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Computers and Chemical Engineering 94 (2016) 18–27

Contents lists available at ScienceDirect

Computers and Chemical Engineering

journa l homepage: www.e lsev ier .com/ locate /compchemeng

evelopment of an equilibrium theory solver applied to pressurewing adsorption cycles used in carbon capture processes

abriel Oreggioni, Daniel Friedrich, Mauro Luberti, Hyungwoong Ahn, Stefano Brandani ∗

cottish Carbon Capture and Storage, School of Engineering, The University of Edinburgh, The King’s Buildings, Edinburgh EH9 3FB, UK

r t i c l e i n f o

rticle history:eceived 21 March 2016eceived in revised form 30 June 2016ccepted 18 July 2016vailable online 21 July 2016

a b s t r a c t

An equilibrium theory simulator (Esim) for the simulation of cyclic adsorption processes is presented.The equations are solved with a Godunov upwind flux scheme that does not require either the evaluationof characteristics or shock equations or the imposition of a numerical entropy condition to track shocks.Esim is able to simulate non-trace and non-isothermal adsorption systems with any adsorption isotherm.Esim has been validated against gPROMS based simulations that use the full set of governing equations(including mass and heat transfer resistances and axial dispersion) carried out under conditions close

eywords:dsorption dynamicsquilibrium theoryyperbolic systemsodunov methodarbon capture

to the limits where equilibrium theory is valid. Esim enables the establishment of bounds for the opti-mal performance of an equilibrium driven separation and requires only the measurement of adsorptionisotherms.

© 2016 Elsevier Ltd. All rights reserved.

ressure swing adsorption

. Introduction

Pressure Swing Adsorption (PSA) cycles are widely used as gaseparation unit operations; examples of industrial applicationsnclude hydrogen purification and air separation for the produc-ion of both nitrogen and oxygen (Ruthven, 1984; Ruthven et al.,994). In the case of hydrogen purification and oxygen produc-ion the separation is based on equilibrium selectivity. The moretrongly adsorbed components are held by the solid and a highurity product is obtained. Production of nitrogen is achieved via

kinetic selectivity because oxygen has a slightly smaller criticaliameter and diffuses more rapidly in solids that have criticallyized micropores (Ruthven et al., 1994). Clearly in this case the sep-ration is operated far from equilibrium conditions, which wouldither favour nitrogen due to the larger quadrupole moment orould yield a small equilibrium selectivity if van der Waals forces

revail (Ruthven, 1984). CO2 post-combustion capture from coalred power plants using PSA can be achieved using adsorbents thathow high equilibrium selectivity (see for example Kikkinides et al.,993; Krishnamurthy et al., 2014; Webley, 2014), but in this case

ore complex process configurations are needed because carbon

ioxide is the more strongly adsorbed component (see for examplebner and Ritter, 2002; Luberti et al., 2013; Wang et al., 2013). For

∗ Corresponding author.E-mail address: s.brandani@ed.ac.uk (S. Brandani).

ttp://dx.doi.org/10.1016/j.compchemeng.2016.07.020098-1354/© 2016 Elsevier Ltd. All rights reserved.

these systems, the materials are designed to reduce mass transferresistances, which are primarily in the macropores (Hu et al., 2014).

The design and optimisation of PSA processes are complexbecause the system is intrinsically dynamic, depends on a largenumber of parameters and the simulations have to be carried outuntil the adsorption columns reach cyclic steady state (CSS) (Chenget al., 1998; Friedrich et al., 2013 and Simo et al., 2008). For equi-librium driven separations any mass transfer resistance or otherterms that lead to dispersion of the separation front will reducethe efficiency of the separation. For this reason, simulations whichneglect any effects that lead to dispersion of the separation frontcan establish optimal bounds on the performance of equilibriumdriven separations (Ruthven, 1984; Ruthven et al., 1994). To achievethis, these simulations assume instantaneous mass and heat trans-fer between the gas and solid phase and negligible axial dispersion.Thus equilibrium theory assumes that the adsorbed phase is inequilibrium with the gas phase: the adsorbed phase concentra-tion is calculated directly from the adsorption isotherm and thegas phase concentration. This simplification reduces the number ofdifferential equations since only the mass and the energy balancesin the gas phase have to be solved without mass and heat trans-fer resistances and axial dispersion terms (Ruthven, 1984). Thusequilibrium theory simulations can establish the range of optimal

conditions of the full process from a limited number of experimen-tal equilibrium measurements or molecular simulations (see forexample Banu et al., 2013).

