non-equilibrium stage modeling and non-linear … preliminary column design for the neq model is...

Katariya, Kamath, Mahajani and Moudgalya, Comp. and Chem. Eng., November 06

Non-equilibrium stage modeling and Non-linear

dynamic effects in the synthesis of TAME by

Reactive Distillation

Amit M. Katariya, Ravindra S. Kamath, Kannan M. Moudgalya and

Sanjay M. Mahajani∗

Department of Chemical Engineering,

Indian Institute of Technology Bombay,

Mumbai-400076. INDIA

* Corresponding author:

Prof. Sanjay M. Mahajani,

Department of Chemical Engineering, IIT Bombay

Mumbai-400076.

E-mail: [email protected]

Tel :- +91-22-25767246 Fax : +91-22-25726895.

1


Abstract

Tertiary amyl methyl ether (TAME) is a potential gasoline additive that can be

advantageously synthesized using the Reactive Distillation (RD) technology. This

work emphasizes on non-linear effects in dynamic simulations of reactive distil-

lation column. For certain configurations, dynamic simulation with equilibrium

stage (EQ) model leads to the sustained oscillations (limit cycles) which have

been reported in our earlier work (Katariya et al., 2006b). Feed condition and

Damkohler number are the important parameters that influence the existence of

these effects. To confirm the authenticity of the observed non-linear behaviors, a

more realistic and rigorous dynamic NEQ model for a packed column is developed

which uses a consistent hardware design. The steady state behavior of the NEQ

model is examined by varying the number of segments and the column height.

The dynamic simulation and the bifurcation study with stability analysis indicate

that the parameter space, in which oscillations may be observed, is shifted in the

case of NEQ model.

Keywords: Reactive Distillation, Dynamic Simulation, Continuation analysis, Non-

equilibrium model, Hopf Bifurcation, Oscillations.

2


Introduction

Computer-aided design and simulation of multi-component multistage separation

processes such as distillation, gas absorption and reactive distillation are important

aspects of modern chemical engineering. Currently, such simulations are based on

the very well-known equilibrium stage model. The EQ model assumes that the

vapor and liquid leaving a stage are in equilibrium. Equilibrium stage simula-

tions are frequently termed rigorous, but this appellation is not entirely justified

because in actual operation, columns rarely, if ever, operate at equilibrium. The

degree of separation is, in fact, determined as much by mass and energy trans-

fer between the phases being contacted on a tray or within sections of a packed

column, as it is by thermodynamic equilibrium considerations. The usual way

of dealing with departures from equilibrium in multistage towers is through the

use of stage and/or overall efficiencies or use of height equivalent to a theoretical

plate (HETP) in case of packed towers. Though, this may be a useful approach

for simulating an existing column for which there is a good deal of data available,

it may not be possible to predict safely how the column will perform under quite

different operating conditions (Baur et al., 2000a). Furthermore, it is difficult to

use this approach to simulate new processes in the design stage for which no plant

data exists.

It is advantageous to use NEQ model over the EQ model due to some of the

following reasons. It eliminates the need for efficiencies and HETPs. The op-

erating strategies for the influence of chemical reactions on separations can be

accounted in a better way. The over-design or under-design can also be avoided as

the tray and packed columns are modeled with greater accuracy thereby reducing

the capital and operating costs. Also, as mentioned before the NEQ model is more

realistic as compared to the EQ model and represents a more accurate modeling

of reactive systems.

3


The non-equilibrium (NEQ) model assumes that the vapor-liquid equilibrium is

established only at the interface between the bulk liquid and vapor phases, and

employs a transport-based approach to predict the flux of mass and energy across

the interface. Various authors have presented steady state non-equilibrium stage

models for tray (Higler et al., 1999; Baur et al., 2000a,b, 2003) as well as packed

(Sundmacher and Hoffmann, 1996; Peng et al., 2002; Jakobsson et al., 2004; As-

prion, 2006) reactive distillation columns. For the purpose of design, optimal

operation, and the control of the reactive distillation process, a rigorous theoreti-

cal dynamic model is required. The modeling and simulation with NEQ model is a

computationally rigorous activity as it involves large number of highly non-linear

equations like pressure drop correlations, packing holdup correlations etc. Hence,

there are very few publications on dynamic simulation of tray (Baur et al., 2001;

Schenk et al., 1999) and packed (Kreul et al., 1998; Peng et al., 2003; Noeres et al.,

2004; Xu et al., 2005) reactive distillation columns using NEQ model. They differ

in the way the mass and heat transfer resistances are incorporated in the model.

The main differences are 1. the use of driving force for the mass transfer: some

use concentration gradient, whereas others use the correct gradients of chemical

potentials and fugacities. 2. the diffusivity models: Fick’s law or Stefan-Maxwell

approach and 3. Number of phases involved: two phase or pseudo homogeneous

model and three phase heterogeneous model.

TAME, a popular fuel additive, is commercially produced by Reactive Distilla-

tion through the reaction of methanol with isoamylene coming from C5-stream of

the refinery. It is a widely studied model system to understand the complex be-

havior of reactive distillation. A few case studies of TAME synthesis in RD using

both EQ and NEQ models have appeared in the literature (Subawalla and Fair,

1999; Mohl et al., 1999; Baur et al., 2000b, 2003; Peng et al., 2003; Ouni et al.,

4


2004; Katariya et al., 2006a). Most of these except that by Peng et al. (2003) are

restricted to steady state analysis. Peng et al. (2003) have compared the dynamic

rate-based and equilibrium models for a packed reactive distillation column for the

production of tert-amyl methyl ether (TAME) and proposed a new approach to

simplify the dynamic rate-based model by assuming the mass transfer coefficients

to be time invariant. It can reduce the number of equations by up to two-third

and still accurately predict the dynamic behavior. A high-index problem in the

models may arise if the pressure drop is not related to vapor and liquid flow rates

(Kreul et al., 1998).

