ࡱ>  r5@ Tibjbj22 wXXJ     PPPd8|d2Dȉ(3l8{}}}}}}$ƴRP5  6NH P{{+ HP;8 `T0l/d0n ;dd    P;|ddDt"ddISPITIVANJE TO NOSTI METODE ELEMENTARNIH BILANCI* U PROVOENJU TOPLINE Dr.sc. Neven NINI, Dipl.ing. Dinko JURJEVI Fakultet elektrotehnike strojarstva i brodogradnje, Sveu iliate u Splitu (Faculty of Electrical Engineering, Mechanical Engineering & Naval Architecture, University of Split) R. Boakovia b.b., 21000 Split, Croatia UDK-broj: 536.62:621.165 Klju ne rije i: metoda elementarnih bilanci, provoenje topline Keywords: control volume method, heat conduction Uvod Obzirom na raairenost problema provo]enja topline u tehnici i na ograni ene mogunosti njihova analiti kog rjeaavanja, uloga numeri kih metoda u ovom podru ju je zna ajna. Meu ostalim metodama, metoda elementarnih bilanci - uvedena u Patankar [1] (1980) - odlikuje se najveim izravnim udjelom fizikalnog sadr~aja u matemati kom modeliranju procesa. Razlog tome je u fizikalnom modeliranju realnog procesa, a koje zapo inje geometrijskom podjelom tijela na djelie elementarnih obujama. U njima elementarni djelii tijela imaju homogene temperature  one koje bi se dobile njihovom zamialjenom izolacijom i uravnote~avanjem. Te su temperature referentne za definiranje njihove unutarnje energije u svakom trenutku. Ovim se realno stanje modelira kao stepeni asto  ravnote~no, a neravnote~ni se procesi koncentriraju na grani nim plohama elementarnih obujama. Sa ovako nastalim skokovitim razlikama temperatura susjednih djelia treba izraziti toplinske tokove kroz njihove zajedni ke plohe. Zato referentne temperature djelia treba pripisati odreenim referentnim ili vornim to kama unutar djelia, kako bi se temperaturne razlike spregnule sa svojim toplinskim otporima, u smjeru okomitom na razdvojne plohe. Ra unski je to najjednostavnije, ako se uspostavi bilo kakav odnos referentnih to aka sa grani nim plohama djelia, u kome su spojnice tih to aka okomite na plohe koje ih dijele. U prilog ovoj metodi elementarnih bilanci (MEB) ide i to, ato se svakim izborom referentne to ke unutar djelia mo~e pogrijeaiti u pojedinim smjerovima, dok je ukupna greaka u sumi (paralelnih) otpora na elno nula. Uobi ajena je i dodatna pretpostavka, da su toplinski tokovi kroz pojedine meuplohe takvi, kao da su rasporedi temperature meu referentnim to kama linearni, te da va~i Fourierov zakon provoenja. U ovako definiranom fizi kom modelu realnog procesa provoenja topline primjenjuje se prvi zakon termodinamike u obliku bilanci, po kojima je suma svih toplinskih tokova u pojedine djelie jednaka promjeni njihovih unutarnjih energija. Za sve unutarnje djelie tijela nepoznanice su isklju ivo njihove temperature, a bilanci ima koliko i djelia. Za rubne djelie bilance su samo prilagoene rubnim uvjetima. To nost ra unanja toplinskih tokova ovisi o tome koliko su dobiveni gradijenti temperature bliski realnim projekcijama gradijenata na meuplohama susjednih djelia, i to tijekom kona nih vremenskih intervala. Odstupanje ra unskih gradijenata od njihovih stvarnih projekcija na meuplohama ovdje se interpretira kao prostorna i vremenska neadekvatnost modela u datom vremenskom intervalu, na datom mjestu. _________________________ * Ovaj naziv metode nam se ini znatno boljim od doslovnog prijevoda sa emgleskog kontrolnih volumena Tijekom 1980 tih i 1990 tih godina postali su dostupni mnogi komercijalni ra unalni programi s razli itim metodama za rjeaavanje problema provoenja topline. No neovisno od primjene takvih programa, poznavanje metode elementarnih bilanci i njene prakti ne primjene ima smisla, jer in~enjera bolje dr~i u kontaktu sa fizikalnim sadr~ajem problema koji rjeaava i smanjuje njegovu zavisnost od gotovih standardnih programa. U nastavku se daju rezultati prora una nestacionarnih temperaturnih polja u duga kom homogenom valjku s razli itim grani nim uvjetima. Prora uni su izvraeni analiti kom metodom [2], a zatim razli itim varijantama metode elementarnih bilanci (MEB). Usporedbom i analizom tih rezultata sagledavaju se razli iti izvori i veli ine greaaka karakteristi nih za primjenu MEB u tipi nim procesima provoenja. Referentno tijelo, proces i analiti ko rjeaenje Kao referentni proces i tijelo odabrani su hlaenje odnosno grijanje beskona nog valjka homogene po etne temperature, u grani nim uvjetima tree vrste (a, T() i s homogenim rasporedom koeficijenta prijelaza topline po povraini. Analiti ko