Introducing the Concept of Roving Pumps in the Synthesis of

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Introducing the Concept of Roving Pumps in the Synthesis of Multipurpose Batch Plants Shaun Engelbrecht, and Thokozani Majozi Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b04865 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 29, 2018

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Industrial & Engineering Chemistry Research

Introducing the Concept of Roving Pumps in the Synthesis of Multipurpose Batch Plants

Shaun Engelbrecht and Thokozani Majozi*

NRF/DST Chair: Sustainable Process Engineering, Johannesburg, South Africa School of Chemical and Metallurgical Engineering, University of the Witwatersrand, Johannesburg, South Africa *

Corresponding author: e-mail: [email protected] (T. Majozi); Telephone: +27 (0) 11 717 7384

1 ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Abstract Presented in this manuscript is a novel technique for the exploitation of a unique feature of multipurpose batch plants, so called roving pumps. The discretized nature of batch processes allows for the allocation of pumps to different units and tasks depending on pump availability over the time horizon of interest. A MILP formulation is proposed to determine the optimal number and allocation of pumps within the batch plant while simultaneously synthesizing the plant and optimizing the production schedule with variable material transfer times. It is shown by way of an example that it is possible to decrease the number of pumps required for operation from seven to two pumps, while another example demonstrates a decrease in the number of pumps required from eleven to three. It is, therefore, possible to reduce capital and maintenance costs by implementing the formulation while still achieving or improving on existing throughputs. Keywords: Pump, Batch, Synthesis, Multipurpose, Optimization, Modelling


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1. Introduction Batch processes are commonly used to produce high value products in low volumes. Multipurpose batch plants afford great flexibility by producing a number of different products using the same equipment. This flexibility allows for improved equipment utilization while adapting to rapid market fluctuations. In the selection of plant equipment, the processing requirements and production material properties dictate the design. Most batch chemical plants, as traditionally encountered in the food, pharmaceutical and agrochemical industries, predominantly require liquids for processing. The liquids need to be transferred between processing units using pumps. Batch plant optimization and scheduling have been studied since the 1970s and gained considerable attention in the field of mathematical modelling during the 1990s. The use of mathematical modelling has made it possible to determine the optimal processing time and material allotted to processing units in order to manage a batch plant efficiently. Sparrow et al.1 presented an initial formulation to address the general scheduling problem of multiproduct batch plants by looking at heuristic and branch and bound techniques. Grossmann and Sargent2 presented an initial MINLP model to design sequential multiproduct batch processes. Suhami and Mah3 elaborated on this work and presented a method to simultaneously schedule and design multipurpose batch plants using heuristics. A number of configurations were generated, from which the most profitable was chosen as the optimal. However, global optimality cannot guaranteed for heuristic based approaches 4. Batch production requires that a series of independent processing steps be performed based on a batch recipe. Various techniques to represent the recipes of batch plants have been discussed in literature. Kondili et al.5 introduced the State Task Network (STN) representation which makes use of nodes and arcs to show the relationship between various states and tasks. A MILP formulation for the short term scheduling of multipurpose batch 3 ACS Paragon Plus Environment


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plants based on the STN representation was then developed. However, this formulation required a large number of binary variables in order to yield an accurate result, due to the discrete time representation employed and the number of binary variables associated with task processing. Shah et al.6 addressed computational issues associated with the model of Kondili et al.5 and presented a formulation with a simpler set of reformulated equations and solution techniques, yielding a MILP model with a smaller integrality gap. The Resource Task Network (RTN) , developed by Pantelides7, differed slightly from the STN representation by distinguishing between raw materials, intermediates and products. It also included aspects such as utilities, other essential materials and manpower. Tasks are defined similarly to that of the STN representation and are extended to include aspects such as cleaning, storage and transportation. The State Sequence Network (SSN) representation introduced by Majozi and Zhu8, allowed the introduction of a single-type binary variable for scheduling batch plants. The binary variable, , , , relates to whether a state, s, will be used in a unit, j, at a given time point, p. Binary variables associated with tasks and units are implicitly incorporated, allowing the binary dimension of the scheduling problem to be significantly reduced. Furthermore, if more than one state is required for a given processing task, only a single effective state is required to be defined to represent a task. In order to obtain truly optimal results, the continuous time representation has been employed extensively. Pinto and Grossmann9 as well as Schilling and Pantelides10 increased the accuracy of scheduling models by using time slots and time points within a continuous time representation, respectively. Pinto and Grossmann9 presented a model that was able to meet intermediate orders during the time horizon while employing cyclic scheduling. Karimi and McDonald11 expanded on intermediate orders within a batch schedule by employing time slots in short term scheduling problems. An extended MINLP mathematical formulation


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based on the RTN representation and a continuous time representation was presented by Zhang and Sargent12. A continuous-time MILP mathematical formulation that decoupled task events from unit events was presented by Ierapetritou and Floudas13 which allowed a significant decrease in the number of binary variables. Lin and Floudas14 proposed a mathematical model which addressed the integrated design and synthesis of batch plants based on a continuous time formulation. Shaik and Floudas15 presented a MILP formulation that is based on a three index binary variable which allowed tasks to continue over multiple event points. The formulation required that an iterative procedure be used to determine the number of event points. An additional iteration was required to find the optimal duration of tasks over multiple event points. The SSN representation was further employed by Seid and Majozi16 in a formulation which addressed crucial omissions when scheduling batch plants requiring the finite intermediate storage (FIS) policy. Literature models evaluated were found to be suboptimal and exhibited unlimited intermediate storage (UIS) characteristics in the final schedule. The model also required fewer event points by basing the continuous representation on unit specific slots, as shown by Susarla et al.17. This allowed for better solving performance and longer batch time horizons to be evaluated. Seid and Majozi18 developed a continuous time design and synthesis model for multipurpose batch plants with the objective of maximizing the annualized profit, while enforcing the correct sequencing of tasks when the FIS policy is required. Sanmartí et al.19 developed the graphical S-graph approach to schedule batch operation without the use of time points; however, it is limited to no intermediate storage (NIS) and UIS policies. Notably, within these scheduling models, it was mainly the major processing equipment of the batch plant, such as the reactors, separators and heaters which were scheduled. No



