GORE: Modeling Filter Compression, Fluid Generation and Parallelizing the Potts Model

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1 GORE: Modeling Filter Compression, Fluid Generation and Parallelizing the Potts Model Mathematical Problems in Industry, June 12 17, 2016 June 17, 2016 (Mathematical Problems in Industry, June 12 17, 2016) GORE June 17, / 38

2 GORE: Modeling the Movement of Gas and Fluid through a Diffusive Filter S. Bohun, C. Breward, P. Dubovski, D. Schwendeman, H. Yaple, S. Ahmed, J. Batista, M. Hennessey, T. Hueckel, S. Iyaniwura, F. Meng, M. Mohebujjaman, Y. Qian, P. Sanaei, D. Serino, O. Shonibare MPI June 17, 2016 Bohun, Breward, et al. (MPI) GORE June 17, / 38

3 Our Problem Develop a mathematical model for characterizing fluid production and transport inside of a porous medium. Quantify the time scale and spatial profile of flooding of the pore space by the fluid. Bohun, Breward, et al. (MPI) GORE June 17, / 38

4 A Coupled Multiscale Approach Small Scale - Analyze the formation of sulfuric acid on the surface of a single filter pellet. Large Scale - Analyze the transport of sulfur dioxide in the gas channel and filter. Bohun, Breward, et al. (MPI) GORE June 17, / 38

5 Small Scale analysis Gas diffuses into the filter from the gas channel. Sulfur Dioxide (bad) reacts with Oxygen and Water on the surface of the Activated Carbon to form Sulfuric Acid (good). 2SO 2 + O 2 + 2H 2 O 2H 2 SO 4 Bohun, Breward, et al. (MPI) GORE June 17, / 38

6 Outer Small Scale Problem S i is the concentration of SO 2 in layer i. C i is the concentration of O 2 in layer i. C 1 = D 1 2 S 1 C 1, = d 1 2 S 1, Outer Layer t t C 1 C, S 1 S, r, Boundary Condition C 1 = k 1C 2, S 1 = k 2S 2, r = R + a Interface Condition Bohun, Breward, et al. (MPI) GORE June 17, / 38

7 Inner Small Scale Problem C 2 t = D 2 2 C 2, S 2 t = d 2 2 S 2, Inner Layer D 2 C 2 r = λs2 2C 2, d 2 S 2 r = 2λS2 2C 2, r = R Boundary Condition C 1 D 1 r = C 2 D2 r, S 1 d1 r = S 2 d2, r = R + a Interface Condition r Bohun, Breward, et al. (MPI) GORE June 17, / 38

8 Area Increase Problem At r = R, v = 2δλS 2 2C 2 and v = 0 At r = R + a, v = a t + ua x and a t = 2δλR2 S2C 2 2 (R + a) 2 Bohun, Breward, et al. (MPI) GORE June 17, / 38

9 Reduced Micro Scale Model The nondimensionalized reduced model is with boundary conditions and interface conditions Here S = S 2, T = S 1. S t = DS xx, T t = T xx, (1) DS x = 2λS 2 x=0, T = T x=b, (2) T = χs x=a, DS x = T x x=a. (3) Time scale: The flux in each layer is approximately constant. Quasi-Steady Diffusion: Diffusion is much faster than liquid layer growth. Solutions take the form S = S + S, T = T + T. Bohun, Breward, et al. (MPI) GORE June 17, / 38

10 Graphs (1D cartesian) Bohun, Breward, et al. (MPI) GORE June 17, / 38

11 Graphs (1D spherical symmetric) Bohun, Breward, et al. (MPI) GORE June 17, / 38

12 Zooming Out Bohun, Breward, et al. (MPI) GORE June 17, / 38

13 Macroscale Model W is the concentration of a particular gas species in the channel. T is the concentration of a particular gas species in the filter. Outer U h 2 W z = d 1 T y y=0, 0 < z < L (4) W (0, t) = W in Inner d 1 T yy = 3 F (a, T ), 0 < y < H/2, 0 < z < L Rφ (5) a t = δf (a, T ), t > 0. (6) T y y=h/2 = 0, T y=0 = W, a t=0 = 0. Bohun, Breward, et al. (MPI) GORE June 17, / 38

