Rajvanshi, S C

English

IACS Institutional Repository

Im h an  J Phy^ 47, l(i-2U (1973)Heat transfer from two parallel coaxial disks rotating at different speeds with a source on the axis o f rotationS (1 Hajvansiul)eparbncnl of Mathenmtic,s,'M. R. Ihiginverinp (lolUgc, Jaipur {Rmnved 7 March 1972, ravised l(i May 1972)Keat Ifttiitil'or Iruni two i)aralld cjoaxiul disks j ol-iiting iil dirfcicnt Bpeeds 111 tho pruHcnce of a soiucc on 1h(i axis of rolaiion lias l)oon investigated The solution has boon obtained in the form of double series expansion The effect of rotation on temperature piotile and Nusselt’s number has htjoii discuHSedlNTKOnU(JTJOJSiSoiirec Ilow between two paralkd disks rotating \Nitli tln^  Ram(^  velocity Jia,s been stiidic'd by lfreitnej‘ iV: Pohlhausen (19(12) and Kr(‘ith tS: ikaibe (19(i5,1960). Kridth & Viviand (1907) consideied Lh(> axisymmetiic How betwet'u two disks, rotating at different angular velocities witli a line souiec' at the centre Tlui equations of motion are solved by doubk*. series expansion about ii known solution at a large ]-adius. Tho results are valid tor small rotational Ttiylor numbers ot the disks and at a distance r ^  (Rc)^.In the present paper the nature of heat transfer has been investigated between two parallel coaxial disks rotating at- different speeds. A liiui source has been assumed to be present on the axis of rotatimi Tlu^  surfaces of the two disks arc taken to be at constant, tomiieratures. H’lie. Iluid is incomjiressible, so that th(! moinentum (ujuations are indcqieudent of tlu^  lu'at. transfer phenoinwia. Temperature distribution has been obtained as a double scries expansion The energy (’quation is simjililied by expanding tmnperature in powers of downstream coordinate. The resulting equations have been solved for small Prandtl number The (jffe.cl. of rotation on tempiwature distribution has been shown grapliically. Nussolt's numbers for both the disks have also been calculated.Statement oe ProblemConsider tlie How of a viscous fluid between tvAo parallel rotating disks with a source on the axis ol rotation. We shall Avork with cylindrical polar coordinates (r, 0, 2). Let tlu‘ middle ])oiut of the axis of rotation bc5 tlie origin. Hhe .surfaces , ' ■ »ve Hiiali \k are delinod by z — -\-a and ~ — — a, respectively. The upjier propagation. constant angular velocity wo and tlie lower one with oii. TheHeat transfer from two parallel etc. 17flow rate of tJie source is Q. The boundary conditions on the velocity profile are it — it? — 0, at s =  ± a ,V ~  rwi, at  ^ ™ —a, v =  fcD.^ , at g =  -\-a,  ^ ... (1)J 2nru dz — Q,- a  Jwhere u, v,w  are the compononls of velocity along r, 0,2 directions.Tlii^  axisymineti'io form of energy equation in oylinclT-ical polar coordi­nates is(IT , - d T  \ , (  d'^ T , \ OT , d'^ T \ , r. . .  (2)where Cp, h, p and T are specific heat, thermal conductivity, density and tem­perature, respectively. Viscous dissipation (^) of the fluid in axisymmotric case is given bywhere p is the coefficient of viscosity. The boundary conditions for temperature areandT =  at g =  —u, T ^  f.i_ at i  H: } ... (4)Appropriate dimensionless variables are defined by the folloAving relations‘tnv— U v  -  V vr =  ar, z =  az, u =  ----  , v —a aa^ Cp... (5)where .03in« (2), (8) mid (5) giveI _  J_ I Lrir j 1-.  ^ -hS. C- Rajvanshir>ry +-(r; j + 1  } - -^['d  ^ ~ r ) ^(t3)(Prm idtJ im m bor) - -  .KliHed hoim dury coii(Uti(jii,s uii tlie fluid and fejnperat-urc- an'u, — 10 -"- 0, at 2: — 1 I,(7)(«)V --  oc^ r, ab s =  — 1, v — a^r, at — -] 1,T  - ,s, at , T -  .s^  at =. 1-],a ,  - -  ( H « ) -  ,V " V 4:7rav' f , ,  imilNj =  TaSolution or Equationsing KicUli Viviaiid (1907) tlie forms of 11, v and w are taken as ther f  ,(.^)-|-(li.0i[ + ..........] ,/,(*)-) ............, .. (9),y ,(=) + ....]. J‘line deiKjbeH diffcu iLlial coefficient with rewjiect to s. /„(«) «-ud j/„(~) an*, nless fuiictioiih lu be detenniiiod from momentum equations Tlieee hav<; l)e(m delerniiiiod by Kreith I'c Viviand (1907) for small valuesHeat transfer from two parallel etc. 1*)of and Casal (1950) states that the expansions in tlie ease =  0, arc con- \rergont for |aol < 0  17. Equations (6) and (9) give(IT^  dz r  \ dr^  ^  dr ^  dz^  j ~ Mf/ -i I./ -i )+ m '~ x ^ + ( ^ ) g ' - j o ' i + m r - j \ ) +  .......... (i»>)\vherc terms upto (l/r^) have hccn retained on right hand side. Equation (10) readily suggests that the form ()f 7' sliould heT{r,z) =  r^T_^{z)^[-T,{z)^-^7\{z) +  ... ( 11)We suhstitute (11) into (10) and equate the coofiieients of equal poM ers o/ i on hoth sides This gives us a set of ordinary differential equations The firsl tuo dif­ferential et(uations arij2'"_, =  P(2f 3 - 2 / _ , ... (12) n = - 7 ’-2+l"(2(Rf‘) f ,7 ’.5 ~ 2 /,i7 ''„ -1 2 f_ /-2 (R « )! , '_ ,7 ',-2 (R o ) /"_ ,f ,)  ...(i;t) T’he modified boundary conditions for temperatuie are^ U (± l)  ^^ o ( - l ) - 'h ^  ^ o (-l-l)-^ 2 } iU)Ecpiation (J2) is a non-linear ordinary differential equation Its solution is oh lained by a perturbation method in powers of Vrandtl number P in ihe form(15)Wo substitute (16) in (12) and equate like powins of P on both sides This gives a set of linear ordinary differential equations. These equations together uith modified boundary eonditions giveT’-iS =  -gP(«i+aa)»(l-*»)+0(P»).Proceeding in a similar manner, the solution of equation (13) is(16)^0 =  iC(-«2—■Si)-+(«2+*’ i)H-I*r(ai+a2)“ ?5i+('’ a--«i)02+(Re)(aj*0,+aia,0,+a3*0,)l+P=[(Re)(ai4-a3)VoH(l(l"'') • ■ (17)'20 S. C. Rajvanshiwiidro<j>, -- - ^ , , a | “(5s’ -352«-2U «+ l75s* f3523-525s2-]0s+385)-f- a22(5z’ -f-352;‘'-2 1 » ‘i-175;:*+3''^=®"h525s2-19g-385)125200 OL^OL i^^ Z  ^-2 1 2 “ -I- 352» -192),=:= -  .