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consideration was given to the optimal synthesis and scheduling of pumping equipment within multipurpose batch plants. Loonkar and Robinson20 noted that the capital cost of a batch plant is dependent on the rating of pumping equipment, heat exchanger duty and the size of the processing vessels. A general procedure for sizing interdependent equipment was also given. Barbosa-Póvoa and Macchietto21 stated that transfer connections must be present between vessels in order to facilitate transfer of material. A general MILP model was presented that restricted transfers based on the availability of relevant connections. Hegyhati and Friedler22 highlighted that material transfers were generally simplified to fixed durations when scheduling plant production, regardless of the batch size and costs associated with the transfers. Hence, they proposed discrete transfer times and associated energy costs based on the pumping requirement. Lee et al.23 considered fixed material transfer times in their work in order to allow for heat integration during the transfer. Each transfer time was a fixed parameter which was independent of the amount of material transferred. Hegyháti et al.24 noted that many models overlooked the subtle transfer infeasibility that arose when various operational philosophies were employed. More, specifically, it was noted that mathematical models allowed material to be transferred to and from a processing unit at the same time, so called cross-transfer. Hegyháti et al.24 further demonstrated that the S-graph approach was not susceptible to this infeasibility. Majozi and Friedler25 extended the S-graph method to maximize throughput over a given time horizon. A guided algorithm was presented to find the guaranteed global optimum. The algorithm was applied only for the NIS policy and when the batch sizes were fixed. Models found in literature have, therefore, steadily increased the accuracy of batch plant scheduling while also reducing the time required to solve these problems. Nevertheless, there is little consideration for the required transfer times that are inherent when any form of


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material transfer occurs. This omission can lead to significant inaccuracies when these models are applied in practice. The discrete operation of tasks within batch plants allows for the possibility of reusing processing equipment for various tasks. Since pumping equipment is used intermittently it can also be reused for various transfer tasks. It is, therefore, possible to reuse pumping equipment more frequently and also reduce the total number of pumps required within the plant. Therefore, a new mathematical model is required that can sufficiently synthesize and schedule both the major and pumping equipment simultaneously.



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2. Motivation Conventionally, pumps are used to transfer material between fixed sources and sinks in a batch plant, as seen in Figure . This results in a large number of pumps needed throughout the plant. These pumps will only be active for limited periods of time when they are required to transfer specific material between sources and sinks. Previous formulations neglected the inclusion of the transfer equipment in the analysis, synthesis and optimization.

Feed

Storage Unit 1

Pump 1

Unit 2

Pump 2

Pump 3

Figure 1. Conventional pump allocation within a batch plant The allocation of pumps is a crucial point to consider, especially when noting that pumps are generally expensive components of the plant in terms of capital, operational and maintenance costs. Additionally, in order to schedule a batch plant as realistically as possible, the inclusion of pumping allocation and the associated transfer times is paramount. This inclusion is of even greater importance when considering the nature of multipurpose batch plants, wherein material may be transferred between a number of shared processing units. In this manuscript, the concept of roving pumps is introduced. These roving pumps are not necessarily dedicated to particular units or sections in the plant but can, instead, be reassigned to different processing units and sections as required. By using roving pumps, flexibility in the batch plant is increased since various pumps can perform the same tasks, provided the materials are compatible. Additionally, where pumping equipment is limited, roving pumps allow plant production to continue at the best possible efficiency. 8 ACS Paragon Plus Environment

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It is evident from Figure 2 that the proposed allocation of pumps would result in increased pumping equipment utilization. This is favorable since prolonged disuse of pumps may increase maintenance costs due to clogging or blockages and structural degradation. Figure 2 (a)–(d) depict that a reduced number of pumps may be used to transfer material between various units when allowing the pumps to be flexibly allocated. Notably, this can be achieved while preserving or even improving on the production throughput of cases where flexible pump allocation is not taken into consideration.

Storage 1

Storage 1

Unit 1

Unit 1

Feed

Feed

Pump 1

Pump 1 Unit 2

Unit 2

(a)

(b)

Storage 1 Storage 1

Unit 1

Unit 1

Feed

Feed

Pump 1

Pump 1 Unit 2

Unit 2

(c)

(d)

Figure 2. Proposed pump allocation 9 ACS Paragon Plus Environment


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This synthesis method can be applied for plant design where it is desirable to limit the number of equipment units required, in order to save on capital investment and reduce the space occupied by the plant. It can also be applied in retrofit design in order to eliminate unnecessary processing and pumping equipment in a plant, thereby saving on maintenance costs. Notably, multipurpose batch plant synthesis models are inherently incomplete if the basic auxiliary equipment, such as pumping equipment and piping, are not included in the formulation. The focus for this paper is, therefore, to develop a mathematical model that enables the synthesis and scheduling of multipurpose batch plants along with the flexible allocation of pumps throughout the plant over a given time horizon of interest, such that a particular performance index is optimized.


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3. Problem statement The problem that is addressed in this manuscript can be stated as follows. Given: (i)

The production recipes of various products,

(ii)

The available processing units and storage vessels,

(iii)

The available pumps and respective pumping rates, and

(iv)

The time horizon of interest

Determine: (i)

The optimal plant production schedule,

(ii)

The optimal number of processing units and piping connections,

(iii)

The optimal allocation of pumps to processing units within the time horizon,

(iv)

The required material quantity to be processed in each processing unit, and

(v)

The time required to transfer the material between areas within the plant

This should be accomplished in such a way that the performance of the batch processing plant is maximized.



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4. Mathematical formulation The constraints for the mathematical model that allow for pump allocation and transfer times are shown below. Constraints (1) - (15) address the allocation of pumping equipment to transfer tasks, whereas Constraints (16) - (41) address the scheduling of the various processing tasks based on pump availability. The scheduling model is reformulated from the models presented by Lee et al.23 and Seid and Majozi18 to allow for pumping allocation and variable transfer times.

4.1. Pumping constraints Constraints (1) - (15) address the assignment of pumps to transfer material between units and the transfer times required for those transfers. Constraint (1) states that pumping into a vessel may occur, provided a piping connection is present between the pump and the destination vessel. However, the presence of a piping connection does not necessarily mean that pumping must occur at a given time point.

′, , , ≤ ,

∀ ∈ , , ∈ , ∈

(1)

Similarly, Constraint (2) states that pumping from a vessel may occur, provided a piping connection is present between the source vessel and the pump. Again, the presence of a piping connection does not necessarily mean that pumping must occur at a given time point.

