Friedrich Workshop on Modelling and Simulation of Coal-fired Power Generation and CCS Process

Eﬃcient simulations of general adsorption cycles

Daniel Friedrich, Maria-Chiara Ferrari, Peter Reid and Stefano
Brandani

Institute for Materials and Processes
University of Edinburgh

Mathematical Modelling and Simulation
of Power Plants and CO2 Capture

d.friedrich@ed.ac.uk

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 behaviour
Estimation of device and process parameters
Predicting the behaviour of future experiments
Design of experiments
Optimisation 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 micropore
diffusion, adsorption, heat transfer

Efficient simulations Daniel Friedrich


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



Example system

Simulation of the DP-PSA system
Lots of data from this experiment so need a fast and eﬃcient
dynamic simulator for the analysis of the experiments and to
estimate adsorbent parameters

CO2 mole fraction in the column
Simulation tool requirements
1
Conservation of mass, energy
Mole fraction

0.5
and momentum
Fast simulation of one cycle
0
1
Colu 40
Acceleration of convergence
mn 0.5
leng 20
th 0 0
Time Robustness and ease of use
Test case starting with uniform CO2 /N2 mixture and inducing
a CO2 concentration gradient along the column length



Modelling of adsorption processes

Adsorption process model hierarchy
Column Pellet
Intercrystalline
Macropores
0 1
External fluid film
Rp
z=0 z=Lc rp Idealised spherical
crystallites



Modelling of adsorption processes

Simple adsorption column model
Axial dispersed plug ﬂow Isothermal system
LDF mass transfer Langmuir isotherm
No frictional pressure drop Ideal gas

Gas phase material balance
∂ci 1 − ∂¯i
q ∂(ci u) ∂Ji
+ + + =0
∂t ∂t ∂z ∂z
∂ci u + |u|
D = (ci0+ − ci0− )
∂z z=0 2

Solid phase material balance
∂¯i
q qsi bp Pxi
= ki − qi
¯
∂t 1 + bp Pxi



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 on
Finite Elements Method (OCFEM)

Simulation to Cyclic Steady State
Differential Algebraic Equations are solved by state-of-the-art
solver SUNDIALS
Accelerate the convergence to CSS with extrapolation

Benefits
Spatial discretisations tailored to the corresponding phase
SUNDIALS is robust and simplifies the development



Spatial discretisation

Flux-limiting Finite Volume Method for convective term
Sharp fronts require specialised discretisation schemes
Finite Volume Method is inherently conservative: simulations
with a low number of grid points are possible
Flux-limiting scheme guarantees the correct behaviour of the
solution
Use higher order methods in smooth regions
First-order upwind methods close to the shock
Comparison of discretisations
Flux−limited
1.2
Upwind
1 Central differences
Mole fraction

0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
Column length



Spatial discretisation

Orthogonal collocation on ﬁnite element method

The adsorbed concentration in an adsorbent pellet changes
considerably over one cycle but only a small part of the pellet
actively takes part in the process.

Pellet concentration
Uniquely suited to stationary
Adsorbed concentration

gradients in the solid phase
0.7
Split the domain into small
0.6 elements to handle steep
gradients
0.5
1 24
Par
ticle 0.5 22
High order of accuracy for a
leng
th 0 20 T ime small number of grid points



Time integration

SUNDIALS

Solver for Differential/Algebraic equation systems
F (t, y (t), y (t)) = 0
˙
Based on variable-order, variable-step size Backward
Differentiation 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.


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 of
cycles

Acceleration schemes
1 Non-sequential cycle calculation: extrapolation algorithms

2 Model and discretisation switch
3 Interpolation of the starting conditions




extrapolation
Extrapolate the next solution from the previous 2m solutions
1 Set x0 = yi algorithm:
r
2 Generate xi+1 = F (xi ) −1 =0
r
0 = xr
3 Apply algorithm
−1
r r +1 (r +1) (r )
4 Set yi+1 =
(0) s+1 = s−1 + s − s
2m

0
0
0 m depends on the problem eigenvalues
1
1 0
0 2 For the DP-PSA m = 2 or 3
1 0
1 3
2 1 0 Switch to subsequent substitution close
0 2 4
2 1 to CSS
1 3
3 2
0 2 Skelboe, IEEE Transactions on Circuits
3
1 and Systems, 27, 3, 1980
4
0



extrapolation: Convergence to CSS

Convergence to CSS
Subsequent substitution: linear convergence
extrapolation: quadratic convergence
In the studied cases 35% faster convergence

Comparison of the approach to CSS Approach to CSS in PSA process
Subsequent substitution
0.6
ε extrapolation
−2
Cyclic difference

10

Mole fraction
0.5

−3
0.4
10

0.3
Subsequent substitution
−4
ε extrapolation
10 0.2
0 20 40 60 80 100 0 20 40 60 80 100
Cycle number Cycle number



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 which
accurately describe the problem
Interpolate the coarse solution to the accurate description
Simulate accurate model to CSS
Linear interpolation
1 Last grid

0.8
New grid Reduces simulation time
by at least an order of
Mole fraction

0.6
magnitude
0.4
Simpler than an adaptive
0.2
scheme
0
0 0.2 0.4 0.6 0.8 1
Column length



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 many
runs of the simulation tool have to be performed

Comparison of the approach to CSS Subsequent substitution
1
Cold start
Extrapolation to CSS
Restart

Relative simulation time
Cyclic difference

0
10 Restart from last solution

0.5 Simulation with
−2 variable grid size
10

−4
10
0 20 40 60 80 100
Cycle number Different simulation schemes



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


Simulation of general adsorption cycles

Cycle simulator interface

Simulation of 2-bed, multi-step VSA/PSA cycles
Arbitrary number and schedule of cycle steps



Conclusion

Model hierarchy describing the adsorption process in
diﬀerent levels of detail
Simulation tool for cyclic adsorption processes with
tailored discretisation schemes for the ﬂuid and solid phase
Several acceleration schemes which accelerate the
convergence 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



Acknowledgement

Financial support
EPSRC grant ’Innovative Gas Separations for Carbon
Capture’, EP/G062129/1
EPSRC grant ’Fundamentals of Optimised Capture Using
Solids’, 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 develop
the laboratories



Questions?

Thank you for your attention!
Questions?


Friedrich Workshop on Modelling and Simulation of Coal-fired Power Generation and CCS Process

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