Synthesis of TAME by reactive distillation is known to exhibit non-linear dy-

namic effects such as multiple steady states and relevant literature is reviewed

in our earlier work (Katariya et al., 2006b). We showed for the first time that

under certain conditions, the EQ model based dynamic simulation of the reactive

distillation column exhibits another type of non-linear effect i.e. sustained oscil-

lations or limit cycles (Katariya et al., 2006b). In order to further examine the

authenticity of this observed non-linear dynamic effect, here we present a rigorous

dynamic non-equilibrium model for the synthesis of TAME in packed RD columns.

The model includes all the essential dynamic terms comprising vapor and liquid

holdups. Since the TAME synthesis is carried out at high pressure (4.5 bar), it is

important to consider the vapor mass and energy holdups in the modeling equa-

tions, which are otherwise neglected in the earlier studies due to index issues (Peng

et al., 2003) and low pressure operations (Noeres et al., 2004). Also, time variant

mass and heat transfer coefficients are considered in our model, which are made

time invariant in the earlier studies due to computational difficulties and simu-

lation time. We also present a comparative study of steady state and dynamic

simulations using both EQ and NEQ models. Detailed index analysis of the NEQ

model is carried out and the variables responsible for the higher index in each

5


case are identified and accordingly, model simplifications are made without com-

promising on essential dynamic terms. The work by Reepmeyer et al. (2004) may

be referred to understand and handle some of the numerical issues involved in the

dynamic simulations. Also to systematically investigate the non-linear dynamics

of the system, bifurcation behavior of the simplified NEQ model with stability

analysis has been carried out in some cases which, to the best of our knowledge,

has not been reported till date.

Model description and hardware specification

A rigorous NEQ model has been developed to examine the effect of column hard-

ware and heat and mass transfer resistances on the non-linear behavior of the

RD column. The purpose is to compare the performance and behavior with that

obtained by the EQ model in our earlier studies (Katariya et al., 2006b). We re-

fer to Powers et al. (1988) for detailed model implementation and computational

aspects. In case of NEQ models, the specification of hardware design information

such as column diameter, tray or packing type and geometry etc., is mandatory. A

packed column has been selected for the NEQ simulations. Each continuous sec-

tion of the packed column is divided into a number of segments, each of which acts

as a non-equilibrium stage. The packing selected for the reactive and non-reactive

rectifying and stripping sections are KATAPAK-S and Sulzer-BX, respectively.

The hydraulic and mass transfer correlations for the selected packing are obtained

from Rocha et al. (1993, 1996) and Kolodziej et al. (2004), respectively and are

given in appendix A.

The preliminary column design for the NEQ model is derived from the steady

state results of the EQ model. The column diameter is estimated by applying the

fractional approach to flooding. The height of each packed section is calculated

6


Packing height = 8.41 mNumber of segments = 71

Number of segments = 88

Rectifying section4−theoreticalstages

Reaction section19−theoretical stages

Catalyst loading = 29518 eq[H+]

Stripping section10−theoretical stages

Packing height = 5.58 m

TAME

Column pressure = 4.5 bar

Isopentane + MeOH

Reflux ratio = 1.5

Pure methanol feed215 kmol/h305KStage location = 24

Pre−reacted feed

1196.1 kmol/h

Stage location = 29 Methanol 0.13042 2M1B 0.00798 2M2B 0.07018

isopentane 0.66016 TAME 0.13126

325K

Packing height = 2.43 mNumber of segments = 30

Reboiler duty = 20.5 MW

Column diameter = 3.87 m

Figure 1: The conceptual column configuration used for EQ and NEQ simulations

along with the hardware design derived from it.

by multiplying the HETP with the corresponding number of theoretical stages.

The hardware design for the selected conceptual column configuration is shown in

Figure 1.

Kinetics and Thermodynamics

The following three reactions have been considered while modeling the process for

the synthesis of TAME, which includes two synthesis reactions for TAME from

the isomers of isoamylene and one isomerization reaction. The side reactions such

as dimerization of methanol and formation of TAA have been neglected. The rate

equations are as given below. The temperature dependent rate constants and the

equilibrium constants for the reactions are obtained from Rihko and Krause (1995)

and Faisal et al. (2000).

MeOH + 2M1B ⇐⇒ TAME R1 = kf1(a2M1B

aMeOH−

1

Ka1

aTAME

a2MeOH

)

MeOH + 2M2B ⇐⇒ TAME R2 = kf2(a2M2B

aMeOH

−1

Ka2

aTAME

a2MeOH

)

7


2M1B ⇐⇒ 2M2B R3 = kf3(a2M1B

aMeOH−

1

Ka3

a2M2B

aMeOH)

As the system consists of mixture of polar and non-polar components, it is highly

non-ideal and the use of activity based kinetics and thermodynamics is justifiable.

The UNIQUAC model has been used for describing non-ideality of the liquid phase,

with binary interaction parameters taken from HYSYS. All the thermodynamic

and kinetic parameters used in the study have been also reported in our earlier

work (Katariya et al., 2006b). The process design of the column and the input

conditions have been obtained from Subawalla and Fair (1999). Figure 1 shows

the column configuration along with operating and design parameters used for

the study. Here, methanol is fed in excess, which is required to form a minimum

boiling azeotrope with inerts (e.g.isopentane) and separate them efficiently from

the top of the column. Escess methanol also helps to maintain the desired tem-

perature (330-350K) in the reactive section of the column. Also, Subawalla and

Fair (1999) have observed in their analysis that if the methanol used is less than

the amount required to form an azeotrope then the conversion of amylene and the

purity of the TAME are adversely affected.

