The Open Fuels & Enegy Siene Jounl, 9,, -6 Open Aess Stey Flow Cuve Pipe with Ciul Coss-Setion. Compison of Numeil n Expeimentl Results M.A. Petkis *, G.T. Khlios n S. Kplnis Tehnologil Eutionl Institute of Pts, Deptment of Mehnil Engeeg, M. Alexnou 1, Pt, Geee, Abstt: Numeil solution hs been obte of the equtions of motion of visous ompessible flui uve nnul onuit with iul oss-setion. These equtions e ppoximte by fite-iffeene equtions whih e of the seon-oe uy with espet to the gi sizes. The ompute esults e pesente fo the nge 96 D 8, whee D is the Den numbe of the flow n fo vious sizes of oe ii, the limitg ses of vey lge n vey smll oe beg lso stuie. It is shown tht the se of smll oe ius, the vition of the Den numbe ffets signifintly the flow popeties, sitution whih is not obseve when the oe ius is lge. INTRODUCTION Stey flow to uve pipes hs been stuie extensively ue to the pplitions engeeg. Annul flow is n impotnt fetue ouble-pipe het exhnges, hemil mixg n yg mhey. The theoetil explotion of the stey flow pipe of slight uvtue ws fist onsiee by Den [1, ]. It ws shown tht the effet of uvtue is to eue the flux, n tht the non-imensionl volume flux epens only on the so-lle Den numbe K = whee n R R( mx / ) e, espetively, the ius n the ius of uvtue of the pipe, mx is the mximum xil veloity stight pipe of the sme ius n pessue gient n is the kemti visosity of the flui. Den s solution is vli fo K 576. Den s wok ws extene by MConlogue n Sivstv [3] who opte the pmete 1 D = G 3 R μ ple of K, whee μ is the oeffiient of visosity n G is the onstnt pessue gient. The eltion between D n K is given by D = 4K 1. By mens of Fouie nlysis they eue the equtions of motion to oy iffeentil equtions whih e solve numeilly the nge 96 D 65.7. In tems of the Den numbe D efe by MConlogue n Sivstv [3], expeimentl eviene ite tht the uppe limit of the lm-flow nge ws ppoximtely D=5. Tuesell n Ale [4] bige ptilly the gp by obtg esults up to D=3578 though numeil poeue, wokg on helilly oile pipe of moete pith. An othe ppoh ws given by Geenspn [5]. A ombtion of entl, fow n *Aess oesponene to this utho t the Tehnologil Eutionl Institute of Pts, Deptment of Mehnil Engeeg, M. Alexnou 1, Pt, Geee; E-mil: petkis@teipt.g bkw iffeene sheme ws use to ppoximte the equtions of motion by iffeene equtions. The metho ws pplie fo viety of vlues of D between zeo n 5. The spg the il n the ngul ietion ws the sme fo ll D (=.1 n =/18, espetively). Among the most elible stuies, is tht Colls n Denis [6] who eple ll ptil eivtives the equtions of motion by entl-iffeene ppoximtions. They use thee pis of gi sizes (=.1 n =/18, =.5 n =/36, =.5 n =/7) fo 96 D 5. An epte iteion of uy ws opte s temtg onition to the solution poeue of the iffeene equtions whih wee of seon-oe uy ws opte s temtg onition to the solution poeue of the iffeene equtions t ll gi pots. In subsequent ppe Dennis [7] ppoximte the Nvie-Stokes equtions of motion by fite-iffeene equtions whih wee of seonoe uy with espet to the gi sizes. Annul flow uve pipes volvg het tnsfe hs been stuie by Khlios [8], n Choi n Pk [9], while possible effets of tey theteiztion on theosleosis hs been stuie by Khlios n Petkis [1]. In the pesent wok the fully evelope flow of visous ompessible flui uve nnul pipe is stuie numeilly. The poeue oves the nge 96 D 8 n efes to both smll o lge ius tio k (the numbe k is efe s the tio of the ius k of the pipe to the ius of the oe). The equtions of motion e ppoximte tems of fite iffeenes t the gi pots of the nnul om. In ptiul, the ppoximtion is bse on elxtion metho esibe by Allen n Shwell [11]. Five pis of gi sizes e use fo 96 D 8 n the egee of uy of the lulte esults is eteme by test whih temtes the itetive poeue. Results oneng the xil veloity hve been obte fo et ius tios k, lug the ses of k 1 n. In the fist se (k=1.1) the flow is eue to the 1876-973X/9 9 Benthm Open
