ANALYTICAL STUDY OF NATURAL CONVECTION IN A HEATED VERTICAL CHANNEL

Transcription

1 EFT 9 th Interntionl Conference on et Trnsfer, Fluid Mechnics nd Thermodynmics 6 8 July Mlt NYTIC STUDY OF NTUR CONVECTION IN ETED VERTIC CNNE Duovsky V. uthor for corresondence Dertment of Mechnicl Engineering, Ben-Gurion University of Negev, Beer-Shev, Isrel, E-mil: vdimd@gu.c.il BSTRCT In the resent work, nturl convection of ir in heted verticl chnnels is studied. On the sis of the concetion of induced-forced nlogy, simle nd icient nlyticl method of het trnsfer nd flow clcultions is develoed. Generl het-trnsfer correltions in the form = (Re,Pr) re otined for the ir flow. The velocity (Reynolds numer) is otined y solving momentum eution for induced convection tking into ccount energy lnce for this cse. It is shown tht the develoed method is simle nd cler when lied to the rolem under considertion. In this clcultion, the initilly known het inut (het trnsfer rte or wll temerture) nd hydrulic rmeters re used. The develoed method is lied to the rolem nturl convection of ir in symmetriclly or symmetriclly heted verticl chnnel, which hs llowed us to mke comrison with the literry dt. The nlogy nd difference etween the nturl convection in heted chnnel nd heted verticl lte in n infinite volume is demonstrted. criterion of trnsition etween the two is resented, nd it is shown tht this trnsition is smooth. INTRODUCTION In the resent work, nturl induced convection of ir is studied. We use the term "induced" for the nturl convection in chnnels with hysiclly ovious direction of vented flow, s distinct from the free nturl convection or, shortly, free convection in n infinite volume. concetion of induced-forced nlogy is offered in the resent er. In generl, we otin induced het-trnsfer correltions in the form = (Re,Pr) (Re-sed nlysis). The induced flow rmeters must e defined, ecuse they re initilly unknown, in contrst to the cse of forced convection. For this nlysis, we hve to consider the fct tht n induced flow is initited y the heting surfce itself, nd not y n externl source. This fct determines the resence of n dditionl connection etween therml chrcteristics nd ir flow. Velocity (Reynolds numer) clcultion is relized y solving momentum eution for induced convection tking into ccount the energy lnce for this cse. The constructive roerties of the whole th of the flow re ccounted for in hydrulic clcultion similrly to those of forced convection. Therefore, the vlues of hydrulic coicients ly n imortnt role in clcultions, esecilly since the friction coicient for the induced convection cn differ from the cse of forced convection. It should e ointed out tht the clcultions of hydrulic chrcteristics of the system re not the suject of our study. In our clcultions we use dt of Zvirin [] for the friction coicients for the induced convection flow, nd recommendtions of Idelchik [] regrding locl resistnce fctors. Next, universl het-trnsfer correltions (Re,Pr) re utilized. This rocedure is siclly the sme s tht for the forced convection. Note tht het-trnsfer correltions (Re) for induced convection do not coincide with het-trnsfer correltions for forced convection, too. Bsic ides of the resent roch were first resented y Duovsky et l. [3]. The roch hs een thoroughly develoed nd verified y Duovsky []. Severl investigtions were devoted to induced convection in two-dimensionl comletely heted verticl chnnel. It is the simlest, lthough rther imortnt cse, which deserved greter ttention. The results of gret numer of exeriments comrehensively reflecting the rolem under study were resented y ung et l. [], Br-Cohen nd Rohsenow [6], Mnc nd Nrdini [7], ulett nd Mnc [8]. The generliztion of exerimentl dt, s mentioned y ulett nd Mnc [8], ws relized in - R coordintes, where is the chnnel scing nd gβ ρ is kµ R 7

