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Simulated Moving Bed Reactor for the Synthesis of 2‑Ethylhexyl Acetate. Part II: Simulation Based Design Bhoja Reddy, Vivek Chandra Gyani, and Sanjay Mahajani* Department of Chemical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India ABSTRACT: A simulated moving bed reactor (SMBR) is a multifunctional reactor wherein the reaction and chromatographic separation take place simultaneously. The positions of the inlet and outlet ports are switched at periodic intervals to simulate a true moving bed system. The design exercise for these units is a challenging task due to the large number of degrees of freedom such as the column volume, SMBR configuration (i.e., number of beds in each section), sectional flow rates, switch time, desorbent concentration and feed concentration. In this work, we propose a simulation based design methodology for the SMBR process that is employed for heterogeneously catalyzed reversible reactions, in which one of the reactants works as a solvent and the catalyst also plays the role of adsorbent. The objectives are productivity (PR) maximization and desorbent consumption (DC) minimization which have a direct impact on the total annualized cost of the process. The performance parameters such as conversion, raffinate purity, and extract purity are chosen as constraints. The model reaction considered is the esterification of acetic acid with 2-ethylhexanol catalyzed by cation exchange resin.

1. INTRODUCTION The integration of chemical reaction and chromatographic separation into a single unit is called reactive chromatography. Instant removal of product during the course of the reaction suppresses the backward reaction to give near-complete conversion for reversible reactions. This process may offer benefits such as reduction in capital and operating costs. The continuous operation on a large scale can be performed advantageously in a simulated moving bed reactor (SMBR) wherein the movement of the solid is simulated by appropriately switching the positions of inlet and outlet ports simultaneously.1 An SMBR consists of four sections with each section having a certain number of beds. The catalyst used for this reaction, i.e., an ion exchange resin, also acts as an adsorbent. The working principle of the SMBR is similar to that of a true moving bed reactor (TMBR). Hence, it becomes easier to interpret, if not all, most of the simulations results by visualizing the unit as a TMBR. The mathematical model, method of solution, performance criterion, and effect of each operating variable on the performance of the SMBR were studied in detail, in our previous work, for the synthesis of 2-ethylhexyl acetate.1 The fact that performance depends on several operating and design parameters poses a difficulty in the selection of optimum conditions. Based on the understanding of the effect of all the operating conditions on the performance as described in part I of this work,1 we propose a systematic design algorithm to obtain near-optimal conditions that meet our targets in terms of purity of the product streams and the conversion. The method not only exploits some of the conceptual design tools reported in the literature; it also makes use of the SMBR simulator at various stages and, hence, we call this exercise the simulation based design of the SMBR. The complete process for the production of ester involves the SMBR unit followed by the recovery of desorbent from both raffinate and extract streams as shown in Figure 1. Much of the energy is consumed in the recovery process. Hence, © 2014 American Chemical Society

while the capital cost mainly depends on the productivity (PR) of the SMBR, the operating cost is strongly governed by the amount of desorbent used in the SMBR operation. PR is the moles of desired product obtained in the raffinate stream per kilogram of adsorbent/catalyst per unit time (see eq 1a). PR =

QR ∫

t

t+t*

C 2R‐EHAc dt

(1 − ε)VcolNcolt *ρp

(1a)

where QR is the volumetric raffinate flow rate, C is the liquid phase concentration, ε is the voidage, Vcol is the volume of a single column, Ncol is the total number of columns in the SMBR unit, t* is the switch time, t represents the beginning of the switching time, and ρp is the density of dry adsorbent particle. For a near-complete conversion and near-complete separation of the products, one can approximate PR as the moles of the limiting reactant processed per kilogram of the adsorbent/ catalyst per unit time (see eq 1b). PR ≈

Q FC2F‐EH (1 − ε)VcolNcolρp

(1b)

where QF is the volumetric feed flow rate. Desorbent consumption (DC) is defined as the moles of desorbent consumed per mole of the product realized in the raffinate stream (see eq 2a). DC =

D F (Q DCAcH + Q F(CAcH − XC2F‐EH))t *

QR ∫

t

Received: Revised: Accepted: Published: 15824

t+t*

C 2R‐EHAc dt

(2a)

May 21, 2014 July 26, 2014 September 23, 2014 September 23, 2014 dx.doi.org/10.1021/ie502091w | Ind. Eng. Chem. Res. 2014, 53, 15824−15835

Industrial & Engineering Chemistry Research

Article

Figure 1. Complete SMBR process for the production of 2-EHAc ester. D, distillation column.

raphy (SMBC) and is mainly based on the flow rate ratio parameter (mj).3 The flow rate ratio corresponding to a section (j) is defined as the ratio of the net fluid flow rate to the solid flow rate and is given by eq 3.

where QD is the volumetric desorbent flow rate and X is the per pass conversion of the limiting reactant. In the system that we have studied here, the desorbent happens to be one of the reactants. The concentration of this reactant is always in excess at any position in the SMBR. Thus, while calculating DC we have deducted the amount of desorbent/reactant consumed due to the reaction. For an equimolar feed concentration, and for complete conversion and separation, one can assume DC to be the moles of the fresh desorbent added at the desorbent node per mole of the limiting reactant processed (see eq 2b). DC ≈

mj =

( ) (3)

where Qj is the fluid flow rate in section j. The triangle theory of SMBC, without reaction, defines the boundaries for the feasible flow rate ratio in each section for the separation of the two-component mixture. For example, the bounds on flow rate ratio parameters for the mixture consisting of 2-EHAc and water are given by eqs 4a−4c.3 In this case, 2EHAc has the lowest affinity and water has the highest affinity toward the adsorbent.

