OUTCOME 2 TUTORIAL 2 VISCOSITY

Transcription

1 Uit 41: Fluid Mechaics Uit code: T/601/1445 QCF Level: 4 Credit value: 15 Uderstad the effects of viscosit i fluids OUTCOME TUTORIAL VISCOSITY Viscosit: shear stress; shear rate; damic viscosit; kiematic viscosit Viscosit measuremet: operatig priciples ad limitatios of viscosit measurig devices e.g. fallig sphere, capillar tube, rotatioal ad orifice viscometers Real fluids: Newtoia fluids; o-newtoia fluids icludig pseudoplastic, Bigham plastic, Casso plastic ad dilatet fluids CONTENTS 1. VISCOSITY Basic Boudar Laer Uits. VISCOSITY METERS 3. NON NEWTONIAN FLUIDS Mathematical Models Let's start b examiig the meaig of viscosit. D.J.DUNN FREESTUDY.CO.UK 1

2 1. VISCOSITY 1.1 BASIC THEORY Molecules of fluids exert forces of attractio o each other. I liquids this is strog eough to keep the mass together but ot strog eough to keep it rigid. I gases these forces are ver weak ad caot hold the mass together. Whe a fluid flows over a surface, the laer ext to the surface ma become attached to it (it wets the surface). The laers of fluid above the surface are movig so there must be shearig takig place betwee the laers of the fluid. Fig..1 Let us suppose that the fluid is flowig over a flat surface i lamiated laers from left to right as show i figure.1. is the distace above the solid surface (o slip surface) L is a arbitrar distace from a poit upstream. d is the thickess of each laer. dl is the legth of the laer. dx is the distace moved b each laer relative to the oe below i a correspodig time dt. u is the velocit of a laer. du is the icrease i velocit betwee two adjacet laers. Each laer moves a distace dx i time dt relative to the laer below it. The ratio dx/dt must be the chage i velocit betwee laers so du = dx/dt. Whe a material is deformed sidewas b a (shear) force actig i the same directio, a shear stress is produced betwee the laers ad a correspodig shear strai is produced. Shear strai is defied as follows. sidewasdeformatio height of the laer beig deformed dx d The rate of shear strai is defied as follows. shear strai time take dt dx dt d du d D.J.DUNN FREESTUDY.CO.UK

3 It is foud that fluids such as water, oil ad air, behave i such a maer that the shear stress betwee laers is directl proportioal to the rate of shear strai. costat x Fluids that obe this law are called NEWTONIAN FLUIDS. It is the costat i this formula that we kow as the damic viscosit of the fluid. DYNAMIC VISCOSITY = shear stress rate of shear d du FORCE BALANCE ad VELOCITY DISTRIBUTION A shear stress exists betwee each laer ad this icreases b d over each laer. The pressure differece betwee the dowstream ed ad the upstream ed is dp. The pressure chage is eeded to overcome the shear stress. The total force o a laer must be zero so balacig forces o oe laer (assumed 1 m wide) we get the followig. dp d d dl 0 d d dp dl It is ormall assumed that the pressure declies uiforml with distace dowstream so the dp pressure gradiet is assumed costat. The mius sig idicates that the pressure falls with dl distace. Itegratig betwee the o slip surface ( = 0) ad a height we get du d dp d d dl d d dp dl d u d...(.1) Itegratig twice to solve u we get the followig. dp du A dl d dp dl u A B A ad B are costats of itegratio that should be solved based o the kow coditios (boudar coditios). For the flat surface cosidered i figure.1 oe boudar coditio is that u = 0 whe = 0 (the o slip surface). Substitutio reveals the followig. 0 = B hece B = 0 At some height above the surface, the velocit will reach the maistream velocit u o. This gives us the secod boudar coditio u = u o whe =. Substitutig we fid the followig. D.J.DUNN FREESTUDY.CO.UK 3

4 dp u o A dl dp u o A hece dl dp dp u u dl dl u dp dl u o o Plottig u agaist gives figure.. BOUNDARY LAYER. The velocit grows from zero at the surface to a maximum at height. I theor, the value of is ifiit but i practice it is take as the height eeded to obtai 99% of the maistream velocit. This laer is called the boudar laer ad is the boudar laer thickess. It is a ver importat cocept ad is discussed more full i chapter 3. The iverse gradiet of the boudar laer is du/d ad this is the rate of shear strai. 1.. UNITS of VISCOSITY Fig DYNAMIC VISCOSITY µ The uits of damic viscosit µ are N s/m. It is ormal i the iteratioal sstem (SI) to give a ame to a compoud uit. The old metric uit was a de.s/cm ad this was called a POISE after Poiseuille. It follows that the SI uit is related to the Poise such that 10 Poise = 1 Ns/m This is ot a acceptable multiple. Sice, however, 1 CetiPoise (1cP) is N s/m the the cp is the accepted SI uit. 1cP = N s/m. is also commol used for damic viscosit. There are other was of expressig viscosit ad this is covered ext. D.J.DUNN FREESTUDY.CO.UK 4