rjeaenje temperaturnog polja je rjeaenje Fourierove jednad~be koja u cilindri nim koordinatama glasi:  EMBED Equation.DSMT4  (1) Jednad~ba (1) se u jednodimenzijskom nestacionarnom obliku svodi na parcijalnu diferencijalnu jednad~bu paraboli kog tipa s radijusom i vremenom kao varijablama  EMBED Equation.DSMT4  (2) Kako je lijeva strana (2) funkcija geometrijske varijable, a desna strana vremenske, njihova e jednakost biti zadovoljena ako su im vrijednosti jednake  EMBED Equation.DSMT4  (3)  EMBED Equation.DSMT4  je separacijski parametar koji prema dvjema diferencijalnim jednad~bama u (3) ima viae rjeaenja ( EMBED Equation.DSMT4 , n = 1, ... , (). Posljedica ovakvog razdvajanja varijabli je analiti ko rjeaenje (2) kao produkt funkcije vremena i funkcije polo~aja, jer uvratenjem  EMBED Equation.DSMT4  u (2) slijedi (3). Zamjenjujui temperaturu T razlikom  EMBED Equation.DSMT4 , karakteristi nom za problem, modificirana diferencijalna jednad~ba dobiva oblik  EMBED Equation.DSMT4  (4) Uz poznate grani ne i vremenske po etne uvjete za r = 0  EMBED Equation.DSMT4  i za r = R  EMBED Equation.DSMT4  te po etne uvjete  EMBED Equation.DSMT4  analiti ko rjeaenje kao produkt vremenske i prostorne funkcije glasi  EMBED Equation.DSMT4  (6) gdje su: EMBED Equation.DSMT4 Besselova funkcija I. vrste, nultog reda za EMBED Equation.DSMT4 EMBED Equation.DSMT4 Besselova funkcija I. vrste, I. reda za EMBED Equation.DSMT4 EMBED Equation.DSMT4 modificiran Fourierov broj Periodi nost Besselovih funkcija daje parcijalna rjeaenja ija suma predstavlja ukupno rjeaenje, pri emu parcijalna rjeaenja zadovoljavaju transcendentalnu jednad~bu: EMBED Equation.DSMT4 za EMBED Equation.DSMT4  modificiran Biotov broj Prema ovoj jednad~bi se korijeni diferencijalne jednad~be (n = (nR nalaze u podru jima EMBED Equation.DSMT4 kada su Besselove funkcije samo pozitivne ili negativne. Simultanom integracijom i iteracijom pronaeno je prvih 60 korijena diferencijalne jednad~be pomou kojih su ra unate promjene temperature analiti kim putem. Dakle, analiti ki se nepoznata temperatura prora unava prethodnim prora unom funkcija J0 i J1 (za radijus r) i Fo*  broja (za trenutak (). Teoretski, za beskona an broj lanova niza (parcijalnih rjeaenja) u nultom bi trenutku sve to ke valjka imale po etnu temperaturu. Pri tome lanovi niza s velikim periodama imaju utjecaj na bezdimenzijski omjer (/(po  samo na po etku hlaenja. Npr. ukoliko se tra~i raspored temperatura od po etka hlaenja nakon 0,01 s utjecaj na rezultat ima prvih 60 lanova niza nakon tri sekunde prvih desetak lanova niza utje e na rezultat nakon 30 s prvi lan niza je dovoljan za odreivanje rasporeda temperatura (sve sa greakama ispod 10-4 C). U odnosu na ostale greake, analiti ko rjeaenje smatrat e se to nim rjeaenjem. Numeri ka rjeaenja metodom elementarnih bilanci Homogenost koeficijenta prijelaza topline po vanjskoj povraini uvjetuje osno  simetri an raspored temperatura, stoga je dovoljno promatrati proces u radijalnom pravcu. Bit metode elementarnih bilanci je u tome, da se za tvar u svakom kona nom elementarnom obujmu, na koje je tijelo podjeljeno, napiae bilanca energije izra~ena preko temperatura Ti u vornim to kama i tih obujama. Pri tome se cijela tvar u obujmu smatra ravnote~nom na temperaturi u vornoj to ki. Temperaturna neravnote~a, u stvarnosti kontinualno rasporeena unutar tijela, koncentrira se prema tome u MEB na granicama elementarnih djelia. Toplinski tokovi na tim granicama onda se odreuju pomou temperatura u vornim to kama s obje strane granice, u pravcu okomitom na nju. Openito, odgovarajua bilanca energije glasi:  EMBED Equation.DSMT4  (7) Slika 1. U samom segmentu pojavljuju se tri karakteristi na kontrolna obujma: centralni s konduktivnim utjecajem na samo jednoj granici (s elementarnim obujmom 2), proizvoljni unutarnji s konduktivnim utjecajem na obje granice i rubni s konduktivnim utjecajem na jednoj i konvektivnim na drugoj granici (sl.1): Izra~avajui temperaturne gradijente na granicama elementarnih obujama linearnom interpolacijom izmeu temperatura u referentnim - vornim - to kama dobiva se: za centralni volumen:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (8) gdje je EMBED Equation.DSMT4  za unutarnje djelie:  EMBED Equation.DSMT4   EMBED Equation.DSMT4 za radijus i te vorne to ke definiran kao:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (9) za rubni