D Q DCAcH

Q FC2F‐EH

Vcolε t* Vcol(1 − ε) t*

Qj −

(2b)

For an optimum design of the SMBR, we need maximum PR and minimum DC to minimize the capital cost and operating cost, respectively. As explained later, these two are conflicting requirements and one has to cleverly manipulate the design and operating parameters to arrive at close to optimum conditions. The algorithm presented in this work is restricted to only SMBR, for the given feed flow rate, feed concentration, and solvent concentration. The next level of process optimization, which also considers the distillation columns and the recoveries, is out of the scope of this work. The mathematical optimization of the SMBR process based on simulations clubbed with a numerical optimization technique is computationally intensive.2 Alternatively, a simplified approach based on conceptual design may be adopted to reduce the computational burden. The estimates of a few key parameters are obtained initially so as to achieve the performance close to the desired one. This is done by making a few reasonable assumptions while retaining the important features of the system. One such design tool is triangle theory, which is well established for the nonreactive separations performed in simulated moving bed chromatog-

m1 > HWater

(4a)

H2‐EHAc < m2 < m3 < HWater

(4b)

−ε < m4 < H2‐EHAc 1−ε

(4c)

where Hi is the Henry’s adsorption constant of component i, when the isotherm is linear. Triangle theory is further extended to the SMBR, to determine the reactive separation region. The reactive separation region helps in selection of flow rates to give the desired conversion and the separation of the products. The reactive separation region can be generated either by the analytical solution of the TMBR model4−6 or by numerical methods.7−9 The main limitation of the triangle theory is that it gives only the range of feasible flow rates for the given column volume, switch time, and SMBR configuration. However, it does not give guidelines on the selection of the column volume and 15825

dx.doi.org/10.1021/ie502091w | Ind. Eng. Chem. Res. 2014, 53, 15824−15835


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which reaction takes place becomes the most influential section in SMBR. The design methodology developed in this work starts with minimum residence time required in the reactive section of the SMBR. Thus, it is important to know in which section of the SMBR the reaction takes place. In the SMBR, the reaction takes place in either section II or section III. In some cases, it takes place in both sections, i.e., II and III. The locale of the reaction depends on sectional flow rate ratios in sections II and III. The triangle theory provides useful inputs in this regard and helps identify the locale of the reaction under the given operating conditions. Working with linear adsorption isotherms is easier than working with nonlinear adsorption isotherms, and hence, we propose to linearize the isotherms of interest. 2.1. Linearization of Adsorption Isotherms. Let us first get an insight into the variation in the concentrations of individual components in the SMBR under the representative conditions. The simulator described in part I of this work1 is used to obtain the concentration profiles when a periodic steady state is obtained in a configuration that has two columns in each section. Figure 2 shows the concentrations of the

switch time, required for the complete design of the SMBR. Also, the theory is based on the TMBR model. Though the SMBR and the TMBR have similar attributes, they differ from each other as regards their residence time distribution (RTD).4 The studies by Lode et al.4 show that the RTD may have a considerable effect on the conversion. The RTD effect on conversion is negligible when the SMBR either has a large number of columns in each section or has a very short switch time. Biressi et al.10 proposed a simulation based design algorithm for SMBC, without reaction, to determine the minimum column volume and the optimum values of all the operating conditions for a given SMBC configuration. Subsequently, Azevedo and Rodrigues11 extended this algorithm to the SMBR for the inversion of sucrose, wherein conversion mainly depends on the enzyme concentration. In the sucrose inversion case, the catalyst and reactants are in the same phase; the conversion can be controlled independently by manipulating the enzyme concentration of the eluent. However, in the present case, the catalyst is solid and it also acts as an adsorbent. Hence, the catalyst concentration or loading cannot be varied independently to obtain the desired conversion without affecting the separation. Further, the assumption in both reported algorithms10,11 is that maximum PR is obtained when the pressure drop in section I is maximum. The basis for their assumption is that the pressure drop is proportional to throughput. However, for an SMBC process, Jupke et al.12 proved that the maximum PR does not necessarily correspond to the point of maximum pressure drop in section I. They showed that the optimum process performance can also be achieved at conditions different from the ones for the maximum pressure drop. Furthermore, in both reported algorithms,10,11 the objective was only to maximize productivity by optimizing the column length and other conditions, but minimization of eluent requirement was not considered in their work. Thus, based on the forgoing discussion, it is necessary to develop a new algorithm for the design of an SMBR. In the present work, we have extended the design algorithm of Biressi et al.10 to the SMBR, wherein solid acts as both adsorbent and catalyst. The developed algorithm can be applied to any reversible heterogeneous catalytic reaction that uses one of the reactants as the solvent for regeneration. Further, we have relaxed the pressure drop assumption. As mentioned, it is not possible to achieve both objective functions, i.e., maximum PR and minimum DC, in one design. Hence, we propose a systematic methodology giving different designs corresponding to a range of objectives, i.e., PR and DC. One has to weigh the importance of these objectives for the case in hand to arrive at the best possible design. The methodology developed in this work is illustrated by considering the example of the synthesis of 2-ethylhexyl acetate. The catalyst also plays the role of adsorbent and selectively separates water formed in the reaction to push the reaction in the forward direction. The reaction kinetics and the isotherm parameters are reported in our earlier work.13

Figure 2. Typical concentration profile at the end of switch time for all sections of SMBR with operating conditions QF = 0.74 mL/min, QR = 2.76 mL/min, QD = 13.79 mL/min, Q3 = 3.17 mL/min, t* = 1000 s, feed concentration AcH:2-EH = 1:1 (mol/mol), temperature = 353 K, length of reactor = 30 cm, and SMBR unit configuration 2−2−2−2.

species in all the sections at the end of the switch time. It can be seen that the desorbent (i.e., acetic acid) exists in large proportions everywhere, and all the other components are present in relatively dilute concentrations. The adsorption isotherm reported by Gyani et al.13 is of Langmuir type. In order to obtain the initial estimates of the flow rates, we consider only the data in the dilute region and thus linearize the isotherm over the composition range of interest. The linear form of the adsorption isotherm is given by eq 5. qi = HiCi

