Note: All costs are in $100 000.

Note: All costs are in $100 000.

7.3.1. Representation of Design, Operations and Separations Duties 7.3.1.1. Design and Operation Decision Variables

It is aimed to optimally design a batch column capable of processing M number of feed mixtures, each mixture constituted of NCm number of components {Multiple Separation Duties, Figure 7.3). The most important variables characterising a batch column design are the still capacity (B0), the number of theoretical stages (N) and the vapour boilup rate (V). The latter can be used to obtain both the column diameter and reboiler and condenser duties (as described in section 7.2), hence it represents a convenient design variable. Here storage issues are ignored and therefore the initial set of design decision variables considered is Dd={N, B0, V}.

Mixture

L^LJTl

Main-Cut Off-Cut Main-Cut or

Main-cut Main-cut Off-cut or

Main-cut Main-cut etc...

Main-Cut Off-Cut Main-Cut or

Main-cut Main-cut Off-cut or

Main-cut Main-cut etc...

Mixture

Mixture 1

Mixture 2

Mixture 1

Mixture 2

Fraction of Total Plant Time Allocated to each Mixture

Operational Alternatives

"M

Number of Batches

Figure 7.3. Batch distillation Column with Multiple Separation Duty and Multiple Operational Alternatives. [Mujtaba and Macchietto, 1996]c

The procedure for processing a given batch charge of mixture m (operation m), can be viewed as a sequence of NTm distillation tasks to produce one or more maincuts, possibly some intermediate off-cuts and a final bottom residue or product (Figure 7.1). For a ternary mixture this can be represented in the form of a STN shown in Figure 7.2. Each state s is characterised by a name (e.g. Dl), an amount Ss (e.g. SDi) and a composition vector xs (e.g. xD1). The molar fraction of an individual component j in state 5 is denoted by x/s. The sets of external feed states, main-cuts and off-cuts states in operation m are defined as EFm, MPm, and OPm, respectively. For example, Figure 7.2 shows operation 1 for a ternary mixture distillation with NT2=4 tasks, EF,={F0], MP,={D1, D2, Bf] and OP,={Rl, R2}. Several feed states could occur, for example in the preparation of a mixed charge or in the subsequent addition of material to the still. These mixing tasks can be considered but are ignored in the following. Overall, a batch charge B0m of feed mixture m produces a quantity ©s, Vse [MPm, OPm}, of each product. A material in state s is assigned a unitary value Cs. Off-cut materials may have a positive residual value, zero value or a negative value accounting for disposal cost.

A name and a set of input and output states (S/; and SOh respectively) characterise each distillation task i. Each task is modelled by an associated set of DAEs of the form f(t,xi(t),xi(t),ui(t),v) = 0, where utft) is a vector of time varying control variables (e.g. reflux ratio, actual vapour boilup rate). With each task an initialisation procedure and an integration procedure are associated. The initialisation procedure permits the calculation of a consistent initial set of all model variables ( *,■(?) and i, (i) ), given the amounts and compositions of all input states fed to the task. From these initial conditions and given a profile of all control variables, u^, the integration procedure permits calculation of all trajectories up to some termination condition (e.g. final time), and hence all final column states, product amounts, utility requirements, etc. and output states for the task. The time for processing each batch (batch time), tbm, is assumed to consist of the processing time tpm (the sum of all task times) and a set up time tsu. A fixed set up cost Csu per batch is assumed.

Mujtaba and Macchietto (1996) assumed that the structure of each operation (a specific sequence of tasks and states and choice of main and off-cut states) is given a priori but the operating control variables, (in particular reflux ratio profile) and termination conditions for each task are to be optimised. It is usually best to operate at the maximum available energy input rate, unless hydraulic problems are encountered. For this reason, the vapour load is assumed to be always at its design value V during operation (Diwekar et al., 1989, 1991) and is dropped from the optimisation variables vector.

Chapter 6 shows that-

• given the feed state (amount and composition) and V, and

• specification of a set of two additional operations decision variables for each task i, denoted by the vector d°i (typically the mole fraction of a key component in a distillate or bottom product and a key component recovery, two product purities for the same component, one purity and a product amount, etc.)

the optimisation problem permits calculation of the amount of distillate, intermediate or bottom product and their composition, recoveries, energy requirements, etc. for the task, as the solution of a well defined dynamic optimisation problem (Mujtaba and Macchietto, 1993). This is discussed in detail in Chapter 6.