G. Oreggioni et al. / Computers and Chem

Nomenclature

Notation Coefficient defined in Eqs. (10) and (11) (s−1)b0 Pre-exponential Arrhenius coefficient for Langmuir

isotherm linear constant (bar−1)C Total fluid phase concentration (mol/m3)c Vector of fluid phase concentrations (mol/m3)ci Concentration of component i in the fluid phase

(mol/m3)cfi

Feed concentration for component i (mol/m3)c0i

Initial concentration in the bed for component i(mol/m3)

Cp Molar heat capacity (gas phase) (J/mol K)cps Mass solid heat capacity at constant pressure

(J/kg K)D Axial diffusion coefficient (m2/s)Dc Column diameter (m)ε Bed porosityf Conservative fluxfi Conservative flux for component balance equation

(mol/m2s)Fnj− 1

2Numerical flux on the left boundary of the cell.

Fnj+ 1

2Numerical flux on the right boundary of the cell.

hf Volumetric enthalpy (J/m3)�Hi Enthalpy of adsorption for component i (J/mol)hmolar Molar enthalpy (J/mol)hrefi Reference enthalpy for component i (J/mol)kLDF Linear Driving Force constant (m/s)kTE Thermal axial dispersion (W/m K)L Column length (m)NC Number of components.Pfeed Feed pressure (bar)Pvacuum Vacuum Pressure (bar)qi Concentration of the adsorbed phase (mol/m3)qi∗ Concentration of the adsorbed phase calculated

using isotherm function (mol/m3)qi Average concentration of adsorbed phase (mol/m3)qs Adsorbent saturation capacity (mol/m3)�s Solid density (kg/m3)s Characteristic speed (m/s)�t Time step size (s)tads Adsorption step time (s)tblow Blowdown step time (s)tpress Pressurisation step time (s)tpurge Purge step time (s)T Temperature (K)Tf Feed temperature (K)T0 Initial temperature in the bed (K)Ug Volumetric internal energy (gas phase) (J/m3)Us Volumetric internal energy (adsorbent) (J/m3)v Interstitial velocity (m/s)vf Interstitial velocity at feed conditions (m/s)W Generic variable of interest.

Spatial average value for the variable of interest�x Grid size (m)f

112

non trace binary ET adsorption systems but the extension to mul-

yi

Component feed mole fraction.

Traditional PSA cycle solvers (see for example Ruthven et al.,

994; Kumar et al., 1994; Nilchan and Pantelides, 1998; Sun et al.,996; Da Silva et al., 1999; Reynolds et al., 2006; Haghpanah et al.,013) can be used to mimic the limiting ET case by including

ical Engineering 94 (2016) 18–27 19

very fast mass transfer kinetics and reducing axial dispersion. Thismethod does not take full advantage of the ET simplification butrequires the solution of the complete, larger set of equations forboth gas and solid phases and furthermore being based on the solu-tion of parabolic PDEs may become unstable, ie some dispersion inthe shocks is necessary to stabilise the numerical solution.

Equilibrium theory has been employed by several authorsto solve adsorption dynamics and the main results are wellsummarised by Ruthven (1984) and Ruthven et al. (1994). Theimportance of equilibrium theory in understanding adsorptiondynamics starts from the pioneering work of Glueckauf (1949).Subsequent major contributions can be found in the papers pub-lished by Helffreich (1967), Rhee et al. (1970, 1971), Klein (1984)and Basmadjian and Coroyannakis (1987) and in the monographby Rhee et al. (1989). In most of these articles, simplifying assump-tions such as trace adsorption separation and linear or Langmuirequilibrium based systems were considered in order to allow thesolution of the system of equations using the method of character-istics. The development of a characteristic based ET solver requiresa check for the condition of shock formation, a detailed analy-sis is needed when using different adsorption isotherms (see forexample Mazzotti, 2006). The use of equilibrium theory to PSAwas developed mainly by Knaebel and Hill (1985), Kayser andKnaebel (1989) and Pigorini and Le Van (1997a,b) who publishedanalytical solutions for isothermal trace PSA systems. Ebner andRitter (2002) modified the analytical solution presented by Kayserand Knaebel (1989) to simulate PSA cycles aimed at the concen-tration/production of the strongest adsorbing species with linearequilibrium isotherms.