14 Breakthrough Curve Bohun, Breward, et al. (MPI) GORE June 17, / 38

15 Critical Time Estimate for Surface Clogging Chemical reaction at droplet perimeter: q 1 2πr Imbibing liquid from pores: q 2 A c(x, y, t)dxdy Absorbing H 2 O (hygroscopicity): q 3 S Droplet coalescence: da f p Use this formation to determine critical time for filter surface clogging. Bohun, Breward, et al. (MPI) GORE June 17, / 38

16 Progress Determined that the concentration profile around the filter particles is governed by Laplace s equation. Physical chemistry results in ODE nonlinearity (messy but solvable) that can be easily translated to fluid formation in the fluid. Generated reasonable breakthrough curve with our model. Estimated surface clog time as a function of the area of the absorbing wall covered by a droplet. Bohun, Breward, et al. (MPI) GORE June 17, / 38

17 Models for Filter Membrane Compression C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, B. Tilley MPI June 17, 2016 C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

18 A Membrane under Compression and Flow 2r f C=C * P + z=0 δ C z =0 P_ z=l 2r s C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

$for transport parameters Darcy permeability void fraction species fraction effective tracer diffusivity retardation$ coefficient Examine how changes in filter compression affect parameters 1 M. Faccini 2012. doi:10.

coefficient Examine how changes in filter compression affect parameters 1 M. Faccini 2012. doi:10.

19 Gore Mission Statement Goal: Mathematically characterize an idealized fibrous structure Use fiber matrix structure for transport parameters Darcy permeability void fraction species fraction effective tracer diffusivity retardation coefficient Examine how changes in filter compression affect parameters 1 M. Faccini doi: /2012/892894, 2 C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

20 Model Darcy s Law Continuity Equation This gives Advection-Diffusion K peff u = p (7) µ u = 0 (8) ( ) Kpeff p = 0. (9) µ c t + 1 γ (fc u ) = D eff 2 c (10) c - concentration of solute p - pressure u - velocity Note: K peff, µ, γ, f, D eff are functions of void fraction (ɛ). C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

21 1D Model For uniform ɛ: p zz = 0 c t + f γ c zu = D eff c zz (11) Nondimensionalize p zz = 0 c t + bc z p z = P e 1 c zz (12) where b = fk p γµ. (13) C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

22 Concentration Profile C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

23 1D Model with Compression Compression reduces ɛ (void fraction). Conservation of solid mass: AL u (1 ɛ u ) = AL c (1 ɛ c ) (14) Change in L c L u = change in c(z, t). C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

24 Results of Compression T = 5, Lc/Lu = 0.75 Final solute concentration: uncompressed Final solute concentration: compressed T = 5, Lc/Lu = 0.5 Final solute concentration: uncompressed Final solute concentration: compressed Concentration c Concentration c Location z T = 25, Lc/Lu = 0.75 Final solute concentration: uncompressed Final solute concentration: compressed Location z T = 25, Lc/Lu = 0.5 Final solute concentration: uncompressed Final solute concentration: compressed Concentration c Concentration c Location z Location z C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

25 Mechanical Model for Compressed Fiber Structure Goal: Simulate fiber deformation due to compression Method: Stiffness method Solve [k]d = f [k] - stiffness matrix d - vector of deformation of nodes (Cartesian coordinates) f - force due to compression Treat fibers as springs 1 Truss: angle varies; fibers straight simple geometry case collapse under shear force 2 Frame: angle constant; fibers can bend C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

26 Results: See Videos Compression of fiber bundle in cube More complex geometry C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

27 Next Steps 1D Compression Model Prescribe u(z = 0) instead of p(z = 0) Introduce clogging: where ξ is bound solute. Stiffness Method-Simulated Model dξ dt = kc Randomly distribute fibers versus using mesh structure Consider how to derive porosity, etc. from results Remove rigidity imposed by boundary conditions C. Bi, V. Ciocanel, S. Hill, V. Mikheev, D. Rumschitzki, GORE B. Tilley (MPI) June 17, / 38

28 Cellular Potts Model Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, M. Zyskin MPI June 17, 2016 Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

Problem Introduction: Cellular Potts Model Lattice model with cells Extensively used in solid-state physics Recently used in biology Avascular tumor growth Build parallel

29 Problem Introduction: Cellular Potts Model Lattice model with cells Extensively used in solid-state physics Recently used in biology Avascular tumor growth Build parallel simulation framework Studying growth of cells and influence of nutrient transport Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

