optimal design of multipurpose batch facilities with ... · the optimal plant design can be...

Optimal design of multipurpose batch facilities with direct,indirect and solar heat integration

Pedro Miguel de Almeida Simão

Thesis to obtain the Master of Science Degree in

Industrial Engineering and Management

Supervisor: Prof.Tânia Rute Xavier de Matos Pinto Varela

Examination Committee

Chairperson: Prof. José Rui De Matos Figueira

Supervisor: Prof.Tânia Rute Xavier de Matos Pinto Varela

Member of the Committee: Prof.Nelson Chibeles Martins

November 2017

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Abstract

Sustainability is currently one of the main trends among consumers and companies. With the increasing

price of fossil fuels the opportunity to leverage renewable energies keeps getting higher, as they are

getting more efficient, more reliable and less costly.

Therefore, based on the previous work of Pinto et al. (2003), a discrete MILP model is presented, where

solar thermal and heat storage technologies are addressed, with variable heat storage volume and

number of solar collectors, with the aim of profit maximization, providing a valuable information for de-

cision-making process.

Key Words: Multipurpose batch facilities; Heat integration; Solar thermal energy; Energy storage; De-

sign optimization; Scheduling; Costs savings;

Resumo

A sustentabilidade é um dos temas mais presentes entre consumidores e empresas. Com o aumento

do preço dos combustíveis fóssies a oportunidade para alavancar energias renováveis continua a cres-

cer, uma vez que estas são cada vez mais eficientes, consistentes e económicas.

Nesse sentido, baseado no trabalho de Pinto et al. (2003), é apresentado um modelo MILP discreto

com consideração de energia termosolar e armazenamento de calor, onde é calculado o número de

panéis solares a instalar, bem como o volume dos termoacumuladores, com o objectivo final de maxi-

mizar do lucro.

Palavras-chave: Fábricas multiprocesso; Integração de calor; Energia termosolar; Armazenamento de

calor; Optimização; Escalonamento; Redução de custos;

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Table of contents

Abstract............................................................................................................................ ii

Resumo ............................................................................................................................ ii

List of Figures.................................................................................................................... v

List of Tables..................................................................................................................... v

1. Introduction ............................................................................................................ 1

1.1. Background and Context ................................................................................................. 1

1.2. Problem Statement......................................................................................................... 1

1.3. Purpose and Goals .......................................................................................................... 2

1.4. Project Structure............................................................................................................. 2

2. Literature review..................................................................................................... 3

2.1. Introduction ................................................................................................................... 3

2.2. Optimal design................................................................................................................ 3

2.3. Heat integration.............................................................................................................. 4

2.4. Solar heat integration ..................................................................................................... 5

2.5. Conclusions..................................................................................................................... 6

3. Mathematical Formulation ...................................................................................... 7

3.1. Introduction ................................................................................................................... 7

3.2. Sets, parameters and variables........................................................................................ 7

3.2.1. STN design sets described in Kondinli et all. (1988) .......................................................... 7

3.2.2. Heat sets ............................................................................................................................ 7

3.2.3. STN parameters described in Kondinli et all. (1988) ......................................................... 8

3.2.4. Heat integration parameters............................................................................................. 8

3.2.5. Heat storage parameters................................................................................................... 9

3.2.6. Solar thermal parameters.................................................................................................. 9

3.2.7. Objective function parameters.......................................................................................... 9

3.2.8. Binary variables ............................................................................................................... 10

3.2.9. Continuous variables ....................................................................................................... 10

3.3. STN model .................................................................................................................... 11

3.4. Heat-integration ........................................................................................................... 12

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3.4.1. Heat-integration constraints ........................................................................................... 13

3.5. Thermal heat storage .................................................................................................... 15

3.5.1. Thermal heat storage constraints ................................................................................... 16

3.6. Solar thermal energy..................................................................................................... 18

3.6.1. Solar thermal energy constraints .................................................................................... 20

3.7. Conclusions................................................................................................................... 23

4. Case Studies .......................................................................................................... 24

4.1. Introduction ................................................................................................................. 24

4.2. Case studies general data.............................................................................................. 24

4.3. Case 1........................................................................................................................... 27

4.3.1. Characterization .............................................................................................................. 27

4.3.2. Case study 1 results ......................................................................................................... 29

4.3.3. Case 1 summary............................................................................................................... 33

4.4. Case 2........................................................................................................................... 34

4.4.1. Characterization .............................................................................................................. 34

4.4.2. Case study 2 results ......................................................................................................... 36

4.4.3. Case 2 summary............................................................................................................... 40

4.5. Conclusions................................................................................................................... 40

5. Conclusions and Future Work ................................................................................ 41

6. References ............................................................................................................ 42

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List of Figures

Figure 1. Heat integration ...................................................................................................................... 12

Figure 2. Heat-integration task allocation.............................................................................................. 14

Figure 3. Indirect heat integration.......................................................................................................... 15

Figure 4. TES charge and discharge tasks ........................................................................................... 15

Figure 5. TES thermal resistance.......................................................................................................... 16

Figure 6. Solar heat source ................................................................................................................... 16

Figure 7. Fluid temperature rise inside solar collector .......................................................................... 19

Figure 8. Solar collectors in series arrangement................................................................................... 20

Figure 9. Collectors in paralelle arragement ......................................................................................... 20

Figure 10. Product recipe STN.............................................................................................................. 27

Figure 11. Superstructure..................................................................................................................... 28

Figure 12. Optimal capacities and plant scheduling.............................................................................. 29

Figure 13. Energy demand.................................................................................................................... 29

Figure 14. Optimal plant topology ......................................................................................................... 30

Figure 15. Optimal plant scheduling...................................................................................................... 30

Figure 16. Energy balance .................................................................................................................... 31

Figure 17. Optimal plant topology ......................................................................................................... 33

Figure 18. Product recipe STN.............................................................................................................. 34

Figure 19. Superstructure, case 2 ......................................................................................................... 36

Figure 20. Optimal plant scheduling...................................................................................................... 37

Figure 21. Energy balance .................................................................................................................... 38

List of Tables

Table 1. Economic characteristics......................................................................................................... 24

Table 2. Thermodynamic characteristics............................................................................................... 25

Table 3. Size paremeter ........................................................................................................................ 26

Table 4. Task characteristics................................................................................................................. 27

Table 5. Unit characteristics ................................................................................................................. 28

Table 6. Optimal installed capacities..................................................................................................... 30

Table 7. Summary ................................................................................................................................. 33

Table 8. Task characteristics................................................................................................................. 34

Table 9. Unit characteristics, case 2 ..................................................................................................... 35

Table 10. Optimal installed capacities................................................................................................... 37

Table 11. Summary ............................................................................................................................... 40

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Glossary

CST – Concentrating Solar Thermal

HEN – Heat Exchange Network

MEN – Mass Exchange Network

MILP – Mixed Integer Linear Programing

MINLP – Mixed Integer Non-Linear Programing

MOGA – Multi-objective Genetic Algorithms

RTN – Resource Task Network

SSN – Sequence State Network

STN – State Task Network

TAM – Time Average Model

TES – Thermal Energy Storage

TSM – Time Slice Model

1

1. Introduction

At operational level, most of the published literature is focused in heat integration option to reduce en-

ergy in the production process, however other technologies may be studied, such as solar thermal en-

ergy and heat storage, which could result in important operational cost savings. As the cost of this

technologies may account for a significant share of capital investment within the plant budget, they also

should be considered at the design level.

Based on the aforementioned, a formulation for detailed design of multipurpose batch facilities it is pre-

sented, addressing simultaneously direct, indirect and solar heat integration. A comprehensive and ge-

neric MILP formulation model was developed to support the decision making and characterize the opti-

mal solution, which maximize the profit and minimize the cost.