For the solution of the ET models under non-linear conditions,the typical approach is the use of the method of characteristics(Ruthven, 1984). The difficulties in the application of the method ofcharacteristics for the hyperbolic problem lie in the need to derivespecific expressions for the analysis of the condition of the forma-tion of shocks. In the case of multi-component and non-isothermaladsorption systems efficient schemes are needed for the numericalsolution of the system under study, which do not require this anal-ysis and are applicable to any adsorption isotherm. Several authorssuch as Guinot (2003), Leveque (2002) and Toro (2009) have devel-oped the use of finite volume based approximate Riemann schemesin the solution of hyperbolic equations arising from the modellingof different engineering problems particularly in the field of wavepropagation in fluid dynamics. The Godunov scheme (Godunov,1962) is the simplest of these approaches since it assumes that thevariables of interest are constant along the finite volume (cell) andduring each simulation time step thus it is a first order in spaceand time scheme; higher order schemes have also been developed(see for example Beam and Warming, 1976 and Lax and Wendroff,1960). However, for higher order methods oscillations are observedwhen tracking shocks (Leveque, 2002).

In this contribution we focus on the development of a numer-ical Godunov type scheme for the solution of the pure hyperbolicequation arising from the mass and energy balance in PSA cyclesusing the equilibrium theory. The novel tool is capable of simu-lating multi-component, non-isothermal and non-trace adsorptiveseparation units. This approach has been previously used in theworks published by Loureiro and Rodrigues (1991), Lim et al. (2001)and Leveque (2002) for dilute single component isothermal sys-tems which show a single transition. In this particular case, thegas velocity remains constant and the application of the Godunovtype scheme is straightforward. Bourdarias et al. (2006) employeda wave propagation form of the Godunov method to simulate

tiple transitions (multicomponent non-isothermal systems) andarbitrary adsorption isotherms is missing. PSA processes includevelocity gradients in the adsorption and desorption steps due to

2 Chem

bap(cvt

2

ga2a

ε

ε

ε

t

a

b

de

accii

ε

ε

ε

U

p

U

s

0 G. Oreggioni et al. / Computers and

ulk adsorption LeVan et al. (1988) and industrial systems are usu-lly closer to adiabatic than isothermal operation. In this work, weresent the development of a new Godunov based numerical toolEsim) to solve dynamics associated with non-isothermal multi-omponent PSA cycles. This tool resolves the issues related to theariable velocity in the gas phase and allows the use of any adsorp-ion isotherm.

. Mathematical model

The general equations for the mass and energy balances for theas phase (assuming plug flow and not considering kinetic energy)re presented in Eqs. (1)–(3) (Ruthven, 1984; Walton and LeVan,005). Eq. (4) gives the mass balance for the solid phase assuming

Linear Driving Force (LDF) mass transfer model.

∂C∂t

+ ε∂ (vC)∂z

+∑Nc

i=1(1 − ε)

∂qi∂t

= 0 (1)

∂ci∂t

+ ε∂ (vci)∂z

+ ε∂(−DC ∂xi

∂z)

∂z+ (1 − �)

∂qi∂t

= 0 (2)

∂Ug∂t

+ ε∂(vhf

)∂z

+ ε∂(−kT ∂T∂z

)∂z

+ (1 − ε)∂Us∂t

= 0 (3)

∂qi∂t

= kpi

ApVp

(qi∗ − qi) (4)

These equations can be simplified due to the following assump-ions:

) Negligible axial mass and thermal dispersion (equilibrium the-ory).

) No mass or heat transfer resistances (equilibrium theory).c) The solid density and the heat capacity for the solid phase

are independent of temperature and adsorbed concentration(Ruthven, 1984).

) The gas phase is considered ideal.) No pressure drop along the column.

