compartmental modeling of high purity air … modeling of high purity air separation columns ......

Computers and Chemical Engineering 29 (2005) 2096–2109

Compartmental modeling of high purity air separation columns

Shoujun Biana, Suabtragool Khowinija, Michael A. Hensona,∗,Paul Belangerb, Lawrence Meganb

a Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003-9303, USAb Process and Systems Research and Development, Praxair, Inc., Tonawanda, NY 14051-7891, USA

Received 6 October 2004; received in revised form 27 May 2005; accepted 6 June 2005Available online 12 July 2005

Abstract

High purity distillation columns are critical unit operations in cryogenic air separation plants. The development of nonlinear controltechnology is motivated by the need to frequently change production rates in response to time varying utility costs. Detailed column modelsbased on stage-by-stage balance equations are too complex to be incorporated directly into optimization-based strategies such as nonlinearmodel predictive control. In this paper, we develop reduced order dynamic models for the upper column of a cryogenic air separation plantby applying time scale arguments to a detailed stage-by-stage model that includes mass and energy balances and accounts for non-idealv ithdrawals a physicallyb nificantlyf ics (AspenT nts is varied.©

K

1

ccLtacsttpltMs

-therial

in-tureim-

hichtized

iza-

s de-on-eat-blyller

d-ation

0d

apor–liquid equilibrium. The column is divided into compartments according to the locations of liquid distributors and feed and wtreams. The differential equations describing each compartment are placed in singularly perturbed form through the application ofased coordinate transformation. Application of singular perturbation theory yields a differential–algebraic equation model with sig

ewer differential variables than the original stage-by-stage model. A rigorous column simulator constructed using Aspen Dynamechnology) is used to access the tradeoff between reduced order model complexity and accuracy as the number of compartme2005 Elsevier Ltd. All rights reserved.

eywords: Air separation; Distillation column; Dynamic modeling; Nonlinear control

. Introduction

Linear model predictive control (LMPC) is the most suc-essful multivariable control technology in the chemical pro-ess industries (Qin & Badgwell, 1997). The success ofMPC is largely attributable to its optimization formula-

ion that allows process constraints to be accommodated insystematic fashion. The inherent assumption of linear pro-

ess dynamics restricts the application of LMPC to relativelymall operating regimes. Nonlinear control technology is of-en required for highly nonlinear processes that are subjecto large and/or frequent changes in operating conditions. Arototypical example of this small but important class of non-

inear processes is continuous polymerization reactors usedo produce multiple polymer grades (Debling et al., 1994;cAuley & MacGregor, 1992). While other controller design

trategies such as feedback linearization (Henson & Seborg,

∗ Corresponding author. Tel.: +1 413 545 3481; fax: +1 413 545 1647.E-mail address: [email protected] (M.A. Henson).

1997, chap. 4;Kravaris & Kantor, 1990) are available, nonlinear model predictive control (NMPC) appears to bemost promising nonlinear control technology for industapplications (Allgower & Zheng, 2000).

NMPC is an extension of LMPC in which a nonlear dynamic model is used to predict and optimize fuprocess performance. NMPC controllers are commonlyplemented using a simultaneous solution strategy in wthe dynamic model equations are temporally discreand posed as equality constraints in a nonlinear optimtion problem (Henson, 1998; Meadows & Rawlings, 1997,chap. 5). Future input and state variables are treated acision variables, thereby producing a potentially large nlinear programming problem that must be solved repedly in real time. Computational complexity is inextricalinked to the nonlinear dynamic model used for controdesign. Commercial NMPC technology (Qin & Badgwell,1998) is effectively restricted to low-order nonlinear moels such as those commonly developed for polymerizreactors.

098-1354/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2005.06.002

S. Bian et al. / Computers and Chemical Engineering 29 (2005) 2096–2109 2097

High purity distillation columns are an important class ofnonlinear processes that are not directly amenable to the ap-plication of NMPC technology due to the high-order nature ofstage-by-stage balance models(Luyben, 1973). Two imple-mentation strategies have been pursued: (1) the developmentof sophisticated solution methods that allow direct incorpo-ration of detailed column models and (2) the derivation ofreduced order controller design models from stage-by-stagemodels to reduce the computational requirements. The firstapproach exploits the large number of algebraic variablestypically present in differential–algebraic equation modelsof distillation columns (Leineweber, 1998). A significantfraction of NMPC computational overhead is attributableto calculation of the Jacobian and Hessian matrices. TheNMPC optimization problem can be decomposed such thatmatrix elements associated with the algebraic variablescan be excluded from these calculations. When combinedwith other computational enhancements including multipleshooting and reduced successive quadratic programming,the decomposition strategy allows the application of NMPCto stage-by-stage balance models of moderate complex-ity (Kronseder, von Stryk, Bulirsch, & Kroner, 2001; Nagyet al., 2000). However, the applicability of this approach isultimately limited by the complexity of the column model.

The derivation of reduced order column models from moredetailed stage-by-stage models has received considerable at-t rderm andc eth-o for-m tionpA bero um-b usedo col-u ,1 ot ollo-c lud-i& ndt lumnl l ac-c in-a teepp wals

miset avef 6M -t qui-l olarh itiont cted

can be derived. Unfortunately, the simplifying assumptionsthat underpin nonlinear wave theory are rarely satisfied inpractice. For example, wave models are incapable of ac-counting for end effects and producing the self-sharpeningbehavior of non-ideal columns. As a result wave models havelimited predictive capability.

