Batch Processing

Home , Batch distillation, Distillation Design, Reactive distillation

CONTENTS 4.1 Early Theoretical Analysis ...... 43 4.1.1 Simple Distillation ...... 45 4.1.2Copyrighted Operating Modes ...... 46 4.2 Hierarchy of Models ...... 53 4.2.1 Rigorous Model ...... 53 4.2.2 Low Holdup Semirigorous Model ...... 53 4.3 Shortcut Model ...... 55 4.4 Optimization and Optimal Control ...... 58 4.5 Complex Systems ...... 59 4.5.1 Azeotropic Distillation ...... 60 4.5.2 Reactive DistillationMaterial...... 61 4.6 Computer Aided Design Software ...... 62 4.7 Summary ...... 63 Notations ...... 63 - Batch distillation1 is the oldest separation processTaylor and the most widely used unit operation in pharmaceutical and specialty chemical industries. The most outstanding feature of batch distillation is its flexibility. This flexibility allows one to deal with uncertainties in feed stock or product specification. In the distillation process, it is assumed that the vapor& formed within a short period is in thermodynamic equilibrium with the liquid. Hence,Francis the vapor composition y is related to the liquid composition x by an equilibrium relation of the functional form y = f(x). The exact relationship for a particular mixture may be obtained from a thermodynamic analysis and is also dependent upon temperature and pressure. Figure 4.1 shows an example equilibrium curve for a system consisting of CS2 and CCl4 at 1 atmosphere pressure. Simple distillation is the simplest form of batch distillation. In this type of distillation, a still is initially filled with a feed mixture, which evaporates after heating and leaves the still in the vapor form. This vapor which is richer in the more volatile component is collected in the condenser at the top. This simple distillation is often referred to as Rayleigh distillation because of Rayleigh’s pioneering theoretical work [35]. The concept of reflux and the use of accessories such as plates and packing materials to increase the mass transfer converts this simple still into a batch distillation column as shown in Figure 4.2a. Because this batch column essentially performs the rectifying operation, it is often called a batch rectifier. The basic difference between batch distillation (rectifier) and continuous distilla-

1This chapter is based on the book by [7].

41 42 Batch Processes: Modeling and Design

1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.2 0.4 0.6 0.8 1 Copyrighted x

FIGURE 4.1 Equilibrium curve for the CS2 and CCl4 mixture at 1 atmosphere pressure tion is that in continuous distillation (Figure 4.2b) the feed is continuously entering the column, while in batch distillationMaterial the feed is charged into the reboiler at the beginning of the operation. This feed is heated to form vapor. This vapor is passed through packing or plates where it meets the liquid coming from top (refluxed back) from the condenser at the top. Mass transfer takes place in contact and one gets a distillate which is richer in the more volatile component. The reboiler in batch distillation gets depleted over time. The process- has an unsteady-state nature. Reflux ratio, which is a ratio of liquid reflux to distillateTaylor is an important operating parame- ter on which operating condition depends. If reflux ratio is kept constant throughout the operation this results in variable product composition. On the other hand, in order to keep the key composition constant, reflux ratio needs to be varied. There is a third policy of operation known as the optimal reflux& policy that is neither the constant reflux policy nor the variable reflux policy. Instead,Francis this operating policy exploits the difference between the two operating modes. Thus, the optimal reflux policy is essentially a trade-off between the two operating modes, and is based on the ability of the process to yield the most profitable operation. Thus, there are three operating conditions for batch distillation. Constant reflux and variable product composition, • Variable reflux and constant product composition of the key component, and • Optimal reflux and optimal product composition. • The flexible and transient nature of batch distillation allows to configure the column in a number of different ways, some of which are shown in Figure 4.3 [36]. The column in Figure 4.3a, as explained, is a conventional batch distillation column with the reboiler at the bottom and the condenser at the top. A single column can be used to separate several products using the multi-fraction operation of batch distillation presented in Figure 4.3b. Some cuts may be desired and others may be intermediate products. These intermediate fractions can be recycled to maximize profits and/or minimize waste generation. Figure 4.3c shows a periodic operation Batch Distillation 43

(a) condenser (b) condenser

distillate distillate reflux reflux section rectifying feed

rectifying section reboil section stripping stripping Copyrighted

bottom product reboiler reboiler

FIGURE 4.2 Types of distillation processes:Material (a) batch distillation and (b) continuous distillation.

in which each charge consists of a fresh feed- stock mixed with the recycled off- specification material from the previous charge.Taylor Figure 4.3d represents a stripping column for separating a heavy component as the bottom product where the liquid feed is initially charged into the top. In 1994, Davidyan et al. [37] presented a batch distillation column that has both stripping and rectifying& sections embedded in it (Figure 4.3e). This column is called the middle vessel column. Although this column has not been investigated completely, recent studies demonstratedFrancis that it provides added flexibility for the batch distillation operation. Recently [38] described a new column configuration called a multivessel column (Figure 4.3f) and showed that the column can obtain purer products at the end of a total reflux operation. These emerging column designs play an important role in separation of complex systems such as azeotropic, extractive, and reactive batch distillation systems. Although all these column configurations are important, we will focus on the conventional batch column in this chapter.

