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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 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
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!
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.
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