Operational Optimization of Batch Distillation Systems - Industrial

Article pubs.acs.org/IECR

Operational Optimization of Batch Distillation Systems Santosh Jain,† Jin-Kuk Kim,*,‡ and Robin Smith† †

Centre of Process Integration, The University of Manchester, Oxford Road, Manchester, M13 9PL, U.K. Department of Chemical Engineering, Hanyang University, 222 Wangshimni-ro, Seongdong-gu, Seoul, 133-791, Republic of Korea

‡

ABSTRACT: The optimal operation of batch distillation systems has been studied in this paper. A new approach termed the limiting gradient approach is introduced for effectively identifying feasible and cost-effective operating profiles for batch distillation systems. A new semirigorous model for batch distillation has been presented for considering holdup of the column, which is applicable for different batch distillation configurations. A novel optimization model incorporating the limiting gradient approach has been proposed, which significantly reduces computational burden to search for a feasible operation of a batch distillation column by considerably reducing effort for carrying out a large number of dynamic simulations required for optimization. Case studies have been presented to demonstrate the applicability of the approach for various batch distillation configurations and to illustrate how the proposed optimization framework systematically addresses different performance indices and objectives.

1. INTRODUCTION Batch distillation is becoming increasingly important as the result of increasing demand of the high-added-value fine and specialty chemicals which have a 15% share of worldwide chemical production with approximately 150 billion US dollars produced annually.1 Batch distillation is a dynamic separation process in which process conditions vary with time. This transient nature of batch distillation allows it to operate and be configured in different ways. Different operating reflux-reboil policies are possible for a batch distillation column by changing operating parameters with time. It is of great interest to rigorously investigate various design and operation options and to systematically identify the most appropriate design of distillation columns with optimal operating strategies. Therefore, it is necessary to develop a reliable and robust modeling and optimization framework for carrying out the operational optimization of batch distillation systems. Various researchers have attempted to optimize batch distillation operation. Most of this research was carried out for the batch rectifier.2−7 Farhat et al.8 used a combination of linear and exponential operating profiles with the aid of a nonlinear programming approach. Stochastic optimization has also been used; for example, Hanke and Li9 used a simulated annealing (SA) technique to obtain the optimal operation profile. The profile-based optimization applied for batch distillation had been limited to some specific types of profiles, for example, piecewise constant, constant, exponential, or linear profiles. A piecewise constant profile can approximate a generalized continuous profile by having a large number of steps, but it increases the size of the optimization problem. This work, therefore, aims to overcome computational difficulties associated with finding a feasible operation of a batch distillation, to screen effectively various general or novel operating profiles and to identify the most appropriate operation profile for the given batch distillation configuration and design constraints. © 2012 American Chemical Society

First, the details of semirigorous batch distillation model used in the research will be presented, together with the validation of the developed model with experimental data. This is followed by a novel automated design method for the operational optimization of batch distillation systems, while a limiting gradient approach is introduced which provides the guiding path for feasible operation. The application of the optimization framework will be discussed in detail for different design options, including rectifier, stripper, and multivessel column, through four case studies.

2. SEMIRIGOROUS MODELING OF BATCH DISTILLATION COLUMN The dynamics of batch distillation has been analyzed by rigorous stage-to-stage models to shortcut models.10,11 Such modeling starts with the simple Rayleigh model.10 Shortcut methods assume zero holdup and constant molar overflow in the column. With the development of computational facilities, more rigorous models were proposed. Shortcut methods assume zero holdup and constant molar overflow in the column. Diwekar and Madhavan12,13 as well as Sundaram and Evans1,2 presented shortcut models for the multicomponent batch rectifiers. More rigorous models for multicomponent batch rectifiers based on stage-by-stage mass balance calculations were used by Noda et al.,4 Farhat et al.,8 Mayur and Jackson,14 Domenech and Enjalbert,15 Galindez and Fredenslund,16 and Bonny.17 Rigorous models of batch rectifiers containing differential mass balance equations and differential or algebraic energy balance equations were used for simulation and optimization studies.3,6,7,9,18,19 Kreul et al.20 presented a rate-based model for multicomponent batch rectifiers. Received: Revised: Accepted: Published: 5749

August 17, 2011 March 21, 2012 March 27, 2012 April 10, 2012 dx.doi.org/10.1021/ie201844g | Ind. Eng. Chem. Res. 2012, 51, 5749−5761

Industrial & Engineering Chemistry Research

Article

According to Diwekar,10 compartment dynamics is given as

Along with the conventional batch rectifier, nonconventional or novel batch distillation columns have also been modeled. Lotter and Diwekar21 and Xu et al.22 presented a shortcut model for batch strippers. Lotter and Diwekar21 also presented a shortcut model for middle vessel columns. Barolo et al.23 and Davidyan et al.24 used rigorous models for the study of middle vessel batch distillation columns. Hasebe et al.,25,26 Wittgens et al.,27 and Skogestad et al.28 used a semirigorous model based on tray-by-tray mass balance equations for multivessel batch distillation columns. Furlonge et al.29 and Low and Sorensen30 used rigorous models for optimization of multivessel batch distillation columns. The selection of the model depends on the objective for which the model is used. For the detailed column design and control studies, rigorous models are required. A shortcut model or a semirigorous model is preferred, when less computational time for obtaining column dynamics is desired. However, the model assumptions must be checked for the system it is used for. The focus of this research is the operational optimization of batch distillation processes, which requires screening and comparing a large number of alternative design and operating options. Thus, the model required in this study should be accurate enough to capture detailed characteristics of batch distillation, and at the same time, simple enough to be computationally efficient. Although the shortcut model requires the least simulation time, it has assumptions of constant relative volatility and zero column holdup. Therefore, a new semirigorous model has been developed in this work, which is able to capture the effect of column holdup, and is not restricted to constant relative volatility mixtures. The concept of compartment analysis31 is employed for the current study and compartment analysis assumes the cumulative holdup of each tray at one sensitive tray. Thus, the dynamics of the tray segment is assigned to one tray only. This reduces the number of differential equations that would otherwise be created in the rigorous model. Figure 1 shows a compartment in a tray segment and holdup of trays 1 to N are accumulated at the sensitive tray s. Thus, the holdup at the sensitive tray s, Hs, can be written as