2 NOMENCTURE [-] iner scle fctor c [m ] Cross-section re s [m ] eting surfce re [m] Chnnel scing C [-] Constnt c [J/kg K] Secific het of ir d [m] ydrulic dimeter of the chnnel f [-] Friction coicient Fr [-] Froude numer g [m/s ] Grvittionl ccelertion Gr [-] Grshof numer Gr [-] Modified Grshof numer h [W/m K] verge het trnsfer coicient sed on temerture difference T [m] eight k [W/mK] Therml conductivity [m] ength of the chnnel n [-] mer of heted sides in two-dimensionl chnnel [-] sselt numer sed on het trnsfer coicient h P [P] Pressure P [P] Pressure difference Pr [-] Prndtl numer [W] et inut " [W/m ] et flux R [-] Ryleigh numer R [-] Modified Ryleigh numer (Elens numer) Re [-] Reynolds numer T [K] Temerture t [s] Time T [K] Temerture difference T v-t υ [m/s] verge ir velocity z [m] Verticl coordinte Secil chrcters β [K - ] Volumetric therml exnsion coicient Θ [K] Reltive ir temerture µ [kg/m s] Dynmic viscosity ρ [kg/m 3 ] ir density ξ [-] Overll locl resistnce fctor ξ [-] Overll resistnce fctor Suscrits ir v verge Chnnel scing D ydrulic dimeter Effective f Friction eted length out Outlet eted mient or overll modified Ryleigh numer (Elens numer). The generl form of the correltions ws. = C R () where the constnt vlue C is eul to.6 [],.73 [6],.68 [7],.77 [8]. Develoing eution (), one cn esily otin. h gβρ = C Pr ; k k where the chnnel scing is µ oviously cncelled. It is esy to check tht these correltions re reduced to well-known correltion of free convection for verticl lte in n infinite volume (Rithy nd ollnds [9]):. =.6 R () with slight increse in the constnt coicient. The mgnitude gβ ρ R is determined s R. Thus, these kµ correltions of eution () do not tke into ccount differences etween the chnnel nd infinite volume. Note tht this rolem is sent in the enchmrk er involving the induced nturl convection in verticl chnnel y Elens []. We demonstrte the simlicity nd clerness of the develoed method liction to the rolem under study nd good greement of the clculted results with exerimentl dt resented in [] [8] nd []. t the sme time, we show the role of chnnel scing. Joint reserch of the licility re of our method nd the trnsition from the induced convection in chnnel to the free convection in n infinite volume hs shown tht the limit of licility of our method nd the limit etween the induced nd free convection re the sme limit. This limit corresonds to the vlue. = R, nd the mgnitude R is 3 gβρ T determined s R. µ GENER PPROC Momentum eution nd energy lnce We hve to find method of clculting the ir flow velocity (Reynolds numer) s function of known vlues with the ccount for energy lnce. ttention is drwn to the fct tht in nturl circultion loos we del with the induced convection nlogously to oen-ended vented chnnels. Zvirin [] clculted fluid velocity y solving momentum eution for nturl loo nd used these flow rmeters in the het-trnsfer nlysis. In our generl roch, we nlyze one-dimensionl momentum eution for induced convection, too, ut with oundry conditions of the mient ir. locl one-dimensionl momentum eution ws written for n induced flow y Zvirin []. This eution for verticl flow is υ P. fρυ ρ = ρg (3) t z d where P is ressure, υ is flow velocity, g is grvity ccelertion, z is the verticl coordinte, nd d is the hydrulic dimeter of the flow chnnel. Sher stress t the wll is exressed y /fρ υ, where f is the friction coicient. It is ssumed tht the Boussines roximtion is vlid, i.e., the density ρ is treted s constnt excet for the uoyncy force term. The density hs reference vlue ρ. ence, the verge velocity υ is uniform over the chnnel height. stedy-stte solution of E. (3) for n oen-ended verticl chnnel, with due regrd for locl hydrulic resistnce is 73

