Interface dynamics in the transcritical flow of liquid fuels into high-pressure combustors

In e ace Dynamics in he T ansc i ical Flow o Liquid Fuels in o High-P essu e Combus o s Llu´ıs Jo e∗and Ja ie U zay† Cen e o Tu bulence Resea ch, S an o d Uni e si y, S an o d, CA 94305, USA Rocke engines and new gene a ions o high-p essu e gas u bines and diesel engines o en imes in ol e a omiza ion, apo iza ion and combus ion o p opellan s injec ed a subc i ical empe a u e in o an en i onmen a a p essu e la ge han ha o he co e- sponding c i ical poin s o he indi idual componen s o he mix u e. This class o a- jec o ies in he he modynamic space ha e been e e ed o as ansc i ical. As a esul , ela i ely sha p in e aces may pe sis in p essu e and empe a u e condi ions whe e hey we e no expec ed o exis . This is pa icula ly ele an in hyd oca bon- ueled mix u es ha display a c i ical-poin ele a ion p ope y by which he wo-phase egion ex ends up o p essu es much la ge han he c i ical p essu es o he indi idual componen s. As a consequence, linea he modynamic ajec o ies emula ing ypical injec ion condi ions e- quen ly pass h ough he wo-phase egion, hus indica ing ha he mix u e may become sepa a ed he e in o liquid and apo phases by an in e ace. In his s udy, a se o mod- i ica ions o he Na ie -S okes equa ions o mul i-componen lows is p oposed based on di use-in e ace heo y in o de o ea he eme gen and anishing in e aces in he same low ield. This equi es app op ia e al e a ions o he s ess enso and di usi e luxes o hea and species. The esul ing anspo o mula ion is pa icula ized o bina y mix- u es, as well as collapsed o single-componen g adien heo y o s a iona y quasi-plana apo -liquid in e aces. Nomencla u e a, b Equa ion o s a e coe icien s cMola densi y, mol/m3 DDi usion coe icien , m2/s ESpeci ic o al ene gy, J/kg eSpeci ic in e nal ene gy, J/kg FHelmhol z ee ene gy, J Speci ic Helmhol z ee ene gy, J/kg GGibbs ee ene gy, J hSpeci ic en halpy, J/kg J,JSpecies di usion luxes, kg/(s·m2) NNumbe o species nNumbe o mols NAA ogad o’s numbe , 1/mol PP essu e, ba q,QHea di usion luxes, J/(s·m2) qcHea conduc ion lux, J/(s·m2) R0Ideal gas cons an , J/(K·mol) sSpeci ic en opy, J/(kg·K) ˙sp od En opy p oduc ion a e, J/(K·s·m3) TTempe a u e, K Time, s eloci y ec o , m/s Mola olume, mol/kg WMolecula weigh , kg/mol XMola ac ion YMass ac ion ZComp essibili y ac o η, ζ Shea and bulk iscosi ies, Pa·s κG adien -ene gy coe icien , m7/(kg·s2) λThe mal conduc i i y, W/(m·K) ¯µMola chemical po en ial, J/mol ρDensi y, kg/m3 σSu ace- ension coe icien , N/m τ,KS ess enso s, N/m2 ϕFugaci y coe icien Subsc ip s cC i ical poin i, j Species index NL Non-local quan i y ∗Pos doc o al Resea che . E-mail: [email p o ec ed] †Senio Resea ch Enginee . E-mail: ju za[email p o ec ed] 1 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 53 d AIAA/SAE/ASEE Join P opulsion Con e ence 10-12 July 2017, A lan a, GA 10.2514/6.2017-4940 Copy igh © 2017 by he Ame ican Ins i u e o Ae onau ics and As onau ics, Inc. All igh s ese ed. AIAA P opulsion and Ene gy Fo um I. In oduc ion The cha ac e is ic p essu e and empe a u e o he bu n gases in he combus o o Apollo’s Sa u n-V F-1 ocke engines we e 77.5 ba and 3572 K. The engine ope a ed a mix u e o liquid oxygen (c i ical p essu e Pc= 50 ba ; c i ical empe a u e Tc= 155 K) and RP-1 uel (Pc= 21 ba ; Tc= 662 K). The ypical injec ion empe a u es o RP-1 and liquid oxygen we e 311 K and 97 K, espec i ely. The p opellan s we e injec ed sepa a ely in o he combus ion chambe h ough a mul i-pe o a ed pla e ha o ced hei a omiza ion and mixing by deli e ing hem as mu ually impinging je s. The F-1 ocke engine is a classical example o a con empo a y gene a ion o high-powe ae o-p opulsion de ices ha pushed he ope a ing condi ions abo e he c i ical poin s o he p opellan s. The echnical command o such high combus ion p essu es ep esen ed a co ne s one in enabling he la ge powe and speci ic