1.1. Background and Context

Industry nowadays faces a large share of high value products with shorter life cycles, requiring flexible

production processes that are resilient to address volatile market needs. Therefore, batch processes

are ideally suited for this situation, characterized by small volume and wide diversity of products that

can be produced through sharing the same equipment in the plant (Halim & Srinivasan, 2009).

Due to the continuing global competition for energy, together with stringent environmental legislations,

batch industries are looking to renewable energies as a mean to reduce both footprint impact and oper-

ational costs.

The main challenge of this solutions is the higher priority for industrial companies to have a reliable

performance of the production process rather than the energy savings reached. Therefore, research to

overcome this barrier and create solid and trustworthy models is required.

1.2. Problem Statement

The optimal plant design can be obtained by solving the following problem:

Given:

The product recipes describing the production of one or more products over a single campaign

structure.

The plant flowsheet with all possible equipment units to be installed.

The units suitability to perform the process/storage tasks.

Task operating temperature.

The utilities requirements and heat-integration options.

The plant heat-transfer units.

The operating and capital cost data involved in the plant/process installation and operation.

The time horizon of planning.

The production requirements over the time horizon.

2

Allowed minimum temperature differences.

Design capacity limits of heat storage unit and solar field.

Solar irradiation profile.

All data related with the solar collector (i.e. type of collector, optical and losses performance

coefficients, concentrating ratio, price).

Determine:

The optimal plant configuration (i.e. number and type of equipment items and their connectivity

including the auxiliary equipment and associated circuits).

A process schedule making use of the selected resources to achieve the required production

(i.e. timing of all tasks, storage policies, batch sizes, amounts transferred and allocation of tasks

to equipment).

Heat-transfer policies.

The optimal direct and indirect plant integration using external fluids.

Utilities requirements.

Number of solar collectors to install.

Heat storage capacity.

So as to optimize the economic performance of the plant, measured in terms of the capital expenditure

and the operating costs and revenues.

1.3. Purpose and Goals

The purpose of this thesis is to fulfill a literature gap by leveraging renewable energy opportunities for

multipurpose batch plants. The goal was to develop a comprehensive optimization model that takes into

account heat storage and solar thermal energy, to design more sustainable production processes in an

accurate way.

1.4. Project Structure

This project is composed of five major chapters. This first chapter describes the background and context

of the problem under study striving to set the baseline for this work. Second chapter provides a literature

review regarding three subjects: design of multipurpose batch facilities; heat integration on multipurpose

batch plants and design of integrated solar thermal energy systems in industrial facilities. Third chapter

outlines the mathematical model and specifies the characteristics and assumptions for each energy

type. Two application examples of the model are provided in chapter four, with key performance metrics

and results.Finally, in the last chapter the main conclusions are presented and future work discussed.

3

2. Literature review

2.1. Introduction

In section 2.1. a comprehensive review of design formulations for multi-purpose batch plants is pre-

sented. In section 2.2. a literature review regarding heat integration on multipurpose batch plants is

presented. Section 2.3. presents a literature review regarding the design of integrated solar thermal

energy systems in industrial facilities. Conclusions for this chapter may be found in section 2.4.

2.2. Optimal design

Grossmann and Sargent (1979) firstly introduced the design problem of batch plants. Other works such

as Papageorgaki and Reklaitis (1990); Pantelides et al. (1993); A P Barbosa-Póvoa and Macchietto

(1994); Barbosa-Póvoa and Pantelides (1997) followed the same course on this assignment problem.

The three different unified frameworks that can be used in optimal design of multipurpose batch plants

are State Task Network (STN), Resource Task Network (RTN) and maximal State Task Network (m-

STN).These methodologies may be formulated in a discrete or continuous time approach. A compre-

hensive comparison between both time models was drawn by Floudas and Lin (2004).

Pinto et al. (2005) presented an improved generic formulation for the optimal design and retrofit of mul-

tipurpose batch facilities operating in a periodic mode, initially proposed by Barbosa-Póvoa and Pan-

telides (1997) where an adapted Resource-Task Network (RTN) methodology was applied, resulting in

high CPU time gains.

Recently a review on the design of batch plants was performed by Barbosa-Póvoa (2007) where both

the grassroots and the retrofit design problems are analysed for both multi-product and multipurpose

plant types, and limitations and gaps are discussed.

The three aforementioned frameworks that can be used in optimal design of multipurpose batch plants

are presented and discussed in Pinto et al. (2008a).

The RTN representation presented in Pinto et al. (2003) considers a single mono-criterion objective

such as profit maximization or minimum makespan. Resorting to the − method, this repre-

sentation is extended in Pinto et al. (2008b) to include more than one economic objective simultane-

ously, such as revenue maximization and cost minimization. Advantages of this method is that it can be

used for any arbitrary problem with either convex or non-convex objective spaces and allows the deci-

sion makers to evaluate the relationship between revenue and cost when designing batch facilities.

The design and retrofit of batch plants operating in a periodic mode presented by Pinto et al. (2005) was

extended in Pinto et al. (2009b) to address demand uncertainty using an adapted form of the Resource-

Task Network (RTN). To handle the uncertainty, a scenario-based planning approach is developed,

resorting to three representative scenarios for the best, worst and expected possibilities with respective

probabilistic data. Also this work was extended by Pinto et al. (2009a) to a multi-objective optimization,

also based on the − method.

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2.3. Heat integration

Papageorgiou et al. (1994) extended the discrete-time scheduling formulation of Kondili et al. (1993)

State-Task Network (STN) where opportunities for both direct and indirect heat integration were consid-

ered for the scheduling problem, as well as possible heat losses from the heat storage tank. The result-

ing model was a nonconvex MINLP problem, for which a global optimum could not be guaranteed.

B. Lee and Reklaitis (1995) developed a model where a MILP formulation determined the operating

schedule for maximum heat integration between the batch units. To avoid generating a complex formu-

lation, it was considered cyclically operating single-product campaigns, with nonzero holding time and

no intermediate storage (NIS).

Boyadjiev et al. (1996) applied direct form of energy integration formulations to an existing antibiotics

manufacture and reported a 40% reduction in overall energy cost. However, in this case, the heat inte-

gration and scheduling problem were solved sequentially.

More recently, Barbosa-Póvoa et al. (2001) extended the work of Barbosa-Póvoa and Macchietto (1994)

to consider heat-integration technologies at the plant design, and consequently, minimize utilities con-

sumptions. The problem is formulated in a discrete-time framework based on the STN and the maximal

State-Task Network (m-STN). However, economic savings were not considered. Later, Pinto et al.

(2003) extended the model to address this gap.

X. Zhao et al. (1998) formulated a batch process scheduling method using mixed integer non-linear

programming (MINLP) with heat integration. A similar formulation of the problem was also used to design

a Heat Exchanger Network (HEN). X. Zhao et al. (1998)

In Fritzson and Berntsson (2006), process integration methods are used to minimize the energy usage

in the slaughtering and meat processing industry. The results for a case study at a modern plant show

that 30% of the external heat demand and more than 10% of the shaft work used in the plant can be

further saved.

In De Boer et al. (2006) three different thermal storage systems were designed, technical and economic

feasibility of integrating heat for each system were evaluated in an existing production facility of organic

surfactants.

T Majozi (2006) embedded heat integration in the continuous-time scheduling framework State-Se-

quence-Network SSN proposed by T Majozi and Zhu (2001) as a MILP formulation solving both sched-

uling and direct heat integration problems simultaneously. The formulation compared to the discrete-

time one results in much smaller number of time points, hence fewer binary variables and suitable for

larger problems. Application of the formulation to an agrochemical facility shows an 18.5% improvement

in profit compared to the situation without heat integration.

Chen and Ciou (2008) proposed a generic method for synthesizing an indirect heat exchange network

and its associated thermal storage policy simultaneously. The problem formulated as a mixed integer

nonlinear program (MINLP) based on a superstructure approach. This work was later extended by Chen

and Ciou (2009) where the constraint of fixed storage temperatures of the recirculating heat transfer

medium is relaxed to increase heat recovery potential.