′, , , ≤ ,

∀ ∈ , , ∈ , ∈

(2)


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Constraints (3) and (4) are required to ensure that pumps cannot be used simultaneously by different units at any given time point. Constraint (3) states that if a pump exists and is to be used for pumping from a source to a unit, then that same pump cannot be used to pump to any other unit at that time point. However, the same processing unit may still utilize the same pump to receive different states from various sources.

, , , + , , , ≤

∀ ∈ , , ∈ , , ∈

!"#$ ,

(3)

∈ Ζ, ≠ , ≠ , ≠

Similarly, Constraint (4) states that if a pump exists and is to be used to pump material from a processing unit to a destination storage vessel at a time point, then that pump may not be used to pump material from other processing units. However, the same processing unit may repeatedly utilize the same pump to transfer different states to various destinations.

' , , , + ' , ′, , ≤

∀ ∈ , , ∈ , ' , ' ∈ '$()( ,

(4)

∈ Ζ, ≠ , ≠ ' , ≠ ' ′

Constraint (5) states that a pump must be active between the storage vessel and the processing unit at a time point for any material to be transferred to a storage vessel from the processing unit. The constraint also sets an upper bound on the amount of material that can be sent to storage.

* , , ≤ + , , , , ,

∀ ∈ ., ∈ Ζ, , ∈ (

-

")/$ ,

∈ , ∈

(5)



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Similarly, Constraint (6) states that a pump must be active in order to transfer product material from a unit to the product stockpile. The constraint also sets an upper bound on the amount of material that can be sent to the product stockpile.

* , , ≤ + 0 , , ,

∀ ∈ . 0 , ∈ Ζ, 0 ∈ 0"

-

'!#( ,

∈ , ∈

(6)

Constraint (7) states that a pump must be associated with the transfer of material out of a storage vessel to a processing unit and also sets an upper bound on the amount of material that can be removed from storage.

*

!( ,

, ≤ + , , , , ,

∀ ∈ ., ∈ Ζ, , ∈ (

-

")/$ ,

∈ , ∈

(7)

For the case where material is transferred between different processing units, Constraint (8) states that a pump must be associated with the transfer and limits the transfer to the design capacity of the accepting unit.

*( , , ′, ≤ + , ′, , -

∀ ∈ ., ∈ Ζ, ∈ , , ∈ , ≠ ′

(8)

Constraint (9) states that a pump must be associated with the transfer of material out of a feed stockpile and limits the transfer to the design capacity of the accepting unit.

*

!(

, , ≤ + , 1 , ,

-

∀ ∈ ., ∈ Ζ, 1 ∈ 1$$' , ∈ , ∈

(9)


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Constraint (10) assigns the exact transfer time for a state between two different units using a particular pump at a time point. This constraint, which is only active if there is material being pumped between unit and ′, states that the transfer time is related to the rating of the pump

that is used and the amount of material transferred. In this manuscript, each pump has a fixed pump rating, which is defined as the transfer time per unit material.

2 *( , , ,

−41 − , , , ≤ 6( , , , , ≤ 2 *( , , ,

+41 − , ′, ,

(10)

∀ ∈ , , ∈ , ∈ ., ∈ Ζ, ≠ ′

Similarly, Constraint (11) states that the transfer time between a processing unit and a storage vessel is dependent on the rating of the active pump and the quantity of material transferred into storage.

2 * , ,

−41 − , , , ≤ 6( , , , , ≤ 2 * , ,

+41 − , , ,

∀ ∈ , ∈ , ∈ (

")/$ & 0" '!#( ,

(11)

∈ ., ∈ Ζ

Additionally, Constraint (12) states that the transfer time between a storage vessel and a processing unit is dependent on the rating of the active pump and the quantity of material transferred out of storage.

2 *

!(

, ,

−41 − , , , ≤ 6( , , , , ≤ 2 *

+41 − , , ,

∀ ∈ , ∈ , ∈ (

")/$ & 1$$' ,

!(

, ,

(12)

∈ ., ∈ Ζ 15

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In a situation where two consecutive tasks take place in one unit, the transfer time for material already present in the unit is zero, as captured in Constraint (13).

6( , , , , = 0

∀ ∈ , ∈ , ∈ ∈ .

(13)

Constraint (14) allocates time to transfer material to processing tasks. This formulation allows for simultaneous transfers into units from various sources. It may be the case that the transfer times are different for each pump used, thus Constraint (14) states that transfer duration assigned allows for sufficient time to accommodate all transfers.

6'!" , , ≥ 6( # , , ; , , ,

∗ ∀ ∈ , ∈ , , ∈ ., , , ∈ (

")/$ ,

0

∈ ., ∈ Ζ

(15)


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4.2. Scheduling Constraints (16) - (41) pertain to scheduling and incorporate the pumping constraints. Constraint (16) states that only one task may occur within a processing unit at any given time point, provided the processing unit exists. Some of the constraints are not necessarily new, but are included here for completeness purposes.

, , ≤ e

?@,

A

(16)

∗ ∀ , ∈ ., , ∈ , ∈

Design capacity constraints Constraint (17) states that the amount of material to be used by a task at any given time point must be within the design capacity of the processing unit.

C , , ≤ *! , , ≤ + , ,

∗ ∀ , ∈ ., , ∈ , ∈

(17)

Constraint (18) states that the amount of a state already stored in a storage vessel along with the material transferred to that storage vessel at a given time point, must not exceed the maximum design capacity of that storage vessel. Furthermore, it only allows such storage in the case where the storage vessel exists.

D , + * , , ≤ + , ,

∀ ∈ ., ∈ , ∈ , , ∈ (

(18)

")/$

Material balance for storage Constraint (19) is the material balance for storage and relates the material entering and leaving storage to that stored during the previous time point. The constraint states that the amount of a state stored at a time point relates to the amount that was previously stored and the material quantities which are leaving and entering the storage vessel.



D , = D , − 1 − * ∈EFG

!(

, , + * , , − 1 ∈EFH

∀ ∈ ., ∈ , ∈ , > 1

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(19)

Constraint (20) determines the amount of a state stored after it is initially used during the first time point. The constraint states that the material leaving storage during the first time point must be taken from the initial reserve.

D , = JK − * ∈EFG

!( ,

,

∀ ∈ ., ∈ , ∈ , = 1

(20)

Processing unit material balance Constraint (21) determines the total amount of material consumed by a task in unit j at a time point p. The constraint states that the state, s, to be consumed may either come from a storage vessel or from an upstream unit, .

