On the Statistical Characterization of Sprays

applied sciences Re iew On he S a is ical Cha ac e iza ion o Sp ays Miguel O. Panão 1,* , Ana S. Moi a 2,3 and An ónio L. Mo ei a 2 1ADAI, LAETA, Mechanical Enginee ing Depa men , Uni e si y o Coimb a, Rua Luis Reis San os, 3030-788 Coimb a, Po ugal 2 IN+, Mechanical Enginee ing Depa men , Ins i u o Supe io Técnico, Uni e si y o Lisbon, A . Ro isco Pais, 1049-001 Lisboa, Po ugal; [email p o ec ed] (A.S.M.); [email p o ec ed] (A.L.M.) 3CINAMIL, Depa men o Exac Sciences and Enginee ing, Po uguese Mili a y Academy, Rua Gomes F ei e, 203, 1169-203 Lisboa, Po ugal *Co espondence: [email p o ec ed] Recei ed: 10 Augus 2020; Accep ed: 29 Augus 2020; Published: 3 Sep embe 2020   Fea u ed Applica ion: The wo k es ablishes he g ounds o he sp ay cha ac e iza ion using s a is ical analysis, how o explo e i o imp o e he physical in e p e a ion o sp ay p ocesses, and ad ance he me hods o epo i in a way ha u he ad ances sp ay science. Abs ac : The s a is ical cha ac e iza ion o sp ays is an essen ial way o o ganizing da a on d op size and eloci y o p o ide eliable in o ma ion on he sp ay dynamics. A clea p esen a ion o da a using s a is ical ools p o ides e idence o a clea esea ch ques ion unde lying he sp ay cha ac e iza ion. In his a icle, a e iew o he bes p ac ices o build his og ams is p esen ed, as well as h ee ele an de ails on sp ay cha ac e iza ion: (i) he applica ion o in o ma ion heo y o assess i we ha e enough in o ma ion (no da a); (ii) he link be ween ma hema ical p obabili y dis ibu ions and he physical in e p e a ion o sp ay da a; (iii) and in oducing, o he i s ime, he concep o d op size di e si y, wi h he quan i ica ion o he polydispe sion and he e ogenei y deg ees. Finally, he iew p esen ed is applied o he cha ac e iza ion o nano luid sp ays o he mal managemen . Keywo ds: d op size dis ibu ions; d op size di e si y; in o ma ion heo y; nano luid sp ays 1. In oducing he S a is ical O ganiza ion o Sp ay Da a A sp ay is a wo-phase low o d ople s in e ac ing wi h a gaseous con inuous phase. The physical p ocess o liquid a omiza ion depends on he a omize ype and b eakup p ocess, and once comple ed, he d ople s o med ha e mul iple sizes and eloci ies, and s a is ical his og ams a e he mos common way o o ganizing he la ge amoun o da a on hei cha ac e is ics. In he sense o da a o ganiza ion, he his og ams o ganizing he sizes and eloci ies o d ople s by classes do no ep esen a p obabili y o occu ence, as in con en ional s a is ical analysis, bu a p obabili y o p esence o d ople s in a sp ay, since he a omiza ion mechanisms al eady occu ed. This small language shi allows conside ing each p obabili y alue as ep esen ing he deg ee o ele ance o a ce ain class in he sp ay—a no ion which will be essen ial o unde s and d op size di e si y. In p ac ice, a e so ing da a by classes, he way his og ams ep esen he p obabili y o p esence is di iding he coun s in each class ( nk ) by he o al sample size ( N ), pk=nk/N . Howe e , i he e is a need o inc ease he de ail o he dis ibu ion, one can build he disc e e dis ibu ion in e ms o densi y o p obabili y o p esence by conside ing he bin wid h ( δDk ) in he p obabili y alue as pdk=pk/δDk . The bin wid h is cons an i size classes a e egula ly spaced wi hin he spec um, o can a y he size i i egula ly spaced. Appl. Sci. 2020,10, 6122; doi:10.3390/app10176122 www.mdpi.com/jou nal/applsci Appl. Sci. 2020,10, 6122 2 o 18 In he case o disc e e p obabili y dis ibu ions, he sum o all p obabili y alues associa ed wi h each class k is equal o one, ∑pk= 1, which means ha each p obabili y alue pk co esponds o he weigh a numbe o d ops wi hin a cha ac e is ic class k has in he en i e sp ay. I is why one designa es his way o p esen ing sp ay da a as a numbe -weigh ed p obabili y dis ibu ion. The e a e o he ways as shown la e . The in e p e a ion o he p obabili y as a numbe -weigh ed alue o class k , o example, applied o d op sizes dk , allows o calcula ing he momen s o he size dis ibu ion, such as he a e age size o d ople s, d10 =∑kdkpk=∑kdkpdkδDk , whe he using a p obabili y disc e e dis ibu ion o a p obabili y disc e e densi y dis ibu ion, espec i ely. Al hough his is basic s a is ical knowledge, in se e al esea ch wo ks, i is unclea which is he dis ibu ion