Physics Joural Vol., No. 3, 05, pp. 343-348 http://www.aisciece.org/joural/pj Mathematical Modelig of Self - Oscillatios i the Combustio Chamber of Liquid Rocket Egie with Variable Latecy Combustio B. I. Basok, V. V. otsuleko * Departmet of Thermophysical Fudametals of Eergy-Savig Techologies, Istitute of Egieerig Thermal Physics of NAS of Ukraie, Kiev, Ukraie Abstract Self-oscillatios ad certai of their regularities determied by solutio of a system of differetial equatios with variable delay argumet equatios that is used i cosiderig combustio istability i combustio chambers of liquid-propellat rocket egies are modeled mathematically. Periodic solutios of the system of equatios of ostatioary motio of a medium i a liquid-propellat rocket egie were obtaied, with the aid of which the possibility of lowerig the amplitude of the logitudial self-oscillatios of vibratio combustio or their complete removal has bee substatiated. Aalytically determied critical time delay combustio, above which a statioary combustio becomes ustable ad self-excited oscillatios. Keywords Vibratio Combustio, Istability, Self - Oscillatio, Limitig Cycle, the Time Delay of Combustio Received: September 6, 05 / Accepted: December, 05 / Published olie: December 9, 05 @ 05 The Authors. Published by America Istitute of Sciece. This Ope Access article is uder the CC BY-NC licese. http://creativecommos.org/liceses/by-c/4.0/. Itroductio The ostatioarity of the process of burig of a fuel i differet devices is geerated by the chemical-kietic, diffusio-thermal, ad covective pheomea []. Furthermore, accordig to the first low of thermo- dyamics, the heat released i combustio of the fuel i the flow is coverted to the iteral eergy ad compoets of the total head whose value is depedet o the flow rate. The ecessary coditio of excitatio of self-oscillatios i the case i questio is the presece of the ascedig brach o the head characteristic [, 3], which is formed by the heat-tohead coversio. I [4], it has bee established that, i burig of the fuel i solid-fuel egies, the pressure depedece of the rate of formatio of the gas ca be such that the system will lose stability [5]. I liquid-propellat rocket egies (LPRE, the coditio of itrachamber istability lies i the formatio of the ascedig brach of the p = F ( depedece of the pressure i the combustio chamber o the flow rate. rowth i the head or the pressure p i the combustio chamber with flow rate features promietly amog the causes of excitatio of vibratioal-combustio self-oscillatios; it has ot bee cosidered i theoretical descriptio of the pheomeo of istability of combustio i liquid-propellat rocket egies. The reaso is that the coditios of formatio of the ascedig brach o the p = F ( depedece of the head characteristic of the combustio chamber o the flow rate remaied ukow. Thus, e.g., i [5, 6], the above depedece of the head characteristic has bee represeted as beig mootoically decreasig. I [5], the stability of the statioary regime with such a depedece has bee substatiated. Therefore, combustio istability i liquid-propellat rocket egies * Correspodig author E-mail address: basok@ittf.kiev.ua (B. I. Basok, gosul@ukr.et (V. V. otsuleko
344 B. I. Basok ad V. V. otsuleko: Mathematical Modelig of Self - Oscillatios i the Combustio Chamber of Liquid Rocket Egie with Variable Latecy Combustio was maily determied by the mechaism of pheomeological time delay of the combustio of the fuel [5 7] ad by the maifestatio of differet feedbacks which are iteral ad act uiversally.. Formulatio of the problem I this paper, we cosider the problem of determiig the critical time delay of combustio ad the costructio of the boudaries of the stability regio whe the delay is variable, depedig o the pressure iside the combustio chamber. I Fig. shows a diagram of the combustio chamber rocket egie. 3. System of Equatios of Itrachamber Combustio Istability The equatio of motio for the sectios of the flow of a fuel 0 0 ad of combustio products 3 3 (Fig. i the form of d mw = p h h p S, takig ito accout that the f T fr mass of the combustio is m = ρls, will be writte as [8, 9] where La,c.ch L a,c.ch d = F ( p, ( = l S acoustic mass of the combustio chamber; mass flow rate i the combustio chamber; f p = H pressure characteristic of parallel coectio cetrifugal pumps oxidat ad fuel; F = p h h pressure head characteristic f T fr of fuel flow i a combustio chamber [8-9]; ht ( pressure losses because of heat supply (thermal resistace; hfr ( viscous losses alog the legth of the combustio chamber. The mass equatio i the combustio chamber is represeted, accordig to [8], i the form ( τ ( dm = t p, where = f + Cox, f flow rate of a fuel, C ox flow rate of oxidizer, τ ( p the time delay of combustio of fuel, mass flow rate through the ozzle. Sice dp d ρ is equal to c, where c velocity of soud i the flow, it may be reduced to the form dp Ca,c.ch = t τ ( p φ ( p, ( where φ ( p = characteristic of the ozzle of a liquidpropellat rocket egie, Ca,c.ch = V c acoustic flexibility of the combustio chamber, V the volume of the combustio chamber. Experimetal studies [7] showed that the depedece of τ ( p is mootoically decreasig. Figure shows the experimetal poits ad their approximatio usig the followig formula ( p bexp τ = ap p, where a = 0., b = 0.006. 