Modelling rheological cone-plate test conditions

Transcription

1 ANNUAL TRANSACTIONS OF THE NORDIC RHEOLOGY SOCIETY, VOL. 16, 28 Modellig heological coe-plate test coditios Reida Bafod Schülle 1 ad Calos Salas-Bigas 2 1 Depatmet of Chemisty, Biotechology ad Food Sciece, Nowegia Uivesity of Life Scieces, N-1432 Ås, Noway. 2 Depatmet of Mathematical Scieces ad Techology, Nowegia Uivesity of Life Scieces, N-1432 Ås, Noway. ABSTRACT Cotol with flow coditios is of paamout impotace i elatio to heological tests. Special focus is made o the shea ate distibutio i the flow volume of coe/plate test geometies ad o the shea stess depedece o the cetal gap. Small vaiatios i the cetal gap due to tempeatue vaiatios may cause sigificat measuemet eos especially fo dilatat fluids. INTRODUCTION It is ofte desied to test o-newtoia fluids at coditios of costat shea ate sice the viscosity fo such fluids may be highly depedet o the shea ate. Coe/plate test geometies ae ofte chose fo these types of tests sice this test geomety appoaches the costat shea ate coditio i the whole flow domai. The flow coditios i coe/plate geometies have peviously bee descibed by Waltes 1. Complex flow coditios ca be peset if the coe agle is lage the 4º. Howeve, as will be show i this study, the shea ate is ot costat, ad the effects of this ca be sigificat. The cetal gap cleaace is ofte set at the begiig of a test whe the tempeatue equals oom tempeatue. If the test tempeatue is much highe o iceases duig the test it has bee show 2 that the coe shaft tempeatue also iceases ad this may cause the cetal gap distace to chage. The peset study assumes that the iitial gap is set to 5 μm, ad the effects of gap cleaace vaiatios ae quatified. BASIC THEORY Rheological behaviou fo fluids with o yield stess is ofte descibed by the well kow powe law equatio: τ = Kγ& (1) whee τ is the shea stess ad γ& is the shea ate. K is called the cosistecy coefficiet ad is the powe law idex 3. Diffeetiatio of the equatio with espect to shea ate yields a expessio fo the dyamic viscosity: dτ η = = K & γ d & γ 1 (2) Fo o-newtoia fluids, whee 1, it is clealy see that the viscosity is a fuctio of shea ate. MATERIALS AND METHODS The geomety studied i this wok is a well kow coe/plate system (Paa Physica MK22) whee the coe agle, α, is 1º ad the adius is 25 mm. The default cete distace fom the bottom plate duig measuemets is.5 mm. The zeo gap

2 settig is omally doe duig istumet iitiatio at oom tempeatue. It is ofte stated the viscosity measuemets i coe/plate systems ae pefomed with a costat shea ate eveywhee i the test volume. This is, howeve, ot quite tue. Cosideig the geomety i Fig. 1 we ca wite the followig equatio fo the agula plate velocity, u, ad fo the gap, δ : u = ω (3) δ = δ + taα (4) I = τ 2πd = 2π K & γ d (6) = 2πK ω d δ + taα Aalytical solutio to this itegal is ot kow, but it is possible to use fiite elemet modellig to calculate the flow domai. Howeve, if we let the gap distace ted to zeo, the ideal case with close to costat shea ate, we obtai the simplified appoximatio: I = 2πK = 2πK ω d taα ω d taα ω = πk taα 2 (7) Figue 1: Sketch (ot to scale) of coe plate test geomety. Radius is 25 mm, coe agle is 1 ad the cetal gap is 5 µm. The shea ate ca the be expessed by: u ω & γ = = (5) δ δ + taα The shea ate calculated fom this equatio fo diffeet values of δ is show i Fig. 2 with a otatioal speed of 1 ad/s. The toque o the shaft ca be expessed as the itegal of the shea stess multiplied by the adius ove the coe suface, thus: This equatio ca be solved eadily givig the equied combiatios of K ad fo simila values of toque o the divig shaft, I. The adius,, is 25 mm. The solutio of Eq. 7 is show i Fig. 3. K K 2 1 ω 1 2 = ta α (8) Eq. 8 has bee used to detemie the values of K to give the same toque fo a pseudo plastic ad a dilatat fluid; see Table 1. Table 1: Values fo compaative calculatios givig a toque of.56 Nm with zeo gap cleaace. K Newtoia case Pseudo plastic case Dilatat case.87 2.