30 Goal: Monte Carlo simulation in parallel Model algorithm development based on paper Implement portions of the algorithm using MPI Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

31 Chemical Reaction-Diffusion of Oxygen and Nutrients u O2 t u n t = D O2 2 u O2 + a 0 u O2 u T O 2 u O O 2 u T O 2 = D n 2 u n + b 0 u n u T n u O n u T n u O2 Ω = c O2 u n Ω = c n Hamiltonian H = J τ(s1 )τ(s 2 ) [1 δ (S 1, S 2 )] + γ ( v V T) 2 lattice sites cells 1 H < 0 P Acc = k b T H 0 e H Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

32 Preliminary Numerical Results Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

33 Data Structure for Parallel Description Lattice Cell index S i Cell type τ (S i ) Neighbor list Oxygen: u O2 Cell Cell Type: Proliferating, Quiescent, Necrotic Occupying lattice points list Average Oxygen: mean ( u O2 ) Average Nutrition: mean (u n ) Nutrition: u n Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

34 Parallel Domain Decomposition for PDE and H Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

Equilibrium Gibbs Measure ρ κ ij = 1 Z types cell IDs τ={τmn} S=part({τ}) δ (τij =κ) }{{} ρ κ

S = part({τ}): split A, Q cites into cells with distinct IDs ( coloring ).

Number of colorings of n cites grow quickly: part[2] = {{(1, 2)}, {(1), (2)}}) # = 2 part[3] =

35 Equilibrium Gibbs Measure ρ κ ij = 1 Z types cell IDs τ={τmn} S=part({τ}) δ (τij =κ) }{{} ρ κ ij δ (enough O2 )e βh(τ,s)) τ cite types ={A, Q, D, M} (active, quiescent, dead, medium). S = part({τ}): split A, Q cites into cells with distinct IDs ( coloring ). H penalizes for cell volume not optimal, or long particle boundary. Number of colorings of n cites grow quickly: part[2] = {{(1, 2)}, {(1), (2)}}) # = 2 part[3] = {{(1, 2, 3)}, {(1), (2, 3)}, {(1, 2), (3)}, {(1, 3), (2)}, {(1), (2), (3)} # = 5 part[4] = 15; part[5] = 52;... part[9] = Q: Ensuring that cells are simply connected Thermodynamic /large lattice limit? F = 1 βarea lim Area log Z =??? Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

36 Conclusions & Future Work Consider more complicated reaction and energy terms Parallelize other cell dynamics Construct Monte Carlo simulation Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

37 Many thanks go to... NSF Duke University RPI MPI Our industrial sponsors: Dr. Vasu Venkateshwaran and Dr. Zhenyu He Our patient faculty mentors Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

38 Reduction to Macroscale Model Begin with W t + UW z = d 1 (W y y + W zz) Average over y measured from center of channel to filter surface by integrating to get h 2 (Ŵt + UŴz) = d 1 W y y =h/2 Assume change in time is negligible, then we see h 2 UŴz = d 1 T y y=0 Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

39 1D Problem Method Solve using Crank-Nicholson in time Because of nonlinearity, set up as a Newton s Method problem: F ( x n+1 ) = G( x n ) Solve for x n+1, where c n 1. x n = p n 1. c n N+1 p n N Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

40 Parameters and Variables c - concentration of solute inside filter p - pressure u - velocity K peff - Darcy coefficient/permeability µ - viscosity f - sieving coefficient γ partition coefficient/species volume fraction ɛ - void fraction r f - fiber radius r s - solute particle radius δ - distance between fiber bundles A - surface area L u = uncompressed membrane thickness L c = compressed filter thickness Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

41 Further cell dynamics: Cell growth and division Y. Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

42 Potts Model Reference Slide: breadth first search (BFS). Chen, D. Duffy, J. Duan, T. Gu, J. Sexton, Q. Wang, Gore M. Zyskin (MPI) June 17, / 38

Modeling Filtration with Multiple Layers 33 rd Annual Workshop on Mathematical Problems in Industry. June 23, 2017

Modeling Filtration with Multiple Layers 33 rd Annual Workshop on Mathematical Problems in Industry Shuchi Agarwal Ryan Allaire 2 Manuchehr Aminian 3 Chris Breward 4 Chuan Bi 5 Anqi Chen 6 Jon Chapman