Modelling a Generalized Dialysis Device

Transcription

1 Modelling a Generalized Dialysis Device Brian Wetton Mathematics Department University of British Columbia UPC, November 11, 2015

2 Overview Advertising UBC Institute of Applied Mathematics Prototype Device Quick Estimates Basic Modelling: Asymptotics and Numerical Approximation Model Results Summary Example of Mathematics applied to an engineering problem. I will highlight some generic aspects that might be useful to your careers.

3 Institute of Applied Mathematics University of British Columbia Faculty participation from many departments. Interdisciplinary graduate programme. Not every Mathematics Department appreciates applied mathematics. Choose graduate school carefully.

4 Prototype Device I Generalized Dialysis System Collaboration A group in the UBC Chemical and Biological Engineering department was awarded a grant from an oil industry consortium to study a generalized dialysis system that would simultaneously clean waste water and generate useful byproducts. The CHBE group has members Arman Bonakdarpour, Saad Dara, and David Wilkinson. This group is not very mathematically sophisticated. The grant funded my student, Michael Lindstrom, for a year. Previous Published Work: One-Dimensional Model for a Direct Methanol Fuel Cell with a 3D Anode Structure, Lam, Wetton, Wilkinson, JES 158 B29-B35 (2011)

5 Prototype Device II Device Schematic

6 Prototype Device III Diagnostic Operation Dilute nitric acid HNO 3 H + + NO 3 replaces carbonate. Instead of sodium bicarbonate NaHCO 3, sodium nitrate NaNO 3 is formed.

7 Prototype Device IV Second Round Experiments Industrial Target: 40mA/cm 2 at less than 7V.

8 Quick Estimates I Nitric Acid Diagnostic Stoichiometric Estimates 100mM Nitric Acid inlet can produce at most l/min 0.1 mol/l C/mol 60 s/min 64cm 2 = 12.6 A/cm 2. The maximum observed current is a bit less than this limit. Check Units: Make sure relationships you write down are dimensionally consistent Anomaly: The 1mM experimental current is more than the stoichiometric limit. The experimental group has ruled out some of the obvious other reactions that could produce ions to carry the current.

9 Quick Estimates II Carbonic Acid Stoichiometric Estimates H 2 O + CO 2 H 2 CO 3 H + + HCO3 K 2, K 1 K 4, K 3 Dissolved CO 2 is obtained by bubbling pure (1Atm) CO 2 gas in water. There is a known relationship between CO 2 gas pressure P and dissolved CO 2 concentration [CO 2 ]: [CO 2 ] = P/H. Neglecting hydroxide and carbonate ions, C = [H + ] = [HCO 3 ] Equilibrium K 2 K 4 C 2 = K 1 K 3 P/H.

10 Quick Estimates III Carbonic Acid Stoichiometric Estimates, cont. CO 2 gas pressure P; C = [H + ] = [HCO 3 ]; K 2K 4 C 2 = K 1 K 3 P/H P = 1 Atm gives [CO 2 ] = 34mM, C = 0.12 mm. C value roughly consistent with measured ph (ph = log 10 C). Stoichiometric limit current based on C: ma/cm 2. Stoichiometric limit current based on [CO 2 ]: 4.27 ma/cm 2. Observed current density approximately 1mA/cm 2. Limiting current due to [CO 2 ] dissociation: K 1 [CO 2 ] W 10 ma/cm 2. What limits the current in this case? [Excellent question, our models do not give insight yet].

11 Modelling I Nitric Acid Channel

12 Modelling II Fluxes Model steady state c(x, y) concentration [H + ] = [NO 3 ] and electric potential φ(x, y). Fluxes are by convection by constant velocity v, diffusion and conduction J 1 = (0, vc) D 1 c D 1Fc RT φ J 2 = (0, vc) D 2 c + D 2Fc RT φ Conservation of these species: J 1,2 = 0 Nonlinear, vector elliptic problem for c and φ. Book: Newman, Electrochemical Systems. (protons) (nitrate)

13 Modelling III Scaling { } Consider (0, vc) D 1 c D 1Fc RT φ = 0. Scale: c/c 0 c φf /RT φ x/w x y/l y θc y = c xx + ɛ 2 c yy + (cφ x ) x + ɛ 2 (cφ y ) y with θ = vw 2 /(LD 1 ) and ɛ = W /L Neglect ɛ 2 terms. Always scale a new model to get a sense of the importance of terms and the universality of the results. It makes the resulting numerics better conditioned.

14 Modelling IV Scaled System or equivalently θc y = c xx + (cφ x ) x θc y = D {c xx (cφ x ) x } with D = D 2 /D 1 θc y = c xx + (cφ x ) x 0 = (1 D)c xx + (1 + D)(cφ x ) x with initial conditions c(x, 0) = C 0 and boundary conditions x = 0 φ = 0 and c x cφ x = 0 x = 1 φ = β and c x + cφ x = 0 Nonlinear parabolic (in y) for c coupled at each y to a boundary value problem in x.

15 Modelling V Numerical Scheme Never write your own code for a standard problem but this is kind of nonstandard. Compute approximate C n j c((j 1/2)h, nk).

16 C n j Modelling VI Numerical Scheme c((j 1/2)h, nk), Φ n j ψ((j 1/2)h, nk) Approximate θc y = c xx + (cφ x ) x by θ(c n j C n 1 j )/k = (Cj 1 n 2Cj n + Cj+1)/h n 2 + D (FCD + Φ) where F is forward averaging and D ± is forward/backward differencing in x. Other equation and boundary conditions approximated similarly. Result is a nonlinear system for 2N + 4 unknowns for C n and Φ n, solved with Newton s method, marching from inlet to outlet. Standard staggered finite difference (finite volume) method. Solution errors O(k + h 2 ). Implemented in MATLAB. Good for prototyping

17 Results Polarization Curves The model was modified to add the voltage losses to electrolysis (empirical fit) and other elements. Good agreement to 20-30% loss in limiting current due to mass transport limitations.

18 Results Local Current Density

19 Results Concentration Contours Difficult to get this insight from experiments

20 Results Failure of Model for Carbonic acid channel The model we developed for the carbonic acid channel, including reaction with literature parameters, and diffusion, failed to give agreement with the experimental results (limiting current was much too low).

21 Summary Showed an example of electrochemical modelling work done our group. Some generic features that might be useful for your careers: To impress an application scientist (financial engineer, etc.) you have to answer or give insight into their question. Start with the simplest possible calculations. Use the expertise of your application colleagues, but ask questions if what they say does not make sense to you. An application textbook that is accessible to mathematicians is a fantastic find! Scale and use asymptotics to simplify a problem. Use software packages for computations if at all possible, otherwise prototype in MATLAB. Failure may be useful, in that it may show that there are important effects missing from the current understanding of the phenomena.