5 1.. KINEMATIC VISCOSITY damic viscosit This is defied as follows. desit The basic uits are m/s. The old metric uit was the cm/s ad this was called the STOKE after the British scietist. It follows that 1 Stoke (St) = m/s ad this is ot a acceptable SI multiple. The cetistoke (cst),however, is m/s ad this is a acceptable multiple OTHER UNITS 1cSt = m/s = 1 mm/s Other uits of viscosit have come about because of the wa viscosit is measured. For example REDWOOD SECONDS comes from the ame of the Redwood viscometer. Other uits are Egler Degrees, SAE umbers ad so o. Coversio charts ad formulae are available to covert them ito useable egieerig or SI uits VISCOMETERS The measuremet of viscosit is a large ad complicated subject. The priciples rel o the resistace to flow or the resistace to motio through a fluid. Ma of these are covered i British Stadards 188. The followig is a brief descriptio of some tpes. U TUBE VISCOMETER The fluid is draw up ito a reservoir ad allowed to ru through a capillar tube to aother reservoir i the other limb of the U tube. The time take for the level to fall betwee the marks is coverted ito cst b multiplig the time b the viscometer costat. = ct The costat c should be accuratel obtaied b calibratig the viscometer agaist a master viscometer from a stadards laborator. Fig..3 REDWOOD VISCOMETER This works o the priciple of allowig the fluid to ru through a orifice of ver accurate size i a agate block. 50 ml of fluid are allowed to empt from the level idicator ito a measurig flask. The time take is the viscosit i Redwood secods. There are two sizes givig Redwood No.1 or No. secods. These uits are coverted ito egieerig uits with tables. Fig..4 D.J.DUNN FREESTUDY.CO.UK 5

6 FALLING SPHERE VISCOMETER This viscometer is covered i BS188 ad is based o measurig the time for a small sphere to fall i a viscous fluid from oe level to aother. The buoat weight of the sphere is balaced b the fluid resistace ad the sphere falls with a costat velocit. The theor is based o Stoke s Law ad is ol valid for ver slow velocities. The theor is covered later i the sectio o lamiar flow where it is show that the termial velocit (u) of the sphere is related to the damic viscosit () ad the desit of the fluid ad sphere ( f ad s) b the formula = F gd (s -f)/18u Fig..5 F is a correctio factor called the Faxe correctio factor, which takes ito accout a reductio i the velocit due to the effect of the fluid beig costraied to flow betwee the wall of the tube ad the sphere. ROTATIONAL TYPES There are ma tpes of viscometers, which use the priciple that it requires a torque to rotate or oscillate a disc or clider i a fluid. The torque is related to the viscosit. Moder istrumets cosist of a small electric motor, which spis a disc or clider i the fluid. The torsio of the coectig shaft is measured ad processed ito a digital readout of the viscosit i egieerig uits. You should ow fid out more details about viscometers b readig BS188, suitable textbooks or literature from oil compaies. SELF ASSESSMENT EXERCISE No. No Describe the priciple of operatio of the followig tpes of viscometers. a. Redwood Viscometers. b. British Stadard 188 glass U tube viscometer. c. British Stadard 188 Fallig Sphere Viscometer. d. A form of Rotatioal Viscometer D.J.DUNN FREESTUDY.CO.UK 6