Stey Flow Cuve Pipe with Ciul Coss-Setion The Open Fuels & Enegy Siene Jounl, 9, Volume 1 Couette type, while the seon se (k=1) ompison of the vlues of et flow pmetes of the pesent stuy with the pevious esults shows stisftoy geement. It is lso shown tht the xil oe ffets seiously the seony flow, the fluene beg stonge when k gets lge. In ition to the two well-known voties of the seony flow ppeg pl uve tube, two smlle voties e fome lose to the xil oe, thei oig lote t the w sie of the ben. An teestg fetue of the flow the nnul pipe is the flux. It is shown tht the flux is eue ompe to tht stight pipe when the Den numbe is lge. Flly, the esults oneng the xil flow e ompe with those obte by expeiments. It is shown tht the esults ue to the theoetil nlysis e vey goo geement the se tht the ius tio k ten to. EQUATION OF MOTION e onsie the stey ompessible flow of visous flui with loosely oile uve nnul pipe of iul oss-setion, n of ius k (k>1), beg the ius of the oxil oe. The pipe is oile ile of ius L b the xis Oz. In the pesent wok the tio k/l will be ssume vey smll. The geomety of the pipe suggests the use of tooil ootes *, n (Fig. 1). Let U, V n be the veloity omponents tht oespon to these ootes. In ition, let p enote pessue, the ensity n the kemti visosity. Fig. (1). Tooil oote system. The equtions of motion e U U * + V U V * * os L = p V * * * + V * 1 * U V * + V V * + UV * + s L = 1 p + V * * * + V * 1 * U U U * + V * = 1 p + L * + 1 * * + 1 * (1) () (3) whee p The eqution of ontuity is is the pessue gient t the ens of the pipe. U * + U * + 1 V * = (4) The flow is fully evelope, thus epenent of. Eq. (4) is stisfie by toug the stemfuntion F fo flow the oss-setion suh tht * U = F n V = F *. (5) By oss-iffeentition of Eqs. (1) n () n fte oppg the pessue tems we onlue the followg fom fo the equtions of motion: F * + F * 1 F+ * s + L * os = * 1 4 F 1 F * * + F * = G + 1 (6) (7) whee G = 1 P L n = 1 * + 1 * * + 1 * Intoug ition the non-imensionl vibles *= k, L = k ( ) 3 U = u k, 1/ V = v k, F = f, w (8) n substitutg the peeg equtions we obt f f f+ w w w s + os = f (9) 4 1 whee D = f w + f w 1/ L K G K 3 3 = D+ w (1) μ n = + 1 + 1. The veloity omponents must vnish t the sufe of the tube n of the oe, =1 n =1/k, espetively, n hene w = f = f = t =1 n =1/k. The flow is symmetil b the le =, fom whih it follows tht
The Open Fuels & Enegy Siene Jounl, 9, Volume Petkis et l. f(,-)=-f(,u), w(,-)=w(,), while f = w = on the le itself. Numeil Solution The uppe semiiul nnul egion 1/k 1, (Fig. ), is ivie to gi fome by the il les =jg n the semi-iles =1/k+ih, whee h, g enote the size of the gi the esg il ietion n the esg ngle, espetively. In oe to ppoximte t eh gi pot the xil veloity n the stemfuntion we follow metho popose by Petkis n Khlios [1]. 34 8 15 76 k = : DD 1 : = 35 :D= = 8 - -1. -.5.5 1. 5 375 k =1 : DD 1 : = 35 :D= = 7 5 15 Fig. (). Mesh pots - plne. The fl lgebi system ws solve numeilly usg n itetive poeue, the suessive ove elxtion metho. It is neessy to mention tht the itetive sheme ws epete until the opte iteion of uy fo w, mx 1 w () s (,) w ( s+1) (,) 5 15 ws stisfie. Then the vlues of w t ll gi pots wee kept fixe n the itetive poeue ws epete only fo the stemfuntion f until the peeteme limits to oespong egee of uy fo the stemfuntion ws hieve. One ll quntities h onvege to limits, the itetive sequene ws temte. Fo ius tio k= n k=5, solutions wee ie fo D=96, 5,, 8, eh one obte usg five iffeent pis of gi sizes. Fo k=1 we obte solutions fo D up to 7, g with the sme pis of gi sizes ( h=/8, g=/8, h=/16, g=/16, h=/3, g=/3, () h=/64, g=/64 n (e) h=/18, g=/18). Flly, the se k=1 we employe only the fist fou pis n we obte solutions with D vyg between 96 n 5. It ws elize tht esg the vlue of k, esulte the teneny of the whole poeue to be slowe n to onvege fte bigge numbe of itetions. RESULTS AND DISCUSSION In Fig. (3) we show the vition of the pofiles of the non-imensionl xil veloity w long the le of symmety =, fo vious vlues of the Den numbe D, two ses fo k. In the se of lge oe (k=) (Fig. 3) we see tht the uves e nely pboli, the vlue of the xil -1. -.5.5 1. 