Theoretical and Experimental Study of the Rheological Behaviour of Non-Newtonian Fluids Using Falling Cylinder Rheometer

Transcription

1 Thereical ad Experieal Sudy f he helgical Behaviur f N-Newia Fluids Usig Fallig Cylider heeer S.Gh. Eead,. Bagheri ad S.Zeiali. Heris Cheical Egieerig Depare Isfaha Uiversiy f Techlgy 8454 Isfaha, Ira ABSTACT The prese sudy aeped slve fallig cylider gverig equai fr pwer law del fluids ad als csruc a vel fallig cylider rheeer (F.C..) easure he rhelgical prperies f pwer law del -Newia fluids. Differe fallig cyliders wih predesiged desiies were used deerie he appare viscsiy f Plyviylalchl sluis i waer wih varius ccerais. The resuls idicae ha all P.v.a. sluis bey he pwer law del which pwer law idex as well as csisecy idex chage liearly wih ccerai. Icreasig ccerai f he slui decreases pwer law idex while ehaces csisecy idex ad appare viscsiy. INTODUCTION The purpse f he prese aricle is eply a apprpriae slui fr fallig cylider rheeer a pwer law fluid, ad he desig ad csruci he F.C.. easure he rhelgical prperies f Plyviylalchl wih differe ccerais The sudy f flw pas a rigid bjec is f grea pracical iprace ad because i is he fudai f a brach f fluid echaics aely paricle echaics has bee a subjec f ay hereical, uerical ad experieal ivesigais. A exesive lieraure review ca be fud i Happel ad Breer [], which deals wih paricle is i a Newia fluid a lw eylds uber. The paricle echaics is he hereical basis f a experieal apprach kw as he fallig bjec visceery which csiss i allwig a bjec fall freely i he fluid. Afer iiial accelerai, whe he exeral drag he surface ad buyacy bece equal he dwward frce due he graviy, he bjec aais a csa erial velciy. Fallig balls because f heir fas ad siple perais have bee widely used deerie he viscsiy f he Newia ad -Newia fluids [-5]. The fallig cylider visceer is geig re favured ha a fallig ball visceer because i is easier csruc cylider wih predesiged desiies ad is re aeable aalysis ha he sphere ball fr -Newia fluids. Desig ad evaluai f he fallig cylider rheeers fr -Newia fluids baied by Lhre [6], Lhre e al. [7], Ashare e al. [8] ad Eichsad ad Swif [9]. Their aheaical ivesigais were cduced by assuig ed effecs, i essece, a cylider f ifiie legh. The ipra i he desig f he syse f fallig visceers accu fr ed effecs f Newia fluids was reaed by several ivesigars [-3]. THEOY Whe a cylider is drpped i a viscus fluid, afer iiial accelerai, he cylider aais a csa velciy kw as erial velciy. There are hree frces acig he cylider, frce f graviy (acig dwward), buyacy frce (acig upward), ad drag frce (acig upward). I is assued ha he rai f he legh he radius f he cylider (L/) is sufficiely grea which ed effecs ca be egleced. Whe erial velciy is reached he New s secd law reduces he requiree ha he frces he cylider su algebraically er. Eead e al. [4] baied he fallig equais fr a seady,

2 axisyeric flw f a icpressible, pwer law del -Newia fluids wih egligible cvecive ers. Wih respec figure (7) fllwig relai resuls: r Defiig = Pwer law del relai V τ r = ( ) r () V τ r = ( ) r () Siplificai f equai f i i Z-direci wih respec prble assupi we have : P dp ( = + ρg) L d ( rτ r ) r P = r L (3) Iegrai f (3) wih he budary cdii ( τ r = a r = ) gives: τ r = P( ) / L (4) Cbiai f (), () ad (4) resul: V P / / = ( ) ( ) r L (5) P / / ) ( ) L V = ( r Diesiless velciy ( U ) defie as a fllw: (6) V U = / [ ( P / L) ] (7) The (5), ad (6) ay be wrie as: U / = ( ) (8) U = ( / ) (9) Fr frce balace ver cylider we have: π P + π gρ l L + π L( τ w ) = π gρ s L () Cbiai f (4), ad () cylider wall ( = ) yields: P = ( ) g( ρ s ρl ) L Subsiui () i (4) fr ( = ) gives: τ w = ( ) ( ) g ( ρ s ρl ) / () L A ass balace relaed displaced fluid by vig cylider fluid flwig bewee cylider ad ube gap bece as a fllw: π V d = π V (3) Wih respec diesiless velciy defiii: ()