3 P = g ρ dz+ ξ ρυ + f () d where P is the ressure difference etween the outlet nd inlet, nd ξ is the overll locl resistnce fctor. The ressure difference etween the outlet nd inlet of the chnnel is the weight of the environmentl ir column of the height er unit re (gρ ). Denote the verge ir density in the chnnel s v = dz ρ ρ () with the corresonding verge ir temerture T v. Solving E. () in the frmework of the Boussines ρ ρ = βρ yields: roximtion ( ) v T v T β ξ υ g Θ = + f (6) d where reltive outlet ir temerture Θ = T out T, nd we define the ective height of the chnnel s Tv T = Θ detiled nlysis crried out y Duovsky [8] indictes tht the ective height of the chnnel in generl cse is close to the difference etween the heights of the outlet of the chnnel nd the heted lte centre. υ (8) c Θ = cρ s where c is the cross-section re of the heted chnnel nd s is the re of the heting surfce with n verge flux = for the overll het inut. Thus, s (7) s Θ = (9) c υρ Eution (6) yields: 3 gβ s υ = () ρ c ξ + f c d We use the Reynolds numer nd modified Grshof numer sed on the length of the heted lte (for the cse without υρ ditic extension, is eul to ): Re =, Gr Re gβρ kµ =, nd E. () yields: 3 k s = Gr () ξ µ c µ where we define the overll resistnce fctor ξ s sum of locl nd frictionl resistnce fctors ξ =ξ + f. Finlly, s Gr Re = () ξ Pr The right-hnd side of eution () generlly contins dimensionless hydrulic resistnce term (/ξ ), dimensionless uoyncy term ( / ), nd dimensionless het inut term ( s / c )(Gr /Pr). For the cse without ditic extension, s mentioned ove, the uoyncy term is eul to.. et-trnsfer correltions for the induced convection of ir In this section we resent het-trnsfer correltions s ower deendence of the numer on the Re numer. This nlogy with forced convection nlysis is the centrl tenet of our concet. The existence of -Re correltion with geometric rmeters defined solely for heted lte (heted chnnel) indeendently of other geometricl rmeters (such s, for instnce, rmeters of the chnnel inlet-outlet, height nd scing of ditic extension nd so on) demonstrtes the dvntge of our concet. It should e reclled tht revious well-known correltions sed on nturl convection, such s -Gr correltions, deend on some dditionl geometricl rmeters nd re licle to rticulr chnnel design only. In our roch, generl hydrulic chrcteristic of the whole chnnel is used for the induced flow clcultion only, nd secific het-trnsfer exeriments re not needed for every individul cse. numericl nlysis of more thn cses of induced convection ws erformed for vrious structurl versions of the chnnel with different ditic extensions nd heted ltes, including its rrngement within comlicted enclosures. The ects of chimney height, het losses nd rdition het trnsfer, together with geometric nd therml scling, were nlyzed nd sustntited y these results. The method nd verifiction of the numericl clcultions re descried in the revious work (Duovsky []). Finlly, the following generl -Re correltions for n induced convection of ir (Figure ) were otined:.. =.Re Pr lminr flow (3).7. =.Re Pr turulent flow These generl correltions contin the sselt nd Reynolds numers sed on the length of the heting surfce. Note tht for free convection in n infinite volume, the size used in dimensionless grous is the length of the heting surfce, too. In our revious work [] it is shown tht the lminrturulent limit criterion is Re D (nlogously to forced ie nd tue flow), rther thn Re. It is not surrising, ecuse Re D distinguishes lminr nd turulent flows in chnnel in cse of hydrulic clcultion. Thus, it is not surrising tht in the generl grh in Figure for < Re < 3, the flow cn e oth lminr (for Re D <3), nd turulent (for Re D > 3). 3 d 7

4 cordingly, the imortnce of oth Re D nd Re nd their different menings re demonstrted. This dulism (rtil corresondence to either forced or free convection) is the most secific ulity of nturlly induced flows in chnnels. Figure. Generl grh of het-trnsfer correltion Simlifiction of the eutions nd reference cse usge Eutions () - (3) generlly llow otining the solution of the rolem. By considertion of ech serte cse, the eutions ecome simler. For the lminr irflow we ssume tht the locl resistnce is unimortnt, nd the friction coicient f is inversely relted C f to the Reynolds numer ( f = ). Re. d s = R () C f where the modified Ryleigh numer is sed on the heted lte length, R = Gr Pr. For the two-dimensionl cse without ditic extension, with regrd for hydrulic dt of Zvirin [], eution () yields: =.89 n R () where is the chnnel scing nd n is the numer of heted sides; n = for symmetricl (comlete) heting nd n = for symmetric heting. Modifictions