Brandani, Stefano

Ferrari, Maria-Chiara

Friedrich, Daniel

Patel, L.

English

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Efficient Numerical Schemes for the Simulation of AdsorptionProcesses to Cyclic Steady StateCitation for published version:Friedrich, D, Ferrari, M-C, Patel, L & Brandani, S 2010, 'Efficient Numerical Schemes for the Simulation ofAdsorption Processes to Cyclic Steady State'. in AIChE Annual Meeting, Salt Lake City, UT, USA.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Preprint (usually an early version)Published In:AIChE Annual Meeting, Salt Lake City, UT, USAGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015Efficient Numerical Schemes for the Simulation ofAdsorption Processes to Cyclic Steady StateDaniel Friedrich, Maria-Chiara Ferrari, Leena Patel and StefanoBrandaniInstitute for Materials and ProcessesUniversity of EdinburghAIChE Annual Meeting 2010Salt Lake Cityd.friedrich@ed.ac.ukIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionOutline1 Introduction2 Modelling of adsorption processes3 Simulation of adsorption processes4 Acceleration to Cyclic Steady State (CSS)5 Cycle simulatorSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionWhy do we need efficient simulation tools?Benefits of simulationInterpretation of experimental resultsInsight into different physical effects and device behaviourEstimation of device and process parametersPredicting the behaviour of future experimentsDesign of experimentsOptimisation of the processVery complex problem3D problem for pressure, temperature and concentrationDifferent length scales: column, pellets, crystalsDifferent time scales: convection, macro- and microporediffusion, adsorption, heat transferSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionModelling of adsorption processesPotential simplifications of the modelWe can write the full, complex model but in many cases asimplified model will give the same results with much less effortModel simplificationsAxial dispersed plug flowNo frictional pressure drop or Ergun equationMass transfer: LDF, macro- or micropore diffusion, filmresistanceEquilibrium isotherm: linear, Langmuir, dual-site LangmuirTemperature: isothermal, non-isothermal, wall lossSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionModelling of adsorption processesAdsorption process model hierarchyDeveloping models for the varioussimplifications gives us a model hierarchywith different models which are valid indifferent parameter regimesSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionModelling of adsorption processesAdsorption column: 1D, Linear Driving Force (LDF)Axial dispersed plug flowLDF mass transferNo frictional pressure dropIsothermal systemLangmuir isothermIdeal gasGas phase material balance∂ci∂t+1− ∂q¯i∂t+∂(ciu)∂z+∂Ji∂z= 0D∂ci∂z∣∣∣∣z=0=u + |u|2(ci0+ − ci0−)Solid phase material balance∂q¯i∂t= ki(qsibpPxi1 + bpPxi− q¯i)Simulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSimulating adsorptionDifficulties in simulating adsorption processesDifficultiesDynamic systems with many parameters and variablesMany process configurations possibleLarge Peclet number and shock formationConvergence to cyclic steady state can be slow0 20 40 60 80 10000.20.40.60.8Approach to CSS in Dual Piston−PSACycle numberMole fraction  Piston 1Piston 2Simulation tool requirementsConservation of mass, energyand momentumFast simulation of one cycleAcceleration of convergenceRobustness and ease of useSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSimulating adsorptionSolution strategyDiscretise only the spatial dimension: Method of LinesGas phase discretised by the Finite Volume Method (FVM)Solid phase discretised by the Orthogonal Collocation onFinite Elements Method (OCFEM) when diffusion in theparticles is includedSimulation to Cyclic Steady StateDifferential Algebraic Equations are solved by state-of-the-artsolver SUNDIALSAccelerate the convergence to CSS with extrapolationBenefitsSpatial discretisations tailored to the corresponding phaseSUNDIALS is robust and simplifies the developmentSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSpatial discretisationFlux-limiting FVM for convective term (1/2)Sharp fronts require specialiseddiscretisation schemesUse higher order methods insmooth regionsFirst-order upwind methodsclose to the shock 0 0.2 0.4 0.6 0.8 100.20.40.60.811.2Comparison of discretisationsColumn lengthMole fraction  Flux−limitedUpwindCentral differences∂yi∂t+(Fi+1/2 − Fi−1/2)∆z=yi+1 − 2yi + yi−1∆z2+ RiFi+1/2 =v2[(yi + yi+1) + (1− σi ) (yi+1 − yi )]σ(θ) =θ + |θ|1 + θθ =yi − yi−1yi+1 − yiSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSpatial discretisationFlux-limiting FVM for convective term (2/2)Finite Volume Method is inherently conservativeSimulations with a low number of grid points are possibleFlux-limiting scheme guarantees the correct behaviour of thesolution0 0.2 0.4 0.6 0.8 1−0.04−0.03−0.02−0.0100.010.02Deviation from fine gridColumn lengthAbsolute deviation  CD, CGRID=21Flux limiter, CGRID=210 0.2 0.4 0.6 0.8 100.10.20.30.40.5DPPSA: Different grid sizesColumn lengthMole fraction  CD, CGRID=201CD, CGRID=21Flux limiter, CGRID=21Simulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSpatial discretisationOrthogonal collocation on finite element methodThe adsorbed concentration in an adsorbent pellet changesconsiderably over one cycle but only a small part of the pelletactively takes part in the process.20222400.510.50.60.7TimePellet concentrationParticle lengthAdsorbed concentrationUniquely suited to stationarygradients in the solid phaseSplit the domain into smallelements to handle steepgradientsHigh order of accuracy for asmall number of grid pointsSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionTime integrationSUNDIALSSolver for Differential/Algebraic equation systemsF (t, y(t), y˙(t)) = 0Based on variable-order, variable-step size BackwardDifferentiation FormulasNonlinear systems are solved by Newton iterationLinear