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 
n1
  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