3 U = U d The iegrai f abve relai by pars resuls: du ( ) d = (5) d Cbiai f (8), (9), ad (5) bece as a fllw: d / / ( ) ( ) d = (6) If values f, ad specified he value f aybe calculaed fr (6). The iegrai f (8), ad (9) wih fllwig budary cdii gives: U = a = (7) U U = a = (8) U + U = / ( ) d (9) / U = ( ) d () Values f (9), ad () us be equalled a = he: U = / / ( ) d ( ) d () Tha shw U is a fuci f, k. Cbiai f (7), ad (), a cylider wall where U g s V = U c ( ρ ρ ) l c = U ad V V = yields: The (), is a fial relai ad relaed cylider fallig velciy cylider ad fluid desiy differece via pwer law del. (4) () Deeriai f rhelgical paraeers: The equai () als ca be rearraged i he fllwig fr [4]: lv = gc l( ρ s ρ l ) + l U c (3) The csisecy idex () ad pwer law idex () deeried as a fllw: - Fr a pwer law fluid a pl f l V vs. l( ρs ρl ) is a sraigh lie wih slpe f pwer law idex (). - fr (6) value f () by usig () ad () deeried he fr ( ) value f U specified. 3-by usig () wih ay pair f V ad ( ρ s ρ l ) value f () deeried.

4 EPEIMENTAL The fallig cylider rheeer csiss f a lg cylidrical caier filled wih -Newia fluid, i which he differe cyliders wih predesiged desiies are allwed fall uder he ifluece f graviy (Fig. ). The axes f he fallig cylider is parallel he graviy vecr. The cyliders were ade f cylidrical glass ubes ad he eds f he cyliders were ruded be heispherical isure srealie flw ad reduce he erace ad ed effecs. The diesis f he cyliders used i he prese ivesigai are give i Table. Aluiu pwders ca be isered i he hllw cylider icrease is effecive desiy. Usig f hese pwders i he cylider als sabilies he i f he cylider which ay deviae fr he ceerlie. Als a guide rig assebly a he p f he ceerlie was used faciliae he releasig f he cylider alg he ceerlie f he cylidrical filled caier. The eperaure f he fluid was crlled usig he circulaig fluid i he jacke f he caier. The erial velciy f he cylider was deeried by easurig he ie ierval f ravelig f he cylider bewee w easuree lies iscribed he cylidrical caier. A he b f he caier a exi valve was used he brig u he cylider ad drai he es fluids. The ipra paraeers are he fallig ie ad disace, cylider ad fluid desiies, fallig sabiliy, cylider eccericiy ad eperaure variais. The specific care was ake ge accurae resuls. A he prese ivesigai he rhelgical prperies f waer slui f plyviylalchl (lecular weigh equal 7) a differe ccerais ( 7 weigh perces) were easured ( Table ). Due varius rage f viscsiies he easurees were carried u a uber f fallig cyliders f differe geeries ad desiies. The cyliders were seleced s as have slwly fallig i rder have sall errrs i he al fallig ie as well as sall ieria er i he gverig i equais. Therefre fr lw viscsiy liquids a lw ass fallig cylider was used esure he erial velciy is esablished befre he cylider reaches he b f he caier. Fr each ccerai f Plyviylalchl a uber f repeaed easurees were ade ascerai he uceraiy i he experieal resuls. The accuracy f he apparaus perai was cfired wih viscsiy easuree f disilled waer. Figure : Scheaic f fallig cylider rheeer ESULTS AND DISCUSSION

5 Table shws he resuls f a uber f cylider rus usig waer sluis f plyviyl alchl a ccerais f 7 weigh perces. The rhelgical prperies f he sluis were calculaed usig he experieal daa ad equais 5 baied by Eead e al. [4]. Figures ad 3 prese he experieal daa f shear sress vs. shear rae ad illusrae hw pwer law -Newia del fi he experieal resuls f prese sudy. Figure 3 shws he resuls f figure lgarihic crdiaes ad he liear relai bewee he shear sress ad shear rae ephasies ha he rhelgical behavir f he sluis bey he pwer law del. Figure 4 preses he appare viscsiy vs. shear rae calculaed fr he daa f figure 3. Fr figure 4, icreasig shear rae akes reduci i he quaiies f appare viscsiy f he sluis which apprach he csa value. Figure 5 shws he pwer law idex vs. ccerai f he fluid. As see i figure 5 a sraigh lie (equai 6) is fi daa ad as ccerai f he fluid icreases he pwer law idex decreases. The fllwig equai is applicable he experieal daa f figure 5. =.49x (6) Table : Diesis f he Cyliders ad helgical Prperies f P.V.A. Sluis Weigh perce f P.V.A. = c T c Fluid desiy gr 3 c Csisecy idex () N.sec Pwer law idex () Pwer law equai L c τ = 4.94 γ& τ = 9.7 γ& τ =.56 γ& τ =.7 γ& τ =.66 γ& τ = 3.33 γ&

6 Shear Sress weigh perce % 3% 4% 5% 6% 7% Shear ae Figure : Shear sress vs. shear rae fr varius weigh perce f P.V.A. sluis Figure 6 idicaes he effec f fluid ccerai he csisecy idex f he Plyviylalchl sluis. Based he resuls f figure 6 he csisecy idex icreases by icreasig he ccerai f he fluid. The fllwig sraigh lie relai bewee ad x was deeried. =.53x.56 (7) Shear Sress.. weigh perce.. % 3% 4% 5% 6% 7%... Shear ae Figure 3 : Shear sress vs. shear rae fr varius weigh perce f P.V.A. sluis