impulse necessa y o a manned mission o he Moon. Howe e , he ex eme ope a ing condi ions b ough along se e al enginee ing challenges, among which he mos c i ical one o he Apollo P og am p o ed o be ha o combus ion ins abili ies c ea ed by he injec ion con igu a ion.1Ea ly analyses o his p oblem ecognized he concep ual di icul ies ela ed o desc ibing a omiza ion, apo iza ion and combus ion o p opellan s injec ed a subc i ical empe a u es in o an en i onmen a a p essu e la ge han ha o hei co esponding c i ical poin s.2This class o ajec o ies in he he modynamic space ha e been e e ed o as ansc i ical in mo e ecen li e a u e.3The dele e ious combus ion p essu e wa es p esen in he i s designs o he F-1 ocke engines we e emo ed by modi ying he injec ion pla e wi h a numbe o ba les, since he phenomenon was sensi i e o he cha ac e is ic dis ance om he injec ion pla e o he beginning o he combus ion zone. Pa adoxically, ha dis ance appea s o be a quan i y ha dly possible o p edic e en nowadays wi h cu en heo e ical and compu a ional models o easons explained la e in his sec ion. The s udy o ansc i ical dynamics also inds impo an applica ions in ecen designs o gas u bine engines o je p opulsion. In pa icula , cu en ends in ul a-low emission echnologies o a ia ion indus y a e gea ing combus o s owa d lean bu n and high p essu e a ios. Lean bu n aims a dec easing ni ogen oxides by a oiding hei peak p oduc ion a e a s oichiome y, bu equi es an in ense dilu ion and mixing o he uel wi h he ai en e ing he combus o . Simila ly, high p essu e a ios a e employed o inc ease engine powe and educe emissions o ca bon oxides and unbu n hyd oca bons. These p essu e a ios yield combus o p essu es o o de 45 ba a akeo , while mos je uels ha e c i ical p essu es in he ange 15-22 ba . In his way, ansc i ical condi ions likely de elop in he combus o ha may ha e an impac on he uel-ai mixing cha ac e is ics because o he al e a ion o he classic sp ay a omiza ion dynamics expec ed a lowe p essu es. Howe e , he ex en o hese e ec s emains mos ly unknown. Recen expe imen al obse a ions o simila aspec s ha e been made wi hin he con ex o diesel engines by Dahms and cowo ke s.4 The challenge o p edic i e calcula ions o ansc i ical phenomena is he complexi y o he ansi ional cha ac e o liquid b eakup, dispe sion and apo iza ion as condi ions app oach he c i ical poin . To unde - s and his, conside i s he subc i ical limi in which he liquid is injec ed in o a ho gas en i onmen whose p essu e is lowe han he c i ical p essu e o he liquid. In his limi , he liquid- o-gas densi y a io is la ge, and he liquid a omizes ollowing classic dynamics epo ed in se e al s udies.5The esul ing hickness o he liquid-gas in e ace is clea ly no in he con inuum ange and can be aken o be in ini esimally small in hyd odynamic scales. Addi ionally, a ela i ely la ge amoun o ene gy mus be p o ided by he gas in o de o hea up and apo ize he liquid phase. As a esul , he beginning o he apo iza ion s age is delayed a he downs eam un il he liquid has b oken up in o a su icien ly dilu e cloud o d ople s.6Howe e , as he p essu e is inc eased abo e he c i ical poin o he liquid, he liquid-gas densi y a io dec eases because o an inc ease in he densi y o he gas en i onmen . The in e ace becomes hicke as he liquid ecei es hea om he combus o en i onmen and i s empe a u e nea s he c i ical empe a u e. This is accompanied by a dec ease in su ace ension and apo iza ion en halpy, in a manne ha makes he a omiza ion p ocess o inc easingly esemble one a in ini e Webe numbe s ollowed by apid mixing wi h he gas en i onmen wi hou signi ican ene gy ba ie o apo iza ion. Hea y hyd oca bons equi e inc emen s o empe a u e o o de 300−400 K in o de o each hei c i ical poin s. Fo hese uels, i is concep ually plausible ha ini e su ace- ension and apo iza ion e ec s pe sis longe in he combus o . Howe e , he p oblem becomes exceedingly complex in mix u es o hyd oca bons and ypical oxidize s, in ha he esul ing phase diag am displays c i ical-poin ele a ion p ope ies ha , depending on he local composi ion, may lead o locally subc i ical condi ions e en i he p essu e is much la ge han he co esponding c i ical alues o he sepa a e componen s. This may lead o he pe sis ence o ela i ely sha p in e aces in p essu e and empe a u e condi ions whe e hey we e no expec ed o exis . 