Dynamic Modelling of Batch Distillation Columns Chemical ...

Dynamic Modelling of Batch Distillation Columns

Maria Nunes de Almeida Viseu

Thesis to obtain the Master of Science Degree in

Chemical Engineering

Supervisors: Prof. Dr Carla Isabel Costa Pinheiro

Dr Charles Brand

Examination Committee

Chairperson: Prof. Dr Sebastião Manuel Tavares da Silva Alves

Supervisor: Prof. Dr Carla Isabel Costa Pinheiro

Members of the Committee: Prof. Dr João Miguel Alves da Silva

November 2014

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Para os meus pais,

Com amor.

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Abstract

atch distillation is becoming increasingly important in specialty product industries in which flexibility is a

key performance factor. Because high added value chemical compounds are produced in these

industries with uncertain demands and lifetimes, mathematical models that predict separation times and

product purities thereby facilitating plant scheduling are required. The primary purpose of this study is thus to

develop a batch multi-staged distillation model based on mass and energy balances, equilibrium stages and tray

hydraulic relations. The mathematical model was implemented in gPROMS ModelBuilder®, an industry-

leading custom modelling and flowsheet environment software.

Preliminary steps were undertaken prior to implementing the dynamic multi-staged model: batch distillation

operating policies as well as modelling and tray hydraulic considerations were covered in a broad background

review; a theoretical separation example of an equimolar benzene/toluene mixture was used to validate a

simpler Rayleigh distillation model comprising only one equilibrium stage; tray hydraulic correlations

encompassing column diameter, tray holdup and tray pressure drop estimations were tested in a

methanol/water continuous separation case study.

The multi-staged batch model was validated for a methanol/water separation using literature data from an

experimental pilot plant and from theoretical results given by a model implemented in Fortran language and

by commercial simulator Batchsim of Pro/II. A sensitivity analysis was performed to evaluate the model

robustness, testing the effect of the reflux ratio and the heat duty on the separation time and methanol

recovery.

The results simulated in ModelBuilder for the batch multi-staged model reveal a 6.2% overestimation of the

experimental methanol recovery. A very good agreement is found between the ModelBuilder and Fortran

models: the methanol recovery predicted by ModelBuilder is only 2.3% lower. It is shown that the

ModelBuilder multi-staged batch model is robust with ±10% heat duty variations or ±0.5 reflux ratio

differences both affecting the total experiment time in approximately 12%. Differences of ±0.5 in the reflux

ratio are found to have a 2.2 to 6.4% absolute impact on the methanol recovery whilst this recovery is

practically not affected by 10% heat duty variations.

This work offers a tool that may be applied to the scheduling of batch chemical plants and aid industrial

management at the planning level.

Keywords: Batch distillation; equilibrium stages; computational models; gPROMS

B

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Resumo

importância da destilação descontínua tem vindo a aumentar em indústrias de química fina onde a

flexibilidade é um factor-chave de performance. Na medida em que produzem compostos químicos de

alto valor acrescentado, são necessários modelos matemáticos que permitam determinar o tempo de separação

e a pureza dos produtos, facilitando o planeamento industrial. O objectivo principal deste estudo consiste no

desenvolvimento de um modelo dinâmico para colunas de destilação com vários andares baseado em balanços

mássicos e energéticos e andares de equilíbrio. O modelo matemático foi implementado utilizando o software

gPROMS ModelBuilder®.

Antes de o desenvolver, realizaram-se as seguintes etapas: ampla revisão bibliográfica sobre modos de

operação, modelos e relações hidráulicas de perdas de pressão em colunas batch; validação de um modelo

simplificado Rayleigh de um só andar de equilíbrio, com base num exemplo teórico de separação de uma

mistura equimolar benzeno/tolueno; teste de correlações de estimativa de diâmetro, holdup e perdas de pressão

nos pratos, aplicadas a um estudo de caso de destilação em contínuo para a mistura metanol/água.

Validou-se o modelo para uma separação metanol/água, utilizando dados experimentais publicados na

literatura provenientes de uma unidade piloto, e também resultados teóricos gerados por um modelo

desenvolvido em linguagem Fortran e pelo simulador comercial Batchsim Pro/II. Realizou-se também uma

análise de sensibilidade para avaliar a robustez do modelo, testando o efeito da razão de refluxo e do calor

fornecido ao ebulidor no tempo total de separação e na recuperação de metanol.

Os resultados simulados no software ModelBuilder para o modelo sobrestimam em 6.2% a recuperação de

metanol face ao seu valor experimental. Há uma notável concordância entre os dados gerados pelo

ModelBuilder e pelo Fortran: a recuperação de metanol prevista pelo ModelBuilder é apenas 2.3% inferior.

Demonstra-se que o modelo ModelBuilder é robusto com variações de ±10% no calor fornecido ao ebulidor

ou diferenças de ±0.5 na razão de refluxo a afectar o tempo de separação em cerca de 12%. Diferenças de

±0.5 na razão de refluxo têm um impacto absoluto entre 2.2 a 6.4% na recuperação de metanol, enquanto

aquela não é praticamente afectada por variações de 10% no calor fornecido.

Espera-se com este estudo desenvolver uma ferramenta com aplicação na planificação de indústrias químicas

com processos de destilação em descontínuo contribuindo, assim, para uma melhor gestão e controlo fabris.

Palavras-chave: Destilação em descontínuo; andares de equilíbrio; modelos computacionais; gPROMS

A

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Acknowledgements

I would like to express my gratitude to my supervisors Prof. Carla Pinheiro and Dr Charles Brand for their

continuous technical guidance, constructive criticism and friendly support during the production of this thesis.

I would also like to acknowledge Prof. Dr Costas Pantelides and Dr Maarten Nauta for the opportunity to

work at Process System Enterprise and for their valuable knowledge and availability.

A special thanks to Inês and Prisci, with whom I shared my stay in London. I am sincerely thankful for their

valuable friendship and for the time we spent together. A warm thanks goes to Mariana, Renato and Artur, for

their encouragement and support. I am grateful for all the adventures and fun we experienced.

Thank you João for sharing your life with me during our lovely university years.

I am truly grateful to my parents, brother and sister, for their unconditional love.

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Contents

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

1.1. Motivation ........................................................................................................................................................... 1

1.2. State of the art .................................................................................................................................................... 2

1.3. Original contributions ....................................................................................................................................... 3

1.4. Thesis outline ...................................................................................................................................................... 3

2 Literature review........................................................................................................................................................... 5

2.1. Batch distillation operating policies ................................................................................................................ 5

2.2. Distillation modelling ........................................................................................................................................ 9

2.2.1. Equilibrium stage model .............................................................................................................................. 9

2.2.2. Stage efficiency ............................................................................................................................................ 10

2.2.3. Two-film model .......................................................................................................................................... 11

2.2.4. Rate-based stage model ............................................................................................................................. 12

2.2.5. Maxwell-Stefan formulation ..................................................................................................................... 14

2.3. Tray design and operation ............................................................................................................................. 15

2.3.1. Tray design .................................................................................................................................................. 15

2.3.2. Tray operation ............................................................................................................................................. 18

3 Materials and methods ............................................................................................................................................. 21

3.1. gPROMS platform .......................................................................................................................................... 21

3.2. Physical properties .......................................................................................................................................... 22

4 Rayleigh distillation ................................................................................................................................................... 25

4.1. Model equations .............................................................................................................................................. 25

4.2. Model flowsheet .............................................................................................................................................. 28

4.3. Model validation .............................................................................................................................................. 30

4.4. Conclusion ....................................................................................................................................................... 33

5 Dynamic column modelling .................................................................................................................................... 35

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5.1. Model assumptions ......................................................................................................................................... 36

5.2. Model equations .............................................................................................................................................. 36

5.2.1. MESH equations ........................................................................................................................................ 36

5.2.2. Tray hydraulic equations ........................................................................................................................... 37

6 Tray design and operation ....................................................................................................................................... 43

6.1. Case study data ................................................................................................................................................ 43

6.2. Case study results ............................................................................................................................................ 45

6.2.1. Flow regime ................................................................................................................................................. 45

6.2.2. Column diameter ........................................................................................................................................ 46

6.2.3. Tray holdup ................................................................................................................................................. 47

6.2.4. Tray pressure drop ..................................................................................................................................... 48

6.3. Conclusion ....................................................................................................................................................... 50

7 Multi-staged batch distillation ................................................................................................................................. 51

7.1. Operating conditions and Validation data .................................................................................................. 51

7.1.1. Experimental pilot plant ............................................................................................................................ 51

7.1.1. Fortran and Batchsim models .................................................................................................................. 52

7.2. gPROMS ModelBuilder® model ................................................................................................................. 53

7.2.1. Column ......................................................................................................................................................... 55

7.2.2. Sump ............................................................................................................................................................. 57

7.2.3. Reboiler ........................................................................................................................................................ 59

7.2.4. Condenser .................................................................................................................................................... 60

7.2.5. Splitter .......................................................................................................................................................... 60

7.2.6. Pump ............................................................................................................................................................ 61

7.2.7. Recycle breakers .......................................................................................................................................... 61

7.2.8. Product receiver .......................................................................................................................................... 62

7.3. Total reflux ....................................................................................................................................................... 62

7.4. Model validation .............................................................................................................................................. 66

7.4.1. Column section ........................................................................................................................................... 66

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7.4.2. Reboiler ........................................................................................................................................................ 72

7.5. Sensitivity analysis ........................................................................................................................................... 74

7.5.1. Total separation time ................................................................................................................................. 74

7.5.2. Methanol recovery ...................................................................................................................................... 76

7.6. Conclusion ....................................................................................................................................................... 78

8 General conclusions and Future work ................................................................................................................. 79

8.1. General conclusions ....................................................................................................................................... 79

8.2. Future work ...................................................................................................................................................... 81

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List of figures

Figure 2.1 – The three characteristic periods in the cyclic operation of a batch distillation column [5]. ............... 7

Figure 2.2 – Batch rectification column with off-cut [7]. .............................................................................................. 8

Figure 2.3 – Off-cut recycle strategy in binary batch distillation [7]. ........................................................................... 8

Figure 2.4 - Schematic representation of an equilibrium stage [23]. ......................................................................... 10

Figure 2.5 – The two-film model for transfer in vapour and liquid phases [28]. .................................................... 12

Figure 2.6 - Schematic representation of a nonequilibrium stage [13]...................................................................... 14

Figure 2.7 - Bubble-cap tray [39]. ................................................................................................................................... 16

Figure 2.8 - Sieve tray [40]................................................................................................................................................ 16

Figure 2.9 - Valve tray [49]. .............................................................................................................................................. 16

Figure 2.10 – Stable operating region, plates [16]. ....................................................................................................... 17

Figure 4.1 – Two-phase separator unit. ......................................................................................................................... 26

Figure 4.2 – Rayleigh distillation flowsheet. .................................................................................................................. 28

Figure 4.3 – Instantaneous and average benzene molar fraction profiles. ............................................................... 31

Figure 4.4 – Separator and tank temperature profiles. ................................................................................................ 32

Figure 4.5 – Holdup and liquid benzene molar fraction profiles in the still. ........................................................... 32

Figure 5.1 – Countercurrent cascade of N column stages. ......................................................................................... 35

Figure 6.1 – Liquid and vapour flowrate profiles and Porter & Jenkins froth-spray prediction. ......................... 46

Figure 6.2 – Minimum required column diameter vector. .......................................................................................... 46

Figure 6.3 – Liquid and vapour molar holdup profiles. .............................................................................................. 48

Figure 6.4 – Dry vapour and aerated liquid head loss profiles................................................................................... 49

Figure 6.5 – Clear liquid and surface tension contributions to the aerated liquid head loss. ................................ 50

Figure 7.1 – Batch distillation pilot plant - Bonsfills [45]. .......................................................................................... 52

Figure 7.2 – Batch distillation column flowsheet. ........................................................................................................ 55

Figure 7.3 – Total reflux temperature profiles for column stages 1, 10, 12, 13, 14 and 15 and Q = 933.3 W. . 63

Figure 7.4 – Reboiler pressure profile for Q=933.3 W in total reflux simulation. ................................................. 64

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Figure 7.5 – Distillate flowrate profiles for RR=3 and Q=933 W or Q=736 W.................................................... 65

Figure 7.6 - Distillate methanol composition profiles with RR=3 and 𝐃 = 0.28 mol/min. The right-hand side

figure represents the same profiles with an xx axis shift for each of the 3 functions to G(X) = F(X+T1st

plateau).

.............................................................................................................................................................................................. 68

Figure 7.7 - Distillate temperature with RR=3 and 𝐃 = 0.28 mol/min. .................................................................. 69

Figure 7.8 - Methanol composition for plate 5 with RR=3 and 𝐃 = 0.28 mol/min. ........................................... 70

Figure 7.9 – Methanol holdup profiles for column trays with RR=3 and 𝐃 = 0.28 mol/min (ModelBuilder). 70

Figure 7.10 - Plate 5 temperature with RR=3 and 𝐃 = 0.28 mol/min..................................................................... 71

Figure 7.11 - Temperature profiles for column trays with RR=3 and 𝐃 = 0.28 mol/min (ModelBuilder). ...... 72

Figure 7.12 – Reboiler temperature profile with RR=3 and 𝐃 = 0.28 mol/min. ................................................... 73

Figure 7.13 – Reboiler holdup and vapour fraction profiles for RR=3 and 𝐃=0.28 mol/min (ModelBuilder).73

Figure 7.14 – a) Distillate flowrate profiles for a fixed Q = 736 W and RR = 2.5, 3 and 3.5. ............................. 74

Figure 7.15 – ADF and TST for Q = 663 W, 736 W and 810 W and RR = 2.5, 3 and 3.5. ................................. 75

Figure 7.16 – Distillate methanol composition profiles for a fixed Q = 736 W and RR = 2.5, 3 and 3.5. The

right-hand side figure represents the same profiles with an xx axis shift for each of the 3 functions to G(X) =

F(X+T1st

plateau). ................................................................................................................................................................... 76

Figure 7.17 - Distillate methanol composition profiles for a fixed RR=3 and Q = 663 W, 736 W and 810 W.

The right-hand side figure represents the same profiles with an xx axis shift for each of the 3 functions to

G(X) = F(X+T1st

plateau). ..................................................................................................................................................... 77

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List of tables

Table 3.1 – Multiflash physical properties: inputs and output type. ......................................................................... 22

Table 4.1 – Required separator model specifications. ................................................................................................. 29

Table 4.2 – Benzene/toluene problem data provided by Seader et al [13]. ............................................................. 30

Table 4.3 – Required ModelBuilder inputs for the benzene/toluene separation. .................................................. 30

Table 6.1 – Case study methanol/water data. ............................................................................................................... 44

Table 7.1 – Fortran and Batchsim model specifications. ............................................................................................ 53

Table 7.2 – Column specification data. .......................................................................................................................... 56

Table 7.3 – Column initial pressures and holdup compositions on each tray. ........................................................ 57

Table 7.4 – Controllers LC-119 and FC-119 specifications. ...................................................................................... 59

Table 7.5 - Reboiler, column and sump steady-state holdups for Q=736.4 W. ...................................................... 66

Table 7.6 – 1st plateau and off-cut durations and methanol recoveries. The first plateau duration is defined by

the methanol purity, i.e., the first plateau ends when the methanol purity falls below 0.99. ................................ 68

Table 7.7 – ADF and TST relative variations. Base simulation: 736 W with RR = 3. ........................................... 75

Table 7.8 – 1st plateau duration for Q = 663 W, 736 W and 810 W and RR = 2.5, 3 and 3.5. The first plateau

duration is defined by the methanol purity, i.e., the first plateau ends when the methanol purity falls below

0.99....................................................................................................................................................................................... 77

Table 7.9 – Methanol recovery and associated relative variation. Base simulation: 736 W with RR = 3. .......... 78

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Nomenclature

𝐴𝑎 Tray active area (m2)

𝐴ℎ Tray hole area (m2)

𝐴𝑁 Tray net area (m2)

𝐴𝑇 Column transversal area (m2)

𝛽𝑗 Aeration correction factor on stage 𝑗

𝑐𝑡 Total concentration of fluid mixture (mol/m3)

𝐶𝑣𝐻 Hughmark & O’Connell orifice coefficient

𝐶𝑣𝐿 Liebson orifice coefficient

𝐷 Fick diffusivity (m2/s)

𝐷𝑐 Column diameter (m)

𝑑ℎ Hole diameter (m)

𝑑𝑤 Weir diameter (m)

�̅� Average distillate flowrate (mol/min)

Đ𝑖𝑗 Maxwell-Stefan 𝑖 − 𝑗 pair diffusivity (m2/s)

𝐸𝑀𝑉 Murphree efficiency

𝐸0 Overall stage efficiency

𝑓𝑖,𝑗𝐿 Component 𝑖 fugacity on stage 𝑗 – liquid (bar)

