Using HPC To Advance Water Desalination By

Electrodialysis

Clara DruzgalskiDepartment of Mechanical Engineering

Stanford University

Water Desalination

Distillation

Reverse Osmosis

Electrodialysis

Electrodialysis: Industrial

Electrodialysis water treatment plants in Barcelona, Spain produce 257 million liters of water per day.

Abrera (2007) 200 million litersSant Boi del Llobregat (2009) 57 million liters

Credit: Sant Boi del Llobregat

Electrodialysis: Applications

GrayWhite Black

Portable water treatment

Salt production Biomedical analysis: lab-on-a-chip devices

Electrodialysis

Model Problem

Channel Height 10-6 meters

Smallest Feature 10-9 meters

Applied voltage 1-3 Volts

Example Dimensional Values

Model Problem: Experiments

Well-described by 1D theory

Electroconvective chaos: 1D theory no

longer predictive

“

Should we use a commercial code like Comsol Multiphysics or build

a high performance code from scratch?

?

Commercial Software

◎ Commercial codes often use artificial smoothening for numerical robustness. This dissipates small structures generated by turbulent and chaotic fluid motion.

◎ Commerical codes must be general enough to handle a wide variety of problems, but this limits the user’s ability to take advantage of crucial time-saving algorithms

Commercial Software

Custom HPC Software

EKaos a high performance direct numerical simulation code that simulates electrokinetic chaos.

◎ No artificial smoothening

◎ Over 100 times faster than Comsol on a single node in 2D.

EKaos

2D EKaos SimulationConcentration

Charge Density

Experimental Observation

Joeri C. de Valença, R. Martijn Wagterveld, Rob G. H. Lammertink, and Peichun Amy TsaiPhys. Rev. E 92, 031003(R) – Published 8 September 2015

Simulation vs. Experiment

Experiment:De Valenca, et. al.

Simulation:Davidson, et. al.

Submitted to

Scientific Reports

2D EKaos: Current-Voltage

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2D EKaos: Current-Voltage

Qualitative matching with experiment

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3D EKaos Simulation

165 million mesh pointsThat’s over 1 billion degrees of freedom

11 terabytes of dataPer simulation

100,000 time stepsTo reach converged statistics

Each 3D EKaos simulation…

“

Why is a simulation of just one small section of a desalination channel so

computationally expensive?

?

The computational cost is determined by the range of relevant length and time

scales that must be resolved.

AlgorithmDetails

The mathematical details behind a high performance code

Governing EquationsSpecies Conservation:

Navier-Stokes:

Gauss’s Law:

c+ Concentration of cation

c- Concentration of anion

ϕ Electric potential

u Velocity vector

P Pressure 22

y

x

Governing EquationsSpecies Conservation:

Navier-Stokes:

Gauss’s Law:

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y

x

Reservoir:

Boundary Conditions

Membrane:

Periodic in x and z directions

Dimensionless ParametersParameter Description Range Value

ϵ Screening length, EDL size 10-6 – 10-3 10-3

Δϕ Applied voltage 20-120 120

κ Electrohydrodynamic coupling const. O(1) 0.5

c0+ Cation concentration at membrane >1 2

Sc Schmidt number 103 103

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Spatial Discretization

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◎ EKaos: 2D and 3D Direct numerical simulation (DNS)

◎ 3D has over 165 million spatial grid points

◎ Staggered mesh configuration

◎ Non-uniform mesh is used in the membrane-normal direction to handle sharp gradients

◎ Discretization: 2nd order central finite difference scheme

Time IntegrationSpecies Conservation

Navier-Stokes

Gauss’s Law

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Time IntegrationSpecies Conservation

2nd Order Implicit Scheme

Semi-Implicit: 1st order

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Time IntegrationIterative Algorithm

δ-form

Linearization

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Time IntegrationIterative Algorithm

δ-form

Linearization

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Time IntegrationEquation in δ-form

Remove Directional Coupling

Move non-stiff terms to left hand side

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Time IntegrationEquation in δ-form

Remove Directional Coupling

Move non-stiff terms to left hand side

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Time IntegrationEquation in δ-form

Analytical substitution using Gauss’s Law

Remove Directional Coupling

Time IntegrationFinal Equation

• Left hand side operator is linear and now only involves local coupling between δc+ and δc-

• We need to solve for u*, v*, w*, P*, and ϕ* at each iteration

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Pseudo-spectral SolverConservation of momentum

Pressure equation

Gauss’s Law

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By taking advantage of the geometry andusing physical insight we were able to:

1. Design operators that reduced thematrix bandwidth

2. Use fast and robust math libraries suchas LAPACK and FFTW

3. Reduce communication cost acrossprocessors by designing the algorithmwith parallelization in mind.

Conclusions◎Developed EKaos: a parallel 3D DNS code to simulate electroconvective chaos.

◎Developed a numerical algorithm for efficiently solving the coupled Poisson-Nernst-Planck and Navier-Stokes equations

◎Improved prediction of mean current density that has been observed in experiments

◎Comparison of 2D and 3D simulations show qualitative similarities, but quantitative differences

◎Electroconvective chaos can generate structures similar to turbulence.

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Thanks!Any questions?

You can find me at:cdruzgal@stanford.edu