ( t) n V ( e), (2) where: δ [kg/m 3 ] is the roellant density, n the number of elements forming the roellant load; V(e) [m 3 ] the volume of one roellant element. The analytical derivation of e(ω) function is ossible for simle shaed roellant elements only. For elements with more comlicated shaes (cylinder or rism with multile erforations), equation (2) is solved numerically. The burning surface area S[m 2 ] deends on the thickness of the burned roellant layer e(ω), and is calculated using the relation S( e) n A( e), (3) where A(e) is the total area of one roellant element. The functions which describe the variation of the surface area and the volume of one roellant element with the thickness of the burned roellant layer e for different shaed roellant elements are given in the literature [2], [3]. After an infinitely small time dt, the unburned roellant mass inside the closed vessel will decrease by the amount d S( e) u( ) dt, (4) where u() [m/s] is the roellant linear burning rate, which is defined as the seed of the combustion surface in the direction of the normal to itself. The linear burning rate deendence on the combustion ressure, called the combustion rate law, is exressed by the formula u 1 ( ) u, (5) where: u 1 [m/(pa s)] is the characteristic burning velocity; ν [-] the ressure exonent. Equations (1) - (5) allow for comuting the next six unknown variables: the mass of the unburned roellant ω(t), the ressure (ω) and the temerature T(ω) of the gas mixture, the thickness of the burned roellant layer e(ω), the surface combustion area S(e) and the linear burning rate u(). At the beginning of the combustion, when t =, these values are: ω() = ω, (ω ) =, T(ω ) = T, e(ω ) =, S() = S, and u( ) = u. 3 COMPUTER PROGRAM A comuter rogram has been written based on the above described mathematical model. Initial data necessarily for running the rogram are written in advance by the user in an inut file in the order shown in Table 1. Symbols from no. 3 to 6 of Table 1 reresents the roellant element sizes (L length, D outer diameter, d erforations diameter and e k web thickness), the remaining symbols having meaning given in the descrition of the mathematical model. 122 TERMOTEHNICA 1/213

3 MODELING OF CONSTANT VOLUME COMBUSTION OF PROPELLANTS FOR ARTILLERY WEAPONS After reading the initial data inut file, the comuter code calculates the following constant values: initial area, volume and mass of the roellant element, the total number of roellant elements, air mass inside the closed vessel and the maximum amount of heat lost. The unknown variables calculation are done in the oints i = 1,2,..., at equidistant moments of time, where Δt is the time ste. At every oint i, the unknowns variables are determined by successive iterations until the relative error of gas ressure calculation becomes smaller than a certain value ε r required. CALC(t,,ω,u,S,T,e) i 1 t i t(i) i (i) ω i ω(i) i i+1 t(i) t(i-1)+δt u i-1 u(i-1) ω i-1 ω(i-1) S i-1 S(i-1) c i u i u( i ) e i root[ω i = δnv(e)] S i S(e i ) Δω.5δ(S i u i +S i-1 u i-1 ) Δt ω i ω i-1 -Δω ω> DA ω(i) e(i) e k S(i) S(e k ) (i) max,c u(i) u( max,c ) T(i) T max,c RETURN e i root[ω i = δnv(e)] S i S(e i ) i (ω i ) T i T(ω i ) i c c Δ< ε r DA (i) i ω(i) ω i u(i) u i S(i) S i T(i) T i e(i) e i Fig. 1. CALC subroutine flowchart. TERMOTEHNICA 1/

4 Daniel ANTONIE, Sorin GHEORGHIAN, Nicolae MĂRUNȚELU Table 1 Initial data for M3 roellant combustion modeling No. Symbol MU Value No. Symbol MU Value 1 W cm ω a g 1 2 ω g 4 15 T f,a K L mm R a J/(g K) D mm c v,a J/(g K) d mm.5 18 α a cm 3 /g e k mm δ a g/cm T f K R aer J/(g K) R J/(g K) c v,aer J/(g K) c v J/(g K) α aer cm 3 /g.98 1 α cm 3 /g δ aer g/cm δ g/cm Δt s.1 12 u 1 cm/(s MPa) ε r ν max,c MPa e [mm].5 A [cm2] omega [g] u [cm/s] P [MPa] 2 1 T [K] Fig. 2. The variation of the values which describe the M3 roellant combustion versus time at the,2 g/cm 3 loading density. After reaching the required recision, the unknown variables are assigned to indexed variables (vectors), and then roceed to the next calculation. Integration stos when the unburned roellant mass inside the closed vessel is less than or equal to zero. These calculations are erformed using the subroutine CALC, whose flowchart is shown in figure TERMOTEHNICA 1/213