Hs

dxs = V (yN + 1 − y1) + L(x0 − xN ) dt

(2)

The constant molar overflow assumption is made to derive the above equation. The values of compositions leaving the compartment (y1 and xN) are found using the shortcut method.10 In this work, the compartment model is modified that is applicable for zeotropic mixtures. The following assumptions are made to develop the model: liquid holdups of the trays are constant; vapor holdup is neglected; condenser and reboiler are considered as separate compartments; constant molar overflow, thus the energy balance is not considered in the model; ideal stages, so that the vapor and liquid leaving a stage are in equilibrium; pressure drop is neglected in the column; holdup of the sensitive tray is kept constant. A dynamic mass balance equation is set up across the sensitive tray, and an algebraic mass balance equation is set up across other trays. The model can be extended for simulating batch rectifiers, batch strippers, and middle vessel columns as shown in Figure 2.

Mass balance equation for sensitive tray: Hs

dxi ,s dt

= Viseg(yi ,s + 1 − yi ,s ) + Liseg (xi ,s − 1 − xi ,s)

i = 1, 2, ..., NC − 1

(3)

Mass balance equation for nonsensitive tray:

N

Hs =

∑ Hi 1

(1)

Figure 1. Compartment model for batch distillation.

Figure 2. A general multivessel column. 5750

dx.doi.org/10.1021/ie201844g | Ind. Eng. Chem. Res. 2012, 51, 5749−5761


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Table 1. Problem Data for the Validation of a Semirigorous Model

0 = Viseg(yi , j + 1 − yi , j ) + Liseg (xi , j − 1 − xi , j) i = 1, 2, ..., NC − 1

(4)

number of stages feed (kmol) feed composition (mole fraction) tray holdup (kmol) condenser holdup (kmol) vapor load (kmol/h) column pressure (bar) property package total reflux period (h) finite reflux period (h)

Equations for both sensitive and nonsensitive trays:

yi , j = f(T , P , xi , j)

(5)

∑ xi ,j = 1

(6)

i

Equation for the holdup of the sensitive tray: N tray

∑

Hs =

Hi tray (7)

i tray = 1

20 (including a condenser and a reboiler) 2.93 0.407, 0.394, 0.199 3.093 × 10−3 0.03516 2.75 1.013 Soave−Redlich−Kwong 2.54 2.62

Table 2. Simulation Results of a Semirigorous Model

Mass balance equations for the condenser: dHiv = 1 = V1 − L1 − D1 dt dHiv = 1xi ,iv = 1 dt

= V1yi ,1 − L1xi ,iv = 1 − D1xi ,iv = 1

accumulated distillate (kmol) cyclohexane composition (mole fraction)

(8)

Nad and Spiegel33

Rigorous model11

semirigorous model

shortcut model (FUG method)

1.16

1.16

1.154

1.15

z0.895

0.891

0.898

0.985

Mass balance equations for the reboiler: dHiv = n vessel = −Vn vessel + Ln − Dn vessel dt dHiv = n vesselxi ,iv = n vessel dt

= −Vn vesselyi ,iv = n vessel + Lnxi , j = n

− Dn vessel xi ,iv = n vessel

(9)

Mass balance equations for a product vessel (receiver): dPiv = Div dt

dPivxi ,iv dt

= Div xi ,iv

(10)

The model results in a set of differential algebraic equations. A backward difference method is used to solve the model. FORTRAN routine DASSL32 has been used in this work. Nad and Spiegel33 carried out a batch distillation experiment in a packed column for a ternary mixture of cyclohexane, nheptane, and toluene. The packed column was equivalent to a staged column with 20 theoretical stages including a condenser and a reboiler. Initially, the column is operated at total reflux for 2.54 h. After that three products and two byproducts (off-cuts) are withdrawn. Mujtaba11 presented the rigorous simulation results for the first product period. The Soave−Redlich− Kwong (SRK) equation of state was used for the rigorous simulation. Table 1 provides the problem data for the simulation, and thermodynamic data were taken from Reid et al.34 The column was simulated using the semirigorous model developed in this study with three compartments and the shortcut model proposed by Diwekar and Madhavan12 for comparison. Comparison of different models with experimental values (Table 2 and Figure 3) suggests that the semirigorous model proposed in this paper matches well with the rigorous model results11 as well as experimental values. The reflux ratio is given in Figure 3, as the reflux ratio is one of key design parameters to control the dynamics of a column and meet the product specification. Simulation time required for the semirigorous

Figure 3. Comparison of rigorous model, semirigorous model, and shortcut model with experimental values.

model in this study is less than 1 s. Low and Sorensen30 presented the rigorous model for the separation of an alcohol mixture of methanol, ethanol, n-propanol, and n-butanol in a multivessel column. For this problem, the semirigorous model requires 251 algebraic and 25 differential equations, while 200 algebraic and 110 differential equations are needed for the rigorous model. Although the semirigorous model does not account for the energy balance, 85 fewer differential equations for the semirigorous model than the rigorous model are needed. Thus, the semirigorous model is computationally inexpensive compared with the rigorous one.

3. OPTIMIZATION FRAMEWORK In general the batch distillation model can be represented as f{x(t ), u(t ), p , t } = 0

(11)

where t refers to the time interval, x is the vector of state variables (e.g., holdup, concentration, etc.), u is the vector of control variables (e.g., reflux ratio), and p is the timeindependent parameters. State variables and control variables are time dependent. Certain constraints are imposed on state 5751



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Moreover, what is desired is not just to obtain feasible operation paths for the batch distillation, but to identify a most suitable profile, for example, the operating profile for achieving minimum utility consumption. Finding an optimal and feasible profile will be systematically incorporated in the optimization model, which ensures cost-effective, realistic, and practical design of batch distillation systems.

variables and control variables. Physical limits on state and control variables are imposed as xmin ≤ x ≤ xmax umin ≤ u ≤ umax

(12)

Also, conditions of minimum recovery and minimum purity on products are imposed. Such conditions are defined as

∮ x(t )dt ≥ a

4. LIMITING GRADIENT APPROACH FOR BATCH RECTIFIER The limiting gradient approach has been proposed in this work to deal with end point-constrained variables, for example, product purity and recovery. If the value of the end pointconstrained variable is xo at any time to and the minimum value required is xf at end time tf, then the limiting gradients at the particular time is calculated as shown in Figure 4.

