A unified modeling framework for the optimal design and dynamic simulation of staged reactive separation processes

Computers and Chemical Engineering 30 (2006) 1264–1277

A unified modeling framework for the optimal design and dynamic simulation of staged reactive separation processes Natassa Dalaouti 1 , Panos Seferlis ∗ Chemical Process Engineering Research Institute (CPERI)/CERTH, P.O. Box 361, 570 01 Thermi-Thessaloniki, Greece Received 10 November 2005; received in revised form 17 February 2006; accepted 27 February 2006 Available online 2 May 2006

Abstract A unified modeling approach that combines the rigorous non-equilibrium (NEQ) rate-based balance equations with the model-order reduction properties of orthogonal collocation on finite elements (OCFE) approximation techniques is employed for the optimal design, operation optimization and dynamic simulation of complex staged reactive separation processes. The NEQ/OCFE process model formulation involves the rigorous description of mass and heat transfer phenomena, phase equilibrium relations and chemical reactions in both gas and liquid phases in a limited but sufficient for an accurate representation of the column behavior, number of collocation points. In addition, polynomial approximation as implemented in the OCFE techniques, transforms the staged column domain into a continuous analog. Hence, a significant degree of modeling detail is maintained, while a more compact model formulation than the equivalent full-order (tray-by-tray) model is used. The NEQ/OCFE modeling framework has been proved particularly efficient in the optimal design of reactive absorption and distillation columns especially those with multiple side feed and product streams mainly due to the elimination of binary decision variables associated with the existence of column stages in any given column section. Furthermore, the compact model size and the demonstrated ability of the OCFE approximation to accurately identify the optimal solution and reveal the dynamic characteristics of complex reactive separation columns make the proposed modeling framework particularly attractive for real-time optimization and control applications. Distinctive process examples such as the reactive absorption of nitrogen oxides (NOx ) from a gas stream by a weak HNO3 aqueous solution in the industrial production of nitric acid, as well as the production of ethyl acetate by reactive distillation clearly demonstrate the merits and strengths of the NEQ/OCFE modeling framework. © 2006 Elsevier Ltd. All rights reserved. Keywords: Rate-based models; Orthogonal collocation on finite elements; Design optimization; Reactive separation; NOx removal; Ethyl acetate production

1. Introduction Reactive separation processes are quite attractive for many industrial applications. The strong interest in this type of processes is mainly attributed to the significant reduction in the overall investment costs that can be achieved by process intensification through the combination of chemical transformation with products separation. Reactive absorption is widely used in the manufacture of nitric and sulfuric acid, soda ash, the purification of synthesis gases and the recovery of solvents from

Abbreviations: ACOOH, acetic acid; CP, collocation points; EtAc, ethyl acetate; EtOH, ethanol; FE, finite element; HNO3 , nitric acid; H2 O, water; NO, nitrogen monoxide; NO2 , nitrogen dioxide; NOx , nitrogen oxides ∗ Corresponding author. Tel.: +30 2310 498 169; fax: +30 2310 498 160. E-mail address: [email protected] (P. Seferlis). 1 Present address: Testing Research & Standards Center, Public Power Corporation S.A., 9 Leontariou Str., 15351 Kantza-Pallinis/Attiki, Greece. 0098-1354/$ – see front matter © 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2006.02.018

process streams. Another important application is found in air pollution control and the cleaning of effluent gas streams from pollutants and toxic gases. Common applications of reactive distillation are met on the production of various ether solvents and gasoline additives. Both reactive absorption and distillation are complex unit operations characterized by coupled equilibrium, mass and heat transfer and reactive phenomena. Therefore, the use of detailed models that account for the interactions between chemical reactions, heat and mass transfer, and phase equilibrium in the static and dynamic simulation of reactive separation becomes absolutely necessary (Baur, Higler, Taylor, & Krishna, 2000; Kenig, Wiesner, & G´orak, 1997; Taylor & Krishna, 1993). Subsequently, high model complexity leads to large in size and difficult to solve process models. Detailed non-equilibrium (NEQ) rate-based models for reactive separation processes consider the synergetic and highly interactive effects of simultaneous mass and heat transfer, mixing phenomena (Higler, Taylor, & Krishna, 1999), phase equi-

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Nomenclature aint Acol c d d D Ð f g G h h H k0 K L m n N NC NE NR NT P Q r R Rg s t T u U W x x y

specific gas–liquid interfacial area (m2 /m3 ) column stage cross-section (m2 ) molar concentration (mol/m3 ) vector of design variables molar density (mol/m3 ) column diameter (m) Maxwell–Stephan diffusion coefficient (m2 /s) objective function vector of inequality constraints gas molar flow rate (mol/s) vector of equality constraints column height (m) stream molar enthalpy (J/mol) pre-exponential kinetic parameters equilibrium K value liquid molar flow rate (mol/s) component molar holdup (mol) number of collocation points in a finite element molar flux (mol/(m2 s)) number of components number of finite elements in a column section number of chemical reactions number of stages stage pressure (Pa) rate of heat loss (J/s) rate of reaction (mol/(m3 s)) component rate of reaction (mol/(m3 s)) ideal gas constant (8.314 J/(mol K)) column position coordinate time (s) stage temperature (K) component internal energy (J) stream internal energy (J) Lagrange interpolating polynomial vector of process variables liquid-phase mole fraction gas-phase mole fraction

Greek letters δ film thickness (m) h stage height (m) ␧ vector of model parameter φ phase volumetric holdup fraction (m3 phase/m3 stage) η film coordinate (m) Superscripts G gas phase Gf gas film int interface L liquid phase Lf liquid film