djeli:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  (10) gdje je  EMBED Equation.DSMT4  Specifi nost eksplicitne varijante MEB, na koju se gornje jednad~be i odnose, je u tome da za odreenu vrijednost Fo  broja lanovi uz temperature  EMBED Equation.DSMT4  (i = 1 n+1) mogu imati negativnu vrijednost. U tom bi slu aju postojala konstantna vremenska oscilacija temperatura u vorovima, koje bi u jednom koraku padale ali u narednom rasle (bez obzira ato se toplina odvodi) ato nema nikakve fizikalne utemeljenosti. To je razlog zbog kojeg (ali samo u eksplicitnoj varijanti) ti lanovi moraju uvijek biti pozitivni ato se osigurava takvim vremenskim korakom koji e zadovoljavati uvijet Fo ( Fokrit (pri emu su za Fokrit lanovi uz  EMBED Equation.DSMT4  jednaki nuli i zatim se bira najmanji Fokrit). Na taj se na in definira kriterij stabilnosti numeri kog rjeaenja (iz jed. 7, 8 i 9): centar valjka (DR = R/10):  EMBED Equation.DSMT4  unutarnje vorne to ke (DR = R/10):  EMBED Equation.DSMT4  rub valjka (minimalni Dt je za n = 10 i a = 10 000 W/m2K, tj. za Bi = 0,5):  EMBED Equation.DSMT4  Centar valjka je to ka s najmanjim kriti nim vremenskim korakom ato zna i da usvojen vremenski korak mora biti manji od 0,1136 s. Jednad~be bilanci energije u implicitnoj varijanti MEB su sljedee: centralni djeli:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  (11) Kako postoji puna analogija izmeu jednad~bi u implicitnoj i eksplicitnoj varijanti MEB, ostale se jednad~be bilanci energije u implicitnoj varijanti mogu napisati izravno iz eksplicitne: unutarnji djelii (i = 2 n):  EMBED Equation.DSMT4  (12) rubni djeli:  EMBED Equation.DSMT4  (13) Karakteristi ne greake numeri kih prora una Pristupajui analizi odstupanja numeri kih rezultata od analiti kih korisno je sistematizirati greake karakteristi ne za metodu elementarnih bilanci. Pri tome e se zbog vee odreenosti imati u vidu jedan konkretni i to rubni kontrolni polu(element) jer je numeri ki prora un njegove temperature, zbog brzine procesa i polo~aja vorne to ke, podlo~an najveoj greaki. To na tj. analiti ka temperatura na povraini tog elementa ozna avat e se sa  EMBED Equation.DSMT4 , dok je numeri ki dobivena temperatura vorne to ke na istom mjestu ozna ena kao  EMBED Equation.DSMT4 . Po suatini MEB to je ujedno referentna temperatura cijelog rubnog elementarnog djelia. Karakteristi ne pogreake, odnosno izvori greaaka su sljedei: Temperature u vornim to kama MEB su ustvari termodinami ki srednje (dobivene zamialjenim adijabatskim uravnote~enjem dijelia [3]), te je po ovom izvoru greake, u slu aju hlaenja rubnog poluvolumena valjka  EMBED Equation.DSMT4  >  EMBED Equation.DSMT4 . Temperaturom  EMBED Equation.DSMT4  nije dobro izra~en toplinski tok od unutarnjeg (susjednog) elementa temperature  EMBED Equation.DSMT4 u ovaj rubni. Temperaturna razlika  EMBED Equation.DSMT4  spre~e se naime s konduktivnim otporom cijele debljine rubnog dijelia. Za unutarnje elemente analogno sprezanje je s polovinama debljina, ato je to nije, jer se vorne temperature bli~e sredinama elementarnih obujama bolje poklapaju sa stvarnima na istim mjestima. Vremenski trenutak u kome se uzimaju temperaturne razlike prema susjednim elementima (ili prema okolnom fluidu) kod eksplicitne metode je preran, a kod implicitne na neki na in prekasan. Greaka koja je posljedica ovoga nazvat e se greakom zbog  vremenske neadekvatnosti temperaturnih gradijenata. Zamjena stvarnog gradijenta temperature na granici elementa sa gradijentom pri linearnoj interpolaciji izmeu  EMBED Equation.DSMT4  i  EMBED Equation.DSMT4  dovodi do greake koja e se nazvat greaka zbog  prostorne neadekvatnosti temperaturnih gradijenta. Na ilustraciji, (sl.2.), stvarni je gradijent na granici elementarnih obujama manji od dobivenog iz linearne interpolacije (kao da nije uzet na adekvatnom mjestu  na granici). Slika 2. Kona no treba spomenuti da po prirodi provoenja topline, a i odgovarajuih jednad~bi matemati kog modela MEB, postoji tendencija samoispravljanja greaaka proizaalih iz prethodnih koraka prora una. Naime, svaka ra unski prebrza lokalna promjena temperature u narednom vremenskom koraku dovodi do manjih gradijenata i usporavanja ra unske promjene i obratno*. Kako numeri ki dobiveni rezultati ovise o grani nom uvjetu ((,T(), prostornom koraku (krupnoi podjele  (R) i veli ini vremenskog koraka (((), odstupanja od analiti ki