Diffusion-Refraction Experiment. Mike Hansen ChEn 3603, 01/27/2014 University of Utah
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1 Diffusion-Refraction Experiment Mike Hansen ChEn 3603, 01/27/2014 University of Utah
2 Experimental goal From the lecture notes: Fick s Law J A = cd AB rx A Fick s law is a MODEL for JA (has limitations!) DAB = DBA JA = -JB For C > 2 components, everything changes! (graduate school, anyone?) Planar system with constant NA (or na): ca2 c A1 N A = D AB z 2 z 1 c A 1 c A 2 Determine diffusion coefficients of binary mixtures
3 Wiener Method screen cuvette refracted water acrylic rod laser unrefracted sugar water planar light
4 Wiener Method screen A gradient in sugar concentration determines the refraction refracted Diffusion causes the refraction to change over time We can use this to find a diffusion coefficient unrefracted
5 Start experiment now
6 Refraction Physics Light passes through slower medium Decreased velocity Fermat: light takes the path of least time t = t fast + t slow p a2 + b t = 2 + v fast p (d b)2 + c 2 v slow minimize time =0 Fast Medium Slow Medium Snell s Law Conservation of momentum Direction changes (light bends) f a b c s d-b d n fast n slow = sin( fast) sin( slow )
7 Refractive Index n = speed of light in vacuum speed of light in medium tells us how much the light bends The laser light is refracted by the gradient of the solution s mass transfer
8 Refractive Index of Sugar-Water Dilute solutions of sucrose in water: n =( ) c(g/l) Cecil A. Coutinho, Bijith D. Mankidy, and Vinay K. Gupta. A simple refraction experiment for probing diffusion in ternary mixtures. Chemical Engineering Education, 44: =
9 Modeling Diffusion Fick s Law x Valid? Diffusion occurs mainly in the x direction water Valid? sugar water Bulk velocity established? Nope. Why not?
10 Governing Equation p1 Start with mole balance on = r (c sv M ) r J s + S s no bulk flow no reactions Fickian diffusion J s = Drc s
11 Governing Equation p2 Substitute diffusive flux into mole = Dr2 c s assumed D is constant Ignore y and z = c 2
12 Initial Conditions Choose x=0 as the interface between x water water and sugar water c s (x, t = 0) = ( 0 x>0 c 0 x apple 0 sugar water
13 Boundary Conditions Infinite medium assumptions x - Diffusion never reaches the top of the water - Diffusion never reaches the bottom of the sugar water Valid?! sugar water Process time scale versus Diffusion time scale
14 Solving the = c 2 c(x, t) =c 0 Z p exp 4 Dt (x u) 2 4Dt du uh oh... But wait! We only want = c 0 p 4 Dt exp x 2 4Dt
15 Numerical Solution Classical 1-D diffusion equation Can be solved numerically with finite difference = c 2 I did this in MATLAB with a BTCS method. We ll avoid details, but watch movies!
16 Basic Simulation Concentration Profile Laser Light
17 Verifying Infinite Medium Assumption Concentration Profile % Error of Assumption Use an infinite length (5 times the cuvette height) Several days required to invalidate assumption!
18 Diffusive Time Scale Advection Time scale u L adv = L u Diffusion time scales with length squared Why? Diffusion D L Time scale di = L2 D
19 Diffusion and Thermo Equivalent statements Random walk Uncorrelated molecules Random walk Ideal solution Fickian diffusion You suspect nonideal diffusion. What do you check? Activity coefficients!
20 Back = = c 0 p 4 Dt exp x = M 1 4n p 4 Dt exp M 2 x 2 4Dt M 1,M 2 are magnification coefficients
21 = M 1 4n p 4 Dt exp M 2 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!
22 Computing D Use a webcam to capture the laser profile Use MATLAB s powerful image processing tools Then do nonlinear regression to find D
23 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.
24 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
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