using hpc to advance water desalination by electrodialysis · using hpc to advance water...
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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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