epo ed, conside ing ha each app oach (p obabili y o p obabili y densi y) eac s di e en ly when we inc ease he de ail o a dis ibu ion by changing he bin wid h δDk, as illus a ed in he example o Figu e 1. While a smalle bin size implies a highe numbe o classes, in p obabili y dis ibu ions, i leads, ul ima ely, o a uni o m p obabili y dis ibu ion wi h one class pe sample (Figu e 1a). Howe e , in p obabili y densi y dis ibu ions, i inc eases he de ail o allow iden i ying e en ual mul imodali ies dues o di e en d op clus e wi h simila cha ac e is ics, o inc ease he noise in pdk alues, as illus a ed in he examples depic ed in Figu e 1b. Figu e 1. Example o he e ec o changing he bin size δ in disc e e p obabili y ( a ) and p obabili y densi y (b) size dis ibu ions. The simula ed da a ollow a logno mal dis ibu ion unc ion. The pu pose o ema king he di e ence be ween p obabili y and p obabili y densi y dis ibu ion is pa icula ly ele an when compa ing expe imen al wi h simula ed d op cha ac e is ics. In he case o using a p obabili y dis ibu ion ( pk ), he numbe o classes mus be he same, o he ep esen a i e alues o each class ( dk o d op size and uk o one eloci y componen ). O he wise, i he p esen a ion o sp ay da a op s o he p obabili y densi y dis ibu ion, which is dimensional ( pdk [ µ m −1 ]), i is no necessa y o use he same numbe o classes. Ul ima ely, o a oid e oneous compa isons be ween expe imen al and simula ed da a, i is essen ial o be clea abou he app oach ollowed when p esen ing sp ay da a in s a is ical o ma . The ques ion is why one should change o une he numbe o classes when desc ibing he sp ay cha ac e is ics. The unde lying idea o inc easing he numbe o classes is o ob ain a g ea e de ail o he p obabili y dis ibu ions and de ec e en ual mul imodali y associa ed wi h clus e s o da a wi h dissimila cha ac e is ics. In he case o d op sizes, an example gene a ing such mul imodali y would be he p esence o mul iple a omiza ion mechanisms (e.g., ae odynamic and/o hyd odynamic). In addi ion, in he case o he eloci y, di e en wo-phase low e en s like he impac o d ople s on solid su aces con ain in o ma ion abou he axial eloci y componen wi h posi i e alues om hose impinging on he su ace, bu he seconda y d ople s esul ing a e impac ha e a nega i e eloci y componen . The e o e, wha is he c i e ion o choosing a gi en numbe o classes, k ? In addi ion, should he spacing o hese classes be egula o i egula ? Conside ing egula ly spaced classes, one o he basic p inciples in oduced by S u ges [1] s a ed ha he numbe o classes k>log2(N) , wi h N as he o al sample size. Doane [2] u he elabo a ed on S u ge’s ule, bu he p oblem is he o e -smoo hing o his og ams p oduced and i s applicabili y Appl. Sci. 2020,10, 6122 3 o 18 limi ed o a numbe o da a samples below 200, as analyzed by Hyndman [3] . The al e na i es o he S u ge’s and Doane’s ules a e he: •Sco ’s ule o he bin wid h as δD=3.49sN−1/3, whe e sis he s anda d de ia ion [4]; • F eedman–Diaconis’ ule, also o he bin wid h as δD= 2 (IQR)N−1/3 , wi h IQR as he in e qua ile ange [5]; •Rice’s ule o he numbe o classes is k=2N1/3 [6]; • and a ule based on J = 6 in e laced Fibonacci se ies wi h a numbe o classes de ined as k=Jln(N)/ ln(1.618)[7]. Conside ing a simula ed example o wo clus e s o d ople s, each ollowing a logno mal dis ibu ion unc ion LN(dg,γ) = 1 dγ√2πexp −ln(d/dg)2 2γ2!(1) wi h dg as he geome ic diame e and γ as he geome ic s anda d de ia ion, he inal dis ibu ion unc ion is a mix u e be ween he wo as (d) = w1 LN( 40, 0.5 /√6)+( 1 −w1) LN( 70, 0.5 /√6) wi h w1= 0.3. Figu e 2shows he e ec o using di e en c i e ia o o ganize d op size da a wi h (a) N = 10 4 and (b) N= 105measu emen s in he o m o p obabili y densi y dis ibu ions o d op size. Figu e 2. Example o he e ec o changing he numbe o classes in de ails ob ained on he p obabili y densi y dis ibu ion o a mix u e be ween wo clus e s desc ibed by dis inc logno mal dis ibu ions, conside ing di e en sample sizes o (a)N=104and (b)N=105d ople s. Fo he wo sample sizes es ed, he S u ges’ ule clea ly o e -smoo hs he dis ibu ion’s bimodali y. The Sco ’s ule gene a es less classes bu is enough o cap u e he mul imodali y o he d op size dis ibu ion while p oducing he minimum numbe o emp y classes. The F eedman and Diaconis p oposal, and