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In Chen and Chang (2009), a mixed-integer linear program (MILP) is formulated to model short-term

and periodic scheduling problems with direct heat integration. The scheduling formulation is an exten-

sion from the continuous time formulation Resource Task Network (RTN), originally proposed by Castro

et al. (2005), and the direct heat integration formulation is transplanted from the work by T Majozi (2006).

Thokozani Majozi (2009) extended his previous work by including predefined heat storage for more

energy savings and flexibility in process scheduling. Later Stamp and Majozi (2011) addresses the op-

timization of heat storage size and initial temperature of heat storage medium. The resulting model

exhibits MINLP structure, however, a procedure is presented that seeks to find a globally optimal solu-

tion.

Tokos et al. (2010) studied the possibility of energy exchange between batch operations in a large

beverage plant. Heat integration is performed simultaneously by rescheduling an initial operating sched-

ule generated by a MILP formulation of B. Lee and Reklaitis (1995) for maximum heat integration be-

tween the batch units.

Most of the published literature regarding batch plants, only considers heat integration between pro-

cessing units. In Lee et al. (2015) opportunities for energy recovery through heat exchange between

process material streams during transfer are analysed. This approach has the added benefit of short-

ened processing time, which invariably improves throughput, as more batches are likely to be processed

within a given time horizon, compared to conventional heating and cooling in situ. Application of the

proposed model to a case study shows improvements of more than 30% in energy savings and up to

15% in product output. Based on a robust scheduling framework adapted from Seid and Majozi (2012),

the overall model is a mixed integer non-linear program (MINLP), for which global optimality cannot be

guaranteed.

2.4. Solar heat integration

Schnitzer et al. (2007) examined in a case study in an Austrian diary the potential for using solar ther-

mal, with an operating temperature range between 60 and 80ºC. Return on investment is realized after

less than 3 years of implementation, thus, indicates promising technical and economic feasibility of using

solar thermal energy for industrial processes.

Struckmann, Fabio (2008) formulated a mathematical model which describe the thermal performance

of the collector in a computationally efficient manner.

Atkins et al. (2010) analysed, a batch dairy activity supported by solar thermal energy to provide some

of the process heat demand up to 80º of a New Zealand milk powder.

The Muster-Slawitsch et al. (2011) presented a methodology using combinatorial approach for solar

process heat integration and storage system design in batch facilities. To address the variable availa-

bility of solar energy, Nemet et al. (2011) presented a methodology for maximizing solar thermal use. A

MILP formulation is presented, resorting to heat integration methodologies for batch processes based

on Time Slice Model - TSM and considering heat storage.

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A method to design and select banks of solar energy collectors for thermal applications is presented in

Picón-Núñez et al. (2014). The design variables considered in the optimization routine are the number

of collectors in series, number of collector rows, row spacing, and storage volume.

Quijera et al. (2011) evaluates the viability of integrating a solar thermal system to the conventional

energy structure of a dairy plant, operating at low and middle temperatures. This work was extended by

Quijera and Labidi (2013) to consider a combination of pinch and exergy approaches. A payback time

of 8 years was obtained.

In the same year the authors Quijera et al. (2013) applied pinch and exergy analysis methodology in a

tuna fish tinning industrial plant, operating at low and middle temperatures, evaluating different options

of thermal solar energy integration in combination with a heat pump. A payback time of 6 years was

obtained.

Salcedo et al. (2012) addresses the optimal design of desalination plants that integrate a parabolic

trough solar collector. A multi-objective mixed-integer nonlinear programming model (MINLP) is devel-

oped to model such an integrated system and optimize its design.

Picón-Núñez et al. (2014) presented a method for designing medium-temperature solar plants for in-

dustrial process heat from a thermo-economical optimization point of view.

Carpaneto et al. (2015) created an optimisation procedure that can be used at the planning level to find

out the best sizing proportions of solar and conventional sources and for defining the optimal capacity

of storage.

Omu et al. (2016) developed a mixed integer linear programming (MILP) model to facilitate the design

of a solar domestic hot water system.

Wallerand et al. (2016) created a MILP model for optimal design and operation of a solar heated indus-

trial process with constant heating requirements,

2.5. Conclusions

At a design and scheduling level, it is possible to conclude that three different unified frameworks can

be used in optimal design of multipurpose batch plants, namely, State Task Network (STN), Resource

Task Network (RTN) and maximal State Task Network (m-STN).

This methods may be formulated either in a discrete or continuous time approach.

At the heat integration level, since the solutions of the scheduling problem and the heat integration

problem affect each other, researchers started to address both of them simultaneously at the design

level through the above mentioned mathematical formulations, resulting in MILP or MINLP problems.

At the solar energy level, several design models and case study applications for batch facilities are

presented where it is concluded that solar thermal systems have a great potential for low-medium tem-

perature processes, such as those in the food, beverage, diary and some chemical sectors. However,

the integration of solar thermal energy in batch industry is still problematic because as both the demand

and the supply are unsteady and non-continuous.

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3. Mathematical Formulation

3.1. Introduction

This chapter is the main focus of the present work. In section 3.2 all sets, parameters and variables are

defined. Section 3.3 describes the STN model, followed by heat integration model in section 3.4, thermal

heat storage model in section 3.5, solar thermal model in section 3.6. and the objective function in

section 3.7. Conclusions may be found in section 3.8.

3.2. Sets, parameters and variables

For a more comprehensive understanding of the present model, refer to Pinto et al. (2003), as it is the

baseline of the present work. In order to be consistent with the notation previously used, the indices: i,

s, u and j are respectively: tasks, states, utilities and units of STN formulation.

3.2.1.STN design sets described in Kondinli et all. (1988)

Task ∈ = set of processing, material storage, charge and discharge tasks,

Unit ∈ = set of processing, material storage and Thermal Energy Storage (TES) units,

State ∈ = set of states,I = {i: set of tasks which can be performed in unit j},J = {j: set units suitable for task i},I = {i: set of processing tasks},J = {j: set of processing units},J = {j: set of dedicated material storage units},J = {j : set of dedicated storage units suitable for storing state s},K = {k: set of k types of unit j},I /I = {i: set of tasks producing/receiving material to state s},S /S = {s: set of product/feed states}.

3.2.2.Heat sets

Utility u ∈ U = set of utilities,

Heat-transfer unit h ∈ HE = set of units suitable to transfer heat,HI ´, = {h: set of units suitable to transfer heat between tasks (i´, i) ∈ I },I = {u: set of tasks that can use utility u},U = {u: set of utilities suitable for task i},

8

I = {i: set of processing exothermic tasks suitable for integration},I = {i: set of processing endothermic tasks suitable for integration},I = {i: set of tasks that charge TES unit j},I = {i: set of tasks that discharge TES unit j},J = {j: set of TES units},I = {(i´/j´, i/j): set of heat-integrated tasks},Z = {z: set of z volumes of TES unit},Z = {z: set of z numbers of solar panels},

3.2.3.STN parameters described in Kondinli et all. (1988)

M = Big M,p = processing time of task i,ρ , /ρ , = proportion of material of state s entering/leaving task i,∅ , /∅ , = maximum/minimum utilization factor of task i on unit j,V , /V , = maximum/minimum capacity of unit j of type k,∅ , = size factor of state s in storage unit j,Q /Q = minimum/maximum production requirements,

3.2.4.Heat integration parametersB , = maximum availability of utility u at time t,cp = heat capacity of utility u,∆T = temperature difference assumed for utility u,ξ = predefined time offset of task i before integration,Ɋ /Ɓ = fixed/variable heat demand factor of task i at processing time θ,A /A = maximum/minimum heat-transfer area of unit h,p ´, = integration time of tasks (i´, i) ∈ I ,∆Tln ´ = mean fluid logarithmic temperature difference inside unit h when performing integra-

tion of tasks (i´, i) ∈ I ,CU = overall heat transfer coefficient of unit h,

9

3.2.5.Heat storage parameterscp = Heat capacity of storage fluid,T = process temperature of task i,T /T = maximum/minimum TES temperature,V = possible z volumes of TES unit,R = thermal resistance for TES unit with volume z,ρ = density of storage fluid,Y = TES unit overnight heat loss coefficient,∆T = minimum allowable thermal driving force for integration,T = ambient temperature,

