An Improved Model for the Droplet Size Distribution in Sprays Developed From the Principle of Entropy Generation maximization

Transcription

1 ILASS Amercas, 9 th Annual Conference on Lqud Atomzaton and Spray Systems, oronto, Canada, May 6 An Improved Model for the Droplet Sze Dstrbuton n Sprays Developed From the Prncple of Entropy Generaton maxmzaton Xanguo L * and Meshen L Department of Mechancal Engneerng Unversty of Waterloo Waterloo, Ontaro NL G Canada Abstract he maxmum entropy prncple (MEP), whch has been popular n the modelng of droplet sze and velocty dstrbuton n sprays, s, strctly speakng, only applcable for solated systems n thermodynamc equlbrum; whereas the spray formaton processes are rreversble and non-solated wth nteracton between the atomzng lqud and ts surroundng gas medum. In ths study, a new model for the droplet sze dstrbuton has been developed based on the thermodynamcally consstent concept the maxmzaton of entropy generaton durng the lqud atomzaton process. he model predcton compares favorably wth the expermentally measured sze dstrbuton for droplets, near the lqud bulk breakup regon, produced by an ar-blast annular nozzle, a planar nozzle and a practcal gas turbne nozzle. herefore, the present model can be used to predct the ntal droplet sze dstrbuton n sprays. * Correspondng author

2 Introducton Snce the later 98s, Maxmum Entropy Prncple (MEP) method has been appled popularly n the spray feld to predct droplet sze and velocty dstrbuton and obtaned reasonable success. he MEP approach can predct the most lkely droplet sze and velocty dstrbutons under a set of constrants expressng the avalable nformaton related to the dstrbuton sought. he applcaton of MEP to spray modelng was poneered by Sellens and Brzustowsk [] and L and ankn []. When only frst prncples are used as constrants, the droplet sze pdf does not go to zero n the formulaton as the droplet sze approaches zero. An addtonal constrant, partton of surface energy, s added by Sellens. L solved ths problem by defnng a probablty dstrbuton as the number probablty of fndng a droplet n a spray wthn a volume bn. he number-based droplet sze dstrbuton n the resultant formulaton goes to zero as the droplet sze approaches zero. hs s consstent wth physcal ntuton and expermental observaton. Other nvestgators have also mplemented and developed ths method n the past decade. Ahmad and Sellens [] reached a concluson that predcton of the droplet sze dstrbuton s ndependent of the velocty dstrbuton and the constrants on momentum and knetc energy carry only velocty nformaton. However, MEP s not physcally consstent wth the real atomzaton process. Strctly speakng, MEP method s only applcable for solated systems n thermodynamc equlbrum, whereas the spray formaton process s nonsolated and rreversble. hs leads to the dffculty of agreement between MEP-based dstrbutons wth varous expermental data. he success of MEP method may be explaned as the devaton from equlbrum condton s small for the cases nvestgated. he major objectve of the current study s to formulate a new model on the predcton of droplet sze dstrbuton based on the thermodynamcally consstent concept the maxmzaton of entropy generaton (MEG) durng the spray formaton whch s actually a non-solated and rreversble process. he valdaton of the model s made by comparng the theoretcal predcton wth the expermental data measured for three types of nozzles, an ar-blast annular nozzle, a planar nozzle and a practcal gas turbne nozzle. he comparson ndcates good agreement between the model predcton and the measured data. Model Formulaton Consderng an atomzer, such as an ar-blast atomzer, that produces flat or concal lqud sheets at the atomzer ext, the atomzaton process starts at the atomzer ext and ends at some downstream locaton where droplets are formed. Here the spray s assumed steady and sothermal. he control volume, as shown n Fgure, s taken from the atomzer ext to the dropletformaton plane. Accordng to the second law n thermodynamcs, we have, ds dt cv m s m s S m s S source gen () For the steady state steady flow process, the rate of entropy ncrease wthn the control volume vanshes. Here, the mass flow rates may be dfferent between the two states (.e., the state at the nozzle ext and at the droplet formaton plane downstream) consderng the possble mass exchange between the system and the surroundng (condensaton or evaporaton). he dfference can be denoted as S m m m, whch s commonly referred to as the mass source term, and the correspondng specfc entropy s denoted as ssource. hen Eq. () can be recast as S gen S S S m s source m s m s S m s source () At the nozzle ext, the flow at state s n bulk lqud form wth free surface produced. he entropy at the nozzle ext s due to the lqud bulk and the surface tenson effect, hence, S S m s ( l) ( l) () In the breakup regon (state ), a multtude of droplets forms and the total surface area s ncreased dramatcally. Lqud phase exsts nsde each of the droplets. he total entropy s composed of two parts. One part s assocated wth the lqud bulk and s smlar to that at the state ; therefore may be denoted as S. he other part s due to the exstence of the numerous ndvdual droplets, represented as m s ( l ) ( l ) S ( d ), whch s drectly assocated wth the partcle nature of the droplets. Hence, S S S S m s ( d ) ( l ) ( d ) ( l) (4) he formaton of droplets s not determnstc but stochastc. he entropy S ( d ) at state should be assocated wth the probablty dstrbutons of the droplet szes. Dfferent dstrbuton results n dfferent amount of entropy. o evaluate entropy quanttatvely, an analogy between the present case and that of Gbbs ensem-

