Transfer-based models implementation in an equation oriented package

Computers and Chemical Engineering 25 (2001) 1493– 1511 www.elsevier.com/locate/compchemeng

Transfer-based models implementation in an equation oriented package G. Pagani *, A. d’Arminio Monforte, G. Bianchi Enichem Research Center, ‘Instituto Guido Donegani’, 6ia G. Fauser 4, I-28100, No6ara, Italy Received 27 July 2000; received in revised form 14 May 2001; accepted 14 May 2001

Abstract A general transfer based model for L-V and L-L operations was developed in our Equation-Oriented simulator CheOpe and used to analyze several EniChem processes. By the practical use we confirm that these models generally improve usability of a general-purpose simulator. Thanks to the cell approach of CheOpe we can easily analyze different equipment structures or change the level of detail of equipment. Several solution methods of Maxwell-Stefan equations and several effective methods have been implemented. By the analysis of the model behaviour on the basis of many industrial applications, we have found not very significant differences in simulation and design of some non reactive distillation columns using Maxwell-Stefan equations or simpler models that neglect interactions in the mass transfer rate equations. In simulation and design of a reactive system with fast reactions, reaction-transfer interactions cannot be neglected, but often approximate models analytically solved give results as good as more complex models. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Transfer-based models; Equation oriented package; Reaction– transfer interaction

Nomenclature a Ci ’ C fL i C Ii ct d dp D eff i Dik em e Im e $m F hL hV H HF ki,k

interfacial area [L2] molar concentration of component i in liquid bulk [ML − 3] local molar concentration of component i along the liquid film [ML − 3] molar concentration of component i in liquid interface [ML − 3] molar density [ML − 3] column diameter [L] packing particle diameter [L] effective diffusivity of component i in multicomponent mixture [L2t − 1] Maxwell – Stefan diffusion coefficient for the component pair i-k [L2t − 1] enhancement degree of bulk equilibrium reaction m [Mt − 1] enhancement degree of interface equilibrium reaction m [Mt − 1] Cheope variable: enhancement degree of reaction m [Mt − 1] molar flow rate of a general stream entering a cell [Mt − 1] heat transfer coefficient in the liquid phase [QL − 2T − 1t − 1] heat transfer coefficient in the vapor phase [QL − 2T − 1t − 1] molar enthalpy [QM − 1] molar enthalpy of a general stream entering a cell [QM − 1] binary mass transfer coefficient for the component pair i-k [Lt − 1]

* Corresponding author. Tel.: + 39-321-44-7557; fax: + 39-321-44-7506. E-mail address: [email protected] (G. Pagani). 0098-1354/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved. PII: S0098-1354(01)00713-X

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ki,k k eff i kL kV s am s Pm* Ki Ji L L$ Lb Ll$ nai Ni Ncl Ncliq Ncp Ncv Ne Nedg Np N cr N er N fc r N rfcL fcL N r,a i N rfe i Nr Ns Nt Nu Nz P P° Pedp Pez PF Q rm R Ri R if R ’i Ri T T$ TI TL TV TF U V V$ Vb V

multicomponent mass transfer coefficient for the component pair i-k [Lt − 1] effective mass transfer coefficient for the component i [Lt − 1] multicomponent mass transfer coefficient matrix in the liquid phase [Lt − 1] multicomponent mass transfer coefficient matrix in the vapor phase [Lt − 1] activity based constant for liquid or light liquid chemical equilibrium m fugacity based constant for gas chemical equilibrium m [Depending on reaction order] vapor – liquid or light liquid– liquid interface equilibrium ratio of component i molar diffusion transfer rates of component i [Mt − 1] molar flow rate of liquid phase leaving a cell [Mt − 1] Cheope variable: molar flow rate of the liquid phase [Mt − 1] rate of the liquid phase back flow to the superior cell [Mt − 1] Cheope variable: molar flow rate of the second liquid phase [Mt − 1] order of a fast reaction in the ai reagent molar flux of component i [ML − 2t − 1] non-volatile components number Ncp − Nc6 components number non-condensable components number total number of model equations total number of enhancement degrees for a CheOpe cell cell number for a whole process total number of kinetic based reactions in the bulks total number of equilibrium reactions in the bulks total number of kinetic based film reactions total number of kinetic based liquid film reactions (‘fast reactions’) number of kinetic based liquid film reactions (‘fast reactions’) involving component aI total number of equilibrium reactions in the films total number of instantaneous reactions number of a cell model equations tray number of a column cell number for a process unit number of cell internal variables cell pressure [ML − 1t − 2] Cheope variable: cell pressure [ML − 1t − 2] longitudinal Peclet number with reference to the packing particle diameter longitudinal Peclet number with reference to the column height pressure of a general stream entering a cell [ML − 1t − 2] heat rate entering a cell [Qt − 1] rate of reaction m [Mt − 1L − 3] circulating ratio m = 1,N rfcL wi,mrm :conversion rate of component i in the bulk [Mt − 1] m = 1,N rfcL wi,mr mf:conversion rate of component i in the film [Mt − 1] conversion rate of component i [Mt − 1] specific conversion rate of component i [Mt − 1L − 3] cell temperature [T] Cheope variable: cell temperature [T] temperature of a cell interface [T] temperature of a cell liquid bulk [T] temperature of a cell vapor or light liquid bulk [T] temperature of a general stream entering a cell [T] molar flow rate of liquid side stream leaving a cell [Mt − 1] molar flow rate of vapor phase or light liquid phase leaving a cell [Mt − 1] Cheope variable: molar flow rate of the vapor phase or light liquid phase [Mt − 1] molar flowrate of the vapor or light liquid phase back flow to the inferior cell [Mt − 1] volume [L3]

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

W xi x $i x Ii xl$i yi y $i y Ii zi z zc Zj

molar flow rate of vapor or light liquid side stream leaving a cell [Mt − 1] molar fraction of component i in liquid bulk of a cell Cheope variable: mole fraction of component i in liquid phase mole fraction of component i in liquid interface of a cell Cheope variable: mole fraction of component i in second liquid phase mole fraction of component i in vapor or light liquid bulk of a cell Cheope variable: mole fraction of component i in vapor or light liquid mole fraction of component i in vapor or light liquid interface of a cell mole fraction of component i in a general stream entering a cell column height [U] cell height [L] internal variable for cell j [depending on the variable]

Greek letters hm [P] − |m [ML − 1t − 2] ki activity coefficient of component i in liquid or in light liquid phase Y Ncp − 1 dimensional matrix of thermodynamic factors l film thickness [L] li,j Kronecker delta p n/l: adimensional coordinate n Coordinate [L] u*ai depth of penetration of component ai in liquid film [L] wi,m stoichiometric coefficient of component i in reaction m |m i = 1,N€ wi,m / correction factors matrix J correction factor fugacity coefficient of component i in vapor phase [Mt − 1] €i bi molar mass transfer rate of component i [Mt − 1] I bi molar mass transfer rate of component i at the interface [Mt − 1] bT total molar mass transfer rate entering liquid phase [Mt − 1] c heat transfer rate [Qt − 1] xj Ns dimensional vector input– output state variables in cell j [depending on the variable] Subscripts F

general entering stream values

Superscripts c e f fL fV fL% fV% I L V $ %

kinetic reactions equilibrium reactions film values liquid film values vapor or ligh liquid film values local liquid film values local vapor or ligh liquid film values Interface values liquid phase values vapor phase or ligh liquid phase values CheOpe variables local film values