46 345 3 115 k =1 : DD = 5 : = 1 : D= : D= 35 : D = 5-1. -.5.5 1. 8 6 4 ().91.93.95.98 1. Fig. (3). Vition of the xil veloity pofiles with. k =1.1 : D =96 : D =5 : D =1 : D = D =35 DD =5 : :: =6 : D =7 : D =8
Stey Flow Cuve Pipe with Ciul Coss-Setion The Open Fuels & Enegy Siene Jounl, 9, Volume 3 veloity beg eue ompe with its oespong vlue fo the sme vlue of D n smlle oe ius. The egions of the flow e not istt n this is ue to the smll with of the nnul gp. In the limit k 1(Fig. 3), it is eviene tht the flow is simil to the Couette-type flow. hen the oe is smlle (k=1) (Fig. 3b), we see tht the flow onsists of entl visi egion tht the vlue of the xil veloity is vible, while the mximum of the vlue of the xil veloity beomes lge s D eses. In Fig. (3) we show the se fo n even smlle oe ius (k=1). Cuves of onstnt xil veloity e plotte Fig. (4) fo k= n fo vious vlues of the Den numbe. The veloity pofiles ite tht the isoveloity uves e iles onenti to the bounies. As D eses the isoveloity uves fom septe loop the e pt of the ben ue to the entifugl foes. In Fig. (5) we hve plotte the isoveloity uves fo smll oe ii (k=1). The fom of these uves ites the evelopment of entl visi egion s D eses. The xil isoveloity les beome pllel ove signifint pt of the flui whih pt the effets of visosity my be neglete. The seony flow ptten fo k= is shown Fig. (6). The flow ptten ites the effet of the Den numbe n of the size of the oe on the seony flow itself. The fomtion of the system of fou voties is the esult of the visous hte of the Stokes bouny lyes tht e fome long the two bounies. hen k=1 (Fig. 7) n fo smll vlues of Den numbe two lge voties e fome, exteng to the whole nge of the flui, n two smlle voties lose to the e sie of the ne bouny. The mximum xil veloity ous on the le of symmety =,. In Fig. (8) the vition of the position of w with D is shown fo vious vlues of the ius tio k. In the sme figue we hve e expeimentl esults oneng the flow uve pipe, whee enotes the non-imensionl istne of the position of w mx fom the ente of the oss-setion of the tube. It is obvious tht the geement between expeimentl n numeil esults is vey goo. In Fig. (9) we show the vition of w mx with D. It n be seen tht s k eses ou numeil esults tent to oie with expeimentl ones. CONCLUSIONS In the pesent wok we ppoximte the Nvie-Stokes equtions of motion by mens of elxtion metho. Fo lge oe ii k=, ou metho is vli fo vlues of D up to 8, while fo smlle ius tios the vlue of D fo whih onvegene is possible ws less thn 7 fo k=1 n 5 fo k=1. hen the ius of the oe is smll (k=) the flow ptten esembles to the Couette-type flow, while the - k=.d=35 : 95.11 b: 73.946 : 34.634 - - - - b k=.d=5 : 48.676 b: 13.7.17 : 133.5 () Fig. (4). Isoveloity uves fo k=. b - - - - k=.d=7 :115.368 b:138.937 :19.85 :199.947 e:6.419 b - - - - k=.d=8 :18.335 b:155.84 :1.775 :1.933 e:9.44 b - - - e e
4 The Open Fuels & Enegy Siene Jounl, 9, Volume Petkis et l. k=1.d=5 :3.46 b:46.9 :49.371 :5.639 e:55.331 b e k=.d=7 :.75 b:.13 :-.49 :-9.486x1 - b - - - - - k=1.d= :63.794 b:8.53 :9.674 :9.674 - - - k=.d=5 :.556 b:. :-.184 :-.481 - b - - - - - - - - k=1.d=35 :1.367 b:18.15 :145.3 :156.469 b k=.d=7 :-.31 b:-.494 :-.833 :-.963 e:-.493 f: 1.793 g:.18 g f e - - - - - - - - k=1.d=5 :151.74 b:188.848 :7.53 :16.8 b () k=.d=8 :-.397 b:-1.3 :-1. :-.963 e:.57 f:.71 () b f e - - - - - - - - Fig. (5). Isoveloity uves fo k=1. Fig. (6). Seony flow ptten fo k=.