4. Calculatio of Hydraulic Characteristics i the LPRE Combustio Chamber The viscous losses occurrig over the combustio chamber legth i the smoke gas motio segmet were calculated usig the Darcy - Weisbach formula l w hfr ( = λ ρ, (3 d where λ is the coefficiet of hydraulic losses. Heat iput to the fuel flow with a mass rate of flow = ρws i the chael of the LPRE combustio chamber with a costat ormal cross-sectio area (Fig. leads to a decrease i the desity ρ of the flow of combustio products ad a icrease i its velocity w. Therefore, the heat resistace to such flow [0] is determied by the local hydraulic resistace i the regio of heat iput. I [0], we obtaied a fuctio for determiig the eergy loss i the combustio chamber ht ( at polytropic heat iput to the fuel flow. Let us defie the fuctio ht ( at isobaric heat iput to the fuel flow i the LPRE combustio chamber. We write the eergy equatio for sectios ad (Fig., ad from this equatio we determie the thermal resistace, which is the local oe, appearig i the regio of heat supply [0] p w p w q + u u h ρ + + = + + +, (4 T ρ where ht is the eergy loss i the flow due to the heat iput. Sice at isobaric combustio heat iput to the ideal gas flow q = сp( T T, the chage i the iteral eergy of the u u = с ( T T, ad cp cv = R, the gas v q u = RT ( T. Further, i view of the relatio q u = p ρ p ρ from Eq. (4 it follows that
Physics Joural Vol., No. 3, 05, pp. 343-348 345 h = w w, or i pressure uits, assumig h T = ρ h, we have T T h w w = ρ. w T Makig use of the cotiuity equatio ρw = ρw for the gas flow i the combustio chamber with a costat ormal cross-sectio area ad of the relatio betwee the isobaric process parameters T T = ρ ρ, we obtai [] shows the graphics data, whe the pressure characteristic F ( of the combustio chamber is mootoically decreasig. 5. Calculatig the Boudaries of the Stable Combustio The parameters of statioary combustio mode are determied from the system of equatios ( ( believig i it T T = ρs T h. (5 Determie the formula for the hydraulic characteristics of the jet ozzle. The maximum mass flow rate of gas i the critical sectio of ozzle or where φ = + Smi k k = + p Smik k ( p k + k p w, = φ, (6 k + ( k p krt ; R gas costat of combustio products; k adiabatic expoet; T absolute temperature of the flowig gas. Foud above depedecies (3, (5, (6 makes the system of equatios ( ( fully defied. Itegratig this system of equatios allows to determie the limit cycles ad selfoscillatios i the combustio chamber rocket egie. I Fig. 3 shows the limit cycles ad their correspodig forms of self-oscillatios i a saddle-ode discharge characteristics F ( of the combustio chamber. Respectively i Fig. 4 d dp = 0, = 0. (7 = p = p Usig coditios (7, we obtai p F ( = φ ( p. Also Usig the Taylor decompositio d ( t τ = ( t τ + O ( τ, = ad the system of equatios ( - (, accurate to quatities of order O ( τ, ca be writte i the followig form L a,c.ch d = F ( p, ( p ( φ dp τ C = F p p, (8 a,c.ch La Further, it is more coveiet to switch to dimesioless variables: x =, y p p =. (9 p I the ew variables (9 the system of equatios (8 ca be writte i the followig form: dx La,c.ch = F ( + x p ( y +, dy F ( + x p ( y + pc a,c.ch = ( x + τ ( p ( y + φ ( p ( y +. La,c.ch (0 Thus, the ature of stability of statioary combustio mode is reduced to the study of stability of zero equilibrium of the dyamic system (0. Usig Lyapuov's first method of stability aalysis let us cosider the Jacobia matrix of the system (0 is calculated i the zero positio of equilibrium F p La La,c.ch J =. ( τ ( p τ ( p F ( φ ( p pc a,c.ch La,c.ch Ca,c.ch La,c.ch