(5)

Here, Hi is the pseudo-Henry’s adsorption constant of component i, while qi and Ci are the solid and bulk phase concentrations, respectively. Since acetic acid is always present in excess, the corresponding constants for 2-ethylhexanol (H2‑EH), water (HWater), and 2-ethylhexyl acetate (H2‑EHAc) are determined with reference to acetic acid. Hi is calculated using eq 6 and is given by

2. RESIDENCE TIME IN THE REACTIVE SECTION The SMBR has four sections, and every section has a specific role to play. In most of the cases, under the given conditions, it is only one section that governs the overall performance of the SMBR, and hence, the flow rate and volume of this section need more attention. If the reaction is slow and reversible, and the desorbent flow rate is sufficiently large, then the section in

Hi = 15826

ρb K i Γ ∞ i 1 + KAcHρAcH

(6)



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Here, Ki and Γ∞ i are the equilibrium constant and capacity of the adsorbent for component i, respectively, ρb is the bulk density of the adsorbent, and ρAcH is the molar density of acetic acid. The estimated constants for the components 2-EH, 2EHAc, and water are given in Table 1.

further calculate the solid loading and switch time. There is a minimum limit on the liquid residence time in the reactive section, which can be determined by assuming the separation to be instantaneous. In such a case, the reaction is first order irreversible and the residence time required for the reaction is given by eq 7 for a specific conversion (say 99%) assuming that the reactive section of the SMBR acts like an ideal plug flow reactor.

Table 1. Estimated Pseudo-Henry Constants of Linearized Adsorption Isotherm component

Hi

2-ethylhexanol (2-EH) 2-ethylhexyl acetate (2-EHAc) water

0.119 0.028 3.151

τrxn = −

1 ln(1 − X ) kR

(7)

where τrxn is the minimum residence time for the liquid phase in the reactive section, and kR is the overall reaction rate constant. The residence time available in the SMBR, depending on where the reaction takes place, is given by eqs 8a−8c:

2.2. Reactive Section. The function of each section in SMBC/SMBR mainly depends on the sectional flow rate ratio (mj) in the respective section. Even though the reaction takes place in either section II or section III, the simultaneous separation of the products takes place in both sections. The necessary conditions, i.e., H2‑EHAc < m2 < m3 < HWater, should be satisfied for the complete separation of the products.2 As acetic acid is present in excess, the esterification reaction can be considered as the pseudo-first-order reaction. From an adsorption viewpoint, it is a pseudoternary mixture with three solutes (2-ethylhexanol, ester, and water) and a solvent (acetic acid). As shown in Figure 3, the region of complete separation

region 1: τrxn =

sVcolε Q2

(8a)

sVcolε pV ε + col Q2 Q3

(8b)

pVcolε Q3

(8c)

region 2: τrxn =

region 3: τrxn =

where p and s are the numbers of columns in sections III and II, respectively. This is an important step in the algorithm as it gives an initial estimate of the residence time and allows one to calculate the switch time and sectional volume as described in section 2.4. 2.4. Switch Time and Column Volume. Once the minimum residence time is known, we can calculate the corresponding switch time (t*) for a point in the m2−m3 plane using eqs 3 and 8a−8c. The corresponding expressions for t* are given by eqs 9a−9c. region 1: Figure 3. Division of complete product separation triangle for first order reaction and linear adsorption isotherms.

t* =

on the m2−m3 plane can be divided into three parts when the adsorption constant of the reactant (2-ethylhexanol) lies between the Henry’s adsorption constants of the two products i.e., HWater > H2‑EH > H2‑EHAc. The reaction takes place only in section II, if the operating point lies in region 1. On the other hand, it takes place only in section III if the operating point lies in region 3. The reaction occurs in both sections if the operating point corresponds to region 2.2 It may be noted here that the boundaries of the regions, in reality, are not as sharp as shown in Figure 3 because the isotherm may not be exactly linear. Nevertheless, this exercise helps us identify the locale of reaction from this plot. Hence, to confirm the predictions of the triangle theory and further finetune the results, the simulations are performed with an actual nonlinear adsorption isotherm. 2.3. Minimum Residence Time in the Reactive Section. Once the locale of the reaction is known, we need to determine the residence time for the liquid in this section to

1 ⎛⎜ 1 − ε ⎞⎟ τrxn + m2τrxn s⎝ ε ⎠

(9a)

region 2: ⎞−1 p τrxn ⎛ s + t* = ⎜ ⎟ m2(1 − ε) + ε ⎠ ε ⎝ m3(1 − ε) + ε

(9b)

region 3: t* =

1 ⎛⎜ 1 − ε ⎞⎟ τrxn + m3τrxn p⎝ ε ⎠

(9c)

Now based on the mass balance at the feed port and from the definition of m2 and m3 (eq 3), the column volume of the SMBR for a given feed flow rate can be calculated as Vcol =

Q Ft * (m3 − m2)(1 − ε)

(10)

As mentioned before, in most of the design methods reported in the literature (e.g., refs 10 and 11), the minimum value of the 15827



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column length is assumed arbitrarily, which makes the procedure lengthy. The procedure given in the present work provides a good starting point for further calculations.