Stey Flow Cuve Pipe with Ciul Coss-Setion The Open Fuels & Enegy Siene Jounl, 9, Volume 5 vition of the Den numbe oes not ffet seiously the fom of the xil n seony flow. As the ius of the oe eses (k=1) the flow epens on the Den numbe. hen D eses the both the bouny lyes on the pipe n on the oe beome thne. Thee is n visi egion of the flow ue to the smll ius of the oe n the well known system of fou voties is ppe. Flly, numeil n expeimentl esults e vey goo geement, the se of k 1. 1..5.5 Colls n Dennis (6) Akiym n Chen (13) Ale (14) * k = 1 15 5 375 5 Fig. (8). Vition of the position of w mx with the Den numbe. D k= 1.D=1 : 4.4 b:.785 :.436 :.5 b 5 mx 375 5 Colls n Dennis [6] Akiym n Cheng [13] Ale [14] =1 k =1 k = 15 k =1.D=35 :.143 b: 5.681 :.847 : 1.475 () Fig. (7). Seony flow ptten fo k=1. b Fig. (9). Vition of w mx with the Den numbe. REFERENCES 4 6 8 D [1] Den,.R. Note on the motion of flui uve pipes. Philos. Mg., 197,, 8-3. [] Den,.R. The stemle motion of flui uve pipe. Philos. Mg., 198, 3, 673-693. [3] MConlogue, D.J.; Sivstv. Motion of flui uve tube. R.S. Po. Royl So. Lonon, Se. A, 1968, 37, 37-53. [4] Tuesell, L.C.; Ale, R.J. Numeil tetment of fully evelope lm flow helilly oile tubes. A I Ch E J, 197, 16, 11-115. [5] Geenspn, A.D. Seony flow uve tube. J. Flui Meh., 1973, 57, 167-176. [6] Colls,.M.; Dennis, S.C.R. The stey motion of visous flui uve tube. Qut. J. Meh. Appl. Mth, 1975, 8, 133-156. [7] Dennis, S.C.R. Clultion of the stey flow though uve tube usg new fite-iffeene metho. Qut. J. Flui Meh., 198, 99, 499-467.
6 The Open Fuels & Enegy Siene Jounl, 9, Volume Petkis et l. [8] Khlios, G.T. Mixe onvetion flow hete uve pipe with oe. Phys. Fluis, 199, A, 164-175. [9] Choi,H.K; Pk, S.O. A numeil stuy of mixe onvtive het tnsfe uve nnulus. Po. n JSME-KSME, 1, 85-88, 199. [1] Khlios, G.T.; Petkis, M.A. Fully evelope stey flow slightly uve nnul pipe. At Mehni, 1991, 88, 1-1. [11] Allen, D.N.; Shwell, R.V. Relxtion methos pplie to eteme the motion. Qt. J. Meh. Appl. Mth., 1955, 8, 19-145. [1] Petkis, M.A.; Khlios, G.T. Flui flow behvio uve nnul onuit. Int. J. Non Le Meh., 1999, 34, 13-5. [13] Akiym, M; Cheng, K.C. Bouny votiity metho fo lm foe onvetion het tnsfe uve pipes. Int. J. Het Mss Tnsfe, 1971, 14, 1659-1675. [14] Ale, M. Stomung gekummten Rohen. Z. Angew. Mth. Meh., 1934, 14, 57-75. Reeive: Deembe 15, 8 Revise: Febuy 6, 9 Aepte: Febuy 11, 9 Petkis et l.; Liensee Benthm Open. This is n open ess tile liense une the tems of the Cetive Commons Attibution Non-Commeil Liense (http://etiveommons.og/lienses/byn/3./) whih pemits unestite, non-ommeil use, istibution n epoution ny meium, povie the wok is popely ite.