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librium relations and chemical reactions in both gas and liquid phases. Separation columns either in staged or packed configuration are treated as modules with a number of equivalent stages (Kenig et al., 1997). According to the two thin-film model (Taylor & Krishna, 1993), mass and heat transfer resistance is considered concentrated within a thin film region adjacent to the vapor–liquid interface. The Maxwell–Stephan equations (Taylor & Krishna, 1993) describe the coupled multicomponent mass transfer mechanism, while chemical reactions in the bulk and film regions introduce significant non-linearity in the behavior of the system. An excellent review on modeling and design of reactive distillation processes can be found in Taylor and Krishna (2000). The design of staged reactive absorption and distillation processes involves the determination of design parameters and operating conditions that optimize an economic criterion while satisfying all imposed safety, environmental and operating constraints. The operating conditions involve decisions about the flow rate and concentration of feed streams in the column (e.g., solvent amount, reflux and boil-up streams) and the temperature profile in the column in cases where heat exchange in the stages becomes necessary. Decisions related to the column operating conditions can then be usually represented as continuous variables. On the other hand, structural decisions basically associated with the column stage configuration (e.g., total number of stages in a column or a column section, location of the side feed and product draw streams in a plate column) are variables of discrete nature. The design optimization of staged separation columns with binary variables representing the number of stages in individual column sections (Ciric & Gu, 1994) would result in substantial increase of the computational effort for the solution of the mathematical program. The issue of reducing the total number of binary decision variables becomes even more important when the simultaneous synthesis of column networks and design of multiple columns is considered. Orthogonal collocation on finite elements (OCFE) technique transforms the column sections into a continuous analog. Thus, the discrete number of stages in the column sections becomes a continuous variable and hence the need for integer variables associated with the number of column stages is eliminated. Composition and temperature are treated as continuous functions of position in the column and approximated using piecewise polynomials. OCFE formulations were shown to successfully identify the optimal operating point of the full-order tray-by-tray model with significant savings in computational time (Seferlis & Hrymak, 1994a). Furthermore, the OCFE formulation allows the use of conventional non-linear programming (NLP) algorithms for the calculation of the optimal column configuration as only continuous variables are involved in the synthesis of separation networks (Proios & Pistikopoulos, 2004) and the design of reactive distillation column trains (Seferlis & Grievink, 2001). Finally, as the OCFE formulation uses a smaller set of modeling equations than the full-order tray-by-tray model, the solution effort is substantially decreased thus making the use of the NEQ/OCFE process models in real-time optimization and control applications extremely attractive.

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In most of the previous research efforts regarding the use of OCFE techniques for the modeling of separation units (Huss & Westerberg, 1996; Seferlis & Grievink, 2001; Seferlis & Hrymak, 1994a; Stewart, Levien, & Morari, 1985; Swartz & Stewart, 1986) the associated process assumed instant equilibrium between phases. Torres, Martins, and Bogle (2000) used the non-equilibrium balance equations in conjunction with the OCFE technique for a conventional (i.e. non-reactive) packed distillation column. The present work combines the sophisticated rigorous non-equilibrium rate-based modeling equations for reactive separation systems with the model reduction properties of the OCFE formulation. Moreover, the OCFE formulation provides a unified modeling framework for a wide variety of unit operations ranging from staged to packed separation columns and from reactive absorption to reactive distillation as will be demonstrated with the simulated examples presented in the text. In perspective, the NEQ/OCFE modeling framework greatly facilitates the optimal design, real-time optimization, control and operator training applications for separation units through its accurate steady-state and dynamic simulation, and optimization capabilities. The text is organized as follows: the general formulation of the modeling framework is presented in the following section. The unified NEQ/OCFE formulation is then implemented on the modeling of the reactive absorption of NOx gases by a weak nitric acid aqueous solution as used in the industrial production of nitric acid, as well as on the production of ethyl acetate via reactive distillation. Optimal design and dynamic simulation studies are performed and discussed.

The proposed formulation combines the rigorous NEQ modeling equations with the model reduction properties of the OCFE technique. Following the OCFE formulation by Seferlis and Hrymak (1994a), the column is separated into sections, with each section defined as the part of the column between two streams entering or leaving the column. Each column section is divided into smaller sub domains, namely, the finite elements. For each finite element, a specific number of collocation points is specified. The main feature of the OCFE formulation is that material and energy balances as resulted from the NEQ ratebased equations are satisfied exactly only at the collocation points. The collocation points are chosen as the roots of the discrete Hahn family of orthogonal polynomials. Such a selection ensures that the collocation points coincide with the location of the actual stages in the limiting case that the order of the polynomial (e.g., number of collocation points) becomes equal to the number of actual stages for any given column section. Hence, the complete model can be fully recovered. Lagrange interpolation polynomials are used within each finite element to approximate the liquid and vapor component flow rates, as well as the liquid and vapor stream enthalpies, as follows: ˜ i (s) = L

n


˜ i (sj ), WjL (sj )L

j=0

0 ≤ s ≤ NT, with s0 = 0, i = 1, . . . , NC

˜ i (s) = G

n+1


(1)

˜ i (sj ), WjG (sj )G

j=1

2. Modeling framework for staged reactive separation processes The NEQ model for conventional and reactive separation processes (Kenig, Schneider, & G´orak, 2001; Kenig et al., 1997; Krishnamurthy & Taylor, 1985; Taylor & Krishna, 1993) involves the rigorous description of mass and heat transfer phenomena, phase equilibrium relations and chemical reactions in both the gas and liquid phase calculated in a number of equivalent stages. Mass transfer is described by the thin-film model (Taylor & Krishna, 1993), which assumes that mass transfer resistance is limited in two film regions adjacent to the gas–liquid interface. Gas and liquid bulk phases are in contact only with the corresponding films, while thermodynamic phase equilibrium is assumed to occur only at the interface. Chemical reactions are considered to take place in both the film and bulk phase regions. The basic assumptions of the model are: (i) no axial dispersion along the column, (ii) one-dimensional mass transport normal to the interface, (iii) perfectly mixed gas and liquid bulk phases with uniform temperature and composition and (iv) no liquid entrainment in the vapor phase. The validity of the aforementioned assumptions basically depends on the flow regime that persists within the column stages. For design purposes the assumptions can be safely considered as valid for normal operating conditions. However, for real-time applications (simulation, control and optimization) caution should be taken regarding the hydrodynamic conditions that prevail in the column stages.