4.1 Early Theoretical Analysis

This section presents early theoretical analysis of simple distillation, which was ﬁrst analyzed by [35]. 44 Batch Processes: Modeling and Design

2 3 2 3

Copyrighted

1 1 (a) (b) (c)

Material

- Taylor

& Francis

(d) (e) (f)

FIGURE 4.3 Various configurations of batch distillation column. Batch Distillation 45 4.1.1 Simple Distillation The analysis of simple distillation presented by Rayleigh in 1902 marks the earliest theoretical work on batch distillation. Simple distillation, also called Rayleigh distillation or differential distillation, is the most elementary example of batch distillation. In this distillation system, the vapor is removed from the still during each time in- terval and is condensed in the condenser. The vapor is richer in the more volatile component than the liquid remaining in the still. Over time, the liquid remaining in the still begins to experience a decline in the concentration of the more volatile component, while the distillate collected in the condenser becomes enriched in the more volatile component. No reflux is returned to the still, and no plates or packing materials are provided inside the column. Therefore, various operating policies are not applicable to this distillation system. The earlyCopyrighted analysis of this process for a binary system, proposed by Rayleigh, is given below. Let F be the initial binary feed to the still (moles), and xF be the mole fraction of the more volatile component A in the feed. Let B be the amount of compounds remaining in the still, xB the mole fraction of component A in the still, and xD the mole fraction of component A in the vapor phase. The differential material balance for component A can then be written as:

xD dB = d(B xB ) = B dxB + xB dB, (4.1) Material giving: B dB xB dx = B , (4.2) B xD xB ZF ZxF − or: B xB dx ln = - B . (4.3) F xTaylorD xB ZxF − In this simple distillation process, it is assumed that the vapor formed within a short period is in thermodynamic equilibrium with the liquid. Hence, the vapor composition xD is related to the liquid composition xB&by an equilibrium relation of the form xD = f(xB ). The exact relationship for a particular mixture may be obtained from a thermodynamic analysis depending on temperatureFrancis and pressure. For a system following the ideal behavior given by Raoult’s law, the equilibrium relationship between the vapor composition y (or xD) and liquid composition x (or xB ) of the more volatile component in a binary mixture can be approximated using the concept of constant relative volatility (α), and is given by: αx y = (4.4) ()α 1 x + 1 − Substitution of the above equation in Equation 4.3 results in:

B 1 xB (1 xF ) 1 xF ln = ln − + ln − (4.5) F α 1 x1F ( xB ) 1 xB − − − Although the analysis of simple distillation historically represents the theoretical start of batch distillation research, a complete separation using this process is impossible unless the relative volatility of the mixture is inﬁnite. Therefore, the application of simple distillation is restricted to laboratory scale distillation, where high purities are not required, or when the mixture is easily separable. 46 Batch Processes: Modeling and Design

Example 4.1: A mixture of components A and B with 0.4 mole fraction of A and relative volatility of 2.0 is distilled in a simple distillation. The feed is 100 moles, and 40% of mixture is distilled. Find the distillate composition at the end of operation. Solution: Since 40% of feed is distilled, the residue amount in the reboiler is 60 moles. The bottom composition can be found using Equation 4.5 as given below.

60 1 xB (1 0.4) 1 0.4 ln = ln − + ln − xB = 0.3138 100 2 1 0 .4(1 xB ) 1 xB ⇒ − − − Then the distillate composition can be obtained using Equation 4.4, resulting in distillate composition of 0.4780. Initial distillate composition corresponding to xF = 0.4 (again from the same Equation 4.4) is 0.5700. Therefore, the average distillate compositionCopyrighted is between 0.5700 and 0.4780. As can be seen that this distillate composition is very low for a separation process, simple distillation cannot be used for separation in real practice. To obtain products with high purity, multi-stage batch distillation is used.

4.1.2 Operating ModesMaterial As stated earlier, the two basic modes of batch distillation are (1)constant reflux, and (2)variable reflux, resulting in variable distillation composition and constant distillate composition of the key component, respectively. The third operating mode, optimal reflux or optimal control is the trade-off between the two operating modes. It is easy to understand the two basic- modes using binary distillation. This involves use of McCabe-Thiele graphical methodTaylor described below.

McCabe-Thiele Graphical Method Although simple distillation marks the first analysis of batch& distillation process, the graphical analysis presented by [39] provided the basis for analyzingFrancis batch distillation operating modes. The difference between simple distillation and batch distillation operations is the relation between the distillate composition xD and the bottom composition xB due to the presence of reflux and column internals as shown in Figure 4.4. If one assumes constant molal overflow (Lj = Lj−1 = .. = L0 = L and Vj = Vj−1 = .. = V1 = V ), then McCabe-Theile’s method can be used. In the McCabe-Thiele method, the overall material balance (with no liquid holdup) on the plate is considered from the condenser to the j-th plate. This leads to the following operating line equation (Equation 4.6). Note that in this procedure, we use the theoretical number of stages or plates. If you divide the number of theoretical plates with the plate efficiency, we get the actual plates in a plate column, and if you multiply these theoretical stages with the Height Equivalent to a Theoretical Plate (HETP) then you get the actual height of a packed column. R 1 y = x − + x (4.6) j R1 + j 1 R1 + D

dD dD where R = L/ dt and V = (R + 1) dt . This operating equation represents a line through the point yj (xj−1 = xD) Batch Distillation 47

Condenser

L0 1 dD/dt xD

j-th plate

Copyrighted Vj Lj-1 N xj-1 yj

y xj j+1 N+1 V L Reboiler B, xB Material j+1 j

FIGURE 4.4 - Schematic of a batch distillation column. Taylor

with a slope of R/(R + 1). From (x0 = xD, y1 = xD) draw a horizontal line to equilibrium curve to ﬁnd the x1, for this x1, yj + 1 can& be found out from the operating line (vertical line, Equation 4.6). So the procedureFrancis is, starting from point (xD, xD), Equation 4.6 and the equilibrium curve (e.g., Equation 4.4) between yj and xj can be recursively used from the top plate 1 to the reboiler (the reboiler can be considered as the (N+1)-th plate) as shown in Figure 4.5 (where N + 1 is equal to 6). This procedure relates the distillate composition xD to the still composition xB through the number of stages, N.