𝑓𝑖,𝑗𝑉 Component 𝑖 fugacity on stage 𝑗 – vapour (bar)

𝐹𝑗𝐿 Liquid feed flowrate entering stage 𝑗 (mol/s)

𝐹𝑗𝑉 Vapour feed flowrate entering stage 𝑗 (mol/s)

𝛾𝑖,𝑗 Activity coefficient of component 𝑖 on stage 𝑗

ℎ𝑐,𝑗𝐵

Bennett clear liquid height on stage 𝑗 (m)

ℎ𝑐,𝑗𝑆 Jeronimo & Sawistowski clear liquid height on stage 𝑗 (m)

ℎ𝑑,𝑗𝐻

Hughmark & O’Connell dry vapour head loss on stage 𝑗 (m)

ℎ𝑑,𝑗𝐿 Liebson dry vapour head loss on stage 𝑗 (m)

ℎ𝑓,𝑗 Froth height on stage 𝑗 (m)

ℎ𝑗𝐹𝐿

Liquid feed specific enthalpy on stage 𝑗 (J/mol)

ℎ𝑗𝐹𝑉

Vapour feed specific enthalpy on stage 𝑗 (J/mol)

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ℎ𝑗𝐿 Liquid specific enthalpy for stage 𝑗 (J/mol)

ℎ𝑗𝑉

Vapour specific enthalpy for stage 𝑗 (J/mol)

ℎ𝑙,𝑗𝐵

Bennett aerated liquid head loss on stage 𝑗 (m)

ℎ𝑙,𝑗𝐹

Fair aerated liquid head loss on stage 𝑗 (m)

ℎ𝑤 Weir height (mm)

𝐽𝑖 Mass transfer flux of component 𝑖 (mol.s-1.m-2)

𝐿𝑗 Liquid flowrate leaving stage 𝑗 (mol/s)

𝑙𝑡 Plate spacing (m)

𝑙𝑤 Weir length (m)

𝑀𝑖,𝑗 Component 𝑖 holdup on stage 𝑗 (mol)

𝑀𝑖𝑛𝑣,𝑗,𝑘 Invariant component 𝑘 holdup on stage 𝑗 (mol)

𝑀𝑗𝐿 Liquid holdup on stage 𝑗 (mol)

𝑀𝑗𝑉 Vapour holdup on stage 𝑗 (mol)

𝜇𝑖,𝑗𝐿 Component 𝑖 chemical potential on stage 𝑗 – liquid (J/mol)

𝜇𝑖,𝑗𝑉 Component 𝑖 chemical potential on stage 𝑗 – vapour (J/mol)

𝑁𝑒𝑞𝑢𝑖𝑙 Number of theoretical equilibrium stages

𝑁𝑟𝑒𝑎𝑙 Number of column stages

𝜔𝑖 Component 𝑖 acentric factor

𝑃𝑐 Critical pressure (bar)

𝑃𝑗 Stage 𝑗 pressure (bar)

∅̂𝑖,𝑗𝐿 Fugacity coefficient of component 𝑖 on stage 𝑗 - liquid

∅̂𝑖,𝑗𝑉

Fugacity coefficient of component 𝑖 on stage 𝑗 - vapour

𝑞𝑗 Liquid volumetric flow leaving stage 𝑗 (m3/s)

𝑅 Ideal gas constant (J/K/mol)

𝜌𝐿 Liquid mass density (kg/m3)

𝜌𝑉 Vapour mass density (kg/m3)

𝜌𝑚,𝑗𝐿 Liquid molar density on stage 𝑗 (mol/m3)

𝜎𝑗 Surface tension on stage 𝑗 (dyne/cm)

𝑇𝑐 Critical temperature (K)

𝑇𝑗 Stage 𝑗 temperature (K)

𝑡𝑡 Tray thickness (m)

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𝑈𝑗 Energy holdup on stage 𝑗 (J)

𝑣𝑎,𝑗 Vapour velocity through active area on stage 𝑗 (m/s)

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐹 Fair stage 𝑗 flooding velocity (m/s)

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐾 Kister & Haas stage 𝑗 flooding velocity (m/s)

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐿 Lowenstein stage 𝑗 flooding velocity (m/s)

𝑣ℎ,𝑗 Hole vapour velocity on stage 𝑗 (m/s)

𝑉𝑗 Vapour flowrate leaving stage 𝑗 (mol/s)

𝑉�̂� Vapour volumetric flowrate leaving stage 𝑗 (m3/s)

𝑉𝑚 Molar volume (m3/mol)

Ѵ Volume (m3)

𝑥𝑖,𝑗 Component 𝑖 liquid molar fraction for stage 𝑗

𝑦𝑖,𝑗 Component 𝑖 vapour molar fraction for stage 𝑗

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1

1

Introduction

1.1. Motivation

Batch distillation is becoming increasingly important in industries such as fine chemical, food, biochemical

and pharmaceutical companies. The flexibility of the process is a key issue in the separation performance of

these industries where a wide range of chemical products with high added value are produced with uncertain

demands and lifetimes. Furthermore, the separation of multiple components from a product mixture is one of

the major difficulties in the production processes of these industries. By employing a single batch column a

multicomponent mixture is separated into its pure components whereas to serve the same purpose, the

continuous process counterpart requires a train of distillation columns.

Because the demand for efficiency and productivity in the chemical industry is increasing, reliable

mathematical models that easily predict the separation degree of mixtures are needed. Hence, the main

objective of this work is to provide a dynamic distillation model in gPROMS ModelBuilder® based on mass

and energy balances, equilibrium equations and tray hydraulic correlations. To achieve the desired goal, a

simple one stage equilibrium separator model is first validated for a benzene/toluene separation. Then, tray

hydraulic correlations comprising column diameter, tray holdup and tray pressure drop estimations are

implemented and simulated for a methanol/water case study. Finally, having tested the one stage separation

model and the hydraulic equations, a multi-staged dynamic model is developed and validated against both

experimental data and theoretical models implemented in Fortran and Batchsim of Pro/II for a

methanol/water separation. A further objective of this work is to analyse the robustness of the multi-staged

dynamic model. This goal is attained by performing a sensitivity analysis, evaluating the effect of key variables

such as the reflux ratio and the heat duty on separation performance indicators, namely the product recovery

and the separation time.

The dynamic distillation model developed in this work is a tool that may be applied to the scheduling of batch

chemical plants, since the separation time is actually contemplated. An improved industrial management at the

scheduling and planning level is thus facilitated by the model that is here proposed.

2

1.2. State of the art

A literature review is presented in Chapter 2 of this work to cover the main innovations related to batch

distillation. Batch operating policies, modelling and tray hydraulic issues are discussed giving a broad

background of this field.

Batch distillation operation strategies are well known in the literature. For example, Kister [1] and Mujtaba et

al. [2] studied the effect of the constant distillate composition, constant reflux ratio and optimised reflux ratio

policies on the total distillation time and product purity. Diweker [3] and Noda et al. [4] analysed optimisation

problems in batch distillation concluding that the optimal reflux ratio is, for most cases, close to the constant

reflux ratio. SØrensen [5] compared the optimal reflux ratio policy to the cyclic operating policy for different

column configurations, discussing the practical implementation of a cyclic operating policy. Mujtaba &

Macchietto [6], Milady & Mujtaba [7] and Bonny et al. [8] investigated the benefit of using off-cut recycling

policies. All these optimal operation strategies can be simulated using different dynamic distillation models.

Shortcut models for handling multicomponent mixtures under the assumptions of constant molar overflow

and negligible vapour and liquid tray holdups are useful if a preliminary study of batch distillation is required.

Luyben [9] studied a shortcut method to design batch distillation columns, based on the assumption of

constant relative volatility. A shortcut model with constant relative volatility was used by Barolo & Botteon

[10] for a column at total reflux to separate pure components from a binary mixture. Diweker & Madhaven

[11] developed a shortcut method for the two operating policies of constant distillate composition and

constant reflux ratio. The shortcut method of Sundaram & Evans [12] is also easy to employ; however, it

applies only to the case of constant reflux. In both latter cases the Fenske-Underwood-Gilliland (FUG)

shortcut method for continuous distillation is used by treating batch rectification as a sequence of continuous,

steady-state rectifications at successive time steps. Additional information regarding shortcut methods is

available in the literature (see [13]).

Stage-by-stage composition, temperature and flowrate profiles are needed to perform a detailed simulation of

a batch multicomponent rectification. The first rigorous batch rectification model based on the assumptions

of negligible vapour holdup, constant liquid holdup and perfect mixing of the phases on equilibrium trays was

developed by Meadows [14]. Distefano [15] extended this model by incorporating a computer-based-solution

procedure used to successfully simulate commercial batch-distillation columns. The development and

application of rigorous equilibrium-stage models for batch distillation is described in several textbooks (see,

for example, [16]).

3

A rigorous dynamic multi-staged model based on equilibrium stages and tray pressure drop correlations is

here implemented in gPROMS® ModelBuilder and validated for a multi-tray column operating under a

constant reflux ratio to separate a methanol/water mixture.

1.3. Original contributions

The contributions of this work are the following: development of tray hydraulic correlations in gPROMS

ModelBuilder® which may be applied at the industrial level to aid in the design and operation of distillation

columns; implementation of a dynamic multi-staged dynamic model in ModelBuilder validated against

experimental and theoretical data; understanding of the impact of key performance factors in the aforesaid

model. The multi-staged model developed in this work is a tool that may be applied to the scheduling of batch

chemical plants, since the separation time is actually contemplated.

1.4. Thesis outline

This thesis is structured as follows:

Chapter 2 presents a background review on batch distillation. The main batch operating policies are

discussed and modelling techniques concerning mass and heat transfer phenomena are addressed.

Tray hydraulic design and operation issues are also introduced by analysing column diameter, tray

holdup and tray pressure drop correlations;

In Chapter 3, the gPROMS ModelBuilder® software and the Multiflash™ physical property package

used to develop the models described in this work are introduced;

Chapter 4 describes a Rayleigh distillation, i.e., a simple one-staged equilibrium dynamic separator.

The model equations and flowsheet are presented and a benzene/toluene separation is used to

validate the Rayleigh model;

Chapter 5 focuses on the MESH and the tray hydraulic equations used to develop a multi-staged

dynamic model in ModelBuilder;

In Chapter 6 a continuous multi-staged methanol/water distillation case study is used to test the tray

hydraulic correlations introduced in the ModelBuilder multi-staged model. Column diameter, tray

holdup and tray pressure drop results are presented in this section;

Chapter 7 focuses on the development, validation and testing of the dynamic multi-staged distillation

model. The components of the model are presented separately and the model is validated against

both experimental and theoretical data from a methanol/water separation. The key performance

4

factors of the dynamic model are identified and their impact is evaluated by performing a sensitivity

analysis;

Finally, in Chapter 8, the general conclusions of this work are presented and future work directions

are suggested.

5

2

Literature review istillation is the operation used in the chemical industry to separate a liquid mixture into products

based on their different volatilities. The process has an ancient background history dating back to at

least the 1st century A.D. By the 11th century batch distillation was used in Italy to concentrate the alcoholic

content of beverages [13]. At that time, one-staged batch distillations were carried out by evaporating part of

the liquid feed in a heated vessel, condensing and dripping the vapour into a receiver tank, in a simple process

which is today known as Rayleigh distillation. Cellier-Bluementhal developed the first multistage vertical

column for continuous distillation in 1813 [1], discovering that the separation could be significantly improved

by providing multiple vapour-liquid contact stages. During the 20th century distillation was used as a means of

separating crude oil into various products. Today, multistage distillation is the prime separation technique used

in the chemical industry.

The use of a batch distillation unit is particularly important for processes where flexibility is needed, for

example, when production demand is varying. Moreover, batch distillation is useful in the production of small

amounts of products with high added value and for the separation of material high in solids content [16].

Although batch distillation offers operational flexibility and low capital investment, potential drawbacks

include long batch times, high temperatures in the still and a high wastage of energy [17]. In order to achieve

an energy-efficient operation and improve productivity decisions on batch operating policies should be made.

2.1. Batch distillation operating policies

Two modes of operating a batch distillation column are most frequently used: constant reflux ratio and

constant distillate composition [18]. With the former, the reflux ratio is set at a specified value at which it is

maintained through the run. Initially, the distillate contains the highest concentration of lightest component.

As the distillation continues the distillate composition gets heavier. When the average compositions of the

collected distillate or the bottoms meet the required specification the distillation process is stopped. With a

constant distillate composition, the reflux ratio is constantly increased during the course of the distillation to

maintain a constant overhead composition. In the case of a binary distillation, when the reflux ratio attains a

D

6

very high value the overhead is withdrawn to another receiver, the reflux ratio is reduced and an intermediate

cut is taken. The distillation run ends when the pot liquor meets the required specification.

The characteristics of the system, the product specifications and the ease of implementation affect the choice

of the operating mode. The implementation of a constant reflux ratio policy is simple because of the

availability of rapidly responding flow sensors [16]. Indeed, operating at a constant reflux ratio is the most

common control method. Operation can be controlled using simply a timed reflux splitter, a ratio controller

or a pair of rotameters. However, the average accumulated distillate composition must be estimated by some

method. Note that this is a challenge when the cut point is difficult to define (for example, when the distillate

composition changes slowly with time). For a constant distillate composition policy the control must be linked

to a concentration-sensitive physical variable.

As stated previously, the choice of the operating mode depends upon the materials being separated and upon

the number of theoretical column stages. For example, Kister [1] showed that for a separation of an equimolar

benzene/toluene feed in a column with three theoretical stages the constant distillate composition policy

resulted in a 13% higher product concentration, 23% higher heat input and a 25% lower product recovery

than the constant reflux ratio policy. Therefore, the chemical engineer must choose between the different

alternatives generated by the different operating policies (e.g., high product compositions and low recoveries

vs. low product concentrations and high recoveries).

Maximization of the amount of accumulated distillate or minimization of the operation time can also be

achieved by using an optimal reflux ratio policy [16]. In this case, the reflux ratio varies with time according to

a particular objective function such as the product recovery, operation time, or profit. However, the savings

obtained from using an optimal reflux ratio policy may not be justified for relatively easy separations.

Mujtaba et al. [2] evaluated the performance of a batch reactive distillation to produce lactic acid by hydrolysis

reaction of methyl acetate. A minimum time problem was developed within the gPROMS® software

specifying the amount of bottom product (acid lactic) and product purity as constraints bounds and

optimizing the reflux ratio profile in a piecewise constant manner. The operating time was saved by an average

of 37%, 46% and 48% using 2, 3 and 4 time intervals, respectively, for experiments with product purities

ranging from 0.8 to 0.925.

A column can also operate using a cyclic operating policy (see Figure 2.1) [16]. In this case, three periods of

operation are repeated: filling, total reflux and dumping. The filling period is equivalent to the normal startup

of a batch column, i.e., the reflux drum is filled up and the boilup rate determines the filling time. During the

second period the column operates under total reflux until it reaches equilibrium or a steady-state regime.

Finally, the product is withdrawn by dumping the drum. The duration of the dumping period depends on the

7

maximum distillate flow out of the reflux drum. These three steps are repeated until the distillate product in

the accumulator or the residue in the reboiler meets the required specifications.

The cyclic policy has several advantages in comparison to other conventional schemes. First, the cyclic policy

is easy to operate since minimal control is needed. For example, because no flow measurements are needed,

the cyclic operation becomes particularly suitable for small laboratory batch columns where accurate

measurements of flow may be difficult. Second, the cyclic policy achieves the maximum attainable separation

in the column under total reflux operation. Last, it is safer to operate since it is less sensitive to disturbances.

The main practical disadvantage of using a cyclic policy is that the feed composition must be known

accurately since the reflux drum holdup is determined based on this value. In particular, if the holdup is too

large it may be difficult to meet the product specifications.

Figure 2.1 – The three characteristic periods in the cyclic operation of a batch distillation column [5].

SØrensen [5] compared the cyclic operation to the optimal reflux ratio policy for different column

configurations and discussed the practical implementation of a cyclic operating policy. The cyclic policy

resulted in a 32% saving of the operating time when compared to an optimal reflux ratio policy (linear profile

over two control variables) for a binary separation with a relative volatility of 1.5 in a regular batch column

with 10 trays. The cyclic policy was found to be favourable for close boiling mixtures where a small amount of

light product was to be recovered. However, the benefits from this mode of operation largely depend on the

mixture, dimensions of the column and product specifications.

An additional common strategy for improving the performance of batch distillation processes is the collection

of an off-cut produced between two successive valuable product cuts [19]. For example (see Figure 2.2) a

conventional binary batch distillation usually employs one main-cut and one off-cut. The main-cut is

withdrawn first to the main-cut receiver and meets the light product specification. The off-cut distillate is

withdrawn subsequently until the residue in the reboiler meets the required bottom product specification.