^^^(3b_]23r,^-924_4023-2l2H52z+ll),0,, ^  l-122“+02M-402»-2l22-52r;H-lJ),0«-= 3 ^ -5 2 ;^Ill obtaining the results given in (15), (16) the values of/_](z), flr_i(J2), /i(z) and {/^ (z) Imve been taken from Kreith Viviand (1967).D iscussionsWe nilrodnce anothei- dimensionless temperature in the form■'‘■j (19)Equations (11), (10), (17) and (19) give T* =  L E jr,'2 (a+ l)i> (l-s* )+ ^  (l+2;)+PrE ,(a+l)i‘^ ,+ ^ H -hE2(Re)(a20;j+a04+9i5)l-f-P%(Re)(a~|-l)®06 (20)whore Ea(Eckert number) =.S'2—A'l Cj,(ra—r ,)and oL ^  -1-Heat transfer from two parallel etc. 21TI\o crmiciisionless tcmporaturo in ubeoinco of Bour<M‘ {T*) can bo obtained by talcing (Re) =  0 in equation (20) Hence avo Imvorp^__T* ^  PEa(Re)(aV3“l-a04 f  0r,)+l^%(Ro)(a4 J )¥«• (21)The variation of {T*~~T*)IE2,{Rg) against ;^bas been sliov'ii in liguro 1, for difforoni values of a. Ror numerical work we have taken P — 1 We noU‘ fT'om tlioFigure 1figure iliat T *—f *  is symmetrical, when the two disks rotate with equal angular velocity in the same or in opposite directions. Por a — 1, T* <  T* at every point of tlic region. For a -  —l ,T * > T *  near ))otb the disks, and at other points of the region 2'* <  f *  For a — 0, —0.0, T *>  5'*near the upper disk only and at other points of the region T* <  T* This iiidieat.es that if the two disks rotate with different angular velocities, then T*—T* is positive only near the disk ^^ 0noh rotates with greater angular velocity.The Nusselt’s number of the lower disk is given liy (Na)-^ ^  2a{Q *)-^_Jk{f,^T ,), . .  (22)whore 7T(f®- T'o^ )dTj  27Tf{g)~^_df,and (q)- =  —h(such that is the distance of a given point on the dislf from the axi9 of rotation,li'rom cqualimi.s (5), (IJ), (l(>) (17) and (22) mc have :-M fi ■■■ l-fi -*“'] + 5 (23)(N u);=+„— I + »„“)■! P [ J.Ra(« M)^22 S. C. R ajvanshi- p(Be)P=Ej(a+l)“. (24)Tli(‘ (dTo(;l< of diffcroTit spc-udp ol‘ rotation on tlic NnssoJt’s iiuinbor of tlio difik 2 — 1 has boon shown in hf»;nrcs 2 and lbFor immorical Avork wo Inivc assumed (Re) =  1000, P — l,Eo -  0.02, a, == O.l 1’h(‘ f>;raphs have been drawn for (rQ/(Re)^ ) -  1 and 10. We note from figures 2 and J* tliat Avith decrease in the value of the slope of the curve decrease's At — — 0.1, il Avill be almost parallel to the axis representing (r/(Re)^ i),Heat transfer from two parctUel etc. 2SACKNOWLTCDGJfiMENTSThe author is grateful to J)r. P, 1). Veniia for suggestions and enoom’agemeiil during this investigation. Thanks are also due to the referee of the paper for suggestions.RefebenoesnivitiiL'i' M & i’ohlhdufliM) K, 19()2 Report No ARL-Q'l 'iV .^ Acrospau Hoyewch Labotutorii,Wj'ight-l’aUerBuii At"B, Ohiu.Caaal IDoO Coinpies Hemius dc V Acaddinie dels HcitnceH 230, 178Kreith h\ k  Fenbo J . L 1965 Compters Rnidun dc I'Acadahir ilcs , '^u\ii(rt, 260 518-1IvTOiih F. & VivjaMLl }|, 1967 Trans, JoLunal of Applied Atevhmnivi^  34 511,Wi’iibo -I. i.1. & Kroilli F, 1966 Joiinnd dr Mcrannpir 5 201

Heat Transfer From Two Parallel Coaxial Disks Rotating at Different Speeds with a Source on the Axis of Rotation

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/2950/1/Heat%20transfer%20from%20two%20parallel_S%20C%20Rajvanshi.pdf

Heat Transfer From Two Parallel Coaxial Disks Rotating at Different Speeds with a Source on the Axis of Rotation

Abstract

Similar works

Full text

Available Versions

IACS Institutional Repository