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1. Introduction Computer-aided design and simulation of multicomponent multiphase processes is conventionally performed by equilibrium stage models. This approach is not completely justified because actual equipment rarely, if ever, operates at equilibrium. Moreover, in the enthalpy and K-values calculations, the use of tray efficiency leads to inconsistencies that cause most of computational difficulties (Powers, Vickery, Arehole, & Taylor, 1988; Seader, 1989; Gorak, Wozny, & Jeronim, 1991). In particular, in reactive columns design, the use of stage efficiency becomes very problematic and should be avoided at all (Higler, Taylor, & Krishna, 1998). Transfer-based models allow to overcome these and other drawbacks. Furthermore they are very powerful in correctly solving design and scale-up problems; in fact, their behavior depends on structural equipment parameters and then equipment configuration has an exact correspondence within the model. For the last 15 years several researchers have been working on the development of transfer-based models (Krishnamurthy & Taylor, 1985; Seader 1989; Grottoli, Biardi, & Pellegrini, 1991; Yuxiang and Xen 1992; Taylor, Kooijman, & Hung, 1994Higler et al., 1998; Powers et al., 1998). Yuxiang and Xen (1992) and Lee and Dudukovic (1998) have developed models for reactive operations, assuming that reactions take place only in the bulks. Kenig and Gorak (1995) and Wiesner, Witting, and Gorak (1996) take into account reactions –transfer interactions for linear kinetics; more recently Higler, Taylor, and Krishna (1999) and Kreul, Gorak, and Barton (1999) developed models for reactive separation processes where is considered the coupling between mass transfer and non-linear kinetics. Transfer-based models can greatly and generally improve usability of a general purpose simulator, in particular for the following classes of problems (Taylor & Krishna, 1993):

“

packed column simulation; strongly non-ideal systems distillation; systems with trace components simulation; reactive systems simulation. Very few of the available flowsheeting programs allow transfer-based operation modeling; in reactive operations none of them takes into account the interaction between mass transfer and reactions, assuming that reactions occur only in the bulks. In the following section we will discuss implementation and applications of reactive and non-reactive L–V and L– L transfer-based models in our Equation-Oriented package CheOpe (Pagani & d’Arminio Monforte 1987; Pagani, d’Arminio Monforte, & de Mitri, 1988, 1989; Pagani & d’Arminio Monforte, 1991; Pagani & de Mitri 1991; de Mitri, d’Arminio Monforte, & Pagani, 1992; Pagani, d’Arminio Monforte, & Bianchi, 1993, 1996). “ “ “

2. Guest simulator architecture CheOpe is an EniChem proprietary Equation-Oriented (E-O) simulator based on an interconnected cells approach.

2.1. Problem formulation CheOpe refers to lumped sub-units called ‘cells’; each cell has Ns ‘input–output–internal state variables’ associated1: x° y° x1° T° L° V° L1° Po e°

liquid mole fractions (Ncp variables) ‘vapor’2 mole fractions (Ncp variables) second liquid mole fractions (Ncp variables) cell temperature (one variable) output liquid flow rate (one variable) output ‘vapor’ flow rate (one variable) output second liquid flow rate (one variable) cell pressure (one variable) enhancement degrees Nedg variables

Behavior of non-lumped units can be described by Nu cells (Fig. 1). Interconnected cells models have a great flexibility; they can be easily modified or widened to analyze different equipment structures or to change the level of detail. By this approach, state variables of each unit are the Ns × Nu input–output variables of its cells. A whole

1

A cell can also have some further design variables (Q, U, W…). In the following description, by the term ‘vapor’ we generally mean the light phase (vapor or light liquid phase). 2

Fig. 1. Interconnected cells model.

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

Fig. 2. BQT Jacobian.

process is described by Np cells, if Ne =Ns ×Np is the set of the process model equations.

2.2. Problem solution CheOpe collects all the Ne model equations in a system whose Jacobian is a Np ×Np block matrix where each block is a Ns dimensional matrix. Owing the nature of the flowsheeting problems, Jacobian has a Block Quasi-Tridiagonal (BQT) structure with a zerofree diagonal (Fig. 2). A Newton-Raphson method is used to solve non-linear equation system. The resulting BQT linear system can be efficiently solved using a modified block-Thomas algorithm. In particular, by this approach, the zero–non-zero block structure is well defined and during the solution there is a little fill-in; then the algorithm can be easily implemented (Pagani & d’Arminio Monforte, 1985) with several numerical advantages. For the last 20 years we have worked on tuning Newton– Raphson algorithm (Pagani et al., 1989; Pagani et al., 1996) by solving a very large number of simulation and optimization industrial problems (Pagani & d’Arminio Monforte, 1987; Pagani et al., 1988; Pagani & d’Arminio Monforte, 1991; Pagani, d’Arminio Monforte, & Bianchi, 1998). Nowadays our Newton-Raphon approach is quite stable and robust and it can overcome numerical difficulties arising from highly non-linear equations like them we have to solve in transfer-based models. Anyway, we are considering to develop a continuation method for multiple steady-state analysis.

Vapor and liquid streams can exchange mass and energy across their interface, where equilibrium is assumed. For all VL and LL operations we have developed a sub-unit (Fig. 4) called transfer-based ‘macro-cell’. A model of a transfer-based unit can be obtained by suitably connecting several ‘macro-cell’. For example, if we assume perfectly mixed phases on a tray, a schematic diagram of a transfer-based Nt trays column is as follows (Nu = 3Nt) (Fig. 5). This flow-chart minimizes the Jacobian (Fig. 6) fill-in during solution. More complex cell connection patterns can simulate non-ideal residence time distributions of the vapor and liquid. Given: “ cell ‘geometry ’ (internals type; d, Zc); V L “ the values of input variables (F L, z L, T L F, P F, F , V L V V V z ,T F , P F , Q , Q ), flow rates of the side streams (U, W) and cell pressure (P)3; “ phase equilibrium constant functions Ki = Ki (x I, y I, T I, P) “

i= 1, Ncp;

mole enthalpy functions

Fig. 3. Transfer-based film model.

3. Two fluids phase transfer-based cell model To introduce VL and LL transfer-based models in CheOpe simulator it is enough to build the relative transfer based cell (Fig. 3) and to link CheOpe with a package for fluid dynamic calculation and with a data base of internal geometric parameters.

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Fig. 4. Transfer-based ‘macro-cell’.

3

CheOpe can handle all these variables as unknown.