(13)

For recovery,

x(tf ) ≥ b

(14)

where tf is the final batch distillation time. Various objectives can be used for batch distillation optimization, for example, maximization of profit, minimization of batch distillation time, minimization of energy requirement, maximizing product yield. A general optimization problem for a batch distillation column is defined as, min objective function = f obj(x) x

(15)

subject to,

h(x) = 0 g (x ) ≤ 0

(16)

The model equations required for the optimization, represented as eq 16, includes equality equations of eqs 3−10, and inequality equations of eqs 12−14. For a batch rectifier (stripper), the optimization variables are reflux ratio (reboil ratio), vapor flow rate (heat input to reboiler), batch distillation time, recoveries of the main products and byproducts, and purities of the byproducts. Reflux ratio (reboil ratio) and vapor flow rate are the control variables that can follow a profile during the batch distillation operation. Middle vessel and multivessel columns have additional degrees of freedom to optimize. For such columns, optimization variables are liquid and vapor flows in column sections, product withdrawal rates from column vessels, feed distribution to column vessels, batch distillation time, recoveries of main products and byproducts, purities of byproducts. Also, opportunities can also be explored by allowing vapor and liquid flow between two column sections to pass or to bypass intermediate vessels. It is difficult to satisfy the constraints of eqs 13 and 14, as these constraints can only be checked after the completion of batch distillation simulation for the whole operation period. The selection of an operating policy to provide a feasible processing is not a trivial task in batch distillation systems, and it is always computationally expensive to find a set of optimal decision variables, due to difficulties existing in nonlinear formulation of the model and providing initial settings of decision variables. Therefore, a methodology to be implemented in this study is required to overcome these computational difficulties, such that either a feasible operating policy should be provided, prior to a simulation, or a guiding path that can determine a feasible operation is to be provided. In this work we propose a methodology termed the limiting gradient approach that provides a path for feasible operation.

Figure 4. Limiting gradient.

A schematic flowsheet of a batch rectifier is shown in Figure 5, which separates the lightest product first at the top of the column. Usually, a batch rectifier is operated at maximum vapor flow rate for the utilization of a full capacity of column. By taking the mass balance around the product receiver, dP =D dt

dPx P, i dt 5752

= Dx D, i

(17)



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The concept behind the approach is to manipulate the operating strategy such that the target value is achieved throughout the whole operation, before the end point is reached. Also, the assumption is made that the potential (in terms of meeting the target) available at the point ti is greater than that at the next point ti+1, which facilitates that extra work (in terms of achieving the target), if required, can be done early rather than later. For the batch distillation, each column segment has a maximum allowable vapor and liquid flow rate. At the start of the operation when the feed is rich in the desired component, a lower reflux rate is required, compared with later operation. So, to avoid unnecessarily high reflux requirements in the later stage of operation, maximum or considerable use of the potential available in the beginning is desired. Just before the final batch distillation time, eq 17 can be written in a discretized format as Px P, i|n = Dx D, i|n − 1 × Δt |n + Px P, i|n − 1

(20)

Equation 20 can be written for other time steps as

Figure 5. Conventional batch rectifier column.

Px P, i|n − 1 = Dx D, i|n − 2 × Δt |n − 1 + Px P, i|n2 ⋮

where P is the holdup of the product receiver, xP,i, is the composition of component i in the product receiver, D is the distillate flow rate, and xD,i is the composition of component i in the distillate. Product purity and recovery can be determined by variables P (total product holdup) and PxP,i (holdup of component i in the product). If required minimum purity is puri and recovery is ri, then for a feasible operation to satisfy final product purity and recovery, P and PxP,i should be rFz i F, i ≤ Px P, i|n ≤ Fz F, i

rFz i F, i puri

≤ P|n ≤

Px P, i|n puri

Px P, i|1 = Dx D, i|0 × Δt |1 + Px P, i|0

The summation of equation set 21 and eq 20 gives Px P, i|n − Px P, i|0 = Dx D, i|n − 1 × Δt |n + Dx D, i|n − 2 × Δt |n − 1 + Dx D, i|n − 3 × Δt |n − 2 + ··· + Dx D, i|0 × Δt |1

P|n − P|0 = D|n − 1 × Δt |n + D|n − 2 × Δt |n − 1 + D|n − 3 × Δt |n − 2 + ··· + D|0 × Δt |1

F zF , i puri

(22)

Similarly, eq 22 can be discretized as:

(18)

≤

(21)

(23)

Final recovery of component i (PxP,i|n) should follow the constraint of eq 18. So, using eqs 18 and 22:

(19)

where |n denotes the final value of the variable. Index i is for the desired component in the product. The limiting gradient for P and PxP,i can be shown as Figure 6. For P (total product holdup), minimum final value should be riFzF,i/puri and for PxP,i (holdup of component i in the product) minimum final value should be riFzF,i. If a batch rectifier is operated at maximum vapor flow capacity, then D can be used to control the reflux ratio.

Px P, i|n − Px P, i|0 ≥ rFz i f, i − Px P, i|0

(Dx D, i|n − 1 × Δt |n ) + (Dx D, i|n − 2 × Δt |n − 1) + (Dx D, i|n − 3 × Δt |n − 2 ) + ··· + (Dx D, i|0 × Δt |1) ≥ rFz i f, i − Px P, i|0

(24)

Figure 6. Limiting gradient for P and PxP,i . 5753



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used: (1) At the start of the operation, product vessel composition is less than the purity requirement. Distillate flow rate, D, will be zero until the top product composition is greater than or within tolerance of purity requirement. (2) If distillate composition is higher than the purity requirement, use the equation that provides the higher value of distillate flow rate. (3) If the concentration of the desired component in the product vessel is higher than the purity required, and distillate composition is lower than both the purity requirement and product vessel composition, then use the equation accordingly to provide lower value of distillate flow rate. Limited capacity of the column should also be considered as, for a given column shell, either eqs 32 or 33 can be used to regulate the distillate flow rate within the physical limit for the operation of the column.