2 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 No e ha o single-componen sys ems consis ing o a liquid a omizing in i s own apo , he desc ip ion o he dynamics becomes much simple , in ha he e is p ac ically no dis inc ion be ween he wo phases a p essu es abo e he c i ical poin . Acco dingly, he in e ace disappea s, and he su ace ension anishes. The single-componen case, howe e , is no he one ound in mos p ac ical applica ions. Addi ional ac o s ha p e en u he unde s anding o ansc i ical dynamics a e he la ge unce ain ies in high-p essu e physical p ope ies o complex mix u es o eac an s and combus ion p oduc s, and he lack o quan i a i e expe imen al diagnos ics o model alida ion in such ex eme en i onmen s. This wo k add esses basic heo e ical aspec s o low anspo unde ansc i ical condi ions. I is o ganized as ollows. In Sec ion II, a phase diag am o a ypical hyd oca bon- ueled sys em is desc ibed ha illus a es he he modynamic space o solu ions a high p essu es and desc ibes c i ical-poin ele a ion p ope ies. Sec ion III is de o ed o a de i a ion o a se o conse a ion equa ions ha simul aneously conside su ace- ension e ec s along wi h ela i ely pe meable in e aces. Las ly, conclusions a e p o ided in Sec ion IV . This epo builds on ecen analyses om Re .7by p o iding di e en pe spec i es o he o mula ion and addi ional conside a ions ha may be o p ac ical use in he compu a ion o hyd oca bon- ueled ansc i ical lows. II. Phase diag ams and c i ical-poin ele a ion p ope ies o hyd oca bon- ueled mix u es Some insigh can be gained in o sys em ajec o ies leading o ansc i ical egimes by s udying apo - liquid equilib ium cu es. These co espond o bounda ies o he egion wi hin which he sys em sepa a es in o wo o mo e di e en phases ac oss an in e ace. The phase diag am illus a es he he modynamic space o solu ions o he p oblem, bu does no p o ide any in o ma ion abou he dynamics. Fo ins ance, he liquid- uel s eam and he ho gas en i onmen may be bo h ep esen ed by wo di e en poin s in he phase diag am, bu he he modynamic ajec o ies o he mix u e elemen s a e solu ions o he conse a ion equa ions and he associa ed bounda y condi ions. In his sec ion, mix u es o n-dodecane (Pc,1= 18 ba , Tc,1= 658 K) and ni ogen (Pc,2= 34 ba , Tc,2= 126 K) a e s udied since hey a e commonly conside ed as su oga es o high-p essu e uel/ai mix u es o gas u bines and diesel engines.8, 9 The de ails o he compu a ion o phase en elopes will be omi ed he e as hey a e a classical subjec ea ed in e e ence ex books.10 In gene al, apo -liquid equilib ium cu es a e ob ained by compu ing so- lu ions o he ze oed second-o de a ia ion PN i=1 PN j=1(∂2F/∂ni∂nj)T,ρ,nk6=i,j ∆ni∆nj= 0 o he Helmhol z ee ene gy F, whe e ∆nia e non-ze o pe u ba ions o moles o species i,ρis he densi y, and Nis he numbe o componen s o he mix u e (N= 2 in his example). Simila ly, he calcula ion o he c i ical line, which co esponds o he cu e connec ing he c i ical poin s o di e en mix u e composi ions, ollows he me hodology in oduced in Re .11 The c i ical poin o mul i-componen mix u es is he he modynamic s a e a which he bubble poin and he dew poin con e ge (which does no necessa ily occu a in lec ion poin s o isoba s), and he e o e co esponds o a s able poin a he limi o he modynamic s abili y. They a e ob ained om he solu ions o he equa ion PN i=1 PN j=1 PN k=1(∂3F/∂ni∂nj∂nk)T,ρ,n`6=i,j,k ∆ni∆nj∆nk= 0 o he hi d-o de a ia ion o F. The equa ions desc ibed abo e a e supplemen ed wi h he Peng-Robinson12 equa ion o s a