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H L =H L(x, T L, P),

H V =H V (y, T V, P),

H LF = H LF (z L, T LF , P LF ), “

V V V V HV F =H F (z , T F , P F );

reaction rate functions r Lm = r Lm(x, T L) V m

V m

or

V

r = r (y, T , P) for m= 1, N cr I L I fL r fL m = r m (x, x , T , T )

“

b Li = (k L, a, x, x I, T L, T I, ’ L)

or

I V I fV r fV m = r m (y, y , T , T , P)

the cell behavior depends on 6Ncp + N er + N rfe +7 variables {x, y, x I, y I, T L, T V, T I, L, V, b L, b V, „ L, „ V, I e, e }. Intraphase transfer rates (b L, b V, c L, c V) depend on temperature and concentration gradients and on several other factors (Krishnamurthy & Taylor, 1985) generally lumped into mass (k V, k L) and heat (h V, h L) transfer coefficients

for m = 1, N rfc;

chemical equilibrium constants s am =s am(T L) or s Pm* =s Pm*(T V)

for m = 1, N er ;

L I aI P*I P*I fe s aI m =s m (T ) or s m =s m (T ) for m =1, N r

b = (k , a, y, y , T , T , ’ ) V i

I

V

V

I

V

(a1) i= 1, Ncp − 1

(a2)

„ L = (h L, a, x, T L, T I, ’ L)

(b1)

„ V = (h V, a, y, T V, T I, ’ V)

(b2)

By ‘locally’ solving (see Appendix A) Eq. (a1), Eq. (a2) in b Li and b V i for i= 1, Ncp − 1 and Eq. (b1) and Eq. (b2) in „ L and „ V, we can eliminate these 2Ncp variables. One more variable can be eliminated by posing bN cp = b LN cp = − b V N cp or bT =

% i = 1, N cp − 1

b Li + bN cp = −

% i = 1, N cp − 1

bV i + bN cp.

By this last position cell behavior depends on the 4Ncp + N er + N rfe + 6 variables {x, y, x I, y I, T L, T V, T I, L, V, bT, e, e I}. The mathematical model of the transfer-based ‘macro-cell’ is as follows.

Fig. 5. Cell interconnected model of a transfer-based Nt trays column.

L Cell (L): Ncp + N eL “ r + 2 variables {x “ x°, T L T°, L “ L°,e “ e°}, being {y°= 0, V°= 0, P°=P}. Component mass balances in liquid phase:

− F L·z Li + L·xi − {b Li }− R ’Li = 0

i= 1, Ncp − 1

cp − 1 {b L})− R ’L − F L·z LN cp + L·xN cp − (bT − %N k=I k N cp =0.

Sum equations in liquid bulk: cp %N i = 1 (xi − 1)= 0.

Energy balance in liquid phase: − H LF ·F L + H L·L− Q−{„ L}= 0. Chemical equilibrium in liquid phase: Ncp

Ncp

i=1

i=1

wi,m 5 x wi i,m = s am 5 k − i

Fig. 6. BQT Jacobian of the column.

m= 1, N eL r

Cell (V):Ncp + N eV variables {y “ y°, T V “ r +2 V T°, V “ V°,e “ e°}, being {x° =0, L°= 0, P° = P}.

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

Component mass balances in ‘6apor ’phase: V ’V − F V·z V i + V·yi −{b i } − R i =0.

4. Multicomponent mass transfer relations

i=1, Ncp −1





Ncp − 1 V ’V − F V·z V N cp + V·yN cp + bT −%k = 1 {b k } − R N cp

= 0. Sum equations in ‘6apor’ bulk:

4.1. No reacti6e operations Molar mass transfer can be defined (Krishna & Standart, 1979) by the following Ncp − 1 dimensional matrix notations ’ L = J L + x·bT = c Lt ak L(x I − x)+ xbT V I ’ V = J V + x·’T = c V t ak (y − y)+ ybT

cp %N i = 1 (yi −1) = 0.

Energy balance in ‘6apor’ phase: V V V −HV F ·F + H ·V−{„ } = 0.

Chemical equilibrium in ‘6apor ’ phase: Ncp

Ncp

i=1

i=1

wi,m 5 y wi i,m =s Pm* hm 5 € − i Ncp

Ncp

i=1

i=1

wi,m 5 y wi i,m =s am 5 k − i

or (for light liquid phase) m =1, N

eV r

where k L, k V are the matrices of multicomponent mass transfer coefficients. Molar mass transfer rates and then k L, k V values can be obtained by solving Maxwell–Stefan equations, with reference to a film model. We have implemented in CheOpe the linearized theory by Toor (1964) and Stewart and Prober (1964) and the explicit methods of Krishna (1979, 1981) and of Taylor and Smith (1982). By all these approaches the values of the mass transfer coefficients in ideal gases k V are obtained in terms of binary mass transfer coefficients k V i,k corresponding to zero flux conditions (Bird, Steward, & Lightfoot, 1960), and in terms of JV, correction factors matrix or JV, correction factor (Taylor & Krishna, 1993). For liquid mixtures we can take into account nonideality by posing

Cell (LV)I: 2× Ncp +N rfe +2 variables {x I “ x°, y I “ y°, T I “ T°,bT “ L1°, e f “ e°}, being {V° =0, P° = P}. Sum equations in liquid and ‘6apor’ interfaces:

being

%

Yi, j li, j +

Ncp i=1

(x Ii −1)=0.

Energy balance around the interface: {„ L}+ {„ V}= 0. Component mass balances around the interface:

5 x Ii

wi,m

i= 1, Ncp −1

Ncp

− wi,m =s aI m 5 ki

i=1 Ncp

m =1, N rfeL

i=1 wi,m

5 y Ii

Ncp

wi,m = s Pm*I hm 5 € − i

i=1

i=1 Ncp

wi,m

or (for light liquid phase) 5 y Ii i=1

m =1, N

few r

i("m)= 1, Ncp; j(" m)=1, Ncp

L

(x Ii − xi )+ xi bT b Li = c Lt ak eff i V

Chemical equilibrium in liquid and ‘6apor’ interfaces: Ncp

(ln ki (xj

V eff bV (y Ii − yi )+ yi bT i = c t ak i

i=1, Ncp −1

Interface equilibrium relations: y Ii − Ki ·x Ii =0

id

k L = k L ·Y

By a less rigorous treatment (Hougen & Watson, 1947), mass transfer rates can be defined with reference by the followto effective mass transfer coefficients k eff i ing scalar notations (Kubota, Yamanaka, & Dalla Lana, 1969; Geankoplis, 1972)

I cp %N i = 1 (y i −1)=0.

{b Li }= − {b V i }

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Ncp

− wi,m =s aI m 5 ki i=1

(i= 1, Ncp − 1)

eff eff can be calculated from where k eff i = D i /l and D i Maxwell–Stefan multicomponent diffusion coefficients Dik (Bird et al., 1960). In our package we have implemented the method of Burghardt and Krupikzca (1975), the method of Wilke (1950) and a simple method where k eff i = ki,m, being m a component in large excess. A correction factor J is used in these methods too. The effective diffusivity approach can fail to account for the component interactions (Smith & Taylor, 1983; Taylor & Krishna, 1993; Kooijman & Taylor, 1995); however it has a large employment in engineering design calculations (Vanni, Valerio, & Baldi, 1995; Kreul et al., 1999).