Thus with given B0m and V, specification of the set of operations decision variables D°m = {cf\, i = 1, .., NTm\ (a total of 2* NTm decision variables) and control variables Um = {ut, i = 1,.., NTm } for all distillation tasks in operation m, it will be possible to calculate the overall performance measures for the batch (total distillation time, overall separation, products and intermediates amounts, recoveries, energy, etc.)- The same decision variables may be optimised to achieve some overall objective for the operation, (e.g. overall profit) subject to overall constraints (e.g. overall time, energy, etc.).

It is necessary to specify the relative importance of each duty in order to fully characterise the design requirements. The total time horizon for completing all separation tasks is denoted by H and the available plant operation time over a year by Hyr. For each mixture m, a total amount Fm is processed in NBmJI number of batches (all equal) of size Bom according to operation m. This requires a total operation time Tm = <Pm*H (<Pm is the fraction of production time allocated to mixture m, Figure 7.3). Since it is always optimal to use the column at its maximum capacity (full charges), the amount processed in each batch (feed charge or batch size) for all batches are equal to the still capacity. A capacity utilisation factor may be easily introduced to account for any density differences between mixtures. Hence the mixture index can be dropped and B0 is used instead of B0m to denote both the pot capacity and the batch size for each mixture. The following constraints given by Equations 7.14-7.18 therefore hold (for m - 1,.., M).

The multiple separation duty specifications can be made in several ways. Two of which are:

H and Fm given - Larger B0 will always be favoured (for finite set up times), because the idle time for the plant is reduced and because the capital costs typically increase with capacity by an exponent less than 1 (economy of scale). The optimum value of B0 will depend on constraints 7.14-7.18.

H and &m given - In this case more material per batch can always be processed just by increasing B0 indefinitely, up to the rather useless bound of constraint 7.17. Therefore a specification of the pot size is required a priori.

Mujtaba and Macchietto (1996) considered a general profit function P (the annual profit) which includes net revenues (products, raw materials), operating costs and annualised capital costs. The functions given by Logsdon et al. (1990) are used here for the latter two terms, so as to enable a direct comparison. Other cost models could easily be utilised. Operating costs include utilities (mainly steam) and capital costs include a column with N intermediate plates, a reboiler (with pot) and a condenser. The capital and operating costs of storage for intermediate products are ignored.

operating costs ($/batch) is:

annualised capital cost ($/yr) is:

with V in kmol/hr, K, = 1500, K2 = 9500, K3 = 180 and A = 8000. Cr< is the price for the feed ($/kmol).

7.3.3. Optimisation Problem Formulation and Solution The optimisation problem is expressed as follows:

given:

M feed mixtures, each with NCm components to be separated according to a predefined operation structure (STN), a set of product specifications and constraints for each separation duty (purities of key components in main products, amounts, etc.), product specifications and constraints for the multiple duty operation (production horizon and fraction allocated to each mixture)

determine:

the optimum set of design decision variables Dl> = [B0, N, V}, operating decisions D = {D /,..., D M), (total number of operating decision variables, NV = 2*Zm NTm ), and operating control variables U = {Uh .., Um} (reflux ratios and times) for all tasks in all operations so as to maximise: subject to:

the objective function P any constraints

Mujtaba and Macchietto (1996) formulated a two-level optimisation problem with Dd and D° optimised in an outer loop and U optimised in the inner loop, using an extension of the method of Mujtaba and Macchietto (1993). The outer loop problem can be written as:

Pl-0

h(D°, D°) = 0 g(DD, D°) < 0 Ddi <Dd< DDu D01 <D°< D°u

(overall equality constraints) (overall inequality constraints) (bounds on design variables) (bounds on operating variables)

Equality constraints h(D°, D°) = 0 may include, for example, a ratio between the amounts of two products, etc. Inequality constraints g(D°, D°) < 0 for the overall operation include Equations 7.14-7.18 (the first two of which are easily eliminated when &m and H are specified) and possibly bounds on total batch time for individual mixtures, energy utilisation, etc. Any variables of D° and D° which are fixed are simply dropped from the decision variable list. Here, Strategy II was adopted for the multiple duty specification, requiring B0 to be fixed a priori. Similar considerations hold for V, the vapour boilup rate. The batch time is inversely proportional to V for a specified amount of distillate. Also alternatively, for a given batch time, the amount of product is directly proportional to V. This can be further explained through Equations 7.24-7.26):

Reflux ratio (r) definition:

Rate of distillate, LD (kmol hr):

Batch time, t (hr)for a given amount of distillate ( SD , kmol):

An increase of V will increase LD (Equation 7.25) and vice versa for a given reflux ratio profile r. An increase in LD will decrease batch time t and vice versa

(Equation 7.26) for a given amount of distillate to be produced (SD).