Model Equations

A schematic representation of the NEQ stage is shown in Figure 2. This NEQ

stage may represent a tray or a cross-section of a packed column. The stage

equations are the traditional equations for mass and energy balances for individual

phase, in which mass and heat transfer rates are also included. Bulk variables

(compositions, flow rates, molar fluxes, energy fluxes, temperatures) are different

from the interface variables. Equilibrium is assumed to be only at the interface

and temperatures of vapor and liquid streams are not identical. Condenser and

re-boiler are treated as equilibrium stages.

8


Figure 2: The typical NEQ stage representing tray or section of packed column.

Total material balance equation for the NEQ stage are as below.

dMLk

dt= Lk−1 + FL

k − (Lk + SLk ) + Ac

C∑

i=1

NLi,k +

r∑

m=1

C∑

i=1

γi,mRm,kǫk (1)

dMVk

dt= Vk+1 + F V

k − (Vk + SVk ) −Ac

C∑

i=1

NVi,k (2)

Component material balance equation are written as:

dMLk xi,k

dt= Lk−1xi,k−1 + FL

k xfi,k − (Lk + SLk )xi,k + AcN

Li,k

+r∑

m=1

γi,mRm,kǫk (3)

dMVk yi,k

dt= Vk+1yi,k+1 + F V

k yfi,k − (Vk + SVk )yi,k −AcN

Vi,k (4)

Ac is the interfacial area for vapor-liquid mass transfer and NVi,k and NL

i,k are vapor

and liquid mass transfer fluxes respectively. Only (C−1) component material bal-

ance equations are independent, summation constraint on vapor and liquid phase

9


compositions is used to get the composition of the remaining components.

Energy balance equation:

dELk

dt= Lk−1hk−1 + FL

k hfk − (Lk + SLk )hk −Qk

+ Ac[hLtk(T

Ik − TL

k ) +C∑

i=1

NLi,kH̃

Li,k] (5)

dEVk

dt= Vk+1Hk+1 + F V

k Hfk − (Vk + SVk )Hk

− Ac[hVtk(T

Vk − T I

k ) +C∑

i=1

NVi,kH̃

Vi,k] (6)

H̃Vi,k and H̃V

i,k are partial molar enthalpies of vapor and liquid. Vapor liquid equi-

librium at interface can be as given below.

yIi,k = KIi,kxIi,k (7)

Mass and energy conservation equations for interface can be written as below. It

is assumed that reaction does not take place in the liquid film.

NVi,k = NL

i,k (8)

hLtk(T

Ik − TL

k ) +C∑

i=1

NLi,kH̃

Li,k] = hV

tk(TVk − T I

k ) +C∑

i=1

NVi,kH̃

Vi,k (9)

Summation constraints for the mole fractions in bulk vapor and liquid as well as

at the interface are written as:

C∑

i=1

xi,k = 1.0 (10)

C∑

i=1

yi,k = 1.0 (11)

C∑

i=1

xIi,k = 1.0 (12)

C∑

i=1

yIi,k = 1.0 (13)

10


Fick’s law approach described earlier (Peng et al., 2003) is used to calculate the

mass transfer fluxes.

NLk = CL

tjkV(xIk − xk) + xk

C∑

i=1

NLi,k (14)

NVk = CV

tjkL(yk − yIk) + yk

C∑

i=1

NVi,k (15)

NLk and NV

k are the vectors of mass transfer fluxes of the order (C − 1) for each

stage. Only (C − 1) mass fluxes are independent, summation equation of the in-

terface mole fractions are used to find the mass flux of the last component. kL

and kV are the mass transfer matrices of order (C-1) × (C-1) for each stage.

We used a method suggested by Krishna and Standart (1976) which involves re-

lating [k⋆] to the binary pairs of mass transfer coefficients through solution of

Maxwell-Stefan equations for film model. Matrices are calculated using follow-

ing relations with assumption that the matrices accounting the influence of mass

transfer on the mass transfer coefficients are identity.

[kV]

=[BV

]−1

(16)

[kV]

=[BL

]−1 [

ΓL]

(17)

The elements of the matrix [B] have been calculated using the following equations.

Bii =zi

κi,C

+C∑

k=1

k 6=i

zk

κi,k

(18)

Bij = zi

(1

κi,j

−1

κi,C

)(19)

where zi is the mole fraction of vapor or liquid phase and κi,j is the mass trans-

fer coefficient of the binary pair in an appropriate phase. The packing selected

for the non-reactive and the reactive sections are Suzler-BX and KATAPAK-S,

respectively. Correlations for calculating the binary mass transfer coefficients are

given in Appendix A. Binary diffusion coefficients in the correlations are calculated

using method given by Wilke and Chang (1955) for liquid phase and correlation

11


given by Fuller et al. (1966) for gas phase. Maxwell-Stefan diffusivities are derived

from these infinite dilution diffusivities (Doij) using equation 20.

Dij = (Doij)

(1+xj−xi)/2(Doji)

(1+xi−xj)/2 (20)

[Γ] is a matrix of the thermodynamic factor, calculated using following relation.

Γi,j = δi,j + xi∂lnγi

∂xj(21)

Damkohler number is a key parameter which is the ratio of characteristic residence

time to characteristic reaction time. In the present work Damkohler number is

defined based on the total feed to the column and total amount of the catalyst

used in the column. Boiling point of the lowest boiling component is used as the

reference temperature.

Da =WTkf,ref

FTotal

(22)

Steady state Analysis

Steady state simulations with the help of developed NEQ model are carried out for

the design and operating parameters given in Figure 1. This step is mandatory for

getting the initial steady state required for carrying out the dynamic simulations.