e, which is o mally in oduced la e in Eq. (10). The coe icien s aand bo he equa ion o s a e depend on he c i ical empe a u es, c i ical p essu es and acen ic ac o s o he indi idual mix u e componen s, as well as on he local empe a u e and mix u e composi ion. They a e ob ained by i s compu ing he indi idual alues o he coe icien s o each species, aiand bi, as speci ied in Re .,13 which a e combined using an de Waals mixing ules as a= N X i=1 N X j=1 XiXjaij, b = N X i=1 Xibiwi h aij = (1 −ηij)√aiaj,(1) whe e Xiis he mola ac ion o species i, and η1,2= 0.1561 is a bina y in e ac ion pa ame e i ed o expe imen al da a in Re .14 The apo -liquid equilib ium cu es esul ing om he compu a ions a e shown in Figu e 1 in a h ee- dimensional space o med by P,Tand he mass ac ion o n-dodecane Y. The h ee-dimensional wo- phase egion, which is enclosed unde he su ace en eloping he cu es, eaches much la ge p essu es han he c i ical p essu e o each componen . In p ac ical e ms, his is ansla ed in o he ac ha an n-dodecane liquid je lowing in o a ni ogen ambien a p essu es much la ge han Pc,1= 18 ba may 3 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 Figu e 1. Vapo -liquid equilib ium cu es (solid lines) o n-dodecane/ni ogen mix u es colo ed by p essu e, wi h Yindica ing he mass ac ion o n-dodecane. Pu e-subs ance boiling lines o ni ogen (dashed ed) and n-dodecane (dashed blue) a e shown, along wi h hei co esponding c i ical poin s (squa es). Expe imen ally measu ed c i ical poin s a e deno ed by iangles. unde go ansc i ical ajec o ies ha c oss he wo-phase egion. As a esul , such low may display emnan e ec s o su ace ension and a omiza ion cha ac e is ics simila o lowe p essu e je s ha in p inciple we e no expec ed o be obse ed in hese he modynamic condi ions, as shown in expe imen s in Re .8This c i ical-poin ele a ion p ope y is also illus a ed by he di e gence o he he c i ical line a i ing o he ni ogen side, as obse ed in Figu e 2, which indica es ha he wo-phase egion is unbounded in p essu e. Con e sely, he c i ical line s a ing a he ni ogen c i ical poin mee s a liquid-liquid-gas phase-equilib ium line (indis inguishable om he ni ogen boiling line in he scales o Figu e 2) a an uppe c i ical end poin . The h ee-phase equilib ium line con inues owa d lowe p essu es and empe a u es be ween he boiling lines o he wo pu e componen s, he eby sugges ing ha he c ossing o he h ee-phase equilib ium egion is only ele an a p essu es lowe han Pc,2= 34 ba and ac oss a e y na ow ange o empe a u es a ound Tc,2= 126 K. The phenomena o di e gence o he c i ical line and occu ence o h ee-phase equilib ia a e ypical in mix u es classi ied as class-II/ ype-III acco ding o he analysis in Re .15 This g oup o mix u es, o which o he n-alkane/ni ogen sys ems also belong, is cha ac e ized by indi idual componen s wi h e y di e en c i ical empe a u es. I should be s essed ha he compu a ion o c i ical poin s in complex mix u es in ol es a numbe o assump ions and model pa ame e alues ha ind li le jus i ica ion on physical g ounds. Fo ins ance, he mixing ules (1) co espond o an ad-hoc mola weigh ing o he indi idual coe icien s aiand bi, whose ex- p essions depend on calib a ed in e ac ion pa ame e s and measu ed c i ical poin s o he pu e subs ances.13 Howe e , i is o some in e es o no e ha he esul ing di e gen end o he c i ical line compu ed om he apo -liquid equilib ium ag ees well wi h he alues expe imen ally ob ained in Re .,14 as shown in Figu es 1 and 2. The phase diag am acili a es he unde s anding o he he modynamic ajec o ies in ol ed in he in- jec ion o hyd oca bon uels in o high-p essu e en i onmen s. As an illus a ion, conside he examples o linea he modynamic ajec o ies ollowed by mix u e elemen s in he p oblem o a liquid n-dodecane je in- jec ed in a ni ogen en i onmen a 900 K, which a e p o ided in Figu e 3. Two uel injec ion empe a u es, co esponding o 363 K (case 1) and 563 K (case 