The dispersion terms in Eqs. (2) and (3) can be removed bypplying assumption (a). By using assumption (b) the averageoncentration in the adsorbed phase qi can be replaced by theoncentration of the adsorbed phase calculated directly from thesotherm qi

∗. The resulting system of hyperbolic PDEs implementedn Esim is given by:

∂C∂t

+ ε∂ (vC)∂z

+ (1 − ε)∑Nc

i=1

∂q∗i

∂t= 0 (5)

∂ci∂t

+ ε∂ (vci)∂z

+ (1 − ε)∂qi∗

∂t= 0 (6)

∂Ug∂t

+ ε∂(vhf

)∂z

+ (1 − ε)∂Us∂t

= 0 (7)

The internal energy of the solid is calculated from

s = �scpsT −∑Nc

i=1(−�Hi)qi∗ (8)

While the enthalpy and the internal energy for the gaseoushase can be written as:

hf = hmolar C =(∑Nc

k=1

(yi

∫ T

Tref

Cpi (T) + hrefi

))C (9)

g = (hmolar − RT)C (10)

For the non-isobaric steps (pressurisation, evacuation and pres-ure equalisation), it is assumed that the pressure change with

ical Engineering 94 (2016) 18–27

time is proportional to the difference between the pressure in thecolumn and the sources/sinks:

∂P∂t

= ˛ (P − Pvacuum) (11)

∂P∂t

= ˛(Pfeed − P

)(12)

The inlet flow is a function of the difference between the pres-sure in the column and the source/sink during the pressurisationor blowdown step. The interstitial inlet velocity is one of theunknowns to be determined from the global mass balance for thefeed (pressurisation) or product (blowdown) side of the adsorp-tion column (schematics for the column can be found in Fig. 1). Themodel equations include the adsorption isotherm, which in generalterms can be expressed as

qi∗ = f (T,

¯c) (13)

and the ideal gas law

P = CRT (14)

Since the mass and the energy balances are a hyperbolic sys-tem of equations, only one sided boundary conditions are neededand they correspond to the values of the feed concentrations andtemperature.

3. Numerical schemes for the simulation of adsorptiondynamics under equilibrium theory

3.1. Finite volume method applied to hyperbolic systems

The simplifying assumptions introduced in section 2 lead toa system of hyperbolic equations which need to be solved bynumerical methods. The set of hyperbolic equations that rep-resent the mass and energy balances in the adsorption columnand its associated piecewise initial conditions are an exampleof a Riemann-Cauchy problem (Leveque, 2002); thus numeri-cal methods such as Finite Volume Riemann solvers of the REA(reconstruct-evolve-average) type can be used for the simulationof the equilibrium theory model of PSA processes. The adsorptioncolumn is divided into cells each of which is centred on a node zj.Along each cell the variable of interest is defined by a polynomialfunction (reconstruction). In the simplest case, this polynomial is aconstant function with the average of the variable of interest alongthis cell. The Riemann problem is solved by using the values at theboundary of the cells at the current time step (Wn

j− 12

) (evolution) to

obtain the average value along the cell at the next time step (). Fig. 1 illustrates how the REA type algorithms work. In the toppart of the figure, the column profile at time t is displayed: a con-

stant function is assumed and consequently Wnj− 1

2is equal to ;

the wave front evolves (as shown by the lines overlapping the cellboundaries in the bottom part of the figure) and a new average forthe variable of interest can be obtained at t + dt as shown by thedashed lines.

The Riemann solvers can use a wave propagation or a numericalflux approach (Leveque, 2002). A deeper analysis of the applicationof these two approaches for the problem under study is discussedin Sections 3.2 and 3.3.

3.2. Wave propagation approach

In the wave propagation approach, the temporal evolution of the

average value of the variable of interest ( ) is due to waves thatpropagate along the column. The wave speed (sn

j− 12

) is calculated

G. Oreggioni et al. / Computers and Chemical Engineering 94 (2016) 18–27 21

⁄⁄

⁄ ⁄0

Feed end

Product end

Feed flow

Product flow

Ini�al average valu e

New averagevalu e

Cell Centre

Reconstruc�on

Evolu�on and averaging

e solv

ucd

iedoi

a

ε

w

EaTdotwsstc

ftetei

Fig. 1. Illustration of REA typ

sing characteristics (smooth solution) or the Rankine-Hugoniotondition (shocks) plus an entropy condition which is needed toetermine when a shock takes place (Leveque, 2002).