Constant Reﬂux Mode

Smoker and Rose[40] presented the ﬁrst analysis of the constant reﬂux operation of a binary batch distillation with no holdup. They used the Rayleigh equation in conjunction with the McCabe-Thiele graphical method to capture the dynamics of the batch distillation column. In their procedure, the relationship between xD and xB is recursively determined by the McCabe-Thiele graphical method. Then, the right hand side of the Rayleigh equation (Equation 4.3) is integrated graphically by plotting 1/(xD xB ) versus xB . The area under the curve between the feed − composition xF and the still composition xB now gives the value of the integral, which is ln (B/F ). The average composition of the distillate can be obtained from 48 Batch Processes: Modeling and Design

1 1 A 3 ( x x ) 2 D, D 5 0.8 4

7 0.6 6 9 y 8 0.4

10 Copyrighted 0.2 R Slope = R+1 0 x 0 0.2B 0.4 0.6 0.8 1

Material x

FIGURE 4.5 McCabe-Thiele method - Taylor the following equation: F xF B xB xD,avg = − (4.7) F B − Although Smoker and Rose presented the calculation method& independent of time, the time can be introduced through the vapor boilup rateFrancisV of the reboiler. The resulting equation for determining total batch time is given by: V R + 1 R + 1 dB = dt T = (F B) = D (4.8) R + 1 ⇒ V − V This operation policy is the easiest one and is commonly used. The following example illustrates this policy.

Example 4.2: Consider the same mixture presented in Example 4.1 but use batch distillation column with ﬁve theoretical plates and reﬂux ratio equal to 5.630. a) Find the distillate and still composition when 40 percent of the mixture is distilled similar to Example 4.1. b) If the initial feed is 100 moles and vapor boilup rate is 143.1 moles/hr, what is the total time required to complete the distillation operation? Solution: The distillation operation requires 40 percent of the mixture to be distilled, which corresponds to 60 percent of the original feed F remaining in the reboiler, i.e., B = 0.6F . The problem is that of determining the distillate and still Batch Distillation 49

composition, for which the Rayleigh equation (Equation 4.3) applies. To attain the speciﬁed requirements, the left hand side of the Rayleigh equation can be calculated as: F 1 ln = ln = 0.5108 B 0.6 For various values of xD the operating lines are drawn and the bottom composition is calculated using the procedure shown in Figure 4.5. From trial and error, we discover that the initial value of xD corresponding to the still composition of xF = 0.4 is 0.9471. The rest of the xD are chosen below this composition, and 1 various values of xB , xD xB , are obtained. − xD − xB 1 xB dxB Values of xB versus are plotted in Figure 4.6 from which xD − xB xF xD − xB is obtained for each value of x . The operation is stopped when the integral is equal B R to 0.5108.Copyrighted a) The distillate composition xD at the end of the operation is 0.5681, and the still composition xB is 0.1000. The average distillate composition is obtained using the following material balance equations. 2.2 2.1 Material 2

) B 1.9

D 1.8 1/(x - x - 1.7 Taylor 1.6 1.5 0 0.1 0.2 0.3& 0.4 0.5 x B Francis

FIGURE 4.6 Graphical integration for Example 4.2

D B xDdD = xB dB Z0 ZF D xDav dD = F xF BxB − Z0 F xF BxB xDav = − D 100 0.4 60 0.1 xDav = × − × = 0.8500 40 The average distillate composition is 0.85 for the key component which is much larger than obtained in simple distillation. 50 Batch Processes: Modeling and Design b) The time required for the distillation is given below.

R + 1 T = D = 1.8532 hrs V ×

Variable Reflux In 1937, Bogart [41] presented the first analysis of the variable reflux policy for a binary system. The steps involved in the calculation procedure for the variable reflux mode are similar to those in the case of the constant reflux mode; however, in the variable reflux case, the reflux ratio is varied instead of the distillate composition at each step. Moreover, the Rayleigh equation, though valid for the variable reflux condition,Copyrighted takes a simplified form. Since the distillate composition remains constant (remember that we are considering binary systems here) throughout the operation, the Rayleigh equation reduces to the following equation.

B xD xF = − (4.9) F xD xB − The second step is to establish the relation between R and xB using the McCabe- Thiele graphical method. SeveralMaterial values of R are selected, operating lines are drawn through the fixed point (xD, xD) with the slope of R/(R + 1), and steps are drawn between the operating line and the equilibrium curve to get the bottom composition (xB ). This recursive scheme is repeated until the desired stopping criteria is met, and thus B and xB can be found at each value of the reflux ratio. The time required for this operation at a given product purity- is calculated by plotting the quantity (R+1) F (xD −xF ) 2 versus xB in the followingTaylor equation and then finding the area V × (xD −xF ) under the curve. xF R + 1 F (xD xF ) T = − 2 dxB (4.10) V (xD xB ) ZxB − & Example 4.3: Rework the problem in Example 4.2 for theFrancis variable reflux mode.