dobivenih temperatura analizirat e se u ovisnosti o tim parametrima. Rezultati prora una i analiza Referentni proces u kojem se usporeuju analiti ka i numeri ka rjeaenja je hlaenje beskona nog valjka u grani nim uvjetima tree vrste i u airokom rasponu Bi  brojeva. Kao konkretni fizi ki parametri usvojeni su: R = 25 mm , l = 50 W/mK , a = 1,375 10-5 m2/s , Tpo  = 1000C , T( = 25C U prora unu su koriatene eksplicitna i implicitna varijanta MEB, s variranjem vremenskog i prostornog koraka. Rezultati samog analiti kog prora una nisu posebno interesantni, jer su odgovarajua rjeaenja klasi na, a u standardnim ud~benicima data i grafi ki [2]. U odnosu na analiti ka rjeaenja, apsolutna odstupanja raznih varijanti MEB data su u dijagramima na sl.3. 13. Prva serija dijagrama (sl.3. 6.) prikazuje odstupanja rezultata eksplicitne i implicitne varijante MEB od analiti ki to ne temperature u po etnom periodu hlaenja (( < 3 sek). Rezultati se odnose na: tri vremenska koraka: (( = 0,025 s ; (( = 0,05 s ; (( = 0,1 s dva prostorna koraka: n = 5 ; n = 10 (odnosno (R = R/5 ; (R = R/10) etiri koeficijenta prijelaza topline (25 ; 100 ; 2000 ; 10000 W/m2K). Po etna temperatura, polumjer valjka i fizi ka svojstva materijala u svim su slu ajevima isti. U dijagrame su uneseni rezultati nakon svake desetinke sekunde hlaenja, tj. za vremenski korak (( = 0,025 s unose se rezultati iz svakog etvrtog ra unskog koraka, za korak (( = 0,05 s svakog drugog, a za (( = 0,1 s iz svakog koraka. Slova uz karakteristi nu liniju u svim dijagramima ozna avaju eksplicitnu (E) ili implicitnu (I) varijantu, broj 05 ili 10 krupnou prostorne podjele, a " predstavlja vremenski korak. Apscisa je vremenska os, dok se na ordinati unesena apsolutna temperaturna razlika DT = Tnum  Tan odnosi samo na rubnu vornu to ku (dakle, EMBED Equation.DSMT4 ). Sl.3 prikazuje odstupanja pri najslabijem intezitetu hlaenja (( = 25 W/m2K). Samo na prvi pogled iznenauje pojava najvee greake u eksplicitnoj varijanti za oba prostorna koraka pri najmanjem vremenskom koraku, ve od prvog ra unskog koraka. Ukupne greake ovdje su prema datoj sistematizaciji uglavnom tipa  1 i  3 . Naime, za sve tri vrijednosti vremenskog koraka (( treba o ekivati da e temperature  EMBED Equation.DSMT4  kao prostorno srednje rubnog poluelementa u fazi brzog hlaenja biti viae od njegove minimalne temperature na vanjskoj povraini, a prema tome i od  EMBED Equation.DSMT4 . Slika 3. Najsporiji pad temperature  EMBED Equation.DSMT4  i najvee odstupanje od analiti kog rjeaenja bit e ono ra unato za najmanji vremenski korak jer e se na taj na in (vremenski) najprije uzeti u obzir dovod topline iz susjednog unutarnjeg elementa (pored stalnog odvoda u okolni fluid). Time se mo~e zaklju iti da bi u po etnom periodu hlaenja odstupanja numeri kih od analiti kih rezultata u eksplicitnoj varijanti MEB bila minimalna za najvei tj. kriti ni vremenski korak. Nasuprot eksplicitnoj, u implicitnoj varijanti greaka je najmanja pri najmanjem vremenskom koraku jer su tako vremena u kojima se ra unaju temperaturni gradijenti  smjeataju ranije, kada su gradijenti vei, a onda i ra unato hlaenje br~e. Ipak, ne toliko brzo kao kod eksplicitnih varijanti, koje u po etnom periodu pokazuju prednost, posebno varijante s veim vremenskim koracima. Usporedbom eksplicitnih i implicitnih rezultata vidi se da je bolja ona varijanta koja osigurava br~i pad temperature, a to je eksplicitna i to s najveim vremenskim korakom. Utjecaj prostornog koraka je vidljiv na sl.3. kada prednost ima sitnija, iako joa uvijek relativno krupna podjela. Ona daje 4 5 puta manju apsolutnu greaku od dvostruko veeg broja podjela. Izvori greake kod vrlo grube podjele su tipa  1 i  4 . Vei u inak usrednjavanja temperature i odstupanja od realne u vornoj to ki kod krupnijih elementarnih obujama (n = 5) vodi malom padu temperature i porastu greake u prvih cca 0,25 s. Ovdje se nee obraivati, ali sa ekstremnim prostornim korakom (n = 2), koji dijeli segment valjka na centralni, jedan unutarnji i rubni elementarni obujam, ta greaka raste u cijele prve dvije sekunde. S daljnjim usitnjavanjem (n = 15, 20 i 25) odstupanja meu podjelama n = 10 25 su zanemariva. Na sl.4. 