he in e laced Fibonacci se ies, can p o ide g ea e de ail, bu he esul s wi h he lowe numbe o samples (Figu e 2a) p o e he cos o inc easing he numbe o classes by a la ge noise Appl. Sci. 2020,10, 6122 4 o 18 obse ed in p obabili y densi y alues. I is no ewo hy ha he bes app oach o ep esen d op size dis ibu ions o compa ison pu poses is a p obabili y densi y ( pd k [ µ m −1] ), since he ampli ude o dis ibu ions wi h a di e en numbe o classes does no change as he ampli ude o p obabili y dis ibu ions (pk). The second app oach o de ine classes in d op size s a is ics (less so in eloci y) is he use o i egula bin wid hs. Fo example, in lase di ac ion measu emen sys ems, like Mal e n’s Sp ay ec, he a io (duppe Bounda y −dlowe Bounda y)/dk is cons an , leading o b oade classes o la ge diame e s ep esen ing each class. Howe e , a sys ema ic me hod o using i egula bin wid h o imp o e he desc ip ion o d op size and eloci y dis ibu ions is s ill open o u he esea ch. These de ails seldom appea epo ed in he li e a u e when au ho s p esen he esul s o sp ay cha ac e iza ion and compa e sp ays ob ained in di e en ope a ing condi ions. In his sense, he app oach less p one o e o would be o p esen he s a is ical da a o he sp ay cha ac e is ics in e ms o cumula i e p obabili y dis ibu ions, as explo ed la e in his in oduc ion. A inal ema k on he p esen a ion and analysis o sp ay da a in he o m o s a is ical dis ibu ions is o conside he weigh gi en o each class. In he case o he eloci y o sp ay d ople s, a numbe -weigh ed p obabili y dis ibu ion is he mos adequa e. Howe e , o his og ams o d op size, o he weigh ing ac o s, such as he •a ea-weigh ed pa,k=sk S wi h sk=πnkd2 kand S=∑sk; •and olume-weigh ed p ,k= k V wi h k= (π/6)nkd3 kand V=∑ k; applied o p obabili y dis ibu ions imp o e he in e p e a ion o i s momen s, as epo ed in he wo k o Sowa [8] . The eason o o ganizing sp ay da a wi h o he weigh alues besides he numbe -weigh ed case ela es o he physics associa ed wi h he esea ch ques ion. Namely, i he in es iga ion in ol es phenomena occu ing a he d ople su ace a ea, such as hea and mass ans e e en s, an a ea-weigh ed d op size dis ibu ion is mo e adequa e. Howe e , i he sp ay liquid olume is mo e impo an , such as sp ay cooling applica ions, he mos adequa e is he olume-weigh ed d op size dis ibu ion. The cha ac e iza ion o a sp ay o en in ol es momen s o he measu ed disc e e p obabili y dis ibu ion using single-poin diagnos ic echniques, like he Phase-Dopple In e e ome y, o ield diagnos ic echniques using imaging. In he case o d op size, he calcula ion o each momen om d op size aw da a co esponds o dab = ∑N i=1da i ∑N i=1db i!1 a−b ,∀a>b,{a,b}∈Z+(2) whe e di is a measu emen o d op size in he sample acqui ed. I hese momen s use, ins ead, he numbe -weigh ed p obabili y dis ibu ion alues, he exp ession is dab =∑knk(dk)a ∑knk(dk)b1 a−b,∀a>b,{a,b}∈Z+(3) wi h nk as he numbe o measu emen s coun ed wi hin he size class k whe e dk ep esen s he mid-poin in he in e al be ween a lowe and an uppe bound. The cha ac e iza ion o he sp ay d ople s using he a e age size ob ained om a numbe -weigh ed p obabili y dis ibu ion— d10 —means conside ing all d ople s ha e, on a e age, he same size o d10 . Fo example, when compa ing d10 o di e en loca ions in he sp ay, o he same loca ion o di e en sp ays, any inc ease implies mo e d ople s o la ge sizes. When analyzing he physics o d ople anspo while in e ac ing wi h he su ounding en i onmen , he a i hme ic Appl. Sci. 2020,10, 6122 5 o 18 mean diame e migh be a aluable cha ac e is ic measu e o conside . Howe e , when he esea ch aims a combus ion applica ions, wi h e apo a ion and mass di usion phenomena occu ing a he su ace a ea o each d ople , he bes cha ac e is ic size is d32 , since i is he a e age o an a ea-weigh ed p obabili y dis ibu ion—o else, i he esea ch poin s o sp ay cooling applica ions, whe e he mass deposi ed on he su ace is he ele an pa ame e due o i s con ibu ion o he o ma ion o liquid ilms and hei dynamic beha io , he bes cha ac e is ic size o analyzing he hea and mass ans e in ol ed would be d43, he a e age o a olume-weigh ed p obabili y dis ibu ion. The main poin when choosing he bes way o p esen sp ay da a is he awa eness ha each cha ac e is ic pa ame e , ob ained s a is ically, has an