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On several pilot and industrial applications we have found not very significant differences in simulation and design of some non-reactive distillation columns using Maxwell– Stefan equation or simpler models that neglect interactions in the mass transfer rate equations.

Ni = − D eff i

L

dC fL% i . dn

Continuity equations in liquid film become D eff i

L

d2C fL% i + R fL% i =0 dn 2

(15 i5 Ncliq)

(1)

with the following boundary conditions 5. Reactive operations

n=0

5.1. Reactions only in the bulks

dC fL% i 0 for non-volatile i (15i5 Ncl) dn I C fL% i = C i for volatile i (Ncl + 15i 5Ncliq)

Assuming the reactions occur only in the bulks (reactions in the films are ignored), we can neglect reactions– transfer interaction and molar transfer rates can be expressed as in non-reactive systems.

5.2. Reactions in the bulks and in the films ( fast reactions) With fast reactions operating, mixing time can be equal or higher than reaction time (Astarita, 1967). In these cases the differential balances for mass continuity in liquid and vapor films have to include generation terms too. Rigorous formulation of film continuity equations based on Generalized Maxwell– Stefan relations requires numerical solution methods (Higler et al., 1998; Higler et al., 1999; Higler, Krishna, & Taylor, 1999a) that are very time consuming and, generally, can introduce instability on the whole model solution (Kreul et al., 1999). Furthermore, by neglecting reactions in the films, no significant differences in the results were found (Higler et al., 1998) if reaction velocities are rather low (Suzuki, Yagi, & Hirata, 1971); moreover, Kreul et al. (1999), on the basis of four very different test systems, with very fast reactions too, concluded that reaction– transfer interaction may be neglected because the differences on results can be smaller than the overall model accuracy. In developing our model we have taken into account the coupling between chemical reactions and mass transfer. In particular we have developed two models (‘analytical’ and ‘numerical’ film models) with different description levels. In both models kinetics, chemical equilibrium and instantaneous reactions are considered; we assume that reactions take place in liquid and/or in vapor phase, and that (Pagani & d’Arminio Monforte, 1986): a. a film model can be used; b. film temperature is uniform; c. vapor–liquid interface is flat.

5.2.1. Analytical film model Besides previous hypotheses, we assume that: d. kinetics are low in vapor phase and equilibrium reaction can take place only in vapor phase; e. molar flux Ni for each component i reacting in the film can be expressed as

n= l fL C fL% i = Ci (15i 5Ncliq) being Ncl the number of reactive and non-reactive components in liquid phase only and Ncliq the number of reactive and non-reactive components in liquid phase. Eq. (1) can be solved by introducing the following further assumptions: f. In each of the N rfcL fast reactions there is just one volatile reagent that is absorbed (not stripped); we call ‘a’ the set of these Na components a1, a2 …,aNa ; the fcL ]1 fast same component ai can be involved in N r,a i reactions. g. Concentration of components ‘a’ in liquid phase film is very low with respect to the other Ncliq −Na components (" a) in the film; we call ‘c’the subset of the Nc components c1, c2 …, cNc reacting in the liquid film, ‘s’the subset of the Ns components s1, s2, …, sNs non-reacting in the liquid film. We have Ncliq =Na + Nc + Ns. So, in specific conversion rate expressions R fL% in i Eq. (1), we can assume that concentrations of components ‘c’ and ‘s’ are constant and equal to the interface values, all over the film. fcL h. All the N r,a fast reactions involving the same compoi nent ai are ofthe same order nai in ai. Eq. (1) becomes ’

D eff ai

L

d2C fL ai na + K acii (C fL% ai ) i = 0 dn 2

for components ai (i= 1, Na ) L

D eff ci

2

dC dn

fL% ci 2

%

+

(1a)

j = 1, Na

na K acij (C fL% aj ) j = 0 j

for components ci (i= 1, Nc ) I C fL% Si = C Si = CSi

for components si (i= 1, Ns )

(1b) (1c)

being K aii =

% fcL m = 1,Nr,a i

wi,m

r Im (C Iai)nai

with wi,m : stoichiometric coefficient of component i in fast reaction m; i can be an ‘a’ or a ‘c’ component;

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

C Iai: molar concentration at the interface of component ai ; ai is an ‘a’ component; fcL r Im: rate of the fast reaction m =1, N r,a ,at the interface i conditions. Hikita and Asai (1963) obtained a general analytical approximate solution of Eq. (1a), with the following further position: fcL i. N r,a reactions take place in a pure fast reaction i regime, i.e. components ‘a’ are completely consumed in liquid film; in particular each component ai is consumed in a depth of liquid (Astarita, 1967) u*ai  l fL

 

being u*ai = −

(1+ nai ) 2

 n L

D eff ai K aaii

1/2

with feL

fL% wI,m 5I = 1, N cliq [k fL% − s am = 0 I xI ]

m= 1, N rfeL

To deal with a particular class of instantaneous reactions too we assume that: j. In each of the N ir instantaneous reactions there is just one volatile reagent that is absorbed (not stripped); we call ‘b’ the set of these Nb components b1, b2, …, bNb ; each component bi can be involved in only one instantaneous reaction. A ‘b’ component cannot be an ‘a’ component too. k. Solubility of components ‘b’ in liquid phase is low with respect to the concentrations of the other reacting components in liquid phase; we call ‘d ’the set of these Nd components d1, d2, …, dNd. So we can assume that instantaneous reactions take place on the interface; here and in liquid phase there are no components ‘b’ and concentrations of components ‘d’ are constant and equal to the interface values. The reaction rate r Lm of each instantaneous reaction m is b IV r = L bi,m V wbi,m L m

where bbIV,m is the vapor mass transfer rate at the interface i of component bi involved in reaction m with stoichiometric coefficient wbi,m. If component bi is not reactive in vapor phase, or its generation term in the vapor film is negligible, we have V

eff V b IV ybi bi,m = − c t a k bi

Otherwise, if component bi significantly reacts in vapor film we have (assuming that reaction velocities are constant along the vapor film) 1 effV V b IV ybi − R bfV . bi,m = − c t a k bi 2 i

%k( " m) = 1,N cliq Yk, j

L% dx%k x%ib L% j − x%jb i = %j( " i) = 1,N cliq L L dp ac t k i, j

(15i(" m)5 Ncliq)

db L% i L feL% fcL% =0 −%m = 1, N feL [w i,feL m em ]− al R i r dp (2a)

(2c)

being (2d)

Eq. (2a), Eq. (2b), Eq. (2c), Eq. (2d) are a set of 2Ncliq + N rfeL algebraic-differential equations in the unknowns x%, b L% and e feL% with the following boundary conditions x%i (p= 0)= xi x%i (p= 1)= x Ii

i(" m)= 1, Neliq i(" m)= 1, Ncliq

%j = 1,N cliq b L% j (p= 1)= − bT Solution of this algebraic-differential problem can be achieved by a numerical method, where spatial derivatives are discretized by finite differences obtaining an algebraic system. Film thickness is calculated by average values of the eigenvalues of Maxwell–Stefan diffusivity coefficients and mass transfer coefficients matrices.

