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Quiz 6 – 2014.01.10 Quiz 7 – 2014.01.10 Question (15 mins) A small capillary with an inside diameter of 2.22 10-3 m and a length 0.317 m is being used to continuously measure the flow rate of a liquid having a density of 875 kg/m3 and = 1.13 10-3 Pa∙s. The pressure drop reading across the capillary during flow is 0.0655 m water (density 996 kg/m3). What is the flow rate in m3/s if the end-effect corrections are neglected? What is the Fanning friction factor for this capillary system? TIME IS UP!!! Frictional Losses for Non-Circular Conduits Instead of deriving new correlations for f, an approximation is developed for an equivalent diameter, Deq, which may be used to calculate NRe and f. D eq 4 R H 4 where S Pw RH = hydraulic radius S = cross-sectional area Pw = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid Equivalent Diameter (Deq) D eq 4 R H 4 S Pw Determine the equivalent diameter of the following conduit types: 1. Annular space with outside diameter Do and inside diameter Di 2. Rectangular duct with sides a and b 3. Open channels with liquid depth y and liquid width b Non-Newtonian Fluids Newtonian Fluids water air ethyl alcohol Non-Newtonian Fluids blood ketchup toothpaste Non-Newtonian Fluids grease polymer melt cake batter Non-Newtonian Fluids paint molten metal whipped cream Non-Newtonian Fluids • Foods – Emulsions (mayonnaise, ice cream) – Foams (ice cream, whipped cream) – Suspensions (mustard, chocolate) – Gels (cheese) • Biofluids • Electronic and Optical Materials – Suspension (blood) – Liquid Crystals (monitor displays) – Gel (mucin) – Melts (soldering paste) – Solutions (spittle) • Pharmaceuticals • Personal Care Products – Gels (creams, particle precursors) – Suspensions (nail polish, face scrubs) – Emulsions (creams) – Solutions/Gels (shampoos, – Aerosols (nasal sprays) conditioners) • Polymers – Foams (shaving cream) Non-Newtonian Fluids Why are these fluids non-Newtonian? Non-Newtonian behavior is frequently associated with complex internal structure: • The fluid may have large complex molecules (like a polymer), or • The fluid may be a heterogeneous solution (like a suspension)... Non-Newtonian Fluids Why are these fluids non-Newtonian? Fluid systems may be non-ideal in two ways: 1. The viscosity may depend on shear rate 2. The viscosity may depend on time Some (many) may have both Classification Time-Independent Fluids • The relation between shearing stress and rate is unique but non-linear • The viscosity of the fluid at a given temperature depends on the rate of shearing Classification Time-Independent Fluids Classification Time-Independent Fluids 1. Bingham plastics h depends on a critical/yield shear stress (t0) and then becomes constant Ex. sludge paint blood ketchup Classification Time-Independent Fluids 1. Bingham plastics Classification Time-Independent Fluids 2. Power law fluids Classification Time-Independent Fluids 2. Power law fluids Pseudoplastic fluids : h decreases as the shear rate increases (shear rate thinning) Ex. polymer melts paper pulp in water clay solutions molasses whipped cream Classification Time-Independent Fluids 2. Power law fluids Dilatant fluids : h decreases as the shear rate increases (shear rate thickening) Ex. Quicksand Starch suspension Wet sand Classification Time-Dependent Fluids Shear rate depends on the shearing time or on the previous shear rate history Classification Time-Dependent Fluids 1. Thixotropic fluids : shear stress decreases with time at constant shear rate; alternatively, the apparent viscosity decreases with time : the change is reversible; the fluid “rebuilds” itself once shearing is removed Ex. gelatin shortening cream Classification Time-Dependent Fluids 2. Rheopectic fluids : shear stress increases with time at constant shear rate; the apparent viscosity increases with time : the change is reversible Ex. highly concentrated starch solutions gravy beating and thickening of egg whites inks Classification Viscoelastic Fluids The shear stress is determined by the shear strain and the rate of shear strain • when applied stress is removed, the material does not instantly vanish since the internal structure of the material can sustain the stress for some time (relaxation time) • due to the internal stress, the fluid will deform on its own, even when external stresses are removed Shear Stress Behavior Non-Newtonian Fluids • For Newtonian fluids: t rz d z dr • For Non-Newtonian fluids: t rz h d z dr where h is the apparent viscosity and is not constant for non-Newtonian fluids. Shear Stress Behavior Modeling Power Law Fluids n t rz du z du z K K dr dr n1 du z dr where: K = flow consistency index n = flow behavior index eff Shear Stress Behavior Shear Stress Behavior Modelling Bingham Plastics t rz t 0 du z dr 0 (rigid) t rz t 0 t rz du z dr t0