of the reltion () for the cse of het inut considered s the known wll temerture insted of the known het trnsfer rte, s well s for the cse of dimensionless grous sed on chnnel scing insted of lte length, my e very useful: D =.867n R (6) =.89n R (7).867 = n R (8) 3 gβρ T where R. Elens numer R µ nd the Ryleigh numer R were defined ove. Note tht E. (7) demonstrtes tht correct rocessing of exerimentl results in - R coordintes should e crried out with resect to R vlue. Now we consider reference cse tht my e otined from the study of exerimentl or numericl clcultions. For this set of conditions, it is ossile to erform direct reclcultion of nother cse with different het inut or/nd geometriclly scled system. The generl forms of Es. (), (7) nd Es. (6), (8) in terms of hysicl rmeters re, resectively: h ~ ; h ~ where is liner scle fctor. Turulent flow ( T ) (9). s = R () ξ For the sme two-dimensionl cse without ditic extension, eution () yields: n =. R () ξ If we ssume tht the locl resistnce is unimortnt (similrly to the lminr cse), eution () yields n =.3 R () f This result is indeendent of the chnnel scing if the friction coicient f is constnt. If the overll resistnce fctor ξ is indeendent of the liner scle fctor, the generl form of the reltions for turulent flows ( ) ; h ~ ( ) 3 h ~ T, (3) is lso indeendent of the liner scle fctor. Vlidtion of the results The eutions () - (3) generlly llow receiving correct solution of the rolem. Now we comre our nlyticl results with exerimentl dt of the two-dimensionl cse without ditic extension []-[8]. 7

5 cording to ulett nd Mnc [8], the generliztion of exerimentl dt of [], [6], [7] nd [8] ws relized in - R coordintes. The comrison of these results with clcultions using eutions ()-(3) in the sme coordintes is demonstrted in Figure. Figure. Comrison of our nlyticl redictions to exerimentl dt of fully heted two-dimensionl chnnel In the enchmrk er [], Elens otins the correltion for symmetricl heting cse in the following form: 3 ex = R () R comrison with our results in R coordintes is shown in Figure 3..7 Figure 3. Comrison of sselt-ryleigh reltionshis of Elens with our clcultions frictionl resistnce without locl hydrulic one. Nevertheless, the comrison demonstrtes good ulittive nd untittive corresondence. IMIT OF TE PPROC ND TRNSITION FROM INDUCED CONVECTION TO FREE CONVECTION s shown in our revious work (Duovsky []), criterion for the limits of the method in the generl cse is the vlue of the secific Froude numer υ Fr = > () gβ d Θ Now we nlyze two-dimensionl verticl chnnel involving heted lte nd n oosite lte. The oosite lte locted t the distnce my e ditic (n = ) or with the sme heting (n = ). Oviously, with incresing distnce, the induced convection in the chnnel finlly sses into free convection in n infinite volume. The following rolems re studied: to define the limit etween the induced convection in chnnel nd free convection in n infinite volume; to concretize the licility limit of the nlyticl roch (eution ()) for this cse; to find out whether these limits re the sme or not. mericl clcultions were crried out in the following rnge: =..6m, =..m, T = o C, " = W/m. The results re rocessed with resect to the vlue /n/ R. nd resented in Figures nd. In Figure the vlue /(Re Pr). is shown in Y-direction. It is shown tht this vlue is ctully constnt close to., s it should e ccording to the eution (3) for the induced convection under the conditions defined y the reltion (). In Figure the vlue / free is shown in Y-direction, where free is determined y the well-known reltion for heted lte in n infinite volume []:. =.9R (6) free One cn esily see tht for /n/ R. >, the het trnsfer corresonds to free convection in n infinite volume. The rnge 3 n. < R (7) corresonds to the induced convection in chnnel oth in Figure nd in Figure. In this rnge, the numericl results nd the suggested nlyticl clcultions for the induced convection re in very good greement. For /n/ R. < 3, the clcultion of the het trnsfer coicient using temerture difference T s for free convection, is hysiclly incorrect. Summing u, it is ossile to ssert tht the licility rnge of the suggested nlyticl method extends u to the trnsition of the induced convection into free convection. Such trnsition occurs t /n/ R. =. Note tht in the strict sense we comre two different cses the generl cse exmined y Elens with the cse of only 76