Similarly, the total amount of product (P|n) must satisfy the constraint of eq 19. Thus, using eqs 19 and 22: ⎛ rFz ⎞ ⎛ Px P, i|n ⎞ i f, i − P|0 ⎟ ≤ (P|n − P|0 ) ≤ ⎜ − P|0 ⎟ ⎜ ⎝ puri ⎠ ⎝ puri ⎠

⎛ rFz ⎞ i f, i − P|0 ⎟ ≤ (D|n − 1 × Δt |n ) + (D|n − 2 × Δt |n − 1) ⎜ ⎝ puri ⎠ + (D|n − 3 × Δt |n − 2 ) + ··· + (D|0 × Δt |1) ⎛ Px P, i|n ⎞ ≤⎜ − P|0 ⎟ ⎝ puri ⎠

(25)

The operating characteristics of batch distillation can be interpreted as D|0 ≥ D|1 ≥ D|2 ≥ ··· ≥ D|n

(26)

x D, i|0 ≥ x D, i|1 ≥ x D, i|2 ≥ ··· ≥ x D, i|n

(27)

5. LIMITING GRADIENT APPROACH FOR MIDDLE VESSEL AND MULTIVESSEL COLUMNS Middle vessel and multivessel columns provide two or more products simultaneously. For all the products, purity and recovery constraints must meet. Therefore, it is more difficult to decide feasible operating policy for such columns. Figure 7 shows a typical multivessel column. Vapor from a column section can be bypassed or passed through intermediate

Using eqs 26 and 27, eq 24 can be written as Dx D, i|0 × (Δt |1 + Δt |2 + Δt |3 + ··· + Δt |n ) ≥ rFz i f, i − Px P, i|0

(28)

The second term in eq 28 can be written as (Δt |1 + Δt |2 + Δt |3 + ··· + Δt |n ) = tf − t0

(29)

where tf is final batch distillation time and t0 is the initial batch distillation time. Using eq 29 in eq 28 gives Dx D, i|0 ≥

rFz i f, i − Px P, i|0 (tf − t0)

(30)

Similarly, using eqs 26 and 29 in eq 25: ⎛ rFz ⎞ ⎛ Px P, i|n ⎞ i f, i − P|0 ⎟ ≤ D|0 (tf − t0) ≤ ⎜ − P|0 ⎟ ⎜ ⎝ puri ⎠ ⎝ puri ⎠

(31)

To satisfy eq 30, the limiting gradient is multiplied by a factor, termed the multiplication factor, mf(t), of which the value is greater than 1 and which is a function of the independent variable time. If the value of the multiplication factor for any time t between t0 and tf is mf|t, then the distillate flow rate can be calculated as: ⎞ ⎛ rFz i f, i − Px P, i|t ⎟⎟ D|t = mf|t ⎜⎜ ⎝ x D, i|t (tf − t ) ⎠

(32)

Similarly, using a multiplication factor, distillate flow rate for any time t can be calculated as ⎞ ⎛ rFz i f, i ⎜ puri − P|t ⎟ D|t = mf|t ⎜ ⎜ (tf − t ) ⎟⎟ ⎠ ⎝

(33)

At the start of the operation (t = 0), the composition of product receiver is assumed to be at feed composition and the holdup of the product receiver is assumed negligible. For subsequent time points, these values are predicted by solving the batch distillation model and updated during solving the model. The following guidelines, based on the column dynamics, are proposed to decide which equation (eqs 32 or 33) should be

Figure 7. Multivessel column without vapor passing through intermediate vessels. 5754



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vessels. In a multivessel column controlled variables are liquid and vapor flow rate in column sections, and product withdrawal rate from vessels. The limiting gradient approach explained in the previous section cannot be applied directly to multivessel columns, as the limiting gradient approach is for open loop systems. Determination of distillate flow rate in a batch rectifier is open loop systems because there is no vapor−liquid flow from a product vessel to the column. But, in a multivessel column, different column sections are interconnected and form a closed loop system. Thus, the limiting gradient approach discussed in the previous section cannot be directly applied for multivessel or middle vessel batch distillation columns. The approach should be modified to apply to middle vessel and multivessel columns. First, the mass balance equations for the column vessels and product vessels of a multivessel column are developed. For a product vessel of a multivessel column, the total mass balance and component balance equations are dPiv = Div dt

dPivx Pi ,iv dt

Product withdrawal rates from the vessels of a multivessel column are determined using the methodology discussed for a batch rectifier. Each product vessel of a multivessel column is used to collect a particular product, and each product has a key component of a desired purity, and the key component is used in the calculation of the limiting gradient. Segments of a multivessel column interact with each other. Thus, changes of liquid−vapor flow in a segment affect the performance of segments above and below it. Liquid from a segment is fed to the segment below, and vapor from a segment is fed to the above segment. So, to determine the liquid−vapor flow rate in a multivessel column, the equations need to be solved simultaneously. The limiting gradient is calculated for each vessel for corresponding product recovery and purity. The limiting gradient for the amount of total product for vessel iv (iv = 1 for condenser; iv = nvessel for reboiler; nvessel is the total number of vessels in the column) at time t is calculated as: rFz i f, i

(34)

= Div x Di ,iv

grad1 =

(35)

d(Pn vesselx Pnvessel, i + Hn vesselxn vessel, i) dt

(36)

= Ln vessel − 1

xin , i − Vn vessel − 1yn vessel, i

(37)

For the condenser and its product vessel, the total mass balance and component mass balance are d(H1 + P1) = −L1 + V1 dt d(Px 1 P1, i + H1x1, i) dt

= −L1x1, i + V1y1, i

grad 2 =

d(Pivx Piv , i + Hivx iv, i) dt

= L iv − 1xin , i − L iv x iv, i

(39)