2), a e s udied, along wi h h ee ni ogen-en i onmen p essu es, namely, 50,100,and 200 ba . The ajec o ies a e supe imposed on maximum- empe a u e cu es bounding he wo-phase egion a each p essu e. No e ha he ajec o ies esul ing om in eg a ion o he conse a ion equa ions may no be gene ally linea .16 Fo all p essu e alues conside ed in case 1, he mix u e elemen s s a as comp essed liquids in he n-dodecane s eam. As hea is supplied om he su ounding gas, he mix u e elemen s en e he wo-phase egion whe e hey necessa ily sepa a e in o liquid and apo phases by an in e ace whe e su ace- ension o ces ope a e. The mix u e elemen s e en ually exi he wo-phase egion and change phase o a supe c i ical s a e while mixing wi h he su ounding ni ogen gas. On he o he hand, in case 2, he in e sec ion wi h he wo-phase egion is comple ely a oided o he la ges p essu e alue conside ed in he ni ogen en i onmen . 4 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 Figu e 2. Two-dimensional p ojec ions o he apo -liquid equilib ium cu es on he empe a u e-p essu e plane o a n-dodecane/ni ogen sys em, including desc ip ion o he diag am o Y= 0.5(le panel) along wi h a ays o apo - equilib ium cu es o di e en alues o Y( igh panel). Re e o cap ion in Figu e 1 o he emaining symbols and lines. The esul ing he modynamic pa h in ol es no sus ained sepa a ion o he wo componen s h ough an in e ace. In hese condi ions, a omiza ion and mixing a e solely limi ed by he a e o supe c i ical mass di usion. Figu e 3 demons a es ha he luid dynamical desc ip ion o p oblems whe e ansc i ical condi ions a e a e sed equi es he ea men o eme ging and anishing in e aces in he same low ield depending on he local he modynamic condi ions o he mix u e elemen s as hey mo e ac oss he low ield. A se o modi ica ions o he Na ie -S okes equa ions a e p esen ed below ha a emp o enable his ea men . III. Conse a ion equa ions o ansc i ical lows This sec ion desc ibes a gene al o mula ion o he conse a ion equa ions and associa ed anspo luxes based on di use-in e ace heo y o mul i-componen lows. The de elopmen begins by ou lining he main cha ac e is ics o he heo y, and con inues wi h he gene al conse a ion equa ions along wi h de i a ions o he anspo luxes om he modynamic conside a ions. III.A. Non-local he modynamic e ec s The heo e ical ounda ions o he di use-in e ace app oach we e i s es ablished o single-componen sys ems in he modynamic equilib ium by an de Waals.17 I was la e ex ended o s udy bina y mix u es nea he c i ical poin by Cahn and Hillia d.18 Mo e ecen ly, he app oach has been coupled o he equa ions o luid mo ion o single-componen sys ems.19 The modynamic in es iga ions o ansi ion be ween wo- and single-phase s a es using he di use-in e ace app oach o ine and chemically eac ing mul i-componen sys ems ha e also p o ided unde s anding o he beha io o p opellan s in high-p essu e en i onmen s.7,20 The di use-in e ace heo y es s upon he ac ha he wo-phase egion wi hin he liquid- apo equi- lib ium cu e in he phase diag am is he modynamically uns able, in ha no s able he modynamic s a e exis s ha desc ibes a spa ially homogeneous mix u e o liquid and apo . This is pe haps easily isualized by b inging a single-componen luid in a closed essel o i s apo p essu e. The subs ance ends o sepa a e in wo phases o unequal densi y bounded by a hin ansi ion laye whe e capilla y o ces become impo an . The gene al esul s o he di use-in e ace heo y a e aimed a desc ibing he mechanics o he ansi ion laye as well as he luxes o ene gy and mass ac oss i . The o me co espond o amilia su ace- ension o ces eme ging om he esis ance o he in e ace o ge de o med, while he la e ep esen apo iza ion and di usi e mixing. 