unde lying physical meaning, depending on he he mo luid phenomena in ol ed. Finally, i is no ewo hy ha p o iding he momen s o d op size and eloci y dis ibu ions may limi he use o sp ay da a in u u e wo ks and sp ay simula ions because o he inabili y o econs uc he o iginal dis ibu ions om he momen s epo ed. The e o e, he nex sec ion discusses he implica ions o sp ay science o he a emp o i a p obabili y dis ibu ion unc ions o he his og ams o d op size. 2. D op Size Dis ibu ion Func ions and Sp ay Science Sp ay cha ac e iza ion p o ides ele an in o ma ion o d ople s dynamics o he applica ion o i s mean quan i ies in empi ical co ela ions ela ed wi h hea ans e and luid low p ocesses. The se e al op ical diagnos ic echniques acqui e la ge amoun s o da a and he challenge is o en how o p ocess i . The c i e ion o he sample size ( N ) o sp ay da a ecu s o en o he no ion o s a is ical unce ain y. When i s alue is below a p e-de ined h eshold, he measu emen s ops because he expe imen alis has enough da a. Howe e , his c i e ion o en in e p e s sp ay da a as a andom p obabilis ic e en , while sp ay s a is ics is mo e a me hod o o ganizing da a. The e o e, he igh ques ion is no whe he he e is enough da a o pos -p ocess and cha ac e ize a sp ay, bu whe he he e is enough in o ma ion. Sec ion 2.1 e iews an app oach based on in o ma ion heo y o make his assessmen . Secondly, he meaning o using an a e age quan i y o desc ibe a sp ay, whe e d ople s ha e mul iple sizes, is o assume ha wha e e physical phenomenon a ec s an a e age size ep esen s wha occu s o he en i e sp ay. The limi a ion o p esen ing he sp ay da a based solely on mean quan i ies is he loss o in o ma ion o he local o global d op polydispe sed sizes o eloci ies in he sp ay and i s po en ial use ulness in he de elopmen o nume ical models ha simula e sp ays. Howe e , while i is di icul o e ie e in o ma ion o he o iginal s a is ical dis ibu ions om hei mean quan i ies, he abili y o econs uc d op size o eloci y dis ibu ions om cha ac e is ic pa ame e s o he ma hema ical p obabili y dis ibu ion unc ions ( pd ), ins ead o i s momen s, allows o ob aining he mean quan i ies wi hou losing he in o ma ion o he o iginal dis ibu ions. The e o e, app oaching sp ay cha ac e iza ion om he poin o iew o econs uc ing p obabili y dis ibu ions has signi ican ad an ages o e he app oach ha uses solely momen s e ie ed om disc e e p obabili y dis ibu ions ha desc ibe a sp ay (e.g., see Panão and Radu [9] ). Sec ion 2.2 e iews he i ing o p obabili y dis ibu ion unc ions o his og ams o d op da a o e ie e he cha ac e is ic pa ame e s allowing he econs uc ion o sp ay da a dis ibu ions, bu in oduces he a gumen o whe he o no such i ing can p o ide some insigh in o he physics o he a omiza ion p ocess. The inal Sec ion 2.3 in oduces, o he i s ime, he no ion o d op size di e si y, dis inguishing he polydispe sion deg ee om he he e ogenei y deg ee and p esen ing he bes pa ame e s o hei cha ac e iza ion. An accu a e cha ac e iza ion o d op size di e si y is ele an o he design o sp ays. 2.1. De ining Enoughness in La ge Da a Samples An expe imen alis s ops a measu emen based o c i e ia ela ed o a s a is ical unce ain y. The de ini ion o s a is ical unce ain y con ains h ee condi ions: 1. i is maximum o a uni o m dis ibu ion whe e all classes ha e he same p obabili y; 2. a small a ia ion in he p obabili y o a class gene a es a small a ia ion in he unce ain y; Appl. Sci. 2020,10, 6122 6 o 18 3. and, inally, i depends on he dis ibu ion i sel . The common measu e used o he s a is ical unce ain y conside s he s anda d de ia ion ( sx ) and he sample size (N) as ε=Zcsx √N(4) wi h Zc as he coe icien associa ed wi h he con idence in e al conside ed (e.g., Zc= 1.96 o a 95% Con idence In e al). When he mean alue ( x ) is di e en om ze o, di iding ε by he mean p o ides he unce ain y in pe cen age. Since he s anda d de ia ion depends on he sp ay cha ac e is ics, educing he s a is ical unce ain y implies adding mo e da a o inc ease N . Howe e , as a gued by Panão [10] , i he sp ay begins o ope a e in a di e en way and he dis ibu ion changes, he s a is ical unce ain y as de ined in Equa ion (4) con inues o dec ease, wi hou p o iding any e idence o he expe imen alis abou he changes occu ing in he sp ay. In a ce ain