Same design and operating parameters as in EQ stage simulations (Katariya et al.,

2006b) have been used to compare the behavior (P = 4.5 bar, Qreb = 20.5 MW,

R = 1.5). Pure methanol is fed at the bottom of the reactive zone whereas the

pre-reacted feed was supplied at the midspoint of non-reactive stripping section.

The NEQ model equations are implemented in large scale equation oriented sim-

ulator DIVA (Kroner et al., 1990). DIVA uses the equation oriented approach

for solving all the differential and algebraic equations simultaneously. This comes

with an inbuilt package for continuation and stability analysis for the DAEs sys-

tems.

12


Initially the column height in NEQ model was divided in the same number of

slices as the number of equilibrium stages, i.e. 33 (4 slices in non-reactive recti-

fying, 19 slices in reactive section and 10 slices in non-reactive stripping section).

Following attempts have been made to arrive at the initial steady state of the

NEQ simulations which is required for starting the dynamic simulation.

1. Steady state simulation of NEQ model: The results from equilibrium stage

model were used as initial guesses to the non-linear algebraic equation solver.

The guess values for bulk and interface variables were assumed to be same.

Convergence failed in this case.

2. Steady state simulation of NEQ model with infinite mass and heat transfer

coefficients: The model when solved with infinite mass and heat transfer

coefficients, is equivalent to the EQ model. This model with the initial

guesses same as in an attempt one above is used for the simulation and a

continuation approach was used to reach the finite values of mass and heat

transfer coefficients. In this case also the convergence could not be obtained

due to the non-linearity and interaction of the pressure drop and holdup

equations.

3. Dynamic simulation of rigorous NEQ model: The integration of the rigorous

dynamic model for relatively large time to arrive at the steady state has

been carried out. This attempt was also failed due to large number of stiff

DAEs.

4. Dynamic simulation of constant holdup NEQ model: The dynamic model

was simplified with certain assumptions like constant molar holdup of liquid

and negligible vapor and energy holdup on each segment of the column and

the required steady state was obtained. This steady state was then used in

further simulations. This approach was found to work well in most of the

cases. The number of segments in each section of the column was increased

13


such that there was no further change in the column profiles, conversion of

isoamylene and purity of the TAME in the bottom.

Influence of number of segments

The effect of the number of segments in the packed sections of the RD column on

the steady state results using the NEQ model is shown in Figure 3. When the

number of segments in a particular section is chosen to be same as the number of

corresponding theoretical stages in the EQ model, a significant difference in the

composition profile predicted by the two models is seen but only in the stripping

section. As the number of segments in the stripping section increases, the NEQ

profile in the stripping section moves in the direction towards the EQ profile,

crosses it and continues to move away from it. Finally a stage is reached when

a further increase in the number of segments does not significantly influence the

composition profiles. Peng et al. (2002) have correlated the effect of NEQ segments

with the extent of back-mixing. At very large number of segments, back-mixing

in liquid and vapor phases is virtually absent and there is no effect of further

change in the number of segments. Thus Figure 3 shows that the number of

slices i.e the extent of back-mixing in the packed columns strongly influences the

composition profile. In real columns, back-mixing and other non-ideal conditions

cannot be eliminated and hence an appropriate number of segments should be

used. However, this number cannot be determined a priori. For steady state

simulations it was observed that for the number of segments greater than 189 the

composition profiles do not change significantly.

As discussed before, the objective of the present work is to confirm whether the

oscillations observed in the EQ model predictions still persist in the case of NEQ

simulations. In other words, we examine whether the consideration of mass and

heat transfer limitations would influence the presence of non-linear dynamic be-

14


0 0.2 0.4

0

2

4

6

8

10

12

14

16

Mole fraction

He

igh

t o

f th

e c

olu

mn

[m

]

0 0.5 1

0

2

4

6

8

10

12

14

16

Mole fraction340 360 380 400

0

2

4

6

8

10

12

14

16

Temperature [K]

EQ (35 stages)NEQ (33−segments)NEQ (47−segments)NEQ (189−segments)

Isoamylene

Methanol

TAME

Isopentane

Figure 3: Comparison of the steady state composition and temperature profiles

along the height of the column for the EQ model and NEQ model with various

number of total segments.

15


havior. The oscillations being a non-linear dynamic effect, may originate from the

nonlinearity in the vapor-liquid equilibrium relation, reaction kinetics or the func-

tional dependence of the physical properties on the compositions and/or tempera-

ture (Kienle and Marquardt, 2003). Hence, the probability of realizing oscillations

with NEQ model will be more if we work in the same region of composition and

temperature space for which the oscillations were observed in the case of the EQ

model predictions. As mentioned before, in the case of NEQ predictions, the com-

position profiles, conversion of isoamylene and TAME purity in the bottom are

significantly different from that obtained by the EQ model and are very sensitive

to the change in column height and number of segments. Hence, we present here

two different column designs as mentioned below. Further we perform dynamic

simulations and the bifurcation analysis in some cases, to explore the possibility

of the presence of oscillations.

• In the first case, we vary the column height, especially that of the stripping

section, such that the composition and temperature profiles and isoamylene

conversion/TAME purity are close to the EQ predictions. The number of

segments used here is such that the back mixing is absent (i.e. 189 segments)

and there is no further change in the composition profile with increase in

number of segments.

• In the second case, the number of segments, which represent the extent of

back-mixing in vapor and liquid phases, is varied such that composition and

temperature profiles of EQ and NEQ model match reasonably well. This is

the case with partial back-mixing.