7 . NON-NEWTONIAN FLUIDS Cosider figure.6. This shows the relatioship betwee shear stress ad rate of shear strai. Graph A shows a ideal fluid that has o viscosit ad hece has o shear stress at a poit. This is ofte used i theoretical models of fluid flow. Graph B shows a Newtoia Fluid. This is the tpe of fluid with which this book is mostl cocered, fluids such as water ad oil. A Newtoia fluid obes the rule = µ du/d = µ. The graph is hece a straight lie ad the gradiet is the viscosit. There is a rage of other liquid or semi-liquid materials that do ot obe this law ad produce strage flow characteristics. Such materials iclude various foodstuffs, paits, cemets ad so o. Ma of these are i fact solid particles suspeded i a liquid with various cocetratios. Graph C shows the relatioship for a Dilatet fluid. The gradiet ad hece viscosit icreases with ad such fluids are also called shear-thickeig. This pheomeo occurs with some solutios of sugar ad starches. Graph D shows the relatioship for a Pseudo-plastic. The gradiet ad hece viscosit reduces with ad the are called shear-thiig. Most foodstuffs are like this as well as cla ad liquid cemet.. Other fluids behave like a plastic ad require a miimum stress before it shears. This is plastic behaviour but ulike plastics, there ma be o elasticit prior to shearig. Graph E shows the relatioship for a Bigham plastic. This is the special case where the behaviour is the same as a Newtoia fluid except for the existece of the ield stress. Foodstuffs cotaiig high level of fats roximate to this model (butter, margarie, chocolate ad Maoaise). Graph F shows the relatioship for a plastic fluid that exhibits shear thickeig characteristics. Graph G shows the relatioship for a Casso fluid. This is a plastic fluid that exhibits shearthiig characteristics. This model was developed for fluids cotaiig rod like solids ad is ofte lied to molte chocolate ad blood. Fig..6 D.J.DUNN FREESTUDY.CO.UK 7

8 MATHEMATICAL MODELS The graphs that relate shear stress ad rate of shear strai are based o models or equatios. Most are mathematical equatios created to represet empirical data. Hirschel ad Bulkele developed the power law for o-newtoia equatios.this is as follows. K K is called the cosistec coefficiet ad is a power. I the case of a Newtoia fluid = 1 ad = 0 ad K = (the damic viscosit) For a Bigham plastic, = 1 ad K is also called the plastic viscosit p. The relatioship reduces to For a dilatet fluid, = 0 ad >1 For a pseudo-plastic, = 0 ad <1 p The model for both is K The Herchel-Bulkele model is as follows. K This ma be developed as follows. K K dividig b K K 1 sometimes writted as K K 1 1 For a Bigham plastic 1 so The ratio is called the aret viscosit For a Fluid with o ield shear value 0 K so p where K p 1 is called the plastic viscosit. The Casso fluid model is quite differet i form from the others ad is as follows. 1 1 K 1 Note that fluids with a shear ield stress will flow i a pipe as a plug. Withi a certai radius, the shear stress will be isufficiet to produce shearig so iside that radius the fluid flows as a solid plug. D.J.DUNN FREESTUDY.CO.UK 8

9 WORKED EXAMPLE No. 1 The Herchel-Bulkele model for a o-newtoia fluid is as follows. K. Derive a equatio for the miimum pressure required drop per metre legth i a straight horizotal pipe that will produce flow. Give that the pressure drop per metre legth i the pipe is 60 Pa/m ad the ield shear stress is 0. Pa, calculate the radius of the slug slidig through the middle. SOLUTION Fig..7 The pressure differece p actig o the cross sectioal area must produce sufficiet force to overcome the shear stress actig o the surface area of the clidrical slug. For the slug to move, the shear stress must be at least equal to the ield value. Balacig the forces gives the followig. p x r = x rl p/l = /r 60 = x 0./r r = 0.4/60 = m or 6.6 mm WORKED EXAMPLE No. A Bigham plastic flows i a pipe ad it is observed that the cetral plug is 30 mm diameter whe the pressure drop is 100 Pa/m. Calculate the ield shear stress. Give that at a larger radius the rate of shear strai is 0 s -1 ad the cosistec coefficiet is 0.6 Pa s, calculate the shear stress. SOLUTION For a Bigham plastic, the same theor as i the last example lies. p/l = /r 100 = /0.015 = 100 x 0.015/ = 0.75 Pa A mathematical model for a Bigham plastic is K = x 0 = 1.75 Pa D.J.DUNN FREESTUDY.CO.UK 9

10 WORKED EXAMPLE No. 3 Research has show that tomato ketchup has the followig viscous properties at 5 o C. Cosistec coefficiet K = 18.7 Pa s Power = 0.7 Shear ield stress = 3 Pa Calculate the aret viscosit whe the rate of shear is 1, 10, 100 ad 1000 s -1 ad coclude o the effect of the shear rate o the aret viscosit. SOLUTION This fluid should obe the Herchel-Bulkele equatio so 1 K Evaluatig at the various strai rates we get. = 1 = 18.8 = 10 = 3.48 = 100 = = 1000 = 0.1 The aret viscosit reduces as the shear rate icreases. SELF ASSESSMENT EXERCISE No. Fid examples of the followig o- Newtoia fluids b searchig the web. Pseudo Plastic Bigham s Plastic Casso Plastic Dilatet Fluid D.J.DUNN FREESTUDY.CO.UK 10