systems are solved by direct or iterative solversForward and adjoint sensitivity analysishttps://computation.llnl.gov/casc/sundials/main.htmlBenefitsBased on robust and efficient algorithmsModular implementation in CUser control over most aspects of the integrationSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionExample systemDual Piston Pressure Swing Adsorption (DP-PSA)Low sample requirementRapid testing of adsorbentsMany different configurationsEfficient numerical simulation tool requiredClosed system: needs conservative schemeDynamic system with many parameters and variablesLarge Peclet number: shock formation and propagationLong computation to cyclic steady stateSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionExample systemSimulation of the DP-PSA systemLots of data from this experiment so need a fast and efficientdynamic simulator for the analysis of the experiments and toestimate adsorbent parameters0204000.5100.51TimeCO2 mole fraction in the columnColumn lengthMole fraction0 10 20 30 4000.20.40.60.8CO2 mole fraction in the pistonsTimeMole fraction  Y1P1Y1P2Test case starting with uniform CO2/N2 mixture and inducinga CO2 concentration gradient along the column lengthSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionAcceleration to Cyclic Steady StateAcceleration to Cyclic Steady StateCyclic Steady StateMany adsorption systems will reach Cyclic Steady StateAt 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 ofcyclesAcceleration schemes1 Non-sequential cycle calculation: extrapolation algorithms2 Model and discretisation switch3 Interpolation of the starting conditionsSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionAcceleration to Cyclic Steady State extrapolationExtrapolate the next solution from the previous 2m solutions1 Set x0 = yi2 Generate xi+1 = F (xi )3 Apply  algorithm4 Set yi+1 = (0)2m algorithm:r−1 = 0r0 = xrrs+1 = r+1s−1 +((r+1)s − (r)s)−1000110 0211 0320 12 0421 1330 223140m depends on the problem eigenvaluesFor the DP-PSA m = 2 or 3Switch to subsequent substitution closeto CSSSkelboe, IEEE Transactions on Circuitsand Systems, 27, 3, 1980Simulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionAcceleration to Cyclic Steady State extrapolation: Convergence to CSSConvergence to CSSSubsequent substitution: linear convergence extrapolation: quadratic convergenceIn the studied cases 35% faster convergence0 20 40 60 80 10010−410−310−2Comparison of the approach to CSSCycle numberCyclic difference  Subsequent substitutionε extrapolation0 20 40 60 80 1000.20.30.40.50.6Approach to CSS in PSA processCycle numberMole fraction  Subsequent substitutionε extrapolationSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionModel switch and interpolationModel and discretisation switchStart with the simplest case from the model hierarchy, e.g.isothermal and LDF, and coarse discretisationAt CSS switch to the model and discretisation whichaccurately describe the problemInterpolate the coarse solution to the accurate descriptionSimulate accurate model to CSS0 0.2 0.4 0.6 0.8 100.20.40.60.81Linear interpolationColumn lengthMole fraction  Last gridNew grid Reduces simulation timeby at least an order ofmagnitudeSimpler than an adaptiveschemeSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionModel switch and interpolationInterpolation of the starting conditionsRestart from old solutionsInterpolate the initial conditons from previous runsCan reduce the simulation time to CSS by ∼ 50%Especially important for parameter estimation because manyruns of the simulation tool have to be performed0 20 40 60 80 10010−410−2100Comparison of the approach to CSSCycle numberCyclic difference  Cold startRestartRela tive s imulation  time10.5Subsequent substitutionExtrapolation to CSSRestart from last solutionSimulation withvariable grid sizeDifferent simulation schemesSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionSimulation of extended Skarstrom cyclePreliminary results for the cycle simulatorColumn code is integrated intoa cycle simulatorArbitrary sequence of stepsPreliminary results arecomparable with AdsimSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionConclusionModel hierarchy describing the adsorption process indifferent levels of detailSimulation tool for cyclic adsorption processes withtailored discretisation schemes for the fluid and solid phaseSeveral acceleration schemes which accelerate theconvergence to CSS by at least an order of magnitudeApplied the simulation tool to the simulation of the DualPiston - PSA systemFuture workImplement all models from the model hierarchyAutomated choice of the correct modelParameter estimation from experimental dataSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionAcknowledgementFinancial supportEPSRC grant ’Innovative Gas Separations for CarbonCapture’, EP/G062129/1EPSRC Science and Innovation Award ’Carbon Capture fromPower Plant and Atmosphere’, EP/F034520/1Experimental work275d, Total Reflux Dual Piston-PSA: a Novel Apparatus forTesting Adsorbent MaterialsWenli Dang, Daniel Friedrich, Maria-Chiara Ferrari and StefanoBrandaniSimulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionQuestions?Thank you for your attention!Questions?Simulation of Adsorption Processes Daniel FriedrichIntroduction Modelling adsorption Simulating adsorption Acceleration Cycle simulator ConclusionOrthogonal collocation on finite element methodApproximate solution by trial functions yi =∑Nk=0 y(i)k zkSubstitute trial solution into original equations R = F (yi )Set residual R to zero at the roots of orthogonal polynomialsSplit the domain into small elements to handle steep gradientsin the solutionImplementation:Arbitrary number of elementsElement order between 3 and 7Arbitrary position of elementboundarySimulation of Adsorption Processes Daniel Friedrich

Efficient Numerical Schemes for the Simulation of Adsorption Processes to Cyclic Steady State

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