8

Figure 2.2 – Batch rectification column with off-cut [7].

Off-cuts must be either disposed of safely (meeting environmental constraints) or recycled thus establishing a

campaign mode operation. The latter case is shown in Figure 2.3 for a binary batch distillation. Note that the

initial fresh feed stock is mixed with off-cut recycled material from the previous batch. As the batch cycle is

repeated the amount and composition of the off-cut from the current batch is identical to those from the

previous batch creating a quasi-steady mode of operation.

Figure 2.3 – Off-cut recycle strategy in binary batch distillation [7].

Bonny et al. discussed optimal off-cut recycle policies for multicomponent mixtures considering three

different criteria: production rate, recovered products and processing time [8]. Miladi & Mujtaba [7] evaluated

the effect of off-cut recycling on an annual profit objective function using a close boiling cyclohexane/toluene

separation and a wide boiling n-heptane/toluene separation. The off-cut recycling was found to be more

beneficial for the difficult (close boiling) separation and for this case an improvement of 14% in the objective

function was noticed. Note that raw material costs are lower when recycling off-cuts; however, the total

storage costs increase. Hence, both the raw material and the storage costs can be determining factors for the

production and recycling of off-cuts.

9

2.2. Distillation modelling

Dynamic models that integrate the above-mentioned operating policies are a key factor in improving the

performance of batch distillation separations at the planning and scheduling level. The developments

concerning vapour-liquid modelling behavior in the column stages are highlighted in this section. First, the

equilibrium stage model is described and connected with stage efficiency correction factors. The two-film

model is then presented as a way to describe phase hydrodynamics and subsequently linked to rate-based

models which account for bulk transport, coupling effects and species interactions. The Maxwell-Stefan

formulation is finally presented as the most fundamentally sound way to model mass transfer in

multicomponent systems and thus the appropriate choice for incorporation in rate-based models.

2.2.1. Equilibrium stage model

The modelling of distillation processes has been based on the concept of equilibrium stage since 1950 [20].

The equilibrium stage model for distillation processes has been described by several authors, namely King

[21], Kister [1], Seader & Henley [13] and Perry & Green [16].

A general schematic representation of an equilibrium stage 𝑗 is shown in Figure 2.4. Liquid and vapour feeds

can enter stage 𝑗 at flowrates 𝐹𝑗𝐿 and 𝐹𝑗

𝑉, respectively. Interstage liquid from the above stage 𝑗 − 1 with a

mole fraction of component 𝑖 of 𝑥𝑖,𝑗−1and interstage vapour from the adjacent stage 𝑗 + 1 with a molar

composition 𝑦𝑖,𝑗+1 of 𝑖 are also entering stage 𝑗. The heat transfer rates from (+) and to (-) stage 𝑗 are

designated by 𝑄𝑗𝐿𝐹 and 𝑄𝑗

𝑉𝐹 for the liquid and for the vapour phase, respectively.

The liquid and vapour phases leaving each stage 𝑗 at temperature 𝑇𝑗 and pressure 𝑃𝑗 are assumed to be in

equilibrium with each other. Phase equilibrium demands the simultaneous occurrence of thermal, mechanical

and thermodynamic equilibrium. Firstly, the condition of thermal equilibrium establishes an identical

temperature for each phase on the basis that there are no heat transfer fluxes between the two phases

(𝑇𝑗𝑉 = 𝑇𝑗

𝐿). Additionally, a force balance considering mechanical equilibrium yields equal pressures for the

vapour and liquid phases (𝑃𝑗𝑉 = 𝑃𝑗

𝐿). Finally, the condition of thermodynamic equilibrium considers an

identical chemical potential for both phases and for each component 𝑖 of the mixture, i.e.,

𝜇𝑖,𝑗𝑉 (𝑇𝑗

𝑉 , 𝑃𝑗𝑉 , 𝑦𝑖,𝑗) = 𝜇𝑖,𝑗

𝐿 (𝑇𝑗𝐿 , 𝑃𝑗

𝐿 , 𝑥𝑖,𝑗). This last condition is determined assuming that the rate of vaporization

of component 𝑖 is identical to its rate of condensation [22]. For this reason, there are no composition

gradients in each phase for all the components of the mixture.

10

The fugacity term 𝑓𝑖 is generally used instead of the chemical potential. The two concepts relate to each other

as follows:

𝑑𝜇𝑖 = 𝑅𝑇𝑑𝑙𝑛𝑓𝑖

(2.1)

Therefore, the vapour-liquid equilibrium can also be expressed considering identical vapour and liquid

fugacities for component 𝑖. Equations (2.2) and (2.3) show that the vapour fugacity varies with fugacity

coefficient ∅̂𝑖𝑉

which measures the deviation from the gas ideal behavior. The deviation to an ideal solution in

the liquid phase is measured by the activity coefficient 𝛾𝑖 .

𝑓𝑖,𝑗𝑉(𝑇𝑗

𝑉, 𝑃𝑗𝑉 , 𝑦𝑖,𝑗) = 𝑓𝑖,𝑗

𝐿 (𝑇𝑗𝐿, 𝑃𝑗

𝐿 , 𝑥𝑖,𝑗) (2.2)

𝑦𝑖,𝑗. ∅̂𝑖,𝑗𝑉(𝑇𝑗

𝑉, 𝑃𝑗𝑉 , 𝑦𝑖,𝑗). 𝑃𝑗

𝑉 = 𝑥𝑖,𝑗. 𝛾𝑖,𝑗(𝑇𝑗𝐿 , 𝑃𝑗

𝐿 , 𝑥𝑖,𝑗). 𝑓𝑖(𝑝𝑢𝑟𝑒),𝑗𝐿 (2.3)

The NRTL activity coefficient model and the Soave-Redlich-Kwong equation of state were used in all

simulations presented in this work to determine the activity and fugacity coefficients for the liquid and gas

phases, respectively. The equations of an equilibrium stage are presented in Chapter 5 being referred to as the

MESH equations: material, equilibrium, summation and energy balances.

Figure 2.4 - Schematic representation of an equilibrium stage [23].

2.2.2. Stage efficiency

The equilibrium model equations consider that equilibrium with respect to both heat and mass transfer is

attained in each stage. The assumption of thermal equilibrium is generally valid unless significant temperature

changes occur from stage to stage. However, equilibrium with respect to mass transfer is typically not a valid

assumption primarily due to insufficient contact time between the vapour and liquid phases and deficient

11

mixing. For real separations the effects of mixing, entrainment, flow configuration and mass and heat transfer

should be analysed.

For the case where all feed components have the same mass transfer efficiency the number of actual stages is

related to the number of equilibrium stages by the overall stage efficiency, 𝐸0:

𝐸0 =𝑁𝑒𝑞𝑢𝑖𝑙

𝑁𝑟𝑒𝑎𝑙

(2.4)

When equilibrium data are available it is common to employ a stage efficiency to account for the deviation

from the equilibrium behavior [21]. The most frequent stage efficiency used for the description of individual

stages is the Murphree efficiency, defined in the following equation with respect to the vapour phase:

𝐸𝑀𝑉 =𝑦𝑖,𝑗 − 𝑦𝑖,𝑗+1

𝑦𝑖,𝑗∗ − 𝑦𝑖,𝑗+1

(2.5)

where 𝑦𝑖,𝑗+1and 𝑦𝑖,𝑗 refer to the gross inlet and outlet vapour streams of stage 𝑗 and 𝑦𝑖,𝑗∗ is the vapour

composition that would be in equilibrium with the actual outlet composition 𝑥𝑖,𝑗of the liquid phase.

The Murphree vapour phase efficiency can be combined with the MESH equilibrium model equations when it

is desired to compute actual rather than ideal equilibrium stages. The use of such efficiency is particularly

convenient in multistage separation processes for feeds containing components of a wide range of

concentration and where all components are not sharply separated [16].

2.2.3. Two-film model

Mass and heat transfer at a vapour-liquid interface can be described using a variety of theoretical models (see,

for example, Brodkey & Hershey [24]). The two-film model is often used since there is a broad spectrum of

experimental correlations available in the literature for several internals and systems that provide suitable

estimations for model parameters. Detailed descriptions of the two-film model are provided by Lewis &

Whitman [25], Taylor & Krishna [26] and Krishna & Standart [27].

In the film model (Figure 2.5) the entire resistance to mass and heat transfer is concentrated in a hypothetical

film with thickness δ adjacent to the phase boundary. Mass transfer occurs through the film in the direction

normal to the interface. For the case where liquid and gas come into contact, this phenomenon is described by

considering the existence of two films of gas and liquid on the two sides of the interface. That is, on the

liquid side of the interface there exists a layer of liquid which is free from mixing by convection and similarly,

on the gas side of the interface there is a layer of gas where transfer occurs by molecular diffusion alone.

Outside these films, in the fluid bulks, the turbulence is high and consequently perfect mixing occurs with

uniform compositions at all points. The mass transfer rates 𝑁 within the films can be determined from quasi-

12

stationary transfer relations since the storage capacity for mass and energy in these films is negligible

compared to that in the bulk phases.

Multicomponent diffusion in the films can be described using the Maxwell-Stefan equations described in

section 2.2.5, where diffusion fluxes of the components are related to their chemical potential. In this stage

model equilibrium stage exists only at the interface, i.e., there is a negligible resistance to mass transfer across

the interface.

Figure 2.5 – The two-film model for transfer in vapour and liquid phases [28].

2.2.4. Rate-based stage model

Although the application of Murphree efficiency 𝐸𝑀𝑉 is adequate for binary close-boiling and ideal mixtures,

deficiencies for multicomponent mixtures and for cases where efficiencies are low have been detected [13]. In

fact, Toor & Burchard [29] showed that 𝐸𝑀𝑉 could present all possible values from minus to plus ∞ for

multicomponent systems.

To avoid the difficulties associated with applying tray efficiencies to equilibrium models, a nonequilibrium

transport model or rate-based model was developed by Waggoner & Loud [30] in 1977 for a close-boiling

system. However, an energy transfer equation was not considered since thermal equilibrium is a valid

assumption for close-boiling mixtures.

A detailed explanation of a rate-based model including rigorous mass and heat transfer is presented by

Koojman & Taylor [31]. The stage model uses the two-film theory and considers different possibilities for

vapour and liquid flow configurations including perfectly mixed and plug flow on each tray.

Multicomponent mixtures in reactive distillation frequently exhibit large thermodynamic non-idealities.

Moreover, the occurrence of chemical reactions can significantly affect interphase mass transfers. For this

reason, efficiencies are devoid of any physical meaning. Therefore, the use of a rate-based model for reactive

13

distillation is particularly important since efficiencies are not considered and the mass transfer process has to

be directly modeled. Several examples of reactive distillation processes using a rate-based model are available

in the literature, namely Higler et al. [32], Górak et al. [28] and Lee & Dudukovic [33].

Figure 2.6 shows a schematic diagram of a nonequilibrium stage 𝑗. The nonequilibrium model equations are

referred to as the MERSHQ equations (Material, Energy balances, Rate of mass and heat transfer, Summation

of compositions, Hydrodynamic equation of pressure drop and eQuilibrium) [16]. Contrary to the equilibrium

case, rate-based mass and enthalpy balances consider component mass transfer rates 𝑁𝑖,𝑗 and heat transfer

rates 𝐸𝑗 across the phase boundary from the vapour phase to the liquid phase (+) or vice versa (-). The

conventional mass and energy balances around stage 𝑗 are replaced by two separate balances for the vapour

and the liquid phases, coupled by the terms 𝑁𝑖,𝑗 and 𝐸𝑗 . Phase equilibrium for each component is assumed to

exist only at the phase interface. Furthermore, the fluxes of mass and energy are continuous across the

interface. The transfer rates 𝑁𝑖,𝑗 are generally related to the chemical potential gradients using the Maxwell-

Stefan equations [34]. The terms 𝐸𝑗 include convective and enthalpy-flow contributions, where the former are

based on interfacial area and heat transfer coefficients from the Chilton-Colburn analogy for the vapour phase

and the penetration theory for the liquid phase [35].

Physical reality is more accurately captured through the nonequilibrium model which accounts for bulk

transport, coupling effects and species interactions. Rate-based models are particularly suitable when

simulating packed columns, systems with strongly nonideal liquid solutions, systems with trace components,

columns with rapidly changing profiles and systems where tray-efficiency data are not available [16]. However,

a limitation of a rate-based model compared to the equilibrium-stage model is that it cannot be computed

independently of the geometry of the column. Additionally, rate-based models require physical properties such

as surface tension, diffusion coefficients, viscosities, etc. for calculation of mass and heat transfer coefficients

and interfacial areas.

14

Figure 2.6 - Schematic representation of a nonequilibrium stage [13].

2.2.5. Maxwell-Stefan formulation

The phase hydrodynamics can be described by a variety of models such as the film, penetration and turbulent

boundary layer models. For mass transfer within a fluid phase the equations of continuity need to be solved

along with the Maxwell-Stefan diffusion equations [34]. The Maxwell-Stefan formulation is the most

fundamentally sound way to model mass transfer in multicomponent systems. Thus, the vapour-liquid mass

transfer can be modeled combining the film model presentation and the Maxwell-Stefan diffusion theory (see,

for example, Higler et al. [36] or Baur et al. [37] ).

Diffusional coupling effects may occur in distillation systems where thermodynamic non-idealities are

significant or where strong differences in diffusivities for different pairs of components can be observed. For

these cases, the concentration gradient of a component strongly influences the flux of a different component.

For example, Górak provided data on the vapour phase driving force (concentration gradient) for 2-propanol

for distillation of methanol-propanol-water in a packed column [38]. The system in question exhibited

diffusional coupling effects such as diffusion of a component despite the absence of a driving force, diffusion

of a component in a direction opposite to that dictated by its driving force and no diffusion despite a large

driving force (osmotic diffusion, reverse diffusion and diffusion barrier, respectively). Because these

phenomena occur frequently in multicomponent distillation, the use of the Maxwell-Stefan formulation

instead of Fick’s law for describing diffusion is of vital importance. Indeed, Fick’s law postulates a linear

dependence of the flux 𝐽𝑖 with respect to Fick’s diffusivity 𝐷 and the composition gradient 𝑑𝑥𝑖

𝑑𝑧. Hence, the use

of the Fick formulation to describe diffusional coupling effects would be inadequate since Fick’s diffusivity

would be 𝐷 → ∞ for osmotic diffusion, 𝐷 < 0 for reverse diffusion and 𝐷 = 0 for diffusion barrier

phenomena.

15

The Maxwell-Stefan formulation is obtained considering that the force exerted per mole of diffusing species 𝑖

is balanced by friction between species 𝑖 and each of the other species in the mixture. For unidirectional

transport of component 𝑖 in a multicomponent mixture the Maxwell-Stefan formulation relates the chemical

potential of component 𝑖 to the molar fluxes 𝑁 in the following manner [34]:

−𝑥𝑖𝑅𝑇

𝑑𝜇𝑖𝑑𝑧 𝑅,𝑇

=∑𝑥𝑗𝑁𝑖 − 𝑥𝑖𝑁𝑗

𝑐𝑡Đ𝑖𝑗

𝑛

𝑗≠𝑖

(2.6)

where 𝑐𝑡 is the total molar concentration of the fluid mixture, 𝜇𝑖 the chemical potential of component 𝑖, 𝑥 the

mole fraction and Đ𝑖𝑗 the Maxwell-Stefan 𝑖 − 𝑗 pair diffusivity.

The Maxwell-Stefan diffusivity Đ has the physical significance of an inverse drag coefficient and does not

consider thermodynamic non-ideality effects. For this reason, it is more easily predictable than the Fick

diffusivity where the drag effects and the thermodynamic non-ideality effects are coupled.

2.3. Tray design and operation

To obtain satisfactory stage holdup estimations and pressure profiles, tray hydraulic equations may be added

to a multi-staged dynamic column model. A background review on tray design considerations, namely column

diameter correlations, and tray operation relations such as holdup and pressure drop estimations is here

presented.

2.3.1. Tray design

The primary phase of the hardware design sets the major equipment requirements, namely the type of tray and

the column diameter. The present subsection examines the common tray types used in distillation columns

and focuses on minimum diameter estimation by examining well-known flood correlations.

Before the 1950s the bubble-cap tray was the predominant type of gas disperser unit for distillation columns

[16]. This device is a perforated plate with centered risers around the holes and caps in the form of inverted

cups over the risers. Gas passes through the riser, flows under the cap and is dispersed into the liquid through

several slots in the lower part of the cap. The ability to operate at low vapour and liquid rates avoiding liquid

drainage is a unique advantage associated to this type of tray. Bubble caps were largely superseded by sieve

and valve-type perforations primarily due to their high cost. A lower capacity, higher entrainment and higher

pressure drop are also reasons which explain the popularity decrease of bubble-cap trays. These types of trays

are currently employed only in special applications, in particular operations handling exceptionally low liquid

flowrates [1].