6. Back-flow model for packed column In packed columns, a ‘macro-cell’ does not match a physical part of the column itself, as in a tray column; then we have to use a suitable macro-fluid dynamic model and to develop a rule to define the relative parameters. If we neglect radial gradients and consider drop or bubble composition in each ‘macro-cell’ as uniform, axial mixing effect can be described, for each phase, by a suitable number of ‘macro-cells’ in cascade. So, if longitudinal dispersion values are similar in both phases, macro-fluid dynamic model consists of Nt adjacent ‘macro-cells’ (Kramers & Aldberda, 1953), being Nt = 0.5

5.2.2. ‘Numerical’ film model Continuity equations in liquid film become

(2b)

molar flow rates bi for Ncliq − 1 components reacting in the film can be expressed by the generalized Maxwell–Stefan equations for a film model

x%m = 1−%j( " m) = 1,N cliq x%j

[C Iai](1 − nai )/2

i = 1, Na

(1 5i 5Ncliq)

1501



Z Pedp + 1 dp

where Pedp is longitudinal Peclet number of one of the two phases (Fig. 7). When longitudinal dispersion of each phase is quite different, we can suppose that material exchange of each ‘macro-cell’ j with its superior ( j− 1) and inferior ( j+1) ‘macro-cell’ is due to net flows (V and L) and to back flows (V b and L b).

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

1502

RL=

L [(N c P − 1)/Pe z ]− 0.5 L [(N c P − 1)/Pe z ]+ 0.5

RV= being Pe Lz = Pe Lz = Fig. 7. Flow configuration for a ‘macro-cell’ in a packed column. Table 1 BU–PE–ES distillation column Components

  

Z Pe Ldp dp Z Pe V dp dp

Nc p = 0.5

Z Pe c dp dp

L V where Pe c dp ] max{Pe dp, Pe dp}.

Butane (BU), pentane (PE), hexane (ES)

No. of stages Pressure (atm) Condenser Partial (vapor product) Pressure (atm) Reflux rate (kmoles/h) Reboiler Partial (liquid product) Pressure (atm) Bottom product flow rate (kmoles/h) Feed to stage Column internals Column diameter (m) Active area (m2) K-model Excess enthalpy

V [(N c P − 1)/Pe z ]− 0.5 c [(N P − 1)/Pe V z ]+ 0.5

10 7

7 46.1

7 200 5 Sieve tray 2.8 1–50 RKS RKS

Column design and other pertinent data.

With this position, net liquid and ‘vapour’ molar flow rate leaving ‘macro-cell’ j are b c b Lc j = Lj − L j + 1 V j =Vj −V j − 1

7. Model analysis Our goal is to develop an industrial package of practical (and reliable) use, avoiding approaches that introduce several numerical problems on a simulator performance and great difficulties in hydrodynamics and thermodynamics data retrieval, with a negligible improvement in model accuracy (Kreul et al., 1999). Because a process engineer has to deal with serious time constraints, it is not always possible nor useful consider the higher model complexity. Unfortunately, in many cases, it is not possible to know in advances which simplifications are suitable. By the practical use of our package we have analyzed if some model simplifications lead to a significant accuracy change of the model itself. Analysis results are not general, but just restricted to compare, on the basis of some significant problems, model sensitivity to: “ diffusional interactions; “ reaction–transfer interactions. On this subject, more general results will be presented in a paper to be published.

We define circulating ratios

R Lj =

L bj Lb = c j b Lj L j +L j + 1

Vb Vb R = j= c j b Vj V j +V j − 1 V j

By posing L bj =L b

for j=2, Nt

V bj =V b

for j=1, Nt −1

8. Model applications Several industrial and pilot plant scale processes have been analyzed on the basis of our transfer-based model. We show some results obtained in liquid–liquid and in liquid–vapor operations to prove CheOpe behavior on transfer-based model simulation and to analyze how model simplifications can change accuracy of the model itself. We cannot describe some processes in a great detail owing to secrecy.

we have 05 R Lj B1 and 05 R V j B1.

8.1. Distillation columns

Miyauchi and Vermeulen (1963) show the anology between back-flow and dispersed model. In particular, with constant net molar flow rate L c and V c we have

8.1.1. Butane (BU), pentane (PE), hexane (ES) distillation (Goal: analysis of numerical performance) As reported in Table 1, we have forced active area

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

1503

sponding profiles obtained in an equilibrium column (area= ). Equilibrium column model solution is, obviously, the asymptotic solution of transfer-based column model. In Table 2 are provided SQRT monitoring function values (Pagani et al., 1996) during iterative solution for each trial, where



NE

NGE

1

1

n

SQRT =100 % f( 2i + % g¯ 2i

Fig. 8. BU – PE– ES distillation column. Calculated temperature (°C) profiles for four area values (1, 10, 20 and 50 m2) and in an equilibrium column (area = ).

Fig. 9. BU –PE– ES distillation column. Calculated butane molar fraction profiles for four area values (1, 10, 20 and 50 m2) and in an equilibrium column (area = ).

values from 1 to 50 m2 to analyze as column model behavior changes. In Fig. 8 and Fig. 9 profiles of temperature and butane composition for four area values (1, 10, 20, 50 m2) are compared with the corre-

being f( i and g¯ i the scaled residuals of model and constraint equations: the trend of this function can be useful as a measure of the numerical troubles. From Table 2 it results that transfer-model converges to the solution similarly as a corresponding equilibrium stage model if column design specification are consistent with a satisfactory operation of a real column (Taylor et al., 1994; Lee & Dudukovic, 1998).

8.1.2. Methyl-cyclohexane and n-heptane distillation in a laboratory column (Goals: agreement between model and experiments) To verify our model we have consider some distillation runs in laboratory packed column with two different system: methyl-cyclohexane/n-heptane and acetonel/methanol/water (see Section 8.1.3) The column has an internal diameter of 70 mm and it has four beds of structured packing (Sulzer DX), with a total height of 3.135 m. Column design and other pertinent data are included in Table 3. Liquid samples can be drawn out at the end of each column section. We have chosen the mixture methyl-cyclohexane/n-heptane because it is usually used to test a column efficiency. The distillation was carried out at total reflux in a batch way; instead, in the simulations, the reboiler is fed by a flow F equal to the known reflux, the distillate flow is set equal to 2×10 − 3 F. In Fig. 10 we compare calculated and experimental (continuous lines) compositions in the column.

Table 2 BU–PE–ES distillation column SQRT at iteration no.

