Diffusion-Refraction Experiment

Mike Hansen ChEn 3603, 01/27/2014

University of Utah

Experimental goal

• Fick’s law is a MODEL for JA (has limitations!) • DAB = DBA • JA = -JB • For C > 2 components, everything changes!

(graduate school, anyone?)

JA = �cDABrxAFick’s “Law” Planar system with constant NA (or nA):

NA = �DAB

✓cA2 � cA1

z2 � z1

◆

cA1 cA2

From the lecture notes:

Determine diffusion coefficients of binary mixtures

Wiener Method

cuvette

water

sugar water

laseracrylic rod

screen

refracted

unrefracted

planar light

Wiener Methodscreen

unrefracted

refracted

• A gradient in sugar concentration determines the refraction

• Diffusion causes the refraction to change over time

• We can use this to find a diffusion coefficient

Start experiment now

Refraction PhysicsLight passes through

slower medium

Conservation of momentum

Decreased velocity

Direction changes (light bends)

Fast Medium

Slow Medium

a

b

c

d✓f

✓s d-b

Fermat: light takes the path of least time

t = tfast

+ tslow

t =

pa2 + b2

vfast

+

p(d� b)2 + c2

vslow

minimize time over b!

@t

@b= 0

Snell’s Lawnfast

nslow

=sin(✓

fast

)

sin(✓slow

)

tells us how much the light bendsn =

speed of light in vacuum

speed of light in medium

experiment

Refractive Index

The laser light is refracted by the gradient of the solution’s refractive index

@n

@x

=@n

@c

@c

@x

mass transfer

Refractive Index of Sugar-Water

Dilute solutions of sucrose in water:

n = (1.6013 · 10�4) · c(g/L) + 1.3318Cecil A. Coutinho, Bijith D. Mankidy, and Vinay K. Gupta. A simple refraction experiment for probing diffusion in ternary mixtures. Chemical Engineering Education, 44:134–139, 2010.

@n

@c= 1.6013 · 10�4

experiment

@n

@x

=@n

@c

@c

@x

mass transfer

Modeling Diffusion

sugar water

water

x

Fick’s Law

Diffusion occurs mainly in the x direction

Bulk velocity established?

Valid?

Valid?

Nope. Why not?

Governing Equation p1Start with mole balance on sugar

@cs@t

= �r · (csvM )�r · Js + Ss

no bulk flow no reactions

Js = �Drcs

Fickian diffusion

Governing Equation p2

Substitute diffusive flux into mole balance

@cs@t

= Dr2csassumed D is constant

@cs

@t

= D

@

2cs

@x

2

Ignore y and z contributions

Initial Conditions

sugar water

water

x

Choose x=0 as the interface between water and sugar water

cs(x, t = 0) =

(0

c0

x > 0

x 0

Boundary Conditions

sugar water

water

x

Infinite medium assumptions- Diffusion never reaches the top of the water

- Diffusion never reaches the bottom of the sugar

Valid?!

Process time scale versus

Diffusion time scale

Solving the PDE@cs

@t

= D

@

2cs

@x

2c(x, t) = c0

Z +1

0

1p4⇡Dt

exp

✓� (x� u)

2

4Dt

◆du

uh oh...

But wait! We only want the derivative!

experiment

@n

@x

=@n

@c

@c

@x

mass transfer

@c

@x

=

c0p4⇡Dt

exp

✓� x

2

4Dt

◆

Numerical Solution

@cs

@t

= D

@

2cs

@x

2

• Classical 1-D diffusion equation

• Can be solved numerically with finite difference methods

• I did this in MATLAB with a BTCS method.

• We’ll avoid details, but watch movies!

Basic Simulation

Concentration Profile Laser Light

Verifying Infinite Medium Assumption

Concentration Profile % Error of Assumption

Use an “infinite” length (5 times the cuvette height)

Several days required to invalidate assumption!

Diffusive Time Scale

L

u

Advection Time scale

⌧adv =L

u

L

Diffusion

⌧di↵ =L2

D

D

Time scale

‘Diffusion time’ scales with length squared Why?

Diffusion and ThermoRandom walk

• Uncorrelated molecules

• Random walk

• Ideal solution

• Fickian diffusion

Equivalent statements

You suspect nonideal diffusion. What do you check?

Activity coefficients!

Back to Experiment

@n

@x

= M14np4⇡Dt

exp

✓�M2

x

2

4Dt

◆

experiment

@n

@x

=@n

@c

@c

@x

mass transfer

@c

@x

=

c0p4⇡Dt

exp

✓� x

2

4Dt

◆

@n

@c= 1.6013 · 10�4

M1,M2 are magnification coefficients

Now what?!@n

@x

= M14np4⇡Dt

exp

✓�M2

x

2

4Dt

◆

what do we do with this?

For a given time, laser profile, and refractive index of solution, we know everything in this expression but the diffusion coefficient.

This is a data-fitting problem!

Computing DUse a webcam to capture the laser profile

Use MATLAB’s powerful image processing tools

Then do nonlinear regression to find D

Senior Lab

My 2nd semester project was to write an improved GUI for this experiment.

The lab computer has since crashed… so this is a neat and open senior project if you’re interested.

Acknowledgements

• Tony Butterfield

• Senior Lab professor (award-winning!)

• Kyle Branch

• Senior lab teammate, MATLAB image processing guru

• Colin Young

• Author of ChE Outreach demo about the Wiener Method