ISSN MECHANIKA Volume 22(2):

Transcription

1 9 ISSN 97. MECHANIKA. Voue (): 9 Modeng of het nd ss trnsfer proesses n phse trnsforton ye of spryed wter nto gs:. Ther stte nyss of dropet sppng n hud r fow G. Musks M. Mzukenė A. Bčus J. Gudznsks Kuns Unversty of Tehnoogy Studentų LT- Kuns Lthun E-: g@ktu.t onk.zukene@ktu.t gnts.us@ktu.t juozs.gudznsks@ktu.t Noenture - ther dffusvty /s; p - ss spef het J/(kg K); - urer nuer; g - evporton veoty kg/s; L - tent het of evporton J/kg; - vpour ss fux kg/( s); p - pressure P; - het fux W/ ; r - rd oordnte ; Pe - Peet nuer; T - teperture; λ - ther ondutvty W/( K); - oeur ss kg/ko; - densty kg/ ; - te; w - veoty /s; susrpts C - dropet entre; o - ondenston; - onveton; e - evporton; f - phse hnge; g - gs; - te ndex n dgt shee; t - nuer of terton; I - ndex of ontro te; j - ndex of rd oordnte; J - ndex of dropet surfe; k - onduton; k+r - onduton nd rdton; - ud; - ss verge; η - non-denson rd oordnte; r - rdton; - dropet surfe; v - vpor; vg - gs-vpor xture; - nt stte; - fr fro dropet; supersrpts + - extern sde of dropet surfe; - ntern sde of dropet surfe.. Introduton Hot gses tht re produed n orgn sod gsfton proess s effetvey ooed down y wter njeton nto trnsportton systes ondtoned r ouston proess s ontroed nd ented fues preters. r enton-ove nd other wter spryed ppton ses ther energy onsuptons s portnt for proeedng phse trnsfortons on the dropet surfe nd for ud hetng n the dropet. Therefore reserhes of wter dropets het trnsfer nd evporton [-7] s we vpour ondenston on ts surfe [8-] s reevnt. The hnge of gs xture oposton tht rry out dropets s used y ongong phse trnstons on the surfe of the dropet nd teperture vrtons re defned y dropet het trnsfer proesses. Het fow tht s provded y the gs stutes ud evporton fro dropet surfe nd so wrs ud n dropet. Lud wrng rte s defned y ntensty of het offtke to dropet. In oned hetng se the het s eng provded y onveton nd rdton: r. Het sup- pyng for dropets y onveton nd rdton hs ts own peurtes tht s defned y trnsfer proess nture. A fowng round fud provdes het for the surfe y onveton whe t se-trnsprent dropet rdton het s provded y sorng eetrognet wves of ght n the nfrred spetru. Therefore the rdton het wrth the ud n the dropet drety nd extern onveton het fro dropet surfe ust e eded off nsde the dropet y ntern het exhnge. Therefore tot het fow densty of dropet r defnes fud hetng ntensty. Se-trnsprent uds sors spetr rdton y surfe poory [] so ssupton s popur. A onvetve oponent r r of tot het fow s defned y dropet ruton nd het ondutvty proesses. Intern yers of dropet re heted uneveny therefore dropet ther stte s desred y vre funton T(x y z τ) n te nd spe. A dropet unstedy teperture fed s desred y te nd rd oordnte funton T(r τ) n sphery syetr hetng ssupton se. A ther stte of non-sother dropet s defned y ss verge teperture funton of dropet: T r r T r. () r r dr A dropet hetng ondtons n gs fow [] hs strong pt for peurtes of dropet unstedy teperture funton T(r τ). When extern hetng provdes het spred y rdton nd ondutvty funton T(r τ) s defned ordng to ntegr type ode [7] tht s onvenent for nuer odeng. Then het tht s eded off to the dropet s desred y urer s ow. In se of onvetve hetng dropet sps n gs fow nd frton fores rses on dropet surfe. On ther pt ud s rutng nd ntensfes ntern het trnsfer. In