16


Case 1: NEQ model without back-mixing

Influence of height of the stripping section

The steady state result using the NEQ model, with total number of slices 189,

showed a much lower isoamylene conversion (67.6 %) compared to 84.6% obtained

in the EQ model. Since the primary objective was to compare the non-linear

dynamic effects of EQ and NEQ models, getting similar conversions and end com-

positions is essential. So, an attempt was made to change the hardware design

(diameter and heights of sections) estimated from the EQ model to a new design

such that EQ and NEQ models give similar results, and the composition and tem-

perature profiles roughly lie in the same domain. From the previous analysis, it

is clear that the stripping zone plays a crucial role in the column behavior. The

effect of height of the stripping section on isoamylene conversion was investigated

using continuation analysis and is shown in Figure 4. Surprisingly, conversion

of isoamylene increases with decrease in height of the column. This is clearly a

counter-intuitive effect since we expect that a larger packed height should result

in a better separation and as per the principles of RD, a better separation of the

product TAME from the reactants should result in enhanced amylene conversion.

However, an optimum in conversion was observed beyond which conversion of

amylene again decreases as height is decreased. This is because an increase in the

number of stages in the stripping section results in better separation of not only

TAME but C5 olefins also, which are the reactants. C5-olefins under otherwise

similar conditions find the way out from the bottom thereby causing a reduction in

their concentrations in the reactive zone. Hence, the overall isoamylene conversion

decreases with an increase in the height of the stripping section. This particular

effect has also been confirmed through EQ stage simulations as well. The optimum

stripping height in this case was found to be about 1.62 m. A height of 1.98 m was

selected for the new design since it not only gives a conversion of isoamylene close

17


0 1 2 3 4 5 60.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

Height of the stripping section [m]

Isoa

myl

ene

conv

ersi

on

Figure 4: Effect of height of the stripping section on the isoamylene conversion.

to the optimum but also the conversion and end composition are very similar to

that given by the earlier design with EQ model.

The steady state composition and temperature profiles of the NEQ model using

this new design are plotted along with that of the EQ model in Figure 5. Even

though the top and bottom compositions and temperature profiles are similar,

certain sections of the stripping zone show different compositions.

Dynamic Simulation

For the NEQ model, the liquid and vapor flow rates in the packed sections are not

responsible for the higher index as algebraic equations for these variables in terms

of pressure drop and holdup correlations are incorporated in the model. However,

the liquid and vapor flows associated with the condenser and the re-boiler can pose

high-index problems as those are modeled using equilibrium assumptions. One will

arrive at index two DAEs when the equations for the holdup as a function of vapor

and/or liquid flows e.g. controller equations are not explicitly considered in the

18


0 0.2 0.4

2

4

6

8

10

12

Mole fraction

Hei

ght o

f the

col

umn

[m]

0 0.4 0.8Mole fraction

350 400Temperature [K]

4.5 4.55Pressure [bar]

0 1 2Holdup [Kmol]

Met

hano

l

Isoa

myl

ene

TA

ME

Isop

enta

ne

Vap

or

Liqu

id

EQ model NEQ model

Figure 5: Comparison of composition and temperature profile for the EQ and the

NEQ model with the new design.

model. If these equations are not available (open loop column) then some of the

differential equations need to be converted to algebraic equations by neglecting

the dynamics to eliminate the index problem. It can be proved with the help of

a detailed index analysis that at least the following differential equations need to

be converted to algebraic equations:

1. Energy balance for the condenser: this is because condenser load does not

appear in any other algebraic equation.

2. Energy balance for the re-boiler: this is to account for re-boiler duty, bot-

tom flow rate or vapor flow from the re-boiler depending upon the bottom

specification.

3. Total material balance for the re-boiler and condenser: this is to account for

either vapor or liquid flow.

19


Apart from this rigorous model, we define constant holdups dynamic NEQ model

as the one in which differential equations for all the total material and energy

balances are converted to algebraic equations, as was done in the case of the EQ

model (Katariya et al., 2006b). Starting from a steady state with the same operat-

ing conditions, the dynamics of EQ model, rigorous NEQ model and the constant

holdups NEQ model for a 2% step increase in the pre-reacted feed flow rate have

been studied. As seen from Figure 6, both the NEQ models show a slightly dif-

ferent dynamics but reach the same steady state while the EQ model reaches a

different steady state as expected. Both the NEQ models take almost equal com-

putation times since the total number of equations (differential and algebraic) is

the same. However, the rigorous NEQ model was much more difficult to converge

for larger step changes because of the stiffness issues. The convergence properties

of the constant holdups NEQ model were very similar to that of the EQ model

with almost no convergence problem up to ± 5% step changes in operating pa-

rameters. The computation time for the NEQ models was observed to be almost

15 times higher than that of the EQ model.

Figure 7 shows the response of the average values of liquid and vapor side heat

and mass transfer coefficients to a step increase in the pre-reacted feed flow rate

using rigorous NEQ model. It has been seen that for very small changes in the

feed there are significant changes in the heat and mass transfer coefficients. This

justifies the fact that time variant heat and mass transfer coefficients have to be

considered while simulating the NEQ model for reactive distillation.

To confirm the authenticity of the sustained oscillations observed in the dynamic

EQ model, similar analysis was repeated with constant holdup NEQ model us-

ing the new hardware design (i.e 189 total number of segments, stripping section

height = 2.43m, reactive section height = 8.41m and rectifying section height =

20


0 5 10 15 200.78

0.79

0.8

0.81

0.82

0.83

0.84

0.85

0.86

0.87

0.88

TA

ME

pur

ity in

Bot

tom

Time, [h]

NEQ (Constant holdup)NEQ (rigorous)EQ model

Figure 6: Dynamic response of EQ, rigorous and ‘constant holdups’ NEQ model

for a 2% step change in feed flow.

2.98m). As seen in Figure 8, unlike the EQ model, oscillations were not observed

and the system always reaches the corresponding steady state. Thus, the oscil-

latory behavior that existed in the EQ model disappears in the NEQ model for

the desired isoamylene conversion and TAME purity in the bottom. However, it

must be noted that the parameter space wherein the non-linear dynamic effects

are observed in EQ model, is likely to shift in the case of NEQ model simulations.