As the operating and capital costs grow with V by an economic factor less than 1, a column with large V will always be profitable (Diwekar et al., 1989) and the problem becomes unbounded (also proven in Logsdon et al., 1990). Hence V is also fixed a priori. This leaves just N (the number of internal ideal separation stages or plates) as the only design variable to be optimised (D° = {N}). Out of all possible operation decision variables, it is common to specify the mole fraction of key components in main-cuts and sometimes some recoveries or amounts for off-cuts. Assuming NSP such specifications are made, there are (NV + 1 - NSP) outer optimisation problem decision variables.

Apart from N (an integer) the optimisation problem is a standard Nonlinear Programming (NLP) problem, with the inner optimisation problem providing the values of the outer objective function and constraints. In the outer problem Mujtaba and Macchietto (1996) proposed to use N as a continuous variable (within integer bounds). When N is passed to the inner loop problems its value is rounded off to the nearest integer (e.g. N= 10.66 is rounded to 11, but 10.45 is rounded to 10), as required for plate-to-plate calculations. The gradient with respect to N is calculated by finite difference using function and constraint values at the rounded off value of N and at its next higher integer value. At the final iteration the immediately lower and upper integer values are checked and the best solution is taken as the optimum. Eliceche and Sargent (1981) also considered a similar strategy.

Having design parameters fixed in the outer problem and with a specific choice of D° (discussed in section 7.2) the inner loop optimisation can be partitioned into M independent sequences (one for each mixture) of NTm dynamic optimisation problems. This will result to a total of ND = Em NTm problems. In each (one for each task) problem the control vector m^ for each task is optimised. This can be clearly explained with reference to Figure 7.3 which shows separation of M (=2) mixtures (mixture 1 = ternary and mixture 2 = binary) and number of tasks involved in each separation duty (3 tasks for mixture 1 and 2 tasks for mixture 2). Therefore, there are 5 (= ND) independent inner loop optimal control problems. In each task a parameterisation of the time varying control vector into a number of control intervals (typically 1-4) is used, so that a finite number of parameters is obtained to represent the control functions. Mujtaba and Macchietto (1996) used a piecewise constant approximation to the reflux ratio profile, yielding two optimisation parameters (a control level and interval length) for each control interval. For any task i in operation m the inner loop optimisation problem (problem Pl-i) can be stated as:

given: an initial charge from task i-1

determine: an optimal reflux ratio profile r(t) to obtain the required product ( S*Di, xDi )

so as to minimise: the task time subject to: any constraints

Mathematically it can be written as:

Pl-i min

DAE model equations etc.

where Sp;, xDi are the amount of distillate and its composition in task i at the end of task time i„ S*Di, x*Di are the specified amount of distillate product and its purity.

The solution of the inner loop problems is achieved using rigorous dynamic optimisation algorithms detailed in Chapter 5 and 6 and also by Mujtaba and Macchietto (1993) (except for minor nomenclature changes). However, the solution of the outer loop optimisation problem especially calculating gradients with respect to the decision variables are slightly different and therefore the method will be explained here with reference to Figure 7.4.

A complete solution of the sequence of inner loop problems (Figure 7.5a) is required for each "function evaluation" of the outer loop problem. Mujtaba and Macchietto (1996) obtained the gradients of the objective function with respect to the decision variables by finite difference technique but in an efficient way. This is achieved by utilising the solution of each inner loop problem from the function evaluation stage.

During the function evaluation step, the solution (reflux profiles, column profile, duration of task) of each inner loop is stored as A, B, C, D and E respectively, as shown in Figure 7.5a. For the gradient with respect to design decision variable N, the variable is perturbed and the inner loop problems are solved as shown in Figure 7.5b. The new solutions Al, Bl, Cl, D1 and El are used to calculate the gradient of the objective function with respect to the design variable (N).