se of fored ud ruton t the dropet ts ther stte s desred y energy nd Nver-Stokes euton syste. The tter nyt souton s not posse nd the nuer souton s opted []. Therefore nuer shee of onveton het fow n dropet woud e susepte for hne oputng nd woud opte tertve ethod ppton wherey dropet surfe teperture defnton s sed on. To defne unstedy teperture fed funton T (η τ) T (η τ) n se of sppng dropet s ute dffut. In ths work sppng dropet ther stte defnng ethodoogy s deveoped nd sppng ntensty pt for wter dropet wrng n hud r fow s so nysed. dr

2 97. eserh ethod nd resuts r dropet ther stte defnton the onedenson effetve ondutvty epr ode s provded [] nd het trnsfer y onduton nd rdton n the dropet ntegr odes [7]. To defne dropet surfe teperture te funton T (τ) ne of fuxes tht fows n nd off on the surfe of dropet s provded. Ths s defned y expressons () nd () for ondensng nd evporton reges respetvey. p v =. nd nysed n urer te se = / Prry dropet sppng n hud r hs sgnfnt pt for uted ther ondutvty preter tht defnes wter ruton ntensty n dropet (Fg. ). k... ; () f o. () f e At expressons () nd () potent rdton fow sorpton y dropet surfe s dened. Extern onvetve het fow s desred y the ethod [ ]. Phse trnsforton fow on the surfe of the dropet s desred y wter vpour fow densty f v L whe for onvetve het fow desrpton tht s eded to dropet odfed urer ow s pped: k T r. () r r At opound hetng se y ondutvty nd rdton n unstedy teperture fed grdent s desred y nfnte ntegr eutons seres [7]: dt T r n r d r n p n r n r n r sn os r dr n exp d. Lud ruton potent nput to ne of het spred n dropet s evuted y effetve het ondutvty preter where t expresson () Peet nuer s defned ordng to ud fow xu veoty w : Pe = ρ w / μ [] on the surfe of dropet. Ths fow s used y frton fores. n () Pe k.8.8tnh. g. () Itertve nuer shee s de fro expnded () nd () expressons. A dropet surfe te funton T (τ) defnton of ths shee s ppe for dropet hetng ses. However t ust e reeer tht surfe teperture funton T (τ) defnes ony rgn vue of unstedy teperture fed T(r ) funton of dropet T(r ) = T (). Therefore o teperture T(r < ) ordng to dsussed ethodoogy n e defned ony t ssupton of ste ud t the dropet. In hud r sppng wter dropet phse trnsforton ye s odeed ordng to oundry ondtons T g = K p =. MP =.7 T = 78 K Fg. Dropet prry sppng n r fow pt for effetve ther ondutvty preter. e : 8; / o " k" A pont tht defnes eynods nuer nt vue (Fg. ) so shows phse trnsfortons of ondensng rege retve durton n spet of dropet hetng y onduton. It s ery seen tht ondenston rege te nreses nd onveton het trnsforton t the dropet s ore prevent f dropet sps ore ntensvey. Wter ruton ntensty s senstve for dropet sppng nd dspersty. A dropet growth n ondenston rege s sute ftor for ruton n t whe dropet sppng wekenng s ftor tht redues wter ruton n dropet. At the egnnng of ondensng rege n ntensfton of onveton het trnsfer oserves (Fg. ). Ths ens tht for soe te dropet growth ftor tht s fvoure for dropet ruton s stronger thn sppng wekenng ftor tht repress ruton. At oth ftors nng pont k extree pont oserves n funton grphs (Fg. ) fter whh the wter ruton onsstenty suffotes n the dropet. The dropet nt sppng sgnfnty nfuenes the dropet surfe wrng proess (Fg. ). The dropet nt sppng sgnfnty nfuenes the dropet surfe wrng proess (Fg. ). In unstedy phse trnsforton rege dropet surfe