3. DESIGN ALGORITHM AND RESULTS A typical SMBR design problem is to determine the nearoptimum values of the parameters such as the column volume, number of beds, and sectional flow rates for a given production rate with specific constraints on conversion and purities. In this work, this exercise is divided into two steps. In the first step, the objective is to find out all the feasible points in the m2−m3 plane and determine the minimum residence time required in the reactive section(s) to meet the required constraints. In the second step, we identify all possible ways to maximize the PR and minimize the DC. As mentioned earlier, the maximization of PR leads to minimum capital cost, whereas minimization of DC helps in reducing the load on downstream processing and hence the energy consumption. For a given case, the design may be finalized based on the relative importance of PR and DC. The methodology proposed here is applied to the synthesis of 2-ethylhexyl acetate for a given feed flow rate (4 mL/min), desorbent concentration (99.9% pure acetic acid), and equimolar feed concentration, with specific limits on conversion and purity. The required conversion and purities of product streams are chosen as 99%. If the conversion or purities are low, then the product streams are required to be subjected to multistep separations and the purpose of using reactive chromatography is not served. The number of degrees of freedom for SMBR is 10. These are the column diameter, column length, SMBR configuration (i.e., number of beds in each section), four sectional flow rates, switch time, desorbent concentration, and feed concentration. The objective is to satisfy constraints on per pass conversion of the limiting reactant and the purities of the two product streams, viz., raffinate and extract. The design methodology developed in this work takes input from the triangle theory of SMBC, which is well established for the pure separation case. Further, we work in the framework of the algorithm suggested by Biressi et al.10 The following outcomes of these studies are used while developing the algorithm in the present case. 1. The cross section of the column is the scale-up parameter. For higher capacities it is increased proportionately. 2. From the adsorption isotherms, it is possible to predict a priori the values of m1 and m4 which guarantee complete regeneration of the adsorbent and the desorbent in sections I and IV, respectively. In order to account for the nonlinearity of the adsorption isotherms and model inaccuracies, proper safety margins on m1 and m4 are included. On the basis of the above considerations, we fix the column cross section and the two sectional flow rates (m1 and m4). Initially, each section is assumed to contain two beds. The number of beds in each section is varied later to maximize the PR. For the given feed flow rate, feed concentration, and desorbent concentration, by fixing m1, m4, column area, and configuration (i.e., 2−2−2−2), the number of degrees of freedom is reduced to three. One can choose any three variables out of m2, m3, column length, and switch time. The fourth parameter can be calculated using eq 10. Our design algorithm starts with the determination of right values of these three variables. 3.1. Feasible Points in m2−m3 Plane. To start with, we consider a few representative points in the m2−m3 plane which are well spread over the separation triangle (Figure 4). To

Figure 4. Representative points on m2−m3 plane with their corresponding PR values.

account for the nonlinearity of adsorption isotherms, some points outside the separation triangle are also considered. For a particular representative point, eqs 9a−9c and eq 10 allow us to calculate the switch time and the column volume for the minimum residence time in the reactive section. The minimum residence time is calculated by assuming the reactive section of the SMBR is a plug flow reactor and by assuming the reaction to be irreversible. The SMBR simulator is then used to predict the performance with the known kinetics and the Langmuir adsorption isotherm. As anticipated, the conversion constraint is not satisfied at the minimum residence time because, in the actual case, the reaction is reversible and the separation of water is not instantaneous. The residence time of the reactive section is thus gradually increased under otherwise similar conditions to step up the conversion. The increment in residence time is stopped when the required constraints are achieved. For a few representative points, the target constraints are not achieved and the sensitivities of purity and conversion toward the change in the residence time are very small. Such points are discarded and considered to be infeasible points. Thus, for all the feasible points we arrive at a certain PR value. Figure 4 shows the selected representative points on the m2−m3 plane and the corresponding PR values for the feasible points. The design algorithm to identify these feasible points on the m2−m3 plane is summarized in Figure 5. Effect of Residence Time on the Performance of SMBR. The effect of variation in the residence time on the performance parameters such as purities, conversion, and PR is shown in Figure 6 for one of the representative points. Similar trends are observed for the other points as well. The conversion and raffinate purity increase with an increase in residence time. While the extract purity is almost insensitive to the residence time, PR decreases with an increase in residence time. The reason for these trends is as follows. It is known that the separation is not instantaneous; the reaction is not complete at lower residence time causing the unconverted reactant to dilute the raffinate stream. The extract purity depends only on the concentration of the eluent entering section I and the concentration in the solid phase entering section I. These are not affected by the residence time in section III. PR decreases with an increase in residence time as the volume of each bed increases with the residence time, and this effect is much more dominant than the one due to a rise in conversion. The selected residence time (see Figure 6) is such that the conversion and purity constraints are just satisfied, and 15828



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Figure 5. Design algorithm to locate the feasible points in m2−m3 plane and determine the corresponding PR values.

Figure 4 are to be short-listed further based on the PR and DC. Let us now understand how PR and DC vary over the m2−m3 plane. An insight into this aspect would help us identify the right m2−m3 combinations giving the desired performance. It is necessary to define one more parameter here, i.e., the pseudo solid flow rate (QS), which makes a strong impact on the PR and DC. It is nothing but the virtual solid flow rate calculated as the bed volume divided by the switch time (eq 11). QS =

Vcol(1 − ε) t*

(11)

The pseudo solid flow rate (QS) can thus be expressed in terms of m3 and m2 as Figure 6. Effect of residence time on purities, conversion, and PR for a point (1, 2.5) on m2−m3 plane with 2−2−2−2 SMBR configuration.

QS =

QF m3 − m 2

(12)

Hence, for the given feed flow rate, QS is inversely proportional to m3 − m2; i.e., when we move away from the diagonal in the feasible region of the m2−m3 plot, the value of QS decreases. The DC can now be related to QS by eq 13,

further increase in the residence time does not significantly improve the results. 3.2. Short-Listing Feasible Points Based on the Objective Functions. The several feasible points realized in 15829



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Figure 7. (a) Variation in PR with flow rate in section II for a constant pseudo solid flow rate and for a given feed flow rate. (b) m2−m3 plot showing constant pseudo solid flow rate line for a given feed flow rate.