sense, i ails o comply o he hi d condi ion de ining a s a is ical unce ain y because i is mo e sensi i e o he sample size han he dis ibu ion i sel . Fo his eason, an app oach based on in o ma ion heo y is a be e op ion. In in o ma ion heo y, he Shannon en opy complies o all he a o emen ioned cha ac e is ics o a s a is ical unce ain y. Conside ing he p obabili y alues o any disc e e dis ibu ion ( pi ), he exp ession o calcula ing he Shannon en opy His H=−∑ i (piln(pi))(5) As an example, i we conside d op size, he minimum alue co esponds o a monosize sp ay wi h all d ople s ha ing he same size, hus, p= 1, and H= 0. The maximum alue occu s o a hypo he ical sp ay whe e all d ople s ha e he same p obabili y o being p esen , co esponding o a uni o m dis ibu ion wi h k classes, each o a di e en d op size. The e o e, he maximum Shannon en opy is max(H) = 1 /k . In sp ays, while measu ing size and eloci y, conside ing he Shannon en opy no malized by i s maximum alue, Hn=H/max(H) , i ends o s abilize (see Panão [10] o de ails). The meaning is ha adding mo e da a does no mean adding mo e in o ma ion because he shape and scale o he dis ibu ion s abilized. Howe e , as a gued in Panão [10] , he enoughness equi es a c i e ion wi h in e p e a i e alue and p oposes he excess en opy ( EE ) because, as de ined in Feldman e al. [ 11 ], i is wha bes cap u es he na u e o con e gence o he en opy a e as he amoun o memo y gained, o he cos o amnesia i all da a would suddenly be los . To e alua e he e olu ion o EE while measu ing, • one calcula es he en opy a e ha quan i ies he di e ence be ween he no malized Shannon en opy wi h adding one alue he N samples and he Shannon en opy wi h N samples- ˙ Hn= |Hn(N+1)−Hn(N)|; •conside s he limi when N→∞, which is ˙ h; •and he excess en opy EE is o mula ed as EE =∑∞ N=1˙ Hn(N)−˙ h In he case o sp ay da a, he s abiliza ion implies ˙ h = 0, hus, i simpli ies EE . The me hod p oposed is se ing a con e gence c i e ion– εEE and s op measu ing when |EE(N)−median(EE)|<εEE , conside ing he numbe o samples co esponding o he median as he minimum equi ed (see Panão [10] o mo e de ails). Once he expe imen alis has enough in o ma ion and o ganizes he sp ay da a wi h his og ams o d op eloci y and weigh ed dis ibu ions o d op sizes, depending on he physical p ocess unde analysis, one o he me hods o a oid losing in o ma ion o he measu ed dis ibu ions is i ing da a o known ma hema ical o empi ical dis ibu ion unc ions. Howe e , he ques ion is whe he such i ing can p o ide u he insigh in o liquid a omiza ion mechanisms. This is he opic explo ed in he nex sub-sec ion. Appl. Sci. 2020,10, 6122 7 o 18 2.2. Unde lying Physics o P obabili y Dis ibu ion Func ions Applied o Sp ays An impo an conside a ion when cha ac e izing sp ays is he e ec o he in e ac ion be ween d ople s and he con inuous phase on he local (o e en o e all) size and eloci y dis ibu ions. This in e ac ion in ol es di e en anspo phenomena, wi h momen um and ene gy exchanges be ween he dispe sed phase (sp ay) and he ca ie phase (su ounding en i onmen ), gene a ing e en ual seconda y b eakup o he sp ay d ople s leading o changes in he shape and scale o d op size dis ibu ions, o accele a ion (posi i e o nega i e) cap u ed by changes in local eloci y p obabili y dis ibu ions. Howe e , he physical eason why a ce ain dis ibu ion unc ion migh i be e han ano he is s ill open o u he esea ch. This wo k ad ances an a gumen in a o o dis inguishing be ween modeling and cha ac e izing d op size dis ibu ions. The pu pose o modeling d op size dis ibu ions is o p edic hem om he in o ma ion o he a omize geome y and ope a ing condi ions. The simula ion o sp ays using a nume ical app oach [ 12 ], o a s a is ical o s ochas ic app oach [ 13 ] can p oduce da a on he d ople s cha ac e is ics, bu i is di e en om s a is ically p ocessing such da a. The e o e, acco ding o Déchele e e al. [14], he e a e ou me hods o modeling d op size dis ibu ions: • he empi ical; • he Maximum En opy Fo malism (MEF); • he Disc e e P obabili y Func ion (DPF); •and he S ochas ic. Howe e , he pu pose o cha ac e izing a sp ay is o desc ibe, as accu a ely as possible, he polydispe sion o sizes and eloci ies o i s d ople s. This desc ip ion aims a ob aining mean quan i ies o analyzing hea ans e and low p