hets up rpdy (Fg. ) however wrng rte onsstenty suffotes unt eoes zero t fn stge of unstedy evporton (Fg. ). In euru evporton rege teperture of dropet hetng y ondutvty s stedy. A sppng dropet teperture dereses t euru evporton rege (Fg. ): t the egnnng of euru evporton dropet oong down even eertes however t rpdy rehes xu oong down rte nd fter tht s onsstenty wekenng (Fg. ). Therefore t euru evporton proess when dropet sppng redues pprohng to ther stte of dropet heted y ondutvty s gettng oser (Fg. ). Deeper nner y-

3 98 ers hetng proess nyss s reured to defne sppng dropet ther stte. A dropet sppng uses onveton "" het trnsfer n theren. Its ntensty s defned y o het fow densty funton T 9 (r τ). desrpton nsde dropet: F grd T " " " " " k ". (7) Lo teperture grdent s desred y ntegr euton: T r n n r n r os r n r r n n dt exp d. n d Condton r s pped for expresson (8) nd teperture grdent s defned for eded off onveton het fow desrpton t the dropet () n expresson (Fg. ). (8) 8 7 T 7... T K s Fg. Dropet prry sppng n r fow pt for surfe teperture: - unstedy phse trnsforton rege; - unstedy evporton fn nd euru evporton prry sttes. e : 8 The tter s dffut to desre due to ft tht dropet deter s hngng t phse trnsforton ye. Therefore densoness rd oordnte η = r / () of dropet s ntrodued. By ts spet dropet densoness rdus s unvers euse ts unt vue rens n dropet phse trnsforton ye o nf f. It s onvenent for dropet unstedy teperture fed T η () o grdent grdt η () nd o onveton het fow densty " " () funtons desrpton. It s ssued tht funton F " " defned y effetve ther ondutvty o preters s exstng. Effetve ther hetng ondutvty theory s pped for onvetve trnsfer ntensty T K s Fg. Dropet prry sppng n r fow pt for ts surfe wrng rte: - unstedy phse trnsforton rege; - unstedy evporton fn nd euru evporton prry sttes. e : 8; T ( T T ) / T ; F o

4 99 oposte onvetve ondutve fud: A onveton het o fow densty funton s " " " " k " " (). It s reted fro oponents of () tht s used y ruton nd () tht s defned y het ondutvty. (9) " " " " k " " Het ondutvty oponent of dropet s defned y het ondutvty urer ow t (9) expresson: grdt grdt k " " " " r " k ". ().. Fg. Dropet prry sppng n r fow pt for teperture grdent tht s uted ordng to ste ud ode nsde the dropet. e : 8 Conveton het fow oponent tht s used y ud ruton n the dropet n e defned ordng to dfferene of o onveton het fow nd ts ondutvty oponent: grdt grdt " " " " k " " " " F.() " k " " " It s ssued tht t unstedy teperture fed n o grdent " k" nd "" het exhnge ses proporton of the effetve ther ondutvty s: grd T grdt K " k " " " F " ". () Then onveton het fow oponent tht s used y ud ruton n the dropet s defned ordng to expresson: " " F " " " k ". () F " " In odeed phse trnsforton ye te hnge step Δ I / (I-) s defned y provdng fnte nuer of ontro ponts n free hosen urer nuer hngng nterv I. Provdng fnte J nuer of ontro ponts unt rdus of dropet s dvded to J- nterv Δη j / (J-). Preserved ondtons: ; I I kon J J. j j j j j () When defnng teperture fed grdent n dropet y teperture dfferene nd rd oordnte hnge rto grdt j " " T j " " T j " " / j j dropet o teperture when wter rutes n t n e defned y shee: T T j " " J " " grdt grdt J j " " j " " j j. () j A strong nd ner reedng ruton ses n dropet s provded for preter funton defnton. In se of strong ruton: j k " " " " F