Compartmental models are derived directly from stage-by-stage balance models by dividing the column into a smallnumber of sections termed compartments (Benallou, Se-borg, & Mellichamp, 1986; Horton, Bequette & Edgar, 1991;Levine & Rouchon, 1991). A dynamic model of each com-partment is developed by combining stage-by-stage balanceswith overall balances over the entire compartment. If thenumber of stages in the compartment is sufficiently large andeach stage has a comparable liquid holdup, then the overalldynamics of the compartment are much slower than the dy-namics of any individual stage within the compartment. Thistime scale separation allows the compartment dynamics to beapproximated with the differential equations for a represen-tative stage whose holdup is equal to the total compartmentholdup (Levine & Rouchon, 1991). The balance equationsfor other stages within the compartment are reduced to alge-braic equations due to their relatively fast dynamics. Advan-tages of the compartmental modeling approach include: (1)perfect steady-state agreement with the associated stage-by-stage model is ensured; (2) a suitable tradeoff between modelc just-i tureo

fewd entd raice MPCs braicv rallyd ed ase van-t raice ientN raice ,1 .,2 isp eallys er ofa atrix.

earm ionc atedb onset f thee ont non-l -p inp ased

ention. While other methods are available, reduced oodeling techniques based on collocation, wave theory

ompartmentalization are representative. Collocation mds involve the construction of a spatial mesh and theulation of dynamic balance equations at the collocaoints (Levien, Stewart, & Morari, 1985; Yang & Lee, 1997).reduced order model is obtained by choosing the num

f collocation points to be considerably less than the ner of separation stages. Many studies are strictly focn steady-state collocation models used for distillationmn design (Seferlis & Grievink, 2001; Seferlis & Hrymak994a, 1994b; Swartz & Stewart, 1986). As compared t

he compartmentalization approach discussed below, cation techniques suffer from a number of limitations incng (Carta, Tola, Servida, & Morbidelli, 1995a, 1995b; Yang

Lee, 1997): (1) the collocation points may not correspoo actual separation stages; (2) state variables at other coocations must be determined by interpolation; (3) modeuracy is limited by the number of collocation points; (4)ccurate predictions can be obtained for columns with srofiles and/or discontinuities due to feed and withdratreams.

The wave modeling approach is based on the prehat the column profile can be described by a moving wront with constant pattern (Kienle, 2000; Marquardt, 198;arquardt & Amrhein, 1994). Under simplifying assump

ions including binary separation, ideal vapor–liquid eibrium, constant equimolar overflow and constant moldups, a single differential equation for the wave pos

hat allows the entire composition profile to be reconstru

omplexity and dynamic accuracy can be achieved by adng the number of compartments; (3) the triangular strucf the original stage-by-stage model is retained.

Compartmental models are comprised of relativelyifferential equations representing the overall compartmynamics and a potentially large number of algebquations derived from the dynamic stage balances. Nimultaneous solution methods require that the algeariables be included as decision variables and tempoiscretized versions of the algebraic equations be posquality constraints. Consequently the computational ad

age of reducing most of the differential equations to algebquations is not readily apparent. Recently, highly efficMPC solution methods for sparse differential–algebquation (DAE) models have been developed (Leineweber998) and applied to distillation columns (Kronseder et al001; Nagy et al., 2000). Although beyond the scope of thaper, we envision that compartmental models are iduited for these solution methods due to the large numblgebraic variables and the sparseness of the Jacobian m

We are particularly interested in developing nonlinodeling and control technology for cryogenic distillat

olumns in air separation plants. This research is motivy the need to frequently adjust production rates in resp

o time varying electricity costs caused by deregulation olectrical utility industry. Our previous work has focused

he derivation of reduced order nonlinear models usinginear wave theory (Zhu, Henson, & Megan, 2001) and comartmentalization (Khowinij, Henson, Belanger, & Megan,ress) as well as the development of a NMPC strategy b

2098 S. Bian et al. / Computers and Chemical Engineering 29 (2005) 2096–2109

on the wave model (Bian, Henson, Belanger, & Megan, 2005)for a nitrogen purification column. The purpose of this paperis to evaluate reduced order compartmental models for theupper column of a double column plant used to produce bothnitrogen and oxygen products. A longer term goal is to uti-lize these compartmental models to design NMPC controllersthrough the application of customized solution methods thatexploit the DAE model structure (Leineweber, 1998).

The remainder of the paper is organized as follows. In Sec-tion2, our previous research on nitrogen purification columnsis briefly reviewed to provide context for the present work.The development of an Aspen Dynamics simulator, a stage-by-stage balance model and reduced order compartmentalmodels for the upper column is discussed in Section3. Sec-tion4contains simulation results that demonstrate the relativeperformance of the various models. Finally, a summary andconclusions are presented in Section5.

2. Previous work on nitrogen purification columns

Our previous research on reduced order modeling of cryo-genic air separation plants focused on the nitrogen purifica-tion column shown inFig. 1. The column has 42 equilibriumstages, a distributor located in the middle of the column toimprove liquid flow characteristics, a sump located at theb orya tiallyfl apors ducet ,2 uids n ast col-uw logy)a lations

ls oft the-o le om-p res-s wavep tob ngess entiald penc duet tionn larged bilityt com-b roacht avem

Recently we have developed reduced order compartmentalmodels that provided a considerably more detailed descrip-tion of the nitrogen purification column dynamics than waspossible with the wave modeling approach (Khowinij et al.,in press). The stage-by-stage model used for compartmentalmodel derivation included ternary component balances andliquid hydraulic relations. The condenser, liquid distributorand feed stage were treated as single stage compartmentsdue to their relatively large liquid holdups. Two reduced or-der models were formulated by dividing the column sectionsabove and below the distributor into two compartments (fivecompartment model) or four compartments (seven compart-ment model). For example, the stage-by-stage model con-sisting of 130 differential equations was approximated bythe seven compartment model with 19 differential equationsand 111 algebraic equations. The two compartmental modelswere dynamically simulated using the DAE solver availablein MATLAB. Each model produced close agreement with thestage-by-stage model and satisfactory predictions as com-pared to the considerably more detailed Aspen simulator. Atypical simulation time for the seven compartment model wasapproximately 20% of that required for the stage-by-stagemodel.

