Non-equilibrium stage modeling and Non-linear. dynamic effects in the synthesis of TAME by. Reactive Distillation

Transcription

1 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 INDIA * Corresponding author: Prof. Sanjay M. Mahajani, Department of Chemical Engineering, IIT Bombay Mumbai sanjaym@che.iitb.ac.in Tel : Fax :

2 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 distillation 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, Nonequilibrium model, Hopf Bifurcation, Oscillations. 2

3 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 simulations 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 transfer 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 operating 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

4 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; Asprion, 2006) reactive distillation columns. For the purpose of design, optimal operation, and the control of the reactive distillation process, a rigorous theoretical 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 Distillation 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 behavior 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

5 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 dynamic 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 oscillations 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 equations, 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 simulation 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

6 case are identified and accordingly, model simplifications are made without compromising 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 hardware 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 refer 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 section 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

7 Column pressure = 4.5 bar Isopentane + MeOH Reflux ratio = 1.5 Rectifying section 4 theoreticalstages Packing height = 2.43 m Number of segments = 30 Pure methanol feed 215 kmol/h 305K Stage location = 24 Reaction section 19 theoretical stages Catalyst loading = eq[h+] Packing height = 8.41 m Number of segments = 71 Pre reacted feed kmol/h 325K Stage location = 29 Methanol M1B M2B TAME isopentane Stripping section 10 theoretical stages TAME Reboiler duty = 20.5 MW Column diameter = 3.87 m Packing height = 5.58 m Number of segments = 88 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 MeOH + 2M2B TAME R 1 = k f1 ( a 2M1B 1 a TAME a MeOH K a1 a 2 MeOH R 2 = k f2 ( a 2M2B 1 a TAME a MeOH K a2 a 2 MeOH ) ) 7

8 2M1B 2M2B R 3 = k f3 ( a 2M1B a MeOH 1 K a3 a 2M2B a MeOH ) 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 temperature ( K) 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

9 Figure 2: The typical NEQ stage representing tray or section of packed column. Total material balance equation for the NEQ stage are as below. dm L k dt C r C = L k 1 + Fk L (L k + Sk L ) + A c Ni,k L + γ i,m R m,k ǫ k (1) i=1 m=1 i=1 dm V k dt C = V k+1 + Fk V (V k + Sk V ) A c Ni,k V (2) i=1 Component material balance equation are written as: dm L k x i,k dt = L k 1 x i,k 1 + Fk L x fi,k (L k + Sk L )x i,k + A c Ni,k L r + γ i,m R m,k ǫ k (3) m=1 dm V k y i,k dt = V k+1 y i,k+1 + F V k y fi,k (V k + S V k )y i,k A c N V i,k (4) A c is the interfacial area for vapor-liquid mass transfer and N V i,k and N L i,k are vapor and liquid mass transfer fluxes respectively. Only (C 1) component material balance equations are independent, summation constraint on vapor and liquid phase 9

10 compositions is used to get the composition of the remaining components. Energy balance equation: de L k dt = L k 1 h k 1 + Fk L h fk (L k + Sk L )h k Q k C + A c [h L tk (T k I T k L ) + Ni,k H L i,k L ] (5) i=1 de V k dt = V k+1 H k+1 + Fk V H fk (V k + Sk V )H k C A c [h V tk (T k V T k I ) + Ni,k H V i,k V ] (6) i=1 H V i,k and H V i,k are partial molar enthalpies of vapor and liquid. Vapor liquid equilibrium at interface can be as given below. y Ii,k = K Ii,k x Ii,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. N V i,k = NL i,k (8) C h L tk(tk I Tk L ) + N L H C i,k i,k] L = h V tk(tk V Tk I ) + Ni,k H V i,k V (9) i=1 i=1 Summation constraints for the mole fractions in bulk vapor and liquid as well as at the interface are written as: C x i,k = 1.0 (10) i=1 C y i,k = 1.0 (11) i=1 C x Ii,k = 1.0 (12) i=1 C y Ii,k = 1.0 (13) i=1 10

11 Fick s law approach described earlier (Peng et al., 2003) is used to calculate the mass transfer fluxes. N L k N V k = Ctj L C kv (x Ik x k ) + x k N L i,k (14) i=1 = Ctj V C kl (y k y Ik ) + y k N V i,k (15) i=1 N L k and N V 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 interface mole fractions are used to find the mass flux of the last component. k L and k V 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 relating [k ] to the binary pairs of mass transfer coefficients through solution of Maxwell-Stefan equations for film model. Matrices are calculated using following relations with assumption that the matrices accounting the influence of mass transfer on the mass transfer coefficients are identity. [ k V ] = [ B V] 1 (16) [ k V ] = [ B L] 1 [ Γ L] (17) The elements of the matrix [B] have been calculated using the following equations. B ii = z i κ i,c + C k=1 k i B ij = z i ( 1 κ i,j 1 κ i,c z k κ i,k (18) ) where z i is the mole fraction of vapor or liquid phase and κ i,j is the mass transfer 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, (19) 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