It is very important to note that the operating decision variables are selected in such a way that the variables associated with one separation duty do not affect the operations involved in other separation duties. Also operating decision variables associated with any tasks within any separation duty do not affect preceding tasks but only affect the following distillation tasks. Therefore, within any separation duty, the evaluation of gradients with respect to operating decision variables of a particular task requires the solution of the inner loop problem concerning that task and the tasks ahead. For example, the evaluation of the gradient with respect to d°2 in Task 2 of separation duty 1, solutions of inner loop problems PI-2 and PI-3 concerning Task 2 and 3 respectively are required (B2, C2, Figure 7.5c). The new solutions B2, C2 together with A, D and E from function evaluation step are used to calculate the gradient with respect to d°2■ Similarly solutions D3, E3 together with A, B and C are used to calculate the gradient with respect to d°j of separation duty 2 (Figure 7.5d).

(foVH TaSk7}WBl)MTask zfM^H Task 3 I-

Initial Charge Int. Res. 1 Int. Res. 2 Bottom

Mixture 1 Product

Operating Sequence for Ternary Mixture

Operating Sequence for Ternary Mixture

Initial Charge Int. Res. 1 Mixture 2

Bottom Product

Figure 7.4. Operating Sequences of Batch Distillation Column with Two Separation Duties. [Mujtaba and Macchietto, 1996]d

Decision Variables passed Solution of inner loops

Decision Variables passed Solution of inner loops

Separation Duty 1 - Mixture 1 Separation Duty 2 - Mixture 2

Figure 7.5a. Function Evaluation. [Mujtaba and Macchietto, 1996]e

Figure 7.5a. Function Evaluation. [Mujtaba and Macchietto, 1996]e

Figure 7.5b. Gradient with respect to design variable Nc

Separation Duty 1 - Mixture 1 Separation Duty 2 - Mixture 2

Figure 7.5c. Gradient with respect to operating variable (separation duty 1). [Mujtaba and Macchietto, 1996]f

Figure 7.5c. Gradient with respect to operating variable (separation duty 1). [Mujtaba and Macchietto, 1996]f

Separation Duty 1 - Mixture 1 Separation Duty 2 - Mixture 2

Figure 7.5d. Gradient with respect to operating variable (separation duty 2). [Mujtaba and Macchietto, 1996]f

The solution procedure described above is generally applicable for single and multiple separation duties. The M sequences of inner loop problems for different mixtures may be solved in parallel if desired. The single separation duty optimal design and operation problem is simply obtained by setting M= 1. The single separation duty optimal operation problem presented in Chapter 6 can simply be obtained by setting M= 1 and fixing variables in

7.3.4. Examples

13 A. 1. Single Separation Duty, M= 1

The example is taken from Mujtaba and Macchietto (1996). As described in Chapter 4 (section 4.2.4.2.3), Nad and Spiegel (1987) considered a single duty ternary separation in an 18-plate (excluding the condenser and reboiler) batch distillation column using the operation sequence shown in Figure 7.2. The total operation time was 8.86 hr. Mujtaba and Macchietto developed the optimal design and operating policy for this ternary separation using the operation sequence and product specifications used by Nad and Spiegel. The results of Nad and Spiegel are used here as the base case for comparison.

In this separation, there are 4 distillation tasks (NT=A), producing 3 main product states MP={D1, D2, Bf) and 2 off-cut states OP={Rl, R2} from a feed mixture EF={FO}. There are a total of 9 possible outer decision variables. Of these, the key component purities of the main-cuts and of the final bottom product are set to the values given by Nad and Spiegel (1987). Additional specification of the recovery of component 1 in Task 2 results in a total of 5 decision variables to be optimised in the outer level optimisation problem. The detailed dynamic model (Type IV-CMH) of Mujtaba and Macchietto (1993) was used here with non-ideal thermodynamics described by the Soave-Redlich-Kwong (SRK) equation of state. Two time intervals for the reflux ratio in Tasks 1 and 3 and 1 interval for Tasks 2 and 4 are used. This gives a total of 12 (6 reflux levels and 6 switching times) inner loop optimisation variables to be optimised. The input data, problem specifications and cost coefficients are given in Table 7.1.

The optimum number of plates, the optimum values of the decision variables for both outer and inner loop optimisation problems, and optimal amounts and composition of all products are shown in Table 7.2. Typical composition profiles in the product accumulator tank are shown in Figure 7.6. Bold faced mole fractions in Table 2

are the specifications (all satisfied) and underlined mole fractions are decision variables which were optimised. Although the optimum number of plates is almost close to that of the base case, the optimal total operation time is 14% lower than the base case. The profit with the optimal design and operation is 35% higher than that for the base case (calculated using the same cost model). This is obtained with only 6 reflux intervals while the base case operation required about 20 reflux changes. There is no need for initial total reflux in the optimal operation as required in the base case (for 2.54 hrs). It is reasonable to assume that the use of large total reflux operation period (2.54 hrs) affects the overall profit for the base case.