Such a possibility can be ascertained only by studying the bifurcation behavior

with respect to all possible parameters and their combinations, using continuation

method coupled with stability analysis, which is computationally an intensive task.

The presence of non-ideality in terms of the partial back-mixing may also influence

the column performance. Hence the model with partial back-mixing has been

considered in the next section for the realistic comparison of non-linear dynamics.

21


0 5 10 15 20

32.5

33

33.5

Vap

. sid

e M

.T. c

oef.

[m/h

r]

0 5 10 15 20

1.3

1.32

1.34

1.36

Time, [h]

Liq.

sid

e M

.T. c

oef.

[m/h

r]

(a)

2 4 6 8 10 12 14 16 18 20150

155

160

165

170

Vap

. sid

e H

.T. c

oef.

[W/m

2 K]

2 4 6 8 10 12 14 16 18

1455

1460

1465

1470

1475

Time, [h]

Liq.

sid

e H

.T. c

oef.

[W/m

2 K]

(b)

Figure 7: Dynamic response of (a) Mass transfer coefficient and (b) Heat transfer

coefficients, in the rigorous NEQ model to a 2% step change in feed flow.

Case 2: NEQ model with Partial back-mixing

As mentioned earlier, it is difficult to perform the bifurcation and stability anal-

ysis of the model with 189 NEQ segments. Also to have a realistic comparison of

non-linear behavior observed in the EQ stage simulations (Katariya et al., 2006b),

NEQ model with partial back-mixing is considered. For studying the detailed

bifurcation behavior with stability analysis, simulations were carried out with re-

duced number of segments such that the steady state column profiles with both

EQ and NEQ models are close to each other, with approximately same conver-

sion of isoamylene and the purity of TAME in the bottom. Figure 9 shows the

steady state composition and temperature profiles when the column is divided

in 47 segments (6 segments in non-reactive rectifying section in 2.43 m height,

23 segments in reactive section in 8.41m height, and 18 segments in non-reactive

stripping section in 3.4m height), with almost same conversion of isoamylene and

TAME purity in the bottoms. A very good match with the base case EQ profiles

is observed.

The response of the rigorous NEQ model with partial back-mixing, constant

holdups NEQ model with partial back-mixing and constant holdup EQ model,

to a change in 2-methyl-1-butene concentration can be seen in Figure 10. Signif-

22


0 50 100 150 200 250 300

402

404

406

Bot

tom

tem

p, [K

]

0 50 100 150 200 250 300

0.86

0.88

0.9

TA

ME

pur

ity in

Bot

tom

0 50 100 150 200 250 3007

8

9

10x 10

−3

Fee

d co

mpo

sitio

n

Time, [h]

Da = 3.0

Figure 8: Dynamic response for change in amylene feed composition for Da = 3.0

using the NEQ model with new design.

icant differences are observed in the responses. As mentioned before, sustained

oscillations are realized only in the case of EQ model. Whereas oscillations disap-

pear in both the NEQ models under similar operating conditions.

Bifurcation analysis of NEQ model with partial back-mixing

Comparison of the bifurcation diagrams for EQ and NEQ models is shown in

Figure 11. Both the curves almost overlap quantitatively but they have different

stability behaviors. The EQ model shows unstable solution branch with the pres-

ence of Hopf bifurcation whereas NEQ model under similar condition shows the

stable solution. This comparison shows that modeling assumptions have a signif-

icant impact on the observed oscillations in case of EQ model. The difference in

the stability of the two curves in Figure 11 does not imply that the oscillations

have disappeared in NEQ model. Figure 12 shows the bifurcation diagram with

re-boiler duty as a parameter. The presence of the Hopf bifurcation point is re-

23


0 0.2 0.4

2

4

6

8

10

12

14

Mole fraction

Hei

ght o

f the

col

umn

[m]

0 0.5 1

2

4

6

8

10

12

14

Mole fraction350 400

2

4

6

8

10

12

14

Temperature [K]

EQ NEQ

Met

hano

l

Isoa

myl

ene

Isop

enta

ne

TA

ME

Figure 9: Steady state composition and temperature profiles: comparison of EQ

and NEQ model with partial back-mixing. Da = 3.0 Q = 20.5 MW, R = 1.5, P

= 4.5 bar. Amylene conversion (EQ) = 0.8457; (NEQ) = 0.8461. TAME purity

in bottom (EQ) = 0.8782; (NEQ) = 0.8776.

alized in this case. This probably implies that the parameter space wherein the

oscillations were observed in the case of EQ model has been shifted while dealing

with the NEQ model. Hopf bifurcation is observed at higher reboiler duty. Fig-

ure 13 shows the bifurcation diagram with respect to Damkohler number at the

corresponding higher reboiler duty. Upto certain value of the Da (Da = 4.39),

stable steady state is observed, which then converts to unstable steady state with

possible oscillations. This behavior is qualitatively similar to that of the EQ model

reported earlier (Katariya et al., 2006b).

From the foregoing discussion, it is clear that the oscillations observed in the EQ

model are not because of ignoring the transport processes. To understand the

cause behind this effect, it may be useful to study separately each of the model-

ing entities, such as column stages, condenser and reboiler. Methanol- isopentane

24


0 5 10 15 20 25 30 35 40 45

0.88

0.89

0.9

TA

ME

pur

ity in

Bot

tom

0 20 40 60 80 1000.75

0.8

0.85

0.9

0.95

Time, [h]

TA

ME

pur

ity in

Bot

tom

Da = 3.0 NEQ (Constant holdups)NEQ (rigorous)

EQ Model

Figure 10: Comparison of the step (10 % increase) response in 2M1B concentration

in feed for EQ, rigorous NEQ and constant holdup NEQ models: Plot of TAME

purity in bottom vs time. (Operating and design parameters: Da = 3.0, Q= 20.5

MW, R = 1.5, P = 4.5 bar.)

mixture, which is realized as a distillate, has been observed to exhibit phase split-

ting under certain conditions. Also the work by Zayer et al. (2007) identifies the

role of energy balance formulation in the dynamics of CSTR and reactive flash.