In this paper, compartmental models for the upper columnof the double column air separation plant shown inFig. 1are developed and evaluated. This work is an essential stept linga mna archo sentw

1 mnams

2 el forp-

tantces

3 tal-and

3

owni ilera umn( s twoa stage3 s thec upperc atedc from

ottom of the column which holds a large liquid inventnd an integrated condenser/reboiler in which the parashed bottom stream is vaporized and the nitrogen vtream from the top of the column is condensed to prohe reflux and liquid nitrogen product streams (Zhu et al.001). The feed is introduced immediately above the liqump, and a portion of the overhead stream is withdrawhe gaseous nitrogen product. A dynamic simulator of themn and the integrated condenser/reboiler (Bian et al., 2005)as developed using Aspen Dynamics (Aspen Technond used as a surrogate for the nitrogen plant in our simutudies.

We have developed reduced order nonlinear modehe nitrogen purification column using nonlinear wavery (Bian et al., 2005; Zhu et al., 2001). A single differentiaquation was used to track the position of the nitrogen cosition wave front. When combined with a profile expion that approximates the wave pattern, the predictedosition allowed the entire nitrogen composition profilee constructed. Simulation tests for production rate chahowed that the nonlinear wave model captured the essynamic behavior of the Aspen model. However, the Asomposition profiles exhibited self sharpening behavioro nonlinear equilibrium relations as well as wave distorear the ends of the column. The wave model producedynamic and steady-state prediction errors due to its ina

o capture such non-ideal effects. We have shown thatined state and parameter estimation is a feasible app

o improve the predictive capability of the nonlinear wodel (Bian et al., 2005).

owards our long-term goal of developing nonlinear modend control technology for double column and triple coluir separation plants. With respect to our previous resen nitrogen purification columns, novel aspects of the preork include:

. Construction of an Aspen simulator of the upper coluthat accounts for the multiple feed and withdrawal streand liquid distributors located along the column.

. Development of a more accurate stage-by-stage modcompartmental model derivation. Simplifying assumtions including constant equimolar overflow and consrelative volatility are relaxed by adding energy balanand non-ideal vapor–liquid equilibrium relations.

. A more detailed investigation of different compartmenization strategies on reduced order model complexitycomputational efficiency.

. Upper column modeling

A schematic representation of the upper column is shn Fig. 1. The column has 59 stages including the rebond seven liquid distributors located throughout the colstages 1, 10, 18, 26, 35, 43 and 51). The column receiveir feed streams (liquid air on stage 18 and turbine air on5) following compression and expansion that achieveryogenic temperatures necessary for separation. Theolumn is coupled to the lower column through the integrondenser/reboiler, the reflux stream on stage 1 obtained


Fig. 1. Schematic diagram of a typical double column air separation plant.

the overhead of the lower column and kettle liquid feed ob-tained from the bottom of the lower column. The nitrogenproduct is withdrawn from stage 1 and a gaseous nitrogenwaste stream is withdrawn from stage 10. Because the focusof this paper is upper column modeling, the lower column andthe compression, expansion and heat exchange equipment areneglected. Therefore, the scope is limited to the upper columnwith reboiler, and the various feed and withdrawal streamsare treated as independent inputs.

3.1. Aspen simulator

A detailed dynamic simulator of the upper column wasdeveloped using Aspen Dynamics (Aspen Technology). TheAspen column modelRadFrac was used to solve the dy-namic component balances and steady-state energy balancesfor each stage. Modeling of the lower column was avoidedby treating the integrated condenser/reboiler as a simple re-boiler with a constant condenser side temperature. The ther-modynamic models used were NRTL for the liquid phaseand Peng–Robinson for the vapor phase. Proprietary ther-modynamic property data for the air components (nitrogen,

oxygen and argon) were provided by Praxair. PID controllerswere implemented for regulation of the reboiler level and theoverhead pressure. Equipment specifications and the steady-state operating conditions listed inTable 1were obtainedfrom a typical Praxair double column air separation plant.The Aspen simulator was used to represent the upper col-umn in our modeling studies. To evaluate the compartmentalmodels over a reasonable range of operating conditions, twoother steady states corresponding to liquid air feed flow ratechanges of±25% from the nominal value were also investi-gated. This feed stream was chosen for manipulation becauseit represents the major source of air to the column and is in-dependent of the unmodeled lower column. A more realisticdisturbance caused by a production rate change would requirean integrated model of the lower and upper columns that isbeyond the scope of this work.