L#?@,A *! , , = *

!(

, , + *( , , ′ M ∈EFH

∗ ∀ , ∈ , ∈ ., ∈ , , ∈ .,

(21)

Similarly, Constraint (22) states that the amount of material produced within a unit is the amount of material that is transferred to storage and to other units.

L0 *! , , = * , , + *( , , , + 1 ?@,A ∈EFG

∗ ∀ , ∈ , ∈ ., ∈ , , ∈ .,

(22)

Constraint (23) stipulates that the amount of material that can be produced by a task at any given time point, must be less than the combined available storage capacity and the design capacity of the processing unit. This is required for the case when only a single unit can produce the intermediate state within the batch process at a given time point.


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L0 *! , , − 1 ≤ D , + + N , ?@,A ∀ ∈ , ∈ , , ∈

∗ ., ,

∈ ., > 1

(23)

Constraint (24) states that the total amount of material that can currently be consumed in a task must be available from storage or from a producing task during the previous time point. This allocation ensures that downstream tasks may receive material from multiple upstream tasks. Constraint (24) is nonlinear due to the presence of the bilinear term, however, it can be

FG ?@,A ∈;?@,A

L#?@,A *! , ,

≤ D , − 1

+

FH ?@,A ∈;?@,A

L0 *! , , ?@,A

− 1 N ,

(24)

∗ ∀ ∈ , ∈ , , ∈ ., , ∈ ., > 1

exactly linearized using the Glover26 transformation.

Task sequencing Constraint (25) describes the end time for a task and states that the processing duration depends on of the batch size.

60 , , ≥ 6! , , + O, , ,

+ P, *! , ,

(25)

∗ ∀ ∈ , , ∈ .,

Constraint (26) states that a consuming task may commence in a unit after the required upstream tasks, as well as all the necessary transfers from and to that unit, have been completed.



0 0 # # !( 6! , , ≥ 60 , , − 1 + 6'!" , , − 1 + 6'!" , ,

0 # , , , ,

∀ ∈

∈

∗ .,

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(26)

Similarly, Constraint (27) states that the same task in a unit may only commence again after processing has concluded and the necessary transfers from the unit have been performed during the previous time point. !( 6! , , ≥ 60 , , − 1 + 6'!" , , − 1 + 6'!" , ,

∀ ∈ , , ∈

∗ .,

(27)

Constraint (28) pertains to scheduling over multiple time points such that interim material storage is accounted for, as noted by Seid and Majozi18. When all produced material is stored during the time point following its production, the constraint states that it can only be used after two time points. This allows sufficient time for the material to fully enter storage and later be removed from storage. # # 6! , , ≥ 60 , , − 2 + 6'!" , ,

0

0 − 4 R1 − , , − 2 S

(28)

# ∗ ∀, ∈ , , , , ∈ ., , > 2 0

Constraint (29) is required for the case where the FIS policy is implemented18. Used in conjunction with the above sequencing constraints, Constraint (29) states that the consuming task begins when all transfers have been completed. # # 6! , , ≤ 60 , , − 1 + 6'!" , ,

0

# + 4 R2 − , , − , , − 1 S 0

(29)

# ∗ ∀ ∈ , ,A , , ∈ ., , ∈ ., > 1 0


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If a task is to recommence during a subsequent time point, it may only do so if all the material from the task of the preceding time point has been removed from the unit and fresh material for processing has been transferred to the unit during the current time point. While material is being transferred to the downstream task, material is still present in the source vessel and it is, therefore, not available for use until the transfers have concluded. The downstream task commences when all transfers to it have concluded. The start of the downstream task must, therefore, precede that of the recommencing upstream task in order to avoid pumping to the upstream task before the source unit has been emptied. This eliminates the possibility of a new task commencing before the unit has finished emptying. Constraint (30), therefore, states that a task, , , may be initiated again once its dependent downstream

tasks, , M , have started and the required material for the current task has been transferred

into the processing unit. 6! , , ≥ 6! R, M , S + 6'!" , ,

− 4 R2 − , , − ′, , , S

(30)

∗ ∀ ∈ , , ∈ , , , , M ∈ ., , ∈

Pump sequencing constraints Constraint (31) states that the end time for pumping into a unit corresponds to the start time of a processing task, provided that the pump in question is allocated between the relevant vessels and the processing task is active during the time point.



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6! , ,

, , ≤ 6! , , −4 R2 − , , , − , , S ≤ 6-0

+4 R2 − , , , − , , S

∀ ∈ , ∈ , ∈

!"#$ , ,

(31)

∗ ∈ ., , ∈ Ζ

Constraint (32) states that processing for a task in unit j may commence after sufficient time for pumping tasks to the task is given. 6! , , − 6'!" , , − 4 R2 − , , , − , , S ≤

, , ≤ 6! , , − 6'!" 6-! , , + 4 R2 − , , , −

, , S

∀ ∈ , ∈ , ∈

!"#$ , ,

(32)

∗ ∈ ., , ∈ Ζ

Constraint (33) states that the starting time for pumping out of a unit starts at the end time of a processing task, provided that the pump in question is allocated between the relevant vessels and the producing task is active during the time point.

60 , , − 4 R2 − ' , , , − , , S ≤ 6-!!( , ,

≤ 60 , , + 4 R2 − ' , , , − , , S

(33)

∀ ∈ , ∈ , ' ∈ '$()( , , ∈ . ,∗ , ∈ Ζ

Constraint (34) states that the end time for pumping out of a unit occurs such that sufficient time for pumping is given. !( 60 , , + 6'!" , ,

−4 R2 − ' , , , − , , S ≤ 6-0!( , , ≤ 60 , ,

(34)

!( +6'!" , , + 4 R2 − ' , , , − , , S


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∗ ∀ ∈ , ∈ , ' ∈ '$()( , , ∈ ., , ∈ Ζ

Constraints (35) - (39) are required for the correct scheduling of pumping tasks such that there is no using the same pump. Constraint (35) states that a particular pump may be used again to pump material into a unit after it has concluded pumping material into a unit during the previous time point. , , ≥ 6-0 , ′, − 1 6-!

∀ ∈ , , ′ ∈ , ∈ Ζ

(35)

Similarly, Constraint (36) states that a particular pump may be used again to pump material out of a unit after it has concluded pumping material out of any unit during the previous time point.