5 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 Figu e 3. Examples o linea he modynamic ajec o ies (solid lines) supe imposed on maximum- empe a u e cu es (dashed colo ed lines) bounding he wo-phase egion o n-dodecane/ni ogen mix u es a ni ogen-en i onmen p es- su es o 50,100,and 200 ba . Symbols deno e he he modynamic condi ions o he ni ogen en i onmen (le -hand-side squa e) and n-dodecane s eam a 363 K (lowe igh -hand-side squa e) and 663 K (uppe igh -hand-side squa e). The diamond symbols ep esen c i ical poin s a he co esponding p essu e. The desc ip ion o he s uc u e o he ansi ion laye equi es conside a ion o non-local he modynamic po en ials, whe e he non-locali y is ep esen ed by g adien s o selec ed a iables. Addi ional conside a ions based on he second p inciple o he modynamics ypically p eclude non-locali y o be exp essed only in e ms o composi ion o densi y g adien s (e.g., see Re .21 o de ails on he ma hema ical jus i ica ion). The analysis is acili a ed when he deg ee o non-locali y is assumed o be small, wi h he cha ac e is ic leng h o he composi ion g adien s being la ge compa ed o in e molecula dis ances, which ypically limi s he heo y o si ua ions when he in e ace is ela i ely hick compa ed o he molecula mean ee pa h, as in condi ions nea and abo e he c i ical poin . In his limi , he dis u bances o he local he modynamic s a e a e p opo ional o he squa e o he composi ion g adien s in he i s app oxima ion.18 Fo ins ance, he non-local co ec ions o he speci ic alues o Helmhol z ee ene gy , in e nal ene gy eand en opy s a e NL = +1 2ρ N X i=1 N X j=1 κij∇ρi∇ρj, eNL =e+1 2ρ N X i=1 N X j=1 κe ij∇ρi∇ρj, sNL =s+1 2ρ N X i=1 N X j=1 κs ij∇ρi∇ρj,(2) whe e ρis he mix u e densi y, ρiis he pa ial densi y o species i, and Nis he numbe o species. Addi ionally, κij,κe ij and κs ij a e g adien -ene gy coe icien s, which can be compu ed di ec ly as a unc ion o collision pa ame e s om kine ic- heo y conside a ions o in e ac ions be ween molecules in egions subjec ed o mac oscopic densi y g adien s (e.g., see Re .22 and Chap e 1 in Re .23). Since he g adien -ene gy coe icien s κij a e ela ed o he in e ace hickness and su ace ension, hei p ecise cha ac e iza ion is cen al o he p edic ions o he di use-in e ace heo y. Howe e , app op ia e o mula ions o his pa ame e s a e lacking, and mos in es iga ions u ilize ela ions o he ype κij = √κiiκjj =κji o he c oss-in luence coe icien s i6=j, along wi h empi ical co ela ions o he indi idual coe icien s κii such as24 ln κii aibi2/3N8/3 A=κ0,i +κ1,i ln 1−T Tc,i +κ2,i ln 1−T Tc,i 2 (3) 6 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 o T/Tc,i ≤0.95, whe e Tc,i is he c i ical empe a u e alue, and NAis he A ogad o’s numbe . In Eq. (3), he co ela ion coe icien s κ0,i,κ1,i and κ2,i a e usually calib a ed based on expe imen al measu emen s o su ace ension, while he pa ame e s aiand bico espond o coe icien s o he equa ion o s a e, as desc ibed in Sec ion II. No e ha models such as Eq. (3) ypically yield κii = 0 abo e he c i ical empe a u e o he co esponding species, as sugges ed by he ac ha he su ace ension anishes o single-componen sys ems abo e he c i ical poin . Fo ins ance, in he n-dodecane/ni ogen sys em desc ibed in Sec ion II, he ele an g adien -ene gy coe icien becomes ha o he n-dodecane, κ1,1, since he empe a u e in he low is la ge han he c i ical empe a u e o ni ogen e e ywhe e (i.e., κ2,2=κ1,2=κ2,1= 0). Exac exp essions ela ing he g adien -ene gy coe icien s κij,κe ij and κs ij can be easily de i ed by subs i u ing he ela ions (2) in o he de ini ion o he local Helmhol z ee ene gy, =e−Ts, wi h s=−(∂ /∂T)ρ,ni, he eby yielding κe ij =κij +T∂κij ∂T ρ,nk and κs ij =−∂κij ∂T ρ,nk .