6. su dijagrami odstupanja od analiti kih rjeaenja pri ja im intezitetima hlaenja, takoer tijekom prve tri sekunde. Slika 4. Slika 5. Slika 6. Vidljiva je gruba proporcionalnost apsolutne temperaturne razlike i koeficijenata prijelaza topline. Najbolja ilustracija osjetljive isprepletenosti i naizmjeni nog prevladavanja razli itih izvora greake su rezultati koji se odnose na implicitnu metodu za grublju podjelu (n = 5). Za vremenski korak Dt = 0,05 s i a = 25 W/m2K greaka  iako nikad ne prelazi 0,8C  raste od 20-te sekunde (sl.7.) i nastavlja rasti i dalje, pa i nakon 30 te minute (sl.8). Bli~e objaanjenje ovoga fenomena bilo bi u slijedeem. U odmakloj fazi hlaenja valjka, pri relativno malom koeficijentu prijelaza topline, toplinski tok s povraine valjka je ve toliko malog intenziteta da greaka zbog linearne interpolacije temperature na unutarnjoj granici rubnog volumena ne daje gradijent kakav je stvarno na toj granici, ve toliko vei da je rezultirajua  EMBED Equation.DSMT4  pr*,:bd & R f  ( X 48$*46BFrb»´­¦¦ h[c5CJ h=s5CJ ht 5CJ h8;u5CJ h~A5CJ h>25CJ h*}5CJ h95CJ hPF5CJ hr5CJ hrCJh9 hrCJ] hr6 hrCJ hr5hr hPFCJ h9CJ hrCJ 1 L R H | L!$d`a$gd $da$gd$da$ $ & Fda$d 4d^4` $dha$$da$d $d`a$d`bccRibd| L!f""#.###&$Z$$$$\&&~'''''*(,(\)^)d)f)*++>+@+B+D+++,,,<->-l-n-p-r-୥ΘέΊΥ}sjhrEHUjwA hrCJUVhw%jhrEHUjMB hrCJUVjhrUh} jhr5CJH*hr5CJOJQJ hrCJhr h}5CJ hDDm5CJ hr5CJ h95CJ h5CJhh5CJ-L!P!R!$$''*(,(*+,-/034556&7`d^``d d`$d`a$ $ & Fda$ $d`a$r---.../x/z///// 0 000@0B0D0F000 1 1:1<1>1@1^1`111r2t22222222233H3J3Ⱦvj.B hrCJUVjIhrEHU$jtB hrCJUVmHnHu jhrj hrEHUjCA hrCJUVj hrEHUj,B hrCJUVj]hrEHUj,B hrCJUVjhrUhw%h}hr.J3L3N3`4b4444444<5N5z5|55555560626`6b6d6f666666677777788齳馜鏅r$jB hrCJUVmHnHujhrEHUj.B hrCJUVjhrEHUjA hrCJUVjhrEHUjA hrCJUVh8;uhw%j hrEHUj#A hrCJUVhrjhrUjUhrEHU(&77 8)88899P;;;<<BxBBCrDtDvDxDd & Fd & Fd$d`a$@ ^@ ` d`88888)8*8>8?8@8A8p8q88888888888888888888888X;Z;;;Ż鰦雑~tijCB hrUVj5hrEHU$jxB hrCJUVmHnHuj3hrEHUjPA hrUVj.hrEHUjPA hrUVj,hrEHUjPA hrUVj(hrEHUjPA hrUVh8;uhrjhrUju"hrEHU&;;;;;;;;;j<l<n<t<v<x<<<<<<<2?4????BADAFAHANACCrDvDxDDDDDBFjFFFGGG鴪铎y h8;uhr h8;u5CJ hr5CJ hr5h8;u hrH* jqhr jthrj?hrEHUj5/B hrCJUV jlhr hrH* jdhrj<hrEHUjPB hrUVhrjhrUj9hrEHU/xDDD~HHKKKKK>N@NOO$PzPP&QQBR & Fd & Fdd d`d`d`gd8;u & FdGG HHVIZIvIIFJJzK|KKKKKKKKZLfLLLL M*N:NNNNNDOJOXO\OOOOOOOOOPP P"P$P&PTPVPjmA hrCJUVjJhrEHUjVA hrCJUVjGhrEHU$jb2B hrCJUVmHnHuh){ hr56jrChrEHU$jA1B hrCJUVmHnHujhrUhi|hrh8;u2VPXPZPvPPPPPPPPPPPQQ Q"Q&Q(QVQXQZQ\QQQQQQQQQRR R"R>RTR廱垔wj`jbhrEHUjA hrCJUVj_hrEHU$j2B hrCJUVmHnHujYhrEHU$jO3B hrCJUVmHnHujVhrEHU$j2B hrCJUVmHnHujRhrEHUj}A hrCJUVhrhi|jhrUj.OhrEHU%TR`RdRfRRRRRRRRRRRRRSSS S0S6SJSLSzS|S~SSSSTTTTTTJXLXNXPXƼ{qlel jhr hr6jxhrEHU$j$5B hrCJUVmHnHuj~uhrEHUj0A hrCJUVjphrEHUjA hrCJUVj?khrEHUjA hrCJUVjLghrEHU$jw3B hrCJUVmHnHujhrUhrhi|'BRRR:SSSYZbZZZ[[\\Z]]].^__N` & Fd & Fd & F & F & F` d` & FdPXTX\XXXXXXXXXX,Y4YZZZ ZNZPZRZTZZZZZZZZZ[ [@[B[\[^[[[[[[[sijhrEHU$jܺB hrCJUVmHnHu hrH*jhrEHU$jպB hrCJUVmHnHuj!~hrEHU$jB hrCJUVmHnHuhrOJQJj{hrEHU$j5B hrCJUVmHnHujhrU hr6 hrH*hr)[:]>]n]z]]]]]]]]]]]]](^*^^^____``` `H`Z`f`n`p```````abDbccddddzjoA hrCJUV hr5CJjřhrEHUj1A hrCJUVj;hrEHU$j7B hrCJUVmHnHujhrEHUj0A hrCJUVj3hrEHUjA hrCJUVjhrUhi|hr0N`n````aa\f^fffhhlloorrrrhuju & Fdd & Fd & Fd d` & Fddddreteeeeeee0fXf~ffgg|h~hhhhhhhhhhhii>i@iBiDiiijjjjbjdjj齳馜鏅xnjhrEHUj(pA hrCJUVjhrEHUjoA hrCJUVj{hrEHUjoA hrCJUVj٣hrEHUjoA hrCJUVh=l2hi|j7hrEHUjoA hrCJUVhrjhrUjhrEHU+jjjj$llJloopppp p"pPpRpTpVpqqrrpstsbuduuuuuvvvvJwwNyPyyyܴܣܝܖ܎܇܇yql hrH*hrOJQJ hr5CJ jthr jDhr jhrH* jahr hr0J hr56jhrEHUj A hrCJUVj{hrEHUjA hrCJUVh=l2hrjhrUjhrEHUjupA hrCJUV*juHwJwww6yyy||~N~~Tz@D $d`a$ & Fdd`d & Fd d`yyyyyyyJzNzn|r|}}~~z~|~~~~~~~~~(*>@bdf>@Bz468:@BD…jhrEHU$j{B hrCJUVmHnHujhrUhrOJQJhi? hr6 jDhr jthrh=l2 jhrH* hrH* hrH*hr?…ąօ؅$&(np̉Ή VXnrܖ:ěƛ,VXğƟ͝h){hrOJQJ hr56hi?h=l2j*hrEHU hr5jhrEHUjA hrCJUVjhrEHUjA hrCJUVjhrU jthr jDhr hrH*hr jahr1 BD(:<$d`a$ d`HJ  24 .2JN.0 8h\^׻׻ױש׻ hrCJ hr5CJ jhrH* hrH* hi?5 hrH* hr5hi?hrOJQJh){UhrjhrUjhrEHU$jJB hrCJUVmHnHu=imjetno viaa od stvarne, odnosno analiti ke. Rije  je dakle, o utjecaju izvora greake usljed  prostorne neadekvatnosti izbora gradijenta . Po istoj varijanti metode, greaka pada na nulu i zatim zadobiva suprotan predznak ako se samo promijeni vremenski korak na Dt = 0,1 s. To zna i da bi stagnaciju ili stalan pad greake dao tek neki odreeni srednji vremenski korak pri kojemu bi se  prostorna neadekvatnost izbora gradijenta kompenzirala  vremenskom neadekvatnoau . Slika 7. Slika 8. Slika 9. Slika 10. Na dijagramima hlaenja za izrazito visoke koeficijente prijelaza topline a = 2000 i 10000 W/m2K (sl.11. 