12 given by Fuller et al. (1966) for gas phase. Maxwell-Stefan diffusivities are derived from these infinite dilution diffusivities (D o ij) using equation 20. D ij = (D o ij )(1+x j x i )/2 (D o ji )(1+x i x j )/2 (20) [Γ] is a matrix of the thermodynamic factor, calculated using following relation. Γ i,j = δ i,j + x i lnγ i x j (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 = W Tk f,ref F Total (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, Q reb = 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 simulator 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 systems. 12

13 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 rectifying, 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

14 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

15 EQ (35 stages) NEQ (33 segments) NEQ (47 segments) NEQ (189 segments) 4 Isoamylene 4 TAME 4 Height of the column [m] Methanol Isopentane Mole fraction Mole fraction Temperature [K] 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

16 havior. The oscillations being a non-linear dynamic effect, may originate from the nonlinearity in the vapor-liquid equilibrium relation, reaction kinetics or the functional dependence of the physical properties on the compositions and/or temperature (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 composition 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

17 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 compositions 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 temperature 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

18 Isoamylene conversion Height of the stripping section [m] 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

19 Height of the column [m] Isoamylene Methanol TAME Isopentane EQ model NEQ model Vapor Liquid Mole fraction Mole fraction Temperature [K] Pressure [bar] Holdup [Kmol] 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, bottom 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

20 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 operating 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 different 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 computation 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 parameters. 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 using 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

21 TAME purity in Bottom NEQ (Constant holdup) NEQ (rigorous) EQ model Time, [h] 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 oscillatory 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

22 Vap. side M.T. coef. [m/hr] Vap. side H.T. coef. [W/m 2 K] Liq. side M.T. coef. [m/hr] Liq. side H.T. coef. [W/m 2 K] Time, [h] Time, [h] (a) (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 analysis 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 reduced number of segments such that the steady state column profiles with both EQ and NEQ models are close to each other, with approximately same conversion 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

23 Bottom temp, [K] TAME purity in Bottom Feed composition x Da = Time, [h] 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 disappear 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 presence of Hopf bifurcation whereas NEQ model under similar condition shows the stable solution. This comparison shows that modeling assumptions have a significant 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

24 2 2 2 NEQ EQ Height of the column [m] Isoamylene Methanol TAME Isopentane Mole fraction Mole fraction Temperature [K] 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) = ; (NEQ) = TAME purity in bottom (EQ) = ; (NEQ) = 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. Figure 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 modeling entities, such as column stages, condenser and reboiler. Methanol- isopentane 24

25 TAME purity in Bottom Da = 3.0 NEQ (Constant holdups) NEQ (rigorous) TAME purity in Bottom EQ Model Time, [h] 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 splitting 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

26 Stable solution Unstable solution Turning point Hopf bifurcation point Bottom temparature NEQ MODEL EQ MODEL Damkohlar number Figure 11: Comparison of the bifurcation diagrams of EQ and NEQ models: Plot of Bottom temperature vs Damkohler number as continuation parameter. (Operating and design parameters: Q= 20.5 MW, R = 1.5, P = 4.5 bar.) Bottom temparature [K] Stable solution Unstable solution Turning point Hopf bifurcation point Reboiler Duty [MW] 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

27 Bottom temparature Stable solution Unstable solution Turning Point Hopf bifurcation point Damkohlar number 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 dynamic 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 section 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

28 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 explanation to this phenomenon. 28

29 Appendix A: Calculation of Mass transfer coefficients Non-reactive SULZER BX packing (Rocha et al., 1993, 1996) void fraction: ǫ = 0.9 packing area (m 2 /m 3 ): a = 492 channel flow angle: θ = 60 channel side (mm): 8.9 Gas phase calculations: k g.s D g = 0.054( (U ge + U Le )ρ g S ) 0.8 ( µ g ) 0.33 (23) µ g D g ρ g U ge = U Le = U gs ǫ(1 h L )sinθ U Ls ǫh L sinθ (24) (25) U ge and U Le : effective gas and liquid velocity in m/s respectively. U gs and U Ls : Superficial gas and liquid velocity in m/s respectively. k g : mass transfer coefficient in m/s (for binary pair) S : characteristic length i.e side dimension of the corrugation crass-section (m) h L : fractional liquid holdup. µ g and µ L : gas and liquid viscosity in Pa.s D g and D L : gas and liquid Diffusion coefficient in m 2 /s (for binary pair) Liquid phase calculations: k L = 2( D LC E U Le ) (m/s) (26) πs C E : Factor slightly less than unity to account for those part of the packed bed that do not encourage the rapid surface renewal. C E = 0.9 Hydraulic calculations: h L = (4 F t 3µ L U Ls S )2/3 ( ) 1/3 (27) ρ L sinθǫg eff 29