Figure 8. (a) PR with pseudo solid flow rate. Region I, points of lower DC, considered as candidate points for further analysis. Region II, points of higher DC not considered for further analysis. (b) m2−m3 plot showing maximum PR points on the line of constant pseudo solid flow rate for a given feed flow rate.

indicating that for higher QS one would need high DC. That means as we move away from the diagonal, the DC decreases. DC ≈ =

D Q DCAcH

Q FC2F‐EH

=

purities are satisfied. It is observed that, initially, PR increases with an increase in sectional flow rate and then decreases at the higher values of the sectional flow rate. At a low sectional flow rate, the elution of the low affinity component becomes difficult. On the other hand, at a very high sectional flow rate, the adsorption of the high affinity component is adversely affected. Hence, the volumes of the bed required for achieving complete separation at the lower and higher sectional flow rates are relatively high. Hence, the PR goes through a maximum when the sectional flow rate is increased for a constant pseudo solid flow rate and the feed flow rate, i.e., when m3 − m2 is constant. For a particular pseudo solid flow rate, the point of maximum PR is of further interest as the other points have lesser values of PR. 3.2.2. Effect of Pseudo Solid Velocity. In section 3.2.1, the pseudo solid flow rate was kept constant and the individual flow rate ratio in section II or III was varied. A maximum in PR is realized at a certain flow rate. In this section, we examine the effect of QS on the maximum PR that can be obtained. As we know, QS is inversely proportional to m3 − m2, and hence as we move away from the diagonal, QS would decrease. Figure 8 shows that the maximum PR increases with QS and then decreases at higher values of QS. At a very low solid flow rate the water in the reactive section is not removed efficiently, which decreases the PR. At very high solid flow rates, ester is carried away by the solid, thereby leading to a drop in PR. Hence, there is an optimum value of the pseudo solid velocity at which PR is maximum.

D (m1 − m4 )CAcH

(m3 − m2)C2F‐EH

D Q S(m1 − m4 )CAcH

Q FC2F‐EH

(13)

PR, on the other hand, is not directly related to pseudo solid flow rate. Hence, we discuss the variation in PR in section 3.2.1. First, we consider the effect of the absolute flow ratio, either m3 or m2, and then we examine the effect of the pseudo solid flow rate on the difference m3 − m2. 3.2.1. Productivity as a Function of Flow Rate Ratio (m3 or m2). For the reaction of interest in the present case, the reactive section is section III for almost all the operating conditions. Hence, we have considered only the flow rate ratios in region 3 in Figure 3, for further calculations. For a particular feed flow rate and pseudo solid velocity, the effect of the flow rate in section II (Q2) on PR is shown in Figure 7a. The effect of m2 is same as that of Q2 for the given pseudo solid velocity and the feed flow rate (see eqs 3 and 11). As m3 − m2 is constant for a given pseudo solid velocity and feed flow rate (eq 10), the effect of m3 is also same as that of m2. Thus, the PR values shown in Figure 7a correspond to operating conditions on a line parallel to the diagonal on the m2−m3 plane (Figure 7b). At each operating point, the constraints on the conversion and 15830



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to be five with 2−1−1−1 (order of sections is III−IV−I−II) configuration.

From eq 13, it is known that DC is proportional to the pseudo solid velocity. Hence, for the minimum DC, the pseudo solid velocity should be as low as possible. The maximum value of the pseudo solid velocity is the value governed by the maximum PR point (Figure 8a). These points correspond to the maximum PR point on the constant pseudo solid flow rate line (Figure 8b). Thus, increasing QS is advantageous only from the point of view of increasing PR. Hence, all the other points in Figure 8a having a pseudo solid velocity higher than the maximum PR point (region II in Figure 8a) are directly rejected for further computations. These points will have lesser PR values than the maximum PR and also have higher DC values than that at the maximum PR point. The points on the m2−m3 plane, which are of interest for further computations, are shown in Figure 9.

Table 2. Increasing the PR by Reducing the Number of Columns in Each Section for a Selected Feasible Point (1, 2.5) on m2−m3 Plane

a

SMBR configna

wt of catal (g)

conv (%)

raffinate purity (%)

extract purity (%)

PR (mol of 2EHAc/kg of adsorbent·day)

2−2−2−1 2−2−1−1 2−1−1−1 1−1−1−1

828.9 710.5 592.1 473.6

99.28 99.28 99.28 90.05

99.23 99.23 99.23 86.78

99.89 99.89 99.89 99.88

33.23 38.77 46.52 52.75

Order of sections is III−IV−I−II.

Reducing Total Catalyst Loading. The PR can be further increased by reducing the amount of catalyst in each bed of the SMBR without compromising on the performance targets. In the previous step, it was observed that the catalyst/adsorbent volume in certain sections does not affect the performance significantly (e.g., sections I, II, and IV). Now the catalyst loading in these sections is reduced by reducing the bed volume. Since all the beds in each section have the same volume, the section that controls the performance (i.e., section III) would run short of catalyst. To maintain the same catalyst loading in section III, the number of beds in section III is increased. This results in a reduced amount of total catalyst/ adsorbent loading, as desired, in the other sections that do not control the overall performance and have only one bed in them. As an example, consider the SMBR configuration 2−1−1−1 (order of sections is III−IV−I−II), with a specific catalyst loading in each column for which section III controls the performance. Now, let the configuration be n−1−1−1 (n > 2). The amount of catalyst in a single bed for the new configuration is calculated as the amount of catalyst in section III in the old configuration divided by n, where n is the number of columns in section III in the new configuration. Further, the switch time is varied in order to maintain the same pseudo solid velocity. This ensures that the catalyst loading in section III and the pseudo solid velocity are the same as before and all the other operating conditions remain unchanged. The performance is examined through simulations by increasing the value of n until any of the other sections starts showing its influence on the performance of the SMBR. This procedure for reducing the amount of catalyst in a single bed is stopped when the increment in the PR is not significant. The sequence of the steps thus followed is depicted in Table 3. The maximum number of beds in a section mainly depends on the residence time required in a reactive section. If there is a practical limit on

Figure 9. Short-listed operating points in m2−m3 plane.