(40)

(41)

Total mass balance and component mass balance around an intermediate vessel (if vapor is passing through) and its product vessel are d(Hiv + Piv) = L iv − 1 − L iv + Viv − Viv − 1 dt

(42)

d(Pivx Piv , i + Hivx iv, i) dt = L iv − 1xin , i − L iv x iv, i + Vivyiv, i − Viv − 1yiv, i

(t f − t )

(44)

rFz i f, i − (Pivx p, i + Hivx iv, i)|t (t f − t )

(45)

The term riFzf,i is the maximum recovery of component i in the product and the term (Pivxp,i + Hivxiv,i)|t determines the recovery of component i in the product at time t. Equations 44 and 45 determine the limiting gradient for product recovery and purity at the vessel iv. 5.1. Vapor Bypassing the Intermediate Vessels. In a multivessel column, if vapor is not passing through the intermediate vessels then vapor flow is assumed constant in the column (constant molar flow is assumed). Thus, for an n segment column there are n + 1 control variables (n liquid flow rates in n column segments and one vapor flow rate) to be determined. To determine n + 1 control variables, then n + 1 equations are to be solved. In an n segment column, n + 1 equations are generated for n + 1 column vessels (including the condenser and the reboiler). First, grad1 and grad2 are calculated at the column vessels and are used in eqs 36−43, then the following criteria are applied for using these equations to generate the n + 1 equations: If both signs of grad1 and grad2 are the same, add the equations of the vessel to form one equation. If signs of grad1 and grad2 are different, use the equation of the gradient with a negative sign. Thus, for each vessel, one equation is generated, and the corresponding limiting gradient (or summation of limiting gradients) is multiplied by a multiplication factor. The solution of the n + 1 linear equations provides liquid and vapor flow rate in the column at time t.

(38)

Total mass balance and component mass balance around an intermediate vessel (if vapor is bypassing) and its product vessel are d(Hiv + Piv) = L iv − 1 − L iv dt

− (Piv + Hiv)|t

where ri is the desired recovery of desired component i in the product. The first term in the numerator, riFzf,i/puri, determines the maximum holdup of the product and the second term in the numerator (Piv + Hiv)|t is the product holdup at any time t. After a distillation operation in a multivessel column, holdup of the column vessels is mixed with the corresponding product vessels. The term (Piv + Hiv) is used to determine total product holdup at any time t. Similarly, the limiting gradient for the amount of component i in the product is calculated as

Total mass balance and component mass balance around the reboiler and its corresponding product vessel are d(Hn vessel + Pn vessel) = Ln vessel − 1 − Vn vessel − 1 dt

puri

(43) 5755



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5.2. Vapor Passing through the Intermediate Vessels. If vapor is passing through an intermediate vessel, this results in different vapor flows in the segments above and below the intermediate vessel. If, for an n segment column, vapor is passing through m intermediate vessels (m ≤ n − 1, for n − 1 total intermediate vessels), n liquid flow rates and m + 1 vapor flow rates are to be determined. If vapor is not passing through an intermediate vessel iv, then the segments above and below it have the same vapor flow rate. Thus, n + m + 1 equations are to be solved to determine n + m + 1 flow rates. For the condenser, two equations can be generated. The first equation is based on the limiting gradient of the total holdup of the product (grad1) and the amount of desired component in the product (grad2). When either grad1 or grad2 is negative, only one equation can be generated from grad1 and grad2. To generate the second equation at the condenser, a component balance at condenser is taken as H1

dxi ,1 dt

= V1(yi ,2 − xi ,1)

When vapor bypasses the intermediate vessel just below the condenser, vapor flow is constant in the two segments below the condenser, assuming the constant molar flow (i.e., V1 = V2). As the intermediate vessel below the condenser is not fully interconnected with the vessel below it, the product purity and recovery required at this vessel cannot be controlled by manipulating the purity and recovery at the other vessels. Thus, one equation in terms of grad1 and grad2 (using the criteria given for a rectifier) is generated at the intermediate vessel below the condenser. Also, the purity equation for the vessel (the second vessel below the condenser) below to the intermediate vessel (just below the condenser) is not considered. Another case is when vapor bypasses the intermediate vessel other than the intermediate vessel just below the condenser, and the second equation for the intermediate vessel is replaced by Viv = Viv‑1.

6. APPLICATION OF A PROFILE GENERATION TOOL FOR THE OPTIMIZATION Using the limiting gradient approach, variables for identifying optimal operation of a batch rectifier (or stripper) are overall batch distillation time, product recovery, byproduct recovery, and purity, and multiplication factor for the limiting gradient. The multiplication factor, mf(t), is a time-dependent control variable that follows a profile during the batch distillation operation. The shape of the profile, in this work, will be determined using the profile generation tool developed by Choong and Smith,35 which is described in Appendix A. This profile generation tool has been successfully applied for the optimization of batch crystallization35 and the optimization of catalytic reactors.36 The optimization of a batch distillation operation is a dynamic optimization problem, which is formulated as a nonlinear programming (NLP) problem. The NAG FORTRAN 77 library routine E04UCF, based on sequential quadratic programming (SQP), was used to solve the NLP. Sometimes, the multiplication factors provide large oscillations in vapor−liquid flow in the column. To reduce the oscillations, a constraint is imposed such that only maximum change per minute in flow rates (typically 10%) is allowed. However, the value of this constraint is problem-dependent and case-specific.