6-!!( , , ≥ 6-0!( , , − 1 ∀ ∈ , , ′ ∈ , ∈ Ζ

(36)

Additionally, Constraint (37) stipulates that a pump may be used to pump material into a unit after it has concluded pumping material out of any unit during the previous time point. , , ≥ 6-0!( , ′, − 1 6-!

∀ ∈ , , ′ ∈ , ∈ Ζ

(37)

Constraint (38) states that pumping into a particular unit may commence only after all pumping out of the same unit has concluded during the previous time point. , , ≥ 6-0!( ′, , − 1 6-!

∀ ∈ , ∈ , , ′ ∈ Ζ

(38)



Page 24 of 53

Constraint (39) states that pumping from a vessel into another unit may only occur after the vessel has concluded receiving its material from all pumping tasks during the previous time point. , , ≥ 6-0!( , , − 1 6-!

− 42 − ' , , , − 1 − , , ,

∀ ∈ , , ∈ , , ∈ Ζ, ∈

!"#$ , '

(39)

∈ '$()( , = '

Constraint (40) states that all pumping and, therefore, processing must conclude within the time horizon of interest.

6-0!( , , ≤ T

∀ ∈ , ∈ , ∈ Ζ

(40)

Objective function The objective function is the maximization of annualized profit available from the batch plant. In this work it is assumed that 7200 working hours are available per annum. Profit is described as the difference between the revenue from the products and the cost of all feed material, equipment and piping connections.


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*UV*W XY6

= Z L0 *! , , [ . 0 ?@,A 0 ?@,A ∈ ; H

− L#?@,A *! , , [ . 1 ] T^" ⁄T 0 ?@,A ∈ ; \

− [ + [`a ⁄2 − [

-

− , [ ,

(41)

,>

− , [-b + , [b-

,

∀ ∈ , ∈ , , ∈ (

-

")/$ ,

∗ ∈ , , ∈ ., , . 0 , . 1 ∈ ., ∈



Page 26 of 53

5. Applications In order to assess the proposed MILP model, examples found in literature were evaluated. All results were obtained using a computer with 16 GB of RAM, 2.59 GHz, Intel I7 Processor with Windows 10 operating system and the GAMS 24.8.3 software suite.

5.1. First illustrative example The first example is based on the scenario given by Lin and Floudas14, which involves a multipurpose batch plant capable of performing 4 distinct tasks. The production recipe also allows a product state, S5, to be consumed to produce another product, S6. Three major processing units are available to perform the processing tasks, along with a storage vessel that is available to store state 4 (S4). The recipe representation for the scenario is given in Figure 3. Data pertaining to the equipment may be found in Table 1 and economic data in Table 2, as adapted from Lin and Floudas14. The data pertaining to pumping rates for the various available pumps are given in Table 3. The illustrative pumping rates were calculated such that no pump will be active for more than five minutes when the pumps are placed conventionally with the plant operating at full capacity. A time horizon of 12 hours was considered.

Figure 3. STN representation for the first illustrative example


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Table 1. Equipment data for the first illustrative example

Unit / Storage Vessel

Capacity

Suitability (Task/Storage)

Unit J1

50-150 mu

Unit J2

50-150 mu

Unit J3

50-200 mu

Vessel 4

100 mu

T1 T2 T1 T2 T3 T4 S4

Task Duration (Hours) 2 2 2 2 4 2 -

Cost (Cost Units) 100 150 120 15

Table 2. Economic data for the first illustrative example

Material/Equipment S1, S2 S5 (Product 1) S6 (Product 2) Pump base cost Pump rating cost Piping cost per connection

Price (Cost units) 0.1 /mu 0.4 /mu 0.6 /mu 10 0.001 5

Table 3. Pump ratings for the first illustrative example

Pump A1 A2 A3 B1 B2 B3 B4

Pump Rating (seconds / unit material) 2.00 2.00 1.50 2.00 2.00 1.50 1.50



Page 28 of 53

Figure 4 shows the typical flowsheet for the first illustrative example and depicts the pumps which would conventionally be used to transfer material between different areas within the plant. In order to simplify the flowsheet, it is deconstructed such that the pumps required to pump material out of processing units and the pumps required to pump material into processing units are separated, as shown in Figure 5 and Figure 6, respectively. These deconstructed views are useful for larger plants, where the piping connections are difficult to identify and follow. A total of seven pumps are required to pump material throughout the plant when the plant is conventionally operated.

Feed State 1 S1f

Storage V1 Pump B1

Product 1 Unit J1

Pump B3 Pump A1

Feed State 2 S2f

Pump B4

Unit J3

Unit J2 Pump B2

Pump A3

Pump A2

Product 2

Figure 4. First illustrative example: conventional pump allocation and plant flowsheet


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Storage V1 Unit J1

Product 1 Pump A1

Unit J3

Unit J2

Product 2

Pump A3

Pump A2

Figure 5. First illustrative example: conventional pump allocation for pumping from units

Storage V1

Feed State 1 S1f

Product 1

Pump B1 Pump B4 Unit J1

Pump B3

Feed State 2 S2f

Pump B2

Unit J2

Unit J3

Figure 6. First illustrative example: conventional pump allocation for pumping to units 29 ACS Paragon Plus Environment


Page 30 of 53

In solving the example, Constraints (3) and (4) are to be applied. These constraints are key for the assignment of pumps for transfer tasks. For the first illustrative example, Constraint (3) is employed as follows, when the destination unit is taken as unit J1 from source vessel J2, using a pump A1 at a given time point, p. This explicit statement dictates that no other pumping transfer may occur elsewhere in the plant that is not feeding unit J1, without compromising possible feeds to other units.

1, 2, c1, + 2, 1, c1, + 3, 1, c1, ≤ c1

1, 2, c1, + 3, 2, c1, ≤ c1

1, 2, c1, + 2, 3, c1, ≤ c1

1, 2, c1, + 2, 1, c1, + 3, 1, c1, ≤ c1

1, 2, c1, + 2, .1Y, c1, + 3, .1Y, c1, ≤ c1

1, 2, c1, + 2, .2Y, c1, + 3, .2Y, c1, ≤ c1

1, 2, c1, + 3, Xe 1, c1, ≤ c1

Constraint (3) is similarly employed to cater for any other destination unit j, from source

vessel vs, using pump , at time point p. Similarly, Constraint (4) is employed as follows to

transfer material to destination storage vessel V1, from source unit J1, using pump A1 at time point p.