3.3. Increasing Productivity and Reducing Desorbent Consumption. In this section, we describe subsequent steps to increase the PR further and reduce the DC and arrive at a near-optimal design. As mentioned before, this exercise is performed only on the selected feasible points, which were obtained in section 3.2.2 (see Figure 9). While section 3.3.1 describes the steps to maximize the PR, section 3.3.2 deals with minimization of the DC. 3.3.1. Increasing the Productivity. From the definition of PR (eqs 1a and 1b), it is known that the PR is inversely proportional to the amount of catalyst used in the SMBR unit. The exercise presented in section 3.2 is for an SMBR with all sections containing equal numbers of beds and equal catalyst loadings. However, it is not necessary that each section should have an equal amount of catalyst/adsorbent to perform the expected duty. Hence, as the next step, the option of unequal loading in each section is explored. Reducing Number of Beds in SMBR. The number of beds in each section is reduced and the performance of the SMBR is examined through simulations. It is observed that the performance of the SMBR is sensitive to the number of beds in section III, wherein much of the reaction takes place. In other words, section III governs the overall performance of the SMBR under these conditions. Hence, when the number of beds in section III is decreased, the conversion and raffinate purities decrease significantly. The sequence of steps and the performance variation with reduction in the number of beds in each section are shown in Table 2. The minimum number of beds without affecting the performance of the SMBR was found

Table 3. Increasing the PR by Reducing the Amount of Catalyst Loading in Each Bed for a Selected Feasible Point (1, 2.5) on m2−m3 Plane

a

15831

SMBR configna

wt of catal (g)

conv (%)

raffinate purity (%)

extract purity (%)

PR (mol of 2EHAc/kg of adsorbent·day)

2−1−1−1 4−1−1−1 6−1−1−1 8−1−1−1

592.1 414.4 355.2 325.6

99.28 99.51 99.3 98.97

99.23 99.76 99.82 99.83

99.89 99.7 99.43 99.06

46.52 68.16 81.04 89.8

Order of sections is III−IV−I−II. dx.doi.org/10.1021/ie502091w | Ind. Eng. Chem. Res. 2014, 53, 15824−15835


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the number of beds in a given section, the designs violating the practical constraint will obviously be ruled out. 3.3.2. Reducing the Desorbent Consumption. In all previous calculations (sections 3.1 and 3.2), m1 was assigned a higher value while m4 was assigned the minimum limit given by eq 4c, so as to achieve complete regeneration of the solid and the eluent in sections I and IV, respectively. The DC is directly proportional to the amount of fresh desorbent stream introduced, which in turn is directly proportional to m1 − m4 (see eq 13). Hence, there is a scope to minimize DC by reducing m1 and increasing m4, making sure that the conversion and purity targets are not disturbed. The value of m1 is decreased until the constraint on raffinate purity is within a permissible limit. Similarly, the value of m4 is increased until the constraint on extract purity is not overruled. Effect of Sectional Flow Rates in Sections I and IV on the Performance. As discussed before, here we vary m1 and m4 and examine their effects on the performance targets (Figures 10−13). For the given pseudo solid velocity and the feed flow

Figure 11. Effect of flow rate in section IV on purities and conversion with operating conditions m1 = 10, m2 = 1, m3 = 2.5, L = 36.09 cm, t* = 3330 s, feed concentration AcH:2-EH = 1:1 (mol/mol), temperature = 353 K, and SMBR unit configuration: 2−2−2−1.

Figure 12. Effect of flow rate in section I on PR and DC with operating conditions m2 = 1, m3 = 2.5, m4 = −0.5, L = 36.09 cm, t* = 3330 s, feed concentration AcH:2-EH = 1:1 (mol/mol), temperature = 353 K, and SMBR unit configuration 2−2−2−1.

Figure 10. Effect of flow rate in section I on purities and conversion with operating conditions m2 = 1, m3 = 2.5, m4 = −0.5, L = 36.09 cm, t* = 3330 s, feed concentration AcH:2-EH = 1:1 (mol/mol), temperature = 353 K, and SMBR unit configuration 2−2−2−1.

rate, m1 and m4 indirectly represent Q1 and Q4, respectively. Figure 10 shows the effect of Q1 on purities and conversion. The raffinate purity and conversion increase with increase in Q1 and become insensitive at higher values of Q1. At lower values of Q1, the adsorbent in section I is not regenerated completely. In such a case, adsorption of the highest affinity component (i.e., water) in sections II and III is adversely affected, thereby reducing the conversion and also the raffinate purity. The extract purity is independent of Q1 and depends only on the eluent and solid flow rate coming to section I. The selected value of Q1 is just sufficient to regenerate the solid in section I completely. A further increase in Q1 does not have much effect on the purities and conversion (see Figure 10). As regards Q4, Figure 11 shows that the raffinate purity and conversion are almost independent of Q4 and the extract purity decreases with an increase in Q4. At higher values of Q4, eluent in section IV is not regenerated properly and the ester which is not adsorbed in section IV enters section I and finds outlet through the extract. The selected maximum value of Q4 is the one for which eluent is regenerated completely in section IV. With further increase in Q4, the regeneration is adversely affected (see Figure 11). Figures 12 and 13 show the effect of variation in Q1 and Q4 on DC and PR, respectively. As expected, the DC increases with Q1 and decreases with Q4. As regards PR, it is indirectly

Figure 13. Effect of flow rate in section IV on PR of ester and DC operating conditions m1 = 10, m2 = 1, m3 = 2.5, L = 36.09 cm, t* = 3330 s, feed concentration AcH:2-EH = 1:1 (mol/mol), temperature = 353 K, and SMBR unit configuration 2−2−2−1.