(46)

Here component i is the desired component in the product at the condenser. By applying the limiting gradient for the desired component purity: H1(pur1 − xi ,1) tf − t

= V1(yi ,2 − xi ,1)

(47)

For the reboiler, the first equation is based on grad1 and grad2 for the reboiler product, and the second equation is for the desired component purity in the product for the reboiler. Hn vessel

(purn vessel − xkey, n vessel) (t f − t )

= Ln vessel − 1(xkey, n vessel − 1 − xkey, n vessel) − Vn vessel − 1 (ykey, n vessel − xkey, n vessel)

(48)

If the column contains only one intermediate vessel (middle vessel column), then these four equations are sufficient. If a multivessel column contains m (m > 1) intermediate vessels, then 2(m − 1) equations are required to determine liquid and vapor flow in the column sections. If two equations are generated at each intermediate vessel (as with the condenser and reboiler), a total of 2m equations are formed and the system becomes overspecified. Therefore, two redundant equations are to be avoided. Usually, the intermediate vessel just below the condenser is not used to generate the equations. For this vessel, product purity and recovery are controlled by controlling the purity and recovery at other vessels, because the system is a closed loop system. For an intermediate vessel, the following equation can be used for calculating the desired component purity in the product from the intermediate vessel: Hiv

7. CASE STUDIES 7.1. Case Study 1. A general optimization problem in batch distillation operation is to maximize the distillate throughput for a given batch time. A simple toluene/cyclohexane system has been optimized by Logsdon and Biegler6 using both a shortcut method and a rigorous method. Problem data for the case study are given in Table 3.

(puriv − xkey,iv)

Table 3. Problem Data6 for Case Study 1

(t f − t )

number of trays tray holdup (kmol) condenser holdup (kmol) vapor flow rate (kmol/h) reboiler charge (kmol) feed composition (cyclohexane/toluene) time (h)

= L iv − 1(xkey, in − xkey,iv) − Viv − 1(ykey,iv − xkey,iv) + Viv(ykey, in − xkey,iv)

(49)

A multivessel column may have mixed-vapor-pass arrangements; that is, vapor is allowed to pass through some intermediate vessels and to bypass other intermediate vessels. 5756

10 1.0 1.0 120 100 0.55/0.45 1



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Table 5. Column Properties for Batch Rectifier4 for Case Study 2

Thermodynamic behavior of vapor and liquid phases are assumed ideal, and K-values for vapor−liquid relations are modeled by ideal vapor pressure relations using the Antoine Equation and data taken from Reid et al.34 Tray segments of the batch distillation column are divided into three compartments of size 4, 3, and 3, and the sensitive trays in the segment are 2, 6, and 9 (from the top). The initial multiplication factor profile is fixed to be a constant value of 1.5. Figure 8 compares reflux ratio profiles obtained by Logsdon and Biegler6 and those by using the profile-based approach in

total number of stages feed (kmol) tray holdup (kmol) reflux drum holdup (kmol) reboiler holdup (kmol) vapor flow rate (kmol/h) reflux flow rate (kmol/h)

presented here leads to higher performance indexes compared with the results given by Noda et al.4 Case B requires higher product purities and is the most difficult separation and the performance index is the lowest in this case. Case A is the easiest separation and has the highest performance index. 7.3. Case Study 3. The optimal operation of a batch rectifier is studied for the separation of cyclohexane, n-heptane, and toluene mixture. A profit-based objective function is used for optimization. Cost data and other problem data are given in Tables 8 and 9. Details of the profit function are given in Appendix B. The optimized results presented by Mujtaba and Macchietto3 are considered here as the base case. Separation of cyclohexane, n-heptane, and toluene is a difficult process as relative volatilities are low, and two byproducts (or waste products or off-cut) are required between the separations of cyclohexane and n-hetane, and n-heptane and toluene. Operation of a batch rectifier is optimized with the aid of the limiting gradient approach, and the profile generation algorithm. Four separation tasks are to be performed to separate three products and two byproducts. Different multiplication factors for limiting gradients are used for each separation task. Table 10 provides the initial parameters for the profiles of the multiplication factors, while Table 11 presents the optimized product compositions and amount, and batch distillation time. It can be noted that a higher recovery of toluene product is obtained by reducing the byproduct between the separation of n-heptane and toluene. From Table 12, the optimization provides a higher profit, due to lower batch distillation time and lower byproduct production, which enables a greater number of batches to be processed. Table 13 provides the optimal profile parameters for the multiplication factors, which shows that a different multiplication factor profile is used for each separation task. Figure 10 compares the reflux ratio profile of the optimal case against the reflux ratio profiles of Mujtaba and Macchietto3 (shown as base case in the figure) and the initial case. 7.4. Case Study 4. A feed of methanol, ethanol, n-propanol, and n-butanol is to be separated into four relatively pure component products of purity 95 mol % for methanol and ethanol and 99% purity for n-propanol and n-butanol. Ideal gas and ideal solutions are assumed, and vapor pressure data are taken from Reid et al.34 The multivessel column illustrated in Figure 7 is used. Vapor flow is not allowed to pass through the intermediate vessels of the column, and no continuous product withdrawal is allowed at the reboiler, at the intermediate vessels, at the condenser, and the reflux drum. Tables 14 and 15 provide the problem data. The adjacent volatility data suggests that the separation of methanol and ethanol is the most difficult separation and the separation of n-propanol and n-butanol is the least difficult. Thus, the largest segment of 20 trays is used

Figure 8. Reflux ratio profile for Case Study 1.

this work. The profile obtained in this work shows that, for most of the batch operation, a relatively lower reflux ratio is required in comparison with values obtained by Logsdon and Biegler.6 With the optimization framework presented in this paper, a better objective value with 39.61 mol of distillate product is obtained, compared to 37.03 of base case,36 when 99.8% of purity specification is desired. The profile-based approach optimizes batch distillation operations more efficiently in comparison with previous approaches. Table 4 provides the optimal values of the profile for the multiplication factor. Table 4. Optimal Profile Parameters for Multiplication Factor of Case Study 1 inlet value intermediate value outlet value intermediate location A1 A2

initial value

optimization bounds

optimal value

1.5 1.5 1.5 0.0 1.0 1.0

1.0−10.0 1.0−10.0 1.0−10.0 0.0−1.0 1.0−10.0 1.0−10.0

2.56 2.86 1.33 0.34 1.0 2.4

7.2. Case Study 2. The optimization of multicomponent batch distillation is carried out. Data taken from Noda et al.4 are presented in Table 5. Table 6 provides feed and product specification and relative volatilities for the components for three different cases. Noda et al.4 optimized the column to maximize the performance index defined as: perfomance index (P.I.) =

total products (mol) batch time (h)