ocesses—o i s aim is o imp o e ou unde s anding o he na u e unde lying he a omiza ion mechanisms. The e a e wo ca ego ies o p obabili y dis ibu ion unc ions used o desc ibe d ople s’ cha ac e is ics: ma hema ical and empi ical. Le eb e and McDonell [15] p o ide a syn hesis o he main p obabili y dis ibu ions in each ca ego y. Excep o he Rosin–Rammle o Weibull, he Nukiyama–Tanasawa and Uppe -Limi empi ical dis ibu ion unc ions a e complex and p oblems a ise when de e mining he bes - i alues o hei pa ame e s. As o he ma hema ical dis ibu ion unc ions, he simples is he Log-No mal, while he Log-Hype bolic is also complex and p oblems a ise wi h inding he bes i ing pa ame e s. One dis ibu ion absen om Le eb e and McDonell [15] and o he e iew wo ks is he Gamma dis ibu ion unc ion which Ville maux e al. [ 16 ] associa ed wi h he dis ibu ion o d ople s esul ing om he agmen a ion o ligamen s, gene a ing a sp ay, e iewed la e in his sec ion. While mos esea ch on sp ay cha ac e iza ion ocuses on he i ing p ocess, ew wo ks such as Ville maux [17] and Ville maux e al. [ 16 ] add ess he meaning o he ma hema ical dis ibu ion unc ion used and he physical backg ound o such i ing. As men ioned in he In oduc ion, ins ead o ocusing ou a en ion on p obabili y o p obabili y densi y unc ions, we p opose a g ea e ocus on cumula i e dis ibu ion unc ions ( F(d) ). The e o e, any compa ison be ween di e en F(d) becomes uni e sal and i such dis ibu ion p ope ly desc ibes he local o global expe imen al esul s, i s digi iza ion o simula e a sp ay is ela i ely accessible. In he case o he Log-No mal dis ibu ion, i s cumula i e o m gi en by FLN(d,dg,γ) = 1 21+e ln(d/dg) √2γ (6) includes dg as he scale pa ame e , he geome ic mean diame e , and γ as he shape pa ame e o he dis ibu ion. Applied o sp ay cha ac e iza ion, he eason o using a Log-No mal dis ibu ion unc ion is ela ed o he mul iplica i e e ec o subsequen s ages o d ople s b eaking up du ing a omiza ion, as a cascade p ocess, whe e one d op b eaks in o wo o mo e and so on. Howe e , i we conside he Appl. Sci. 2020,10, 6122 8 o 18 in e ac ion be ween d ople s and he con inuous gaseous phase, in ime, he d agging o d ople s leads o seconda y lows which e en ually p oduce a o ical e ec on he anspo o subsequen d ople s. Namely, smalle d ople s, wi h lowe esponse imes, d agged by seconda y lows, may e u n o upwa d loca ions, edis ibu ing he coun s o ce ain d op size classes in loca ions u he downs eam o he sp ay ajec o y. The e o e, e en i i is no he esul o a mul iplica i e b eakup p ocess, he p esence o hese smalle d ops a ec s he p obabili y dis ibu ions desc ibing he sp ay, ha ing an e ec simila o ha o a cascade o mul iple b eakup s ages. One could e en specula e whe he o no he eason o a Log-No mal dis ibu ion bes i ing expe imen al esul s exp esses he way he mul iphase low o ganizes he anspo o d ople s acco ding o hei size in a cascade pa e n. Besides se e al b eakup s ages and anspo phenomena, some a omiza ion p ocesses esul om he disin eg a ion o liquid shee s o je s in o ligamen s, and hose ligamen s u he agmen ing in o d ople s. In his case, Ville maux e al. [ 16 ] a gues each ligamen cons i u ed o se e al blobs, and when i agmen s in o se e al d ople s, he size dis ibu ion ha easonably i s is a Gamma p obabili y dis ibu ion unc ion, which cumula i e o m is exp essed by FΓ(d,a,b) = 1 baΓ(a)Zd 0xa−1exp(−x/b)dx (7) wi h a and b as he shape and scale pa ame e , espec i ely. In his case, a cha ac e is ic size co esponds he p oduc o bo h: dc=a·b. Conside ing empi ical dis ibu ion unc ions, his wo k ocuses on he Weibull dis ibu ion, i s applied o desc ibe he dis ibu ion o d op sizes by Rosin and Rammle , FWB(d,dc,q) = 1−exp −d dcq(8) whe e dc is a scale pa ame e ela ed o a cha ac e is ic d op size and q is he shape pa ame e and conside ed a measu e o he sp eading in d op sizes. The accu acy o his empi ical p obabili y dis ibu ion is bes ela ed o d op size dis ibu ions wi h ewe smalle d ople s o na owe size dis ibu ions [15]. A inal no e conside s he Nukiyama–Tanasawa empi ical dis ibu ion ha is o en used o i expe imen al da a. He e, pa icula a en ion is gi en o he wo k o Li and Tankin [18] ha de i ed he exp ession using an in o ma ion- heo y app oach, and o he sp ay liquid olume, i s cumula i e o m