affected by Q1 and Q4 as Q1 influences conversion and Q4 influences the extract purity. Thus, the drop in Q1 and rise in Q4 result in lower values of PR. It may be seen that the performance declines in terms of both PR and DC for the values of Q1 and Q4 higher than the selected ones. 3.4. Design Configurations. 3.4.1. Number of Paths Leading to Different Designs. From the foregoing discussion, 15832



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Table 4. SMBR Designs Showing the Sequence in Which Different Paths Are Performed for a Feasible Point (1, 2.5) on on m2− m3 Plane stepsa

design results

design

R-I

R-II

R-IV

R-CL

R-m1

I-m4

PR

DC

confign

D1 D2 D3 D4 D5 D6 D7

− 1 − 1 1 − 1

3 2 1 2 2 1 2

− − 2 3 − 2 3

− − − − 3 3 4

1 3 3 4 4 4 5

2 4 4 5 5 5 6

32.77 38.22 38.74 46.17 51.31 52.02 80.88

12.38 17.33 13.19 17.46 17.78 13.33 19.18

2−2−2−1 2−2−1−1 2−1−2−1 2−1−1−1 6−6−1−1 6−1−6−1 6−1−1−1

a

R-I, R-II, and R-IV correspond to reducing the number of columns in sections I, II, and IV, respectively. R-CL, reducing the catalyst loading. R-m1, decreasing m1; I-m4, increasing m4.

configurations will have PR values less than the PR obtained from design D7 and DC values higher than the DC obtained from design D1. It is possible that, in some cases, DC alone influences the total cost. Thus, the objective becomes the minimization of DC for a given feed flow rate and feed concentration. In such a case, it is not necessary to go through all the steps of the complete design algorithm described before. Minimization of Desorbent Consumption Alone. As mentioned in section 3.2, the DC is lowest when the pseudo solid velocity is minimum, i.e., when m3 − m2 is maximum for a feasible point. This helps us to select the suitable points in the m2−m3 plane. From the separation triangle derived for the linear isotherms (Figure 14), the point corresponding to the maximum

it is clear that there are four different steps (reduction in number of beds in sections I, II, and IV and reduction in bed loading) to increase the PR and two steps (decreasing m1 and increasing m4) to decrease the DC for a feasible point in the m2−m3 plane. In this section, it is explained how the different paths going through these six steps lead to different designs with different values of PR and DC. All the possible paths involving the six steps, performed in different orders, converge to seven designs (D1−D7), which are shown in Table 4. For each design, the order in which the six steps is performed is also shown in Table 4. The corresponding values of PR and DC and the SMBR configuration for a particular feasible point are listed in the last three columns. All the other paths not shown in Table 4 are either impractical or redundant because of the following reasons: • Reduction in the catalyst loading should be performed only after we identify the section that controls the performance (e.g., section III in this case). • The column volume of the SMBR is selected based on the minimum volume required in the reactive section. This is the reason why decreasing the number of beds or loading in section III is not possible as the reaction takes place mostly in section III. • The reduction in number of beds or loading in section I is not possible once the minimum value of m1 is employed. This is because the least value of m1 selected is such that purity constraints are just satisfied at that adsorbent loading. Decreasing the number of beds or loading in section I adversely affects the raffinate purity. • Similarly, the reduction in number of beds or loading in section IV is not possible after we work with the maximum value of m4, which influences the extract purity. • The reduction in number of beds or loading in section II may be performed at any stage as these steps do not have much effect on either the purities or conversion. • The order in which the number of beds in sections I and IV are reduced does not make a difference. This is because sections I and IV have an impact on the raffinate purity and extract purity, respectively, and they are independent of each other. • Similarly, the DC is reduced by decreasing m1 and by increasing m4. It may be noted that the result is independent of the order in which these two steps are executed. Design D1 corresponds to a design with minimum DC as the reduction in m1 and increase in m4 are performed before any other step. Similarly, design D7 corresponds to the maximization of PR design in which the reduction in catalyst loading is performed before any other step. All the other design

Figure 14. Representative points on (m2, m3) plane for DC minimization.

difference between m2 and m3 [HWater − H2‑EHAc] is identified, which is 3.0 in the present case. To account for the effect, if any, of the nonlinearity in isotherms, the maximum difference between m2 and m3 is increased by a small factor. Figure 14 shows a few representative points on the line parallel to the diagonal and with an intercept of 3.5. Simulations are performed and constraints are examined as described in section 3.1. If all the points on the selected line are infeasible then, the intercept is decreased by a small factor and the same procedure is repeated until we get at least one feasible operating point. The feasible and infeasible points on the m2−m3 plane are shown in Figure 14. As mentioned above, the path D1 gives the minimum DC. Once the feasible points are known, then one 15833



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conversion and purity, is considered to illustrate the algorithm. The best operating conditions obtained by following the proposed algorithm exhibit a much improved performance over the one obtained by one parameter continuation parametric studies in part I.1 The different designs are identified satisfying the desired constraint so that further selection is made based on the cost considerations. The work therefore presents a complete algorithm for the design of an SMBR to be used for a solid catalyzed reversible reaction in which one of the reactants acts as a solvent.

may only perform the steps given in design D1 to reduce the DC further. 3.4.2. Productivity vs Desorbent Consumption. For the optimum process design, one needs the operating point which leads to high PR and low DC. For all the points of interest (shown in Figure 9), if we perform the paths shown in Table 4 then we get different values of the objective functions (i.e., DC and PR) as shown in Figure 15. The curve joining the least

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■ Figure 15. PR vs DC for all possible designs originated from the shortlisted operating points. Dashed line represents the curve of minimum DC and maximum PR.

value of DC for a given PR represents the set of designs which are of maximum interest. This is analogous to a Pareto curve obtained in multiobjective optimization. Further, the weightage assigned to DC and PR would decide which design finally becomes the best choice. By considering one of the points on the Pareto curve, the PR obtained is 80.18 mol of 2-EHAc/(kg of adsorbent·day), which is much higher than the PR obtained from parametric studies (32.01 mol of 2-EHAc/(kg of adsorbent·day)) in part I of this work.1 The DC for the corresponding point is 19.18 mol of AcH/mol of 2-EHAc, which is much less than the DC obtained from parametric studies (46.35 mol of AcH/mol of 2-EHAc).1 Though we show the application of this algorithm to the synthesis of 2-EHAc in the present work, the algorithm can be very well used for other similar reactions in RC. The methodology illustrated in this work assumes that the reactive section of SMBR is section III. This methodology can be extended to the case wherein the reaction takes place either in section II or in both sections II and III. The steps to increase PR and DC may be modified appropriately, but the overall procedure remains similar.