10 103.0 0.3 1.0 1.0−99.0 50.0 20.0−50.0

(50)

Table 7 provides optimized results using the optimization addressed in this paper and compares them with the results presented by Noda et al.4 Figure 9 shows the reflux rate profile for Case A of the rectifier. For all the cases, the approach 5757



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Table 6. Feed and Product Composition and Relative Volatility of Mixture for Different Cases4 in Case Study 2 feed composition (light/intermediate/heavy)

product composition (light/intermediate/heavy)

relative volatility (light/intermediate/heavy)

0.3/0.4/0.3 0.3/0.4/0.3 0.3/0.4/0.3

0.95/0.95/0.95 0.99/0.99/0.99 0.95/0.95/0.95

9.0/3.0/1.0 9.0/3.0/1.0 6.25/2.5/1.0

Case A Case B Case C

Table 11. Optimization Results of Case Study 3 (I)a

Table 7. Optimization Results of Case Study 2 performance index cases

optimal results

Noda et al.

Case A Case B Case C

22.3 15.8 17.6

22.2 15.2 17.1

composition (cyclohexane/nheptane/toluene)

4

Product 1 Byproduct 1 Product 2 Byproduct 2 Product 3 a

0.893/0.105/0.002 (0.893/0.100/0.005) 0.346/0.598/0.056 (0.373/0.601/0.026) 0.013/0.863/0.124 (0.012/0.863/0.125) 0.000/0.360/0.640 (0.000/0.362/0.638) 0.000/0.009/0.991 (0.000/0.010/0.990)

amount (kmol)

time (h)

1.11 (1.12)

2.14 (2.56)

0.53 (0.48)

1.67 (2.27)

0.78 (0.78)

1.86 (2.22)

0.12 (0.22)

0.87 (0.63)

0.39 (0.33)

Note: Results of Mujtaba and Macchietto3 are shown in brackets.

Table 12. Optimization Results of Case Study 3 (II) Mujtaba and Macchietto3

25780.0 7.64 1223.2

15168.0 6.54 1074.1

profit ($/yr) total batch distillation time (h) no. of batches

Figure 9. Reflux rate profile for Case 1 of rectifying column of Case Study 2.

Table 13. Optimization Results of Case Study 3 (III)

Table 8. Problem Data3 for Case Study 3 feed (kmol) vapor flow rate (kmol/h) condenser holdup (kmol) total trays holdup (kmol) number of trays total plant operation time (h/yr) number of byproducts allowed cost of byproducts ($/kmol)

optimal value

2.93 2.75 0.035 0.056 17 8000 2 1.0

profile parameters

column 1

column 2

column 3

column 4

inlet value intermediate value outlet value intermediate location A1 A2

2.56 2.86 1.33 0.34 1.0 2.4

4.24 2.56 2.23 0.27 1.0 2.9

1.66 1.66 1.90 0.13 1.0 2.5

2.24 2.30 1.00 0.33 1.0 2.0

Table 9. Feed and Product Composition and Cost Data for Case Study 3 cyclohexane n-heptane toluene cost ($/kmol)

feed

product 1

0.407 0.394 0.199 2.0

0.895

product 2

product 3

0.863 30.0

26.0

0.99 24.0

to separate methanol and ethanol, and the smallest segment of 10 trays is used to separate n-propanol and n-butanol. Figure 10. Comparison of reflux ratio profile for optimal case and base case for Case Study 3.

Table 10. Initial Parameters of Profiles for Multiplication Factors for Case Study 3 profile parameters

column 1

column 2

column 3

column 4

inlet value intermediate value outlet value intermediate location A1 A2

1.5 1.5 1.0 0.2 1.0 2.0

1.5 1.5 1.0 0.2 1.0 2.0

1.5 1.5 1.0 0.2 1.0 2.0

1.5 1.5 1.00 0.2 1.0 2.0

The objective of the case study is to maximize the annual product revenue. The initial operation of the base case column is determined by using the profile parameters for the multiplication factors (mfi) as given in Table 16. Batch distillation time is 4.0 h for the base case column and initial feed distribution is given in Table 17. Optimization bounds for feed distribution are also given in Table 17. 5758



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Table 14. Feed and Product Data for Case Study 4 methanol ethanol n-propanol n-butanol cost ($/kmol)

feed

product 1

0.45 0.35 0.1 0.1 2.0

0.95

product 2

product 3

product 4

0.95 0.99 7.7

17.2

52.8

0.99 81.4

relative volatility of feed 8.02 4.86 2.31 1.0

Table 15. Problem Data for Case Study 4 feed (kmol) no. of column segments number of trays in segments tray holdup (kmol) batch distillation time (h) cleaning time (h) total plant operation time (h/yr) maximum vapor flow rate (kmol/h)

100 3 20/15/10 0.01 1.0−10.0 0.5 8000 250.0

Table 16. Optimal Parameters of Profiles for Multiplication Factors for Case Study 4a vessel 1 inlet value intermediate value outlet value intermediate location A1 A2 a

1.24 1.31 1.66 0.07

(1.5) (1.5) (1.0) (0.2)

2.1 (1.0) 7.9 (5.0)

vessel 2

vessel 3

vessel 4

6.71 (1.5) 1.17 (1.5) 1.21 (1.0) 0.0 (0.2)

3.11 (1.5) 1.7 (1.5) 2.1 (1.0) 0.25 (0.2)

1.24 (1.5) 1.63 (1.5) 1.0 (1.0) 0.2 (0.2)

1.42 (1.0) 2.36 (5.0)

1.0 (1.0) 8.18 (5.0)

1.0 (1.0) 7.3 (5.0)

Figure 11. Liquid and vapor flows for optimal operation of Case Study 4.