3 ble n statstcal thermodynamcs may be made here. he entropy due to the droplet nature can be derved as, wth detals gven elsewhere [4] S ( d ) k N B total P ln P (5) Accordng to the Gbbs equaton for a smple compressble substance wth free nterfaces, the lqud bulk entropy change can be expressed as ds( l ) du pdv da pvdp da (6) Snce the atomzaton process s assumed sothermal, the nternal energy, only a functon of temperature, s unchanged. Entropy s a thermodynamc property that depends only on the state of the system. Hence ntegratng Eq. (6) over the sothermal process results n, s ( l ) s ( l ) v p p a a (7) At the atomzer ext, a thn lqud sheet forms from the annular ar-blast atomzer, and the lqud pressure s almost the same as the surroundng ar pressure snce the curvature effect s small and neglgble. However, a crcular lqud jet forms for sold-cone sprays produced by small orfce atomzers, the lqud pressure s larger than the ambent ar pressure due to the surface tenson effect. However, the specfc value of p wll not affect the determnaton of the droplet sze dstrbuton as shown later. herefore, for smplcty, we assume p = p g. For lqud pressure at state, let s consder only one lqud droplet. For a lqud droplet n the ar, the pressure dfference across the free nterface s related to the surface tenson effect. herefore, the entropy generaton per unt mass flow rate n the atomzaton process s therefore s gen k B D 6 E D F D S m a P ln P P (8) where D D D, v E 8 D, and v F 6 4 p D. At state, the droplet sze dstrbuton s n realty the ntal dstrbuton for droplets just formed n a spray. here are nfnte sets of the probablty dstrbuton P that can satsfy the global constrants on the atomzaton process such as the conservaton laws. In accordance wth rreversble thermodynamcs, the least based (or the most realstc) dstrbuton s the one that maxmzes the amount of entropy generated for the naturally occurrng atomzaton process. For the present study the constrants mposed on the atomzaton process are the conservaton of lqud mass and the normalzaton condton. he normalzaton condton s P (9) he expresson for the mass conservaton under steady state can be wrtten n non-dmensonalzed form as: P D S m () where S m S m m denotes the dmensonless mass source term. o maxmze the entropy generaton s gen under the constrants of Eq. (9) and Eq. (), the Lagrange s method s adopted here. hat yelds, P exp D D D () o obtan the probablty of fndng the droplets whose volume s between V n and V n, we have to evaluate Vn Vn n V V n P exp D D D P V Vn Vn () It s generally regarded that the droplet sze and velocty n sprays vary contnuously rather than dscretely. herefore, the subscrpt can be dropped, and the summaton form of Eq. () becomes an ntegral over the droplet sze; that s Vn Vn f d V Dn Dn f d D Dn Dn D exp D D D d D () where D n and D n are the droplet dameters correspondng to the droplet volume V n and V n respectvely; and f s the contnuous droplet sze probablty densty functon (pdf). hus, f D exp D D D (4) he above formulaton s equvalent to the dstrbuton from the MEP usng an extended set of constrants. In addton to the normalzaton condton and the conservaton of mass, two other constrants are wrtten on the bass of the defnton of the mean dameters D and D. All the constrants that are needed to determne the unknown coeffcents s are lsted below.