16

Figure 2.7 - Bubble-cap tray [39].

The sieve tray is a flat plate with round orifices where vapour velocity prevents liquid from weeping through

the holes. Sieve trays are simple and inexpensive. Additionally, maintenance and fouling tendencies as well as

corrosion effects are very low. However, liquid drainage may occur at low gas flowrates resulting in liquid

bypassing and reduced efficiency.

Valve trays are plates with movable valves that provide variable orifices of round or rectangular shapes, with

or without a caging structure. The valve plate minimizes liquid weeping by reducing the disk openings as the

vapour rate decreases. Hence, the total orifice area is varied to retain a satisfactory pressure balance across the

plate.

Figure 2.8 - Sieve tray [40].

The minimum diameter of a distillation column is calculated from the maximum allowable capacity of a plate

for handling gas and liquid flow [1]. Because the maximum allowable capacity is fixed according to flooding

predictions, understanding the different flooding mechanisms is of vital importance.

Flooding is excessive accumulation of liquid inside the column. Two main types of flooding occur, as follows:

1) Entrainment flooding. Increasing the gas rate for a constant liquid rate causes excessive carry-up of

liquid by vapour entrainment to the tray above. At the flood point it is difficult to maintain net

Figure 2.9 - Valve tray [49].

17

downward flow of liquid. Consequently, as the column liquid inventory increases there is a sharp drop

in plate efficiency and increase in pressure drop. While spray entrainment flooding is predominant at

low liquid flowrates, at higher liquid flowrates froth entrainment flooding becomes the prevailing

flooding mechanism. In the first case, most of the liquid is in the form of liquid drops and the bulk of

these drops are entrained into the tray above. In the second case, the dispersion on the tray is in the

form of a froth and the froth envelope approaches the tray above when tray spacing is small.

2) Downcomer flooding. Increasing the liquid rate while holding a constant gas rate causes liquid flow

to overtax the capacity of the downcomer with an associated liquid accumulation and pressure drop

increase. In the case of downcomer backup/downflow flooding, the backup of aerated liquid in the

downcomer exceeds the tray spacing and liquid accumulates on the tray above because of tray

pressure drop, liquid height on the tray and frictional losses in the downcomer apron. Downcomer

choke flooding may also occur when frictional losses in the downcomer and downcomer entrance

become excessive due to a high velocity of aerated liquid in the downcomer. In this case the frothy

mixture cannot be transported to the tray below and therefore liquid accumulates on the tray above.

A typical qualitative capacity diagram is shown in Figure 2.10 for all plate devices. The area of stable operation

is bound by the tray stability limits. The upper limit to vapour flow is set by the aforementioned flooding

mechanisms. As the vapour rate is lowered the limit of excessive weeping is reached, meaning the vapour flow

is insufficient to maintain a level of liquid on the plate.

Figure 2.10 – Stable operating region, plates [16].

As previously stated the column diameter should be selected according to flooding predictions. More

specifically, the column diameter can be calculated from the vapour flooding velocity which is based on the

column net area. For this reason, the Lowenstein [18], Fair [16] and the Kister & Haas [1] flooding

correlations were implemented in this work using gPROMS ModelBuilder®. These correlations can be

consulted in the Dynamic column modelling section of this work.

18

Sinnott [18] recommends the Lowenstein correlation for maximum allowable vapour velocity calculation and

hence column area and diameter estimation. The correlation is based on the Souders and Brown equation in

which entrainment is controlled by the carry-up of liquid droplets [1]. The approximate estimate of the

diameter should be revised when the detailed plate design is undertaken.

The Fair flood correlation gives flooding gas velocities to ±10% and has been the standard of the industry for

entrainment prediction [16]. However, the correlation applies only to non-foaming systems where weir height

is less than 15% of plate spacing. The Fair correlation can be used for sieve-plate perforations with a

fractional hole area (ratio of perforation area to active area) of 0.1 or greater and when holes are 13 mm or less

in diameter. Similarly, the correlation also applies to bubble-cap and valve trays when the ratio of slot (bubble-

cap) or full valve opening (valve) area to active area is 0.1 or greater. Fair’s correlation is known to predict

most entrainment data well, although slightly conservatively.

Kister & Haas reported a recent correlation for entrainment flooding which was shown to predict a large data

of sieve and valve tray flood points to within ±20% [1]. The correlation applies to non-foaming systems with

a plate spacing, hole diameter, fractional hole area and weir height between 36-91 cm, 0.32-2.5 cm, 0.06-0.2

and 0-7.6 cm, respectively. Contrary to the Fair correlation, the Kister & Haas method provides a suitable

approximation to the effects of physical properties, operating variables and tray geometry on entrainment

flooding. The correlation was also obtained from a much larger base of industrial and laboratory-scale

columns data. However, although the Fair correlation can be used both for froth and spray entrainment flood

predictions, the Kister & Haas correlation applies exclusively to spray entrainment flood estimations.

2.3.2. Tray operation

If stage holdups are to be included in a dynamic column model, the first step is to estimate the effective liquid

holdup on the trays. The vapour holdup is then calculated considering the total tray geometry, the volume

occupied by the liquid holdup and the vapour density. Hence, in this study two correlations were added to the

dynamic multi-staged ModelBuilder model: the Bennett and the Jeronimo & Sawistowski effective clear liquid

height correlations. The Bennett clear liquid height calculation applies to froth-type regimes. This clear liquid

height calculation is based on the weir height, the liquid flow and the froth density [16]. The Jeronimo &

Sawistowski correlation has been successfully used as a building block for correlating entrainment flooding

and spray regime entrainment, strictly predicting clear liquid heights at the froth to spray transition. However,

it has been shown that the clear liquid height in the spray regime is similar to the clear liquid height at this

transition [1] and thus the correlation is applicable to spray-type regimes.

19

To implement a pressure-driven dynamic system in a batch column the total pressure drop across a tray has to

be taken into consideration. Hence, in this work several tray pressure drop correlations were developed and

tested using ModelBuilder. Note that the pressure drop equations are listed in section 5.2.2.3 of this thesis.

There are two main sources of pressure loss: the pressure drop calculated for the flow of vapour through the

dry holes ℎ𝑑 and the pressure drop through the aerated mass on the plate ℎ𝑙 [18].

Vapour pressure drop in a tower is generally from 0.35 to 1.03 kPa/tray [13]. Several published correlations

are available for evaluating ℎ𝑑. In this study, three correlations were implemented in ModelBuilder: the

standard orifice equation, Liebson et al. correlation [1] and the Hughmark & O’Connell correlation [41]. The

correlation by Liebson et al. is preferred by Fair et al. and Van Winkle. Ludwig and Chase recommend the

Hughmark & O’Connell correlation [1]. The vapour velocity in the holes, the liquid and vapour densities and

the orifice coefficient are the prime variables affecting the vapour pressure drop. For both the Liebson and

the Hughmark & O’Connell correlations, the orifice discharge coefficient is a function of the ratio of tray

thickness to hole diameter and the fractional hole area. The standard orifice equation is deducted from

Bernoulli’s principle and in this case, a constant orifice discharge coefficient of 0.75 is used.

In this work, two liquid pressure drop correlations were developed in ModelBuilder: the Fair and the Bennett

aerated liquid pressure drop correlations [1].

The approach followed by Fair for pressure drop prediction uses a dimensionless tray aeration factor, 𝛽 [16].

In this context, ℎ𝑙 is determined multiplying 𝛽 by the clear liquid height of liquid on the tray, ℎ𝑐. For sieve

and valve plates, ℎ𝑐 is the sum of the weir height ℎ𝑤, the hydraulic gradient ℎℎ𝑔 and the liquid head over the

outlet weir ℎ𝑜𝑤.

The hydraulic gradient is the head of liquid necessary to overcome the frictional resistance to liquid passage

across the tray. For a significant gradient the resistance to gas flow near the liquid inlet to the tray may become

excessive, resulting in an inoperative upstream portion of the plate. The hydraulic gradient term was not

considered in this thesis since this term is negligible for sieve trays and the usual practice is to omit it from the

pressure drop calculation [16].

To determine the liquid head over the outlet weir the corrected Francis weir formula for segmental and

circular weirs was also included in ModelBuilder. For segmental weirs ℎ𝑜𝑤 may be determined as a function of

the liquid flow, the weir length and a wall correction factor. The wall correction factor is proportional to the

weir crest and considers liquid flow constriction at the approach to the weir. For circular weirs ℎ𝑜𝑤 may be

estimated using the liquid flow and the weir diameter.

20

A more recent and fundamental relationship to determine the pressure loss through the aerated mass was

recommended by Bennett [1]. Indeed, Bennett developed a model of froth flow across the weir consequently

avoiding the correction of the clear liquid flow for aeration effects. In this case, the pressure drop through the

aerated liquid is the sum of two separate terms: the effective clear-liquid height ℎ𝑐 used to determine the

liquid holdup and the residual pressure drop due to surface tension ℎ𝑅. The residual pressure drop can be

interpreted as the excess pressure which bubbles must overcome due to the difference between the pressure

inside the bubble and that of the liquid.

For 302 experimental data points covering a wide range of systems the Bennett pressure drop correlation gave

an average relative error of ±0.35%. An error of approximately 5% was obtained for Fair’s correlation

considering a similar data base [16]. When an accurate pressure drop calculation is needed or when the

residual pressure drop is substantial the Bennett pressure drop correlation should be used. For example, for

sieve trays with a hole diameter lower than 0.32 cm the surface tension head loss term is significant and

should be determined using Bennett’s correlation [1]. On the contrary, estimating the residual head as a

function of the surface tension, froth density and froth height via Bennett’s correlation is an elaborate method

and its use is not justified when the surface tension head loss term is small. It should also be noted that the

Bennett pressure loss correlation is based on froth regime considerations and is not applicable to the spray

regime.

21

3

Materials and methods

3.1. gPROMS platform

gPROMS® is a custom modelling platform for process industries covering process and equipment

development and design as well as optimisation of process operations [42]. gPROMS products offer several

capabilities: support for multiscale modelling, meaning that micro to full-scale phenomena can be taken into

consideration simultaneously in the same model; execution and maintenance of custom models that apply to a

wide range of equipments; process flowsheet design; steady-state and dynamic modelling implemented within

the same interface and empirical parameter estimation from laboratory or industrial scale data.

One of the main gPROMS products is gPROMS ModelBuilder® where custom modelling capabilities and

process flowsheeting environments are provided. Process model implementation, graphical development and

configuration of hierarchical flowsheets, steady-state or dynamic simulation and optimisation are key

ModelBuilder applications. Thence, all the elements of the model lifecycle are taken into account:

Building custom process models by transcribing the process equations using the library models

supplied by ModelBuilder. Features such as icons, dialogs and reports may be associated to these

models and incorporated into flowsheets;

Constructing flowsheets utilising the gPROMS equation-oriented approach, for example, by

assigning downstream values and calculating upstream values;

Validating models against experimental data using parameter estimation techniques based on

mathematical optimisation algorithms;

Simulating steady-state or dynamic models within the same framework;

Determining optimal answers using dynamic and mixed-integer optimisation;

Exporting models for implementation in other engineering software environments, using, for

instance, gPROMS objects that may be executed in Excel or VBA interfaces.

In this study ModelBuilder was used for both flowsheeting and model development purposes. For example,

the Rayleigh distillation separator was implemented specifying key inputs in a drag-and-drop type flowsheeting

22

activity. Model development was carried out by adding tray hydraulic relations encompassing column

diameter, tray holdup and tray pressure drop correlations and by implementing the dynamic MESH equations

in the ModelBuilder model libraries. Indeed, to build the pressure-driven dynamic column model validated in

Chapter 7, the ModelBuilder libraries needed to include the tray holdup and pressure drop correlations along

with the dynamic material and energy equations for equilibrium column stages.

3.2. Physical properties

The standard gPROMS physical property package is Infochem Multiflash™ supplied by KBC Advanced

Technologies [43]. Multiflash is specifically designed for equation-oriented modelling, as is gPROMS, thereby

generating tight convergence of iterations and of analytical partial derivatives with respect to temperature,

pressure and composition. The phase equilibria is determined in Multiflash for different combinations of

conditions, namely PVT, enthalpy, entropy and internal energy. Furthermore, Multiflash calculates the

fractions of any particular phase at a fixed pressure or temperature, including dew and bubble points.

The physical properties used in this work for the Rayleigh separator (Chapter 4) and the dynamic column

model (Chapter 5) are listed in Table 3.1. Each property has zero or more inputs that may be scalars, such as

the pressure 𝑃 and the temperature 𝑇, or arrays, as is the case of the liquid �⃗� and vapour �⃗� composition

fractions. Additionally, each property has a single output which may be a scalar or an array. For example, the

liquid enthalpy method returns a scalar whereas the liquid fugacity coefficient method returns an array.

Table 3.1 – Multiflash physical properties: inputs and output type.

gPROMS property name Inputs Output Type

MolecularWeight - Array

LiquidDensity 𝑇, 𝑃, �⃗� Scalar

VapourDensity 𝑇, 𝑃, �⃗� Scalar

LiquidEnthalpy 𝑇, 𝑃, �⃗� Scalar

VapourEnthalpy 𝑇, 𝑃, �⃗� Scalar

LiquidFugacityCoefficient 𝑇, 𝑃, �⃗� Array

VapourFugacityCoefficient 𝑇, 𝑃, �⃗� Array

SurfaceTension 𝑇, 𝑃, �⃗�, �⃗� Scalar

The thermodynamic properties are calculated in the Multiflash property package which contains all

commonly-used equations of state and activity coefficient thermodynamic models. Equations of state models

23

include the cubic equations of state, namely the Soave-Redlich-Kwong and the Peng-Robinson equations. If

improved predictions of thermal and volumetric properties are needed the Lee-Kesler and the Benedict-Wee-

Rubin-Starling non-cubic equations of state may also be selected, for example. The Soave-Redlich-Kwong

equation of state given by Equations (3.1) to (3.7) was chosen in this study to account for intermolecular

attractive and repulsive forces occurring in the real gas, since it is adequate for fugacity calculations [44].

𝑃 = 𝑅𝑇

𝑉𝑚 − 𝑏𝑚𝑖𝑥𝑡−

𝑎𝑚𝑖𝑥𝑡(𝑇)

𝑉𝑚(𝑉𝑚 + 𝑏𝑚𝑖𝑥𝑡)

(3.1)

√𝑎𝑚𝑖𝑥𝑡(𝑇) =∑𝑦𝑖√𝑎𝑖(𝑇)

𝑁𝐶

𝑖

(3.2)

𝑏𝑚𝑖𝑥𝑡 = ∑𝑦𝑖𝑏𝑖

𝑁𝐶

𝑖

(3.3)

Note that 𝑉𝑚 is the molar volume and 𝑅 the ideal gas constant. The parameters 𝑎𝑚𝑖𝑥𝑡 and 𝑏𝑚𝑖𝑥𝑡 are

determined using the molar fractions 𝑦𝑖 and the pure component parameters 𝑎𝑖 and 𝑏𝑖 which are calculated

from the critical temperature 𝑇𝑐𝑖, the critical pressure 𝑃𝑐𝑖 and the acentric factor 𝜔𝑖.

𝑎𝑖(𝑇) =0.42747𝑅2𝑇𝑐𝑖

2

𝑃𝑐𝑖𝛼𝑖(𝑇), 𝑖 = 1,…𝑁𝐶. (3.4)

𝛼𝑖(𝑇) = [1 +𝑚𝑖 (1 − √𝑇

𝑇𝑐𝑖)]

2

, 𝑖 = 1,…𝑁𝐶. (3.5)

𝑚𝑖 = 0.48 + 1.574𝜔𝑖 − 0.176𝜔𝑖2, 𝑖 = 1,…𝑁𝐶. (3.6)

𝑏𝑖 = 0.08644𝑅𝑇𝑐𝑖𝑃𝑐𝑖

, 𝑖 = 1,…𝑁𝐶. (3.7)

A number of activity coefficient models such as the UNIQUAC, UNIFAC and NRTL models are available in

Multiflash. The NRTL activity coefficient model given by Equations (3.8) to (3.13) was selected in this work

because it may be used for vapour-liquid equilibrium calculations and it is often useful for non-ideal systems

such as the methanol-water mixtures presented in Chapters 6 and 7. The activity coefficients 𝛾1 and 𝛾2 of the

binary mixture relate to the molar fractions 𝑥 and to the binary parameters 𝐺𝑖𝑗 and 𝜏𝑖𝑗 which can be

determined using the interaction energy 𝑈𝑖𝑗 between molecular surfaces of components 𝑖 and 𝑗 and the non-

randomness parameters 𝛼12 and 𝛼21.