3.2.6.Solar thermal parametersG = hourly solar radiation,τα = solar collector optical efficiency,UL = solar collector 1st order heat loss coefficient,A = surface area of one solar collector,B /B = maximum/minimum flowrate inside solar collector k,N = given number z of solar collectors connected to TES unit,∆T = fluid temperature rise inside the solar collector,

3.2.7.Objective function parametersHoursYr = number of annual working hours,

CCF = capital charge factor,v /p = value/price of state s,OC , /OC , = fixed/variable operating cost of task i in unit j,OC /OC = fixed/variable operational cost of solar task,OC = operating cost of dedicated storage of state s,OC = batch size dependent operational cost of utility u,CC , /CC , = fixed/variable capital cost of unit j of type k,CC /CC = fixed/variable capital costs of TES unit,CC /CC = fixed/variable capital costs of heat transfer unit h,CC /CC = fixed/variable capital costs of solar collector of type k,

10

3.2.8.Binary variablesE , = 1, if unit j of a given type k is installed; 0, otherwise,E = 1, if unit j is installed; 0, otherwise,E = 1, if heat-transfer unit h is installed; 0, otherwise,E , = 1, if TES unit j with volume z is installed; 0, otherwise,E , = 1, if z solar collectors are connected to TES unit j; 0, otherwise,W ´, , , = 1 if unit h is performing heat-integration of tasks (i´, i) ∈ I , at the time t; 0, otherwise,

3.2.9.Continuous variablesB , , batch of task i in unit j at the start time t,V capacity of processing unit j ,S , amount of material in state s at the beginning of period t,D , amount of material delivered from state s at the beginning of period t,R , amount of material received from state s at the beginning of period t,B , , utilization level of external utility u by task i at the beginning of period t,Q , heat supplied to task i by external utility u at the beginning of period t ,A heat-transfer area of unit h,Q ’, , , heat transferred between tasks (i´, i) ∈ I , through heat transfer unit h at the beginning of period

t,Q , heat required by processing task i during time t,Q , amount of energy lost from TES unit j during time t,Q , amount of energy stored in TES unit j at the beginning of time t,T , mean temperature of TES unit j at the beginning of time t,T , , mean temperature of TES unit j with volume z at the beginning of time t,η , efficiency of solar collector connected to TES unit j at time t,T , average fluid temperature inside solar collector connected to TES unit j at time t,T , solar collector inlet fluid temperature coming from TES unit j at time t,T , solar collector outlet fluid temperature going to TES unit j at time t,B , mass flowrate supplied to solar field at time t, connected to TES unit j,Q , solar heat charged to TES unit j at time t,

11

3.3. STN model

For more information regarding the STN model, refer to (Kondinli et all. 1988)

Processing unit existence constraints. A processing unit j, if chosen to be installed in the plant, can

only exist in a single given type:

,∈ =∀ ∈ ∪ (1)

The processing unit j existence is linked to the allocation of the suitable tasks according to the following

rules:

(1) at any time point each unit is either idle or processing a single task,(2) tasks cannot be pre-empted once they have been started.

, ,∈ ∩ ≤ 1∀ ∈ (2)

Capacity and batch size constraints. For each unit j the amount of material processed at any time point

must be either zero or within the equipment capacity.0 ≤ , , ≤ ∙ , ,∀ ∈ , ∈ , = 1,… , (3)∅ , ∙ − ∙ (1 − , , ) ≤ , , ≤ ∅ , ∙∀ ∈ , ∈ , = 1,… , (4)

Using the existence variable , , the range of the available capacities for unit j can be defined by

,∈ ∙ , ≤ ≤ ,∈ ∙ ,∀ ∈ ∪ (5)

where , / are the maximum and minimum capacity of unit j of type k. For a discrete size unit

, = , .

Storage constraints. Storage can be modelled either through dedicated storage vessels, where only one

type of material can be stored, or in multipurpose storage vessels where more than one type of materials

can be stored. The latter are modelled as processing units where no material transformation is per-

formed. As for the dedicated vessels we have0 ≤ , ≤ ∅ , ∙∀ , ∈ , = 1,… , + 1 (6)

therefore, at any time, each vessel capacity must accommodate the amount of material stored.

Mass balances. For each state and at any time point t the amount of material present in that state is

related to the amount produced, consumed and transferred by all incident tasks. Thus we have:

12

, = , + , ∙ , ,∈∈ − , ∙ , ,∈∈ − , + ,∀ , = 1,… , + 1 (7)

Production requirement constraints. The production requirements are allowed to float within given upper

and lower bounds. Considering that we may have different plant locations where the same type of ma-

terial can be stored or delivered, an overall production goal, for each product over the time horizon is

defined. ≤ , − , + , ≤∀ (8)

The same procedure is applied for each intermediate deliver and receipt of material. This allows the

possibility of optimizing the production levels as part of the design problem, taking into account the

trade-off between the capital and operating cost and the production added value.

3.4. Heat-integration

To model the heat-integration, tasks are divided in two types:

hot tasks, which include exothermic and (TES) discharge tasks,

and cold tasks, which include endothermic and (TES) charge tasks.

The heat integration must always be performed between a hot and a cold task as shown in Figure 1, all

possible combinations are as follows:

Exothermic and endothermic ;

Exothermic and (TES) charge;

(Endothermic and (TES) discharge;

Figure 1. Heat integration

The heat can be transferred between two units through a dedicated heat exchanger. The type of fluids

and systems to cool/heat the processing units were not addressed in this model. For more guidance

regarding this issue refer to (Pinto et al. 2003).

13

3.4.1.Heat-integration constraints

Global heat supply constraints. The hourly heat Q , required by or removed from each task is based on

a fixed and variable heat demand factors, respectively Ɋ and Ɓ .

Q , = Ɋ W , , +Ɓ B , ,∀ ∈ , = 1,… , (9)

The heating or cooling of each task is satisfied primarily by heat-integration ´, , , , either in a direct or

indirect form, and additionally if required, by external utilities , , .For exothermic tasks: Q ´, = ´, + ´, , ,∈ ´,Ɐ ´ ∈ , = 1,… , (10)

For endothermic tasks: Q , = , + ´, , ,∈ ´,´Ɐ ∈ , = 1,… , (11)

Adapted from Stamp and Majozi (2011)

External utilities constraint. Given the utility heat capacity as well as its assumed temperature

difference ∆ , necessary quantity , , to cool/heat task i is given by

, = , , ∙ ∙ ∆ ,∀ ∈ , = 1,… , (12)

limits on the utilities availability are ensured by0 ≤ , ,∩ ≤ ,Ɐ , = 1, … , (13)

Heat-integrated task activation constraints. If a task is undergoing a heat-integration is should be

activated according to the following rules:

1) start within a pre-defined time offset before undergoing a heat-integration,

2) if active, it may operate in either integrated or standalone mode,

3) can only be integrated with only one compatible task at any time point.

14

For exothermic tasks: W ´, , ,∈ ´, ≤ W ´, ´, ´´∈ ´∀ ´ , = 1,… , (14)

For endothermic tasks: W ´, , ,´∈ ´, ≤ W , ,∈∪∀ , = 1,… , (15)


Heat-transfer unit allocation constraint. The allocation of heat transfer unit t is done according to the

following rules:

(1) at any time each heat-transfer unit h is either idle or processing a single heat-integration task,

(2) heat-integration cannot be stopped once it has been started.

W ´, , ,´´ ≤ 1

∀ℎ ∈ ´, , ( ´, ) , = 1, … , (16)

In Figure 2 is shown the heat-integration task allocation with time off-sets for processing tasks.