13.) uo ava se, u usporedbi s a = 25 W/m2K, sna~an porast utjecaja greake tipa  5 , koji ima za posljedicu da sve krivulje greaki u obje varijante MEB brzo konvergiraju nuli  ve nakon 3,8 odnosno 0,8 min. Pri manje intenzivnom hlaenju (sl.9. i 10.), nakon prve 3 sekunde prestaje intenzivan utjecaj izvora greake tipa  1 , a dolaze do izra~aja slabiji utjecaji drugih tipova, sa slo~enim i promjenjivim ukupnim utjecajem., zavisnim od svim okolnostima nabrojenim u izvorima greake tipa  1   5 . Slika 11. Slika 12. Slika 13. Radi provjere iznijetih razmatranja o izvorima i utjecaju pojedinih tipova greaki, na sl.14. je prikazan proces grijanja valjka u fizikalno suprotnim po etnim i rubnim uvjetima od onih na sl.4. (Tpo  = 25C, T( = 1000C). Kako svih pet navedenih tipova izvora greake u MEB ovise o prostornom rasporedu temperatura u valjku, a on je u svakom trenutku simetri an u odnosu na ordinatu pri hlaenju, to bi se istim simultanim utjecajem svih tipova izvora greake mogli opravdati samo rezultati potpuno simetri ni svim odgovarajuim varijantama MEB na sl.4. A to je prema dijagramima na sl.4. i sl.14. evidentno to no. Slika 14. Bezdimenzijski prikaz rezultata Na sl.15. i sl.16. predstavljeni su rezultati svih varijanti prora una u bezdimenzijskom obliku [2] radi uopavanja rezultata ispitivanja. Na ordinati je bezdimenzijska relativna promjena temperature povraine valjka pri hlaenju, na apscisi Fo*  broj kao bezdimenzijsko vrijeme, dok je Bi*  broj parametar u airokom rasponu vrijednosti (za a = 25, 100, 2000 i 10000 W/m2K). Svaki od etiri snopa krivulja za etiri Bi*  broja predstavlja rezultate prora una po svim koriatenim varijantama MEB, uklju ujui i to no, odnosno analiti ko rjeaenje. To je rjeaenje predstavljeno debljom krivuljom. Na sl.15. dati su u krupnijoj razmjeri rezultati prora una za period najbr~e promjene temperature (0,002 < Fo* < 0,0672). To se podru je prakti ki poklapa s ordinatom na sl.16 (druga razmjera za Fo*) pa je izostavljeno i uzet u obzir samo prikaz od prve minute nadalje. Za svo vrijeme koje odgovara Fo* > 0,0672 (tj. za t > 3 s), vidi se da je relativno odstupanje od analiti kog rjeaenja u uobi ajenom grafi kom prikazu rezultata nevidljivo (krivulja analiti kog poklapa se s numeri kim rjeaenjima). Slika 15. Slika 16. Odstupanja su vidljiva i treba ih uzeti u obzir za Fo* < 0,02 i za Bi* > 1. Pri tome najvea greaka pripada implicitnoj varijanti MEB s vrlo grubom podjelom na kontrolne volumene (n = 5) i s najveim vremenskim korakom (Dt = 0,1 s) gdje greaka dosti~e do 10% i pozitivna je (za naglo hlaenje). Najbolji pak rezultat u periodu najbr~e promjene temperature daje eksplicitna varijanta MEB, s umjereno sitnom prostornom podjelom (Bi ( 0,1) i krupnijim vremenskim korakom, najbli~im kriti nom (Fokrit = 0,25). Njena greaka u cijelom periodu nigdje ne prelazi 1%. Zaklju ak U radu su prikazani rezultati usporednih prora una temperaturnih nestacionarnih polja analiti kom metodom i raznim varijantama metode elementarnih bilanci. Zbog fizikalne transparentnosti MEB moglo se greake karakteristi ne za tu metodu podijeliti prema njihovim ishodiatima i prepoznati ih u pojedinim rezultatima prora una. U tom pogledu zaklju ak je da utjecaj pojedinih uzroka greaki ovisi o vremenski promjenjivim konkretnim lokalnim okolnostima u tijelu i to na na in da relativna veli ina pojedinih greaki ovisi o koriatenoj varijanti MEB. Zbog toga na~alost nije mogue izvesti openiti zaklju ak o prednosti ove ili one varijante i o granicama ukupne greake u konkretnim problemima. No rezultati iz poglavlja 6. ukazuju na eksplicitnu varijantu MEB s relativno velikim vremenskim korakom (blizu granice stabilnosti rjeaenja) kao na onu koja prakti no tijekom cijelog trajanja procesa daje relativnu greaku ispod 1%. Uspjeanost ove varijante MEB se mo~e povezati sa meusobnom kompenzacijom greake usrednjavanja temperature (greaka tipa  1 )  koja vodi na ra unski viae temperature pri hlaenju (i ni~e pri grijanju) s greakom preranih ra unskih gradijenata (tip  3 )  koja vodi na ra unski ni~e temperature pri hlaenju (i viae pri grijanju). Kod osobito brzih dijelova procesa glavni je izvor greake za grani ne (polu)obujme tipa  1 u poglavlju 4., pa je na elno mogua i posebna ocjena njene veli ine i znaka, u ato se ovdje nije upuatalo. Utjecaj pribli~ne aproksimacije realne granice tijela ovdje takoer nije tretiran, jer je u slu aju valjka ra unska granica ujedno i