24

𝑙𝑛𝛾1 = 𝑥22 [𝜏21(

𝐺21𝑥1 + 𝑥2𝐺21

)2 +𝜏12𝐺12

(𝑥2 + 𝑥1𝐺12)2] (3.8)

𝑙𝑛𝛾2 = 𝑥12 [𝜏12(

𝐺12𝑥2 + 𝑥1𝐺12

)2 +𝜏21𝐺21

(𝑥1 + 𝑥2𝐺21)2]

(3.9)

𝑙𝑛𝐺12 = −𝛼12𝜏12 (3.10)

𝑙𝑛𝐺21 = −𝛼21𝜏21 (3.11)

𝜏12 =𝑈12 − 𝑈22

𝑅𝑇 (3.12)

𝜏21 =𝑈21 −𝑈11

𝑅𝑇 (3.13)

25

4

Rayleigh distillation he Rayleigh separation is the simplest form of a batch distillation process. A liquid mixture is charged to

a still-pot and brought to boiling. The vapour formed is assumed to be in equilibrium with perfectly

mixed liquid in the still and is continuously condensed to produce a distillate. Note that in this case no trays or

packing are provided, that is, there is only one equilibrium stage. Analysing the behavior of one equilibrium

stage dynamic unit is the first step in understanding how a multi-staged batch distillation column works. Thus,

this chapter is in fact an introduction to subsequent section 5 of this work.

In this chapter the basic Rayleigh model equations are presented, given some insights into the phase

equilibrium of the mixture and the characteristics of the vessel where the separation is taking place. A Rayleigh

binary benzene/toluene separation example provided by Seader et al. [13] implemented in ModelBuilder will

also be shown. The validity of the Rayleigh separator model is attested by matching the obtained results with

data supplied by Seader et al.

4.1. Model equations

A flash unit of the form shown in Figure 4.1 is considered. This two-phase separator model available in the

ModelBuilder library was used to simulate a Rayleigh batch distillation, according to the following

assumptions:

The liquid outlet flowrate is set to zero by closing the associated valve (see Figure 4.2), that is, only

the vapour outlet is continuously withdrawn and condensed to produce a distillate, while the liquid

remains in the still;

The feed continuous flowrate is set to zero;

A charge is introduced at time 𝑡 = 0 in the separator according to the specified initial conditions;

The liquid and vapour phases are perfectly mixed in the separator;

The liquid and vapour are assumed to be in phase equilibrium with each other, that is, no distinction

is made between the temperatures, pressures and components chemical potentials of the two phases

in the separator;

A constant heat duty 𝑄 is provided;

T

26

Adiabatic separator, i.e., there is no heat loss through the vessel;

No chemical reactions occur.

Figure 4.1 – Two-phase separator unit.

The two-phase vessel model is essentially a one-staged distillation separation. Hence, the material, equilibrium,

summation and energy balances defined by the MESH equations for a distillation section model are

applicable. A material balance on component 𝑖 yields:

𝑑𝑀𝑖𝑑𝑡

+ 𝐿𝑥𝑖 + 𝑉𝑦𝑖 = 𝐹𝑧𝑖 , 𝑖 = 1,… 𝑐. (4.1)

where 𝑀𝑖 refers to the total holdup of component 𝑖 and 𝐿, 𝑉 and 𝐹 represent the liquid, vapour and feed

molar flowrates with molar fractions 𝑥𝑖, 𝑦𝑖 and 𝑧𝑖 , respectively.

For 𝑐 components, the liquid and vapour fractions obey thermodynamic equilibrium relations of the form:

𝑦𝑖𝑥𝑖=∅̂𝑖𝐿

∅̂𝑖𝑉 , 𝑖 = 1,… 𝑐.

(4.2)

where ∅̂𝑖𝐿 and ∅̂𝑖

𝑉 represent the fugacity coefficients of species 𝑖 in the liquid and vapour phases,

respectively.

The summation equations normalise the liquid and vapour molar fractions:

∑𝑥𝑖𝑖

= 1 (4.3)

27

∑𝑦𝑖𝑖

= 1 (4.4)

The energy balance defines the energy holdup accumulation 𝑑𝑈

𝑑𝑡, as follows:

𝑑𝑈

𝑑𝑡+ 𝐿ℎ𝐿 + 𝑉ℎ𝑉 = 𝑄 + 𝐹ℎ𝐹

(4.5)

The material and energy holdups must be defined considering the total vessel volume Ѵ and the total liquid

and vapour holdups 𝑀𝐿 and 𝑀𝑉 , respectively:

𝑀𝑖 = 𝑀𝐿𝑥𝑖 +𝑀

𝑉𝑦𝑖, 𝑖 = 1,… 𝑐. (4.6)

𝑈 = 𝑀𝐿ℎ𝐿 +𝑀𝑉ℎ𝑉 − 𝑃Ѵ (4.7)

𝑀𝐿

𝜌𝑚𝐿 +

𝑀𝑉

𝜌𝑚𝑉 = Ѵ (4.8)

Note that a pressure-driven system was considered for all the Rayleigh system presented in this work. Hence,

the exit flowrate 𝑉 is a function of the pressure difference between the flash and the downstream units:

𝑉 = 𝑓(𝑃 − 𝑃𝑑𝑜𝑤𝑛) (4.9)

In addition to the above equations, the thermophysical property relations for fugacity coefficients, specific

enthalpies and densities must be considered:

∅̂𝑖𝐿= ∅̂𝑖

𝐿(𝑇, 𝑃, �⃗�), 𝑖 = 1,… 𝑐. (4.10)

∅̂𝑖𝑉= ∅̂𝑖

𝑉(𝑇, 𝑃, �⃗�), 𝑖 = 1,… 𝑐. (4.11)

ℎ𝐿 = ℎ𝐿(𝑇, 𝑃, �⃗�) (4.12)

ℎ𝑉 = ℎ𝑉(𝑇, 𝑃, �⃗�) (4.13)

𝜌𝐿 = 𝜌𝐿(𝑇, 𝑃, �⃗�) (4.14)

𝜌𝑉 = 𝜌𝑉(𝑇, 𝑃, �⃗�) (4.15)

As mentioned previously, the liquid and the vapour thermophysical properties were determined in Multiflash

using the NRTL model and the Soave-Redlich-Kwong equation of state, respectively.

28

4.2. Model flowsheet

Figure 4.2 shows the Rayleigh distillation process flowsheet implemented in ModelBuilder for the binary

benzene/toluene separation.

Figure 4.2 – Rayleigh distillation flowsheet.

The dummy source is a feed with zero flowrate. In the two-phase vessel model a feed source is required to

introduce the Multiflash physical property package, where liquid and vapour densities, specific enthalpies and

the species fugacity coefficients are determined. Note that there is also no liquid stream withdrawn in a

Rayleigh distillation scheme and thus liquid valve V-111 shown in Figure 4.2 is closed.

The feed charge is determined by specifying the following initial conditions in the still: initial pressure, vapour

fraction and holdup compositions. Hence, the total holdup is calculated using these initial conditions and the

specified volume vessel Ѵ. Note that the initial conditions may also be specified using a combination of

different variable sets, listed in Table 4.1.

29

Table 4.1 – Required separator model specifications.

The condenser cools the distillate vapour according to a required thermal specification, namely the subcooling

delta temperature ∆𝑇𝑠𝑢𝑏, outlet temperature, enthalpy, heat duty or the required vapour fraction. Additionally,

the pressure drop or alternatively the outlet pressure should be specified.

The condensed distillate accumulates in the product receiver tank. For the Rayleigh simulation analysed in this

study, an initial temperature and negligible initial component mass holdups were specified, in addition to the

receiver volume. There is no outlet stream from the receiver tank and consequently valve V-121 in Figure 4.2

is closed.

All the Rayleigh simulations were run in a pressure-driven mode and consequently the flowrates are

determined according to pressure differentials. For example, the distillate flowrate is calculated using the still

pressure and the pressure at the liquid surface of the product receiver. Thus, care must be taken when

specifying pressures for all points in the system (sources, sinks, separators, tanks, etc.).

Tab Specifications

Design

Cylindrical vessel

2 of the following geometrical variables: radius, diameter, surface area, cross-section area, length

Flat/Hemispherical heads

No other specifications required Ellipsoidal heads

Ellipsoidal radius Torispherical heads

Crown radius

Knuckle radius

Operation Adiabatic operation or specified heat duty

Initial conditions

Holdup

Composition and 1 intensive variable: pressure, temperature or specific enthalpy

Overall holdup and composition

Component holdup Thermal specification

1 of the following specifications: pressure, temperature, specific enthalpy, saturated liquid, saturated vapour, liquid level fraction, vapour fraction, subcooled liquid

∆𝑇𝑠𝑢𝑏 or superheated vapour ∆𝑇𝑠𝑢𝑝

30

4.3. Model validation

Seader et al. [13] provides a theoretical Rayleigh distillation example based on a binary benzene/toluene

separation. To validate the Rayleigh distillation model implemented in ModelBuilder, the results obtained in

the simulation were matched with the corresponding data presented by Seader et al.

The present example consists of a batch still loaded with 100 kmol of a binary 50 mol% benzene in toluene

mixture (Table 4.2). A constant boilup rate of 10 kmol/hr is assumed at a pressure of 101.3 kPa. A feed

charge of 100 kmol was obtained in ModelBuilder by specifying a vapour fraction of 0.0207 mol/mol at 1 bar

with an equimolar charge and a 158 m3 vessel (Table 4.3). Note that the simulations easily converge to the

required solution when a given vapour fraction is specified at time 𝑡 = 0. However, if vapour is present the

required vessel volume increases to maintain an initial holdup of 100 kmol. The simulations are considerably

harder to execute using a saturated liquid at 𝑡 = 0, in which case the volume vessel would significantly be

reduced. Hence, the vessel volume here is devoid of physical meaning.

Because the distillate rate and, therefore, the liquid depletion rate in the still, vary with the heat input rate, a

heat duty of 93 kW was specified to maintain the required distillate rate of 10 kmol/hr.

Table 4.2 – Benzene/toluene problem data provided by Seader et al [13].

Table 4.3 – Required ModelBuilder inputs for the benzene/toluene separation.

Feed charge (kmol)

Charge composition (mol%)

Boilup rate (kmol/hr)

Pressure (kPa)

100 Toluene – 50 Benzene - 50

10 101.3

Separator

Tank Condenser

Cylindrical vessel with flat heads

D = 4.5 m

L = 4.5 m

Cylindrical vessel with flat heads

D = 4.5 m

L = 4.5 m

Pressure specification

∆𝑃 = 0 bar

Operation

Q = 93 kW

Operation

P = 1 bar

Initial holdup

Toluene – 50 mol %

Benzene – 50 mol %

P = 1.0001 bar Initial thermal specification

Vapour fraction - 0.0207 mol/mol

Initial holdup

Toluene - 10-5 kg

Benzene - 10-5 kg Initial thermal specification

T = 75.9 °C

Thermal specification

∆𝑇𝑠𝑢𝑏 = 10 °C

31

Figure 4.3 shows the instantaneous and average vapour composition profiles obtained according to Seader et

al. and using ModelBuilder. The instantaneous composition is determined in the vapour outlet of the

separator, while the average composition is measured in the product tank receiver. Evidently, for both

simulations the composition of the lower-boiling point component benzene decreases as distillation proceeds.

As expected a sharper decrease is observed in the instantaneous vapour composition profiles, where there is

no product accumulation and prior compositions are not taken into account. In this case, during 8.9 hours the

benzene molar fraction varies from 0.73 to 0.22 and from 0.71 to 0.21 according to Seader et al. and using

ModelBuilder, respectively. After 8.9 hr, both solutions predict a similar average benzene molar fraction, that

is, 0.57 and 0.56 for Seader et al. and for ModelBuilder, respectively.

Figure 4.3 – Instantaneous and average benzene molar fraction profiles.

A comparison of the temperature in the still as predicted by the model developed and the data presented by

Seader et al. is depicted in Figure 4.4. The temperature profile in the separator provided by Seader et al. is in

agreement with the profile obtained in ModelBuilder. Indeed, the temperature increases from approximately

93°C to 105°C (ModelBuilder) or 107°C (Seader et al.) as the separation takes place and the lighter

component benzene is distilled off.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8 9 10

yb

en

zen

e (

mo

l /

mo

l)

Time (hr)

Inst. ModelBuilderInst. SeaderAvg. ModelBuilderAvg. Seader

32

Figure 4.4 – Separator and tank temperature profiles.

Figure 4.5 shows the still holdup and liquid benzene molar fraction profiles given by ModelBuilder and Seader

et al. The still holdup reduction occurs primarily due to benzene depletion, since the benzene molar fraction

decreases in the liquid mixture throughout the simulation (from 0.5 to 0.1, approximately).

Figure 4.5 – Holdup and liquid benzene molar fraction profiles in the still.

Overall good agreement is found between the model prediction and the results obtained in the work from

Seader et al. in relation to temperature, composition and holdup profiles. However, some differences can be

observed, in particular regarding the distillate composition profiles. These differences may be explained

considering vapour-liquid equilibrium data such as fugacity coefficients are calculated in ModelBuilder by

calling the external physical property package – Multiflash. Note that the Multiflash software uses the NRTL

activity model and the Soave-Redlich-Kwong equation of state for the liquid phase and the gas phase,

respectively. This way of estimating thermophysical properties is much more detailed than simply assuming a

constant relative volatility α of 2.41, which is what is done in the example provided by Seader et al. Thus,

reality is most likely better captured by the ModelBuilder prediction.

80

85

90

95

100

105

110

0 1 2 3 4 5 6 7 8 9 10

Sep

ara

tor

T(o

C)

Time (hr)

ModelBuilder

Seader

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10

Ho

ldu

p (

km

ol)

Time (hr)

ModelBuilder

Seader

0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8 9 10

xb

en

zen

e(m

ol/

mo

l)

Time (hr)

ModelBuilderSeader

33

4.4. Conclusion

This chapter showed that a one-staged separation unit can be implemented in ModelBuilder using a two-phase

separator modelled with the MESH equations. The reliability of the model was confirmed by validating the

results obtained from a binary benzene/toluene separation.

The following sections extend the single equilibrium stage concept to a multi-staged distillation column

section. To this effect, the column modelling equations will be presented and implemented in ModelBuilder.

34


35

5

Dynamic column modelling his chapter presents the dynamic column section model developed in this work, introducing the model

assumptions and the model equations. The dynamic model consists of a countercurrent cascade of 𝑁

equilibrium stages (see Figure 5.1). Interstage liquid from the above stage 𝑗 − 1 with molar flowrate 𝐿𝑗−1 and

a mole fraction of component 𝑖 of 𝑥𝑖,𝑗−1and interstage vapour from the adjacent stage 𝑗 + 1 with flowrate

𝐿𝑗+1 and a molar composition 𝑦𝑖,𝑗+1 of 𝑖 are entering stage 𝑗. Liquid and vapour feeds can also enter stage 𝑗

at a total molar flowrate 𝐹𝑗. The liquid and vapour phases are leaving each stage 𝑗 at temperature 𝑇𝑗 and

pressure 𝑃𝑗.

Figure 5.1 – Countercurrent cascade of N column stages.

T

36

5.1. Model assumptions

The mathematical model presented in this work is based on the following assumptions:

Liquid and vapour feeds and products can enter any tray;

Neither of the existing phases entrains the other phase on each stage;

No heat is lost from any stage, i.e., the column is adiabatic;

The liquid and vapour phases are perfectly mixed on each column stage;

Phase equilibrium, that is, thermal, mechanical and thermodynamic equilibrium occurs at each stage

(equilibrium stage model);

The downcomer holdup is negligible;

No chemical reactions occur.

5.2. Model equations

The tray model equations are addressed in this section. The material, equilibrium, summation and energy

balances, known as the MESH equations, are first presented; tray hydraulic correlations covering column

diameter estimations as well as tray holdup and tray pressure drop predictions are listed in the second

subsection.

5.2.1. MESH equations

Each tray 𝑗 is modelled according to the MESH equations, that is, using material balances (M), phase

equilibrium relations (E), mole fraction summations (S) and an energy balance (H). To avoid high index

problems preparing the equations for future reactive term introduction (see Appendix A), the reaction

invariant holdup terms 𝑀𝑖𝑛𝑣 𝑗,𝑘 were considered. These representing the 𝑘 invariant quantities that remain

unchanged by the equilibrium reactions that occur on each stage 𝑗. Thus, for 𝐾 invariants, 𝑐 components and

𝐽 stages the MESH equations are as follows:

𝑑𝑀𝑖𝑛𝑣,𝑗,𝑘

𝑑𝑡+ 𝐿𝑗∑{𝑥𝑖,𝑗 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

+ 𝑉𝑗∑{𝑦𝑖,𝑗 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

= 𝐿𝑗−1∑{𝑥𝑖,𝑗−1 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

+ 𝑉𝑗+1∑{𝑦𝑖,𝑗+1 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

+𝐹𝑗𝐿∑{𝑥𝑖,𝑗

𝐹 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

+ 𝐹𝑗𝑉∑{𝑦𝑖,𝑗

𝐹 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖},

𝑖

𝑘 = 1,…𝐾; 𝑗 = 1,… 𝐽.