0 1 2 3 4 5 SQRT monitoring function values.

Area (m2) 1

10

20

50



304.46 135.52 15.25 5.78 0.08 0.14×10−2

304.46 94.54 5.53 0.15 0.96×10−3

304.46 91.19 4.68 0.07 0.59×l0−2

304.46 90.16 3.80 0.42 0.15×10−2

231.70 42.35 1.55 0.02 0.50×10−3

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

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Table 3 Methyl-cyclohexane and n-heptane distillation Number of components Number of sections Sections height from the top (m) Packed height (m) Column internals Channel side S (m) Channel high H (m) Channel base B (m) Channel flow angle q (°) Equivalent diameter (m) Column diameter (mm) Total reflux Reflux flow rate(g/h) K-model Enthalpy Binary diffusion coefficients (L) Binary diffusion coefficients (V) Density (L) Density (V) Interfacial tension Viscosity (L) Viscosity (V) Binary mass transfer coefficients

2 4 0.55; 0.935; 1.045; 0.605 3.135 Sulzer DX stainless steel 0.0041 0.0029 0.0058 45° 0.0033 70 1000–6000 Ideal Ideal Scheibel–Vignes Lennard–Jones–Brokaw Rackett Hayden Macleod–Sugden Polynomial Polynomial AIChE

Column design and other pertinent data.

We have defined an error function as: ERR% =

x EXP −x CALC i i ×100 x EXP i

with

i =C7, MeC6

where x EXP is the experimental weight fraction of the i component i, and x CALC is the calculated one. i In Table 4 are reported percentage differences (ERR%) between experimental and calculated liquid compositions. Differences are in general similar to the analysis errors.

8.1.3. Acetone, methanol and water distillation in a laboratory column (Goals: agreement between model and experiments; analysis of model sensiti6ity to the method used for sol6ing Maxwell–Stefan equations) To test our model with a polar mixture, where mass transfer phenomena could be more important, we have chosen the system acetone, methanol and water. Laboratory column is the same as in the previous tests Section 8.1.2. Column design and other pertinent data are included in Table 5. The distillation was carried out at total reflux in a batch way. In the simulations the reboiler is fed by a flow equal to the known reflux; an unknown acetone flow is fed on the top to obtain a specified acetone composition at an internal point of the column. Column has been simulated using alternatively Taylor and Smith (1982) and equal diffusivity methods. In Table 6 some predicted and measured liquid composition profiles are compared. It’s interesting observe that both methods are near experimental values. Only in few cases errors for both methods are higher than 10%. These errors are acceptable if we consider analysis errors.

Fig. 10. Methyl-cyclohexane(2)/n-heptane() experimental and calculated (continuous lines) compositions in the columns. xW, weight fraction: z, column height from the top (m).

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

1505

Table 4 Methyl-cyclohexane and n-heptane distillation Packing high (from the top) (m)

0 0.55 1.485 2.53 3.135

Reflux=1000 g/h

Reflux= 3000 g/h

Reflux= 5000 g/h

Reflux= 6000 g/h

n-heptane (ERR%)

M-cyclohex. (ERR%)

n-heptane (ERR%)

M-cyclohex. (ERR%)

n-heptane (ERR%)

M-cyclohex. (ERR%)

n-heptane (ERR%)

M-cyclohex. (ERR%)

1.26098 11.57836 0.12008 1.48243 0.13474

0.54798 7.50423 0.25286 9.70577 1.41405

1.98120 5.48222 0.28937 0.58372 0.11753

1.26228 5.69572 0.85890 4.72640 1.54896

0.76184 7.42091 0.47787 1.16066 0.12945

0.54316 7.97356 1.50049 10.12982 1.71386

0.63205 9.91163 3.92198 0.34404 0.12492

0.45295 9.93147 10.07676 2.65090 1.57780

Percentage errors.

8.2. Extraction column (Industrial problem. Goals: agreement between model and experiments; analysis of model performance on a L– L operation) In the last 15 years the applications of liquid– liquid extraction have had a rapid growth, in particular in petrochemical and biochemical processes. Then it is not surprising if a great attention has been paid to describe some fundamentals as drop size, hold-up, axial mixing, and so on. In particular, Rocha-Uribe, Fair, & Humphrey (1986) have proposed some mass transfer coefficient correlation for sieve trays, recently used by Lao, Kingsley, Krishnamurthy, and Taylor (1989) in developing a non-equilibrium stage model for liquid– liquid extraction in sieve tray columns. Table 5 Acetone, methanol and water distillation Number of components Number of sections Sections height from the top (m) Packed height (m) Column internals Channel side S (m) Channel high H (m) Channel base B (m) Channel flow angle q (°) Equivalent diameter (m) Column diameter (mm) Total reflux Reflux flow rate (g/h) Standard fugacity Activity coefficients Enthalpy (L) Enthalpy (V) Vapor pressure Binary diffusion coefficients (L) Binary diffusion coefficients (V) Density (L) Density (V) Interfacial tension Viscosity (L) Viscosity (V) Binary mass transfer coefficients

3 4 0.55; 0.935; 1.045; 0.605 3.135 Suizer DX stainless steel 0.0041 0.0029 0.0058 45° 0.0033 70 2000–10 000 Viriale UNIQUAC UNIFAC Polinomyal Antoine Scheibel–Vignes Gilliland Lyckman–Eckert–Prausnitz Soave Macleod–Sugden Polynomial Polynomial AIChE

Column design and other pertinent data.

Our attention is concerned with packed column for which Seibert and Fair (1988) and, more recently, Godfrey and Slater (1994) have published several interesting correlation that we have used in our package to predict dispersed phase drop diameter, terminal and slip velocity, binary mass transfer coefficients in continuous and dispersed phase, and so on. In particular we analyze the design of a column to extract phenol from an aqueous solution using cumene as solvent. Molar composition of phenol in purified water phase has to be 5 700× 10 − 6. To solve this design problem we can take advantage of the design mode provided by CheOpe. By this mode the height of each section is determined by solving a design specification equation simultaneously with model equations, avoiding to perform several simulations until we have met the design criteria. We have just to define in advance the number of sections and the relative ratios. Column design and other pertinent data and results are included in Table 7. Stages number (36) in transfer-based model allows to obtain a Pedp value corresponding to the measured longitudinal dispersion in the column. Fig. 11 shows phenol concentration in the column: using CheOpe equilibrium stage model, we obtain Xphenol : 380× 10 − 6, using the rate transfer model with Krishna method (Krishna, 1979, 1981) we have Xphenol : 597× 10 − 6,as in the real column is. In Table 8 are reported some algorithm performances.

Fig. 11. Extraction column. Phenol concentration in the column. × , phenol molar fraction; , equilibrium model; 2, mass transfer model.