1, 1, c1, + Xe 1, 2, c1, + Xe 1, 3, c1, ≤ c1

1, 1, c1, + Xe 2, 2, c1, + Xe 2, 3, c1, ≤ c1

1, 1, c1, + 1, 2, c1, + 1, 3, c1, ≤ c1

Constraint (4) is similarly employed to cater for any other destination storage vessel vd, from

source unit j, using pump , at time point p.


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5.1.1.

Results

The MILP problem was solved using the CPLEX solver. Figure 7 shows the resulting optimal plant pumping structure needed to cater for the example. Deconstructing the optimal pumping structure in a similar fashion as with the flowsheet, yields the simplified view for the pumping inlet connections and the pumping outlet connections, as shown in Figure 8 and Figure 9 respectively. By flexibly allocating pumps, the plant required two pumps to transfer material throughout the batch plant whereas seven pumps were required when flexible pump allocation was not considered. The synthesis results also state that a storage vessel is not required for state 4. The problem was solved using four time points, yielding an objective value of 126 751 cost units in a CPU solution time of 2.45 seconds.

Product 1 Feed State 1 S1f

Feed State 2 S2f Unit J1 Product 2

Pump A1

Pump A2

Unit J2

Unit J3

Figure 7. First illustrative example: optimal roving pump allocation and connections



Page 32 of 53

Product 1 Feed State 1 S1f

Product 2

Unit J1 Feed State 2 S2f

Pump A1

Pump A2

Unit J2

Unit J3

Figure 8. First illustrative example: Optimal pump allocation (pump inlet)

Feed State 1 S1f

Feed State 2 S2f

Unit J1

Product 1

Product 2

Pump A1

Pump A2

Unit J2

Unit J3

Figure 9. First illustrative example: optimal pump allocation (pump outlet)


Page 33 of 53

Figure 10 shows the optimal pumping schedule required for the first illustrative example when pumps are flexibly assigned. As seen on the schedule, a number of pumping tasks performed by the same pump, to the same destination from different sources are highlighted. These correspond to the processing tasks that require multiple feeds to a processing unit. The order in which material is to be pumped into a processing unit is of no consequence, since all raw feeds need to enter a unit before operation may commence. Therefore, the pumping order is left to the discretion of the user. The optimal production schedule using roving pumps may

J1 J2 S2f SOURCE

J3

Prod 1 Prod 2 J2 J3

Prod 1

J3

J3

J1

A1 A1 PUMP

J3

S1f

DEST

A2

A2

be found in Figure 11.

Pumping Tasks A1 A1 A2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

2

4 6 8 Time Horizon (Hours)

10

12

Figure 10. Optimal pumping schedule obtained for the first illustrative example using roving pumps



T4 J3

200

T3 J3

200

Tasks

200

T2 J1

80

T1 J2

120

0

120

120

2

4

6

8

10

12

Time Horizon (Hours)

Figure 11. Optimal production schedule obtained for the first illustrative example using roving pumps Figure 12 is the optimal production schedule for when the plant is not optimally synthesized and translates to an annualized profit of 126 665 cost units. The optimal production schedules for both the case when roving pumps are allocated and for the case when the plant is operated in a conventional manner, are very similar. Both cases yielded the same throughput within the time horizon of interest. Therefore, by allocating pumps flexibly throughout the plant it is possible to achieve at least the same throughput as when the plant is operated conventionally. T4 J3

200

200

T3 J3

200

Tasks

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 53

T2 J2

80

T1 J1

120

0

120

2

120

4

6

8

10

12


Figure 12. Optimal production schedule for conventional plant operation


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In order to assess the scalability of the proposed formulation, the first illustrative example was subjected to varying conditions. Both the number of available pumps and the number of time points were varied in order to evaluate the solution time required. A standard pumping rate of 6 seconds per unit material was allocated for each pump. Table 4 shows the achievable objective value and the required solution time when a varying number of pumps are used. Four time points were used for each entry. To evaluate the effect that a varying number of time points has on the required solution time, three standard pumps were used with an increasing number of time points. The scalability results for a varying number of time points may be seen in Table 5. Table 4. Effect of available pumps on the scalability of the first illustrative example Available number of pumps 1 3 5 7 9 11

Objective value (c.u) 35 709 126 753 126 753 126 753 126 753 126 753

CPU solution time (s) 0.6 0.9 1.5 2.8 3.6 5.1

Table 5. Effect of number of time points on the scalability of the first illustrative example Number of time points 1 2 3 4

Objective CPU solution value (c.u) time (s) 0 0.3 35 564 0.6 81 159 0.7 126 753 0.9

Both an increase in the number of available pumps and the number of time points increases the required solution time. However, as seen in Table 4, the number of pumps significantly impacts on the required solution time. This is mainly due to the increased number of required binary variables in the formulation that makes the problem combinatorically complex. 35 ACS Paragon Plus Environment


Page 36 of 53

5.2. Second illustrative example Originally presented by Kondili et al.5, a multipurpose batch plant superstructure is presented that makes provision for a heating unit, two reactors which can perform three different reactions each, followed by a separation unit. The STN representation of the recipe for the given example may be found in Figure 13. Data pertaining to the equipment may be found in Table 6, and data pertaining to the economic aspects may be found in Table 7 as adapted from Kondili et al.5 Pumping data for the eleven pumps present may be found in Table 8. A time horizon of 12 hours was considered.