(5.1)

37

𝑦𝑖,𝑗

𝑥𝑖,𝑗=∅̂𝑖,𝑗

𝐿

∅̂𝑖,𝑗𝑉 , 𝑖 = 1,… 𝑐. ; 𝑗 = 1,… 𝐽. (5.2)

∑𝑥𝑖,𝑗𝑖

= 1, 𝑗 = 1,… 𝐽. (5.3)

∑𝑦𝑖,𝑗𝑖

= 1, 𝑗 = 1,… 𝐽. (5.4)

𝑑𝑈𝑗

𝑑𝑡+ 𝐿𝑗ℎ𝑗

𝐿 + 𝑉𝑗ℎ𝑗𝑉 = 𝐿𝑗−1ℎ𝑗−1

𝐿 + 𝑉𝑗+1ℎ𝑗+1𝑉 + ℎ𝑗

𝐹𝐿𝐹𝑗𝐿 + ℎ𝑗

𝐹𝑉𝐹𝑗𝑉, 𝑗 = 1,… 𝐽.

(5.5)

Note that U and h denote internal energy holdups and specific molar enthalpies, respectively. For more

information regarding symbol notation, the nomenclature section should be consulted.

If equilibrium reactions do not occur, the MESH equations are simplified. In this case, the identity matrix is

generated if the elements 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖 are grouped in a matrix for each stage 𝑗 in which the rows are defined by

the invariants 𝑘 and the columns by the components 𝑖.

The equilibrium equations (5.2) are obtained defining the fugacity coefficients ∅̂𝑖,𝑗𝐿 and ∅̂𝑖,𝑗

𝑉 of species 𝑖 in

the liquid and vapour mixtures and considering that for phase equilibrium the fugacity of any species 𝑖 in the

mixture is equal in both phases. Note that the fugacity coefficients are determined in Multiflash as described

in Chapter 3. The energy holdup accumulation term 𝑑𝑈𝑗

𝑑𝑡 is calculated using the energy balance equation (5.5)

for each stage and thus the specific liquid and vapour molar enthalpies ℎ𝑗𝐿 and ℎ𝑗

𝑉 must also be determined

in Multiflash.

5.2.2. Tray hydraulic equations

Column diameter 5.2.2.1.

The main factor that determines the required column diameter 𝐷𝑐 is the vapour volumetric flowrate �̂�.

Indeed, a maximum flowrate exists beyond which flooding occurs due to excessive liquid accumulation inside

the column. This flood point is determined by calculating the maximum allowable vapour velocity 𝑣𝑓𝑙𝑜𝑜𝑑 ,

according to a given correlation. Conservative designs call for approaches to flooding of 75 to 85%. For this

reason, a safety factor 𝑆 ≈ 0.8 was included in the model. Given the fractional net area 𝐴𝑁

𝐴𝑇, the minimum

column diameter 𝐷𝑐 is calculated by:

38

𝐷𝑐 = 𝑀𝑎𝑥

{

√4𝑉�̂�

𝜋 × 𝜌𝑗𝑉 × 𝑆 × 𝑣𝑓𝑙𝑜𝑜𝑑,𝑗 × (

𝐴𝑁𝐴𝑇)}

(5.6)

As previously stated two types of flooding are considered separately: entrainment flooding and downcomer

flooding. In either case, at the flood point any liquid fed to the column is carried out with the vapour leaving

the column [1]. If tray spacing is at least 0.6 m and the downcomer cross-sectional area is at least 10% of the

total cross-section area, downcomer flooding seldom occurs [1]. The usual limit of stable operation in a trayed

tower is entrainment flooding. For this reason, the present work considers three distinct entrainment flood

predictions: the Lowenstein, Fair and the Kister & Haas correlations.

For the Lowenstein correlation, the plate spacing 𝑙𝑡 (m), the liquid density 𝜌𝑗𝐿 (kg/m3) and the vapour density

𝜌𝑗𝑉 (kg/m3) are used to estimate the maximum allowable superficial vapour velocity 𝑣𝑓𝑙𝑜𝑜𝑑,𝑗

𝐿 (m/s) in stage 𝑗:

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐿 = −0.171𝑙𝑡

2 + 0.27𝑙𝑡 − 0.047(𝜌𝑗𝐿 − 𝜌𝑗

𝑉

𝜌𝑗𝑉 )

0.5

, 𝑗 = 1,… 𝐽. (5.7)

For the Fair correlation, 𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐹 (m/s) is a function of the flow parameter 𝐹𝑙𝑣,𝑗, plate spacing 𝑙𝑡 (m) and

surface tension 𝜎𝑗 (dyne/cm):

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐹 = √

𝜌𝑗𝐿 − 𝜌𝑗

𝑉

𝜌𝑗𝑉 (

𝜎𝑗

20)0.2 × {0.0105 + 0.1496 𝑙𝑡

0.755𝑒−1.463𝐹𝑙𝑣,𝑗0.842

} , 𝑗 = 1,… 𝐽. (5.8)

𝐹𝑙𝑣,𝑗 =𝐿𝑗

𝑉𝑗(𝜌𝑗𝑉

𝜌𝑗𝐿)0.5, 𝑗 = 1,… 𝐽. (5.9)

Note that in the above equation the flow parameter is based on liquid and vapour mass flowrates 𝐿𝑗 and 𝑉𝑗.

For the Kister & Haas correlation the vapour flooding velocity 𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐾 (m/s) is estimated using the hole

diameter 𝑑ℎ(m) and the effective clear liquid height in the spray regime (m), ℎ𝑐,𝑗𝑆, calculated according to the

Jeronimo & Sawistowski clear liquid height prediction, as listed in the subsequent section.

𝑣𝑓𝑙𝑜𝑜𝑑,𝑗𝐾 = √


𝑉

𝜌𝑗𝑉 0.0277(

{1000 × 𝑑ℎ}2𝜎𝑗

𝜌𝑗𝐿 )0.125(

𝜌𝑗𝑉

𝜌𝑗𝐿)0.1(

𝑙𝑡

1000 × ℎ𝑐,𝑗𝑆)0.5,

𝑗 = 1,… 𝐽.

(5.10)

Tray holdup 5.2.2.2.

To define the accumulation terms in the material and energy dynamic equations the liquid and vapour holdups

must be considered. The effective clear liquid height ℎ𝑐,𝑗 is the height to which the aerated mass on the trays

39

would collapse in the absence of vapour flow. This height gives a measure of the liquid level on the tray and

can be used to calculate the liquid molar holdup for a given stage 𝑗, as follows:

ℎ𝑐,𝑗 × 𝐴𝑎 × 𝜌𝑚,𝑗𝐿 = 𝑀𝑗

𝐿 , 𝑗 = 1,… 𝐽. (5.11)

where 𝐴𝑎 refers to the active area and 𝜌𝑚,𝑗𝐿 and 𝑀𝑗

𝐿 are the liquid molar density and liquid molar holdup of

stage 𝑗, respectively.

In this work, two effective clear liquid correlations have been used: the Bennett clear liquid height correlation

and the Jeronimo & Sawistowski correlation.

The Bennett clear liquid height correlation is based on froth regime considerations and is not applicable to the

spray regime. First, a dimensionless effective froth density ∅𝑒,𝑗 is calculated for each stage using a factor 𝐾𝑠,𝑗:

∅𝑒,𝑗 = exp[−12.55𝐾𝑠,𝑗0.91] , 𝑗 = 1,… 𝐽. (5.12)

𝐾𝑠,𝑗 = 𝑣𝑎,𝑗(𝜌𝑗𝑉


𝑉)0.5, 𝑗 = 1,… 𝐽. (5.13)

where 𝑣𝑎,𝑗 refers to the vapour velocity (m/s) through the active area 𝐴𝑎.

Then the froth height ℎ𝑓,𝑗 (m) is calculated and multiplied by the effective froth density giving the effective

clear liquid height ℎ𝑐,𝑗𝐵

(m):

ℎ𝑐,𝑗𝐵 = ℎ𝑓,𝑗 × ∅𝑒,𝑗, 𝑗 = 1,… 𝐽. (5.14)

ℎ𝑓,𝑗 = 10−3 {ℎ𝑤 + 15330𝐶(

𝑞𝑗

∅𝑒,𝑗)23} , 𝑗 = 1,… 𝐽. (5.15)

𝐶 = 0.0327 + 0.0286 exp (−0.1378 ℎ𝑤) (5.16)

where ℎ𝑤 is the weir height (mm) and 𝑞𝑗 the liquid volumetric flow in each stage (m3/s).

The Jeronimo & Sawistowski correlation predicts clear liquid heights in the spray regime. In this case, ℎ𝑐,𝑗𝑆

(m) is calculated using the total hole area 𝐴ℎ (m2), the active area 𝐴𝑎 (m2) and the hole diameter 𝑑ℎ (mm):

ℎ𝑐,𝑗𝑆 = 10−3 {ℎ𝑐,𝐻2𝑂,𝑗(

996

𝜌𝑗𝐿 )

0.5(1−𝑛)} , 𝑗 = 1,… 𝐽. (5.17)

ℎ𝑐,𝐻2𝑂,𝑗 =0.497

𝐴ℎ𝐴𝑎

−0.791

𝑑ℎ0.833

1 + 0.013𝐿𝑗−0.59 𝐴ℎ

𝐴𝑎

−1.79 , 𝑗 = 1,… 𝐽. (5.18)

40

𝑛 = 0.00091𝑑ℎ𝐴ℎ𝐴𝑎

(5.19)

Finally, the vapour molar holdup 𝑀𝑗𝑉 for a given stage 𝑗 is calculated using the volume available per stage Ѵ

and the liquid volumetric holdup:

ℎ𝑐,𝑗 × 𝐴𝑎 +𝑀𝑗𝑉

𝜌𝑚,𝑗𝑉 = Ѵ, 𝑗 = 1,… 𝐽.

(5.20)

Tray pressure drop 5.2.2.3.

The total pressure drop across a sieve tray is the sum of the pressure drop due to vapour flow friction through

dry tray perforations, ℎ𝑑, and the pressure drop through the aerated liquid mass on a tray, ℎ𝑙.

The dry vapour pressure drop across the disperser unit is estimated using expressions derived for flow

through orifices [16]. In the present work, three distinct dry vapour pressure drop estimations were used: the

standard orifice equation, the Liebson correlation and the Hughmark & O’Connell correlation.

For sieve plates, the pressure drop through the dispersers ℎ𝑑,𝑗 (m) is given by:

ℎ𝑑,𝑗 =50.8

𝐶𝑣2

𝜌𝑗𝑉

𝜌𝑗𝐿 𝑣ℎ,𝑗

2, 𝑗 = 1,… 𝐽. (5.21)

where 𝑣ℎ,𝑗 refers to the vapour velocity through the holes (m/s). Note that SI units should be used when

estimating ℎ𝑑,𝑗 using the above equation. If the simple standard orifice equation is required, a constant value

of 𝐶𝑣𝑆 = 0.75 may be used in the above equation. This standard orifice equation is derived from Bernoulli’s

principle. The more accurate Liebson correlation uses directly the fractional hole area 𝐴ℎ

𝐴𝑎 to estimate the

orifice coefficient 𝐶𝑣𝐿 shown in the above equation. 𝐶𝑣

𝐿 is given by:

𝐶𝑣𝐿 = 0.74

𝐴ℎ𝐴𝑎+ 𝑒

0.29𝑡𝑡𝑑ℎ⁄ −0.56

(5.22)

where 𝑑ℎ is the hole diameter and 𝑡𝑡 the tray thickness. Hughmark & O’Connell’s correlation determines an

orifice coefficient 𝐶𝑣𝐻 fit by the following equation:

𝐶𝑣𝐻 = 0.85032 − 0.04231

𝑑ℎ𝑡𝑡+ 0.0017954(

𝑑ℎ𝑡𝑡)2

(5.23)

The head of liquid required to overcome the pressure drop of gas on a dry tray 𝑗, ℎ𝑑,𝑗𝐻

(m), is then given by:

41

ℎ𝑑,𝑗𝐻 = 5.1204 × 10−5𝑣ℎ,𝑗

2𝜌𝑗𝑉 𝜌𝑤𝑎𝑡𝑒𝑟

𝜌𝑗𝐿 [1 − (

𝐴ℎ𝐴𝑎)2

] 1𝐶𝑣𝐻2⁄ , 𝑗 = 1,… 𝐽. (5.24)

In this dissertation, the pressure drop through the aerated mass ℎ𝑙 was calculated using two distinct

correlations: The Fair correlation and the Bennett aerated liquid pressure drop correlation.

The Fair correlation is based on the concept of correcting clear liquid flows for aeration effects. Hence, the

pressure drop through the aerated mass on stage 𝑗, ℎ𝑙,𝑗𝐹

, is calculated using an aeration factor 𝛽:

ℎ𝑙,𝑗𝐹 = 𝛽𝑗ℎ𝑑𝑠,𝑗, 𝑗 = 1,… 𝐽. (5.25)

𝛽𝑗 = 0.0825 𝑙𝑛𝑞𝑗

𝑙𝑤− 0.269 ln (𝑣ℎ,𝑗𝜌𝑗

𝑉0.5) + 1.679, 𝑗 = 1,… 𝐽. (5.26)

In the above equation, ℎ𝑑𝑠,𝑗 is the sum of the weir height ℎ𝑤 and the height of the crest over the weir ℎ𝑜𝑤,𝑗:

ℎ𝑑𝑠,𝑗 = ℎ𝑑𝑠,𝑗 + ℎ𝑜𝑤,𝑗, 𝑗 = 1,… 𝐽.

(5.27)

The value of ℎ𝑜𝑤,𝑗 is calculated from the Francis weir equation and its modifications for various weir types.

Indeed, in this study ℎ𝑜𝑤,𝑗 is calculated both for segmental (section 6) and circular (section 7) weirs using the

Francis weir equation and its modification for different weir types. For a segmental weir and for height ℎ𝑜𝑤,𝑗𝑆

in meters of liquid:

ℎ𝑜𝑤,𝑗𝑆 = 0.664(

𝑞𝑗

𝑙𝑤)23, 𝑗 = 1,… 𝐽.

(5.28)

An oldershaw laboratory column is modelled in section 7 to validate the dynamic column model. In this case,

the modified Francis equation for circular weirs was used to determine ℎ𝑜𝑤,𝑗 (m), as follows:

ℎ𝑜𝑤,𝑗 = 44.3(𝑞𝑗

𝑑𝑤 × 1000)0.704, 𝑗 = 1,… 𝐽.

(5.29)

where 𝑑𝑤 (m) refers to the weir diameter and 𝑞𝑗 (m3/s) is the volumetric liquid flow. Note that all correlations

listed in this section may apply to columns with circular downcomers considering simply 𝑑𝑤 = 𝑙𝑤.

The Bennett aerated liquid pressure drop correlation departs from the concept of clear liquid flow corrected

for aeration effects. Instead, the correlation uses a model of froth flow across the weir. The pressure drop in

stage 𝑗 through the aerated mass ℎ𝑙,𝑗𝐵

is the sum of the effective clear liquid height ℎ𝑐,𝑗𝐵

(calculated

according to the Bennett effective clear liquid height correlation) and the residual pressure drop due to surface

tension:

ℎ𝑙,𝑗𝐵 = ℎ𝑐,𝑗

𝐵 + ℎ𝑅,𝑗, 𝑗 = 1,… 𝐽. (5.30)

42

ℎ𝑅,𝑗 =472000𝜎𝑗

𝑔𝜌𝑗𝐿 [

𝑔(𝜌𝑗𝐿 − 𝜌𝑗

𝑉)

𝑑ℎ × 𝜎𝑗 × 106 ]13⁄ , 𝑗 = 1,… 𝐽. (5.31)

𝜎𝑗 = 𝑓(𝑇𝑗, 𝑃𝑗, 𝑥𝑗⃗⃗⃗⃗ , 𝑦𝑗⃗⃗⃗⃗ ), 𝑗 = 1,… 𝐽. (5.32)

where 𝜎𝑗 (N/m) refers to the surface tension and ℎ𝑅,𝑗 (m) is the surface tension loss term.

43

6

Tray design and operation n this chapter a methanol/water case study is used to compare the different tray hydraulic correlations

implemented in ModelBuilder. First, the minimum required column diameter is determined using the

Lowenstein, Fair and the Kister & Haas correlations. Secondly, the liquid and vapour holdups on the trays are

estimated using the Bennett effective clear liquid height and the Jeronimo & Sawistowski holdup correlations.