1506

Packing high (m) (from the top)

0 0.55 1.485 2.53 3.135

Experimental data (weight fraction)

Taylor Smith method (weight fraction)

Diffusivity method prevision (weigh fraction)

Taylor Smith method (% errors)

Diffusivity method (% errors)

Acetone

Methanol Water Acetone

Methanol Water Acetone

Methanol Water Acetone

Methanol Water Acetone

Methanol Water

0.85 0.7 0 0 0

0.15 0.34 1 1 0.7

0.16 0.30 0.99 0.99 0.74

0.151 0.295 0.999 0.999 0.741

4.5 12.6 0.1 0.05 0.001

1.25 13.9 0.03 0.05 1.E-3

Predicted an measured liquid composition values.

0 0 0 0 0.27

0.84 0.7 1.E-3 4.E-8 0

0 0 0 5.E-4 0.27

0.848 0.699 3.E-4 4.E-9 0

6.E-4 4.E-3 7.E-6 5.E-4 0.26

1.60 3.E-3 n.c n.c n.c

n.c n.c n.c n.c 0.003

0.64 1.E-3 n.c n.c n.c

n.c n.c n.c n.c 3.E-3

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

Table 6 Acetone, methanol and water distillation

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511 Table 8 Extraction column

Table 7 Extraction column Number of components Number of sections Ratio between sections Packed height (m) Column internals Column diameter (m) K-model Phenol feed stream Phenol in bottom stream Calculated In actual equipment Binary diffusion coefficients Density Interfacial tension Viscosity % flooding Pe continuous phase Pe dispersed phase M–S equations solution

1507

4 5 3:2 7 Raising rings 38 mm 1.1 Proprietary correlation 4000 ppm

No. of stages CPU (s) No. of iterations No. of variables

Simulation mode

Design mode

Equilibrium

36 21.7 4 1008

36 27.6 6 1008

5 15.5 4 75

Algorithm performances. 597 ppm :600 ppm Scheibel–Vignes Proprietary correlation Proprietary correlation Proprietary correlation 43–35 (from top to bottom) 7 112.65 Krishna, 1979, 1981

Table 9 Laboratory L–V stirred tank reactor Number of components

Column design and other pertinent data and results.

7

Reactions in liquid phase

Initial estimates for composition on the top and bottom of the column have been obtained by equilibrium column results. Initial estimates of mass transfer rates have been set to zero values. The interface compositions are initialized with values corresponding to the bulk quantities. The derivatives of mass transfer coefficients have been neglected: by this way CheOpe lightly increases iterations number, but saves about 10 s for the solution. Correction factors matrix J has been approximated by I, given the low mass transfer rates.

1.A+1.B “ 1.C+1.D 1.C+1.E “ 1.E1 1.C+1.E “ 1.E2 Liquid feed Flow rate (kmoles/h) Composition (molar)

86.877 B= 0.042; E= 0.771; D = 0.187

Vapor feed Flow rate (kmoles/h) Composition (molar) K-model Excess enthalpy

10 A = 1.00 NRTL Null (ideal)

Some pertinent data.

Table 10 Laboratory L–V stirred tank reactor Area (m2) 0.92 Liquid product molar components A B C E D El E2 Conversion rate (kmoles/h) A B C E D El E2 Number of iterations Impeller speed and shape effect on mass transfer.

0.009 0.018 0.018 0.746 0.204 0.002 0.002 −1.99 −1.99 +1.66 −0.33 +1.99 +0.15 +0.18 13

1.00

0.009 0.018 0.019 0.744 0.204 0.002 0.002 −2.07 −2.07 +1.74 −0.32 +2.06 +0.14 +0.18 12

2.00

0.022 0.012 0.025 0.732 0.206 0.001 0.001 −2.53 −2.53 +2.29 −0.24 +2.53 +0.11 +0.13 11

3.00

0.030 0.010 0.027 0.724 0.206 0.001 0.001 −2.70 −2.70 +2.49 −0.21 +2.70 +0.09 +0.12 10



0.063 0.007 0.030 0.697 0.201 0.001 0.001 −3.03 −3.03 +2.89 −0.14 +3.03 +0.06 +0.08 5

1508

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

combined effect of complex equilibrium, competing reactions and physical mass transfer (Suchak, Jethani, & Joshi, 1990). The following bulk gas-phase reactions occur when NO and NO2 are mixed in presence of O2 and water vapor (Carta, 1984): (1) (2) (3) (4) Fig. 12. Reactive absorption column.

8.3. Laboratory L–V stirred tank reactor (Goal: analysis of numerical performance) Some system features are reported in Table 9. To analyze impeller speed and shape effect on mass transfer, we have performed a large simulation plan. In Table 10 some results are shown, concerning a few of this trial where interfacial area is changing; results are compared with those of an equilibrium model. In the solution of this problem, we found that transfer-based model is significantly more difficult to converge than a corresponding equilibrium stage model, as iteration number in Table 10 shows. Particularly, when area value is small, the run requires more iterations to converge, because at each iteration Newton corrections on temperatures, flow rates and compositions are applied with a suitable little step length, found by optimization of a monitoring function (Taylor et al., 1994; Lee & Dudukovic, 1998). Suitable starting values are obtained by solving an equivalent equilibrium based model. By these and other procedures, Newton-Raphson method shows a good robustness and reliability (Pagani et al., 1989).

8.4. Reacti6e absorption column (Industrial problem. Goals: agreement between model and experiments; analysis of model sensiti6ity to reaction– transfer interactions) Chemical processes where separation and reactions are combined have many advantages and are very promising (Sundnacher & Hoffmann, 1996; Lee & Dudukovic, 1998). Processes like these are generally quite complex and the traditional equilibrium stage models often fail in their behavior prediction. For example, in catalytic distillation processes stage efficiency values have great uncertainties and can introduce computational difficulties; in several absorption processes, chemism depends on the combined effect of many chemical equilibrium, fast and instantaneous chemical reactions and physical mass transfer that can’t be taken into account with simpler models. The absorption of NOx gases with alkali is an industrial operation used to remove nitrogen oxides from off-gas. Absorption column behavior depends on the

(5)

2NO+O2 “ 2NO2 2NO2 l N2O4 NO+NO2 l N2O3 NO+NO2+H2O l 2HNO2 3NO2+H2O l 2HNO3+NO

(irreversible and slow) (equilibrium) (equilibrium) (equilibrium) (equilibrium)

At the gas interface the following reactions occur (Suchak et al.): (2%) (3%) (4%) (5%)

2NO2 l N2O4 NO+NO2 l N2O3 NO+NO2+H2O “ 2HNO2 3NO2+H2O “ 2HNO3+NO

(equilibrium) (equilibrium) (irreversible) (irreversible)

The following reactions occur in liquid phase(Suchak et al., 1990): (6) (7) (8) (9) (10)

2NO2+2NaOH “ NaNO2+ NaNO3+H2O N2O3+2NaOH “ 2NaNO2+H2O N2O4+2NaOH “ NaNO2+NaNO3+H2O HNO3+NaOH “ NaNO3+H2O HNO2+NaOH “ NaNO2+H2O

(fast) (fast) (fast) (instantaneous) (instantaneous)

To better run some industrial absorption columns, several conditions have been tested (Cainelli, 1996) on a pilot column in the Enichem P. to Marghera Research Center (Fig. 12). The column has a 0.5 m diameter and it is partitioned in three sections of Sulzer Mellapack 750Y; total height is 1.8 m. The column is fed at the top with both liquid and gas; liquid from the bottom is partially recycled to the top. Table 11 shows as model results are in good agreement with experimental results. We have simulated this operation using both ‘numerical’ and ‘analytical’ film models. Moreover, to analyze model sensitivity to reaction–transfer interactions, we have simulated this operation supposing the reaction 6, 7, and 8 as occurring only in the liquid bulks. Results, for a particular run, are provided in Table 12. We can confirm that analytical model is quite reliable and model simplifications not significantly change accu-

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

racy of the model, but dramatically reduce calculation time. By the other side, results obtained by neglecting reaction– transfer interactions are not reliable. We have tuned our model on pilot plant results, and then we are using our model to look for an optimal industrial plant running.