Product 1 S7

40% HEAT Feed A S1

Hot A S5

Intermediate BC 40% S8 60% R1

10% Impure Product S9

60%

Intermediate AB S6

80%

SEP 90%

20% S4

50% S3

50% R2 Feed B S2

R3 Feed C

Product 2 S10

Figure 13. STN representation for the second illustrative example


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Table 6. Equipment data for the second illustrative example Unit / Storage Vessel Unit J1

20-50 mu

Unit J2

50-70 mu

Unit J3

70 mu

Unit J4 Vessel 1 Vessel 2 Vessel 3 Vessel 4

50-80 mu 30 mu 60 mu 70 mu 100 mu

Capacity

Suitability (Task/Storage)

Task Duration (Hours)

Cost (Cost Units)

HEAT R1 R2 R3 R1 R2 R3 SEP S5 S6 S8 S9

1 + 0.0067mu 2 + 0.0267mu 2 + 0.0267mu 1 + 0.0133mu 2 + 0.0167mu 2 + 0.0167mu 1 + 0.0083mu 2 + 0.0033mu -

100 150

120 150 30 15 10 20

Table 7. Economic data for the second illustrative example Price (Cost units) (Thousands) 0.001 /mu S1 0.002 /mu S2 0.0015 /mu S3 0.0015 /mu S4 0.02 S7 (Product 1) 0.03 S10 (Product 2) 10 Pump base cost 0.001 Pump rating cost 5 Piping cost per connection Material/Equipment



Page 38 of 53

Table 8. Pump ratings for the second illustrative example

Pump A1 A2 A3 A4 B1 B2 B3 B4 B5 B6 B7

Pump Rating (seconds / unit material) 6.00 4.29 4.29 3.75 6.00 4.29 4.29 3.75 4.29 4.29 3.75

Figure 14 and Figure 15 are deconstructed views of the complete flowsheet and pumping connections for the second illustrative example when the pumps are placed according to convention. Due to the large number of piping connections, it is difficult to identify the structure clearly, hence the deconstructed views aid with plant layout representation. A total of eleven pumps are required to service all areas of the plant.


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Storage V1

Unit J1 Heater Unit J3 Reactor

Pump A3

Unit J2 Reactor Pump A1

Pump A2

Unit J4 Separator

Pump A4

Product 1

Storage V2

Storage V3

Storage V4

Product 2

Figure 14. Second illustrative example: conventional pump allocation for pumping from units



Feed State 1 S1f

Storage V1

Feed State 2 S2f

Page 40 of 53

Feed State 3/4 S3f/S4f

Pump B4 Pump B3

Pump B2

Pump B1

Unit J4 Separator

Unit J2 Reactor

Unit J1 Heater

Storage V2

Unit J3 Reactor

Storage V3

Pump B5

Storage V4

Pump B6

Pump B7

Figure 15. Second illustrative example: conventional pump allocation for pumping to units


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As shown in the first illustrative example, Constraints (3) and (4) are key for pump allocation. For the second illustrative example, Constraint (3) is employed as follows, when the destination unit is J1 from source vessel J2, using a pump A1 at a given time point, p.

1, 2, c1, + 2, 1, c1, + 3, 1, c1, + 4, 1, c1, ≤ c1

1, 2, c1, + 3, 2, c1, + 4, 2, c1, ≤ c1

1, 2, c1, + 2, 3, c1, + 4, 3, c1, ≤ c1

1, 2, c1, + 2, 4, c1, + 3, 4, c1, ≤ c1

1, 2, c1, + 2, 1, c1, + 3, 1, c1, + 4, 1, c1, ≤ c1

1, 2, c1, + 2, 2, c1, + 3, 2, c1, + 4, 2, c1, ≤ c1

1, 2, c1, + 2, 3, c1, + 3, 3, c1, + 4, 3, c1, ≤ c1

1, 2, c1, + 2, 4, c1, + 3, 4, c1, + 4, 4, c1, ≤ c1 1, 2, c1, + 2, .2Y, c1, + 3, .2Y, c1, ≤ c1

1, 2, c1, + 2, .3Y, c1, + 3, .3Y, c1, ≤ c1

Constraint (3) is similarly employed to cater for any other destination unit j, from source

vessel vs, using pump , at time point p. Constraint (4) is employed as follows to transfer material to destination storage vessel V2, from source unit J2, using pump A1 at time point p.

2, 2, c1, + Xe 1, 3, c1, ≤ c1

2, 2, c1, + Xe 2, 4, c1, ≤ c1 2, 1, c1, + 1, 1, c1, ≤ c1

2, 2, c1, + 2, 3, c1, ≤ c1

2, 2, c1, + 3, 3, c1, + 3, 4, c1, ≤ c1

2, 2, c1, + 4, 3, c1, ≤ c1

Constraint (4) is similarly employed to cater for any other destination storage vessel vd, from

source unit j, using pump, , at time point p.



5.2.1.

Page 42 of 53

Results

The MILP problem was solved using the CPLEX solver. Figure 16 and Figure 17 show the deconstructed view of the resultant optimal allocation of pumps to the batch plant. A total of three pumps are required, which is significantly fewer than the eleven pumps required when the plant is operated in a conventional manner. Four time points were used in order to find an objective value of 1.8 x 106 cost units. in a CPU solution time of 25.1 seconds.

Feed State 1 S1f

Feed State 2 S2f

Unit J1 Heater


Unit J3 Reactor

Unit J2 Reactor

Unit J4 Separator

Pump B1

Pump A1

Pump B2 Storage V3 Product 1

Storage V2 Product 2

Figure 16. Second illustrative example: optimal pump allocation (pump inlet)


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Feed State 1 S1f

Feed State 2 S2f


Unit J2 Reactor

Unit J3 Reactor Unit J4 Separator

Pump B1

Pump A1

Unit J1 Heater Pump B2

Product 1

Storage V2 Product 2 Storage V3

Figure 17. Second illustrative example: optimal pump allocation (pump outlet) The pumping schedule for the second illustrative example may be found in Figure 18. Similarly to the pumping schedule obtained for the first illustrative example, it leaves the order of material transfer using the same pump to the discretion of the user. The resultant production schedule may be found in Figure 19.


J4 V2 Prod 1 V3 J1 Prod 1

J2

J3

V2

J3

J2

J3

J1

J3

J2

J3

S4f

J3

S3f S2f

Page 44 of 53

J3 J3

J4

V3

J3

J2

J1

J2

S3f

J2

S2f SOURCE

J4

DEST

A1

V3 Prod 2

S1f

J3

B2 B2 B2 B2 B1 B1 B1 B1 B1 B1 B1 B1

J3

A1

J3

A1 A1 A1 A1 PUMP

Pumping Tasks

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

B2


0

2

4

6 Time Horizon (Hours)

8

10

12

Figure 18. Optimal pumping schedule obtained for the second illustrative example


Page 45 of 53

SEP J4

70

R3 J2

Tasks

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


70

R2 J3

70

R2 J2

55

R1 J3

70

R1 J2

50

HEAT J1

50

0

70

28

2

4

6

8

10

12


Figure 19. Optimal production schedule obtained for the second illustrative example Similar to the first illustrative example, the scalability of the proposed formulation on the second illustrative example was evaluated. The effect of the number of available pumps on the second illustrative example may be seen in Table 9, while the effect of a varying number of time points may be seen in Table 10. Table 9. Effect of available pumps on the scalability of the second illustrative example Available number of pumps 1 3 5 7 9 11

