Dynamic characteristics of the recessed chamber within a gas liquid coaxial injector

Transcription

1 Reseach Aticle Dynamic chaacteistics of the ecessed chambe within a gas liquid coaxial injecto Advances in Mechanical Engineeing 216, Vol. 8(4) 1 7 Ó The Autho(s) 216 DOI: / aime.sagepub.com Qing-fei Fu and Li-jun Yang Abstact In this aticle, a linea dynamic chaacteistic model of the ecessed chambe in a gas liquid shea coaxial injecto is developed. The gaseous injecto flow befoe the ecessed chambe is consideed to be steady, and we just investigate the esponse of pessue dop acoss the ecessed chambe to the mass flow ate fluctuation of the liquid injecto. The tansfe function of the ecessed chambe is obtained, consideing the distibute chaacteistics of liquid doplets within the ecessed chambe. The esults show that the amplitude of pessue dop pulsation would decease as the liquid flow pulsation fequency inceases, when the vaiation of liquid velocity is not taken into account. While consideing the vaiation of liquid velocity, the esult is opposite. The amplitude of pessue dop oscillation on the bounday of the ecessed chambe inceases with the incease in liquid velocity oscillation fequency. Keywods Gas liquid injecto, dynamic chaacteistics, ecessed chambe, tansfe function, liquid ocket engine Date eceived: 5 Decembe 215; accepted: 8 Febuay 216 Academic Edito: Oonzio Manca Intoduction As the demand foage thust liquid ocket engines inceases, some highly effective popellant such as liquid hydogen, liquid oxygen, and keosene as well as some new techniques such as staged combustion cycle ae widely used. The application of these new techniques facilitates the use and eseach of coaxial injecto technique. Fo a gas liquid coaxial swil injecto, the liquid popellant is injected though a cente swil injecto, and the gaseous popellant is injected with high velocity though an annulaap aound the cente injecto. The hollow cone liquid sheet ejected out of cente injecto impinges on the suounding gas steam, poviding bette atomization and highe pefomance. In industial applications, the cente injecto always etacts with espect to the exit suface of the oute injecto, foming a ecessed chambe. 1 The ecessed chambe of coaxial injecto can enhance the mixing of popellants, subsequently affect the stability of flame and esult in favoable and stable combustion. 2 5 As shown in Figue 1, the oxidant and fuel of ecessed coaxial injecto flow in each tube. They meet in the mixed ecessed chambe, beak up, mix, and eject out of the nozzle. The ecessed length of oxidant nozzle outlet s section to fuel nozzle outlet s is the pincipal section which influences the inteaction of popellant in the entance section of combustion chambe, and the inteaction of gas liquid phase in the ecessed chambe School of Astonautics, Beijing Univesity of Aeonautics and Astonautics, Beijing, China Coesponding autho: Li-jun Yang, School of Astonautics, Beijing Univesity of Aeonautics and Astonautics, Beijing 1191, China. yanglijun@buaa.edu.cn Ceative Commons CC-BY: This aticle is distibuted unde the tems of the Ceative Commons Attibution 3. License ( which pemits any use, epoduction and distibution of the wok without futhe pemission povided the oiginal wok is attibuted as specified on the SAGE and Open Access pages ( open-access-at-sage).