F. () At se of ner reedng wter ruton nsde the dropet the funton shee: F " " () s defned ordng to j F k " ". (7) j J A Modeed dropet sppng reges n r fow s defned y freey hosen eynods nuer e t nterv e. At eh se I = nd J = whe urer rter I s euted for rteron o tht defnes phse trnsforton rege durton I = o(e ) [7]. Up to urer te steps Δ s de n phse trnsforton ye. Te Het fow denstys on the surfe of dropet kw/ e f o ; o f o ; nf f nf ; nf A dropet energy stte s defned y extern onvetve het Q f nd ntern onvetve het Q phse trnsforton het Q fows. Ther uted denstes

5 f nd t the egnnng of phse trnsforton rege nd ondensng rege hnge to evporton rege s we t euru evporton egnnng oents for dfferent e vues s gven n Te. Intern onveton het fow Q spreds y het ondutvty nd s trnsferred y rutng wter n dropet. Therefore ntern het fow ntensty s defned y () nd () expresson whh s desred y onduton nd onveton het fow oponents k " " " " densty oponents re gven n Te. kw. Intern onveton het fow Cuted o het fow spred n dropet depends fro pped ode of het exhnge n the dropet (Fg. ). Het trnsfer ode nfuene s rghtest n ondenston phse trnsforton rege where dropet sppng s the ost ntensve. Te Intern onveton het fow densty nd ts ondutvty nd onveton oponents kw/ k " " " " o k " " o " " o e " k " " " kw k " " " " kw " k " " " kw k " " " "....8 Fg. Dropet sppng pt for het strton ntensty to entr yers. - e = ; :.7;.8;.7; - e = ; :.;.77;.9. Het spred ode nsde the dropet: () "k" () "" strong ruton se () "" ner reedng ruton se....8 Fg. Dropet sppng pt for onveton het fow oponents. - e = ; :.7;.8; - e = 8; :.;.77. Het spred ode nsde the dropet: () "" strong ruton se () "" ner reedng ruton se

6 In odeed sppng dropet ses het fow densty η k uted ordng to se k s ess thn het fow densty uted η ordng to (7) expresson. The dfferene etween the s rghter t ore ntensve wter ruton (Fg. ). Strong nd ner reedng ruton n the dropet odes for s sppng veotes gves ose resut (Fg. ) whe for rger sppng veotes rghter dfferene redy oserves etween uted η het fows (Fg. ). In sppng dropet the dstruton etween spredng o het fow η onvetve η nd ondutvty ηk oponents depends fro dropet sppng veoty (Fg. ). At s sppng veotes onduton het fow ηk s rghter n dropet (Fg. ). When sppng veoty nresng het fow spred onvetve oponent η nfuene so grows (Fg. ). In fster sppng dropet wter rutes ore ntensvey nd het s eded to ts entr yers ore uky too (Fg. ). Therefore dropet hets up euy nd oentu teperture dssouton n "" het trnsfer se s strong dfferent fro "k" het trnsfer se (Fg. 7). In strong ruton ode se uted dropet teperture n entr yers s ghty hgher thn n se of ner reedng ruton ode. It s oserved n ow (Fg. 7 ) nd strong (Fg. 7 ) sppng dropets. T K T T Fg. 7 Dropet sppng pt for o teperture. - e = ; :.7;.9;.8; 7.7; - e ; :.;.;.77; 7.9. Het spred ode nsde the dropet: () "k" () "" strong ruton se () "" ner reedng ruton se....8 T Fg. 8 Lud ruton ode nsde the dropet pt for uted o tepertures devton fro "k" het trnsfer se: ΔT η = T η T η k. - e = ; :.7;.9;.8;.7; - e = ; :.;.;.77;.9. () "" strong ruton () "" ner reedng ruton At the nt stge of dropet hetng the ggest dfferene oserves etween urves nd tht refets nstntneous teperture fed T η (Fg. 7). However ths dfferene s ess thn one nd hf degrees nd n unstedy phse trnsforton rege s onsstenty deresng (Fg. 8).