30 F t = 29.12(We L F RL ) 0.15 S Re 0.2 L ǫ 0.6 (1 0.93cosγ)(sinθ) 0.3 (28) g eff We L F RL Re L P z = g[ ρ L ρ g ρ l ][1 ( P/ z) ( P/ z) flood ] (29) = U2 Lsρ L S σ = U2 Ls Sg (30) (31) = U LsSρ L µ L (32) = ( P d z )[ 1 1 K 2 h L ] 5 (33) ( P d z ) = AU2 gs + BU gs (34) 0.177ρ g A = (35) Sǫ 2 (sinθ) 2 B = µ g S 2 ǫsinθ (36) K 2 = S (37) β = a e a p = ( ( U2 Ls Sg )0.111 ) (38) a e : effective interfacial area (m 2 /m 3 ) a p : area of packing (m 2 /m 3 ) σ: surface tension in N/m cosγ = 0.9 for σ < cosγ = σ for σ > P/ z: pressure drop per unit section height (Pa/m). Reactive KATAPAK packing (Kolodziej et al., 2004) void fraction (m 3 /m 3 ): ǫ = packing area or specific surface area(m 2 /m 3 ): a = corrugation angle: θ = 45 crimp height (mm): 11.5 crimp wavelength (mm): 21.8 thickness of single sandwich (mm) =

31 Mass transfer coefficient: for Re L = for630 < Re g < 2181 Sh L = Re L Sc 0.5 L (39) for2181 < Re g < 5900 Sh L = Re L Re g Sc 0.5 L (40) for Re L = and Re g = Sh g = Re g Re L Sc 0.33 g (41) Sc g = µ g ρ g D g Sc L = µ L ρ g D L Sh g = k gd e D g Sh L = k Lν 2 D L Re g = g ogd e ǫµ g Re L = g old e ǫµ L Re gk = g ogd ek ǫµ g 1 d ek = 4ǫK/a K = 1 + (4/a p D) ν 2 = ( mu2 L ρ 2 L g )1/3 d e = 4ǫ/a p : hydraulic packing diameter (m) for mass transfer. Hydraulic calculations: Re L = and Re gk = h d = Re L (42) h stat = 4F s S ( 2σ(1 cosγ) gρ L (1 ρ g /ρ L )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

32 Re L < 94 P H = P dry H 94 Re L 264 P H (44) = P dry H exp( re L) (45) 264 < Re L < 630 P H P dry H = P dry H exp( re L) (46) = ψaρ gw 2 og 8ǫ 3 K ψ = 6.275Re gk 550 < Re gk < 1550 ψ = 6.561Re gk 1550 Re gk < 6000 (47) (48) w og : Superficial velocity(m/s) of gas F s = Nomenclature Symbol a i Interpretation Activity coefficient of i th component A c Interfacial area for vapor liquid mass transfer, m 2 C L tk, CV tk Total concentration of Liquid and vapor phase on k th stage, mol/m 3 D, B Molar flow rates of distillate and bottom E L k, E V k H L i,k, H V i,k Liquid and Vapor Energy holdup of k th stage, J Partial molar enthalpy liquid and vapor for i th component and k th stage, J/mol H k, h k, h fk Molar enthalpy of vapor, liquid and feed for k th stage, J/mol 32

33 h L tk, h V tk Liquid and vapor side heat transfer coefficients for k th stage, W/m 2 K K Ii,k Interface vapor-liquid equilibrium constant of i th component and k th stage k L, k V k fm K am Mk L, MV k N L k, N V k Liquid and vapor mass transfer coefficient matrices, m/s Forward rate constant of m th reaction, mol/eq.s Equilibrium constant of m th reaction Liquid and Vapor molar holdup of k th stage, mol Vectors of liquid and vapor mass transfer fluxes of the order (C 1), mol/m 2 s N L i,k, NV i,k Liquid and vapor mass transfer fluxes for i th component and k th stage, mol/m 2 s Q k Q r, Q c R R m,k S V k, SL k Heat loss from k th stage, J/s Reboiler and condenser duty, J/s Reflux ratio Rate of m th reaction and k th stage, mol/s Molar flow of vapor and liquid side streams of k th stage, mol/s T L, T V, T I V k, L k, F k Liquid, vapor and interface temperatures, K Molar flow rates of vapor, liquid and feed of k th stage respectively, mol/s W x D, x B x i,k, y i,k Weight of the catalyst, Kg or equivalents Mole fraction of distillate and bottom Liquid and vapor mole fractions of i th component and k th stage x Ii,k, y Ii,k Interphase liquid and vapor mole fractions of i th component and k th stage x fi,k, y fi,k Liquid and vapor feed mole fraction of 33

34 Greek letters i th component and k th stage δ ǫ k γ i,m Dirac-delta function Volume or weight of the catalyst for k th stage Stoichiometric coefficient of i th component and m th reaction γ i Activity coefficient of i th component Abbreviations RD EQ NEQ MeOH 2M1B 2M2B TAME i-pent Da DAE Reactive Distillation Equilibrium stage modeling Non-equilibrium stage modeling Methanol 2-Methyl 1-Butene 2-Methyl 2-Butene Tertiary-Amyl Methyl Ether Iso-pentane Damkohler Number Differential Algebraic Equation 34

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