esul s in FNT(d,dc,q) = 1−(1+qd3)exp −qd3(9) whe e qis also he shape pa ame e . The a e age quan i ies e e ed o so a a e ela ed o momen s in p obabili y dis ibu ions, bu conside ing he cumula i e dis ibu ion, any quan i y co esponds o a ep esen a i e diame e , gene ally exp essed as Dxw , whe e x is he ype o dis ibu ion, and w is he pe cen cumula i e alue ela ed o he ep esen a i e diame e . The e o e, i he cumula i e dis ibu ion is •numbe -based, Dnw ep esen s he size con aining w% o he d ople s in he sp ay; •a ea-based, Daw ep esen s he size con aining w% o he sp ay su ace a ea; • olume-based, D w ep esen s he size con aining w% o he liquid sp ay olume. One o he mos ele an ep esen a i e diame e s co esponds o 50% ( Dn0.5 , Da0.5 , o D 0.5 ) because di iding he classes by ha numbe allows a be e compa ison be ween cumula i e dis ibu ions, use ul o analyzing he e ec o pa ame ic a ia ions in he sp ay. 2.3. In oducing D op Size Di e si y in a Sp ay D op size dis ibu ions a e a way o o ganize he da a acqui ed o cha ac e ize a sp ay. The cha ac e iza ion o he di e si y o d op sizes answe s wo dis inc ques ions: (1) how many Appl. Sci. 2020,10, 6122 9 o 18 di e en sizes a e ele an in a sp ay; (2) and how di e en a e he ele an sizes in a sp ay. The wo d ele an links o he p obabili y o p esence o ce ain d op size classes ela i e o o he s. In he known ex book on A omiza ion and Sp ays, Le eb e and McDonell [15] e e o his di e si y as d op sp ay dispe sion associa ed wi h he size ange o d ople s. On he o he hand, se e al esea ch a icles add ess he di e en sizes o he sp ay d ople s as a polydispe sed sp ay. In he au ho s’ opinion, hese a e wo di e en hings, which is why we in oduce he concep o D op Size Di e si y (DSD) measu ed by wo di e en deg ees: • he polydispe sion deg ee o quan i y he mul i ude o di e en sizes ha a e ele an in a sp ay. Thus, he maximum o he case whe e all di e en sizes ha e he same p obabili y o being p esen in he sp ay, and • he he e ogenei y deg ee o quan i y how di e en a e he ele an sizes in he sp ay. The e o e, i is ela ed wi h he size ange o size dispe sion. The challenge is o de ise he igh indica o s o measu e bo h deg ees. Among he se e al indica o s a ailable and syn hesized in Le eb e and McDonell [15] , he mos known and used indica o is he Rela i e Span ob ained om he ep esen a i e diame e s o a olume-based cumula i e size dis ibu ion (D X wi h 0 <X<1 as he ac ion o he sp ay liquid olume) as ∆ =D 0.9 −D 0.1 D 0.5 (10) Conside ing he no maliza ion o a ep esen a i e diame e by he alue ep esen ing hal o he liquid olume- D 0.5 -as D∗ X =D X/D 0.5 , he in e p e a ion o Equa ion (10) ela i e o he ange o d op sizes co esponds o a di e ence- ∆ =D∗ 0.9 −D∗ 0.1 which is equal o ze o when all d ople s in he sp ay ha e he same size, and maximum when all d ople s ha e he same p obabili y o occu ence (a limi un ealis ic case). Panão [19] p oposed a di e en app oach based on in o ma ion heo y, h ough he concep o he no malized Shannon en opy, al eady de ined in Sec ion 2.1 as Hn=H ln(Nbins)(11) In he in o ma ion heo y e minology applied o sp ay cha ac e iza ion, a sp ay whe e all d ople s ha e he same size, p= 1, esul ing in a null no malized Shannon en opy, Hn= 0, while an un ealis ic sp ay wi h all classes ha ing he same p obabili y o p esence (uni o m dis ibu ion), Hn=1, because he Shannon en opy–nume a o in Equa ion (11)–is maximum. Finally, Ga cía e al. [ 20 ] p oposed a hi d app oach based on he s anda d de ia ion o he olume-weigh ed d op size dis ibu ion exp essed as SD =qd2 53 −d2 43 (12) whe e d53 and d43 a e he second- and i s -o de momen s o he olume-weigh ed d op size dis ibu ion, espec i ely. The au ho s compa ed his s anda d de ia ion wi h Shannon en opy and ound inconsis encies. They s a e ha he Shannon en opy has a main d awback, since he in o ma ion o he d op sizes ep esen ing each class is no explici ly included; hus, i hese p obabili y alues would be andomly ea anged, he H alue would be he same. This is an impo an insigh because i allows o unde s and he di e ence be ween polydispe sion and size dispe sion in a sp ay. To compa e he h ee app oaches, conside he simula ion o a sp ay mixing wo monosize d ople s eams o 10 µ m and 20 µ m. A weigh pa ame e w , a ying be ween 0 and 1, se s he pe cen