View
163
Download
1
Tags:
Embed Size (px)
Citation preview
Efficient simulations of general adsorption cycles
Daniel Friedrich, Maria-Chiara Ferrari, Peter Reid and StefanoBrandani
Institute for Materials and ProcessesUniversity of Edinburgh
Mathematical Modelling and Simulationof Power Plants and CO2 Capture
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Motivation
Why do we need efficient simulation tools?
Benefits of simulation
Interpretation of experimental results
Insight into different physical effects and device behaviourEstimation of device and process parameters
Predicting the behaviour of future experiments
Design of experimentsOptimisation of the process
Adsorption processes are very complex
3D problem for pressure, temperature and concentration
Different length scales: column, pellets, crystals
Different time scales: convection, macro- and microporediffusion, adsorption, heat transfer
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Example system
Dual Piston Pressure Swing Adsorption (DP-PSA)
Low sample requirement
Rapid testing of adsorbents
Many different configurations
Efficient numerical simulation tool required
Closed system: needs conservative scheme
Dynamic system with many parameters and variables
Large Peclet number: shock formation and propagation
Long computation to cyclic steady state
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Example system
Simulation of the DP-PSA system
Lots of data from this experiment so need a fast and efficientdynamic simulator for the analysis of the experiments and toestimate adsorbent parameters
020
40
0
0.5
10
0.5
1
Time
CO2 mole fraction in the column
Column length
Mol
e fr
actio
n
Simulation tool requirements
Conservation of mass, energyand momentum
Fast simulation of one cycle
Acceleration of convergence
Robustness and ease of use
Test case starting with uniform CO2/N2 mixture and inducinga CO2 concentration gradient along the column length
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Modelling of adsorption processes
Adsorption process model hierarchy
Rp
External fluid film
IntercrystallineMacropores
rp Idealised spherical
crystallites
Column
10
z=0 z=Lc
Pellet
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Modelling of adsorption processes
Simple adsorption column model
Axial dispersed plug flowLDF mass transferNo frictional pressure drop
Isothermal systemLangmuir isothermIdeal gas
Gas phase material balance
∂ci∂t
+1− εε
∂qi∂t
+∂(ciu)
∂z+∂Ji∂z
= 0
D∂ci∂z
∣∣∣∣z=0
=u + |u|
2(ci0+ − ci0−)
Solid phase material balance
∂qi∂t
= ki
(qsibpPxi
1 + bpPxi− qi
)
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Simulating adsorption
Solution strategy
Discretise only the spatial dimension: Method of Lines
Gas phase discretised by the Finite Volume Method (FVM)
Solid phase discretised by the Orthogonal Collocation onFinite Elements Method (OCFEM)
Simulation to Cyclic Steady State
Differential Algebraic Equations are solved by state-of-the-artsolver SUNDIALS
Accelerate the convergence to CSS with extrapolation
Benefits
Spatial discretisations tailored to the corresponding phase
SUNDIALS is robust and simplifies the development
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Spatial discretisation
Flux-limiting Finite Volume Method for convective term
Sharp fronts require specialised discretisation schemes
Finite Volume Method is inherently conservative: simulationswith a low number of grid points are possible
Flux-limiting scheme guarantees the correct behaviour of thesolution
Use higher order methods in smooth regionsFirst-order upwind methods close to the shock
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
1.2
Comparison of discretisations
Column length
Mol
e fr
actio
n
Flux−limitedUpwindCentral differences
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Spatial discretisation
Orthogonal collocation on finite element method
The adsorbed concentration in an adsorbent pellet changesconsiderably over one cycle but only a small part of the pelletactively takes part in the process.
2022
24
0
0.5
10.5
0.6
0.7
Time
Pellet concentration
Particle length
Ads
orbe
d co
ncen
trat
ion
Uniquely suited to stationarygradients in the solid phase
Split the domain into smallelements to handle steepgradients
High order of accuracy for asmall number of grid points
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Time integration
SUNDIALS
Solver for Differential/Algebraic equation systemsF (t, y(t), y(t)) = 0
Based on variable-order, variable-step size BackwardDifferentiation Formulas
Nonlinear systems are solved by Newton iteration
Linear systems are solved by direct or iterative solvers
Benefits
Based on robust and efficient algorithms
Modular implementation in C
User control over most aspects of the integration
A. C. Hindmarsh, P. N. Brown, K. E. Grant, S. L. Lee, R. Serban, D. E. Shumaker, and C. S. Woodward. Sundials:Suite of Nonlinear and Differential/Algebraic Equation Solvers. ACM Transactions on Mathematical Software,31(3) : 363396, September 2005.