Figure 2. Heat-integration task allocation

Heat-integration equipment design constraints. The required heat transfer units to be installed in the

plant and their optimized heat-transfer areas are given by:∙ ∙ ∆Tln ´, ⩾ ’, , ,∀ℎ ∈ ´, , = 1, … , (17)∙ ≤ ≤ ∙∀ℎ (18)

1 2 3 4 5 6 7Time t

, ,´, ´, ´, , ,

’ = 2

= 1’ = 2

Hot task i´, unit j´

Cold task i, unit j

Heat-integration, unit h

15

Since it is possible to accomplish the heat-transfer, then it is necessary to guarantee that the associated

integration task occurs.

’, , , − ∙ W ´, , ,´ ≤ 0∀ℎ ∈ ´, , = 1, … , (19)

3.5. Thermal heat storage

As mentioned before, each TES unit can only exchange heat with one processing at each time point,

through a dedicated heat exchanger,Figure 3

Figure 3. Indirect heat integration

contrarily to processing tasks, TES unit has a variable temperature, which depends on the amount of

energy stored. This temperature is assessed at each time point to understand which processing tasks

it can be integrated with. A minimum temperature difference ∆T must exist so that heat can be trans-

ferred, as shown in Figure 4.

Figure 4. TES charge and discharge tasks

One must also consider the heat loss, which depends on thermal resistance Figure 5

∆T

Charge task

Tem

pera

ture

(ºC

)T´ Exothermic task

timet t+p ∗

timet t+p ∗T ∆T

Endothermic taskTem

pera

ture

(ºC

)

Discharge task

16

Figure 5. TES thermal resistance

It is also considered a possible solar heat source, connected to each TES unit, to increase hot utility

savings, Figure 6. The fluid inside the tank, water in most cases, can be pumped through the collectors,

absorbing heat from the sun, and then returning to the storage tank. The complete solar thermal energy

mode will be further discussed in section 3.6 .

Figure 6. Solar heat source

3.5.1.Thermal heat storage constraints

TES unit allocation. Each TES unit j can only be integrated with one exothermic or endothermic task

at any time point. Also the indirect heat-integration cannot be stopped once it has been started.

W ´, , ,∈´ + W ´, , ,∈´∈∈ ´, ,´ ≤

∀ ∈ , = 1,… , (20)


TES thermal resistance

= + + +R

RR

Heat loss

17

Energy balance constraints. At any time point the amount of energy stored in TES unit j is the sum

of all the energy charged, discharged and lost.

, = , + ’, , ,∈´ − ’, , ,∈´∈∈ ´, , − , + ,∀ , = 1,… , + 1 (21)

Adapted from Omu et al.(2016)

The initial heat stored is lower or equal to the previous day minus the overnight heat loss.

, ≤ , ∙ (1 − )∀ (22)

TES temperature constraint. TES temperature is mostly dependent on the heat stored and the volume

of the tank. = ∙ ∙ ∙ ( − )The current work aims to have this volume as a design variable, however it would result in a non-linear

problem duo to the variable temperature. To address this, the following linear transformation was devel-

oped, where the installed volume is introduced as a parameter, and the model selects the optimum

within given multiple possibilities, like a discrete variable.

, = ∙ ∙ ∙ ( , , − ∙ , )∀ , = 1,… , + 1 (23)

, = , ,Ɐ∀ , = 1,… , + 1 (24)= ,

∀ (25)

Where , is a binary variable that indicates which storage tank volume should be installed.

Heat loss constraint. The hourly heat loss is dependent on both temperature and thermal re-

sistance of TES unit j.

, = ( , , − ∙ , )∀ , = 1,… , (26)

18

TES temperature limits constraints. The temperature inside TES unit j at any time point must be

either zero or within the equipment capacity.

, ∙ ≤ , , ≤ , ∙∀ , , = 1, … , + 1 (27)

The minimum thermal driving forces must be obeyed during an indirect heat-integration.

For charge tasks:

T ´ − , ≥ ∆ − ∙ (1 − W ´, , ,∈ ´,´ )

∀ ∈ , , = 1, … , + 1 (28)

For discharge tasks:

, − T ≥ ∆ − ∙ (1 − W ´, , ,∈ ´,´ )

∀ ´ ∈ , , = 1, … , + 1 (29)

In case solar thermal ernergy is also considered, due to the charge effect, the following restriction must

be introduced for the endothermic task :

, − T ≥ ∆ − ∙ (1 − W ´, , ,∈ ´,´ )

∀ ´ ∈ , , = 1, … , + 1 (30)

3.6. Solar thermal energy

This section tackles the selection of solar thermal energy, where the number of collectors connected to

each TES unit j is assessed, as well as how much energy will be collected at each time point.

This approach is to provide some of the process heat demand with solar thermal energy, which is cap-

tured whenever it is available. The solar profile, Graphic 1, is therefore the critical factor for determining

hot utility savings and hence the cost benefit of using a solar collector system. (Atkins et al. 2010)

19

Graphic 1- Hourly solar profile

The collector modeled in this work is the flat plate collector, usually suitable for direct solar energy rather

than diffuse energy, it is a low-medium energy system, most commonly working between 0 and 120ºC

Struckmann (2008). There are other types of technologies, such as concentrating solar, that can be

further studied and adapted.

In literature there are only two ways to operationally run the collector, either a fixed outlet fluid temper-

ature is obtained by manipulating the mass flowrate through the collector, or the most commonly used,

a fixed mass flowrate is supplied with a variable outlet temperature Atkins et al. (2010). The latter focus

on maximizing the amount of solar energy recovered by keeping a fixed mass flowrate high with a vari-

able outlet temperature low, ignoring however the pumping costs. To better address both issues, in this

work an alternative method was considered, where a fluid temperature rise is considered inside the

collector, which will allow both variable mass flowrate and outlet temperature.

Figure 7. Fluid temperature rise inside solar collector

Pressure drop inside the pipes is not considered in this model, further studies can be performed to

assess this effect.

The operating conditions depend on the arrangement of the solar collectors and respective associated

costs. In a series arrangement, Figure 8, the outlet of each collector becomes the inlet stream of the

next one, generally when many collectors are in series the pressure drop through circuit is high and the

collector efficiency of the last panels is low.

4 6 8 10 12 14 16 18 20

kW/m²

0.6

0.5

0.4

0.3

0.2

0.1

0

Sola

rrad

iatio

n

Hour

Collector ∆

20

Figure 8. Solar collectors in series arrangement

In parallel arrangement Figure 9, the fluid would pass through only one collector, leading to a lower

temperature and higher efficiency for the same flowrate.

Figure 9. Collectors in paralelle arragement

As the objective in this model is to maximize the total heat collected by the solar panels instead of

achieving a target temperature, the parallel arrangement is assumed. As such the inlet and outlet tem-

peratures for all the collectors are the same.

3.6.1.Solar thermal energy constraints

Solar technology existence constraint. If the solar technology is installed in TES unit j, then this unit

must exist. ≤∀ (31)

Collector efficiency constraint. The collector efficiency , varies with the hourly solar radiation ,

ambient air temperature and average collector internal fluid temperature ,, = τα − UL ∙ ( , − )

∀ , = 1,… , (32)

The optical efficiency τα and thermal loss coefficients UL are generally determined experimentally and

their relative contributions depend mainly on the physical design of the collector selected. (Atkins et al.

2010)

Collector internal fluid temperature constraints. For a first order analysis, the collector internal fluid

temperature is assumed to be the linear average of the inlet , and outlet , temperatures (Atkins

2010)

, = , + ,2

21

∀ , = 1,… , (33)

Inlet fluid temperature , during a certain time period, must be equal to the average TES temperature

during that period.

