rheology lab experiment

8/12/2019 rheology lab experiment

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CBE 443 - Rheology

Capillary Viscometry

Coaxial Cylinder Rheometry



Basic Units

SI cgs

Mass: m kg g

Velocity: v m/s cm/s

Acceleration: a m/s2 cm/s2

Force: F = m∙a N = kg∙m/s2 dyne = g∙cm/s2

Energy: E = F ∙x J = N∙m

dyne∙cm



Extensions for Rheology

SI cgs

Shear stress:

= F/A

Pa = N/m2 =

kg/(m∙s2)

dyne/cm2

Shear rate: 1/s 1/s

Viscosity: Pa∙s = kg/(m∙s) Poise,

P = g/(cm∙s)

dy

dv x

d

d a ,



Newtonian Fluid Behavior

constant

d

d Consider fluid betweenstationary plate andplate moving at vx

To increase vx, have to

increase xy

But shear rateincreases, too

Plot of shear stress

versus shear rate is aline with slope =

y

x

vx

y

v

y

v

dy

dv x x x

0

0



Non-Newtonian Fluid Behavior

constant

d

d a

Consider fluid betweenstationary plate and platemoving at vx

To increase vx, have toincrease xy

But shear rate increases,too, but varies with y

Plot of shear stress versusshear rate is not linear.

Apparent viscosity varies withshear rate.

y

x

vx

)( y f dy

dv x



Examples of Non-Newtonian

Behavior

Shear stress,

Shear rate,

Dilatant or

Shear thickening

Shear stress,

Shear rate,

Pseudoplastic or

Shear thinning



Examples of Non-Newtonian

Behavior

Viscosity,

a

time

Rheopectic

ThixotropicViscosity,

a

time



Viscosity

Newtonian fluids

μ = f(fluid, T)

Temperature dependence described by ArrheniusEquation

RT

E Aexp

Where T = absolute temperature [K]

R = gas constant = 1.987 [cal/(mol K)]

E = activation energy for viscous flow [cal/mol]

A = frequency factor (or viscosity at T0) [dyne s/cm2]



Finding Arrhenius Parameters

Measure μ at 4 temperatures

Plot ln(μ) versus 1/T

Slope = E/R Intercept = ln(A)

T R

E A

1lnln



Apparent Viscosity

Newtonian fluids μ = f(fluid, T)

Non-Newtonian fluids

a still varies with fluid and T Also can vary with

Shear rate

Shear time, shear history

Sample preparation

Test-system geometry

Test system operation

Composition, additives, microphases



Goals of this Unit

Use 2 methods to study fluid viscosity

Capillary viscometry

Good for Newtonian fluids We measure time to drain a calibrated tube to get μ

Coaxial cylinder rheometry Good for viscous Newtonian fluids and non-Newtonian fluids

We measure torque (M) on a spindle as it spins at differentrates (N rpm)

Plot shear stress versus shear rate, and get a from the slope



Cannon-Fenske Capillary

Fill reservoir ~1/2 way

(Use same vol of fluid each time.)

Use bulb to draw fluid into the test

arm. Remove bulb to let fluid drain.

Measure time for meniscus to flow

from top mark to bottom mark.

Repeat time measurement 3 times.(Looking for repeatability.)



Capillary Theory

Energy balance: friction (ie – viscous

dissipation) leads to loss of energy

Hagen-Poiseuille equation for laminar

flow through a cylinder

P L

RQ

8

4 gh P

t

V Q D

where

Combine and rearrange:t

V L

h g R

D

8

4



Capillary Calibration

So drain time from particular capillary,

with the reservoir filled with a consistent

volume of liquid, is proportional to the

kinematic viscosity of the fluid:

Correcting for end effects:

bt

2t

cbt



Capillary Plan

Each student calibrates a capillary viscometer. Measure t for 4 solutions of known viscosity

Fit points with

Find b and c for their capillary

Each student uses their calibrated capillary to findviscosity of an unknown; uses the viscosity todetermine the glycerol concentration of the unknown.

Each student uses their calibrated capillary to find Arrhenius parameters for a known glycerol solution.

2t

cbt



Capillary Viscometry Questions

Pool results with your team members to

determine the following:

How do b and c vary with the diameter of thecapillary used?

What is the composition of the unknown?

How do the Arrhenius parameters A and E vary

with the glycerol concentration?



Brookfield Coaxial Cylinder

Rheometer

Lower spindle into abeaker containing the

sample

Set the rpm, N

Measure the torque

on the spindle, M

Repeat for at least 3

values of N



Coaxial Cylinder Theory - 1

Equation of motion:

Force bal on spindle

Combine:

constant01 22

2 r r r r

dr

d

r

L

M Rr

i Rr at r ir

2

22

i

r ii R

M L R R F M and

A

F 2



Coaxial Cylinder Theory - 3

Separate Variables and rearrange:

Integrate from r = Ri (ω = ωi) to r = Ro (ω = 0)

dr r m L

M d n

n

21

1

2

n

on

i

n

i R Rnm L

M 22

1

22



Finding n & m for a fluid

Know L, Ri, Ro

Set N rpm, measure M % of full scale Calculate

ω = 2π∙N/60 [rad/s]

M = R/100*MFS [dyne cm]

Find n, m that gives best fit for all pointsOpt 1: Nonlinear least-squares fit

Opt 2: Linearize the equation

no

ni

n

i R Rn

m L

M 221

22



Linearizing to get n & m:

Can linearize by taking natural log of both sides:

Rearrange:

Plot ln(M) versus ln(ωi).

Gives a line with Slope = n

n

on

i

n

i R R

n

m L M

n

221

22

1lnln

1ln

n

o

n

i

n

i R R

n

m Lnn M

221

22

1

lnlnln

no

ni

no

ni

n

R Rn

n L

m

R Rn

m L

n

22

221

2lninterceptexp

2

1

22

1lnintercept



Interpretation of n, m

If n = 1 → Newtonian fluid

Apparent viscosity independent of shear rate

m = apparent viscosity = μ

If n > 1 → Dilatant fluid

Apparent viscosity ↑ as shear rate ↑

If n < 1 → Psuedoplastic fluid

Apparent viscosity ↓as shear rate ↑

1

n

r a

dr d r m

dr

d r



Coaxial Cylinder Rheometry Plan

Each student tests 7 materials

Measure M versus N for at least 3 rpm

settings for a particular combination of

sample/spindle/rheometer Determine m, n

Determine if the apparent viscosity changes

with time.



Coaxial Rheometry Questions

Pool results with your team members to

determine the following:

How reproducible are m,n results? Same machine, different spindles

Same spindle, different machine

Operator

How do the power law parameters, m and n, vary

with the glycerol concentration?

rheology lab experiment

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