1 ≤ s ≤ NT + 1, with sn+1 = NT + 1, i = 1, . . . , NC ˜ L (s) = ˜ t (s)H L

n


(2)

˜ t (sj )H ˜ L (sj ), WjL (sj )L

j=0

0 ≤ s ≤ NT, with s0 = 0

˜ t (s)H ˜ G (s) = G

n+1


(3)

˜ t (sj )H ˜ G (sj ), WjG (sj )G

j=1

1 ≤ s ≤ NT + 1, with sn+1 = NT + 1

(4)

where NC is the number of the components, NT the number of actual stages approximated by a given finite element, n the num˜ i (s) the ˜ i (s) and G ber of collocation points in the given element, L ˜ ˜ component molar flow rates, Lt (s) and Gt (s) the total liquid and ˜ G (s) are the liquid and ˜ L (s) and H gas stream flow rates, and H gas stream molar enthalpies, respectively. It should be noted here that points s0 = 0 and sn+1 = NT + 1 are interpolation points for the liquid and vapor phase approximation schemes, respectively. Functions WL (s) and WG (s) represent Lagrange interpolation polynomials of order n + 1 given by the expressions: WjL (s) =

n  k=0,k=j

s − sk , sj − s k

j = 0, . . . , n

(5)

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WjG (s) =

n+1  k=1,k=j

s − sk , sj − s k

j = 1, . . . , n + 1

(6)

Polynomials WjL (s) and WjG (s) are equal to zero at collocation points sk when k = j and to unity when k = j. Within each finite element, a Lagrange polynomial of different order can be used as dictated by the shape and characteristics of the approximated variable profiles (e.g., linear or irregularly shaped profiles, steep fronts). Stages that are connected to mass or energy streams entering (e.g., feed streams) or leaving (e.g., product draw streams) the column are treated as discrete equilibrium stages so that the effects of discontinuities in the mass and energy flows within the column sections do not influence the continuity and smoothness of the interpolating polynomial schemes within the elements (Eqs. (1)–(4)). Such a requirement can however be relaxed when the side material and/or energy streams are distributed uniformly along the length of a finite element or column section (e.g., similar mass and/or heat flows to successive stages). Hence, the effect of the side streams on the column profiles will be uniformly spread and will not diminish the quality of the polynomial approximation. The size of each column section (equivalent to the number of stages) can vary within real-valued bounds acting as a continuous decision variable (degree of freedom) in the design optimization problem. Subsequently, column section size governs the size of the embedded finite elements. The only constraint that applies on the size of each column section refers to the approximated total number of real column stages, which must be equal to or greater than the number of collocation points used in the given section (i.e. NT ≥ n). The conditions at the element boundaries obey zero-order continuity, which is a plausible assumption for staged units with discrete profiles. However, certain smoothness conditions for the concentration and enthalpy profiles may be employed in the case of packed columns (e.g., first derivative continuity at element boundary). For staged separation columns and assuming zero-order continuity, if nk is the number of collocation points in the kth element, the boundary conditions connecting two consecutive elements yield: nk


L ˜ i (sj,k ) = L ˜ i (s0,k+1 ), Wj,k (NTk )L

j=0

i = 1, . . . , NC, k = 1, . . . , NE − 1

(7)

˜ t (sn +1,k )H ˜ G (snk +1,k ) G k nk+1 +1
G ˜ t (sj,k+1 )H ˜ G (sj,k+1 ), = Wj,k+1 (1)G j=1

k = 1, . . . , NE − 1

(10)

In Eqs. (7)–(10), the first index for symbol s refers to the corresponding collocation point and the second one to the corresponding finite element. According to Eq. (7), the extrapolated value for the liquid flow rate at the endpoint of the kth element (i.e. s = NTk ) is set equal to the flow rate at the interpolation and boundary point in the following (k + 1)th element (i.e. s0,k+1 = 0). Similarly, the vapor flow rate at the endpoint of the kth element (i.e. sn+1,k = NTk + 1), which is also an interpolation point in the approximation scheme, is set equal to the extrapolated value for the vapor flow rate at the beginning of the neighboring (k + 1)th element (i.e. s = 1). Column regions with steep or rapidly changing composition and temperature profiles may require the increase of the density of collocation points which can be achieved with the adaptive placement of the element boundaries (Seferlis & Hrymak, 1994b). Assuming that uniform hydrodynamic conditions prevail within each finite element, the dynamic mass balances employed at the collocation point sj , are described by the following equations: dmLi (sj ) ˜ i (sj − 1) − L ˜ i (sj ) + (φL (sj )RLb =L i (sj ) dt + NiLb (sj )αint )Acol h, i = 1, . . . , NC, j = 1, . . . , n

˜ i (sn +1,k ) = G k

G ˜ i (sj,k+1 ), Wj,k+1 (1)G

−NiGb (sj )αint )Acol h, i = 1, . . . , NC, j = 1, . . . , n

i = 1, . . . , NC, k = 1, . . . , NE − 1 nk


(8)

L ˜ t (sj,k )H ˜ L (sj,k ) = L ˜ t (s0,k+1 )H ˜ L (s0,k+1 ), Wj,k (NTk )L

j=0

k = 1, . . . , NE − 1

(9)

(12)

The terms on the left-hand side of Eqs. (11) and (12) account for the component molar accumulation (i.e. holdup) in the liquid and gas bulk phase, respectively. These are related to the available liquid and vapor volumes corresponding to an equivalent stage by the equations: ˜ i (sj ) col L A h, mLi (sj ) = φL (sj )d L (sj ) ˜ t (sj ) L

G G mG i (sj ) = φ (sj )d (sj )

j=1

(11)

dmG i (sj ) ˜ i (sj + 1) − G ˜ i (sj ) + (φG (sj )RGb =G i (sj ) dt

i = 1, . . . , NC, j = 1, . . . , n nk+1
+1

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(13)

˜ i (sj ) G Acol h, ˜ t (sj ) G

i = 1, . . . , NC, j = 1, . . . , n

(14)

In the above equations, dL (sj ) and dG (sj ) denote the liquid and gas-phase density, φL (sj ) and φG (sj ) the liquid and gas-phase volumetric fraction, respectively, calculated at the conditions pre˜ i (sj ), T˜ (sj ), P(s ˜ i (sj ), G ˜ j )], vailing at the collocation points [L while h stands for the height of the equivalent stage.