realna. Openito e u dvodimenzionom slu aju ta geometrijska aproksimacija za krivocrtnu granicu biti finija ili grublja stepenasta. Takva aproksimacija na elno mijenja i toplinski kapacitet i konduktivni i konvektivni otpor cijelog rubnog sloja tijela. U [4] je predlo~en na in fiktivne (ra unske) ciljane izmjene toplinskog kapaciteta rubnih elemenata tijela i lokalnog koeficijenta prijelaza topline tako, da toplinski kapaciteti i toplinski otpori rubnih elemenata tijela  i pored stepenaste aproksimacije granice  ostanu ravni realnima. No ispunjenje ovoga uvjeta u opem slu aju predstavlja prevelike teakoe. Stoga kao preporuka ostaje da se krivocrtna kontura tako zamijeni sa dovoljno finom stepenastom, da toplinski kapaciteti i konduktivni otpori rubnih dijelova tijela ostanu bliski stvarnima, a da se samo lokalni koeficijenti prijelaza topline podijele faktorom poveanja duljine konture u odnosu na realnu. Literatura / References [1] PATANKAR, S.V.: Numerical Heat Transfer and Fluid Flow, Hemisfere Publishing Corp. and McGraw Hill Book Co., New York 1980. [2] PITTS, D.R.; SISSOM, L.E.: Heat transfer, McGraw Hill Inc., New York 1977. [3] NINI, N.: Elementi prijenosa topline i tvari, Interna skripta, Sveu iliate u Splitu, Fakultet elektrotehnike, strojarstva i brodogradnje, Split 2002. [4] NINI, N.; PODRUG, S.; JELASKA, D.: Prora un nestacionarnog temperaturnog polja metodom kontrolnih volumena s kompenzacijom rubnih uvjeta, Strojarstvo 41 (5,6) 207  214, 1999. Oznake / Symbols  EMBED Equation.DSMT4   koeficijent temperaturne vodljivosti [m2/s]  thermal diffusivity  EMBED Equation.DSMT4   Biotov broj  Biot number  EMBED Equation.DSMT4  modificiran Biotov broj  modified Biot number C specifi ni toplinski kapacitet [J/kgK]  specific thermal capacity  EMBED Equation.DSMT4  Fourierov broj  Fourier number  EMBED Equation.DSMT4  modificiran Fourierov broj  modified Fourier number M masa [kg]  mass N broj prostornih koraka du~ polumjera  number of space steps along radius  EMBED Equation.DSMT4  toplinski tok [W]  heat flux R, DR polumjer, duljina prostornih koraka [m]  radius, space steps lenghts T, DT temperatura, temperaturna razlika [C]  temperature, temperature difference V volumen [m3]  volume a koeficijent prijelaza topline konvekcijom [W/m2K]  convective heat transfer coefficient l koeficijent toplinske vodljivosti [W/mK]  thermal conductivity r gustoa [kg/m3]  density  EMBED Equation.DSMT4  relativna temperatura [C]  relative temperature  EMBED Equation.DSMT4  bezdimenzijska temperatura  nondimensional temperature  t, Dt  vrijeme, ra unski vremenski korak [s]  time, time computation step  Indeksi / Indices 1 temperatura centra [C]  centreline temperature i temperature unutarnjih vornih to aka [C]  internal nodal points temperatures n + 1 temperatura rubne vorne to ke [C]  boundary nodal point temperature po  po etna temperatura [C]  initial temperature t temperatura na po etku ra unskog vremenskog koraka [C]  temperature at the beginning of time computation step t + Dt temperatura na kraju vremenskog koraka [C]  temperature at the end of time step ( temperatura okoline [C]  environment temperature  ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Sa~etak: U radu su numeri ka rjeaenja nestacionarnog temperaturnog polja u valjku, dobivena metodom elementarnih bilanci (MEB), usporeena s postojeim analiti kim rjeaenjima kao to nima. Karakteristi ne razlike, svojstvene raznim varijantama MEB, prikazane su grafi ki u realnom i u bezdimenzijskom obliku. Izvedeni zaklju ci olakaavaju izbor varijante i ocjenu greake MEB u konkretnim primjenama. Oni daju odreenu prednost eksplicitnoj metodi sa vremenskim korakom blizu granice stabilnosti i sa dovoljno sitnom prostornom podjelom. U tom slu aju, u granicama ispitanih Bi i Nu zna ajki relativna greaka nije prelazila 1%. ACCURACY ANALYSIS OF THE CONTROL VOLUME METHOD IN HEAT CONDUCTION Abstract: Numerical solutions of the nonstationary temperature field in the cylinder obtained by control volume method (CVM) are compared to the conventional analytical solution. The differences characteristic of different CVM variants are shown graphically in real and in dimensionless form.. The conclusions based on the analysis facilitate the choice of the specific CVM variant and error estimation in concrete applications. They give explicit variant with time step close up to stability limit a certain advantage over other variants. At that space grid has to be fine enough for the purpose. In that case accuracy for tested Bi and Nu numbers is within plus or minus 1%. Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 1.  Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 2.  Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 3.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 4.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 5.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 6.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 7.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 8.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 9.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 10.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 11.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 12.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 13.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 14.  Legenda / Legend 0,025 ; 0,05 ; 0,1  ra unski vremenski koraci [s] / time computation steps E ; I  eksplicitna ; implicitna varijanta MKV / explicit ; implicit CVM scheme 05 ; 10  broj prostornih koraka du~ polumjera / number of space steps along radius Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 15.  Neven NINI, Dinko JURJEVI ISPITIVANJE TO NOSTI METODE KONTROLNIH VOLUMENA U PROVOENJU TOPLINE Slika 16.  Objaanjenja uz slike / Text under figures Sl.1.  Segment valjka i karakteristi ni kontrolni volumeni Fig.1.  Cylinder segment and characteristic control volumes Sl.2.  Realni i modelni raspored temperature rubnog kontrolnog volumena Fig.2.  Real and modelling boundary control volume temperature distribution Sl.3.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 3 s za a = 25 W/m2K Fig.3.  Differences between numerical and analitical results in time period 0 to 3 s and a = 25 W/m2K Sl.4.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 3 s za a = 100 W/m2K Fig.4.  Differences between numerical and analitical results in time period 0 to 3 s and a = 100 W/m2K Sl.5.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 3 s za a = 2000 W/m2K Fig.5.  Differences between numerical and analitical results in time period 0 to 3 s and a = 2000 W/m2K Sl.6.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 3 s za a = 10 000 W/m2K Fig.6.  Differences between numerical and analitical results in time period 0 to 3 s and a = 10 000 W/m2K Sl.7.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 60 s za a = 25 W/m2K Fig.7.  Differences between numerical and analitical results in time period 0 to 60 s and a = 25 W/m2K Sl.8.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 30 min za a = 25 W/m2K Fig.8.  Differences between numerical and analitical results in time period 0 to 30 min and a = 25 W/m2K Sl.9.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 60 s za a = 100 W/m2K Fig.9.  Differences between numerical and analitical results in time period 0 to 60 s and a = 100 W/m2K Sl.10.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 30 min za a = 100 W/m2K Fig.10.  Differences between numerical and analitical results in time period 0 to 30 min and a = 100 W/m2K Sl.11.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 60 s za a = 2000 W/m2K Fig.11.  Differences between numerical and analitical results in time period 0 to 60 s and a = 2000 W/m2K Sl.12.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 240 s za a = 2000 W/m2K Fig.12.  Differences between numerical and analitical results in time period 0 to 240 s and a = 2000 W/m2K Sl.13.  Odstupanja numeri kih od analiti kih rezultata u intervalu 0 do 60 s za a = 10 000 W/m2K Fig.13.  Differences between numerical and analitical results in time period 0 to 60 s and a = 10 000 W/m2K Sl.14.  Odstupanja numeri kih od analiti kih rezultata pri grijanju valjka u intervalu 0 do 3 s za a = 100 W/m2K Fig.14.  Differences between numerical and analitical results during cylinder heating in time period 0 to 3 s and a = 100 W/m2K Sl.15.  Bezdimenzijska temperatura za vremenski interval 0 do 3 s Fig.15.  Nondimensional temperature in time period 0 to 3 s Sl.16.  Bezdimenzijska temperatura za vremenski interval 0 do 30 min Fig.16.  Nondimensional temperature in time period 0 to 30 min * Naravno, poatujui kriterije stabilnosti pri svakom vremenskom koraku. 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