, = , + ,2∀ , = 1,… , (34)

Solar heat constraints. The solar heat collected by one panel depends on its surface area and

efficiency, as well as the hourly solar radiation. ≤ ∙ ∙In this work, one aimed to have as design variable the number of installed collectors, however this would

result in a non-linear problem duo to the variable collector efficiency. To address this the following linear

transformation was developed, where the number of installed collectors is introduced as a pa-

rameter and the model selects the optimum within multiple given possibilities:

, ≤ ∙ ∙ ∙ , + ∙ (1 − , )∀ , , = 1, … , (35)The binary , , selects number of collectors connected to TES unit j.= ,∀ (36)

The amount of solar heat collected also depends on the mass flowrate , for a given fluid temper-

ature rise ∆ inside the collector.

, = , ∙ ∙ ∆∀ , = 1,… , (37)∆ = , − ,∀ , = 1,… , (38)

Adapted from Stuckmann (2008)

Mass flowrate limits constraints. The mass flowrate at any time point must be either zero or within

the collectors’ capacity. ∙ ∙ , ≤ , ≤ ∙ ∙ ,∀ , = 1,… , (39)

22

Objective Function

The most profitable solution is obtained by subtracting to the yearly product revenue (PR) all operation

costs (OC*), raw material costs (RMC) and the opportunity cost of capital of all installed units (CC*). The

objective function is as follows:max = ( − − − − − − ) ∙ − ( ++ + ) ∙ (40)

Where HoursYr denotes the number of annual working hours and CCF the capital cost factor.

Product revenue. This equation takes into account the total final product produced, which is the

deference between the final and initial quantities , − , , the sold quantity , , and the selling

price ( ).

Raw materials costs. This equation takes into account the total raw material consumed, which is the

deference between the initial and final quantities , − , , the purchased quantity , , and the

purchase price ( ).

STN operational costs. A fix cost factor ( , ) is considered for task activation and two batch depend-

ent cost factors are considered, one for processing task i ( , ) and another for storing material s ( ).

Operational utilities costs. This equation takes into account the total external utility consumed , , and

its cost .

Heat-integration operational costs. A fix operational cost factor ( ´, , ) is considered, related to

the activation of the heat integration task, and a variable operational cost factor is assumed ( ´, , )

related to amount of energy transferred between tasks (í,i).

Solar operation costs. This equation takes into account the total water pumped through the solar pan-

els , and the pumping costs .

= , − , ∙ + , ∙∈ (41)

= , − , ∙ + , ∙∈ (42)

= , ∙ , , + , ∙ , , + ∙ ,∈∈ (43)

= ∙ , , (44)

= ´, , ∙ W ´, , ,∈ ´, + ´, , ∙ Q ´, , ,´ (45)

23

STN capital costs. A fix ( , ) and a volume dependent cost factor is considered to install

unit j of type k.

Heat storage capital cost constraint. A fix ( ) and a volume dependent ( ) cost factor is

considered to install TES unit j.

Heat-integration capital costs. A fix ( ) and an area dependent ( ) cost factor is considered

to install heat transfer unit h.

Solar capital cost constraint. A fix ( ) and a quantity dependent ( ) cost factor is con-

sidered to install the solar collectors.

3.7. Conclusions

A detailed mathematical framework for the design of multipurpose batch facilities with direct, indirect

and solar heat-integrations has been presented. Optimal plant configuration and operation are obtained

as well as optimal plant heat-transfer policies, heat-transfer units associated areas, heat storage vol-

umes, and number of solar collectors. Important aspects are considered simultaneously, such as plant

topology – the choice of the plant equipment – and plant operation, where the final scheduling is per-

formed accounting for a set of heat-transfer policies. The problem was formulated through a mixed

integer linear programming were binary variables define equipment choices and operability, and contin-

uous variables define the equipment capacities and the amounts of material within the overall process.

A new approach to bypass MINLP models was proposed, where size parameters were introduced to

replace unwanted variables. Further work can be developed to assess the performance of this approach

for complex cases.

= , ∙ (46)

= ( , ∙ , + ∙ )∪ (47)

= ( + ∙ ) ∙ , (48)

= ∙ , + ∙ (49)

= , ∙ ( + ∙ ) (50)

24

4. Case Studies

4.1. Introduction

Two illustrative case studies are presented to show the model applicability.

General data such as cost of utilities, price of products, or thermodynamic characteristics is presented

in section 4.2 and will be the same for both cases. In section 4.3 a simple case with two reactors and

one TES unit is presented, followed by a more complex one with five reactors and two TES units in

section 4.4. Conclusions are presented in section 4.5.

The General Algebraic Modelling System (GAMS) was used. All problems were solved with 0% margin

of optimality on an Intel core i5, 2.6 GHz.

4.2. Case studies general data

As aforementioned, the following general data will be the same for both cases. Economic characteristics

are presented in Table 1. Final products have a selling price of 100 currency units per ton (c.u./ton),

while raw materials have a purchase price of 5 c.u./ton. External utilities are steam and water, costing

10 and 2 c.u./kWh, respectively. Additionally, it is considered a heat transfer cost of 1 c.u./ kWh ex-

changed between two units, and for the solar system, a pumping cost of 5 c.u./ton.

It is assumed 3000 working hours per year and a capital cost factor of 33%.

Table 1. Economic characteristics

Parameter Value

Selling price of ∈ 100 c.u./ton

Purchase price of ∈ 5 c.u./ton

Steam cost 10 c.u./kWh

Cooling water cost 2 c.u./kWh

Task activation 20 c.u.

Heat transferred 1 c.u./kWh

Solar pumping costs 5 c.u./(ton)

Hours/Year 3000 h/year

Capital cost factor 33 %

c.u. = currency units

25

Thermodynamic characteristics are presented in Table 2.

Regarding heat integration, a minimum driving force of 10ºC is considered so that heat can be trans-

ferred, the mean logarithmic temperature difference between the two fluids inside the heat exchanger

for all pair of tasks (i´,i) is also 10ºC and all heat exchangers are considered to have a transfer coefficient

of 3.6 kWh/m2.

Regarding heat storage, no hear losses were considered, all TES units have a temperature upper limit

of 100ºC and a storage fluid with a heat capacity of 1.19 kWh/ ton.ºC.

Regarding solar energy, all collectors have a surface area of 1 m2, an optical efficiency τα of 86%, a

heat loss factor UL of 0.048 kWh and an assumed internal fluid temperature rise of 20ºC.

Table 2. Thermodynamic characteristics

Parameter Value∆ ºC 10∆ ´ (all i´,i) ºC 10

kWh/m2 3.6

ºC 100

ºC 25

kWh/ton.ºC 1.19

m2 1

% 86.1

kWh 0.0048∆ ºC 20

Solar profile is presented in Graphic 2

Graphic 2. Solar profile

0,4

0,5

0,6

0,68 0,68

0,6

0,5

0,4

0,3

0,4

0,5

0,6

0,7

0,8

0 1 2 3 4 5 6 7 8

Sola

r rad

iatio

n (k

Wh/

m2 )

time t

26

Table 3. Size paremeter

Size parameter (m³) (un)Z1 1.2 10Z2 1.4 20Z3 1.6 30Z4 1.8 40Z5 2 50Z6 2.2 60Z7 2.4 70Z8 2.6 80Z9 2.8 90Z10 3 100Z11 3.2 110Z12 3.4 120Z13 3.6 130Z14 3.8 140Z15 4 150Z16 4.2 160Z17 4.4 170Z18 4.6 180Z19 4.8 190Z20 5 200Z21 5.2 210Z22 5.4 220Z23 5.6 230Z24 5.8 240Z25 6 250Z26 6.2 260Z27 6.4 270Z28 6.6 280Z29 6.8 290Z30 7 300Z31 7.2 310Z32 7.4 320Z33 7.6 330Z34 7.8 340Z35 8 350Z36 8.2 360Z37 8.4 370Z38 8.6 380Z39 8.8 390Z40 9 400

27

Both size parameters for TES volume and number of solar collectors are shown in

Table 3. As the upper and lower limits for TES volume are respectively 1 m3 and 9 m3, this range wasdivided in 40 intervals with the same length. The same approach was applied to the number of solarcollectors, which can vary between 10 un and 400 un.