age o d ops p esen in he mixed sp ay om each o he monosize sou ces. The e o e, i w=0, all d ople s ha e he size o 20 µ m, and i w= 1, all d ople s ha e 10 µ m. Figu e 3shows he esul o he Rela i e Span ( ∆ ), no malized Shannon en opy based on he olume-weigh d op size dis ibu ion ( Hn, ), Appl. Sci. 2020,10, 6122 16 o 18 Finally, in his wo k, we in oduced he concep o d op size di e si y o be e unde s and he many di e en sizes ele an in a sp ay, as well as how di e en he ele an sizes in a sp ay a e. The polydispe sion deg ee gi en by he no malized Shannon en opy, Hn , and sp ay he e ogenei y deg ee gi en by he olume-weigh ed s anda d de ia ion, SD [ µ m], can desc ibe his di e si y, as depic ed in Figu e 10. Figu e 10. Polydispe sion deg ee gi en by he no malized Shannon en opy, Hn , and sp ay he e ogenei y deg ee gi en by he olume-weigh ed s anda d de ia ion, SD [ µ m], o he planes o Z=10 mm ( op) and Z=20 mm (bo om). In he case o dis illed wa e , he esul s show a highe polydispe sion deg ee a ound =8 mm , whe e he sp ay he e ogenei y deg ee also becomes highe . Howe e , despi e local changes in Hn , he beha io does no seem signi ican ly a ec ed by adding su ac an o p oduce he base luid, and u he adding nanopa icles. Howe e , in e ms o he e ogenei y, he esul s a e di e en . The addi ion o su ac an exe s a majo e ec on he he e ogenei y o he sp ay cha ac e is ics. Howe e , he addi ion o nanopa icles p oduces a negligibly e ec . Consequen ly, one may also expec a mino e ec o he nanopa icles du ing sp ay impac . This can ac ually be a bene icial ea u e, o ins ance, in sp ay cooling applica ions, as i sugges s ha he nanopa icles can be used o al e he he mal p ope ies o he wo king luids, wi hou signi ican ly a ec ing he main sp ay cha ac e is ics. 4. Conclusions The cha ac e iza ion o a sp ay is no me ely acqui ing in o ma ion on he size and eloci y o i s d ople s in se e al loca ions om single-poin measu emen echniques, o in se e al planes om 2D- o 3D-measu emen echniques. Al hough i is essen ial o acqui e enough in o ma ion, as explo ed he e h ough an in o ma ion heo y app oach, he cla i y o he esea ch ques ion associa ed wi h he sp ay cha ac e iza ion is e y much like he cla i y in he display and analysis o da a. The e o e, a well-de ined esea ch ques ion is wha guides he kind o sp ay cha ac e iza ion pe o med. In his a icle, we e iew he s a is ical language used in sp ay cha ac e iza ion, and: • he di e ences be ween o ganizing sp ay da a using p obabili y his og ams o his og ams o p obabili y densi y; •how o choose he numbe o classes in his og ams; • he di e en kinds o p obabili y his og ams conside ing he numbe o d ople s, hei a ea, o he liquid olume, and co esponding momen s; • a me hod based on he excess en opy om in o ma ion heo y o assess i he e is enough in o ma ion o pos -p ocessing; Appl. Sci. 2020,10, 6122 17 o 18 • he physical meaning o i ing ma hema ical p obabili y dis ibu ion unc ions, namely he Log-No mal, Gamma, and Weibull, o sp ay da a; • and in oduce, o he i s ime, he no ion o D op Size Di e si y wi h i s polydispe sion and he e ogenei y deg ees quan i ied by he no malized Shannon en opy and olume-weigh ed s anda d de ia ion, espec i ely. Finally, he opics explo ed on sp ay cha ac e iza ion a e applied o nano luid sp ays o explo e he e ec o in oducing a su ac an and nanopa icles on he sp ay s uc u e and dynamic cha ac e is ics. Au ho Con ibu ions: Concep ualiza ion, in es iga ion, M.O.P.; esou ces, A.S.M. and A.L.M.; w i ing–o iginal d a p epa a ion, M.O.P; w i ing– e iew and edi ing, M.O.P and A.S.M.; p ojec adminis a ion, A.L.M.; unding acquisi ion, A.S.M. and A.L.M. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: Ana S. Moi a would like o acknowledge p ojec No. 030171 unded by LISBOA-01-0145-FEDER- 030171/PTDC/EME-SIS/30171/2017, and p ojec UTAP-EXPL/CTE/0064/2017. Con lic s o In e es : The au ho s decla e no con lic o in e es . Abb e ia ions The ollowing abb e ia ions a e used in his manusc ip : CTAB Ce ylT ime hylAmmonium B omide DSD D op Size Di e si y PDI Phase-Dopple In e e ome e Re e ences 1. S u ges, H.A. The choice o a class in e al. J. Am. S a . Assoc. 1926,21, 65–66. 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This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).