This work may be extended to the multistage columns and more specifically to

TAME synthesis. It has been noticed that the assumption of pseudo-steady state

energy balance, especially for the condenser and reboiler, strongly in-fluences the

dynamic behavior, often resulting in oscillations. A detailed investigation on these

aspects is expected to give a better insight into the nonlinear dynamics of TAME

synthesis in reactive distillation. A preliminary study of these topics is available

in Katariya (2007).

25


0 5 10 15 20 25370

375

380

385

390

395

400

405

Damkohlar number

Bot

tom

tem

para

ture

4.4 4.6 4.8 5

404.1

404.2

404.3

404.4

3 4 5 6 7403

403.5

404

404.5

405

Stable solutionUnstable solutionTurning pointHopf bifurcation point

EQ MODEL

NEQ MODEL

Figure 11: Comparison of the bifurcation diagrams of EQ and NEQ models: Plot

of Bottom temperature vs Damkohler number as continuation parameter. (Oper-

ating and design parameters: Q= 20.5 MW, R = 1.5, P = 4.5 bar.).

20.5 20.6 20.7 20.8 20.9 21403

404

405

406

407

408

409

410

411

Reboiler Duty [MW]

Bot

tom

tem

para

ture

[K]

Stable solution

Unstable solution

Turning point

Hopf bifurcation point

Figure 12: Bifurcation diagrams of NEQ models: Plot of Bottom temperature

vs Re-boiler duty as continuation parameter. (Operating and design parameters:

Da= 3.0, R = 1.5, P = 4.5 bar).

26


1 2 3 4 5 6 7406.9

407

407.1

407.2

407.3

407.4

407.5

407.6

407.7

407.8

407.9

Damkohlar number

Bot

tom

tem

para

ture

Stable solution

Unstable solution

Turning Point

Hopf bifurcation point

Figure 13: Bifurcation diagrams of NEQ models: Plot of Bottom temperature vs

Damkohler number as continuation parameter. (Operating and design parameters:

Q = 20.82MW, R = 1.5, P = 4.5 bar).

Conclusion

A rigorous dynamic NEQ stage model has been formulated and solved for the

synthesis of TAME by reactive distillation. The results of steady state and dy-

namic simulations using both EQ model and NEQ models with and without partial

back-mixing are compared. From the steady state analysis of NEQ model, it is

found that the number of segments (extent of back-mixing) in the stripping sec-

tion strongly influences the performance of the NEQ results. A counter-intuitive

behavior in the form of isoamylene conversion increasing with decrease in the

height of the stripping section is observed. Also the dynamic response of time

variant mass and heat transfer coefficients show significant variation when small

disturbances are introduced. This implies that consideration of time variant mass

and heat transfer coefficient is important. Synthesis of TAME in RD may be

associated with non-linear dynamic effects like limit cycles, which are confirmed

by dynamic simulation using an EQ model in our earlier studies. However, NEQ

27


model simulations in the same parameter space and operating region do not reveal

such phenomena. The oscillatory behavior that existed in the EQ model has been

shifted to a new parametric space in the case of NEQ model that considers partial

back-mixing.

To summarize, in order to explain the oscillations observed in a simple EQ model,

we have studied in detail a complex and computationally rigorous NEQ model that

incorporates concentration dependent heat and mass transfer coefficients. Steady

state and dynamic simulations, along with bifurcations studies, have confirmed

the existence of oscillations in the NEQ model as well. We believe that the effect

of possible liquid phase splitting and dynamic energy balance may provide an ex-

planation to this phenomenon.

28


Appendix A: Calculation of Mass transfer coeffi-

cients

Non-reactive SULZER BX packing (Rocha et al., 1993, 1996)

void fraction: ǫ = 0.9

packing area (m2/m3): a = 492

channel flow angle: θ = 60

channel side (mm): 8.9

Gas phase calculations:

kg.S

Dg

= 0.054((Uge + ULe)ρgS

µg

)0.8(µg

Dgρg

)0.33 (23)

Uge =Ugs

ǫ(1 − hL)sinθ(24)

ULe =ULs

ǫhLsinθ(25)

Uge and ULe : effective gas and liquid velocity in m/s respectively.

Ugs and ULs : Superficial gas and liquid velocity in m/s respectively.

kg : mass transfer coefficient in m/s (for binary pair)

S : characteristic length i.e side dimension of the corrugation crass-section (m)

hL : fractional liquid holdup.

µg and µL : gas and liquid viscosity in Pa.s

Dg and DL : gas and liquid Diffusion coefficient in m2/s (for binary pair)

Liquid phase calculations:

kL = 2(DLCEULe

πS)0.5 ....(m/s) (26)

CE: Factor slightly less than unity to account for those part of the packed bed

that do not encourage the rapid surface renewal. CE= 0.9

Hydraulic calculations:

hL = (4Ft

S)2/3(

3µLULs

ρLsinθǫgeff)1/3 (27)

29


Ft = 29.12(WeLFRL)0.15 S0.359

Re0.2L ǫ0.6(1 − 0.93cosγ)(sinθ)0.3

(28)

geff = g[ρL − ρg

ρl

][1 −(∆P/∆z)

(∆P/∆z)flood

] (29)

WeL =U2

LsρLS

σ(30)

FRL =U2

Ls

Sg(31)

ReL =ULsSρL

µL

(32)

∆P

∆z= (

∆Pd

∆z)[

1

1 −K2hL]5 (33)

(∆Pd

∆z) = AU2

gs +BUgs (34)

A =0.177ρg

Sǫ2(sinθ)2(35)

B =88.774µg

S2ǫsinθ(36)

K2 = 0.614 + 71.35S (37)

β =ae

ap

= (1 − 1.203(U2

Ls

Sg)0.111) (38)

ae: effective interfacial area (m2/m3) ap: area of packing (m2/m3) σ: surface

tension in N/m

cosγ = 0.9 for σ < 0.055

cosγ = 5.211.1016.835σ for σ > 0.055

∆P/∆z: pressure drop per unit section height (Pa/m).