7 Appare Viscsiy weigh perce % 3% 4% 5% 6% 7%..... Shear ae Figure 4 : Appare viscsiy vs. shear rae fr varius P.V.A. ccerais CONCLUDING EMAS The fallig cylider apparaus used i his ivesigai csised f hllw cylidrical glass ubes wih heispherical eds ad isered aluiu pwders ake differe effecive desiies. The erial velciy f each cylider was deeried i w csecuive secis f he fall disace esure aaie f csa velciy. Usig he experieal daa baied by he csruced fallig cylider rheeer (F.C..) ad eplyig he aalyical resuls [4], he rhelgical prperies f he pwer law del - Newia fluids ca be calculaed. As a specific case he experieal resuls f he viscsiy f disilled waer shws very gd accuracy f he f.c.r. apparaus. The rhelgical prperies f he sluis f Plyviylalchl wih differe ccerais deeried by he csruced rheeer ad idicaes he excelle fi f he resuls wih pwer law del. The liear crrelais baied bewee he pwer law idex as well as csisecy idex ad ccerai f he Plyviylalchl sluis..85 Pwer Law Idex ( ) = -.49x = Weigh Perce ( x ) Figure 5 : Pwer law idex vs. P.V.A. weigh perce

8 Csisecy Idex ( ) =.53x -.56 = Weigh Perce ( x ) Figure 6 : Csisecy idex vs. P.V.A. weigh perce Fallig Cylider Wall Tube ceer V Tube Wall V τ r c Fig. 7- Velciy prfile ad shear sress disribui fr pwer law del fluid i fallig cylider rheeer

9 NOMENCLATUE g Graviaial Accelerai, sec c, Diesiless L Legh f cylider, N.sec Csisecy idex, Pwer law idex, Diesiless c Cylider radius, r, Diesiless U V Caier radius, Diesiless erial velciy Terial velciy, sec x Weigh perce f Plyviylalchl, Diesiless Diesiless disace kg ρ c Cylider desiy, 3 kg ρ l Liquid desiy, 3 N p Pressure drp, EFEENCES. Happel, J. & Breer, H., (986), Lw eylds Nuber Hydrdyaics. Marius Nijhff, Bs. Pwell,. L., Mdy, L. A., Sker, G. G., Millike, W. J. & Graha, A. L., (989), Develpe f a Fallig Ball heeer wih Applicais Opaque Syses: Measuree f he helgy f Suspesis f ds, Jural f helgy, 33, Bucher, T. H. & Irvie, T.F. Jr., (99), Use f he Fallig ball Visceer Obai Flw Curves fr Ielasic N-Newia Fluids, Jural f N-Newia Fluid Mechaics, 36, Pha-Thie, N., Zheg,., & Taer,. I., (99), Flw Alg he Ceerlie Behid a Sphere i Uifr Srea, Jural f N-Newia Fluid Mechaics, 4, Ji, H., Pha-Thie, N. & Taer,. I., (99), A Fiie Elee Aalysis f he Flw Pas a Sphere i a Cylidrical Tube, Cpuaial Mechaics, 8, Lhre, J., (96), A Experieal Verified Thereical Sudy f he Fallig Cylider Visceer, Ph.D. Thesis, Uiversiy f asas, Lawrece, U.S.A. 7. Lhre, J., Swif, G. W. & uraa, F., (96), A Experieal Verified Thereical Sudy f Fallig Cylider Visceer, A.I.Ch.E., 6, Ashare, E., Bird,. B. & Lescarbura, J. A., (965), Fallig Cylider Visceer fr N- Newia Fluids, A.I.Ch.E.,, Eichsad, F. J. & Swif, G. W., (966), Thereical Aalysis f he Fallig Cylider Visceer fr Pwer Law ad Bigha Plasic Fluids, A.I.Ch.E.,, Che, M.C.S. & Swif, G.W., (97), Aalysis f Erace ad Exi Effecs i a Fallig Cylider Visceer, A.I.Ch.E., 8, 46-49

10 . Cha,..Y. & Jacks, D.A., (985), A Auaed Fallig Cylider High Pressure Laser- Dppler Visceer, Jural f Physics: Sci. Isu., 8, Gui, F., (99), Thereical ad Experieal Sudy f he Precisi Fallig Tube Visceer, Ph.D. Thesis, Sae Uiversiy f NewYrk a Sey Brk, U.S.A. 3. Fuli, G. & Irvie, T. F., (994), Thereical ad Experieal Sudy f Fallig Cylider Visceer, I. J. Hea Mass Trasfer, 47, Eead, S.Gh., Bagheri,. & Zeiali Haris, S., (), Derivig Applied Equais fr Fallig Cylider heeer, 6h Fluid Dyaics Cf., Tehra, Ira,, 33-43