Solution: Since the distillate composition is held constant throughout the variable reﬂux mode of operation, the distillate composition xD = xDav = 0.85. For the various iterates of R, we obtain the corresponding values of xB . The value of the amount of product distilled at each xB is also calculated using the Rayleigh equation for the variable reﬂux condition (Equation 4.9).

xD xF D = F (1 − ) − xD xB − a) The operation is stopped when the amount of product collected is greater than or equal to 40. The still composition at D = 40 is found to be equal to 0.10. b) The time required for this operation is calculated by plotting the quantity R+1 F (xD − xF ) 2 versus xB and then ﬁnding the area under the curve between V (xD − xB ) xB equal to 0.4 (Figure 4.7) and 0.1. The time required is found to be 1.995 hrs. Batch Distillation 51

30 25 20

B 15

dt / dx 10

5 0 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Copyrighted x B

FIGURE 4.7 Graphical integration for calculation of batch time for Example 4.3

Optimal Reflux Policy Material The optimal reflux mode is a reflux profile that optimizes the given indices of column performance chosen as the objectives. The indices used in practice generally include the minimum batch time, maximum distillate, or maximum profit functions. This reflux policy is essentially a trade-off between the two operating modes, and is based on the ability to yield the most profitable operation- from optimal performance. The calculation of this policy is a difficult problemTaylor and relies on optimal control theory and is discussed in Chapter 13 on optimization and optimal control. For example, consider the following reflux profile for the same separation given in Examples 4.2 and 4.3. & xD .9019 .8980 .8899 .8749 .8455 .8265 .8010 .7635 .7018 R 5.472 5.349 5.237 5.138 4.980 5.348 Francis5.966 6.839 7.623

Remember we are using the same batch and the same column for this operation. Since in this case neither distillate composition nor reflux ratio is constant, the following procedure is used for integration of the Rayleigh equation. For each pair of values of xD and R, operating lines are drawn with a slope equal R to R+1 and passing through the point (xD, xD). Plate to plate calculations are used to obtain the corresponding values of xB . As stated earlier, the basic batch distillation column satisfies the Rayleigh equation. Therefore we can use the same equation to calculate the total amount of distillate, as was done for the constant reflux condition. Values of xB , xD xB , 1/xD xB − − are obtained for each operating line. Values of xB versus 1/xD xB are plotted from − which the right hand side of the Rayleigh equation is obtained as the area under the B curve and is found to be 0.5108, which corresponds to ln[ F ]. Hence the total amount of distillate given by F (1 B ) is equal to 40 moles, which is what we obtained in − F the variable reflux and constant reflux operations in Examples 4.2 and 4.3. We resort to the basic mass balance equation to obtain the average distillate composition used in the case of constant reflux mode. 52 Batch Processes: Modeling and Design

D B xDdD = xB dB Z0 ZF D xDav dD = F xF BxB − Z0 F xF BxB xDav = − D 100 0.4 60 0.1 xDav = × − × = 0.8500 40

The average distillate composition in this case is found to be 0.85, which is again the same asCopyrighted that of the two operating modes. Similarly, the time requirement for this operation is derived from the basic material balance equations as was done in the case with the variable reﬂux condition. The time T required when neither distillate composition nor reﬂux is constant is found to be

xF B R + 1 T = dxB (4.11) V xD xB MaterialZxB − The value of T is found to be 1.686 hrs, smaller than for the constant and variable reflux modes of operation. It can be seen that this reflux profile requires the least amount of time to obtain the same amount of product with the same- purity. This policy is neither variable reflux nor constant reflux but is in betweenTaylor the two as shown in Figure 4.8. The policy which was specified in the above example is the optimal policy obtained by Coward in 1967[42]. & 50 1

45 0.95 Francis

40 0.9

35 0.85 Variable reflux 30 Constant reflux 0.8 Optimal reflux 25 0.75 Variable reflux 20 0.7 Constant reflux Optimal reflux 15 Reflux Ratio 0.65

10 Distillate Composition 0.6

5 0.55

0 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5

Bottom Composition Bottom Composition

FIGURE 4.8 The three operating modes Batch Distillation 53

4.2 Hierarchy of Models

As seen in earlier sections, the earlier models of the batch rectifier were built on assumptions of negligible liquid holdup and ideal binary systems. Computers have played an important role in relaxing these assumptions, especially the negligible holdup assumption. Distefano analyzed the numerical differential equations for multicomponent batch distillation in 1968 for the first time[43]. The rigorous models of batch distillation in current state of the art computer packages are based on his pioneering work. However, it was also acknowledged that due to the severe transients in batch distillation, a hierarchy of models is necessary to capture the dynamics of this flexible operation[44, 36]. This section presents the hierarchy of models ranging from the rigorousCopyrighted model (similar to the one presented by Distefano) to the simplest shortcut model[45].

4.2.1 Rigorous Model

A rigorous model in batch distillation involves consideration of column dynamics along with the reboiler and condenser dynamics. Distefano presented detailed analysis of the characteristics of differentialMaterial mass and energy balances associated with the complete dynamics of a multicomponent batch distillation column. Distefano’s work forms the basis for almost all of the later work on rigorous modeling of batch distillation columns, and this model is presented below. For an arbitrary plate j (Figure 4.4), the total mass, component, and energy balances yield the governing equations are- summarized in Table 4.1. This table lists all the equations involved in the dynamicTaylor analysis of the batch column and the assumptions behind these equations. Note that energy balance equations are difference equations instead of differential equations, this is due to the fact that flowrates are much larger as compared to the enthalpy changes.& The system of equations governing the batch distillation process is difficult to solve as the plate holdup is generally much smaller than reboilerFrancis holdup resulting in severe transients. Stiff equation solver is necessary to solve these type of equations. The stiffness of the system is reduced considerably when one considers zero plate holdup. This results in semirigorous model for batch distillation. This model is similar to what was used earlier with McCabe-Theile method (except with additional energy balance equations whenever necessary).