Objective value (c.u) 0.4 x 106 1.8 x 106 1.8 x 106 1.8 x 106 1.8 x 106 1.8 x 106

CPU solution time (s) 0.6 3.1 7.7 16.2 18.7 31.6

Table 10. Effect of number of time points on the scalability of the second illustrative example Number of time points 1 2 3 4

Objective CPU solution value (c.u) time (s) 0 0.3 6 0.2 x 10 0.8 0.7 x 106 2.1 6 1.8 x 10 3.1 45

ACS Paragon Plus Environment


Page 46 of 53

The second illustrative example is by its nature more complex than the first illustrative example. Therefore, an increase in the number of pumps and the number of time points have a greater impact on the required solution time for the second illustrative example. Nevertheless, while more significant in the second illustrative example, the required solution times are not excessive. Therefore, in general, more complex multipurpose batch plants with a large number of processing equipment, storage vessels, pumps and piping connections will require longer solution times.


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6. Conclusion A MILP mathematical formulation is introduced which is capable of synthesizing multipurpose batch chemical plants while flexibly allocating pumps to transfer different materials to units during the time horizon. Previously, no consideration has been given to the pumping equipment in the synthesis and scheduling of batch plants. The purpose of the pump allocation is to introduce flexibility within the batch plant, allowing for improved transfer time, better unit utilization, decreased capital and maintenance costs as well as improved batch output. The consideration of pumping equipment is crucial due to the impact on the material transfer time, which limits the effective time available for material processing. Notably, the manner in which the units can be scheduled in order to obtain an optimal objective value is dependent on the availability of transfer equipment and piping connections. When flexibly allocating pumps within the batch plant it is possible to significantly reduce capital investment and maintenance costs by completely eliminating the need for some pumps. The first example evaluated in this work showed a decrease in the number of pumps required from seven pumps to two pumps, while the second example showed a decrease in the number of pumps required from eleven to three. However, the proposed formulation has limitations. The decreased number of pumps makes plant operation more complex than when conventional, fixed pump allocation is used. The proposed model, therefore, allows batch plant synthesis in such a manner that less space and fewer equipment pieces are required at the expense of a more complex processing plant. The decrease in the number of pumps and piping connections makes the plant tightly integrated, which may affect the flexibility of the plant when it is required to adapt to large market fluctuations in the future. It may, therefore, become necessary that the synthesis formulation be reapplied to a specific plant whenever it is required to restructure the plant in order to 47 ACS Paragon Plus Environment


Page 48 of 53

adapt to the market fluctuations. Furthermore, if the materials that are used within the plant are not compatible, cross contamination is possible when different materials are transferred using the same pumping and piping equipment. Therefore, additional costs and time allocation will be required when materials are not compatible and cleaning in place is necessary. In the proposed formulation, it is assumed that the materials are compatible, however, literature models that pertain specifically to cleaning, wastewater minimization and resource optimization may be applied alongside the proposed formulation in order to fully cater for this case. The formulation further assumes that the pumping rate is independent of the material properties. If the pumping rate is significantly affected when transferring various materials, this formulation will be limited in its scheduling accuracy. Nevertheless, by optimally pairing pumps to transfer tasks, it is possible to achieve or improve on the throughput of a conventionally run plant, allowing fewer pumps to be used and greater profitability to be realized.


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7. Nomenclature 7.1. Sets {| is a processing unit/piece of equipment} { | /is a time point} {| is any state/material} {| is any state/material that is consumed} {| is any state/material that is produced} { , | , is a task in a processing unit} ∗ ∗ , | , is an effective task in a processing unit} { # # { , | , is a task which consumes state s}

. . # . 0 . , ∗ . , # . ,

. , V

{ , | , is a task which produces state s} {v| v is any vessel and includes processing units , storage vessels , , source vessels , destination vessels ' as well as material feeds 1 and product stockpiles 0 } { | is a pump}

0

0

Ζ

0

7.2. Binary Variables

= 1 if task , occurs during time point = 0 otherwise = 1 if product from unit will be used in the next time point y = 0 otherwise

, ,

g

, ′, ,

= 1 if material is transferred from vessel to using pump at time point y = 0 otherwise

N ,

, ,

= 1 if vessel V6 y = 0 otherwise = 1 if pump V6 y = 0 otherwise

= 1 if piping connection from vessel to pump exists y = 0 otherwise

= 1 if piping connection from pump to vessel exists y = 0 otherwise



Page 50 of 53

7.3. Continuous Variables

Amount of state s used during task A at time point p

*! RA , S

D , * , , * !( , , *( , , ′ 60 , ,

6! , ,

6( , ' , , , Γ, ,

, , 6-0

, , 6-!

Amount of state s stored at time point p Amount of state s transferred from unit j to storage Amount of state s transferred to unit j from storage Amount of state s transferred between unit j and j’ Time a task ends during time point p Time a task starts during time point p

Transfer time for state s from source to destination using pump Glover transform variable Time that pumping into unit j must end using pump

Time that pumping into unit j may start using pump

Time pumping from a unit may start using pump

6-!!( , , 6-0!( , , 6'!" , ,

!( 6'!" , ,

Time pumping from a unit must end using pump Duration assigned to pump into a unit for a task

Duration assigned to pump out of a unit for a task

7.4. Parameters

C + JK L;0?@ A

L;#?@ A

O,

P,

T T^" 4 2 [ 0 [ 1 [ [`a [-b [b[

Lower design capacity bound for vessel v Upper design capacity for vessel v Initial reserve of stored state s Fraction of material produced from used material Fraction of material consumed from used material

Fixed processing time for task ,

Mass dependent variable processing time of task ,

Time horizon of interest Annualized working hours. (7200 hours) Big M constraint, any sufficiently large number. (100 000) Pumping rate of pump measured in hours/amount material used Price of product states Price of feed states Base cost of pump existence Base cost of pump rating Cost for piping connection from pump to vessel v Cost for piping connection from vessel v to pump Cost for vessel v


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Acknowledgements This work was supported by the National Research Fund (NRF) of South Africa. The authors thank the NRF for their financial support. References (1)

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