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Total flow rates and mole fractions in the bulk phases are also defined at the collocation points, as follows: NC


˜ i (sj ) = L ˜ t (sj ), L

xiLb (sj ) =

i=1

˜ i (sj ) L , ˜ t (sj ) L

i = 1, . . . , NC, j = 1, . . . , n

The boundary conditions for Eqs. (20) and (21) are: NiGb (sj ) = NiGf (sj )|ηGf =0 ,

yiGb (sj ) = yiGf (sj )|ηGf =0 ,

i = 1, . . . , NC, j = 1, . . . , n (15)

NiLf (sj )|ηLf =δLf (sj ) = NiLb (sj ),

(22) xiLf (sj )|ηLf =δLf (sj ) = xiLb (sj ),

i = 1, . . . , NC, j = 1, . . . , n NC


˜ i (sj ) = G ˜ t (sj ), G

˜ i (sj ) G , ˜ t (sj ) G

yiGb (sj ) =

i=1

i = 1, . . . , NC, j = 1, . . . , n

(16)

The diffusion molar flux term Ni (sj ) in Eqs. (11) and (12) is estimated by the Maxwell–Stefan equations for multicomponent mixtures, at the conditions prevailing at each collocation point. For ideal gas phase and one-dimensional mass transfer normal to the interface, the Maxwell–Stefan equations for the gas and the liquid phase take the following form: NC
ykGf (sj )NiGf (sj ) − yiGf (sj )NkGf (sj ) ∂yiGf (sj ) = − , ˜ j )/Rg T˜ (sj ))ÐG ∂ηGf (P(s ik k=1,k=i

i = 1, . . . , NC, j = 1, . . . , n, 0 < ηGf ≤ δGf (sj )

(17)

∂xLf (sj ) Γi,k (sj ) k Lf ∂η

i = 1, . . . , NC − 1, j = 1, . . . , n, 0 < ηLf ≤ δLf (sj )

NR


νi,r rr (sj ),

i = 1, . . . , NC, j = 1, . . . , n

(24)

r

where rr (sj ) denote the rates of the reactions taking place. OCFE formulation can be tailored to allow reactive and non-reactive sections in the column. At the gas–liquid interface of each collocation point thermodynamic equilibrium is assumed, described by the following equation: yiint (sj ) = Ki (sj )xiint (sj ),

i = 1, . . . , NC, j = 1, . . . , n (25)

i = 1, . . . , NC, j = 1, . . . , n yiGf (sj )ηGf =δGf (sj ) = yiint (sj ), (18)

where

 ∂ ln γi (sj )  Γik (sj ) = δik + xi (sj ) ∂xk (sj ) T˜ (sj ),P(s ˜ j ),xk (sj ),k=i=1,...,NC−1 (19) The dynamic mass balances in the gas and liquid films, considering the effect of chemical reactions on the mass transfer through the film regions, as well as the boundary conditions connecting the gas and liquid bulk phases with the respective films are also satisfied exactly only at the collocation points. The mass balances in the films can be written in the following form: ∂t

Ri (sj ) =

NiGf (sj )|ηGf =δGf (sj ) = Niint (sj ) = NiLf (sj )|ηLf =0 ,

NC
xkLf (sj )NiLf (sj ) − xiLf (sj )NkLf (sj ) = − , ct (sj )ÐLik k=1,k=i

∂ciGf (sj )

Terms Ri (sj ) in balances (11), (12), (20) and (21) denote the total component reaction rate of the ith component in the gas and liquid bulk and film regions and are estimated at the conditions prevailing at each collocation point by the following equations:

The boundary equations at the interface are:

NC−1
k=1

(23)

+

∂NiGf (sj ) ∂ηGf

i = 1, . . . , NC, j = 1, . . . , n, 0 < η

≤ δ (sj )

(20)

i = 1, . . . , NC, j = 1, . . . , n, 0 < ηLf ≤ δLf (sj )

(21)

Gf

Gf

∂ciLf (sj ) ∂NiLf (sj ) + − RLf i (sj ) = 0, ∂t ∂ηLf

xiint (sj ) = xiLf (sj )|ηLf =0 ,

i = 1, . . . , NC, j = 1, . . . , n

(27)

Neglecting the heat transfer effects along the film regions the overall dynamic energy balance at each collocation point becomes: dU(sj ) ˜ t (sj + 1)H ˜ t (sj − 1)H ˜ L (sj − 1) + G ˜ G (sj + 1) =L dt ˜ t (sj )H ˜ t (sj )H ˜ L (sj ) − G ˜ G (sj ) + Q(sj ), −L j = 1, . . . , n

(28)

where U(sj ) =

NC


G {mLi (sj )uLi (sj ) + mG i (sj )ui (sj )},

i=1

j = 1, . . . , n

− RGf i (sj ) = 0,

(26)

(29)

Q(sj ) is the net heat transferred from the surroundings. Contrary to the basic principle of the OCFE formulation that isolates stages with a mass or energy streams leaving or entering the column and treats them as discrete stages, the systematic and uniform heat exchange that may take place in a given column section can be formulated as in Eq. (28). Deviations from the basic rule are only verified when the heat exchange within a given element is distributed uniformly along the element. For

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Table 1 Reactions in the NOx absorption process Gas phase 2NO + O2 → 2NO2 2NO2 ↔ N2 O4 3NO2 + H2 O ↔ 2HNO3 + NO NO + NO2 ↔ N2 O3 NO + NO2 + H2 O ↔ 2HNO2

Liquid phase N2 O4 + H2 O → HNO2 + HNO3 3HNO2 → HNO3 + H2 O + 2NO N2 O3 + H2 O → 2HNO2 2NO2 + H2 O → HNO2 + HNO3

(RR1) (RR2) (RR3) (RR4) (RR5)