3 The values fo the Newtoia case have bee abitaily chose. The shea stess distibutio o the coe suface ca be detemied by combiatio of Eq. 1 ad Eq. 5, thus: ω τ = K δ + taα Shea Rate (1/s) Coe/Plate Geomety, 1 coe agle Rotatioal speed = 1 ad/s Nomal opeatio (9) Cete cleaace (mm) Radius (mm) Figue 2: Calculated shea ate distibutio i a MK22 coe/plate geomety fo diffeet cetal gap cleaaces. FINITE ELEMENT MODELLING Fiite elemet modellig has peviously bee used to quatify the lack of tempeatue cotol i plate/plate heomete tests 2. The calculatios wee i this study pefomed with the compute pogam COMSOL Multiphysics 3.4. The basic geomety used i the calculatio was the MK22 set up with a.5 mm cetal gap cleaace. All the calculatios wee made with oe otatioal speed, ω = 1 ad/s. Both the coe ad the bottom plate bouday coditios wee specified as o-slip while the fluid to ai bouday coditio was specified as slip. The coe was specified as a movig wall with tagetial velocity, ω.the mai focus was ot effects of vaiatio i Reyolds umbe, Re, beig i the lamia egio with low values of Re. The umbe of degees of feedom solved fo was typically 5 ad the calculatio time was appoximately 1 miute. A figue of the axis-symmetic flow domai is show i Fig. 4. Toque (Nm) Pseudoplastic Case Newtoia Case K =.87 K = K = 5 Dilatat Case Figue 3: The solutio to Eq. 7 ad the cases descibed i fo a commo toque of.56 Nm. The aim of the fiite elemet modellig was to see if vaiatios i K ad affected the flow ad to ivestigate how the shea stess distibutio o the coe suface was affected by itoducig a gap cleaace fo diffeet types of fluids as specified i Table 1. RESULTS The esults show that thee is o lage effect of vaiatios i the cosistecy coefficiet ad the powe law idex o the velocity distibutio ad theefoe oly a distibutio plot fo the Newtoia case is show i Fig. 5 fo the egio ea the peiphey. Calculatios wee made fo a gap cleaace of.5 mm ad fo a gap cleaace equal to zeo. The esults fo zeo gap show that the shea stess was the same at all adii ad the same fo pseudoplastic, Newtoia ad dilatat flow (Fig. 6).

4 Figue 4: Nomal filled geomety (to scale) of axis-symmetic 2D fluid domai with cetelie o the left. The coe adius is 25 mm ad thee is a.5 mm cetal gap cleaace. Howeve, whe a gap cleaace was itoduced, the shea stess distibutios became those show i Fig. 6. The sesitivity to vaiatios i the cetal gap cleaace fo the dilatat test fluid is show i Fig.7. Figue 5: Newtoia iso-velocity lies ea the peiphey of the coe DISCUSSION The legth of the otatig shaft is appoximately 1 mm, ad the pat of this that ca be affected by chages i tempeatue is 5 mm, say. The coefficiet of liea themal expasio of stailess steel at 2 ºC is K -1. Calculatios show that the chages i the cetal gap due to a tempeatue chage of 5 K is 43 μm. The cetal gap cleaace must theefoe ot be set too small to pevet mechaical cotact, ad the stadad value is 5 μm. We see that quite small chages i tempeatue ca give appeciable chage i gap size. The mai focus of this aticle is to make a evaluatio of the shea ate distibutios foud i omal coe/plate heological measuemets. It is omally accepted o stated that the shea ate is costat i the measuig volume if a coe/plate geomety is chose. Calculatios show that this is ot the case (Fig. 2) ad the fiite elemet calculatios pefomed with COMSOL Multiphysics give the same esult. 21 Shea stess (Pa) Newtoia case Pseudoplastic case Dilatat case Pseudoplastic case Newtoia case Dilatat case Zeo gap coditio Radius (m) Figue 6: Wall shea stess distibutio o the coe as a fuctio of adius fo the thee basic cases whe a.5 mm gap is itoduced. Shea stess (Pa) μm 2 μm 1 μm 25 μm 5 μm Radius (m) Figue 7: Sesitivity to vaiatio i cetal gap cleaace fo the dilatat fluid of Table 1.