2 2 Advances in Mechanical Engineeing Figue 1. Schematic diagam of a shea coaxial injecto. plays an impotant ole in completeness and stability of combustion in the combustion chambe. Thee have been many studies on the spay and combustion chaacteistics of the coaxial swil injecto with the ecess configuation; 6 1 howeve, the dynamic chaacteistics of this type of injecto have been less studied. Chen et al. 11 analyzed the popagation law of fluid oscillation in the gas liquid jet; howeve, the calculating wok is lage. Actually, fequency-domain analysis is easie than time-domain analysis while solving dynamic chaacteistics of the injecto, which means calculating the fequency esponse in eal fequency domain is easie than solving diffeential equation. When a sinusoidal input of a cetain fequency is put in the injecto, the amplitude atio of the output and input and the phase shift ae called fequency chaacteistics o fequency esponse. The chaacteistic is the manifestation of dynamic chaacteistics of the injecto in eal fequency domain, and hydomechanics equations can be tansfomed into fequency chaacteistic equations via Laplace tansfomation. Fu and Yang 12 developed a linea dynamic chaacteistics model of a gas liquid coaxial swil injecto. They assumed the gas liquid two-phase flow in the ecessed chambe is homogeneous flow and obtained the tansfe function of the ecessed chambe. This aticle analyzes the inteaction and the dynamic pocess of gas and liquid in the ecessed chambe of gas liquid coaxial injectoinealy, and the distibution of dispesed phase within the ecessed chambe is taken into account, that is, the liquid doplets within the ecessed chambe ae consideed to be inhomogeneous. Theoetical famewok In the ecessed chambe, because the momentum of the gas is much lage than the momentum of the liquid, the liquid film within the liquid nozzle exit will beak up and tun into doplets on account of aeodynamic foce. The doplets will poduce the medium motionesistance foce, which will cause the appeaance of pessue diffeence in the gas in the entance and exit. Doplet distibution within the ecessed chambe is nonunifom, that is the paametes cannot be consideed to be lumped in the ecessed chambe when l l \L ec. Because the fequency coelates with the wavelength, it is necessay to estimate the ange of distubance fequency while consideing the nonunifom distibution of doplets along the ecessed chambe. The liquid entes the mixing zone at the speed of v l, which is slowe than sound velocity appaently, so the distubance wavelength of mass flow ate of liquid phase can be estimated though the fomula below l l = v l ð1þ f l whee l l is the wavelength of oscillation wavelength and f l is the fequency of oscillation wavelength. Let L ec be the length of the ecessed chambe, and if l l \L ec (it means f l.v l =L ec ), the distibution of doplets cannot be ignoed. Suppose that the ejecting velocity of liquid at the liquid nozzle is 3 m/s, and the length of the ecessed chambe is 1 mm, the ange of distubance fequency is f l.3 Hz while the mass distibution chaacteistics of doplets should be taken into account. Thus, it shows that the distibuted chaacteistics of flow paametes cannot be ignoed in medium-high fequency ange. Fo the detailed deivation of the tansfe function, we can efe to the liteatue. 5 Supposing the velocity of liquid in the ecessed chambe is constant The pessue dop of the ecessed chambe whose length is dx can be calculated as dp = F,, v l,... dx ð2þ whee is the density of liquid and is the density of gas, and v l is the velocity diffeence in liquid and gas in the ecessed chambe. The total pessue loss in the ecessed chambe can be obtained by integation of dp fom to x 2 p 1 p 2 = F,, v l,... dx ð3þ Lineaize equation (3) and expess it though elative amount of distubance. The velocity of the gas phase is elatively highe than that of the liquid phase. Accoding to l = v=f, the wavelength fo distubance

3 Fu and Yang 3 of gas phase is geate than that of liquid phase. Fo example, if = 8 m/s, f = 1 Hz, then the distubance wavelength equals to 8 mm, which is much geate than the common ecessed chambeength. Theefoe, it is easonable to conside that and ae only elevant to t (it means that the gas in the ecessed chambe is viewed as lumped paamete), and we can obtain dp 1 dp 2 = k 1 d dx + k 2 d + k 3 d k 4 v l dv l dx ð4þ whee the quantity with oveba denotes the elative quantity, pefix d denotes the petubation quantity, = p 1 p 2, k 1 = F=, k 2 =( F= m : g )( x 2 ), k 3 =( F= v : g )(x 2 ), and k 4 = F= v l. Also, we can obtain equation (5) though = m : l =(Av l) d = d_m l dv l ð5þ In consideation of equation (5), equation (4) can be expessed as dp 1 dp 2 = k l 1 d_m l dx + k 2 ð k l v x 2 l d + k 3 d dv l dx ð6þ The liquid infinitesimal in section x at moment t is ejected out fom the section of nozzle (x = ) at the moment of (t t). Let the velocity of liquid infinitesimal be v l(t t). If the change in tanslation speed of liquid infinitesimal fom to x is ignoed, appaently, delay time t is equal to the time fo the liquid infinitesimal moving fom section to section x t = x v l(t t) ð7þ In the axial diection of the ecessed chambe, the velocity of liquid enteing the ecessed chambe can be hypothesized as v which is a constant. In this situation, on account of equation (7), equation (6) can be expessed as dp 1 = dp 2 + k 1 d_m l(t t) dx + k 2 d + k 3 d ð8þ When the initial conditions ae zeo, conduct Laplace tansfom with equation (8) and obtain dp 1 = dp 2 + k 1 d_m l exp sx v dx + k 2 d + k 3 d ð9þ Calculate the integal tem in equation (9), and we can obtain the elation of elative distubance quantity of the paametes on the bounday of the ecessed chambe dp 1 = dp 2 + W m ðþd_m s l + k 2 d p g + k 3 d d ð1þ In the equation above, W m (s)=k 1 ( = )(v =s) (1 exp ( sx=v )) is the tansfe function of pessue dop oscillation in the ecessed chambe which is caused by liquid flow oscillation. Velocity of liquid in the ecessed chambe changes In actual situation, the velocity of liquid enteing the ecessed chambe changes. At the same time, the delay time t will also change accodingly. Expessing spouting velocity as the sum of the mean value and petubation velocity, we can obtain equation (11) as follows v l(t t) = v (t t) + dv l(t t) ð11þ In this situation, fo velocity fluctuating dv l which is small enough, equation (6) can be expessed as the following equation dp 1 = dp 2 + k 1 d_m + x=v l(t x=v ) ddv l dt dx + k 2 d p g + k 3 d k l v l d dv l(t x=v )dx ð12þ When the initial condition is zeo, we conduct Laplace tansfom with equation (12) and obtain dp 1 = dp 2 + k l 1 p 2 d 3 d_m l exp sx sx dx + dv l exp sx 4 dx5 v v v + k 2 d p g + k 3 d k l v l d dv l exp sx dx v ð13þ

4 4 Advances in Mechanical Engineeing The equation above can be simplified as equation (14) dp 1 dp 2 = W m ðþd_m s l + W v ðþdv s l + k 2 d p g + k 3 d d In the equation above W v ðþ= s v s k l 1 1 e s x v 1 + sx v k l v l ð14þ 1 e s v x ð15þ W v (s) is the tansfe function consideing the oscillation of delay time, which is caused by the oscillation of liquid spouting velocity while enteing the ecessed chambe. Equation (14) manifests that consideing the oscillation of delay time, although thee does not exist flow distubance at the section of nozzle exit, thee is pessue dop oscillation on the bounday of the ecessed chambe. This is because, if liquid spouting velocity while enteing the ecessed chambe is vaiable, at the cetain time t, the total mass of liquid at section x is not only decided by the liquid mass enteing at time (t t), whose aveage velocity is v, but also decided by the paticle enteing befoe time (t t), whose velocity is elatively slow. Theefoe, consideing the oscillation of liquid spouting velocity, although the flow into the ecessed chambe, spouted by the nozzle, is a constant, thee exists the oscillation of flow and density at section x. Results and discussion Figue 2(a) shows the amplitude fequency cuve of W m (s), Figue 2(b) expesses the phase fequency esponse cuve of W m (s), and Figue 2(c) expesses the hodogaph of W m (s). Accoding to this figue, when the fequency inceases, the module value of W m (s) deceases and each to zeo peiodically at the points Figue 2. Dynamic chaacteistics of the ecessed chambe when the velocity change in liquid is not consideed ( = 1 kg=m 3, = :1 MPa, v = 1 m=s, and x = 1 mm): (a) amplitude fequency cuve, (b) phase fequency esponse cuve, and (c) hodogaph.

5 Fu and Yang 5 Figue 3. Dynamic chaacteistics of the ecessed chambe consideing liquid velocity change ( = 1 kg=m 3, = :1 MPa, v = 1 m=s, and x = 1 mm): (a) amplitude fequency chaacteistic, (b) phase fequency chaacteistic, and (c) hodogaph. whee vx=v = 2kp. Also, in the situation of these oscillation fequencies satisfying vx=v = 2kp, the pessue dop oscillation on the bounday of the ecessed chambe obtains the minimum. Accoding to Figue 2(c), when angula fequency of distubance inceases, the module value of W m (s) deceases and eaches zeo peiodically at the points whee vx=v = 2kp. Howeve, only the case k = 1 confoms to the actual situation. Theefoe, only in the situation of these oscillation fequencies when vx=v = 2p, the pessue dop oscillation on the bounday of the ecessed chambe obtains the minimum. In the situation of these oscillations, the amplitude of pessue doeceases because when v is constant, the amplitude of liquid flow oscillation along the diection of the length of the ecessed chambe has simple hamonic chaacteistics. At the point whee vx=v = 2p in the steamwise diection of the ecessed chambe, thee ae intege numbes of distubance waves. Theefoe, the incease in pessue dop at the positive half-wave of oscillation of d _m l and the decease in pessue dop at the negative half-wave of oscillation of d _m l counteact with each othe. In high-fequency oscillation egion, all components of pessue dop oscillation, which have diploid times of peiod, equal zeo. Theefoe, the integal of oscillation of pessue dop will be decided only by the components which cannot exist in integal numbe of peiod in the ecessed chambe. With the incease in oscillation fequency, the wavelength of liquid flow oscillation will decease, and the oscillation amplitude of pessue dop will decease accodingly. Figue 3 shows the dynamic chaacteistics of the ecessed chambe calculated using equation (15), which conside the vaiation of liquid velocity along the ecessed chambe. It shows the dynamic espond of pessue dop to the pulsation of liquid velocity. Figue 3(a) shows the amplitude fequency diagam of W v (s), Figue 3(b) shows the phase fequency diagam of W v (s), and Figue 3(c) shows the hodogaph of W v (s). Accoding to this figue, when distubance fequency inceases, the amplitude of pessue doistubances

6 6 Advances in Mechanical Engineeing on the bounday of the ecessed chambe inceases, and the phase diffeence between pessue dop and liquid spouting velocity also inceases. The hodogaph stats fom positive eal axis. When v inceases, the cuve otates clockwise in spial line. The distance between the endpoint and the oigin is module value of fequency chaacteistic. Compaing the amplitude of pessue dop consideing velocity change and the one iespective of velocity change, when consideing velocity change, the amplitude of pessue dop on the bounday of the ecessed chambe is less than the one when taking no account of velocity change. Conclusion The inteaction of gas phase and liquid phase in the ecessed chambe of gas liquid coaxial injecto is investigated with the fequency method. The dynamic esponse of pessue dop pulsation in the ecessed chambe to the liquid flow pulsation is calculated, in the condition of consideing and not consideing the vaiation of liquid velocity. When distubance fequency is in medium-high fequency ange, the distibution chaacteistic of the dop in the ecessed chambe must be taken into consideation. The calculation esults show that when not consideing the vaiation of liquid velocity acoss the ecessed chambe, the amplitude of pessue dop pulsation on the bounday of the ecessed chambe deceases in geneal, with the incease in liquid velocity oscillation fequency. At points whee vx=v = 2kp = 1, 2,..., the amplitude of pessue dop oscillation becomes zeo peiodically. Also, the phase diffeence between the pessue dop oscillation and liquid velocity oscillation inceases linealy with the incease in distubance fequency. While consideing liquid velocity change in the ecessed chambe, the amplitude of pessue dop oscillation on the bounday of the ecessed chambe inceases with the incease in liquid velocity oscillation fequency. Also, the phase diffeence between them also inceases with the incease in distubance fequency. Declaation of conflicting inteests The autho(s) declaed no potential conflicts of inteest with espect to the eseach, authoship, and/o publication of this aticle. Funding The autho(s) disclosed eceipt of the following financial suppot fo the eseach, authoship, and/o publication of this aticle: This wok was financially suppoted by China National Natue Science Funds (suppot numbes: and ). Refeences 1. Gill GS. Liquid ocket engine injectos. NASA-SP-889, 1976, gov/ pdf 2. Yang V and Andeson WE. Liquid ocket engine combustion instability (Pogess in Astonautics and Aeonautics, no. 169). Reston, VA: AIAA, Bazaov VG and Yang V. Liquid popellant ocket engine injectos dynamics. J Popul Powe 1998; 14: Kim BD and Heiste S. Numeical modeling of hydodynamic instability of swil coaxial injecto in a ecessed egion. In: 42nd AIAA/ASME/SAE/ASEE joint populsion confeence & exhibit, Sacamento, CA, 9 12 July 26, AIAA pape Reston, VA: AIAA. 5. Andeyev AV and Bazaov VG. Gas-liquid injectos dynamics. Moscow: Mashinostoenie Publications, 1991 (in Russian). 6. Yang L, Ge M, Zhang M, et al. Spay chaacteistics of a ecessed gas-liquid coaxial swil injecto. J Popul Powe 28; 24: Li Q, Kang Z, Zhang X, et al. Effect of ecess length on the spay chaacteistics of liquid-centeed swil coaxial injectos. Atomization Spay 215; 26: Im J, Cho S, Yoon Y, et al. Compaative study of spay chaacteistics of gas-centeed and liquid-centeed swil coaxial injectos. J Popul Powe 21; 26: Solle S, Wagne R, Kau H, et al. Combustion stability chaacteistics of coax-swil-injectos fo oxygen/keosene. In: 43d AIAA/ASME/SAE/ASEE joint populsion confeence & exhibit, Cincinnati, OH, 8 11 July 27, AIAA pape Reston, VA: AIAA. 1. Zhang M, Wang L, Zhang Z, et al. Flame chaacteistic test study of coaxial-swil injecto. Combust Sci Technol 28; 14: Chen Z, Wang P and Wang J. The popagation ule analysis of fluid oscillation in ai-liquid injecto with oscillation fluid mechanics method. Missile Space Veh 1998; 4: (in Chinese) 12. Fu Q and Yang L. Theoetical investigation of the dynamics of a gas-liquid coaxial swil injecto. J Popul Powe 211; 27: Appendi Notation A sectional aea (m 2 ) f pulsation fequency (Hz) k coefficient L length (m) m mass flow ate (kg/s) p pessue (MPa) pessue diffeence (MPa) s Laplace opeato t time (s)

7 Fu and Yang 7 v velocity (m/s) W tansfe function x steamwise diection coodinate l distubance wavelength (m) density (kg/m 3 ) t time delay (s) v angula fequency (2pf ) Subscipts g L ec gas phase liquid phase ecessed chambe Supescipts 2 elative quantity