6 Figure. Rtio /(Re Pr). s function of /n/ R. Figure. Rtio /(.9R. ) for different cses SUMMRY concetion of induced-forced nlogy is suggested in the resent er. Generl het-trnsfer correltions in the form = (Re,Pr) (Re-sed nlysis) were otined for the induced ir flow. reliminry clcultion of the velocity (Reynolds numer) is relized y solving momentum eution for the induced convection tking into ccount the energy lnce for this cse. In this clcultion, initilly known rmeters (the Grshof numer) re used, s well s known hydrulic resistnce fctors. Simlified reltions for the lminr nd turulent cses re otined. The existence of reference cse is considered. For tyicl flow, simle formuls for direct reclcultion of nother cse with chnging het inut or/nd geometriclly scled system re demonstrted. The develoed method is lied to the rolem of induced convection in symmetriclly or symmetriclly heted verticl chnnel tht llowed us to mke comrison with the literry dt. Results of the clcultion re in good greement with the exerimentl dt resented in []-[8] nd []. The oint is tht recently suggested het trnsfer correltions []-[8] do not differ, siclly, from the known correltions for free convection in n infinite volume. The one nd only difference consists in the use of dimensionless grous sed on chnnel scing, wheres for free convection the dimensionless grous re sed on the lte height. The difference disers fter the trnsformtion of correltions to the sme sselt nd Ryleigh numers. This ect hs simle hysicl exlntion. tully, it is shown ove tht the limit etween the induced convection nd free convection in n infinite volume occurs t /n/ R. =. For /n/ R. >, the het trnsfer coincides with the free convection in n infinite volume. cordingly, t /n/ R. <, the trnsition to our reltions for the induced convection occurs very smoothly. Thus, it is ossile to recommend dt generliztion of the induced convection in chnnels with resect to the sselt Ryleigh numers sed on the height of heted lte nd in comrison with the well-known free convection in n infinite volume. It is noteworthy tht the results of the resented theoreticl nlysis cn find wider liction nd re not limited y the rolem under study. In rticulr, they cn e lied to the cses of three-dimensionl heted chnnels, chnnels with ditic extensions, nd others. REFERENCES [] Zvirin Y., review of nturl circultion loos in ressurized wter rectors nd other systems, cler Engineering nd Design, Vol. 36, 98,. 3-. [] Idelchik I.E., ndook of hydrulic resistnce, 3 rd edition, CRC Press, Boc Rton, Fl, 99. [3] Duovsky V., Ziskind G., nd etn R., Ventiltion of enclosures y induced convection of ir in turulent flow regimes, th Interntionl et Trnsfer Conference, Grenole, Frnce,. [] Duovsky V Induced convection in enclosures with oenings. Ph.D. Thesis, Ben Gurion University, Beer-Shev, Isrel, 8. [] ung M., Fletcher.S., nd Serns V., Develoing lminr free convection etween verticl flt ltes with symmetric heting, Interntionl Journl of et nd Mss Trnsfer, Vol., 97, [6] Br-Cohen., nd Rohsenow W.M., Thermlly otimum scing of verticl, nturl convection cooled, rllel ltes, Journl of et Trnsfer, Vol. 6, 98,. 6-3 [7] Mnc O., nd Nrdini S., Comosite correltions for ir nturl convection in tilted chnnels, et Trnsfer Engineering, Vol., 999,. 6-7 [8] ulett., nd Mnc O., et nd fluid flow resulting from the chimney ect in symmetriclly heted verticl chnnel with ditic extensions, Int. J. of Therml Sciences, Vol.,,. - [9] Rithy G.D., nd ollnds K.G.T., Nturl Convection, in W.M. Rohsenow et l., Eds. ndook of et Trnsfer Fundmentls, 3 rd ed., McGrw-ill, New York, 999 [] Elens W., et Dissition of Prllel Pltes y Free Convection, Physic, Vol. 9, 9,. -8. [] olmn J.P., et Trnsfer, th ed. McGrw-ill, New York, 77