(4) The conside a ion o non-local he modynamic po en ials, as in Eqs. (2)-(4), leads o he eme gence o in e ace- ela ed anspo luxes and mechanical s esses in he conse a ion equa ions as desc ibed below. III.B. Conse a ion equa ions The desc ip ion o hin in e aces and hei dynamics in conjunc ion wi h he ou e luid mo ion in a single Eule ian ield equi es non- i ial ex ensions o he Na ie -S okes conse a ion equa ions. In p inciple, he de i a ion o hese modi ica ions om molecula conside a ions and i s p inciples is a di icul ask due o he lack o a clea physical unde s anding o he molecula s uc u e o luids ac oss he c i ical poin . In his s udy, a phenomenological app oach is ollowed based on a linea augmen a ion o he de ia o ic pa o he s ess enso , τ, and he hea and species di usion luxes, qand Ji, wi h he co esponding in e acial anspo e ms K,Qand Jide i ed om he di use-in e ace heo y. These, as shown below, can be made o sa is y ce ain condi ions o en opy maximiza ion ha a e in acco d wi h he second law o he modynamics. The esul ing conse a ion equa ions o mass, momen um, species and o al ene gy a e ∂ρ ∂ +∇·(ρ )=0,(5) ∂(ρ ) ∂ +∇·(ρ ⊗ ) = −∇PNL +∇·(τ+K),(6) ∂(ρYi) ∂ +∇·(ρ Yi) = −∇·(Ji+Ji), i = 1, ..., N, (7) ∂(ρE) ∂ +∇·(ρ E) = −∇·(PNL )−∇·(q+Q) + ∇·[(τ+K)· ],(8) which desc ibe he con inuum dynamics o a mul i-phase, mul i-componen luid o Nspecies ha mo es a a mass-a e aged eloci y and has a local densi y ρand o al ene gy E, and which may con ain hin in e aces sepa a ing di e en phases. In his o mula ion, Yiis he mass ac ion o species i,q=qc+PN i=1 hiJiis he sum o he hea conduc ion and he ene gy lux by in e -di usion, hiis he pa ial speci ic en halpy, and PNL a non-local he modynamic p essu e de ined as PNL =P−1 2 N X i=1 N X j=1 κij∇ρi∇ρj.(9) The con enience o ede ining p essu e as in Eq. (9), will become clea in Sec ion III.C. In Eq. (9) he local he modynamic p essu e Pcan be ob ained, o ins ance, om he cubic equa ion o s a e12 P=R0T −b−a 2+ 2b −b2,(10) whose u iliza ion is bene icial a he high p essu es conside ed he e. In he no a ion, =W/ρ is he mola olume, wi h W= (Pi=1 Yi/Wi)−1 he mean molecula weigh . The coe icien s aand b, which co espond o mix u e-a e aged e sions o he pu e-subs ance ones aiand bias in Eq. (1), accoun o eal-gas e ec s 7 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 such as ini e packing and inc eased in e molecula in e ac ions a la ge densi ies and p essu es. I should be s essed ha he choice o Eq. (10) is no cen al o he di use-in e ace o malism inso a as i ep oduces he mul i alued cha ac e o he mix u e densi y in condi ions o phase change. No e ha se e al o he equa ions o s a e a e a ailable in he li e a u e ha ha e simila cha ac e is ics.25,26 Chemical con e sion sou ces ha e been excluded o simplici y om Eq. (7). Gas-phase combus ion eac ions end o occu a om in e aces and in egions whe e he local mass ac ion o uel apo is su icien ly small o wa an s oichiome ic p opo ions. Howe e , his app oxima ion may no be app op ia e i he mal decomposi ion o he liquid uel plays an impo an ole in modi ying he in e ace p ope ies. In he species conse a ion equa ion (7), he di e en Ncomponen s o he mix u e a e desc ibed by hei co esponding mass ac ions i espec i ely o hei phase s a e. No e ha his is in con as wi h adi ional ea men s o dispe sed mul i-phase lows, whe e he gas and liquid mass ac ions a e desc ibed by hei co esponding conse a ion equa ions. In he di use-in e ace o mula ion, he phases a e sepa a ed by in e aces in he modynamic condi ions co esponding o he wo-phase egion. In hose si ua ions, he in e acial s ess enso Kin he momen um equa ion (6) p o ides in o ma ion abou he dynamical equilib ium o he sepa a ing in e ace, while he luxes Qand Jimodi y he anspo o hea and mass ac oss he in e ace acco dingly. The high-p essu e cha ac e is ics o he anspo luxes a e desc ibed in de ail in Sec ion III.C. A comple e desc ip ion o he mix u e s a e equi es speci ica ion o he analy ical o m o he he mo- dynamic po en ials. A high p essu es, inc easing depa u es om he ideal-gas heo y a e obse ed, and consequen ly, de i a ion o mo e complex exp essions a e necessa y. A common app oach o exp ess high- p essu e eal-gas he modynamic po en ials is o decompose hem in o he sum o hei ideal-gas coun e pa s (deno ed below by he supe sc ip 0) and depa u e unc ions ha measu e de ia ions wi h espec o he ideal-gas