instance, heat losses or heat exchange for a group of neighboring stages can be approximated using Eq. (28) within finite elements that exclusively contain the stages where the heat exchange takes place. However, pump-around streams that carry significant material and energy amounts outside and then inside the column at specific locations of the column should be definitely treated as discrete stages. Empirical correlations are employed for the calculation of the pressure drop in the column, the liquid holdup, the specific interfacial area, and the gas and liquid film thickness. These correlations consider column internals and hydraulics and depend on the type of the column plate (e.g., sieve, bubble cap). All calculations take place at the conditions prevailing at the collocation points. 3. Reactive absorption of NOx 3.1. Process description The reactive absorption of nitrogen oxides (NOx ) from a gas stream by a weak HNO3 aqueous solution (Emig & Wohlfahrt, 1979; Joshi, Mahajani, & Juvekar, 1985; Suchac, Jethani, & Joshi, 1991) is an efficient way to remove the nitrogen oxides from gas streams released to the atmosphere and meanwhile to produce a solution of nitric acid. Chemical reactions play an important role in such a system, because they enhance the absorption of the otherwise insoluble in water components (e.g., NO) through their chemical transformation (e.g., oxidation) to more soluble components (e.g., NO2 ). Furthermore, nitric acid is produced through a complex reaction mechanism that involves five gas-phase and four liquid-phase reactions (Joshi et al., 1985) as provided in Table 1. The oxidation of NO to NO2 (RR1) is kinetically the slowest and thus the limiting step in the mechanism (Emig & Wohlfahrt, 1979). The remaining gas-phase equilibrium reactions (RR2)–(RR5) were considered as reversible kinetic reactions. Liquid phase reactions (RR6) and (RR7) are kinetically controlled, while reactions (RR8) and (RR9) were considered as reversible kinetic reactions. The temperature dependency expressions of the equilibrium constants for all the reactions in the kinetic mechanism are shown in Table 2 with data taken from Emig and Wohlfahrt (1979), Joshi et al. (1985) and Suchac et al. (1991). The gas-phase diffusion coefficients used for the calculation of the component diffusion molar fluxes through the Maxwell–Stefan equations were estimated using the Chapman–Enskog–Wilke–Lee model (Reid, Prausnitz, & Poling, 1987), while the liquid-phase diffusion coefficients were

(RR6) (RR7) (RR8) (RR9)

estimated using the method proposed by Siddiqi and Lucas (1986). The NRTL activity model was used for the calculation of the liquid-phase activity coefficients (Infochem, 2000). All other necessary thermodynamic calculations (e.g., stream enthalpy, density and so forth) were based on the SRK equation of state. A gas stream with high concentration of NOx enters the bottom of the staged counter-current reactive absorption column, a liquid water (solvent) stream enters from the top of the column, and a feed stream of weak solution of nitric acid enters the side of the column. The bottoms liquid stream mainly comprises an aqueous nitric acid solution and is partially recycled in the column. The nitric acid concentration in the liquid product that leaves the column as a side draw stream is subject to quality constraints while the concentration of NOx gases in the gas stream in the top of the column is subject to composition constraints due to environmental regulations. The column is consisted of 44 trays with an internal diameter of 3.6 m. The distance between the trays is equal to 0.9 m, except for the five trays closer to the bottom, where the oxidation reaction (RR1) mostly takes place and higher spacing is used to increase the gas-phase holdup and subsequently the extent of (RR1). Cooling was provided in the column stages for the removal of the reactive heat and the control of the column temperature, since reactions are highly exothermal. Furthermore, low column temperatures favor the oxidation reaction (RR1) and absorption of NO2 in water. The empirical correlations provided by Zarzycki and Chacuk (1993) were used for the column pressure drop (p. 514), liquid-phase holdup (p. 513), film thickness (Rocha, Bravo, & Fair, 1996) as well as stage interfacial area (p. 527) for sieve plates. The modular representation of the NOx reactive absorption column in Fig. 1 shows the partition of the column into three sections. One section above the location of the side feed stream, one between the side feed stream and the recycle side stream and one below the recycle side stream. Stages connected to Table 2 Equilibrium and rate constants for the NOx absorption process RR1 RR2 RR3 RR4 RR5 RR6 RR7 RR8 RR9

log10 k1 : 652.1/T − 0.7356 (atm2 s)−1 log10 K2 : 2993/T − 9.226 atm−1 log10 K3 : 2003.8/T − 8.757 atm−1 log10 K4 : 2072/T − 7.234 atm−1 log10 K5 : 2051.17/T − 6.7328 atm−1 log10 k6 : −4139/T + 16.3415 s−1 log10 k7 : −6200/T + 20.1979 (m3 /kmol)2 atm2 /s K8 : 3.3 × 102 kmol/m3 K9 : 3.8 × 109 kmol/m3

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N. Dalaouti, P. Seferlis / Computers and Chemical Engineering 30 (2006) 1264–1277 Table 3 NOx reactive absorption process inlet stream data Gas inlet stream (bottom) NO 21.83 mol/s 58.08 mol/s NO2 N 2 O4 20.11 mol/s O2 82.74 mol/s 1016.44 mol/s N2 T 332 K P 5.6 bar Recycle stream Flow 4.55 mol/s

Liquid inlet stream (top) H2 O 4.55 mol/s T 293 K Side feed stream H2 O 34.62 mol/s HNO3 6.02 mol/s NO2 0.41 mol/s T 306 K

3.2. Steady-state and dynamic model validation

Fig. 1. Modular representation of the NOx reactive absorption column.

material streams entering or leaving the column are treated as discrete stages. Furthermore, the five oxidation stages closer to the bottom of the column are treated as discrete stages because each stage has been designed with a different stage holdup. Each column section is further partitioned into a number of finite elements and for each finite element a number of collocation points is specified. Material and energy rate-based balances (Eqs. (11)–(29)) were then applied at the specified collocation points along the column domain. Gas and liquid inlet stream data for the studied case are provided in Table 3. The solution of the resulting NEQ/OCFE model that comprises a set of differential/algebraic equations was performed by gPROMS® (Process Systems Enterprise, 2003), an integrated process modeling environment. The partial differential equations describing mass transfer in the gas and liquid films (Eqs. (20)–(23)) were discretized using orthogonal collocation on finite elements applied in the film domain. More specifically, fifth and third-order polynomial approximation was selected over two finite elements, respectively.

The steady-state simulation results using the NEQ/OCFE model formulation were compared with those obtained by the corresponding full-order tray-by-tray NEQ (NEQ/FULL) model. The full-order model validation was achieved with steady-state bibliographical data for an industrial NOx removal reactive absorption column (Emig & Wohlfahrt, 1979). It should be noted at this point that in all simulated cases the NEQ/FULL process model is assumed to accurately represent the behavior of the actual unit operation whether validated with experimental data or not. Validation of the NEQ/FULL process model is therefore not in the scope of the current article. For the development of the NEQ/OCFE model a sufficient number of collocation points were placed within each column section to ensure an accurate representation of the process. Three different NEQ/OCFE schemes were tested for their accuracy in representing the full-order (tray-by-tray) model with their specific characteristics given in Table 4. The steady-state temperature profiles, NOx gas-phase and HNO3 liquid-phase profiles (in terms of flow rates) in the column, for the three NEQ/OCFE configurations, are compared to the results obtained by the NEQ/FULL model (Fig. 2). The lines represent the profiles for the NEQ/FULL model, while the data points correspond to the position of the collocation points and the discrete stages in the NEQ/OCFE model formulation. Clearly, the match among all three NEQ/OCFE schemes and the NEQ/FULL model is extremely well (Fig. 2). The maximum deviation from the fullorder profiles was observed in NEQ/OCFE C that had the least total number of collocation points but remained well within acceptable limits. The model size reduction percentage in the