Then, the dry vapour pressure drop is estimated using the standard orifice equation as well as Liebson’s and

Hughmark & O’Connell’s correlations. Finally, the Fair and the Bennett correlations are included in the

simulations to calculate the pressure drop through the aerated mass on the trays. For each of the above-

mentioned cases, a comparison between the different correlations is established and the suitability of each

correlation is analysed considering the operating flow regime in each tray.

6.1. Case study data

Table 6.1 shows the operating and geometrical data for the methanol/water case study. A 20% mass methanol

- 80% mass water mixture of 6.05 kg/s constant flowrate is fed at 1 bar and 95°C to the 6th tray of a

distillation column containing 13 sieve trays, in a continuous distillation system. The geometrical input data

was specified considering typical recommendations suggested by several authors, namely Sinnott [18], Perry

[16] and Azevedo [22]. Hence, the following factors were taken into account:

Fractional active area, 𝐴𝑎

𝐴𝑇: the active area divided by the total column transversal area generally ranges

from 0.7 to 0.9. In the present case study, a value of 𝐴𝑎

𝐴𝑇= 0.82 was specified;

Hole diameter, 𝑑ℎ: the sieve tray hole sizes are usually between 2.5 and 19 mm. The recommended

size for nonfouling systems is 5 mm; larger holes are preferred for fouling systems. In this case, a

usual value of 𝑑ℎ = 4.5 mm was considered;

Fractional hole area, 𝐴ℎ

𝐴𝑎: the ratio of the total area of perforations on the tray to the active area can be

calculated for an equilateral triangular pitch as follows:

𝐴ℎ𝐴𝑎

= 0.907 × (𝑑ℎ𝑙𝑝)2

(6.1)

I

44

where 𝑑ℎ and 𝑙𝑝 are the hole diameter and the hole pitch, respectively. A typical ratio of 𝑑ℎ

𝑙𝑝=

1

2.66

yields the specified fractional hole area 𝐴ℎ

𝐴𝑎= 0.128;

Plate thickness, 𝑡𝑡: typical plate thicknesses are 5 mm for carbon steel and 3 mm for stainless steel.

The ratio of hole diameter to plate thickness generally varies from 1.4 to 2.5. A value of

𝑑ℎ

𝑡𝑡= 2.3 was specified in the present case;

Plate spacing, 𝑙𝑡: the plate spacing is normally between 0.15 m and 1 m and affects the overall height

of the column. For small-diameter columns, close spacing is used [18]. Plate spacings from 0.3 to 0.6

m are generally used for columns above 1 m diameter. As shown later in this chapter, the required

column diameter is above 1 m and therefore a plate spacing of 𝑙𝑡 = 0.6 m is suitable for this

particular case study;

Weir length, 𝑙𝑤: the weir length 𝑙𝑤 is generally between 0.6 to 0.85 of the column diameter. Hence, a

value of 𝑙𝑤

𝐷𝑐= 0.7 was specified;

Weir height, ℎ𝑤: the weir height is a crucial factor in determining the plate efficiency and the volume

of liquid on the plate. Indeed, the plate efficiency increases with a higher weir at the expense of a

higher plate pressure drop. For industrial plates the weir height will normally vary between 40 and 50

mm. As shown in Table 6.1, a 40 mm weir height was chosen for the present case study.

Table 6.1 – Case study methanol/water data.

Operating data

Feed flowrate (kg/s) 6.05

Feed composition (mass %) Methanol – 20

Water – 80

Feed temperature (°C) 95

Feed pressure (bar) 1

Trays 13

Feed tray 6

Geometrical data

Fractional active area 0.82

Fractional hole area 0.128

Hole diameter (mm) 4.5

Hole diameter/pitch length 0.376

Hole diameter/tray thickness 2.3

Plate spacing (m) 0.6

Weir type Segmental

Weir length/column diameter 0.7

Weir height (m) 0.04

45

6.2. Case study results

In the following subsections the column diameter, tray holdup and tray pressure drop equations implemented

in ModelBuilder are analysed for the present case study. Because tray hydraulic correlations, namely holdup

and pressure drop estimations, are chosen according to the type of operating regime, a good understanding of

these flow regimes is required. There are various types of flow regimes that may occur on distillation column

trays under different liquid and vapour flowrates. The froth and spray regimes are the most common

operating regimes in distillation practice [16] and will therefore be analysed in this work.

6.2.1. Flow regime

Figure 6.1 shows the liquid and vapour flowrate profiles for the present case study. It is interesting to note

that the liquid and vapour flowrates remain relatively constant above and below feed tray 6, where a regime

transition is observed. From tray 5 to tray 6 the liquid flowrate is increased from 3.0 kg/s to 5.1 kg/s whereas

the vapour flowrate decreases from 6.6 kg/s to 2.6 kg/s in trays 6 and 7. Figure 6.1 also shows the froth-spray

correlation presented by Porter & Jenkins, where the flow parameter 𝐹𝑙𝑣, the active area 𝐴𝑎 (m2) and the weir

length 𝑙𝑤 (m) are used to determine the froth-spray regime limit, as follows:

𝐹𝑙𝑣,𝑗𝐴𝑎

𝑙𝑤= 0.07, 𝑗 = 1,… 𝐽.

(6.2)

In the rectifying section 𝐹𝑙𝑣,𝑗𝐴𝑎

𝑙𝑤< 0.07 and therefore the column is operating under a spray regime. Indeed, a

spray-type regime is attained under high gas rates and low liquid rates, where jets are formed at each

perforation, with the fraction of the holes that are jetting increasing with vapour velocity. In this case, jetting

is the dominant mechanism as gas passes at high velocity through the pool of liquid held on the tray floor,

carrying and tearing the liquid path.

On the contrary, in the stripping section the inequality 𝐹𝑙𝑣,𝑗𝐴𝑎

𝑙𝑤> 0.07 is observed and consequently the

column is operating under a froth-type regime. Here, the dispersion on the tray is in the form of a bubbly, or

aerated liquid. Vapour bubbles of a wide size range are formed at the tray perforations and circulate at varying

velocities through the liquid, being swept away by the froth. The froth surface is mobile and covered by liquid

droplets.

46

Figure 6.1 – Liquid and vapour flowrate profiles and Porter & Jenkins froth-spray prediction.

6.2.2. Column diameter

Figure 6.2 shows the minimum column diameter estimated using the Lowenstein, Fair and the Kister & Haas

correlations in ModelBuilder for the methanol/water case study. The minimum column diameter vector refers

to all 13 column trays; the maximum value of this vector is the required minimum required diameter.

Figure 6.2 – Minimum required column diameter vector.

The rectifying section of the column is defining the minimum required column diameter: a higher column

diameter is predicted for the first 6 column trays (see Figure 6.2). In this case, a spray-type entrainment

flooding mechanism is more likely to occur since as previously stated the gas rate is high and the liquid rate is

low. Note that all three correlations under study apply to spray entrainment flood predictions.

A required column diameter of 2.1 m, 1.8 m and 1.9 m was estimated according to the Lowenstein, Fair and

the Kister & Haas correlations, respectively. Lowenstein’s correlation does not take into account factors such

0

1

2

3

4

5

6

7

8

1 2 3 4 5 6 7 8 9 10 11 12 13

Flo

wra

te (

kg

/s)

Trays

L

V0

0.02

0.04

0.06

0.08

0.1

1 2 3 4 5 6 7 8 9 10 11 12 13

Flv

.Aa/

Lw

(m

)

Trays

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8 9 10 11 12 13

Min

imu

m d

iam

ete

r (m

)

Trays

Kister & Haas

Fair

Lowenstein

47

as the surface tension and the liquid flowrates in the column and therefore the estimated diameter is most

likely less accurate. This simplified correlation generates the most conservative value for the minimum column

diameter since a relative deviation of +18% from Fair’s correlation is obtained. Note that combining an

adequate accuracy with an acceptable calculation complexity is a key factor to consider when applying these

hydraulic correlations.

As previously mentioned, the Fair flood correlation has been the standard of the industry for entrainment

predictions and may be used for both froth and spray entrainment flood estimations. Because the ratio of

perforation area to active area is greater than 0.1 and the hole sizes are smaller than 13 mm, the correlation

applies to this particular case study. Moreover, since the plate spacing, hole diameter, fractional hole area and

weir height are between 36-91 cm, 0.32-2.5 cm, 0.06-0.2 and 0-7.6 cm, respectively, the Kister & Haas

correlation also applies to this case study. Both the Fair and the Kister & Haas correlations take into account

the surface tension that bubbles must overcome due to the difference between the fluid pressure and the

internal bubble pressure. The minimum required column diameter predicted by both correlations is similar

with the Kister & Haas method estimating a 7% larger column diameter when compared to the Fair

correlation.

6.2.3. Tray holdup

Figure 6.3 shows the liquid molar holdup profiles calculated using Bennett’s and Jeronimo & Sawistowski’s

clear liquid height correlations. The vapour holdup profiles are also shown in Figure 6.3 and were determined

using the volume, the liquid molar holdup and the liquid and vapour molar densities for each stage.

In the rectifying section the column is operating under a spray regime and therefore the Jeronimo &

Sawistowski should be used if precise liquid holdups are to be determined. Nevertheless, in this particular case

the two correlations predict a similar and approximately constant liquid holdup with a maximum relative

difference of 6.9% for tray 1, where the Bennett correlation generates a liquid holdup of 910 moles while the

Jeronimo & Sawistowski correlation predicts a liquid holdup of 977 moles. The vapour holdup profiles in the

rectifying section are practically identical, with a maximum relative difference of 0.06% occurring in tray 1.

As explained previously, in the stripping section a froth-type regime is observed and thus the Jeronimo &

Sawistowski correlation is not applicable. In this case, the Bennett correlation should be used. Note that the

two correlations estimate an approximately constant liquid molar holdup profile. However, for this lower

column section using the correct correlation is crucial since the Bennett holdup correlation generates a

significantly high average liquid molar holdup of 2517 moles whereas the Jeronimo & Sawistowski correlation

estimates an average liquid holdup of only 1658 moles. The vapour holdup profile predicted by both

48

correlations for this section is also relatively constant and in this case, similar vapour molar holdup values are

obtained for each stage, with a maximum relative difference of 1.6% occurring in tray 6.

Figure 6.3 – Liquid and vapour molar holdup profiles.

6.2.4. Tray pressure drop

The two main sources of tray pressure loss are the dry vapour pressure loss through the tray orifices and the

pressure loss generated by the aerated liquid holdup on the tray. The vapour and the aerated liquid head loss

profiles are represented in Figure 6.4, with the former being predicted according to the standard orifice

equation, the Hughmark & O’Connell correlation and the Liebson correlation and the latter calculated using

Fair’s and Bennett’s aerated liquid head loss predictions.

The Hughmark & O’Connell and the Liebson correlations should theoretically predict a more accurate dry

vapour head loss than the standard orifice equation, since the latter assumes a constant orifice coefficient of

0.75 while in the first two cases this same orifice coefficient is estimated using geometrical inputs such as the

fractional hole area, the tray thickness and the sieve tray hole diameter. Nonetheless, for this particular case

study using the simpler standard orifice equation would suffice. Indeed, from Figure 6.4 it can be noted that a

similar and constant dry vapour head loss profile is predicted by the standard orifice equation, the Hughmark

& O’Connell correlation and the Liebson correlation for the two distinct column sections. Because in the

rectifying section the column is operating under a spray-type regime with high vapour velocities, the predicted

vapour head loss is significantly higher than in the stripping section, where a froth-type regime with lower

vapour rates is established. For example, an average dry vapour head loss of 65.7 mm is given by the standard

orifice equation for the rectifying section, whereas this value drops to 10.2 mm in the bottom column stages,

i.e., an 84.5% reduction in the dry vapour head loss is observed.

An approximately constant aerated liquid head loss profile is also predicted for the two column sections

according to the Fair and the Bennett correlations. Here, the effect of the liquid flowrate on the aerated liquid

0

500

1000

1500

2000

2500

3000

3500

4000

1 2 3 4 5 6 7 8 9 10 11 12 13

Liq

uid

ho

ldu

p (

mo

l)

Trays

BennettJeronimo & Sawistowski

50

54

58

62

66

70

1 2 3 4 5 6 7 8 9 10 11 12 13

Vap

ou

r h

old

up

(m

ol)

Trays

BennettJeronimo & Sawistowski

49

head loss is noticeable: for the rectifying section the liquid flowrates are significantly lower and therefore for

both correlations the liquid head loss is almost half of the value calculated for the stripping section.

Theoretically the Fair correlation should estimate liquid head losses more accurately in the upper column

section since using the Bennett correlation is not recommended for spray-type regimes. However, for the

rectifying section, an average aerated liquid head loss of 17.0 mm is given by the Bennett correlation resulting

in only a 7.0% relative difference from the value predicted by the Fair correlation. Thus, for this particular

case study using either correlation would not compromise the liquid head loss profile prediction in the upper

column section. A larger discrepancy can be noted for the bottom column stages where the Fair correlation

estimates an average liquid head loss of 34.1 mm, with a 16.5% relative difference from the 29.2 mm value

predicted by the Bennett correlation. In this case, although both estimations apply to froth-type regimes, the

Bennett correlation should predict liquid head losses more accurately. Indeed, unlike the Fair correlation the

Bennett aerated liquid loss correlation does not correct the clear liquid flow for aeration effects by including a

tray aeration factor. The total aerated liquid pressure loss is calculated by separately summing the clear liquid

height term and the surface tension loss term. Hence, the Bennett correlation should be chosen when the

surface tension loss term has a significant effect on the total liquid pressure loss.

Figure 6.4 – Dry vapour and aerated liquid head loss profiles.

Figure 6.5 shows the clear liquid and the surface tension head loss contributions towards the total aerated

liquid pressure drop in the stripping section, where the Bennett correlation is applicable. The surface tension

loss term consistently contributes 33% to the total aerated liquid head loss in all the column bottom stages.

Consequently, the effect of the surface tension term is substantial and the Bennett correlation should indeed

be used for accurate pressure drop predictions. It should be noted that for cases where the surface tension

contribution is small, the Fair correlation generates accurate pressure drop predictions. In this case the benefit

of simplifying the prediction by avoiding surface tension, froth density and froth height calculation terms

should be considered.

0

10

20

30

40

50

60

70

80

1 2 3 4 5 6 7 8 9 10 11 12

h v

ap

ou

r (m

m)

Trays

Standard orifice equation

Hughmark & O'Connell

Liebson

0

5

10

15

20

25

30

35

40

1 2 3 4 5 6 7 8 9 10 11 12

h a

era

ted

(m

m)

Trays

Bennett

Fair

50

Figure 6.5 – Clear liquid and surface tension contributions to the aerated liquid head loss.

6.3. Conclusion

This chapter showed that the tray hydraulic correlations comprising minimum column diameter calculations

and tray holdup and pressure drop estimations were successfully implemented in ModelBuilder. The

complexity level and the accuracy of these correlations were analysed taking into account the type of regime

operating in the continuous distillation unit. To model batch distillation columns including the energy and

mass holdup variations on each stage, tray holdup and tray pressure drop correlations are required. For this

reason, the subsequent chapter will include a batch distillation column with circular downcomers where the

Francis formula for circular weirs and the standard orifice equation will be used to determine the tray pressure

loss and the liquid and vapour holdups.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

6 7 8 9 10 11 12

Co

ntr

ibu

tio

n a

era

ted

liq

uid

h

ead

lo

ss

Trays

Surface tension

Clear liquid

51

7

Multi-staged batch distillation his chapter shows the implementation of a multi-staged batch distillation system in ModelBuilder based

on a pilot plant experiment lead by Bonsfills [45]. After presenting the problem data given by Bonsfills,

the ModelBuilder distillation flowsheet as well as particular characteristics of the ModelBuilder model, such as

controllability and initial conditions, are assessed. To achieve the same steady-state conditions obtained by

Bonsfills, a section analysing the total reflux ModelBuilder simulations is presented. Then, the simulation

results generated by the finite reflux simulations are validated for the methanol/water system against the

experimental and theoretical data provided by Bonsfills. Finally, the impact of the reflux ratio and the heat

duty supplied to the reboiler is analysed by conducting a sensitivity analysis.

7.1. Operating conditions and Validation data

In this section the pilot experiment lead by Bonsfills for a methanol/water batch multi-staged distillation is

introduced. Bonsfills compared experimental data with two theoretical models that are also presented in this

section: a model implemented in Fortran language and a second model obtained using commercial simulator

Batchsim of Pro/II.

7.1.1. Experimental pilot plant

Bonsfills [45] carried out a pilot plant experiment for a methanol/water batch separation. The pilot plant was

made up of a 15 trays Oldershaw column of 3.75 m height and 50 mm inner diameter. This laboratory

apparatus is a metal sieve-plate column with center-to-side downcomers developed by Oldershaw [13]. The

column worked with an average absolute pressure of 1 atm. To reduce heat losses, an insulation jacket was

added.