NOMENCLATURE 2-EH = 2-ethylhexanol 2-EHAc = 2-ethylhexyl acetate AcH = acetic acid Ci = liquid (bulk) phase concentration of species i, mol/L DC = desorbent consumption, mol of AcH/mol of 2-EHAc Hi = Henry’s adsorption constant of component i Ki = adsorption equilibrium constant of species i, L/mol kR = overall reaction rate constant, s−1 L = length of single column, cm mj = flow ratio parameter of jth section Ncol = total number of columns in SMBR unit NR = representative points number p = number of columns in section III PR = productivity, mol of product/(kg of adsorbent·day) PuR = raffinate purity of 2-ethylhexyl acetate PuX = extract purity of water Qj = volumetric flow rate within jth section, mL/min QD = volumetric desorbent flow rate, mL/min QF = volumetric feed flow rate, mL/min QR = volumetric raffinate flow rate, mL/min QS = pseudo solid flow rate, mL/min qi = adsorbed phase concentration of component i, mol/L s = number of columns in section II S = column cross-sectional area, cm2 SMBR = simulated moving bed reactor t* = switch time, s TMBR = true moving bed reactor Vcol = volume of single column, cm3 X = conversion

Greek Symbols

Γ∞ i = adsorption capacity of component i, mol/g ρb = bulk density of adsorbent, g/L ρp = density of adsorbent particle, g/L ε = void fraction τrxn = residence time for reaction, s

4. CONCLUSION Design of an SMBR is complex because of the large number of degrees of freedom associated with it. Solving it as an optimization problem is mathematically a challenging job, and hence we have proposed a simulation based method that can suggest a near-optimal design. The method is partly based on the learnings from the triangle theory and the parametric studies in part I of this work.1 The SMBR simulator is used at various stages of the algorithm, wherever appropriate. An example of the synthesis of 2-ethylhexyl acetate for a given feed flow rate (4 mL/min), equimolar feed concentration, and desorbent concentration, with specific limits (99%) on

Subscripts and Superscripts

i = component i R = raffinate X = extract F = feed D = desorbent col = column req = required tol = tolerance 15834



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REFERENCES

(1) Gyani, V. C.; Reddy, B.; Bhat, R.; Mahajani, S. Simulated Moving Bed Reactor for the Synthesis of 2-Ethylhexyl Acetate. Part I. Experiments and Simulations. Ind. Eng. Chem. Res. 2014, DOI: 10.1021/ie502090z. (2) Lode, F.; Houmard, M.; Migliorini, C.; Mazzotti, M.; Morbidelli, M. Continuous Reactive Chromatography. Chem. Eng. Sci. 2001, 56, 269. (3) Mazzotti, M.; Storti, G.; Morbidelli, M. Optimal Operation of Simulated Moving Bed Units for Nonlinear Chromatographic Separations. J. Chromatogr., A 1997, 769, 3. (4) Lode, F.; Mazzotti, M.; Morbidelli, M. Comparing True Countercurrent and Simulated Moving-Bed Chromatographic Reactors. AIChE J. 2003, 49, 977. (5) Fricke, J.; Schmidt-Traub, H. A New Method Supporting the Design of Simulated Moving Bed Chromatographic Reactors. Chem. Eng. Process. 2003, 42, 237. (6) Minceva, M.; Silva, V. M. T.; Rodrigues, A. E. Analytical Solution for Reactive Simulated Moving Bed in the Presence of Mass Transfer Resistance. Ind. Eng. Chem. Res. 2005, 44, 5246. (7) Silva, V. M. T. M.; Rodrigues, E. Novel Process for Diethylacetal Synthesis. AIChE J. 2005, 51, 2752. (8) Minceva, M.; Rodrigues, A. E. Simulated Moving-Bed Reactor: Reactive-Separation Regions. AIChE J. 2005, 51, 2737. (9) Strohlein, G.; Mazzotti, M.; Morbidelli, M. Optimal Operation of Simulated Moving Bed Reactors for Nonlinear Adsorption Isotherms and Equilibrium Reactions. Chem. Eng. Sci. 2005, 60, 1525. (10) Biressi, G.; Ludemann-Hombourger, O.; Mazzotti, M.; Nicoud, R. M.; Morbidelli, M. Design and Optimisation of a Simulated Moving Bed Unit: Role of Deviations from Equilibrium Theory. J. Chromatogr., A 2000, 876, 3. (11) Azevedo, D. C. S.; Rodrigues, A. E. Design Methodology and Operation of a Simulated Moving Bed Reactor for the Inversion of Sucrose and Glucose−fructose Separation. Chem. Eng. J. 2001, 82, 95. (12) Jupke, A.; Epping, A.; Schmidt-Traub, H. Optimal Design of Batch and Simulated Moving Bed Chromatographic Separation Processes. J. Chromatogr., A 2002, 944, 93. (13) Gyani, V. C.; Mahajani, S. Reactive Chromatography for the Synthesis of 2-Ethylhexyl Acetate. Sep. Sci. Technol. 2008, 43, 2245.

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