Table 4 Comparison between the full-order model and the NEQ/OCFE model formulation for NOx reactive absorption process Model

Number of stages or finite elements (FE), and collocation points (CP) Top section

Full order OCFE A OCFE B OCFE C

CP/FE 4 4 2

Model reduction ratio

CPU time ratio

Maximum relative deviation (%)

42680

1.0

1.0

–

28790 16720 12580

0.67 0.39 0.29

0.73 0.33 0.14

2.1 4.3 7.6

Middle section

29 FE 4 2 2

Model equations

FE 2 1 1

8 CP/FE 3 2 2

Note: CPU data are for a Pentium 4, 2.0 GHz. gPROMS® 2.3.3 version.

N. Dalaouti, P. Seferlis / Computers and Chemical Engineering 30 (2006) 1264–1277

Fig. 2. Steady-state model validation: (i) temperature and (ii) NOx outlet gases and HNO3 product flow rate profiles.

case of NEQ/OCFE C was 70% resulting also in a sevenfold reduction of the required CPU time for solution. Evidently, the NEQ/OCFE model retains the capacity to accurately approximate the behavior of complex reactive columns with a significantly more compact model formulation. Real-time control applications require the use of reliable, accurate and easy to solve dynamic process models. NEQ/OCFE models are good candidates for such applications mainly due to the significant reduction in the number of states involved in the dynamic model, while keeping the key non-linear features of the process. In addition, the NEQ/OCFE formulation fully preserves the input-output structure of the dynamic system, a particularly desirable situation. The dynamic behavior of the process models with varying model resolution when the inlet gas stream flow rate was subject to a 10% increase is shown in Fig. 3. Clearly, the NEQ/OCFE model dynamic response matches the response calculated using the NEQ/FULL model satisfactorily. As the number of collocation points used for the OCFE approximation decreases its dynamic response becomes slightly faster although deviations in settling times did not exceed a difference of more than 5%. The discrepancy in the initial point is due to the differences at the calculated steady-state points for each model.

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Fig. 3. Dynamic response for a 10% increase in the inlet gas stream flow rate: (i) temperature and (ii) NOx outlet gases flow rate and HNO3 product flow rate.

The comparison between the dynamic behavior of the NEQ/FULL and the NEQ/OCFE model formulations for a 20% decrease of the cooling water flow rates in the column is shown in Fig. 4. As already mentioned, oxidation reactions are highly

Fig. 4. Dynamic response for a 20% decrease in the cooling water flow rate.

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model). In addition, operating (e.g., flooding), safety (e.g., column pressure) and environmental (e.g., emissions) constraints are effective and should be satisfied by the optimal design. The mathematical representation of the design optimization problem for staged separation columns yields: Max

f (x, d, ␧)

s.t.

h(x, d, ␧) = 0 g(x, d, ␧) ≤ 0

x,d

xl ≤ x ≤ xu ,

Fig. 5. Dynamic response for a 10% increase of the NOx content in the inlet gas stream using a NOx concentration PI controller.

exothermal while absorption is favored by low temperature. The reduction of the amount of the cooling water, which acts as the manipulated variable in the temperature control of the column, results in an increase of the NOx content in the outlet gas stream The model formulations corresponding to configurations NEQ/OCFE B and NEQ/OCFE C have the ability of predicting the NOx gases concentration in the gas outlet stream with a deviation in the process gain smaller than 0.2% in absolute terms, while the dynamic characteristics in both cases show no major differences from the behavior predicted by the full-order model. The dynamic behavior of the NEQ/FULL model and the NEQ/OCFE model formulations under closed loop conditions was also studied (Dalaouti & Seferlis, 2005). A PI controller that manipulates the cooling water flow rates in the column sections to maintain the NOx outlet concentration within a desired range was implemented in the model. As shown in Fig. 5, the NEQ/OCFE model formulations retain the ability of predicting the dynamic behavior of the column in closed loop, for a 10% increase in the NOx content in the inlet gas stream. The good agreement between formulations NEQ/OCFE B, C and NEQ/FULL is apparent. The overestimation of the new steadystate in the outlet gas stream is less than 3%. The savings in the computational requirements from the use of the NEQ/OCFE models is similar to those in the steady-state calculations. 3.3. Process design problem statement The design of reactive separation processes involves the determination of the column configuration expressed in total number of stages in the column, the locations of side feed and product draw streams and the operating conditions that optimize an economic criterion, while satisfying the product specifications and the safety and operational constraints for the process. Typically, the economic criterion involves the annualized investment and operating costs for the unit and the revenues from the sale of the products. The behavior of the separation units is obtained using a detailed process model that accurately describes the underlying physical and chemical phenomena (e.g., NEQ

(30) dl ≤ d ≤ du

where x denotes the process (state and control) variables. d is the vector of the design variables that comprises equipment capacity variables (e.g., reboiler and condenser duties), and structural design variables (e.g. number of stages in a column, location of feed streams). Finally, ␧ denotes the vector of model parameters (e.g., kinetic constants) and process disturbances (e.g., feed stream composition and temperature). The process design as derived by the solution of Eq. (30) should ensure that the operating specifications, safety regulations, and environmental constraints are satisfied during plant operation and in addition, good economic performance is maintained for the plant under the presence of model and process related uncertainty (Seferlis & Grievink, 2001). 3.4. Design optimization OCFE techniques eliminate the need for integer variables to describe the number of stages in the column and use instead continuous variables that are associated with the element lengths. The element lengths combined make up the particular column section. Model NEQ/OCFE B was used in the calculation of the optimal design of the NOx reactive absorption column. The optimization problem has seven degrees of freedom that include the number of stages in the top and the middle section of the column, the flow rate of the solvent stream (water) at the top of the column, the flow rate of the side feed stream, and the amount of cooling water in three independent cooling sections of the column. These sections include the cooling water flow rate in the middle section of the column, in the discrete stage where the side stream enters the column, and the overall cooling water flow rate in the bottom section of the column that comprises the discrete oxidation stages and the stage connected to the recycle stream (Fig. 1). The operating and environmental specifications in the column are the nitric acid concentration in the liquid product and the NOx gases concentration in the top of the column, as presented in Table 5. The objective function for minimization includes a term for the column capital cost, the cost for the solvent stream and the cost for the cooling water, as provided below, with cost data taken from Douglas (1981). Total cost = 26.25 × D × h0.805 + 0.5 × L + 0.2 × Ls − 1.24 e−4 Q