3.2. Stage-by-stage balance model

A stage-by-stage balance model was derived to provide thebasis for compartmental model development. Model deriva-tion was based on the following simplifying assumptions:


Table 1Nominal Operating Conditions of the Upper Column

Variable Symbol (Units) Value

N2 product flow rate PN2 (kmol/h) 330.99N2 product impurity yO2,PN2

(ppm) 1.15O2 product flow rate PO2 (kmol/h) 258.443O2 product impurity yO2,PO2

(kmol/kmol) 0.9930Average stage liquid holdup M̄L (kmol) 0.755Average stage vapor holdup M̄V (kmol) 0.0970Distributor liquid holdup M1 (kmol) 5.085Distributor liquid holdup M10 (kmol) 10.151Distributor liquid holdup M18 (kmol) 10.872Distributor liquid holdup M26 (kmol) 11.708Distributor liquid holdup M35 (kmol) 11.442Distributor liquid holdup M43 (kmol) 12.438Distributor liquid holdup M51 (kmol) 12.439Reboiler liquid holdup M59 (kmol) 43.560

(1) negligible vapor phase holdups; (2) ideal vapor phasebehavior; (3) fast temperature dynamics; (4) linear pressureprofiles across the column; (5) complete mixing of the va-por and liquid streams entering the feed stages; (6) constantreboiler medium temperature. The stage-by-stage model iscomprised of dynamic component balances for oxygen andargon, dynamic overall mass balances, steady-state energybalances and an activity coefficient model to account for non-ideal liquid phase behavior. The model equations for thei-thseparation stage are:

MidO2, i

dt= Li−1xO2,i−1 + Vi+1yO2,i+1 − LixO2,i − ViyO2,i − xO2,

MidxAr, i

dt= Li−1xAr,i−1 + Vi+1yAr,i+1 − LixAr,i − ViyAr, i − xAr,

dMi

dt= Li−1 + Vi+1 − Li − Vi

0 = Li−1hLi−1 + Vi+1h

Vi+1 − Lih

Li − Vih

Vi − hL

idMi

dt

1 = yO + yN + yAr

wmth penp de-p pore urec re-l tew

L

w

The vapor–liquid equilibrium relation is written as:

yn,i = κiγn,iKn,ixn,i + (1 − κi)yn,i+1,

n = N2, O2, Ar (3)

where K is the ideal temperature dependent vapor–liquidequilibrium constant,γ the activity coefficient andκ is theMurphee stage efficiency. The ideal vapor–liquid equilibriumconstant was calculated as:

Kn,i = Psatn (Ti)

Pi

(4)

where the Antoine equation was used to compute the purecomponent saturation pressuresPsat

n . A linear increase of thestage pressure from a constant overhead pressure (P1) wasassumed:

Pi = P1 + (i − 1)�P (5)

A constant individual stage pressure drop was correlated tothe reboiler vapor rate (Vr) using Aspen simulation data:

�P = βV 2r (6)

whereβ is a constant. The activity coefficients were calcu-lated using the Margules equation for multicomponent mix-tures (Prausnitz, Lichtenthaler, & de Azevedo, 1986):

γN2,i = exp

γO2,i = exp

γAr,i = exp

w nta ei -u d byP

ed tomi wasm ldupa ther nsers ollerw tion

2 2

hereM is the liquid holdup,L andV the liquid and vaporolar flow rates, respectively,x andy component mole frac-

ions in the liquid and vapor phases, respectively, andhL andV are liquid and vapor phase enthalpies, respectively. Ashysical property data were used to develop temperatureendent correlations for the pure component liquid and vanthalpies. Stream enthalpies were calculated from the pomponent enthalpies using linear mixing rules. A linearationship between liquid holdup and liquid molar flow raas regressed from Aspen simulation data:

i = kiMi (2)

herek is stage hydraulic coefficient.

idMi

dt

idMi

dt

(1)

[AN2,O2x2

O2,i+AN2,Arx

2Ar,i+(AN2,O2+AN2,Ar−AO2,Ar)xO2,ixAr,i

RTi

][

AN2,O2x2N2,i

+AO2,Arx2Ar,i+(AN2,O2+AO2,Ar−AN2,Ar)xN2,ixAr,i

RTi

][

AN2,Arx2N2,i

+AO2,Arx2O2,i

+(AN2,Ar+AO2,Ar−AN2,O2)xN2,ixO2,i

RTi

](7)

hereT is the absolute temperature,R the ideal gas constand the Margules constant,Aj,k, accounts for liquid phas

nteractions between componentsj and k. Proprietary vales of the Margules and Antoine constants were provideraxair.Steady-state mass and enthalpy balances were us

odel mixing of the four feed streams shown inFig. 2withnternal vapor and liquid streams. Each liquid distributor

odeled as a separation stage with negligible vapor hond a very small stage efficiency (5%). Modeling ofeboiler was simplified by assuming a constant condeide temperature as in the Aspen simulator. A PI contras designed to regulate the reboiler level by manipula


of the oxygen product flow rate. The controller tuning pa-rameters were identical to those used in the Aspen simula-tor:kc = −5000 %level/(kmol/h) andτi = 2 s. The completestage-by-stage model consisted of 178 differential equationsand 134 algebraic equations with the following unknowns:xO2,i, xAr,i, Li, Vi andTi. TheMATLAB codeode15s wasused to solve the DAE model.

3.3. Compartmental models

The compartmentalization approach was used to derive re-duced order dynamic models from the stage-by-stage model

described above. To examine the effect of compartmental-ization on reduced order model complexity and accuracy, thefollowing three cases shown inFig. 2were investigated:

• 15 Compartment model: The reboiler (stage 59) and theseven distributors (stages 1, 10, 18, 26, 35, 43 and 51)were treated as separate compartments due to their largeliquid holdups listed inTable 1. Equilibrium stages lo-cated between adjacent distributors were lumped togetheras multistage compartments.

• 9 Compartment model: Distributors located between feedand/or withdrawal points were lumped together with equi-

Fig. 2. Three alternative compartmentalization schemes: (a
) 15 compartments; (b) 9 compartments; (c) 5 compartments.


librium stages such that only a single compartment waslocated between any two feed and/or product streams.