The equilibrium theory introduces an algebraic constraint whichs used to replace ql with the equilibrium amount q∗

iin the mass and

nergy balances (see Eqs. (5) and (6)). Applying the chain rule ofifferentiation for a generic isotherm in which q∗

iis only a function

f the concentration of the species in the gas phase (ci), the changen adsorbed concentration can be calculated from:

∂q∗i

∂t=

∑Nc

j=1

∂qi∗

∂cj

∂cj∂t

= ∂ci∂t

(∂qi∗

∂ci+

∑Nc

j /= i

∂qi∗

∂cj

∂cj∂ci

)(15)

nd the mass balance becomes:

∂ci∂t

+ ε∂ (vci)∂z

+ (1 − ε)∂ci∂t

(∂qi∗

∂ci+

∑Nc

j /= i

∂qi∗

∂cj

)= 0 (16)

hich can be rearranged to

∂ci∂t

+ 1(1 + (1−ε)

ε

(∂qi∗∂ci

+ �Ncj /= i

∂qi∗∂cj

)) ∂ (vci)∂z

= 0 (17)

q. (17) shows that in the case of using the wave propagationpproach, the wave front speed depends strongly on the isotherm.hus for these solvers, the expression multiplying the spatialerivative has to be evaluated for the values of the concentrationn the right and on the left of the cell boundaries in order to checkhat the characteristics do not increase along the column since thisill lead to shock formation. Furthermore, in the case of non-trace

ystems, the interstitial velocity changes along the column due toorption effects and thus the global mass balance has to be solvedo obtain the evolution of the interstitial velocity v with ci in thease of smooth solutions.

Bourdarias et al. (2006) used a Riemann wave propagation solveror a binary, non-trace isothermal adsorption systems in whichhe species obey the Langmuir equilibrium model for an initially

mpty bed. In this case, a single transition is observed and givenhe absence of an initial profile, the entropy condition is definedxclusively by the values of the concentration of the feed and thenitial condition of the bed; thus for this special case Eq. (17) doesn’t

ers and column schematics.

need to be evaluated to check whether a shock is taking place. How-ever, for smooth solutions a differential equation has to be solvedto determine the change of velocity due to the sorption effects.

For a generic multi-component solver applicable to anyisotherm, Eq. (17) as well as the differential equations arisingfrom the global mass balance need to be evaluated in every cell.For isothermal isobaric adsorption systems that obey the Lang-muir equilibrium model where there is an initial profile in thecolumn or for non-isothermal or non-isobaric sorption, Eq. (17)must be evaluated to determine the formation of shocks; thus thewave propagation scheme leads to complicated and computation-ally expensive algebraic-differential equations also for the case ofa smooth solution. Nevertheless, these numerical schemes predictaccurately the shock formation and the convergence is always guar-anteed.

3.3. Numerical flux approach

For hyperbolic equations that can be written in a conservativeform (Eq. (18)), the temporal evolution of the variable of interestis caused by the gradient of a flux of the same or other variablesalong the cells thus it is possible to estimate the average value of thevariable of interest at the next time step using a defined numericalflux function (see below). The updating expressions for the numer-ical approach are capable of tracking shock and smooth transitions;neither the imposition of a numerical entropy condition nor the cal-culation of the characteristics are required thus fewer algebraic oralgebraic-differential equations must be solved. However, a carefulselection of the simulation step and cell size must be carried out toguarantee stability (imposition of stability conditions).

∂W (x, t)∂t

+ ∂f (x, t, W)∂x

= 0 (18)

W is the variable of interest, Wnj

is the spatial average value forW along the cell at one simulation step time and f is the conser-

vative flux which can depend on W . Different expressions for theboundary fluxes Fn

j± 12

as functions of f can be found in the literature,

(for example refer to: Godunov, 1962; Lax and Wendroff, 1960; andBeam and Warming, 1976). Godunov (1962) proposed a numerical