Reactive KATAPAK packing (Kolodziej et al., 2004)

void fraction (m3/m3): ǫ = 0.622

packing area or specific surface area(m2/m3): a = 128.2

corrugation angle: θ = 45

crimp height (mm): 11.5

crimp wavelength (mm): 21.8

thickness of single sandwich (mm) = 21.8

30


Mass transfer coefficient: for ReL = 13 - 320

for630 < Reg < 2181

ShL = 3.846 × 10−3Re0.667L Sc0.5

L (39)

for2181 < Reg < 5900

ShL = 9.457 × 10−5Re0.667L Re0.482

g Sc0.5L (40)

for ReL = 15.2 - 360 and Reg = 610 - 5920

Shg = 0.0476Re0.736g Re0.229

L Sc0.33g (41)

Scg =µg

ρgDg

ScL =µL

ρgDL

Shg =kgde

DgShL =

kLν2

DL

Reg =gogde

ǫµgReL =

goLde

ǫµL

RegK =gogdeK

ǫµg

deK = 4ǫK/a K =1

1 + (4/apD)

ν2 = (mu2

L

ρ2Lg

)1/3

de = 4ǫ/ap : hydraulic packing diameter (m) for mass transfer.

Hydraulic calculations:

ReL = 7.3 − 530 and RegK = 620 − 5900

hd = 0.0273Re0.331L (42)

hstat =4F ′

s

S(

2σ(1 − cosγ)

gρL(1 − ρg/ρL)sinθ)0.5 (43)

Pressure drop calculations:

Limited to 70% of the flooding point i.e for given liquid load, gas velocity does

not exceed 70 % of its value corresponding to flooding.

31


ReL < 94

∆P

H=

∆Pdry

H(44)

94 ≤ ReL ≤ 264

∆P

H= 0.716

∆Pdry

Hexp(0.00357ReL) (45)

264 < ReL < 630

∆P

H= 0.443

∆Pdry

Hexp(0.00539ReL) (46)

∆Pdry

H=

ψaρgw2og

8ǫ3K(47)

ψ = 6.275Re−0.293gK 550 < RegK < 1550

ψ = 6.561Re−0.171gK 1550 ≤ RegK < 6000

(48)

wog : Superficial velocity(m/s) of gas

F ′

s = 0.018

Nomenclature

Symbol Interpretation

ai Activity coefficient of ith component

Ac Interfacial area for vapor liquid mass transfer, m2

CLtk, C

Vtk Total concentration of Liquid and vapor phase

on kth stage, mol/m3

D, B Molar flow rates of distillate and bottom

ELk , EV

k Liquid and Vapor Energy holdup of kth stage, J

H̃Li,k, H̃

Vi,k Partial molar enthalpy liquid and vapor for ith component

and kth stage, J/mol

Hk, hk, hfk Molar enthalpy of vapor, liquid and feed for

kth stage, J/mol

32


hLtk, h

Vtk Liquid and vapor side heat transfer coefficients

for kth stage, W/m2K

KIi,k Interface vapor-liquid equilibrium constant of ith component

and kth stage

kL, kV Liquid and vapor mass transfer coefficient matrices, m/s

kfm Forward rate constant of mth reaction, mol/eq.s

Kam Equilibrium constant of mth reaction

MLk , MV

k Liquid and Vapor molar holdup of kth stage, mol

NLk , NV

k Vectors of liquid and vapor mass transfer fluxes of the

order (C − 1), mol/m2s

NLi,k, N

Vi,k Liquid and vapor mass transfer fluxes for ith component

and kth stage, mol/m2s

Qk Heat loss from kthstage, J/s

Qr, Qc Reboiler and condenser duty, J/s

R Reflux ratio

Rm,k Rate of mth reaction and kth stage, mol/s

SVk , SL

k Molar flow of vapor and liquid side streams

of kth stage, mol/s

TL, T V , T I Liquid, vapor and interface temperatures, K

Vk, Lk, Fk Molar flow rates of vapor, liquid and feed

of kth stage respectively, mol/s

W Weight of the catalyst, Kg or equivalents

xD, xB Mole fraction of distillate and bottom

xi,k, yi,k Liquid and vapor mole fractions of ith

component and kth stage

xIi,k, yIi,k Interphase liquid and vapor mole fractions of

ith component and kth stage

xfi,k, yfi,k Liquid and vapor feed mole fraction of

33


ith component and kth stage

Greek letters

δ Dirac-delta function

ǫk Volume or weight of the catalyst for kth stage

γi,m Stoichiometric coefficient of ith component

and mth reaction

γi Activity coefficient of ith component

Abbreviations

RD Reactive Distillation

EQ Equilibrium stage modeling

NEQ Non-equilibrium stage modeling

MeOH Methanol

2M1B 2-Methyl 1-Butene

2M2B 2-Methyl 2-Butene

TAME Tertiary-Amyl Methyl Ether

i-PENT Iso-pentane

Da Damkohler Number

DAE Differential Algebraic Equation

34

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38

non-equilibrium stage modeling and non-linear … preliminary column design for the neq model is...

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