• 5Compartment model: Each compartment contained a sin-gle feed and withdrawal stream located at the first stage ofthe compartment. The reboiler was treated as a single stagecompartment due to its relatively large liquid holdup.

The compartmental models were derived by transform-ing the stage-by-stage model into singularly perturbed formand neglecting the fast dynamics (Levine & Rouchon, 1991).Single stage compartments cannot be simplified because theassociated model equations do not exhibit a time scale sep-aration. The model-order reduction procedure for multistagecompartments can be summarized as follows: (1) replacethe dynamic balances for a chosen equilibrium stage withdynamic balances derived for the entire compartment; (2)introduce a singular perturbation parameter that representsthe ratio of the liquid holdup of an individual equilibriumstage to the liquid holdup of the entire compartment; (3) re-duce the differential equations for the individual equilibriumstages to algebraic equations by setting the singular pertur-bation parameter to zero. As discussed earlier, we have ap-plied this approach to a simpler nitrogen purification columnmodel (Khowinij et al., in press) under the assumptions ofequimolar overflow and ideal vapor–liquid equilibrium. Herewe outline the compartmental modeling procedure for theu nerb s. Ad

parm ntb

w ub-s tiesf d vap omp ver-a y ot

M

h

A dyn itht g-g doen ccur

investigate this issue further in the next section. By definingMi

Mc= αiε whereαi ≈ 1 accounts for O(1) differences among

individual stage holdups andε � 1, the stage-by-stage modelequation can be placed in singularly perturbed form wherethe average compartment variablesMc andxn,c are the slowvariables and the individual stage variables variablesMi andxn,i are the fast variables (Levine & Rouchon, 1991). The fol-lowing reduced order model equations are obtained by settingε = 0 according to singular perturbation theory (Kokotovic,Khalil, & O’Reilly, 1999):

0 = L0xn,0 + V2yn,2 − L1xn,1 − V1yn,1

0 = L0 + V2 − L1 − V1

0 = L0hL0 + V2h

V2 − L1h

L1 − V1h

V1

...

0 = Lk−2xn,k−2 + Vkyn,c − Lk−1xn,k−1 − Vk−1yn,k−1

0 = Lk−2 + Vk − Lk−1 − Vk−1

0 = Lk−2hLk−2 + Vkh

Vk − Lk−1h

Lk−1 − Vk−1h

Vk−1

Mcdxn,c

dt= L0xn,0 + Vm+1yn,m+1 − Lmxn,m − V1yn,1

−xn,cdMc

dtdMc

dt= L0 + Vm+1 − Lm − V1

0 = L0hL + Vm+1h

V − LmhL − V1hV − hL dMc

musfor

del

if-del.cyn.

pper column stage-by-stage model with steady-state ealances and non-ideal vapor–liquid equilibrium relationetailed derivation is omitted for the sake of brevity.

Consider the dynamic balances for a multistage coment comprised ofm equilibrium stages. Overall componealances about the compartment yield:

dMcxn, c

dt= L0xn,0 + Vm+1yn, m+1 − Lmxn,m − V1yn,1

dMc

dt= L0 + Vm+1 − Lm − V1

hLc

dMc

dt= L0h

L0 + Vm+1h

Vm+1 − LmhL

m − V1hV1

(8)

here the subscriptn denotes oxygen or argon, and the scripts 0 andm + 1 are used to represent liquid properrom the stage immediately above the compartment anor properties from the stage immediately below the cartment, respectively. The total liquid holdup, holdup aged compositions and holdup averaged liquid enthalp

he compartment are defined as:

c =m∑

i=1

Mi, xn,c =∑m

i=1 Mixn,i

dMc

,

Lc =

∑mi=1 Mih

Li

dMc

(9)

time scale separation is introduced by replacing theamic balances for thek-th stage in the compartment w

he overall balances(8). Although previous studies have suested that the location of this so-called “sensitive stage”ot have a significant effect on reduced order model aacy (Khowinij et al., in press; Levine & Rouchon, 1991), we

gy

t-

--

f

-

s-

0 m+1 m 1 c dt

0 = Lkxn,c + Vk+2yn,k+2 − Lk+1xn,k+1 − Vk+1yn,k+1

0 = Lk + Vk+2 − Lk+1 − Vk+1

0 = LkhLc + Vk+2h

Vk+2 − Lk+1h

Lk+1 − Vk+1h

Vk+1

...

0 = Lm−1xn,m−1 + Vm+1yn,m+1 − Lmxn,m − Vmyn,m

0 = Lm−1 + Vm+1 − Lm − Vm

0 = Lm−1hLm−1 + Vm+1h

Vm+1 − LmhL

m − VmhVm

(10)

The 15 compartment model shown inFig. 2a is completedby adding the balance equations and vapor–liquid equilibriurelations for the single stage compartments. An analogoprocedure is used to derive the reduced order equationsthe 9 compartment and 5 compartment models shown inFig.2b and c, respectively. The resulting reduced order moequations are not reported here.Table 2shows that compart-mentalization produces a large reduction in the number of dferential variables as compared to the stage-by-stage moThe effect of model-order reduction on prediction accuraand computational efficiency is explored in the next sectio

Table 2Complexity of the stage-by-stage and compartmental models

Model Differential variables Algebraic variables

Stage-by-stage 178 13415 Compartment 46 2669 Compartment 28 2845 Compartment 16 296


