! = Chemical Engineering Class 35 page 1. I. Non-Newtonian Fluids - Overview A. Simple Classifications of Fluids 1. Shear-Dependent Fluids.

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1 Chemical Egieerig 74 - Class 5 page I. No-Newtoia Fluids - Overview A. Simple Classificatios of Fluids. Shear-epedet Fluids a. Newtoia obeys Newtos Law of viscosity) µ!! = µ dv x dy dv /dy x dv /dy x c. Pseudoplastic molecules lie up oce flow begis)! dv /dy x " dv /dy x Ca be fit with the Power Law " = dvx dy d. ilatat particle suspesios collide whe flow begis)! dv /dy x " dv /dy x O with other models emider: Graduate texts write Newto s Law as τ = -µ dv x /dy e. More complex models i) Powell-Eyrig Model based o Eyrigs theories of the molecular structure of liquids) " = A dv x dy + B sih C dv x ) dy ii) Ellis Model purely empirical)! = A + B! dv x C dy iii) eier-phillipoff Model purely empirical) B A " = A + + " /C) dv x dy iv) For each of these, the three arbitrary costats A, B, ad C) are adjusted to fit the data.

2 Chemical Egieerig 74 - Class 5 page. Time-epedet Fluids heopectic! Thixotropic { { motio causes aggregatio or uravelig of molecules motio causes re-coilig or fragmetig of molecules time a. The chages occur mostly withi the first 6 secods b. escriptio of these fluids is extremely difficult.. Viscoelastic Fluids a. Exhibit elastic recovery from deformatios which occur durig flow b. Also exhibit plastic permaet) deformatio c. Modeled usig simple sprig-dashpot elemets ad combiatios of those elemets Maxwell Elemet Kelvi or Voigt Elemet B. Observatios. No-Newtoia behavior is usually see whe the fluid is composed of small carrier molecules ad larger suspeded molecules or particles. a. Bigham Fluids: Coal slurries, grai slurries, sewage sludge b. Pseudoplastic Fluids: Polymer melts, paper pulp suspesios, pigmet suspesios c. ilatat Fluids: Starch suspesios, mica suspesios, quicksad d. Thixotropic Fluids: Mayoaise, drillig mud, pait, ik e. heopectic Fluids: Gypsum suspesios i water, vaadium petoxide sols f. Viscoelastic Fluids: Polymeric liquids g. Other: Yield Pseudoplastic Fluid: Blood. No-Newtoia fluids exhibit lamiar ad turbulet flow regimes, with a trasitio betwee those regimes.. Equatios for Newtoia fluids cat be used, because there is o sigle value of viscosity for such fluids. e.g. Ca t calculate the eyolds umber usig the Newtoia defiitio

3 Chemical Egieerig 74 - Class 5 page 4. We ofte talk about a apparet viscosity η) where! = " dv x dy a. For example for a power-law fluid " = K dv x dy ) ) so, for such a fluid " = K dv x dy ) ) b. The apparet viscosity varies with shear rate, as plotted o page. II. escribig No-Newtoia Lamiar Flow i Horizotal Cylidrical Pipes A. escribig the Wall Shear Stress all fluids). As i Class 6, we ca begi with the Equatio of Motio to derive " rz = P P ) + gsi + r 4) L *. For horizotal pipes " rz = r P P L = r. At the wall " w = dp 4 which has a egative value, because it acts i the z directio) B. Velocity Profile ad Flowrate. For lamiar flow, we could itroduce the rheological equatio ito The Equatio of Motio or ito Equatio 5) above) ad itegrate to get v z r). Example: For a Power-Law fluid, Çegel s covetio for shear stress gives " rz = K dv z dr dp = K " dv z ) dr Combiig with Equatio 5), itegratig, ad applyig the o-slip coditio gives v z = "dp + + K + " r 7) The volumetric flow rate becomes Q = "dp ) 4K 8 +) 5) 6) 8)

4 Chemical Egieerig 74 - Class 5 page 4 Ad, fially see the Appedices to these otes), + +)! = ad " = ) +) 9). For other types of o-newtoia fluids, the math ofte becomes very complex. C. How ca we apply these cocepts to typical pipeflow data -dp/ vs. v avg )? A Geeral Approach, Usig Newtoia Fluids i Lamiar Flow as a Model. The velocity profile for a Newtoia fluid is " v z = v max r " = v avg r ). So the shear rate is. At the wall " = dv z dr = v avg r! w = 4v avg = 8v avg = 4v avgr ) 4. For a Newtoia fluid, a short-had way to write Newto s Law is! w = µ " w ) 5. But so plottig τ w vs. 8v avg " w = 4 dp gives a straight lie of slope = µ from Equatio 6)! w = 8v avg from Equatio ) So plottig dp 4 vs 8v avg for various diameters gives a straight lie of slope = µ 6. No-Newtoia fluids ca be plotted o that same plot gives a geeralized way of represetig all fluids i lamiar flow). ad Bigham ) 4 )!dp Pseudoplastic Newtoia ilatat 8v avg

5 Chemical Egieerig 74 - Class 5 page 5. How do we desig flow systems for a Power-Law fluid i lamiar flow?. From Equatio 8, Solvig for pressure drop: "dp V = "dp ) 4K 8 +) = V 8 +) ) 4K * " P " P L But the mechaical eergy equatio uses P/ρ, so P " P P = " P ) L L = " + V 8 +)., - * /. Fially, from the mechaical eergy equatio for a horizotal pipe, 4KL P " P = " w f. So for lamiar pipe flow of a power-law fluid, w f = V 8 +) " 4KL ) which ca be used with the mechaical eergy equatio for ay applicable pipig system.

6 Chemical Egieerig 74 - Class 5 page 6 Appedix β for a Power-Law Fluid i Lamiar Pipe Flow Q = v da = " dp ) ** see Equatio 8 of these otes) 4K + // where v da = // ) + + * + "dp K = " dp ) K + ) + * + " r + + r,.. r dr d -. + r dr ) + + " r ) r dr = + " + r + + r + ) r dr which, after some math, gives So v "" da = dp * K ) = + r " + = 4+ = 4+ = " 4 dp ) 4K * +) ) + r ) 4 + ) 4+ +) + +) + + r4+ ) 4 + ) ) see Equatio 7 of these otes) 4 + = 4+ " v 4 dp ) + da Fially, " = A [ v da] = 4K * + dp ) 6 + 4K * + ) +) ) = + +

7 Chemical Egieerig 74 - Class 5 page 7 Appedix α for a Power-Law Fluid i Lamiar Pipe Flow Q = v da = " dp ) ** see Equatio 8 of these otes) 4K + // where v da = // ) + + * + "dp K = " dp ) K + ) + + " r +,.. r dr d -. + r r dr ) + * see Equatio 7 of these otes) + + " r ) r dr = + " + r r + " r + ) r dr which, after some math, gives So v "" da = dp * K ) = + r " + r+ + = 5+ " 5+ + = 5+ " + + = 5+ = " 5 dp ) 4K * +) r " " ) +) 4 + ) 5 + ) 5+ +) ) +) 4 + ) 5 + ) ) ) 5 + ) 5 + " r5+ Fially, " = A v 5 dp ) + da [ v da] = 4 4K * + dp ) 9 + 4K * + ) +) 5 + ) ) = +) +) 5 + )

8 Chemical Egieerig 74 - Class 6 page I. More O No-Newtoia Lamiar Flow i Horizotal Cylidrical Pipes A. etermiig Values of the heological Parameters. We ca measure flow rate ad pressure drop through a pipelie... So how do we determie the values of rheological parameters?. We ca show appedix for these otes) that, for all o-newtoia fluids where γ w = dv z dr w γ w = ξ 4 + dlξ ) 4 dlτ w, ξ = 8v avg, ad τ w = 4 dp. Equatio, called the abiowitsch-mooey equatio, works for most timeidepedet o-newtoia fluids i lamiar flow i cylidrical pipes. a. From data of flow rate ad pressure drop, ξ ad τ w ca be calculated for each coditio data poit) b. A plot of l τ w vs. l ξ will produce d lτ w dlξ which ca be iserted ito the right-had bracket of Equatio. c. The wall shear rate ca the be determied for each data poit. d. Equatio suggests that d l τ w d l ξ provides importat iformatio. 4. The values of rheologic parameters ca also be foud from plots ivolvig τ w ad ξ. a. For Newtoia fluids, d l τ w d l ξ =. i) Why is this so? Hit: How does τ w deped o ξ? To fid out... How does τ w deped o γ? How does γ deped o ξ? γ w = dv z dr = 8v avg = ξ Now, you fiish the story.) ii) So, what kid of plot of the data would tell us µ?

9 Chemical Egieerig 74 - Class 6 page b. For Power-Law fluids, dlτ w dlξ = i) Why is this so? Hit: How does τ w deped o ξ? To fid out... How does τ w deped o γ? How does γ deped o ξ? τ w = K dv z = K γ w dr w τ γ w = w K = dp 4K V = dp π π 4K = γ 8 +) w 8 +) 8 + γ w = ) V 4V = 8 + = 8v avg π π 4 Now, you fiish the story.) = ξ 4 ii) So, what kid of plot of the data would tell us? iii) Would all data sets all pipe diameters, etc.) give the same value of? iv) Oce we kow the value of, ca we determie the value of K? How? v) What are the uits of K?

10 Chemical Egieerig 74 - Class 6 page c. For other No-Newtoias fluids, d lτ w may be a simple fuctio, ad we may be dlξ able to do the same kid of aalysis to determie values of rheologic parameters. B. Whe oes Flow Become No-Lamiar?. For Newtoia fluids: whe eyolds Number vρ/µ) > ~. But, for o-newtoia fluids, there is o µ.. For horizotal, cylidrical pipes, earragig 4. For Power-Law Fluids f = 4 ΔP ρ = f L v ΔP L 8 ρv = τ w ) 8 ρv ) a. By defiitio τ w = K γ w ) = K 8v avg + K 8v avg approximate derivatio) 4) 4 b. We ca substitute ito Equatio to obtai f K 8v 8 ρv 5) c. Fially, if we assume that for lamiar flow, f 64/e from Newtoia fluid) e K 8v 64 8 ρv ρv K8 6) d. It turs out that whe this Geeralized eyolds umber for Power-Law Fluids) exceeds ~, the flow of Power-Law fluids departs from lamiar behavior. dp 4 Lamiar egio 8v avg Turbulet egio l dp 4 Lamiar egio l 8v avg Turbulet egio e. Cautio: The locatio of the turbulet break-off poit i the above graph varies with pipe diameter. 5. For other kids of fluids, Equatios 4-6 ca be re-derived for the particular fluid i questio, resultig i a special formulatio of the eyolds umber for that fluid.

11 Chemical Egieerig 74 - Class 6 page 4 II. escribig No-Newtoia Turbulet Flow i Horizotal Cylidrical Pipes A. Empirical Frictio-Factor Charts Are Used The followig gives f Faig = f arcy /4 for Power-Law Fluids. From N. de Nevers, Fluid Mechaics, Addiso Wesley, 97.).5. Newtoia =.4 =.6 =.8 =. =.5 =.6 = eyolds umber as defied by Equatio 6 B. The solid lies i the above graph are for slurries ad some polymer solutios. From the data of odge ad Metzer, AIChE J. 5:89, 959.) C. The dotted lies are for some polymer solutios ad polymer melts, particularly those which show viscoelastic behavior like rubber cemet), which suppress turbulece From the data of Shaver ad Merrill, AIChE J. 5:8, 959.). With these fluids, much less turbulece is see, ad frictio is reduced.. Apparetly, the log molecules of these fluids absorb eergy ad damp out turbulet fluctuatios.. Tiy amouts as little as. wt) of such polymers i water reduce frictio Such pressure-loss-reductio additives are commoly used i the idustry e.g. the petroleum idustry).. The frictio factor from the above graph is used i a similar way as for Newtoia fluid w f = 4f L/ v /, used i the mechaical eergy equatio as always).

12 Chemical Egieerig 74 - Class 6 page 5 Appedix - erivatio of the abiowitsch-mooey Equatio. From the defiitio of the volumetric flow rate. But x dy = xy y dx V = π v r dθ dr = π v r dr A), ad if we let dy = r dr so y = r ), the V = π vr r= r= r= r dv r= r= = π r dv A) r=. We ca write this as V dv = π r dr = π r dr γ dr A) 4. From the last class, τ rz = r dp ad τ w = dp 4 so r = τ ad dr = dτ A4) τ w τ w 5. Substitutig ito Equatio A, V = π τ w γ τ dτ A5) 8τ w 6. Clearig the pre-itegral fractio o the right side τ w γ τ dτ = 8 V π τ w = 8v avg 4 τ w = 4 ξτ A6) w where we have defied ξ 8v avg b 7. Notig that d F x)dx = d [ Fb) Fa) ] a = F b) db F a) da, we take the derivatives of both sides of equatio A6) to obtai 4 ξτ w dτ w ) + 4 τ w dξ = γ w τ w dτ w 8. Fially, rearragig or where γ w = dv z dr w γ w = ξ dξ ξ τ w dτ w γ w = ξ 4 + dlξ 4 dlτ w, ξ = 8v avg, ad τ w = 4 dp A7) A8) A9)

Chemical Engineering 374

Chemical Engineering 374 Chemical Egieerig 374 Fluid Mechaics NoNewtoia Fluids Outlie 2 Types ad properties of o-newtoia Fluids Pipe flows for o-newtoia fluids Velocity profile / flow rate Pressure op Frictio factor Pump power