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 StefanoBrandani

Institute for Materials and ProcessesUniversity of Edinburgh

Mathematical Modelling and Simulationof Power Plants and CO2 Capture

[email protected]

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


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 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



Modelling of adsorption processes

Adsorption process model hierarchy

Rp

External fluid film

IntercrystallineMacropores

rp Idealised spherical

crystallites

Column

10

z=0 z=Lc

Pellet



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

)



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



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



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



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.



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




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




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



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



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



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 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



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



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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