2 TIME IS UP!!! Question (15 mins) A small capillary with an inside diameter of 2.22 10-3 m and a length 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 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!!!
3 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.where RH = hydraulic radiusS = cross-sectional areaPw = wetted perimeter: sum of the length of the boundaries of the cross-section actually in contact with the fluid
4 Equivalent Diameter (Deq) Determine the equivalent diameter of the following conduit types:Annular space with outside diameter Do and inside diameter DiRectangular duct with sides a and bOpen channels with liquid depth y and liquid width b
9 Non-Newtonian Fluids paint molten metal whipped cream More examples of non-Newtonian fluids.paintmolten metalwhipped cream
10 Non-Newtonian Fluids Foods Emulsions (mayonnaise, ice cream) Foams (ice cream, whipped cream)Suspensions (mustard, chocolate)Gels (cheese)BiofluidsSuspension (blood)Gel (mucin)Solutions (spittle)Personal Care ProductsSuspensions (nail polish, face scrubs)Solutions/Gels (shampoos, conditioners)Foams (shaving cream)Electronic and Optical MaterialsLiquid Crystals (monitor displays)Melts (soldering paste)PharmaceuticalsGels (creams, particle precursors)Emulsions (creams)Aerosols (nasal sprays)PolymersBottomline: many of the fluids we encounter in industrial processes as well as in our everyday lives exhibit non-Newtonian behavior
11 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), orThe fluid may be a heterogeneous solution (like a suspension)...
12 Non-Newtonian Fluids Why are these fluids non-Newtonian? Fluid systems may be non-ideal in two ways:The viscosity may depend on shear rateThe viscosity may depend on timeSome (many) may have both
13 Classification Time-Independent Fluids The relation between shearing stress and rate is unique but non-linearThe viscosity of the fluid at a given temperature depends on the rate of shearing
17 ClassificationTime-Independent FluidsPower law fluids
18 Classification Time-Independent Fluids Power law fluids Pseudoplastic fluids : h decreases as the shear rate increases (shear rate thinning)Ex. polymer meltspaper pulp in waterclay solutionsmolasseswhipped cream
19 Classification Time-Independent Fluids Power law fluids Dilatant fluids : h decreases as the shear rate increases (shear rate thickening)Ex. QuicksandStarch suspensionWet sandDilatancy: at low shear conditions, particles are closely packed. The void spaces between particles is minimal and are filled with solvent (water). As shear stress increases, the total volume increases, increasing the volume of void space. However, the solvent doesn’t fill all of the void space, creating a “dryness” which increases the resistance to shearing stress.
20 Classification Time-Dependent Fluids Shear rate depends on the shearing time or on the previous shear rate history
21 Classification Time-Dependent Fluids 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 removedEx. gelatinshorteningcreamfor a thixotropic fluid, molecules become more and more disentangled over time, thus leading to a decrease in viscosity. If the shear force is removed, the molecules may reaggregate or become entangled again over time
22 Classification Time-Dependent Fluids Rheopectic fluids : shear stress increases with time at constant shear rate; the apparent viscosity increases with time: the change is reversibleEx. highly concentrated starch solutionsgravybeating and thickening of egg whitesinks
23 Classification Viscoelastic Fluids The shear stress is determined by the shear strain and the rate of shear strainwhen 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
24 Shear Stress Behavior Non-Newtonian Fluids For Newtonian fluids:For Non-Newtonian fluids:where h is the apparent viscosity and is not constant for non-Newtonian fluids.
25 Shear Stress Behavior Modeling Power Law Fluids where: K = flow consistency indexn = flow behavior index