beha io .28 Fo ins ance, he depa u e unc ion o he mola en halpy is ¯ h−¯ h0=ZT T0 C0 pdT +ZP 0 −T∂ ∂T PdP, (11) whe e ¯ h0and C0 pa e he ideal-gas e e ence mola alues o en halpy and cons an -p essu e hea capaci y, wi h T0= 298.15 K. Subsequen ly, he mola in e nal ene gy can be ob ained om he en halpy de ini ion as ¯e=¯ h−P . (12) These exp essions a e alid o any equa ion o s a e. Exac o ms o he depa u e unc ions o mul i-species mix u es can be ound in Re .29 o he Peng-Robinson equa ion o s a e. Simila conside a ions apply o he mola chemical po en ial ¯µi=∂G ∂niT,P,nj6=i (13) de ined as he pa ial mola o he Gibbs ee ene gy G. The co esponding decomposi ion is gi en by ¯µi= ¯µ0 i+ R0Tln ϕi,(14) whe e ¯µ0 i(T, P) is he ideal-gas coun e pa . In Eq. (14), he depa u e unc ion in ol es he dimensionless ugaci y coe icien ϕi= i/(XiP), which ep esen s he a io o he ugaci y i o he pa ial p essu e. In pa icula , o he Peng-Robinson equa ion o s a e, he loga i hm o he ugaci y coe icien becomes ln ϕi=bi b(Z−1) −ln (Z−B)−A 2√2B"2PN j=1 Xiaij a−bi b#ln "Z+1 + √2B Z+1−√2B#,(15) whe e A=aP/(R0T)2,B=bP/(R0T), and he coe icien s aand ba e gi en by Eq. (1). Addi ion- ally, Z=P /(R0T) is he comp essibili y ac o , which quan i ies he depa u es om he e e ence alue Z= 1 co esponding o he ideal-gas equa ion o s a e. Figu e 4 shows he ugaci y coe icien s o an n- dodecane/ni ogen mix u e a high p essu e. While depa u es om ideali y a e la ges a low empe a u es, he chemical po en ial esembles ha o he ideal gas o su icien ly la ge empe a u es (e.g., abo e 900 K). Simila ends hold up o p essu es o o de 103ba . The anspo coe icien s also unde go la ge a ia ions ac oss he phase diag am a high p essu es. The ansi ion om liquid-like o gas-like cha ac e is ics p e en he u iliza ion o simple exp essions o he 8 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940 Figu e 4. Loga i hm o he n-dodecane (le panel) and ni ogen ( igh panel) ugaci y coe icien s as a unc ion o empe a u e and uel mass ac ion o an n-dodecane/ni ogen mix u e a P= 100 ba . e alua ion o mix u e’s iscosi y, he mal conduc i i y and di usion coe icien s. Ins ead, he me hod in Re .27 is ypically used o e alua e iscosi y and he mal conduc i i y as unc ion o Tand ρ, whe eas di usion coe icien s can be calcula ed, o example, ollowing he exp essions gi en in Chap e 11 in Re .28 o high-p essu e condi ions. These coe icien s, howe e , a e cu en ly subjec o la ge unce ain ies. III.C. T anspo luxes The sys em o conse a ion equa ions (5)-(8) equi es closu e exp essions o τ,K,qc,Q,Ji, and Ji. This is achie ed h ough he me hod o i e e sible he modynamics by speci ying cons i u i e ela ions such ha he en opy p oduc ion is non-nega i e. This me hodology equi es ha one inds he conse a ion equa ion o en opy guided by he ac ha he sou ce e ms a e w i en as a sum o p oduc s o luxes and he modynamic o ces.31 The o mula ion is g ea ly simpli ied when κij does no depend on empe a u e, in such a way ha κe ij =κij and κs ij = 0, as implied by Eq. (4). In iew o he expe imen al co ela ion (3), his is an app oxima ion ha has an unclea physical jus i ica ion bu has howe e been used in he li e a u e22,30 and will also be ollowed he e. S a ing om he second p inciple o he modynamics o a mul i-componen sys em Tds =de +Pd(1/ρ)− N X i=1 (¯µi/Wi)dYi,(16) and subs i u ing he ela ions (2), he equa ion TdsNL =deNL +PNL d(1/ρ)− N X i=1 (¯µi/Wi)dYi− N X i=1 ψi·d(∇ρi)/ρ (17) is ob ained, whe e ψi= N X j=1 κij∇ρj(18) is an auxilia y a iable. A anspo equa ion o he speci ic en opy sNL can be de i ed by aking he ma- e ial de i a i e o Eq. (17) and subs i u ing he conse a ion equa ions (5)-(8) in o he esul ing exp ession, which yields ρDsNL D +∇·(1 T"qc+Q− N X i=1 {ψi[ρi∇· +∇·(Ji+Ji)] −TesiJi+eµiJi}#)= ˙sp od,(19) 9 o 14 Ame ican Ins i u e o Ae onau ics and As onau ics Downloaded by STANFORD UNIVERSITY on No embe 17, 2017 | h p://a c.aiaa.o g | DOI: 10.2514/6.2017-4940