5 MODELING OF CONSTANT VOLUME COMBUSTION OF PROPELLANTS FOR ARTILLERY WEAPONS Finally, the vectors containing the values t, ω, u, S, T and E are transferred to the main rogram, where they are written to an outut file for rinting or grahics. 3. RESULTS AND CONCLUSIONS In order to verify the mathematical model and for comuting rogram validation, one exerimental firing using M3 trile-base roellant was erformed at.2 g/cm 3 loading density, in the exerimental installation with 2 cm 3 closed vessel [4]. The series of exerimental data was rocessed automatically by the comuter rogram [5], resulting the roellant gases ressure variation versus time. The roellant was comletely burned after t k,c = 18.7 ms, when into the closed vessel the ressure reached the maximum value max, c = = MPa. The initial data used for M3 roellant burning modeling are summarized in Table 1. P [MPa] measured comuted Fig. 3. The M3 roellant gas ressure versus time at the,2 g/cm 3 loading density. rel. error [%] Fig. 4. The variation of relative error of calculated gas ressure versus time. The following are the results based on these initial data. The maximum amount of the heat lost through the walls of the bomb is Q max, c = kj, that is 6.2% of the total heat released during the combustion. The values which described the M3 roellant burning versus time are shown grahically in Figure. 2. The M3 roellant thermo chemical constants were calculated using Corner method [6] and those of black roellant used for igniter were taken from [7]. The M3 roellant burning rate law constants have been determined from the exerimental data for the ressure range of 6-19 MPa. The roellant comletely burns after t k = 14.4 ms, with 4.3 ms less than the time exerimentally measured. In order to comare the calculated gas ressure curve with the measured one, the last was translated with this difference, so that the maximum ressure oints on the two curves should coincide. From the Figure 3 it is observed that the calculated ressure is similar to the measured ressure. To assess the accuracy of the calculation, the relative error of the ressure versus time was comuted. From the Figure 4 it is seen that this error is less than 2% for the times eriods greater than 7.5 ms (namely at the ressures greater than 6 MPa, for which the combustion law coefficients were calculated). In order to obtain good results at lower ressures, it is necessary to modify the comuter rogram in such way to allow the burning rate law defining on two or three ressure sub-ranges. The resented mathematical model may be useful for develoment of new interior ballistic models of artillery guns. REFERENCES [1] VASILE, T., Balistica interioară a gurilor de foc, Vol. 1, Editura ATM, Bucureşti,1993. [2] KRIER, H. and SUMMERFIELD, M. (editors),- Interior Ballistics of Guns, Vol. 66, AIAA, Washington, DC, [3] STANAG 4367 Thermodynamic Interior Ballistic Model with Global Parameters. [4] GHEORGHIAN S., ORBAN, O., Instalaţie cu bombă manometrică entru studiu arderii la volumul constant a roulsanţilor de artilerie., Rev. ACTTM 4/2, Bucureşti. [5] GHEORGHIAN S., Prelucrarea datelor exerimentale obţinute în instalaţia cu bombă manometrică, A XXIX-a Sesiune de comunicări ştiinţifice, Academia Tehnică Militară, Bucureşti, 21. [6] GHEORGHIAN S., Determinarea teoretică a constantelor termochimice utilizate în balistica interioară a gurilor de foc, A XXIX-a Sesiune de comunicări ştiinţifice, Academia Tehnică Militară, Bucureşti, 21. [7] STIEFEL, L., (editor), Progress in Astronautics and Aeronautics: Gun Proulsion Technology, Vol. 19, AIAA, Washington, DC, TERMOTEHNICA 1/