8. CONCLUSIONS Optimal operation of batch distillation columns has been studied. Feasibility of a batch distillation operation is a challenging problem to be dealt with during the optimization. A new semirigorous model has been developed on the basis of a compartmental modeling approach, which is capable of predicting the dynamics of the column without sacrificing too much accuracy and with requiring less computational time. In this work, the limiting gradient approach has been presented that provides the guiding path for feasible batch distillation operation. Product purities determine the feasibility of the operation, and product recoveries have also been considered to determine the feasibility. The limiting gradient approach tries to achieve the target values for the specified purities and recoveries before the final batch distillation time using the available or spare capacity of the column. A multiplication factor is used to optimize the batch distillation operation and the factor follows a profile during the batch distillation operation. A profile generation tool35 was used to generate a wide range of profiles in this study. Six parameters are required to determine the shape of the profile. These six parameters are optimized using an NLP solver, together with other design variables in the design and optimization of batch distillation systems. Case studies have been performed to demonstrate the applicability of the approach. Optimization results have been compared with the published results. The effectiveness of the proposed modeling and optimization approach for identifying optimal and feasible operating profiles of batch operations is clearly demonstrated, as the optimal results provides considerable improvements in objective function, compared with published results.

Note: Initial parameters are shown in brackets.

Table 17. Feed Distribution for Case Study 4 feed fraction optimization results initial values optimization bounds

vessel 1 (condenser)

vessel 2

vessel 3

vessel 4 (reboiler)

0.7

0.06

0.03

0.21

0.27 0.01−0.88

0.27 0.01− 0.88

0.26 0.01− 0.88

0.2 0.1−0.97

Initial and optimal profile parameters for the multiplication factors for the limiting gradients are given in Table 16. Feed distribution is also an optimization variable, and the optimized values are given in Table 17. From Table 18, batch distillation Table 18. Optimal Product Composition and Amount for Case Study 4 methanol ethanol n-propanol n-butanol total amount (kmol)

product 1

product 2

product 3

product 4

0.95 0.05 0.0 0.0 47.2

0.01 0.95 0.04 0.0 34.0

0.0 0.0 0.99 0.01 8.79

0.0 0.0 0.006 0.994 10.01

time is reduced to 1.9 h compared with 4.0 h of the base case. Annual revenue is 7.3 MM$/yr for the optimized case, compared with 2.8 MM$/yr for the base case. Thus, using the limiting gradient approach and the profile generation algorithm, 160% increase in annual revenue than the base case is obtained. Figure 11 provides the optimal profiles for the liquid and vapor flows in the column.

■

APPENDIX A: A PROFILE GENERATION TOOL35 Profiles can be divided into two fundamental profiles, exponential and asymptotic profiles. Type I profile (exponential curve) 5759


Industrial & Engineering Chemistry Research ⎛ t ⎞ A1 z(t ) = z1 − (z1 − z 2)⎜ ⎟ ⎝ t total ⎠

■

(A1)

*Tel.: +82 2 2220 2331. Email: [email protected]. Notes

The authors declare no competing financial interest.

A2

■

ACKNOWLEDGMENTS This research was supported by the International Research & Development Program of the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) of Korea (Grant number: 20110031290), and also supported by Process Integration Research Consortium (PIRC) at the University of Manchester.

(A2)

Both profiles depend on three parameters, the initial value of the profile z1, the final value of the profile z2, and the curvature of the profile (A1 for exponential curve, and A2 for asymptotic curve). Other profiles can be generated by the combination of the above two profiles. Type I + II profile ⎛ t ⎞ A1 z(t ) = z1 − (z1 − z 2)⎜ ⎟ ⎝ t inter ⎠

■

0 < t ≤ t inter

A2 ⎛ t −t ⎞ z(t ) = z 3 − (z 3 − z 2)⎜ total ⎟ ⎝ t total − t inter ⎠

t inter < t ≤ t total (A3)

Type II + I profile A2 ⎛t − t⎞ z(t ) = z 2 − (z 2 − z1)⎜ inter ⎟ ⎝ t inter ⎠

0 < t ≤ t inter

⎛ t − t inter ⎞ A1 z(t ) = z 2 − (z 2 − z 3)⎜ ⎟ ⎝ t total − t inter ⎠

t inter < t ≤ t total (A4)

Combinations of Type I + Type I, and Type II + Type II were not considered because such types of profiles create significant discontinuities where the curves meet and cannot be readily implemented in practice. According to the profile generation algorithm, only six parameters are needed to generate a profile; z1, z2, z3, tinter, A1, and A 2 . Various profiles can be generated with the combinations of Type I and Type II, and different values of profile parameters.

■

APPENDIX B Profit function (f p) for Case Study 3 is given as3 fp = (revb − OC b)NB − ACC

(B1)

revb = PC i i − FCf

(B2)

OC b =

NB =

K3V tb + Csetup A

(B3)

(B4)

t tot = tb + ts

(B5)

ACC = K1V 0.5N 0.8 + K 2V 0.65

(B6)

NOMENCLATURE A = constant used in cost correlation A1 = curvature parameters for profile 1 A2 = curvature parameters for profile 2 ACC = annualized capital cost ($/batch) Csetup = cost of setup ($/batch) Ci = cost of ith product ($/kmol) Cf = cost of feed ($/kmol) D = outlet flow rate from a vessel (kmol/h) F = feed flow rate Hyr = total number of operating hours per year H = holdup of tray (kmol) Ki = constant used in cost correlation L = liquid flow rate (kmol/h) n = number of trays NB = number of batches/yr nc = total number of components nvessel = total number of vessels in a column OCb = operating cost ($/batch) P = product holdup r = product recovery revb = net revenue ($/batch) T = temperature t = time tb = batch distillation time tf = total batch distillation time ts = setup time V = vapor flow rate x = liquid composition y = vapor composition Z = dependent value of a profile zf = feed composition

Subscripts

H yr t tot

AUTHOR INFORMATION

Corresponding Author

Type II profile (asymptotic curve) ⎛t − t⎞ z(t ) = z 2 − (z 2 − z1)⎜ total ⎟ ⎝ t total ⎠

Article

■

where K1 = 1500, K2 = 9500, K3 = 180, A = 8000, Csetup = 0.0, and ts = 0.0.

1 = inlet index of profile 2 = intermediate index of profile 3 = final index of profile i = component index inter = intermediate index of independent variable of profile iseg = segment index j = tray index p = product s = sensitive tray index total = final index of independent variable of profile

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