2 Chemical Engineering 94 (2016) 18–27

so

pfl(s

F

W

Be

Bpls

3a

s(gt

W

W

W

(

f

f

f

at

F

F

su

W

W

2 G. Oreggioni et al. / Computers and

cheme in which the fluxes at the cell boundaries (Eq. (19)) dependn the average value of the variable of interest (Eq. (20)) along the

revious cell (for positive flow direction). and the numericalux are assumed to be constant during the simulation step time�t) thus the resulting numerical scheme is first order in time andpace.

nj−1/2 = 1

�t

∫ tn+1

tn

f(xj− 1

2, t, W

(xj− 1

2, t

))dt

= f(xj− 1

2, t, Wn

j−1

)(19)

˜nj

= 1�x

∫ xj+1/2

xj−1/2

W (x, t)dx (20)

y integrating Eq. (18) and using Eqs. (19) and (20), the followingxpression for the updating formula can be derived:

(21)

y applying Eq. (21) to the governing equations of the system, it isossible to convert the coupled PDEs into a system of algebraic non-

inear equations that must be solved in each cell at every simulationtep time by using a root finding routine.

.4. Godunov numerical flux approach solvers applied todsorption dynamics

The Godunov numerical flux approach must be applied to con-ervative hyperbolic equations; a conservative variable changeEqs. (22)–(24)) has been used to transform the component, thelobal and the energy balances into a suitable form for the applica-ion of Godunov’s method.

i = ci +(

1 − ε

ε

)qi (22)

T = C +(

1 − ε

ε

∑Nc

i=1qi

)(23)

TE = εUg + ((1 − ε)Us) (24)

The conservative fluxes can be defined as shown in Eqs.25)–(27) respectively.

i = vci (25)

T = vC (26)

TE = εvhf (27)

One of the assumptions of the Godunov method is that the aver-ge value of the variables of interest is constant during the time stephus the numerical fluxes can be written:

ni,j−1/2 = vnj−1

˜cnlJ−1

(28)

nT,j−1/2 = vnj−1

˜cnJ−1 (29)

FnTE,j−1/2 = ε v ˜hfJ−1 (30)

By applying Godunov’s method (Eq. (21)) to the conservativeystem defined by Eqs. (22)–(30), it is possible to arrive to thepdating formula defined by Eqs. (31)–(33)(

�t ( ))

n+1lJ

= WnlJ

−�x

Fni j+1/2 − Fni j − 1/2 (31)

n+1TJ = Wn

TJ −(�t

�x

(FnTj+1/2 − FnTj−1/2

))(32)

Fig. 2. Illustration of Riemann problem associated to non-trace adsorption dynam-ics.

˜Wn+1TEJ

= WnTEJ

−(�t

�x

(FnTE,j+1/2

− FnTE,j−1/2

))(33)

The unknowns in each cell are the concentration of the speciesat the next time step (ci

n+1j

) and the velocity (vnj) at the current time

step. As shown in Fig. 2, the interstitial velocity entering the cell j attime t is known as well as the concentrations of the species. Part ofthe molecules of the gas are adsorbed thus the interstitial velocityfor the gas entering to the next cell is an unknown to be determinedas well as the concentration of the species (circled variables) at thetime that the front leaves the cell (t + dt). The Equilibrium Theorysimulator (Esim) obtains the values of these variables by solvingthe nonlinear system of equations given by Eqs. (22)–(33). Esimhas been programmed using the MATLAB software platform andthe non-linear equations arising from the discretisation have beensolved by employing the FSOLVE (MATLAB, 2015) root finding rou-tine. The FSOLVE function is based on quasi newton method for thesolution of algebraic non-linear equations.

4. Breakthrough curve simulations

The results generated by Esim have been validated by compar-ing them with simulations produced using gPROMS in which thefull governing equations for the gas phase and the LDF model for thesolid phase have been implemented (Nilchan and Pantelides, 1998).The axial dimension has been discretised using an orthogonal collo-cation finite element scheme, which is suitable for reversing flows,with a 3rd order polynomial in each element, i.e. a discretisationwith 10 elements has 31 nodes. The assumptions of the equilib-rium theory are approximated in the gPROMS code by using verysmall axial dispersion coefficients (1000 times smaller than thoseestimated from correlations, see Ruthven, 1984) and large masstransfer kLDF values (approx. 10 times larger than the correspond-ing macropore diffusion control values, see Hu et al., 2014). Highervalues lead to numerical instabilities given that a shock is the truesolution in an adsorption step with a favourable type I isotherm(Ruthven, 1984).

