Quiz 6 – 2014.01.10 Quiz 7 – 2014.01.10

Quiz 6 – 2014.01.10Quiz 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!!!

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 4eq Hw

SD RP

Frictional Losses for Non-Circular Conduits

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 b3.Open channels with liquid depth y and liquid

width b

4 4eq Hw

SD RP

Equivalent Diameter (Deq)

Non-Newtonian Fluids

Newtonian Fluids

water

ethyl alcohol

air

Non-Newtonian Fluids

blood

toothpasteketchup

Non-Newtonian Fluids

grease

cake batterpolymer melt

Non-Newtonian Fluids

molten metal

whipped cream

paint

• Foods– Emulsions (mayonnaise, ice cream)– Foams (ice cream, whipped cream) – Suspensions (mustard, chocolate)– Gels (cheese)

• Biofluids– Suspension (blood)– Gel (mucin)– Solutions (spittle)

• Personal Care Products– Suspensions (nail polish, face scrubs)– Solutions/Gels (shampoos,

conditioners)– Foams (shaving cream)

Non-Newtonian Fluids

• Electronic and Optical Materials– Liquid Crystals (monitor displays)– Melts (soldering paste)

• Pharmaceuticals– Gels (creams, particle precursors)– Emulsions (creams)– Aerosols (nasal sprays)

• Polymers

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

Non-Newtonian Fluids

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

Classification

Time-Independent Fluids

Time-Independent Fluids 1. Bingham plastics

h depends on a critical/yield shear stress (t0) and then becomes constant

Ex. sludge paint bloodketchup

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 solutionsmolasseswhipped cream

Classification

Time-Independent Fluids 2. Power law fluids

Dilatant fluids : h decreases as the shear rate increases (shear rate thickening)

Ex. QuicksandStarch suspensionWet 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 removedEx. gelatin

shorteningcream

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 reversibleEx. highly concentrated starch solutions

gravybeating and thickening of egg whitesinks

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

Classification

Non-Newtonian Fluids

• For Newtonian fluids:

• For Non-Newtonian fluids:

where h is the apparent viscosity and is not constant for non-Newtonian fluids.

zrz

ddrt

Shear Stress Behavior

zrz

ddrt h

Modeling Power Law Fluids

where:K = flow consistency indexn = flow behavior index

Shear Stress Behavior

1n nz z z

rzdu du duK Kdr dr dr

t

eff

Shear Stress Behavior

Modelling Bingham Plastics

Shear Stress Behavior

0tt rz 0zdudr

(rigid)

0tt rz

0z

rzdudr

t t