4. Results and discussion

The first set of simulation results were generated by: (1)selecting the sensitive stage of each multistage compartmentto be located in the center of the compartment and (2) adjust-ing stage efficiencies in the top section (stages 2–9) and thereboiler heat transfer coefficient of the stage-by-stage balancemodel such that the nitrogen and oxygen product purities ofthe Aspen model at the nominal steady state were matched.Fig. 3provides a comparison of steady-state oxygen and ar-gon composition profiles obtained from the Aspen simulator,the full-order stage-by-stage model (FOM) and reduced ordermodels with 15, 9 and 5 compartments (Comp15, Comp9 andComp5) at the nominal steady state. The composition pro-files are plotted in physical coordinates and natural log trans-formed coordinates to facilitate comparison of the variousmodels. The stage-by-stage model provides good agreementwith the Aspen simulator over a very wide range of compo-sitions. Although not illustrated here, the inclusion of energybalances and non-ideal vapor–liquid equilibrium relations inthe stage-by-stage model was necessary to achieve satisfac-tory results. Sharp discontinuities occur at the stages wherea feed stream is introduced, while relatively flat compositionprofiles are observed at the locations of the liquid distribu-tors. As predicted by singular perturbation theory (Levine &Rouchon, 1991), each compartmental model provides perfect

steady-state agreement with the stage-by-stage model. Anyfurther improvements in the compartmental models would re-quire the derivation of a more accurate stage-by-stage model.

The motivation for including more compartments in thereduced order model is to improve dynamic prediction accu-racy.Fig. 4 shows the evolution of the nitrogen and oxygenproduct compositions for a 25% decrease in liquid air flowrate att = 1 h. The stage-by-stage model yields a steady-stategain for the nitrogen product composition that is about 65%larger than the value produced by Aspen. By contrast, thesteady-state gain for the oxygen product composition is pre-dicted very accurately. Good dynamic agreement between theAspen simulator and the stage-by-stage model responses isobtained for this step change. The reduced order models yieldslower nitrogen composition responses than the stage-by-stage model, while very good agreement is obtained for theoxygen product purity except for the 5 compartment model.

The evolution of the oxygen and argon composition pro-files for the−25% change in the liquid air flow rate is shownin Fig. 5. The profiles obtained from the Aspen simulatorand the 15 compartment model are plotted at three time in-stants. While the compartmental model is able to track thetwo product compositions (seeFig. 4), the transient com-position profiles are predicted less accurately. Large devia-tions between the Aspen and compartmental profiles are ob-served at the initial steady state (t = 1 h) and immediately af-

Ff

ig. 3. Steady-state oxygen and argon composition profiles for the nominal oull-order stage-by-stage model; Comp15, 15 compartment model; Comp9,

perating point. The various models are denoted as: Aspen, Aspen simulator; FOM,9 compartment model; Comp5, 5 compartment model.


Fig. 4. Dynamic nitrogen and oxygen product composition responses for a−25% change in the liquid air feed flow rate att = 1 h.

Fig. 5. Dynamic oxygen and argon composition profiles for a−25% change in the liquid air feed flow rate att = 1 h.


Fig. 6. Dynamic nitrogen and oxygen product composition responses for a+25% change in the liquid air feed flow rate att = 1 h.

ter the step change (t = 1.1 h) in the upper portion of the col-umn. By contrast the compartmental model yields very accu-rate predictions of the final steady-state composition profiles(t = 5 h).

Fig. 6 shows the dynamic responses of the nitrogen andoxygen products for a+25% change in the liquid air feedstream att = 1 h. The stage-by-stage model generates veryaccurate predictions of the Aspen nitrogen product puritydynamics and steady-state gain. Although the stage-by-stagemodel produces a slightly larger steady-state gain than theAspen simulator for the oxygen product composition, the un-dershooting dynamics caused by the reboiler level controllerare faithfully captured. The 15 compartment model providesthe best agreement with the stage-by-stage model for bothproduct compositions. The 9 compartment model produceslarger errors in the oxygen product purity dynamics, whilethe 5 compartment model exhibits large deviations for bothcompositions responses due to the lumping of equilibriumstages and liquid distributors into the same compartments.These results suggest that 15 compartments are needed toaccurately track the product composition dynamics of thestage-by-stage model.

Oxygen and argon composition profile dynamics pro-duced by the Aspen simulator and the 15 compartment modelfor the +25% change in liquid air flow rate are shown inFig. 7. Despite generating satisfactory predictions of thep -

mental model exhibits large errors in the composition profiledynamics. Particularly large deviations are observed imme-diately after the step change (t = 1.1 h) and at the final steadystate (t = 5 h). These results are partially attributable to theadjustment of the stage-by-stage model to reproduce the As-pen product compositions at the initial steady state. In effect,the stage-by-stage model has been tuned to predict the prod-uct compositions at the expense of the composition profiles.

The previous simulation results were generated by select-ing the sensitive stage of each multistage compartment to belocated in the center of the compartment.Fig. 8 shows theeffect of the sensitive stage locations on product compositionresponses obtained with 15 compartments for a+25% changein the liquid air feed flow rate att = 1 h. The most accuratepredictions of Aspen composition dynamics are producedwhen the sensitive stages are located in the center of the com-partments as in the previous simulation tests. However, verysimilar predictions are obtained when the sensitive stagesare located in either the top or bottom of the compartments.The main advantages of center placement are less inverse re-sponse (see insets) and more accurate dynamic tracking ofthe oxygen product purity response. These results supportthe claim that the locations of the sensitive stages have littleeffect on reduced order model accuracy (Khowinij et al., inpress; Levine & Rouchon, 1991).