4.2.2 Low Holdup Semirigorous Model

For columns where the plate dynamics are significantly faster than the reboiler dynamics (due to very small plate holdups and/or wide boiling components), the stiff integrator often fails to find a solution. The solution to this problem is to split the system into two levels: (a) the reboiler, where the dynamics are slower, can be represented by differential equations (Equations 4.12-4.13), and (b) the rest of the column can be assumed to be in the quasi-steady state. Thus, the composition (i) changes in the condenser and accumulator (dxD /dt), the composition changes on (i) plates (dxj /dt), and the enthalpy changes in the condenser and on plates (δtID 54 Batch Processes: Modeling and Design

TABLE 4.1 The complete column dynamics Assumptions • Negligible vapor holdup, • Adiabatic operation, • Theoretical plates, • Constant molar holdup, • Finite diﬀerence approximations for the enthalpy changes. Copyrighted Composition Calculations Condenser and Accumulator Dynamics i dx( ) (i) (i) D = V1 (y − x ), i = 1, 2, . . . , n dt HD 1 D Plate Dynamics (i) dxj 1 (i) (i) (i) (i) = [Vj+1yj+1 + Lj 1xj 1 − Vjyj − Ljxj ], dt Hj − i1,= 2, . . . , n; j = 1−, 2,...,N MaterialReboiler Dynamics (i) dxB 1 (i) (i) (i) (i) dt = B [LN (xN − xB ) − VB(yB − xB )], i = 1, 2, . . . , n Flowrate Calculations At the Top of the Column L = R dD ; V = (R + 1) dD 0 dt 1- dt On the PlatesTaylor Lj = Vj+1 + Lj 1 − Vj; j = 1, 2,...,N 1 − Vj+1 = [Vj(Jj − Ij) + Lj 1(Ij − Ij 1) + HjδIj] Jj+1 Ij − − − j1,= 2,...,N At the Bottom of the Column& dB dt = LN − VB Francis Heat Duty Calculations Condenser Duty QD = V1(J1 − ID) − HDδtID Reboiler Duty QB = VB(JB − IB) − LN (IN − IB) + BδtIB Thermodynamics Models Equilibrium Relations (i) (k) y( j = f( xj , k = 1, . . . , n),TEj,Pj) Enthalpy Calculations (k) I(j = f( xj , j = 1, . . . , n),TEj,Pj) (k) J( j = f( yj , j = 1, . . . , n),TEj,Pj) Batch Distillation 55 and δtIj ) in Table 4.1 can be assumed to be zero. These equations can be integrated using Runge-Kutta method. This results in a zero holdup model, so this approach can be used for simulating the semirigorous model of batch distillation. dB V = ,B0 = F (4.12) dt − R1 +

dx(k) V B = (x(k) x(k)), x(k) = x(k) (4.13) dt ( R + 1)B B − D B0 F The holdup eﬀects can be neglected in a number of cases where this model ap- proximates the column behavior accurately. This model provides a close approximation to the Rayleigh equation, and for complex systems (e.g., azeotropic systems) the synthesis procedures can be easily derived based on the simple distillation residue curve mapsCopyrighted (trajectories of composition). However, note that this model involves an iterative solution of nonlinear plate-to-plate algebraic equations, which can be computationally less eﬃcient than the rigorous model.

4.3 Shortcut ModelMaterial The rigorous model of batch distillation operation involves a solution of several stiff differential equations and the semirigorous model involves a set of highly nonlinear equations. The computational intensity and memory requirement of the problem increase with an increase in the number of plates- and components. The computational complexity associated with these models doesTaylor not allow us to derive global proper- ties such as feasible regions of operation, which are critical for optimization, optimal control, and synthesis problems. Even if such information is available, the computational costs of optimization, optimal control, or synthesis using these models are prohibitive. One way to deal with these problems associated& with these models is to develop simplified models such as the shortcut model. Francis The shortcut model of batch distillation proposed by [45] is based on the assumption that the batch distillation column can be considered equivalent to a continuous distillation column with changing feed at any instant. Since continuous distillation theory is well-developed and tested, the shortcut method of continuous distillation is modified for batch distillation, and the compositions are updated using a finite- difference approximation for the material balance (based on the Rayleigh equation). The other assumptions of the shortcut method include constant molar overflow and negligible plate holdups. As described earlier, the functional relationship between the distillate composition xD and the bottom composition xB is crucial for the simulation, and the FUG (Fenske-Underwood-Gilliland) method is used for estimating this relation. These equations shown in Table 4.2 along with the material balance equations (Equations 4.12-4.13) are used to simulate the three operating conditions of batch distillation. At any instant of time, Equations 4.12, 4.13, and the differential material balance equation can be used to calculate the bottom composition of all the components. The following procedure is then used to calculate the distillate composition at that instant. The procedure is repeated at each time step until the stopping criterion is 56 Batch Processes: Modeling and Design