4 max D Dmn Dmax Dmn Dmax Dmn Dmax Dmn D exp D D D d D (5) D D D d D D exp D (6) D D D d D 4 D exp D (7) D D D d D S m 5 D exp (8) Results and Dscusson A modfed Newton-Raphson method s used to solve the set of equatons to obtan the Lagrangan multplers. he expermental data are obtaned from the measurements on an annular ar-blast nozzle, a planar nozzle and a practcal gas turbne nozzle usng a commercal phase Doppler Partcle Analyzer (PDPA). he detals on measurement of the annular ar-blast nozzle can be found n [5] and on the planar nozzle and the gas turbne nozzle can be found n [6]. It s assumed that the surroundng gas medum s fully saturated and therefore no mass transfer occurs between the lqud and the ambent gas from the nozzle ext to the breakup regon. herefore the mass source term s set to zero. In the model, the control volume for the theoretcal formulaton s taken from the atomzer ext to the lqud sheet breakup regon. Due to the dffculty encountered n measurng exactly at the breakup regon, all the measurements are taken beyond that regon. It s found that the dscrepancy between the model predcton and measurement data decreases wth the reducton of the dstance from the measurement locaton to the atomzer ext. he comparsons between the predcted and measured sze probablty dstrbuton for the arblast annular nozzle and the planar nozzle are llustrated n Fg. and Fg., respectvely. he comparson for two cases for each nozzle s shown here. Another comparson s made for the actual gas turbne nozzle provded by Pratt & Whtney Canada (PWC) and lsted n Fg. 4. he feature of the former dstrbuton, Eq. (4), s that the mnmum dameter s always equal to zero. However, the measurement for the PWC nozzle ndcates that the mnmum dameter starts wth a non-zero value. By ntroducng the mnmum dameter D nto the equaton as shown below, the dstrbuton s modfed to take ths fact nto account f D D D D (9) exp D he mnmum dameter s taken from the expermental data and normalzed wth mass mean dameter. he unknown parameters, s, are stll determned as before from the constrants equatons (5) - (8). It s seen that the model predcton agrees very well wth the measured droplet sze probablty dstrbuton. he wdth of the dstrbutons matches each other well for the skewed mono-modal dstrbutons, so do the poston and magntude of the peak of the dstrbuton. Conclusons A new model for the droplet sze dstrbuton n sprays has been formulated based on the physcally consstent concept the maxmzaton of entropy generaton durng the non-solated and rreversble lqud atomzaton process. A comparson between the model predcton and expermentally measured droplet sze dstrbuton produced by an ar-blast annular nozzle, a planar nozzle and a practcal gas turbne nozzle ndcates that the model predcton s n satsfactory agreement wth the measurements. he present model can be used to predct the ntal droplet sze dstrbuton n sprays. Further work s under way to extend the present model to the jont droplet sze and velocty dstrbuton. Nomenclature a surface area per unt mass [m /kg] D arthmetc mean dameter [m] D surface area mean dameter [m] D mass or volume mean dameter [m] k B constant [kj/k] m mass flow rate [kg/s] n droplet number produced per unt tme N total total number of droplets produced n a spray per unt tme p pressure [N/m ] s specfc entropy [kg/(kkg)] S entropy flow rate [kj/(ks)] S gen entropy generaton per unt tme [kj/(ks)] temperature [K] u specfc nternal energy [kj/kg] v specfc volume [m /kg] V volume [m ] sothermal compressblty [m /N] surface tenson [N/m] Subscrpts state state c.v. control volume g gas d droplet

5 l lqud References. Sellens, R. W. and Brzustowsk,. A., A Predcton of the Drop Sze Dstrbuton n a Spray from Frst Prncples, Atomzaton Spray echnol., Vol., 89: (985). L, X. and ankn, R. S., Droplet Sze Dstrbuton: A Dervaton of a Nukyama-anasawa ype Dstrbuton Functon, Combust. Sc. and ech., Vol. 56, 65:76 (987). Ahmad, M. and Sellens, R.W. Atomzaton and Sprays, 9: (99) 4. L, X., L, M. and Fu, H, Atomzaton and Sprays, accepted for publcaton 5. Fu, H., Master s thess, Unversty of Waterloo () 6. Km, W.., Mtra, S.K., L, X., Procw, L.A., Hu,.C.J., Part. Part. Syst. Charact., 5:49 () Control Volume Fgure. Schematc of the control volume chosen (a) Fgure. Comparson between the model predcton and measured droplet sze probablty dstrbuton for the annular nozzle (a: est 6; b: est 6) [5], and data measured at the spray centre-lne at a dstance of mm from the nozzle ext (b)

6 (a) Fgure. Comparson between the model predcton and measured droplet sze probablty dstrbuton for the planar nozzle (a: case II; b: case IV) [6] (b) (a) (b) Fgure 4. Comparson between the model predcton and measured droplet sze probablty dstrbuton for the practcal gas turbne nozzle (a: case II; b: case IV) [6]