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Acceleration to Cyclic Steady State
Acceleration to Cyclic Steady State
Cyclic Steady State
Many adsorption systems will reach Cyclic Steady State
At CSS the system state is not changing between cycles, i.e.y(ktc) = y ((k + 1)tc)
Sequential approach to CSS can be very slow, i.e. 1000 s ofcycles
Acceleration schemes
1 Non-sequential cycle calculation: extrapolation algorithms
2 Model and discretisation switch
3 Interpolation of the starting conditions
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Acceleration to Cyclic Steady State
ε extrapolation
Extrapolate the next solution from the previous 2m solutions
1 Set x0 = yi
2 Generate xi+1 = F (xi )
3 Apply ε algorithm
4 Set yi+1 = ε(0)2m
ε algorithm:
εr−1 = 0
εr0 = xr
εrs+1 = εr+1s−1 +
(ε(r+1)s − ε(r)
s
)−1
ε00
ε01
ε10 ε0
2
ε11 ε0
3
ε20 ε1
2 ε04
ε21 ε1
3
ε30 ε2
2
ε31
ε40
m depends on the problem eigenvalues
For the DP-PSA m = 2 or 3
Switch to subsequent substitution closeto CSS
Skelboe, IEEE Transactions on Circuitsand Systems, 27, 3, 1980
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Acceleration to Cyclic Steady State
ε extrapolation: Convergence to CSS
Convergence to CSS
Subsequent substitution: linear convergence
ε extrapolation: quadratic convergence
In the studied cases 35% faster convergence
0 20 40 60 80 10010
−4
10−3
10−2
Comparison of the approach to CSS
Cycle number
Cyc
lic d
iffer
ence
Subsequent substitutionε extrapolation
0 20 40 60 80 1000.2
0.3
0.4
0.5
0.6
Approach to CSS in PSA process
Cycle number
Mol
e fr
actio
n
Subsequent substitutionε extrapolation
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Model switch and interpolation
Model and discretisation switch
Start with the simplest case from the model hierarchy, e.g.isothermal and LDF, and coarse discretisation
At CSS switch to the model and discretisation whichaccurately describe the problem
Interpolate the coarse solution to the accurate description
Simulate accurate model to CSS
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
Linear interpolation
Column length
Mol
e fr
actio
n
Last gridNew grid Reduces simulation time
by at least an order ofmagnitude
Simpler than an adaptivescheme
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Model switch and interpolation
Interpolation of the starting conditions
Restart from old solutions
Interpolate the initial conditons from previous runs
Can reduce the simulation time to CSS by ∼ 50%
Especially important for parameter estimation because manyruns of the simulation tool have to be performed
0 20 40 60 80 10010
−4
10−2
100
Comparison of the approach to CSS
Cycle number
Cyc
lic d
iffer
ence
Cold startRestart
Re
lativ
e s
imul
atio
n tim
e
1
0.5
Subsequent substitution
Extrapolation to CSS
Restart from last solution
Simulation withvariable grid size
Different simulation schemes
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Simulation of general adsorption cycles
General adsorption cycles
Adsorption systems
Multiple adsorption columns
Connected by splitters, mixers,valves and tanks
Series of cycle steps:pressurisation, feed, purge, . . .
Extend DP-PSA simulator to general adsorption cycles
Modular system with different units: adsorption columns,valves, splitters, tanks, ...
Arbitrary number and connection of the units
Simulate different cycle configurations by time events, e.g.switching of valves
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Simulation of general adsorption cycles
Cycle simulator interface
Simulation of 2-bed, multi-step VSA/PSA cycles
Arbitrary number and schedule of cycle steps
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Conclusion
Model hierarchy describing the adsorption process indifferent levels of detail
Simulation tool for cyclic adsorption processes withtailored discretisation schemes for the fluid and solid phase
Several acceleration schemes which accelerate theconvergence to CSS by at least an order of magnitude
Extended the simulation tool to general adsorption cycles
Future work
Implement all models from the model hierarchy
Parallel implementation on multi-core CPUs and GPUs
Integration of PSA/VSA simulator with Unisim
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Acknowledgement
Financial support
EPSRC grant ’Innovative Gas Separations for CarbonCapture’, EP/G062129/1
EPSRC grant ’Fundamentals of Optimised Capture UsingSolids’, EP/I010939/1
ARIC: Adsorption Research Industrial Consortium
Opportunities in our group
2 Post-doc positions on carbon capture
2 Technician/Research Associate positions to further developthe laboratories
Efficient simulations Daniel Friedrich
Introduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator Conclusion
Questions?
Thank you for your attention!
Questions?
Efficient simulations Daniel Friedrich