4.3. Case 1

4.3.1. Characterization

A multipurpose batch plant must be designed at a maximum profit so as to produce two main products

S4 and S3 respectively with 300 to 350 ton and 350 to 400 ton. The STN product recipe may be found

in Figure 10.

Figure 10. Product recipe STN

Task characteristics may be found in Table 4.

Exothermic task T1 operates at 120ºC and requires cooling from external utility (water), with a fix/varia-

ble rate of 7 kWh and 0.5 kWh/ton respectively. Endothermic tasks T2, T4 and T5 require steam with a

fix/variable rate of 4 + 0.3 kWh, 8 + 0.9 kWh and 6 + 0.4 kWh and operate at 100ºC, 60ºC and 70ºC

respectively. One TES unit is considered and may have charge and discharge tasks, C1 and D1, that

operate within its temperature limits (25ºC to 100ºC).

Table 4. Task characteristics

Task Type Heating/Cooling re-

quirements (kWh)

External

utility

Operating temperature

(ºC)

T1 Exothermic 7+0.5 B Water 120

T2 Endothermic 4+0.3 B Steam 100



C1 TES charge - - 25 -100

D1 TES discharge - - 25 -100

28

The equipment design and operational characteristics are shown in Table 5 and the superstructure is

presented in Figure 11.

Processing units R1 and R2 are multipurpose in nature and can perform processing and storage tasks.

Storage vessels V1 to V4 have unlimited capacity and are suitable to store a dedicated state. Heat

storage unit TES1 can perform a charge/discharge tasks C1/ D1. All units may be integrated with each

other, heat exchanger H1 can transfer heat between TES1 and R1, heat exchanger H2 can transfer

heat between TES1 and R2 and heat exchanger H3 can transfer heat between units R1 and R2. Addi-

tionally a solar field SOL1 may be connected to unit TES1.

Table 5. Unit characteristics

Unit Suitability Capacity Costs: Fix: Var

(103 c.u.)

R1 Process task T1/T4 40:300 m³ 5:0.05

R2 Process task T2/T5 70:300 m³ 5:0.05

V1 Store S1 Unlim. m³ 3:0.01




H3 Transfer heat between R1-R2 0:15 m² 5:1

H1 Transfer heat between R1-TES 0:15 m² 5:1

H2 Transfer heat between TES-R2 0:15 m² 5:1

TES Perform task C1/D1 1:9 m³ 5:1

SOL1 Connect to TE1 10:400 un 5:1

Figure 11. Superstructure

29

4.3.2. Case study 1 results

For this case two scenarios are compared, Scenario 1 with no heat integration and Scenario 2 with

direct, indirect and solar heat integrations.

Scenario 1 – No heat integration considered

The optimal plant scheduling is presented in Figure 12, both reactors R1 and R2 are installed with a

total capacity of 287 m³ and 175 m³, respectively, and the total production of state s3 and s4 is respec-

tively 400 and 350 tons. The total heating/cooling requirements from external utilities are 1178 kWh of

steam and 604 kWh of water, Figure 13. The optimal plant topology is shown in Figure 14, where is

also possible to see that storage vessels V1-V4 also were installed.

Figure 12. Optimal capacities and plant scheduling

Figure 13. Energy demand

30

Figure 14. Optimal plant topology

Scenario 2 - Direct, indirect and solar heat integrations considered

A new optimal plant scheduling is presented in Figure 15, where is possible to see which tasks are

allocated to each unit and the size of the processed batch. The total production of state s3 and s4 is

remained 400 and 350 tons respectively.

Figure 15. Optimal plant scheduling

The optimal installed capacities are presented in Table 6.

The capacity of R1 and R2 increased 4% and 67% respectively. Heat exchangers H1 and H3 were

installed, with respective heat transfer areas of 6.48 m2 and 1.32 m2. TES unit was also installed with a

capacity of 4.8 m³, connected to 400 solar units.

Table 6. Optimal installed capacities

Unit ∆ Optimal Capacity

R1 +4% 296 m³

R2 +67% 292 m³

H1 - 6.48 m²

H3 - 1.32 m²

TES1 - 4.8 m³

SOL1 - 400 un

31

In Figure 16 a complete energy balance is shown.

The values associated to arrows represent heat exchanged between units, all energy values with plus

sign (+) represent heat sources, all values with minus sign (-) represent heat sinks and all values with

no sign represent heat stored.

It is possible to see that whenever the total heat transferred to TES unit is positive, the stored energy in

TES increases, and vice-versa. Task T1 provides 293 kWh to TES unit, through heat exchanger H1,

during time points 1-2. In this integration T1 is auto-sufficient and does not require any cooling from

external utilities to operate. Tasks T1 and T2 exchange 95 kWh at time point 5-6, through heat ex-

changer H3, where T2 is auto-sufficient and T1 requires additional 214 kWh of cooling water.

Task T4 is heated by TES unit during time points 3-4 and 7-8, receiving respectively 363 and 285 kWh.

Also, during the first integration, T4 is not auto-sufficient, requiring additional 103 kWh of steam. T5 does

not undergo any integration. In all time periods solar collectors SOL are active, providing between 30

and 128 kWh of heat to TES unit.

Figure 16. Energy balance

32

Graphic 3 shows TES temperature variation.

During the integration with T1, TES temperature does not exceed its limit of 100ºC, also when integrated

with T4, remains above 70ºC, as T4 operates at 60ºC and the minimum driving force is assumed to be

10ºC.

Graphic 3. TES temperature

Graphic 4, shows the efficiency variation of the solar collectors through time. The three high efficiency

peaks are coincident with the lowest TES temperature values. This is because higher temperatures lead

to greater heat losses inside the solar panel.

Graphic 4. Efficiency of solar collectors

The proposed plant topology is shown in Figure 17. Where is possible to see that storage vessels V1-

V4 were also installed.

33

Figure 17. Optimal plant topology

4.3.3.Case 1 summary

The result summary regarding Case 1 is shown in Table 7.

The final objective function took the value of 24 297 ∙ 10 currency units. Total profit improved 11.3%,

while external steam and water requirements dropped 67% and 52% respectively. The problem in study

is characterized by 184 integer variables, 1696 constraints and was solved in 9.85 CPU seconds.

Table 7. Summary

Scenario 1 Scenario 2 ∆Profit (c.u 10³) 21 830 24 297 + 11.3%

External utilities

- Water (kwh) 604 194 - 67%

- Steam (kwh) 1178 564 - 52%

Model statistics

- CPU time (s) 0.23 9.85

- Integer Variables 56 184

- Constraints 336 1696

34

4.4. Case 2

4.4.1.Characterization

A multipurpose batch plant must be designed at a maximum profit so as to produce three main products

S7, S8 and S10 respectively with 300 to 350 ton, 200 to 250 ton and 200 to 250 ton. The STN product

recipe may be found in Figure 18

Figure 18. Product recipe STN

Task characteristics may be found in Table 8

Exothermic tasks T1, T3 and T6 operate respectively at 80ºC, 120ºC and 150ºC, requiring cooling from

external utility (water). Endothermic tasks T2, T4, T5 and T7 require steam and operate at 70ºC, 60ºC,

70ºC and 80ºC respectively. Two TES units are considered and may have charge and discharge tasks

that operate within its temperature limits

Table 8. Task characteristics

Task Type Heating/Cooling re-quirements (kWh)

Operating temperature(ºC)

T1 Exothermic 7 + 0.5 B 80T2 Endothermic 4 + 0.6 B 70T3 Exothermic 9 + 1.3 B 120T4 Endothermic 8 + 0.9 B 60T5 Endothermic 6 + 0.8 B 70T6 Exothermic 9 + 0.9 B 150T7 Endothermic 2 + 0.2 B 80C1 TES1 charge - 25 -100D1 TES1 discharge - 25 -100C2 TES2 charge - 25 -100D2 TES2 discharge - 25 -100

The equipment design and operational characteristics are shown in Table 9 and the superstructure is

presented in Figure 19.