It must be emphasised that this comparison is intended to vali-date the correct implementation of the novel tool, given the lack ofanalytical expressions for the solution of the system under study.The results of the isothermal (Fig. 3) and non-isothermal (Figs. 4–6)

G. Oreggioni et al. / Computers and Chemical Engineering 94 (2016) 18–27 23

Fig. 3. Comparison of concentration breakthrough curves betw

Fig. 4. Concentration and temperature breakthrough curve for the non-isothermalcase.

Fig. 5. Comparison of concentration breakthrough curves between Esim a

een Esim and the gPROMS simulations (isothermal case).

breakthrough simulations show clearly the propagating fronts. Forthe non-isothermal case, two transitions (Fig. 4) can be observed:the first one corresponds to a thermal and concentration shock(Figs. 5 and 6) at approx. 77 s for which the CO2 mole fractionreaches an intermediate plateau. At about 1000 s, a smooth tran-sition takes place because during the front breakthrough there isstill cool gas entering to the column thus a reduction of temperatureoccurs enabling more CO2molecules to be adsorbed. Consequently,after the second transition, the bed is saturated at feed conditions(yfCO2

= 0.15, CfCO2= 9.08 mol/m3, T = 298 K). For both the isothermal

and non-isothermal case Esim manages to track the shocks properlysince the breakthrough curves for the sharp transitions present thesame first order moment. It can be observed that Esim manages tocapture the shock with less smoothing for an equal number of spa-tial domain discretisation units (150 cells in finite volume methodand 51 elements in OCFEM order 3)

The simulation parameters are shown in Table 1, the columndimensions as well as the feed flow correspond to a lab scale PSA rig,the feed compositions replicate the exhaust gas of a coal fired boilerand the adsorbent properties have been taken from Kikkinides et al.

nd the gPROMS simulations (non-isothermal case, first transition).

24 G. Oreggioni et al. / Computers and Chemical Engineering 94 (2016) 18–27

Fig. 6. Comparison of temperature breakthrough curves between Esim and the gPROMS simulations (non-isothermal, first transition).

Table 1Parameters used for the simulation of the breakthrough curve.

Variable Value Variable Value

L (m) 0.5 qs(

mol/m3)

2233

Dc (m) 0.025 boCO2

(bar−1

)8.41e−6

ε 0.4 boN2

(bar−1

)2.79e−4

Feed Flow (mol/s) 0.0051 −�HCO2

(J/mol

)30558

yfCO2

0.15 −�HN2

(J/mol

)15907

yfN2

0.85 CpaCO2

(J/mol K

)a 24.47

Tf (K) 298 CpbCO2

(J/molK2

)0.024

�s(

kg/m3)

708 CpaN2

(J/mol K

)29.41

cps(

J/kg K)

1046 CpbN2

(J/molK2

)−0.00249

tads = tpurge 100 tpress = tblow 100

a

lC

(trR

b

q

spsf

�

w

The gas phase heat capacities from Reid et al. (1987) are approximated with ainear relationship (R = 0.99) in the temperature range of interest: Cpi (T) = Cpai +pbiT .

1993). The Langmuir isotherms have been recalculated imposinghat the saturation capacity is kept constant for all componentsespecting thermodynamic consistency (Rao and Sircar, 1999 anduthven, 1984).

i = b0i e

−(�HiRT

)(34)

i =qsbiPyi

1 + �Ncj=1bjPyj

(35)

Given that the numerical scheme is trying to capture a purehock, the resulting smooth profile is a result of the numerical dis-ersion. The numerical dispersion of the isothermal breakthroughimulations has been quantified by calculating the second momentrom the breakthrough curve (Ruthven, 1984):

2 = 2

∫ ∞

0

(1 − ci (t)

cfeed

)tdt − 2 (36)

here is the first order moment given by

=∫ ∞

0

(1 − ci (t)

cfeed

)dt (37)

Fig. 7. Esim trace system isothermal breakthrough simulations using 50, 100, 150,200 and 400 grids.