(31)

where D is the diameter, h the height of the column, L the solvent flow rate entering the top of the column, Ls (mol/s) the feed side

N. Dalaouti, P. Seferlis / Computers and Chemical Engineering 30 (2006) 1264–1277 Table 5 Design optimization results for the NOx absorption column Design A Column specifications NOx in gas outlet (molar fraction) HNO3 in liquid product (molar fraction)

Design B <0.0125 >0.25

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packing selection. Ethanol and acetic acid form ethyl acetate through an endothermic liquid-phase esterification reversible reaction in the presence of sulfuric acid that acts as the catalyst. C2 H5 OH + CH3 COOH ↔ CH3 COOC2 H5 + H2 O

Number of stages Top section Middle section

16 21

16.69 10

Liquid feed streams flow rates Solvent (top) (mol/s) Side stream (mol/s)

27 37.57

36.38 45

Cooling water flow rates Side stream position Middle section Bottom part

10 157.9 350

80 160 350

Outlet streams concentration NOx in gas outlet (molar fraction) HNO3 in liquid product (molar fraction) Total cost (US$/year 1000)

0.01165 0.25001 1844

0.01249 0.25003 1457

stream flow rate, while Q is the amount of heat removed from the column (negative value). Table 5 shows the design optimization results for two different designs of the NOx reactive absorption column. In Design A, the total number of stages in the column was kept fixed, but the number of stages in the top and middle section of the column were degrees of freedom in the optimization problem. In this way, the optimal location of the side solvent feed stream can be calculated. Therefore, Design A corresponds to the design of the operating policy for an already existing column. It is quite common in industrial practice that columns are equipped with the necessary piping to adaptively feed the column at multiple locations and the selection of the most suitable configuration becomes a viable option for the improvement of the daily plant performance. Design B corresponds to a totally new column design as no constraint on the total number of stages in the column was imposed. Comparison of Designs A and B demonstrates that the size of the column (especially, the middle column section) can be reduced by more than 10 stages for the same set of column specifications. The reduction of the total number of stages was achieved with a considerable increase in the solvent and side stream flow rates. Furthermore, cooling water demands become larger, especially in the parts closer to the top of the column. 4. Ethyl acetate production via reactive distillation 4.1. Process description The modeling NEQ/OCFE framework was also tested on the process of production of ethyl acetate via reactive distillation (Bock, Jimoh, & Wozny, 1997; Seferlis & Grievink, 2001; Suzuki, Komatsu, & Hirata, 1970; Venkataraman, Chan, & Boston, 1990). Kenig, B¨ader et al. (2001) and Kl¨oker et al. (2004) carried out experimental studies for column scale-up and

Pure ethanol and acetic acid are fed separately into the column that operates at atmospheric pressure. The reaction is assumed to take place only in the liquid phase. At these conditions acetic acid as the heaviest of the components moves towards the bottom of the column. Ethyl acetate is the lightest but its relative volatility with respect to ethanol is small. The rate of reaction is generally low and is therefore favored by a large residence time in each column stage. Ethanol has relatively high volatility and prefers the vapor phase rather than the liquid phase where the reaction takes place. Thus, the maintenance of low ethanol composition in the liquid phase reduces the overall production rate of ethyl acetate. The expression for the rate of reaction is given as follows:   A1 r = k01 exp − [ACOOH][EtOH] T   A2 − k02 exp − (32) [EtAc][H2 O] T The system that comprises ethanol, ethyl acetate, acetic acid and water forms a highly non-ideal solution, which may give rise to four binary and one ternary azeotrope mixtures. In addition, Barbosa and Doherty (1988) claimed the existence of a reactive azeotrope when using the Wilson activity model to describe the non-ideality of the liquid phase. Unfavorable physical equilibrium constraints the production of high purity ethyl acetate from a single distillation column. However, the use of a second recovery column operating at a higher pressure leads to the production of the desire purity (Bock et al., 1997). The recovery distillation column is introduced next to the reactive column operating at a higher pressure (350 kPa) to break the azeotrope and produce high purity ethyl acetate as shown in Fig. 6 (Seferlis & Grievink, 2001). The main objective of the current work is to test the proposed NEQ/OCFE formulation in the reactive distillation column under static and dynamic conditions and compare it with the corresponding full-order tray-by-tray model. The reactive column was partitioned into three sections, the rectifying (the section confined between the condenser and the acetic acid feed), the reactive (the section between the two side feed streams) and the stripping section (the section between the ethanol feed and the reboiler). The recycle and the ethanol feed streams enter the column at the same stage. The esterification reaction occurs in all three sections but each column section has a different but uniform, within the section, liquid-phase holdup. The number of stages and the holdups in each column section, the feed steams flow rates, temperature and compositions, as well as the heat duties in the reboiler and the condenser were selected as shown in Table 6 and correspond to the optimized results obtained by Seferlis and Grievink (2001). The two stages, one

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Fig. 6. Ethyl acetate production flowsheet via reactive distillation.

related to the acetic acid feed and the other to the ethanol feed and the recycle streams were modeled as discrete stages. Furthermore, the reboiler and the condenser of the column were modeled as equilibrium stages. The Wilson activity model was used for the liquid phase while ideal gas phase was assumed (Barbosa & Doherty, 1988). Regressed equations were used for the liquid and vapor molar enthalpy. The overall dynamic NEQ/OCFE and NEQ/FULL models were developed and solved using gPROMS® .