TABLE 4.2 Time implicit model equations for the shortcut method

Variable Reflux Constant Reflux Optimal Reflux Differential Material Balance Equation (k) (i) (i) xB xD xB x(i) = x(i) + △ − old , i = 1, 2, . . . , n Bnew Bold (k) (k) xD xB − old Hengstebeck-Geddes Equation C1 (k) (i) αi xD (i) x = k x , i = 2, 3, . . . , n D αk x( ) B B Unknowns (k) (k) R, C1 C1, xD R, C1, xD SummationCopyrighted of Fractions n (i) i=1 xD = 1 (k) C 1 Estimation P x D estimation C1 (k) n αi xD (i) (k) 1 (k) x = 1 x = (i) i=1 αk B D C x xB n αi 1 B i=1 α (k) P k x P B Fenske Equation Material Nmin ≈ C1 Underwood Equations α x(i) α x(i) n i B = 0 ; R + 1 = n i D i=1 αi φ min u i=1 αi φ − − Gilliland CorrelationP - P R Estimation TaylorC 1 E stimation (1 + 54.4X)(X 1) Y1 = − exp − (11 + 117.2X)√X R Rmin h N Nmini X = R1− + ; Y = N1− + & R Estimation R Francis= F (Minimum H)

satisfied. The complete simulation of the shortcut method for the different operating modes is illustrated in the flowchart shown in Figure 4.9. The constant C1 in the Hengstebeck-Geddes equation is equivalent to the minimum number of plates, Nmin, in the Fenske equation. At this stage, the variable (k) reflux operating mode has C1 and R, the constant reflux has xD and C1, and the (k) optimal reflux has xD , C1, and R as unknowns. Summation of distillate composi- (k) tions can be used to obtain C1 for variable reflux and xD for both constant reflux and optimal reflux operation, and the FUG equations to obtain R for variable reflux and C1 for both constant reflux and optimal reflux operations. The optimal reflux mode of operation has an additional unknown, R, which is calculated using the concept of optimizing the Hamiltonian, formulated using the different optimal control methods. The shortcut model is extremely efficient and reasonably accurate for nearly ideal systems and for column with negligible holdup. For columns with severe holdup Batch Distillation 57

Start

Initialization (i) (i) x = F xB Copyrighted i = 2,3,...,n

(k) Initialize: B and xB

Variable Reflux Constant and Optimal Reflux Assume C1 Hengestebeck-Geddes Eqn Material (i) i=2,3,...,n Calculate xD Eqn for Summation

(k) NewC1 - Calculate xD FUGTaylor Eqns Differential Material Balance Eqns (i) Calculate R New C Calculate 1 xB , i=2,3,...,n Optimal Reflux & Calculate R Integration Francis Stopping (k) B and x Criteria Initialize: B met? YES

Stop

FIGURE 4.9 The ﬂowchart for the shortcut method 58 Batch Processes: Modeling and Design eﬀect and highly non-ideal systems, the order of rigorous model can be reduced using collocation approach.

4.4 Optimization and Optimal Control Literature on the optimization of the batch column is focused mostly on the solution of optimal control problems, which includes optimizing the indices of performance such as maximum distillate, minimum time, and maximum profit. However, literature on optimal design of batch distillation for performing specified operations by using the constant reflux or variable reflux policies is very limited[46]. As statedCopyrighted in Chapter 1, control refers to a closed-loop system where the desired operating point is compared to an actual operating point and a knowledge of the error is fed back to the system to drive the actual operating point towards the desired one. However, the optimal control problems we consider here do not fall under this definition of control. Because the decision variables that will result in optimal performance are time-dependent, the control problems described here are referred to as optimal control problems. Thus, use of the control function here provides an open- loop control. The dynamic natureMaterial of these decision variables makes these problems much more difficult to solve as compared to normal optimization where the decision variables are scalar. The indices of performance used in batch distillation optimal control problems are discussed below. - Maximum Distillate Problem – whereTaylor the amount of distillate of a specified • concentration for a specified time is maximized [47, 48, 49, 50, 51]. This problem can be mathematically represented as follows:

T dD T V& max J = dt = Francisdt, (4.14) Rt dt R1 + Z0 Z0 t subject to the material and energy balances. [47] were the first to report the maximum distillate problem for binary batch distillation, which was solved using Pontryagin’s maximum principle, the dynamic programming method, and the calculus of variations. These methods are described in Chapter 14. [48] extended this optimization model to multicomponent systems and used the shortcut batch distillation model along with the maximum principle for the calculation of the optimal reflux policy. [49] used the orthogonal collocation approach on finite elements and nonlinear programming (NLP) optimization techniques over the shortcut model. Further, they extended this method to the rigorous batch distillation model [51] in which they considered the effect of column holdups on optimal control policy. Minimum Time Problem – where the batch time needed to produce a pre- • scribed amount of distillate of a specified concentration is minimized [42, 52]. Although there are several different formulations for the minimum time problem, [52] derived the following formulations to establish a unified theory for all the Batch Distillation 59

optimal control problems: T dt∗ min J = dt. (4.15) Rt dt Z0 where t∗ is a dummy variable as a state variable. Maximum Profit Problem – where a profit function for a specified concentra- • tion of distillate is maximized [53, 49, 54, 55]. Much of the recent research on optimal control problems can be classified into this problem. [53] were the first to use the profit function for maximization in batch distillation , and they solved the optimal control problem. The following simple objective function is given by [53]:

DPr FCF max J = − , (4.16) Copyrighted Rt,T T + ts subject to purity constraints and column modeling equations. [48] used a different objective function to solve the profit maximization problem under the constant and variable reflux conditions. [49] formulated a new profit function and solved the differential algebraic optimization problem for optimal design and operation. [54] developed a detailed dynamic multifraction batch distillation model and discretized the model using the orthogonal collocation method on finite elements, and finallyMaterial solved the maximum profit model using a NLP optimizer. [55] considered a rigorous reactive distillation system for the maximum conversion problem, which can also be classified as the maximum profit problem. The detailed dynamic system is then reduced by using polynomial curve fitting techniques and solved by using a NLP optimizer. A variant of this objective function is to- minimize the mean rate of energy consumption when the market size for the productTaylor is fixed by the current demand. The objective function is given by [56]: T QR(t)dt min J = 0 , (4.17) T + ts R ∗ & s.t. xD,avg x , ≥ Francis D D∗, ≥ where QR is the reboiler heat duty. They used this objective function for optimal control of multivessel columns for the first time. [57] also presented the optimal operation policy based on energy consumption for the multivessel column.

4.5 Complex Systems Thermodynamically and kinetically complex systems like azeotropic, extractive, and reactive systems pose additional bottlenecks in design and operation of batch columns. The following sections describe the methods for analyzing these complex systems. These methods also provide heuristics for synthesis of these columns especially in terms of the diﬀerent cuts obtained in a single column or performance comparison of the complex columns. 60 Batch Processes: Modeling and Design 4.5.1 Azeotropic Distillation In a normal distillation column, the vapor becomes steadily richer in the more volatile component as it passes through successive plates. In azeotropic mixtures this steady increase in concentration does not take place, owing to the so-called azeotropic points. For instance, when a mixture of ethyl alcohol and water is distilled, the concentration of the alcohol steadily increases until it reaches 92.4 percent by moles, when the composition of the vapor equals that of the liquid, and no further enrich- ment occurs (see Figure 4.10). Such a mixture is called an azeotrope and cannot be separated by straightforward distillation.

Copyrighted0.8

0.6

ethanol

y 0.4

0.2 Material

0 0 0.2 0.4 0.6 0.8 1 x ethanol - FIGURE 4.10 Taylor Vapor-liquid equilibrium curve for the ethanol-water system

Azeotropic distillation is an important and widely used separation technique as a large number of azeotropic mixtures are of great industrial& importance. Theoretical studies on azeotropic distillation have mainly centered aroundFrancis methods for predicting the vapor-liquid equilibrium data from liquid solution models and their application to distillation design. However, only during the past two decades has there been a concerted eﬀort to understand the nature of the composition region boundaries. In binary distillation, the barrier is represented by the azeotropic points and in case of a ternary system, they are represented by separatrices. Based on this information, qualitatively similar ideal systems can be represented for these non-ideal azeotropic systems[58, 59]. Doherty & coworkers [60, 61] in their pioneering works proposed several new concepts in azeotropic distillation. They established the use of ternary diagrams and residue curve maps in the design and synthesis of azeotropic continuous distillation columns. In batch distillation, they outlined a synthesis procedure based on the residue curve maps. The residue curve map graphs the liquid composition paths that are solutions to the following set of ordinary diﬀerential equations:

dxi = xi yi i = 1, 2, . . . , n 1, (4.18) dξ − − where n is the number of components in the system, and the independent variable, Batch Distillation 61

Propyl Amine (56 oC) 0.0 1.0

0.2 0.8

0.4 0.6

0.6 0.4

0.8 Copyrighted 0.2

1.0 0.0 H O(100oC) ACN (81.6 oC) 2 0.0 0.2 0.4 0.6 0.8 1.0 Azeotrope (76 oC)

FIGURE 4.11 Material A residue curve map of the propyl amine-acetonitrile-water system. warped time (ξ), is a monotonically increasing- quantity related to real time. One can see that Equation 4.18 is one form of theTaylor Rayleigh equation described earlier. The residue curve map occupies a signiﬁcant place in the conceptual design stage of column sequencing in continuous distillation, and fractions (cuts) sequencing in batch distillation [60, 61, 62]. Despite the advances in the thermodynamics for predicting& azeotropic mixture, feasible distillation boundaries, and sequence of cuts, the azeotropicFrancis batch distillation system is still incipient in terms of design, optimization, and optimal control.

Example 4.4: A residue curve map of the propyl amine-acetonitrile (ACN)- water system are given in Figure 4.11. Find the batch distillation regions and deﬁne the product cuts for each region. Solution: Since the curve from propyl amine to the ACN-water azeotrope is distillation barrier, there are two distillation regions in this system. For the left distillation region, the product sequence is propyl amine, ACN-water azeotrope, and water. For the right region, the sequence is propyl amine, ACN-water azeotrope, and ACN. This example shows that conventional distillation cannot obtain pure water and pure ACN cuts at the same time.