These equations are strictly valid when the concentration of theadsorbing species is very low (trace isothermal system and the fluidvelocity is constant) thus breakthrough curve simulations in whichthe CO2 mole fraction is 0.01 with the balance being an inert gaswere used. The isothermal breakthrough curves for a trace adsorp-tion system have been simulated using 100, 150, 200 and 400 cellsin Esim (Fig. 7). The values for the second order moments for eachcurve are plotted in Fig. 8 showing that the dispersion is inverselyproportional to the square of the number of cells.

5. Cycle simulation and convergence to steady state

Esim has been extended to the solution of PSA cycle dynam-ics. The cyclic steady state results for a non-isothermal non-tracePSA cycle generated by Esim have been compared with the simu-

lations for the same cycle configuration using gPROMS to validatethe correct implementation of all the steps in a cycle. The cycle evo-lution for pressure, CO2 mole fraction and temperature at the feedend of the column at CSS are presented in Fig. 9a–c respectively,

G. Oreggioni et al. / Computers and Chem

F

sctaaFacafi

6

h

determination of the limiting Pareto fronts for the optimization of

ig. 8. Relationship between the second order moment and the number of grids.

howing very good agreement of the results produced by the twoodes. Since Esim simulates a pure equilibrium theory based sys-em cyclic steady state is achieved with a lower number of cycless seen in Fig. 10a and b. For the gPROMS simulations, a realisticxial dispersion and thermal conductivity coefficients (Wen andan, 1975) are used thus smoothing the concentration and temper-ture fronts and reducing numerical instabilities. The comparisononfirms that Esim provides reliable predictions for full PSA cyclesnd that a Godunov type scheme has been implemented for therst time to this complex dynamic system.

. Conclusions

In this work, a new Equilibrium Theory PSA cycle solver (Esim)as been introduced. Esim allows the estimation of the equilib-

Fig. 9. Comparison of pressure (a), temperature (b) and mole fraction (c) tempo

ical Engineering 94 (2016) 18–27 25

rium limiting performance for a given adsorption process and theidentification of a set of high separation efficiency configurationsby just using the isotherm equilibrium function as input data. Thesimplifications associated with the Equilibrium Theory and the factof considering axial dispersion to be negligible convert the massand the energy balances in an adsorption column into a systemof hyperbolic equations. A conservative variable change is definedenabling the use of the Godunov numerical flux. This removes theneed of imposing a numerical entropy condition or of calculatingthe characteristics in each cell.

Esim results have been validated by comparison against sim-ulations using gPROMS in which simulations were undertakenby using micropore equilibrium and macropore LDF model. Theassumption of the Equilibrium Theory has been emulated byemploying very large values for the kLDF constant and very smallvalues for the axial dispersion coefficients. It can also be observedthat Esim manages to properly track shocks and smooth transitionssince the first order moment for the breakthrough curves generatedby Esim are equal to the ones corresponding to the curves producedby the gPROMS based solver; showing the correct implementationof the novel tool.

The convergence to CSS for the new solver has been demon-strated by simulating a non-isothermal non trace binary adsorptionequilibrium based system under conditions of interest in CO2 postcombustion capture technology. Cyclic pressure, CO2 mole fractionand temperature evolution have been compared against the onesproduced by the gPROMS based code and confirm the correct imple-mentation of the new solver to the simulation of full PSA cycles. Thiscomparison also shows that the ET solver requires fewer cycles toreach CSS in comparison with full governing equations thus it indi-cates that ET simulator can in principle be used for the efficient

equilibrium based PSA separations.It can be concluded that the new solver introduced in this work

reduces the complexity of the modelling of adsorption dynamics by

ral evolution in the last simulated cycle in which steady state is reached.

26 G. Oreggioni et al. / Computers and Chemical Engineering 94 (2016) 18–27

2 reco

rtflwatfls

R

B

B

B

B

C

D

E

F

G

G

G

Fig. 10. Comparison of cyclic CO

equiring less empirical data and a simpler code in order to under-ake a quick scan of operating conditions and cycle configurationsor the design of PSA cycles. Its implementation enables the simu-ation of generic adsorption systems with any adsorption isotherm

ithout modifying the main code of the solver. The new tool also is proof of concept of the application of numerical flux method forhe solution of complex adsorption dynamics with rapidly varyingows, presenting a novel approach for the modelling of this kind ofeparation process.

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