9. Conclusion Transfer-based models can solve several chemical engineering problems like design and scale-up of reactors and separator units taking into account structural equipment parameters and equipment configuration. This approach avoids the uncertainties of some empirical parameters like efficiency; equipment configuration (number of trays, feeds and side streams position, an so

1509

on) has an exact correspondence within the model that allows to alter design parameters to improve equipment performance. The amount of data input for this approach is much higher than for an equilibrium model and solution algorithm behavior strongly depends on the consistency of these data. The accuracy of transferbased models greatly depends on the accuracy of the mass and heat transfer correlation; fortunately, recent interest for these models and for algorithm development for solving them have stimulate research in this field. The reliability of transfer-based models is then bound to improve in the next future. We have found not very significant differences in simulation and design of some non-reactive distillation columns using Maxwell–Stefan equation or simpler models that neglect interactions in the mass transfer rate equations. Just with very high reflux ratios, differences can be

Table 11 Reactive absorption column Test n.1

Test n.2

Test n.3

XNaOH XNaNO2 XNaNO3 YNO2 YNO

Feeds 0.01849 0.09124 0.001187 0.000121 0.000122

Feeds 0.01852 0.09121 0.00118 0.000296 0.000297

Feeds 0.1179 0 0 0.00033 0.000331

XNaOH XNaNO2 XNaNO3

Liquid outlet Calculated 0.15 0.084 0.0011

YNO2 YN2O4 YN2O3 YNO

Vapor outlet Calculated 8.2 E−5 6.47 E−8 5.15 E−9 9.21 E−5

SYNOx %errors

Calculated 174 E−6 3.33

Liquid outlet Calculated 0.017 0.089 0.0014

Experimental 0.018 0.095 0.0012

Experimental 0.016 0.098 0.0012

Vapor outlet Calculated 2.1 E−4 4.27 E−7 3.71 E−8 2.5 E−4 Experimental 180 E−6

Calculated 460 E−6 33.5

Liquid outlet Calculated 0.113 0.0009 6.8e−5

Experimental 0.085 0.003 2.1e−5

Vapor outlet Calculated 2.25 E−4 4.86 E−7 3.87 E−8 2.52 E−4 Experimental 345 E−6

Calculated 477 E−6 2.65

Experimental 490 E−6

Comparison between some experimental and calculated gas and liquid stream molar fractions. Table 12 Reactive absorption column Type of model

(A) Analytical

YNO2 YN2O4 YN2O3 YNO T

441.38 0.55 0.06 485.23 1

10−6 10−6 10−6 10−6

(B) Numerical 447.20 0.57 0.06 484.95 21.2

10−6 10−6 10−6 10−6

(C) No interactions

(D) No interactions CI

316.27 10−6 0.28 10−6 0.04 10−6 501.69 10−6 l.7(a)

463.25 0.61 0.07 483.08 0.7

10−6 10−6 10−6 10−6

Comparison between different approaches on the basis of a significant mn. yi =molar fraction of component i on the top of the column; t, simulation time/simulation time for analytical solution. Analytical(A), analytical solution; Numerical(B), numerical solution; No interactions (C) no reaction–transfer interactions; No interactions CI (D), as (C) but using liquid interface composition as liquid composition of diffusing gas. a More iterations are required because starting point for this case was worse than in other cases.

G. Pagani et al. / Computers and Chemical Engineering 25 (2001) 1493–1511

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significant and have some influence in column design, when binary diffusivity coefficients widely differs each others. On this subject, more general results will be presented in a paper to be published. In any case, implementation of effective diffusivity methods is no harder than rigorous ones and computational time of both methods is rather similar; so we in general prefer to use full matrix methods. However, we note that, with many components, numerical troubles can sometimes occur using full matrix methods. In these cases a solution obtained by an effective diffusivity method can be a good starting point for the rigorous solution. In simulation and design of a reactive system with fast reactions, reaction– transfer interactions cannot be neglected, but in our opinion analytically solved approximate models often give results as good as more complex models.

$ $ − 1 in ’ out $ $ − 1 out ’ {d in … j } lx in … j } lx out j d j [… j ] j + {d j d j [… j ] j

= {−ƒjd $j [… $j ] − 1€j }’ For each block, corrections lZj can be obtained by (Eq. (IIb)). Reduced Newton-Raphson behavior is exactly the same as the original algorithm; the algorithm can be represented as O.K.

O

Bm

{x ’, Z ’} [ {f’} [ lx [ lZ O

<

n

If the set of the original (not linearized) Nz ‘local’ equations are locally solved, € ’j = 0 and solution of the linearized system can be found operating on N×N block-matrices, corresponding to the following block of equations o o − 1 in ’ out o o − 1 out ’ {d in … j } lx in … j } lx out j d j [… j ] j + {d j d j [… j ] j

= {−ƒj }’ Appendix A. Reduced Newton-Raphson approach The Ns state variables xj describing a cell j are partitioned between N output variables x out and Nz j internal variabes Zj, where the later are local variables for cell j. Then cell block of equations can be written as out fj (x in j , x j , Zj )=0

(I)

in in where x in j = x 1 , … x n are the Ns ×Nu state variables describing the i(" j ) = 1, Nu cells having streams entering cells j. out ’ In a general point ’ {x in j , x j , z j }, system (Eq. (I)) can be linearized as following in out J in lx out +J $j lZj = − f ’j j lx j + J j j ’

with J in j =

(II)

) ) ) ) ) ) (fj ’ (x in j

(fj ’ (x out j (fj ’

J out = j ’

J $j =

(Zj

Model equations structure allows to partition previous linear equations in two groups ’ ’ in out’ ” in lx out +” $j lZj = −ƒ ’j j lx j +” j j

(N equations) …

in’ j

lx in j +…

out’ j

lx out +… lZj = − € ’j j

(Nz ’local’ equations) ’

(IIa) $’ j

(IIb)

where … $j is not singular. Solution of the linearized system can be found operating on N × N block-matrices, corresponding to the following block of equations.

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