4.5.2 Reactive Distillation Although reactive distillation was acknowledged as a unit operation as early as in the 1920s, it has gained its research interest as an excellent alternative to both 62 Batch Processes: Modeling and Design reaction and separation since the 1980s. For example, most of the new commercial processes of MTBE (methyl-tertiary-butyl ether, an anti-knocking agent) are based on continuous reactive distillation technologies. The analysis of a reactive batch distillation model in a staged column was first published by [63]. Using a stiff integrator for the differential and algebraic equations, they presented a numerical solution technique for the esterification of 1-propanol and acetic acid. They argued that in reactive distillation differential energy balances should be included in the solution. [64] developed a new solution technique based on the orthogonal collocation method on the finite element method for the reactive batch distillation of a packed column. The differential contactor model of a packed column, originally designed by [65], was then reduced to the low-order polynomials with desired accuracy. They compared the results with those from the finite difference method and global collocation method for non-reactive packed-bed batch distillationCopyrighted systems, and showed that their approach was more efficient. [66] extended their previous work to the optimal campaign structure for reactive batch distillation, which can offer reasonably sharp separations between successive cuts and reduce the amount of waste off-cuts. To obtain the optimal reflux policies or profiles for the maximum distillate or minimum time problem, multiperiod reflux optimization [67] can be applied. They showed that for the same production rate, the waste generation can be significantly reduced under the optimal campaign structure. An efficient optimization approachMaterial for reactive batch distillation using polynomial curve fitting techniques was presented by [55]. After finding the optimal solution of the maximum conversion problem, polynomial curve fitting techniques were applied over these solutions, resulting in a nonlinear algebraic maximum profit problem that can be efficiently solved by a standard NLP technique. Four parameters in the profit function, which are maximum conversion,- optimum distillate, optimum reflux ratio, and total reboiler heat load, were thenTaylor represented by polynomials in terms of batch time. This algebraic representation of the optimal solution can be used for online optimization of batch distillation. A dynamic rate-based model for packed-bed batch distillation was recently proposed [68], in which a solid catalyst was used first in the& reactive batch distillation modeling. The pilot-scale experiments were conducted withFrancis strong anion-exchange resins. The results were compared with the experimental data and with the results from its counterpart, the equilibrium-based model. The rate-based model provides more accuracy, much higher physical significance, and more predicability of the experimental data even though the formulation of the rate-based model is complicated.

4.6 Computer Aided Design Software

It is difficult to analyze batch distillation without using computers due to the two reasons stated before: (a) the process is time varying, and one has to resort to complex numerical integration techniques and different simulation models for obtaining the transients, and (b) this ever-changing process also provides flexibility in operating and configuring the column in numerous ways. Based on the current state of the Batch Distillation 63 art in batch distillation techniques and computer simulation technology, Table 4.3 identifies the required functionality and the rationale behind it. There are several commercial software packages for simulations, optimizations, and/or optimal controls of batch distillation (see Table 4.3). These include Bdist- SimOPT (Batch Process Technologies), BatchSim(Simulation Sciences), BatchFrac (Aspen Technology, based on [69]), and MultiBatchDS (Batch Process Research Company). Bdist-SimOPT and MultiBatchDS are derived from the academic pack- age BATCHDIST [45]. Most of these packages except MultiBatchDS are usually limited to conventional systems as they were developed in early or late, 80s.

Copyrighted 4.7 Summary

This chapter presented analysis of batch distillation beginning from the first theoretical analysis in 1902. There are three operating modes of conventional batch distillation column, namely, constant reflux and variable product composition, variable reflux and constant product composition of the key component, and optimal reflux policy. There is a hierarchy of models available for batch distillation. These include shortcut model for feasibility,Material design, optimization, and synthesis, semirigorous model with negligible holdup, rigorous model including all transients with holdup and reduced order model based on collocation method. There are numbers of software packages available for batch distillation design, simulation, and optimization. - Taylor

Notations & Francis B amount of bottom residue [mol] dB dt bottom product flow rate or change of bottom product [mol/time] C1 constant in the Hengstebeck–Geddes equation for conventional batch column D amount of distillate [mol] dD dt distillate rate [mol/time] E entrainer feed rate [mol/time] F amount of feed [mol] Hj molar holdup on plate j [mol] H0,HD condenser holdup [mol] 2 −2 ID enthalpy of the liquid in the condenser [mass length time /mol] 2 −2 Ij enthalpy of the liquid stream leaving plate j [mass length time /mol] 2 −2 Jj enthalpy of the vapor stream leaving plate j [mass length time /mol] Lj liquid stream leaving plate j [mol/time] L0 liquid reflux at the top of the column [mol/time] n number of components N number of plates Nmin] minimum number of plates 64 Batch Processes: Modeling and Design q′ ratio of the top vapor flow rate to the bottom vapor flow rate 2 −2 QR reboiler heat duty [mass length time ] R reflux ratio (= L/D) Rmin g minimum reflux ratio given by the Gilliland correlation Rmin u minimum reflux ratio given by the Underwood equations Rt reflux ratio as a function of time T batch time [time] Vj vapor stream leaving plate j [mol/time] x liquid-phase mole fraction xB mole fraction of liquid in the reboiler xD mole fraction of the distillate xD,avg average distillate mole fraction xF liquid-phase mole fraction of the feed y Copyrightedvapor-phase mole fraction

Greek Letters:

α relative volatility

Material

- Taylor

& Francis Batch Distillation 65 DS Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No No Copyrighted No Yes Yes Yes Yes Yes Yes Yes Yes Yes

Material No No No No No No No No No No Yes No No No No Yes No Yes Yes Yes Yes Yes Yes CHEMCAD ASPEN PLUS CRANIUM

CHEMCAD BATCH BATCHSEP MultiBatch - Taylor

& Francis Features Databank Operations Constant Reflux Variable Reflux Optimal Reflux Optimal Reflux-Fixed Equation Models Shortcut Low Holdup Rigorous/SemirigorousReduced Order Rigorous Configurations Rectifier Yes Semi-batch Recycle waste cut Stripper Middle Vessel Column Options Design Feasibility Optimization Reactive Distillation 3 phase Distillation Uncertainty Analysis TABLE 4.3 Comparison of software packages Copyrighted

Material

- Taylor

& Francis