35

Processing units R1, R2, R3 R4, R5 and R6 can perform processing and storage tasks, from which units

R3 and R6 are multipurpose in nature. Storage vessels V1 to V4 and V7 to V10 have unlimited capacity

and are suitable to store a dedicated state. Heat storage units TES1 and TES2 can perform charge/dis-

charge tasks C1/ D1 and C2/D2, respectively.

The pair of units R1-R2, R3-TES1, R4-TES1, R5-TES2 and R6-TES2, may undergo heat integration

through the respective heat exchanger H1, H3, H4, H5 and H6. Additionally solar fields SOL1 and SOL2

may be connected to unit TES1 and TES2 respectively.

Table 9. Unit characteristics, case 2

Unit Suitability Capacity Costs: Fix: Var (103 c.u.)R1 Process task T1 0:150 m³ 5:0.05R2 Process task T2 0:150 m³ 5:0.05R3 Process task T3/T6 0:400 m³ 5:0.05R4 Process task T4 0:150 m³ 5:0.05R5 Process task T5 0:150 m³ 5:0.05R6 Process task T6/T7 0:200 m³ 5:0.05V1 Store S1 Unlim. m³ 3:0.01V2 Store S2 Unlim. m³ 3:0.01V3 Store S3 Unlim. m³ 3:0.01V4 Store S4 Unlim. m³ 3:0.01V7 Store S7 Unlim. m³ 3:0.01V8 Store S8 Unlim. m³ 3:0.01V9 Store S9 Unlim. m³ 3:0.01V10 Store S10 Unlim. m³ 3:0.01H1 Transfer heat between R1-R2 0:15 m² 5:1H3 Transfer heat between R3-TES1 0:15 m² 5:1H4 Transfer heat between R4-TES1 0:15 m² 5:1H5 Transfer heat between R5-TES2 0:15 m² 5:1H6 Transfer heat between R6-TES2 0:15 m² 5:1TES1 Process task C1/D1 1:8 m³ 5:1TES1 Process task C2/D2 1:9 m³ 5:1SOL1 Connect to TES1 10:400 un 5:2SOL2 Connect to TES2 10:400 un 5:2

36

Figure 19. Superstructure, case 2

4.4.2.Case study 2 results

For this case only Scenario 2 with direct, indirect and solar heat integrations, is presented, and the final

key results are compared to Scenario 1, with no integration. The optimal plant scheduling is presented

in Figure 20, where is possible to see which tasks are allocated to each unit and the size of the pro-

cessed batch. The total production quantites for state s7, s8 and s10 are respectively 300, 250 and 250.

The optimal installed capacities are presented in Table 10. All processing units R1, R2, R3, R4, R5 and

R6 were installed, with the respective capacities of 128, 85, 225, 150, 150 and 200 m³. All heat exchang-

ers H1, H3, H4, H5 and H6 were also installed with a respective area of 0.55, 2.07, 1.43, 1.26 and 1.67.

Also both units TES1 and TES2 were installed with a volume of 5.8 and 9 m3 and are connected to 260

and 390 solar panels. The proposed plant topology results in the superstructure presented in Figure 19,

as all possible units were installed.

37

Figure 20. Optimal plant scheduling

Table 10. Optimal installed capacities

Unit Optimal Capacity

R1 128 m³

R2 85 m³

R3 225 m³

R4 150 m³

R5 150 m³

R6 200 m³

H1 0.55 m²

H3 2.07 m²

H4 1.43 m²

H5 1.26 m²

H6 1.67 m²

TES1 5.8 m³

TES1 9 m³

SOL1 260 un

SOL2 390 un

38

In Figure 21 a complete energy balance is shown. The values associated to arrows represent heat

exchanged between units, all energy values with plus sign (+) represent heat sources and all values

with minus sign (-) represent heat sinks. Task T3 provides 204 kWh and 207 kWh to unit TES1, through

heat exchanger H3, during time points 3-4 and 6-7 respectively. In the first integration T3 is auto-suffi-

cient and does not require any cooling from external utilities to operate. In the second one additional 95

kWh of cooling water are required. Tasks T1 and T2 exchange directly 110 kWh at time points 1-2 and

4-5, through heat exchanger H1, where T2 is auto-sufficient and T1 requires additional 31 kWh of cooling

water. Task T4 is heated by unit TES1 during time points 4-5 and 7-8, it is auto-sufficient and receives

in both times 286 kWh. T5, T6 and T7 integrates with unit TES2 at time points 7-8, 3-4 and 5-6 respec-

tively, are self-sufficient, receiving/providing all required heat to operate, respectively 232, 332 and 244

kWh. SOL1 provides between 23 and 85 kWh to unit TES1 and was not active in time point 6. SOL2

provides between 43 and 114 kWh to unit TES2 and is always active.

Figure 21. Energy balance

39

Graphic 5 show the temperature variation of TES1.

During the integration with task T3, at time points 6-7, TES does not exceed its temperature limit of

100ºC. Also when integrating with task T4 at time points 4-6 and 7-9, TES temperature remains above

70ºC, as task T4 operates at 60ºC.

Graphic 5. Temperature of TES1

Graphic 6 show the temperature variation of TES2.

During the integration with task T6, between time points 3-5, TES temperature does not exceed its limit

of 100ºC. Also when integrating with task T7, during time point 5-7, TES temperature remains above

90ºC, as T7 operates at 80ºC. The same happens at time point 7-9, during the integration with task T5,

that operates at 70ºC and TES stops the integration with 80ºC.

All minimum driving forces were respected.

Graphic 6. Temperature of TES2

40

4.4.3.Case 2 summary

The result summary regarding Case 2 is shown in Table 11.

The final objective function took the value of 24 890 ∙ 10 currency units. Total profit improved 19%,

while external steam and water requirements dropped 61% and 77% respectively. The problem in study

is characterized by 324 integer variables, 3333 constraints and was solved in 180 CPU seconds.

Table 11. Summary

Scenario 1 Scenario 2 ∆Profit (c.u 10³) 21 009 24 890 + 19%

External utilities

- Water (kwh) 1 565 609 - 61%

- Steam (kwh) 1 674 387 - 77%

Model statistics

- CPU time (s) 0.62 180

- Binary Variables 95 324

- Constraints 628 3 333

c.u – currency units

4.5. Conclusions

Two cases were presented in this chapter, in both the model proven to be working accurately and

showed significant optimization opportunities.

Additionally the alternative to use the size parameter to avoid MINP problems seems to be an option to

further explore.

41

5. Conclusions and Future Work

A detailed mathematical framework for the design of multipurpose batch facilities with direct, indirect

and solar heat-integrations has been developed and tested. Optimal plant configuration and operation

are obtained as well as optimal plant heat-transfer policies, heat-transfer units associated areas, heat

storage volumes, and number of solar collectors. Important aspects were considered simultaneously,

such as plant topology – the choice of the plant equipment – and plant operation, where the final sched-

uling is performed accounting for a set of heat-transfer policies. The problem was formulated through a

mixed integer linear programming were binary variables define equipment choices and operability, and

continuous variables define the equipment capacities and the amounts of material within the overall

process. A new approach to bypass MINLP models was proposed, where size parameters were intro-

duced to replace unwanted variables. Further work can be developed to assess the performance of this

approach for more complex models.

For the remaining subjects, further work can be developed to include heat exchangers with one fluid, as

well as the auxiliary batch to better account for associated operational costs. Temperature stratification

of TES unit can also be addressed to improve accuracy, as well as phase-change storage units, with

constant temperature.

For solar thermal energy, the pressure drop across the pipe circuits should be taken into account to

validate the profitability of the solution. New concentrating technologies should also be addressed as

they work with higher temperatures.

42

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