The final set of simulation results were generated with-o e As-
roduct composition responses (seeFig. 6), the compart ut adjusting the stage-by-stage model to reproduce th


Fig. 7. Dynamic oxygen and argon composition profiles for a+25% change in the liquid air feed flow rate att = 1 h.

pen product compositions at the initial steady state.Fig. 9shows nitrogen and oxygen product composition responsesfor a+25% change in the liquid air feed flow rate att = 1 hwhen the sensitive stage of each multistage compartment is

located in the center of the compartment. As compared to thecase where the stage-by-stage model was fine tuned to fit theinitial Aspen product compositions (seeFig. 6), the stage-by-stage model produces a much smaller steady-state gain for

F d oxyg ei

ig. 8. The effect of sensitive stage locations on dynamic nitrogen an
n the liquid air feed flow rate att = 1 h.
en product composition responses of 15 compartment models for a+25% chang


Fig. 9. Dynamic nitrogen and oxygen product composition responses for a+25% change in the liquid air feed flow rate att = 1 h without matching the initialsteady-state product compositions of the Aspen and full-order models.

Fig. 10. Dynamic oxygen and argon composition profiles for a+25% change in the liquid air feed flow rate att = 1 h without matching the initial steady-statep
roduct compositions of the Aspen and full-order models.


the nitrogen product purity but a more accurate steady-stategain for the oxygen product purity using the Aspen modelas the basis for comparison. A comparison of the nitrogenand oxygen composition profiles inFigs. 7(adjusted model)and10 (unadjusted model) generated with 15 compartmentsshows that more accurate prediction of Aspen compositionprofiles is obtained with the unadjusted model. This resultprovides further support for the claim that the adjusted modelhas been tuned to reproduce Aspen product compositions atthe expense of poor composition profile predictions.

As shown inTable 2, the compartmental modeling strategyallows a large reduction in the number of differential variablesas compared to the stage-by-stage model. In our previouswork on nitrogen purification columns, we observed that aseven compartment model produced a five-fold decrease insimulation time as compared to the stage-by-stage modelfrom which it was derived (Khowinij et al., in press). Forthe upper column studied in this paper, we found that com-partmental models produced roughly the same computationtimes as the stage-by-stage model regardless of the numberof compartments used. We believe that these seeminglyincongruous results are attributable to the more complex al-gebraic equations present in the upper column compartmentmodels combined with the inefficiency of theMATLAB DAEsolver. Although not investigated here, we conjecture that thecomputational efficiency of the compartmental models rela-t singa ,C stt ectedt mew mplea f con-s m.T nals . Asd ge oft ltingD PCc thate

5

froma urityd Thes nces,s e relat rousA ucedo ered ught rderr with

relatively few differential variables but a large number ofalgebraic variables. The compartmental models were shownto yield good steady-state and dynamic agreement with thestage-by-stage model as long as a sufficiently large numberof compartments were used. The compartmental models didnot produce a significant decrease in open-loop simulationtime despite the large reduction in the number of differentialvariables. We conjecture that improved compartmentalmodel performance relative to the stage-by-stage model canbe achieved with a more advanced differential–algebraicequation (DAE) solver than those currently available in Mat-lab. More importantly, the order reduction method producessparse DAE models that can be exploited in simultaneoussolution methods developed for nonlinear model predictivecontrol. This direction is the focus of our future research.

Acknowledgements

Financial support from the National Science Foundation(CTS-0241211) and Praxair is gratefully acknowledged. Theauthors also would like to thank Aspen Technology for pro-viding the Aspen Engineering Suite software.

R

A ,

B art-.

B statens.

B

C s of

C e ofunits.

D , H.lefin

H tatus

H l. In

H dy-

K m-

K nent

ive to the stage-by-stage model could be improved by umore sophisticated DAE solver such asDASSL (Brenanampbell, & Petzold, 1989). However, our results sugge

hat the compartmental modeling approach can expo yield significant reductions in open-loop simulation tihen the stage-by-stage model contains relatively silgebraic equations obtained under the assumptions otant equimolar overflow and ideal vapor–liquid equilibriuhis conclusion would need to be verified with additioimulation studies for other types of distillation columnsiscussed earlier, we believe that the primary advanta

he compartmental modeling approach is that the resuAE models can be effectively incorporated into NMontrollers by utilizing customized solution methodsxploit the sparse model structure (Leineweber, 1998).

. Summary and conclusions

Reduced order compartmental models were deriveddetailed stage-by-stage balance model of a high p

istillation column used for cryogenic air separation.tage-by-stage model with dynamic component balateady-state energy balances and non-ideal liquid phasions was shown to provide good agreement with a rigospen simulator over a range of production rates. Redrder models with different numbers of compartments werived directly from the stage-by-stage model thro

he application of singular perturbation theory. The oeduction method produced compartmental models

-

eferences

llgower, F., & Zheng, A. (2000).Nonlinear model predictive control. BaselSwitzerland: Birkhauser.

enallou, A., Seborg, D. E., & Mellichamp, D. A. (1986). Dynamic compmental models for separation processes.AIChE Journal, 32, 1067–1078

ian, S., Henson, M. A., Belanger, P., & Megan, L. (2005). Nonlinearestimation and model predictive control of nitrogen purification columIndustrial and Engineering Chemistry Research, 44, 153–167.

renan, K. E., Campbell, S. L., & Petzold, L. R. (1989).Numerical solutionof initial-value problems in differential–algebraic equations. New York,NY: Elsevier.

arta, R., Tola, G., Servida, A., & Morbidelli, M. (1995). Error analysicollocation models for steady state multistage separation units.Comput-ers and Chemical Engineering, 19, 123–127.