Table 6 Design conditions for the ethyl acetate reactive distillation column Number of stages Rectifying/reactive/stripping section

6/10/6

(m3 )

Stage holdups Rectifying/reactive/stripping section

0.15/1.75/1.634

Acetic acid feed stream Flow rate (kmol/h) Temperature (K)

3.21 363

Ethanol feed stream Flow rate (kmol/h) Temperature (K)

3.145 363

Recycle feed Flow rate (kmol/h) Temperature (K) Composition (molar fraction) EtOH/EtAc/H2 O Heat duty (MJ/h) Condenser/reboiler Kinetic parameters A1 : 7150 K, k01 : 1.74 × 106 m3 /(kmol h) A2 : 7150 K, k02 : 4.428 × 105 m3 /(kmol h)

14.5 381 0.2856/0.4630/0.2514 −530.08/482.58

Fig. 7. Steady-state column profiles: (i) temperature and (ii) EtAc liquid and vapor molar composition profiles.

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Table 7 Comparison between the full-order model and the NEQ/OCFE model formulation for the ethyl acetate reactive distillation column Model

Number of stages or finite elements (FE), and collocation points (CP) Rectifying section

Full order OCFE A OCFE B OCFE C

FE 1 1 1

6 CP/FE 4 2 2

Reactive section FE 3 3 2

10 CP/FE 3 2 2

Model equations

Model reduction ratio

CPU time ratioa

Maximum relative deviation (%)b

11477

1.0

1.0 (69.1 s)

–

9046 6574 4720

0.79 0.57 0.41

0.68 0.44 0.29

0.03 – 0.1

Stripping section FE 2 2 1

6 CP/FE 2 2 2

CPU data are for a Pentium 4, 2.0 GHz. gPROMS® 2.3.3 version. a Refers to dynamic simulation scenario shown in Fig. 8. b Refers to the steady-state error.

4.2. Steady-state and dynamic model validation Simulation results using the non-equilibrium rate-based model for reactive separation processes were much different by the ones obtained by an equilibrium stage model Seferlis and Grievink (2001), since the consideration of resistance in the mass transfer as well the presence of reaction in the liquid film alters the component composition profiles in the column significantly. Three different NEQ/OCFE schemes were tested for their accuracy in representing the steady-state NEQ/FULL

Fig. 8. Dynamic model behavior for step changes in the reboiler heat duty: (i) EtAc distillate composition and (ii) EtAc bottoms stream composition.

(tray-by-tray) model. The steady-state temperature profiles as well as the ethyl acetate composition profiles in both phases for an achieved conversion of acetic acid to ethyl acetate of 49.66% are compared in Fig. 7. The solid line represents the profiles for the full-order rate-based model, while the data points correspond to the collocation points and discrete stages in the NEQ/OCFE approximation model as described in Table 7. The agreement in this case is excellent, while deviations even in the case of OCFE C do not exceed 0.1%. At the same time, the achieved model size reduction percentage is about 60%, while the CPU time required for the solution of the full-order model is most than three times higher than the solution time for case OCFE C. A comparison between the dynamic behavior predicted by NEQ/FULL and the NEQ/OCFE model formulations reveals the excellent prediction properties of the NEQ/OCFE in a dynamic state. The dynamic responses for the ethyl acetate bottoms and distillate composition to multiple step changes of the reboiler heat duty are shown in Fig. 8. The NEQ/OCFE model retains the ability to accurately represent the dynamic behavior of the process, as deviations from the dynamic response predicted by the full-order model were almost negligible. The achieved reduction in the required CPU time for the entire dynamic simulation (Table 7) clearly demonstrates the definite advantage of the NEQ/OCFE model for real-time control applications.

Fig. 9. Dynamic model behavior for a 10% decrease in the EtAc flow rate in the recycle feed for EtAc distillate composition (left ordinate) and EtAc bottoms stream composition (right ordinate).

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The same conclusion can be drawn when comparing the dynamic behavior of the full-order model and the NEQ/OCFE model approximations, for a 10% decrease in the ethyl acetate content in the recycle feed stream flow rate, as shown in Fig. 9. The inverse response in the ethyl acetate bottoms molar composition was exactly calculated. The NEQ/OCFE model retains such approximation accuracy with just a third of the full-order model size when expressed in terms of number of equations. This feature could make the NEQ/OCFE model quite attractive for on-line control applications.

model in terms of balance equations describing the underlying phenomena is fully preserved. Accordingly, further enrichment of the model resolution within a single stage (or its equivalent collocation point) model can be effectively accommodated. Acknowledgement The financial support by the European Commission is gratefully appreciated (GROWTH Program, Project G1RD-CT-200100649).

5. Conclusions A unifying modeling framework, combining a rigorous nonequilibrium rate-based model and orthogonal collocation on finite elements techniques is employed for the optimal design and dynamic simulation of staged reactive separation processes. Non-equilibrium rate-based models are characterized by the accurate representation of the complex behavior of reactive absorption and distillation processes, through the simultaneous consideration of mass and heat transfer phenomena, phase equilibrium relations and chemical reactions in both gas and liquid phases in a number of equivalent stages. The OCFE model approximation transforms the discrete nature of staged columns into a continuous analog. The unifying NEQ/OCFE modeling framework maintains a significant degree of detail that is necessary for the accurate representation of complex reactive separation processes but in a more compact form than the equivalent full-order (tray-by-tray) model thus reducing significantly the computational effort for the solution of the design and simulation problem. The efficiency of the NEQ/OCFE model is enhanced through the elimination of discrete decision variables in the optimal design problem. The proposed unified modeling framework has been tested in steady-state and dynamic simulations for the NOx reactive absorption process as well as for the production of ethyl acetate via reactive distillation. As demonstrated in the two simulated examples, the NEQ/OCFE modeling framework matched the behavior of the relevant full-order model, selected in both cases as the tray-by-tray model, very accurately. The proposed common modeling framework can be used for both staged absorption and distillation columns with or without chemical reactions, thus facilitating process model building and maintenance. Extension of the OCFE technique to packed columns is straightforward, either through the approximation of the packed column as a set of equivalent stages that fully suits the proposed NEQ/OCFE model structure, or through the direct implementation of the OCFE techniques on the discretization of the partial differential balance equations that govern the behavior of the packed column. NEQ/OCFE models manage to maintain the necessary gradient information for optimization purposes and preserve the dynamic characteristics of the process system. The reduction in the process model state variables is compensated through the powerful approximating properties of the OCFE technique in extremely challenging separation systems that involve multiple reactions in both liquid and vapor phases. The key element in this unified modeling framework is that the nature of the full-order

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