346 B. I. Basok ad V. V. otsuleko: Mathematical Modelig of Self - Oscillatios i the Combustio Chamber of Liquid Rocket Egie with Variable Latecy Combustio df where ( + x F ( =, φ ( p dx x= 0 ( + dφ p p y =. dy To determie the critical time delay of combustio is ecessary to pre-compute the roots of the characteristic equatio ( λ y= 0 det J E = 0. ( Calculatig the determiat of (, we obtai ± i tr J 4det J tr J λ, =, where tr ( J trace, ad det ( J the determiat of the Jacobia matrix J. Moreover, accordig to (, obtai the followig represetatios for these characteristics: τ ( p F tr ( J = + φ ( p, La,c.ch Ca,c.ch La,c.ch det ( J = ( F ( φ ( p. L C a,c.ch a,c.ch Thus, the critical delay time of combustio is determied from the coditio: {,} Re λ = 0 ( J ( J tr = 0, det > 0. F τ + = 0, ( p kr φ La,c.ch C a,c.ch L a,c.ch F ( φ ( p <, depedece for the critical time delay from combustiochamber pressure p τ 3 3 L a,c.ch F0 p F0 p kr ( p = 3kC F a,c.ch p kf kf. (4 I Fig. 5 shows a plot of (4 to the combustio chamber legth l = 0.5 m ad diameter d = 0. m, whe as the oxidizig aget used liquid oxyge ad fuel hydroge. 6. Coclusios By umerically itegratig the system of equatios of the ostatioary motio of a medium i a LPRE, i trasitio of it to a degeerate form, for the first time the possibility of cotrollig the amplitude of oscillatios of vibratio combustio at differet pressure head characteristic of a combustio chamber has bee show. The possibility of complete suppressio of self-oscillatios ad attaimet of a statioary regime of combustio i a LPRE has bee established. Aalytically derived the ratio that determies the critical time delay of combustio of gaseous fuel that must be exceeded i the combustio chamber of rocket egies statioary combustio mode becomes ustable ad excited vibratio combustio. Also for the combustio chamber with a mootoically decreasig pressure characteristic was obtaied depedece of the critical delay time of combustio. where fially, we obtai that ( p C F ( τ = L φ. (3 kr a,c.ch a,c.ch Thus, the equilibrium of the dyamic system (0 that determies the parameters of statioary combustio mode becomes ustable if true the followig iequality τ p τ kr >. < Note also that the coditio ( ( p F φ is executed automatically whe pressure characteristic F ( of the combustio chamber is a mootoically decreasig fuctio. For further aalysis of the obtaied formula (3 we well approximate the pressure characteristic of the combustio chamber by a polyomial of third degree, equatig 3 F F k. The from (3 we get the followig 0 F Fig.. Positio of the sectios i the LPRE combustio chamber that are used for costructio of the equatios of motio of combustio products. Fig.. The depedece of the time delay from combustio chamber pressure.
Physics Joural Vol., No. 3, 05, pp. 343-348 347 Fig. 3. Limit cycles ad self - oscillatios of vibratio combustio whe the characteristic c = 3.5 kg sec ; d = 4. kg sec F is a saddle-ode: a = kg sec ; b =.5 kg sec ; Fig. 4. Limit cycles ad self - oscillatios of vibratio combustio whe the characteristic =4 kg sec ; c = 5 kg sec. F is mootoically decreasig: a = kg sec ; b Refereces [] Ya. B. Zel dovich,. I. Bareblatt, V. B. Librovich, ad. M. Makhviladze, Mathematical Theory of Combustio ad Explosio [i Russia], Nauka, Moscow (980. [] V. V. otsuleko, Special modes of the Rijke pheomeo, Izh.-Fiz. Zh., 6, No. 9, 60 64 (005. [3] V. V. otsuleko, Mathematical modelig of the decrease i the amplitudes of vibratios of vibratioal combustio i large idustrial uits, Mat. Modelir., 7, No., 6 4 (005. Fig. 5. The depedece of the critical time delay of combustio τkr ( p. [4] Ya. B. Zel dovich, O. I. Leipuskii, ad V. B. Librovich, Theory of Nostatioary Combustio of upowder [i Russia], Nauka, Moscow (975.
348 B. I. Basok ad V. V. otsuleko: Mathematical Modelig of Self - Oscillatios i the Combustio Chamber of Liquid Rocket Egie with Variable Latecy Combustio [5] M. S. Natazo, Istability of Combustio [i Russia], Mashiostroeie, Moscow (986. [6] K. I. Artamoov, Thermoacoustic Stability [i Russia], Mashiostroeie, Moscow (98. [7] L. Crocco ad Si-I Cheg, Theory of Combustio Istability i Liquid-Propellat Rocket Egies [Russia traslatio], IL, Moscow (958. [8] V. V. otsuleko ad V. N. otsuleko, Mathematical modelig of the self-oscillatios of vibratioal combustio i liquid-propellat rocket egies due to combustio heat release, Mat. Modelyuv. (Deprodzerzhisk os. Tekh. Uiv., No., (5, 6 4 (006. [9] otsuleko V. V. ad otsuleko V. N. O the Problem of Cotrol of the Amplitude of Self-Oscillatios of a Sigig Flame // Joural of Egieerig Physics ad Thermophysics, Volume 87, Issue, 39 34 (04. [0] B. I. Basok ad V.V. otsuleko, Negative thermal resistace i oe-dimesioal steady iviscid flow of a perfect gas, Joural of the Moscow physical-techical Istitute, Vol 6, No. 4(4, 53 57 (04. [] Basok B. I. ad otsuleko V.V. Calculatig the Parameters of Self - Oscillatios i the Vertical Combustio Chamber of the Blast - Furace Air Heater durig Ustable Combustio // Thermal Egieerig, Vol. 6, No., 58 63 (05.