arta, R., Tola, G., Servida, A., & Morbidelli, M. (1995). Performanccollocation models for simulating transient multistage separationComputers and Chemical Engineering, 19, 1141–1151.

ebling, J. A., Han, G. C., Kuijpers, F., VerBurg, J., Zacca, J., & RayW. (1994). Dynamic modeling of product grade transitions for opolymerization processes.AIChE Journal, 40, 506–520.

enson, M. A. (1998). Nonlinear model predictive control: Current sand future directions.Computers and Chemical Engineering, 23, 187–202.

enson, M. A., & Seborg, D. E. (1997). Feedback linearizing controM. A. Henson, & D. E. Seborg (Eds.),Nonlinear Process Control (pp.149–231). Englewood Cliffs, NJ: Prentice-Hall.

orton, R. R., Bequette, B. W., & Edgar, T. F. (1991). Improvements innamic compartmental modeling for distillation.Computers & ChemicalEngineering, 18, 197–201.

howinij, S., Henson, M. A., Belanger, P., & Megan, L. Dynamic copartmental modeling of nitrogen purification columns.Separation andPurification Technology, in press.

ienle, A. (2000). Lower-order dynamic models for ideal multicompodistillation process using nonlinear wave propogation theory.ChemicalEngineering Science, 55, 1817–1828.


Kokotovic, P. V., Khalil, H. K., & O’Reilly, J. (1999).Singular perturbationmethods in control: Analysis and design. Philadelphia, PA: SIAM.

Kravaris, C., & Kantor, J. C. (1990). Geometric methods for nonlinear pro-cess control: 2. Controller synthesis.Industrial and Engineering Chem-istry Research, 29, 2310–2323.

Kronseder, T., von Stryk, O., Bulirsch, R., & Kroner, A. (2001). Towardsnonlinear model–based predictive optimal control of large-scale processmodels with application to air separation plants. In M. Grotschel, & S.O. Krumke (Eds.),Online optimization of large scale systems: State ofthe art (pp. 285–412). Springer-Verlag.

Leineweber, D. B. (1998).Efficient reduced SQP methods for the optimiza-tion of chemical processes described by large sparse DAE models. PhDthesis, University of Heidelberg.

Levine, J., & Rouchon, P. (1991). Quality control of binary distilla-tion columns via nonlinear aggregated models.Automatica, 27, 463–480.

Luyben, W. L. (1973).Process modeling, simulation and control for chemicalengineers. New York, NY: McGraw-Hill.

Marquardt, W. (1986). Nonlinear model reduction for binary distillation.Proceedings of the IFAC control of distillation columns and chemicalreactors (pp. 123–128).Bournemouth, UK.

Marquardt, W., & Amrhein, M. (1994). Development of a linear distillationmodel from design data for process control.Computers and ChemicalEngineering, 18, S349–S353.

McAuley, K. B., & MacGregor, J. F. (1992). Optimal grade transi-tions in a gas-phase polyethylene reactor.AIChE Journal, 38, 1564–1576.

Meadows, E. S., & Rawlings, J. B. (1997). Model predictive control. InM. A. Henson, & D. E. Seborg (Eds.),Nonlinear Process Control (pp.233–310). Englewood Cliffs, NJ: Prentice-Hall.

Nagy, Z., Findeisen, R., Diehl, M., Allgower, F., Bock, H. G., Agachi, S., etrge

scale processes—A case study.Proceedings of the American controlconference (pp. 4249–4254).Chicago, IL.

Prausnitz, J. M., Lichtenthaler, R. N., & de Azevedo, E. G. (1986).Molec-ular thermodynamics of fluid-phase equilibria. Englewood Cliffs, NJ:Prentice-Hall.

Qin, S. J., & Badgwell, T. A. (1997). An overview of industrial model pre-dictive control technology. In J. C. Kantor, C. E. Garcia, & B. Carnahan(Eds.),Chemical Process Control V (pp. 232–256). Austin, TX: CACHE,AIChE.

Qin, S. J., & Badgwell, T. A. (1998). An overview of nonlinear model predic-tive control applications.Proceedings of the nonlinear model predictiveworkshop: Assessment and future directions. Ascona, Switzerland.

Seferlis, P., & Grievink, J. (2001). Optimal design and sensitivity analysisof reactive distillation units using collocation models.Industrial andEngineering Chemistry Research, 40, 1673–1685.

Seferlis, P., & Hrymak, A. N. (1994). Adaptive collocation of finite-elementsmodels for the optimization of multistage distillation units.ChemicalEngineering Science, 49, 1369–1382.

Seferlis, P., & Hrymak, A. N. (1994). Optimization of distillation units usingcollocation models.AIChE Journal, 40, 813–825.

Swartz, C. L. E., & Stewart, W. E. (1986). A collocation approach to distil-lation column design.AIChE Journal, 32, 1832–1838.

Levien, K. L., Stewart, W. E., & Morari, M. (1985). Simulation of frac-tionation by orthogonal collocation.Chemical Engineering Science, 40,409–421.

Yang, D. R., & Lee, K. S. (1997). Monitoring of a distillation column usingmodified extended Kalman filter and a reduced order model.Computersand Chemical Engineering, 21, S565–S570.

Zhu, G. -Y., Henson, M. A., & Megan, L. (2001). Low-order dy-namic modeling of cryogenic distillation columns based on nonlinearwave phenomenon.Separation and Purification Technology, 24, 467–

al. (2000). Real-time feasibility of nonlinear predictive control for la
487.

compartmental modeling of high purity air … modeling of high purity air separation columns ......

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