For each outer loop "function" and "gradient" evaluation 4 and 14 inner loop problems were solved respectively (a total of 124 inner loop problems). For the inner loop problems 12-14 iterations for Tasks 1 and 3 and 5-7 iterations for Tasks 2 and 4 were usually required. For this problem size and detail of dynamic and physical properties models the computation time of slightly over 5 hrs (using SPARC-1 Workstation) is acceptable. It is to note that the optimum number of plates and optimum recovery for Task 1 (Table 7.2) are very close to initial number of plates and recovery (Table 7.1). This is merely a coincidence. However, during function evaluation step the optimisation algorithm hit lower and upper bounds of the variables (shown in Table 7.1) a number of times. Note that the choices of variable bounds were done through physical reasoning as explained in detail in Chapter 6 and Mujtaba and Macchietto (1993).

7.3.4.2. Multiple Separation Duties, Two Mixtures (Af=2): Effect of Different Allocation Time

This example is taken from Mujtaba and Macchietto (1996). The problem is to design a column for 2 binary separation duties. One of the separations is very easy compared to the other one. The fraction of production time for each duty is specified together with the still capacity (B0) and the vapour load (V). Each binary mixture produces only one main distillate product and a bottom residue (states MPj={Dl, Bfl ) and MP2={D2, Bf2}) from feed states EF,={F1} and EF7={F2], respectively, with only one distillation task in each separation duty. Desired purities are specified for the two main-cuts (x'Di and x'D2). Also obtain the optimal operating policies in terms of reflux ratio for the separations.

In this problem, there are 3 outer loop decision variables, N and the recovery of component 1 from each mixture {Re oibq, Re d2,bo)■ Two time intervals for reflux ratio were used for each distillation task giving 4 optimisation variables in each inner loop optimisation making a total of 8 inner loop optimisation variables. A series of problems was solved using different allocation time to each mixture, to show that the optimal design and operation are indeed affected by such allocation. A simple dynamic model (Type III) was used based on constant relative volatilities but incorporating detailed plate-to-plate calculations (Mujtaba and Macchietto, 1993; Mujtaba, 1997). The input data are given in Table 7.3.

Table 7.1. Input Data, Product Specifications and Decision Variables for Ternary Distillation (Detailed Dynamic Model, Single Duty)g. Recoveries Re<a b are defined as: amount of component j in state a / amount of component j in state b. For M= 1, &j= 1. The mixture index is omitted for clarity.

Column: Still Capacity = Batch size, FO, kmol = 2.93

Condenser Vapor Load, V, kmol/hr = 2.75

Column Holdup, kmol:

Condenser =1.2% of FO Internal Plates (total) = 1.9% of FO

Duties: Total Plant Operation Time, Hyr, hr/yr =8000.0

Total time horizon, H, hr = 8000.0

No. of mixtures and fraction of horizon for each: M= 1, = 1

Mixture: Components: Cyclohexane, n-Heptane, Toluene

Feed Composition, xpf), mole fraction = <0.407, 0.394, 0.199>

Specifications:

Cyclohexane mole fraction in D1, x'D1 = 0.895

n-Heptane mole fraction in D2, x'D2 = 0.863

Toluene mole fraction in Bf, x3Bf = 0.990

Cyclohexane recovery in Rl, Re^^i = 0.95

Outer Loop Decision Variables, initial value <lower and upper bounds>: Number of Plates, N =16 <12 22>

Cyclohexane recovery in Dl, Re'Di Fo =0.85 <0.70 0.92>

Cyclohexane mole fraction in Rl, x'R1 =0.40 <0.30 0.45>

Toluene recovery in D2, Re2D2B2 =0.85 <0.70 0.92>

Toluene mole fraction in R2, x2R2 =0.30 <0.30 0.45>

Costs:

CD1 = 30.0 $/kmol, CD2 = 26.0 $/kmol, CR, = CR2 = - 1.0 $/kmol CB/= 24.0 $/kmol, CF0 = 2.0 $/kmol, Csu= 0.0 OC and ACC as described in section 7.3

Table 7.2. Summary of Results - Example lh

Optimal Outer Loop Decision Variables:

N=17 Re'D1_F0 = 0.842 x'R1 = 0.373 ReD2B2 = 0.892 x2R2 = 0.362

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