5 Reductio i wall shea stess (%) Dilatat case Newtoia case Pseudoplastic case Cetal gap (mm) Figue 8: Reductio i shea stess magitude at = 25 mm as a fuctio of cetal gap cleaace fo the thee base cases i Table 1. We see that the shea ate geeally iceases with adius fo all the cases whee the gap is lage tha zeo. The egio of maximum shea ate is located at the coe peiphey ad the shea ate is zeo at the cete. The esults fom this study show clealy that coe/plate test geometies do ot esue costat shea ate i the test volume. This would oly be tue if the cetal gap cleaace was zeo, but i pactise this gap is sigificat to make sue that themal effects o shaft elogatio do ot cause mechaical cotact betwee the coe ad the plate. Whe a cetal gap cleaace is itoduced, the shea stesses o the coe ae geeally educed. This, of couse, is a esult of the educed shea ate esultig fom iceasig the distace betwee the coe ad the bottom plate at all adii. Howeve, it is obseved lage diffeeces betwee the pseudoplastic, Newtoia ad dilatat fluids. The lagest eductio i shea stess is see with the dilatat fluid. The sesitivity to chages i the cetal gap is illustated i Fig. 7 fo a dilatat fluid beig most sesitive to chages i the cetal gap. The chose efeece coditio is fo zeo gap size sice the shea stess is the same at all adii i this special case. The effects would, howeve, be simila if the 5 μm gap was used as the efeece positio. Eve small chages i the gap cleaace cause sigificat chage i coe wall shea stess. Themal expasio effects may cause sigificat vaiatios i the cetal gap, so it seems vital that this is cotolled i ode to miimize measuemet eos. Fig. 8 shows the pecetage eductio i shea stess level as fuctio of cetal gap cleaace fom Eq. 9. The shea stesses closest to the peiphey give the lagest cotibutio to the toque, but it is see that the shea stesses ae also sigificatly affected i this egio whe the gap cleaace is vaied. CONCLUSIONS The coclusios fom this wok ca be summaized as follows: The shea ate is ot costat i omal coe/plate test geometies whe a cetal gap cleaace exists. It is zeo at the cete ad the maximum value is at the peiphey. The cetal gap cleaace ca be affected by a chage i tempeatue of the lowe coe shaft causig measuemet eos. The calculatios show that cotol with the cetal gap cleaace is vey impotat cotollig the shea stess distibutio o the coe. Dilatat fluids ae moe sesitive to vaiatios i the cetal gap cleaace tha Newtoia ad pseudoplastic fluids.

6 REFERENCES 1. Waltes, K., (1975) "Rheomety". Lodo: Chapma & Hall ISBN: Schülle, R.B., R. O, ad C. Salas- Bigas, (27), "Fiite elemet modellig of the fluid tempeatue i a plate plate otatioal heomete i oscillatoy tests", Aual Tasactios of the Nodic Rheology Society, 15: p Steffe, J.F., (1996) "Rheological methods i food pocess egieeig (Secod editio)". East Lacig: Feema Pess ISBN: