CM3110 Transport Processes and Unit Operations I
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1 Lecture 0 F. Morriso CM30 9/8/06 CM30 Trasport Processes ad Uit Operatios I Fluid Mechaics No Newtoia fluids A Itroductio The Weisseberg effect is whe a viscoelastic, o Newtoia fluid will climb a rotatig shaft. Photo by Carlos Arago Sabogal, U. Wiscosi, Madiso Newtoia Fluids Newto s Law of Viscosity Newtoia slope = viscosity v x
2 Lecture 0 F. Morriso CM30 9/8/06 No Newtoia Fluids shear thiig or pseudoplastic Bigham plastic slope = o Newtoia shear thickeig or dilatat v x 3 What is the defiitio of viscosity for No Newtoia Fluids? x v ( x ) v x x (NOTE o coordiate system: Viscosity defiitio is writte for shear flow i x directio ad gradiet i x directio) 4
3 Lecture 0 F. Morriso CM30 9/8/06 Typical polymeric behavior The chages i viscosity with shear rate are so large they must be plotted log log log Viscosity (Greek letter eta) o ero shear viscosity (Newtoia plateau) shear thiig v x shear rate (gamma dot) log 5 I additio, for may polymers there are shear iduced NORMAL (perpedicular) forces. force o surface i directio p ds S x H v ( x ) force o surface i directio ds S x 6 3
4 Lecture 0 F. Morriso CM30 9/8/06 Power Law Model (Ostwald dewaele Model) (does ot model ormal stresses) = m dx dx m or K = cosistecy idex (m = for Newtoia) = power law idex ( = for Newtoia) v x shear rate 7 What does the power law model predict for viscosity? m dx m dx dx dx O a log log plot, this would give a straight lie: log log m log dx Y B M X 8 4
5 Lecture 0 F. Morriso CM30 9/8/06 Power Law Fluid log m shear thickeig, No Newtoia viscosity v x Newtoia m, log shear thiig m, 9 Where do we use the power law expressio? v v t The oe with is for o Newtoia fluids 0 Faith A. Morriso, Michiga 5
6 Lecture 0 F. Morriso CM30 9/8/06 Where do we use the power law expressio? e.g., Poiseuille flow i a tube: r Newtoia r Lg Po L PL r o Newtoia, power law r m solve for directio = r directio = A r cross-sectio A: r EXAMPLE III: Pressureive flow of a Powerlaw fluid i a tube steady state icompressible well developed log tube v (r) L Calculate velocity ad stress profiles fluid R g 6
7 Lecture 0 F. Morriso CM30 9/8/06 Calculate the velocity field for Poiseuille flow of a power law fluid: r Lg P o PL m r L v r r m m r Solve for v (r) 3 Boudary Coditios:? 4 7
8 Lecture 0 F. Morriso CM30 9/8/06 Velocity field Poiseuille flow of a power law fluid: v r R Lg P o PL R r Lm R RP ( PL) v v r rd R R o ( ) R 3 ml Solutio to Poiseuille flow i a tube icompressible, power law fluid v /v,av = r / R 6 8
9 Lecture 0 F. Morriso CM30 9/8/06 Rheology (No-Newtoia Fluid Mechaics) Rheology affects: Processig (desig, costs, productio rates) Ed use (food texture, product pour, motor-oil fuctio) pic/akro%0extruder.jpg mpcm/aamp/examples.html Product quality (surface distortios, aisotropy, stregth, structure developmet) Pomar et al. JNNFM Rheology (No-Newtoia Fluid Mechaics) At Michiga Tech: CM4650 Polymer Rheology (Eve years sprig, MWF 3pm) CM4655 Polymer Rheology Lab (Fall, Fpm usually+lab) 8 9
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