7 Mxu devton of o teperture T η fro ondutvty heted dropet teperture T η k for sppng dropet rehes. K nd. K when e = nd e = n ts entr yers respetvey (Fg. 8). A seeted het trnsfer ode hs nfuene for verge dropet ss teperture whh s uted ordng to expresson () (Fg. 9). Espey rght dfferene oserves etween dropet ther odeng resuts n het trnsfer ses of "" nd "k". Strong nd ner reedng ruton odes gve ose dropet ss verge teperture dyns (Fg. 9 nd urves). r further sppng dropet nyss "" het trnsfer se of strong ruton hs een hosen. T Fg. 9 Het trnsfer ode pt for verge ss teperture of non-sother dropet: () "k" hetng () "" strong ruton () "" ner reedng ruton se. e : ; 8 T Fg. Dropet sppng pt for uted nonsother dropet ther stte: () T (); () T () "k" het trnsfer se; () T () "" het trnsfer se. e : ; ; 8 A uted dropet surfe teperture funton T () grph do not depends fro het trnsfer ode nsde the dropet. However t s strongy ffeted y dropet sppng ntensty n gs (Fg. ). Dropet verge ss teperture funton T () grph tht desres nonsother dropet ther stte hnge depends fro pped het trnsfer ode for dropet nd when dropet sps ore ntensvey ths grph devtes fro "k" se grph (Fg. ). Infuene of pped het trnsfer ode n dropet for uted ther stte of dropet hnges t te: t s sgnfnt n dropet ondensng rege (Fg. 9) s weken t unstedy evporton rege nd n euru evporton rege "k" nd "" het trnsfer odes ensures the se uted dropet ther stte (Fg. ). T Fg. Dropet surfe nd ss tepertures hnge n unstedy fn stge nd n nt stge of euru evportons: () T (); () T () "k"; () T () "" het trnsfer se. e : ; ; 8 Te Sppng pt for dropet yers teperture t the end of ondensng rege o T o T. o T. 8 o T o T e preters re defned ordng to fn tertve ye resuts of ondensng rege. Dropet sppng hs nfuene for uted deter dyns (Fg. ) therefore extern onveton het fow densty tht desres dropet energy stte hnge n unstedy evporton osng stte s dfferent (Fg. ). Wter ruton nsde the dropet ensures ore

8 ntensve het tht s provded to surfe strton to entr yers nd esure opertes dropet yers wrng rte (Fg. ). Ths pts dropet unstedy teperture fed dyns (Fg. 7) nd dropet yers wrs dfferenty (Te ) n ondensng phse trnsforton rege. T K s T K s....8 Fg. Dropet deter hnge n unstedy phse trnsforton rege nd n nt stge of euru evporton. : () ; () ; () ; () ; () 8 e Fg. Dropet onvetve hetng ntensty hnge n unstedy phse trnsforton rege nd n nt stge of euru evporton. e : () ; () ; () ; () ; () 8 When prry sppng veoty n gs nreses then dropet yers verge wrng rte sowdown n ondensng phse trnsforton rege (Te ). Ths eds to durton nresng of ondensng phse trnsforton rege. k....8 Fg. Dropet sppng pt for ts yers wrng rte n ondensng phse trnsforton rege T / T T when. T j j / I : () T T ; () T TC ; () T T ; het trnsfer se: ; ; dropet prry sppng ntensty: - e ; - e " k" "" Te Sppng pt for verge wrng rte ΔT η / Δτ o (T ηo-t ) / τ o of dropet yers t ondensng rege T e T. T.8 T T o o o o o

9 . Conusons Surzng sppng dropet ther stte odeng resuts t n e stte tht uted dropet ther stte ordng to strong ruton ode n wek sppng se s ose to ner reedng ruton odeng resuts whe for ntensve sppng dropet se ordng to ner reedng ruton ode uted dropet ther stte s rght dfferent fro strong ruton odeng resuts. Therefore n dropets sppng se t s reoended to ppy strong ruton ode. Dropet sppng n hud r fow hs n pt for opex het trnsfer proesses nterton n dropet ondensng phse trnsforton rege nd kes preondtons for ntensve het drn to nner yers wth rutng wter. Wter wrs eveny n sppng dropet then surfe yers wrng rte sows down nd ondensng phse trnsforton rege durton nreses. Ths s very portnt ftor for tehnoog proesses optzton of het phse trnsforton utzton fro reove gs. eferenes. Sdf M.H.; Jhn I.; Stgoe A.B.; Hoon.. A theoret ode wth experent verfton for het nd ss trnsfer of sne wter dropets Int. J. Het Mss Trnsfer 8: Wen-ong C.; Hu C.; Le H.; We Z.. Effet of dropet fsh evporton on vuu fsh evporton oong: Modeng Int. J. Het Mss Trnsfer 8: Vokov.S.; Kuznetsov G.V.; Strzhk P.A.. Experent nvestgton of xtures nd foregn nusons n wter dropets nfuene on ntegr hrtersts of ther evporton durng oton through hgh-teperture gs re Internton Journ of Ther Senes 88: Kyoung H.K.; Hyung-Jong.; Kyoungjn.; Horo P.B.. Anyss of wter dropet evporton n gs turne net foggng proess Apped Ther Engneerng -: Srkr S.; ghur S.; Vsudevn.. Trnsent evporton of ovng wter dropets n ste hydrogen r envronent Int. J. Het Mss Trnsfer : -. Tseng C.C.; Vsknt.. Enhneent of wter dropet evporton y rdton sorpton Fre Sfety Journ : Musks G.. egurtes of unstedy rdtve-ondutve het trnsfer n evportng setrnsprent ud dropets Int. J. Het Mss Trnsfer : PII: S 7-9 ( ) hrd W.B III.. Correton for drop wse ondenston het trnsfer: Wter orgn fuds nd nnton Int. J. Het Mss Trnsfer : Jun G.; Chu N.; W H.M.. Poydsperse eroso ondenston wth het nd ss onservton: I. Mode desrpton wth pptons to hoogeneous systes Int. J. Het Mss Trnsfer : Musks G.; Šnkūns S.; Musks G.. Ev-porton nd ondensng ugentton of wter dropets n fue gs Int. J. of Het nd Mss Trnsfer : -. Kortsensteyn N.M.; Suov E.V.; Ystreov A.K. 9. Aout use of ethod of dret nuer souton for suton of uk ondenston of supersturted vpor Int. J. of Het nd Mss Trnsfer : ond F.E.; Mro W.; Eugeny Y.K.. The pt of Mrngon onveton on fud dyns nd ss trnsfer t defore snge rsng dropets A nuer study Che Engneerng Sene : Arzon B.; Srgnno W. A Dropet vporzton ode for spry ouston utons Int. J. of Het nd Mss Trnsfer : -8. ftp://ftp.dee.ufpr.r/cfd/ogrf/propuso/r zon_et 989.pdf. G. Musks M. Mzukenė A. Bčus J.Gudznsks MODELLING OF HEAT AND MASS TANSFE POCESSES IN PHASE TANSFOMATION CYCLE OF SPAYED WATE INTO GAS:. THEMAL STATE ANALYSIS OF A DOPLET SLIPPING IN HUMID AI FLOW S u r y Ths rte presents het trnsfer ode tht s sed on y het ondutvty theory n sppng dropet. Strong nd ner reedng ud ruton ses were nysed. Aordng to reeved suton resuts of ther stte y sppng dropet t n e stte tht ordng to strong ruton ode uted dropet ther stte n wek sppng se s ose to ner reedng odeng resuts whe dropet ther stte s rght dfferent fro strong ruton odeng resuts for ntensve dropet sppng se ordng to ner reedng ruton ode. Therefore t s reoended to ppy strong ud ruton ode for dropet sppng ses. Keywords: wter dropets onveton hetng phse trnsforton ye wter ruton ther stte. eeved Otoer Aepted Mrh