A 4 L 20 mol% methanol/80 mol% water mixture was charged to a 6 L reboiler with a maximum heat duty of

1400 W. The distillation still was heated up with a constant heat duty of 933.3 W under total reflux until

steady-state conditions were attained. Then, the column operated under a finite constant reflux ratio regime

with product being cooled in a total condenser and withdrawn at the top of the column.

T

52

Figure 7.1 shows the pilot plant instrumentation scheme. The reflux ratio was controlled automatically with an

electromagnet. The column had two pressure transducers, one to measure the absolute pressure at the top of

the column and the other one, installed between the top and the bottom of the column, was used to measure

the relative pressure. The plant also included two control valves, one to control the water flowrate supplied to

the condenser and the other to control the absolute pressure at the top of the column. Temperatures were

measured on-line at seven different locations with temperature transducers Pt-100 and liquid phase

compositions were analysed off-line by gas chromatography with 1 mL samples. The distillate flowrate was

also measured off-line by registering the distillate weight in a scale throughout the experiment.

Figure 7.1 – Batch distillation pilot plant - Bonsfills [45].

7.1.1. Fortran and Batchsim models

Bonsfills compared the experimental results with simulations obtained from a batch distillation model

implemented in Fortran language and with results given by commercial simulator Batchsim of Pro/II. In this

thesis the Fortran and Batchsim models will also be used to validate the ModelBuilder column batch model.

Table 7.1 shows the specifications used in the Fortran and Batchsim models. Both models assume 15

theoretical trays with constant liquid and negligible vapour holdups. However, the vapour-liquid equilibrium is

53

determined in the Fortran model using a constant relative volatility of 4.02 and in Batchsim using the

NRTL/Virial thermodynamic models.

Table 7.1 – Fortran and Batchsim model specifications.

7.2. gPROMS ModelBuilder® model

Figure 7.2 shows the flowsheet used to simulate the multi-staged batch system in ModelBuilder. A

methanol/water mixture contained in reboiler E-114 is heated using a constant heat duty. Vapour is formed in

the reboiler and fed to the bottom of column D-110. A vapour-liquid equilibrium process occurs on all

column stages, with liquid flowing from the top of the column to the bottom and vapour ascending

throughout the column section. The top section of the column is enriched with the lighter component

methanol as the bottom section composition in water increases. The liquid leaving the bottom of the column

enters the reboiler, closing the bottom loop. H-119 represents the column sump, i.e., the liquid holdup that is

held on the bottom of the distillation column, below the column stages. The vapour leaving the upper section

of the column is condensed in condenser E-121 and enters the splitter, where the reflux ratio is defined.

For a given reboiler heat duty, two distinct ModelBuilder simulations are scheduled. First, a simulation is

designed to achieve steady-state conditions by operating the column under total reflux. In this case, a reflux

ratio of 𝑅𝑅 =𝐿

𝐷= ∞ is specified via splitter M-123 and no distillate is withdrawn to product receiver F-140.

Note that in the splitter the reflux ratio is indirectly defined by using the distillate split fraction (distillate

flowrate divided by the total inlet splitter flowrate).

Specification Fortran/Batchsim models

Initial reboiler charge (mol) 184.32

Initial reboiler charge composition (mol%) Methanol – 20

Water - 80

Column diameter (mm) 50

Number of column stages 15

Tray 1 pressure (bar) 1.013

Vapour-liquid equilibrium Fortran: α = 4.02

Batchsim: NRTL/Virial

Stage modelling Equilibrium stages

Column liquid flowrate Constant

Column vapour flowrate Constant

Tray liquid holdup Constant

Tray vapour holdup Negligible

Average holdup per stage (mol) 0.175

Column heat loss Adiabatic

Reflux ratio 3

Condenser type Total

Condenser holdup (mol) 0.175

54

All the initial conditions specified in this chapter refer to the initial set of inputs that are given at time 𝑡 = 0 in

the first total reflux simulation. The second simulation uses a particular split fraction based on the desired

reflux ratio. This simulation starts operating in a steady-state mode, using a saved variable set from the first

simulation, that is, all the values assigned to the variables at the end of the first simulation (when steady-state

is achieved) are saved and used to initialise the second simulation. The fact that a saved variable steady-state

set can be used to start a second simulation is an important benefit provided by the ModelBuilder tool: the

time required to run simulations is significantly reduced by saving this steady-state.

Globally the system is pressure-driven, that is, the mass flowrates are calculated in the valve model based on

pressure differentials. For this reason, valves were introduced before and after each equipment.

The input specifications for each sub-model, based on the Bonsfills experiment, are described below.

55

Figure 7.2 – Batch distillation column flowsheet.

7.2.1. Column

The input specifications for the ModelBuilder column model are listed in Table 7.2. The plate spacing of 0.25

m specified in the ModelBuilder simulations was determined by dividing the column height (3.75 m) by the

number of stages. The following input values were calculated considering suggestions given by Seader [13],

Perry [16] and Sinnott [18]: 0.8 active area fraction, 0.1 hole area fraction and 0.7 weir fraction.

56

The equilibrium stages are numbered from top (stage 1) to bottom (stage 15). Feed stage 5 is connected to a

dummy source S-111 with zero flowrate. This source was only added to introduce the Multiflash physical

property package which includes the VLE data for the methanol/water system.

The dry vapour pressure drop and the pressure drop through the aerated liquid were estimated using the

standard orifice equation with an orifice coefficient of 0.75 and the hydrostatic liquid height given by the

Francis formula for circular weirs. Note that the Francis formula defines the liquid flowrate profile in the

column as a function of the difference between the liquid level and the weir height. Thus, an adequate weir

height must be supplied. A weir height of 4.46 mm generates for all the simulations studied in this chapter an

average holdup on the plates of 0.175 moles, within a 9.2% relative error. Indeed, the average molar holdup

specified experimentally by Bonsfills is 0.175 and this value determined indirectly the weir height specified in

the ModelBuilder column model.

Table 7.2 – Column specification data.

The initial conditions specified in the total reflux simulations are listed below. Because the system is pressure-

driven, the liquid and vapour flowrates are a function of the vapour and liquid head losses. At time 𝑡 = 0, all

the mass and energy holdups must be given initial values to account for the M and H differential equations.

These mass and energy holdups were indirectly defined by specifying the set of initial conditions listed in

Table 7.2 and in Table 7.3, that is, the initial liquid level fraction, the initial pressure and the initial holdup

Tab Specifications

Column Number of stages : 15 Dummy feed stage: 5

Design

Tray type: sieve Column diameter: 50 mm Active area fraction: 0.8 Hole area fraction: 0.1 Weir fraction: 0.7 Weir height: 4.46 mm Plate spacing: 0.25 m

Pressure

Dry vapour pressure drop correlation: standard orifice equation Aerated liquid pressure drop correlation: hydrostatic Liquid height correlation: Francis formula for circular weirs

Dynamics

Mode: pressure-driven Initial liquid level fraction trays 1-15: 0.02185 Initial pressure: see Table 7.3 Initial holdup composition: see Table 7.3

Tray efficiencies Tray modelling: equilibrium

57

compositions for each tray. Note that in this case the liquid level fraction is the ratio of liquid level to plate

spacing.

Table 7.3 – Column initial pressures and holdup compositions on each tray.

7.2.2. Sump

H-119 represents the column sump, i.e., the liquid holdup that is held on the bottom of a distillation column,

below the column stages. From a modelling point of view, the sump is of significant importance. Without the

sump, the liquid would flow directly from the last stage of the column to the reboiler, with an intermediate

valve in between. Because the pressure and the flowrate of the stream leaving the column are known variables

calculated in the column model, the flowrate entering the reboiler would not be calculated in the valve model

based on the pressure differential between the last column stage and the reboiler. Both the pressure and

flowrate of the stream entering the reboiler would be known variables and therefore the system would not be

well posed. Hence, to calculate the flowrate entering the reboiler based on the pressure differentials a sump

must be introduced. In this case, the flowrate and pressure entering the sump are known variables, calculated

in the column model. The outlet sump pressure is calculated in the sump model using the hydrostatic pressure

associated to the liquid holdup. However, the flowrate of the stream leaving the sump (or likewise, entering

the reboiler) is an unknown variable and therefore may be calculated in the valve model by introducing a valve

between the sump and the reboiler.

For all the total reflux simulations analysed in this work, the sump specifications are the following:

0.1 L cylindrical vessel with flat heads and 5 cm diameter; this diameter value of 5 cm was based on

the 5 cm column diameter;

Tray Initial Pressure Initial methanol

composition (mol %)

1 1.113 80.8

2 1.121 80.2

3 1.128 80.0

4 1.136 78.7

5 1.143 77.8

6 1.151 76.8

7 1.159 75.7

8 1.166 74.1

9 1.174 71.9

10 1.182 69.0

11 1.189 64.3

12 1.197 64.3

13 1.205 64.3

14 1.213 64.3

15 1.2205 64.3

58

Initial temperature of 76.9 ºC;

0.312 and 4.440 initial molar holdups of methanol and water, respectively;

Constant liquid level fraction of 0.6.

The level controller (LC-119) and the flow controller (FC-119) input specifications are listed in Table 7.4. The

two controllers were added to maintain a constant ratio of liquid level to sump height of 0.6. Hence, during

the simulations the liquid level in the sump is maintained at a constant value of 3.1 cm.

A cascade control scheme was implemented using the level and the flow controllers. In this arrangement, the

constant liquid level in the sump is maintained by varying the sump outlet flowrate, instead of directly

changing the sump outlet valve V-117 stem position, which would be the case in a simple (non-cascade)

control scheme. In the primary loop, the level controller monitors and compares the liquid level fraction to

the specified set-point of 0.6, changing the outlet sump flowrate set-point accordingly. In the secondary loop,

the flow controller reads and compares the flowrate input value to the flowrate set-point, manipulating the

sump outlet valve stem position as necessary.

It is interesting to understand the relation between the outlet sump flowrate and the liquid level in the sump

for this particular pressure-driven system. In this case, lower liquid outlet flowrates are associated with higher

liquid levels and vice-versa. Indeed, if the outlet sump flowrate decreases, the differential pressure between the

reboiler and the sump outlet is also lower. Hence, for a given reboiler pressure, the sump outlet pressure must

also decrease. Since the inlet sump pressure is determined in a flow-driven manner (see above paragraphs), the

hydrostatic pressure, that is, the liquid level must increase to maintain a lower sump outlet pressure.

A proportional-integral type control was implemented for controllers LC-119 and FC-119. In this case, the

proportional steady-state error is avoided dynamically by adding the integral term with an integral time

constant of 0.1 s for both controllers. A maximum and minimum value for the input and output variables was

also specified for both controllers. For example, for controller LC-119 the input value (liquid level fraction) is

set between 0 and 1 and the output variable (outlet sump flowrate) is maintained between 0 and 10 kg/s.

59

Table 7.4 – Controllers LC-119 and FC-119 specifications.

7.2.3. Reboiler

Reboiler E-114 was modelled using a two-phase equilibrium stage separator, that is, using the Rayleigh set of

equations described in section 4.1. The reboiler volume, initial liquid level fraction and initial holdup

compositions were chosen according to the Bonsfills experiment. Hence, the reboiler specifications given

ModelBuilder are the following:

6 L cylindrical vessel with flat heads and 19.5 cm diameter;

Initial pressure of 1.221 bar;

Initial liquid level fraction of 0.631;

80.1 mol% and 19.9 mol% initial holdup compositions of water and methanol, respectively;

Constant heat duty specified in each simulation.

Controller Specifications

LC-119

Controller class : PI Controller action: direct Controlled variable: sump liquid level fraction Manipulated variable: sump outlet mass flowrate Proportional gain: 50 Integral time constant: 0.1 s Set-point: 0.6 Minimum input: 0 Maximum input: 1 Minimum output: 0 kg/s Maximum output: 10 kg/s

FC-119

Controller class : PI Controller action: direct Controller mode: cascade Controlled variable: sump outlet mass flowrate Manipulated variable: outlet sump valve stem position Proportional gain: 100 Integral time constant: 0.1 s Minimum input: 0 kg/s Maximum input: 10 kg/s Minimum output: 0 Maximum output: 1

60

S-116 refers to the reboiler liquid outlet sink and represents a dummy stream with zero flowrate. The

associated valve V-115 is therefore closed. The sink was added merely to comply with the mandatory material

outlet required by the reboiler separator model.

7.2.4. Condenser

The vapour distillate is condensed in total condenser E-121 where a constant sub-cooling temperature

difference of 5ºC and a negligible pressure drop are maintained. For the ModelBuilder simulations the

dynamic mode of the condenser was turned off which in practice means that a negligible holdup was assumed

in E-121.

7.2.5. Splitter

Splitter M-123 is a critical component of the dynamic system. Indeed, the splitter ensures two main functions:

defining the reflux ratio and stabilizing the system by including holdup accumulation, if such balance is

required.

It is interesting to analyse how the reflux ratio is defined in the splitter. The reflux ratio is perfectly controlled

by specifying the required split fraction and un-assigning the pump pressure increase. For a particular reflux

flowrate, reflux valve V-124 generates a given pressure drop which is perfectly compensated by pump L-126,

where the required pressure increase is established. Note that in the Bonsfills experiment the reflux ratio is

automatically controlled with an electromagnet.

Globally the entire flowsheet is run in a pressure-driven mode, i.e., the pressure differentials define the

system’s flowrates. However, the splitter itself is defined in a flow-driven mode. A constant pressure is

assigned to the liquid surface of product container F-140. Consequently, the splitter outlet pressure is

calculated using the liquid surface pressure and the pressure drop generated by valve V-130, placed before

product receiver F-140. Additionally, the inlet splitter pressure is also determined upstream. Because both the

inlet and outlet splitter pressures are calculated in this manner, the inlet and total outlet splitter flowrates may

also be calculated in valve models V-122 (inlet) and V-124/V-130 (outlet), by using the pressure differential

before and after the valves. Hence, both the pressure and the flowrate are well defined in the inlet and outlet

of the splitter and this is the reason why the splitter is run in a flow-driven mode.

As stated previously for a given reboiler heat duty two simulations are scheduled: a first simulation designed to

achieve steady-state conditions operating under total reflux and a second simulation where a constant reflux

ratio is assigned. From a modelling point of view having the splitter operating in a dynamic mode with a

specified volume during the total reflux simulation is beneficial. Indeed, because the splitter stabilizes the

flowrates of the system by containing a non-negligible holdup, the simulation is less likely to fail. Hence, all

61

total reflux simulations include a 1 L cylindrical splitter with flat heads and 10 cm diameter. Note that

introducing a splitter with non-negligible holdup is an artificial tool designed exclusively to help reach the

required steady-state.

In the Bonsfills experiment there is no significant mass holdup in the top portion of the separation system. In

this case, the condenser contains a small holdup and no reflux tank is included. To validate the ModelBuilder

simulations against the results provided by Bonsfills, having a splitter with negligible holdup during the finite

reflux simulations is critical. The products should not be retained in the splitter, compromising the results

validation. For this reason, at the start of the constant finite reflux simulations the splitter was re-assigned a

negligible volume. From a modelling perspective this change does not affect the steady-state conditions at the

start of the finite reflux simulations. Once steady-state is achieved the splitter’s inlet and outlet flowrates,

pressures, temperatures and compositions are identical. Consequently, if the splitter volume is re-assigned a

negligible value all the variables of the entire system remain the same, i.e., the steady-state is not modified. The

only effective change in the system is the splitter holdup which is no longer included. This modification was

taken into account by not considering any splitter holdup when calculating the steady-state molar holdup for

the system.

7.2.6. Pump

The dynamic mode was turned off for pump L-126. As stated above, the pump is working in a perfect control

mode and no characteristic curves are included. The pump pressure increase is not specified and is used to

achieve the desired reflux ratio specified by the splitter split fraction.

7.2.7. Recycle breakers

Recycle breakers are models that facilitate the initialisation procedure of simulations that contain closed loops.

The initialisation procedures help resolve the system, i.e., find a solution for time 𝑡 = 0 by defining the set of

equations that are solved as a first calculation step at time 𝑡 = 0 independently in each model.

In the present case, two recycle breakers are needed: R-128 and R-132. During initialisation all the models are

solved sequentially using the initial guesses provided by the recycle breakers. Hence, in this first step the

reboiler equations are solved first, being followed by the column section and the product receiver equations.

The following initial guesses were assigned to recycle breaker R-128:

Inlet and outlet pressure of 1.15 bar;

Temperature of 62.4 ºC;

80.9 mol% methanol and 19.1 mol% water stream.

62

An initialisation procedure starting with a zero flowrate stream was chosen for Recycle-breaker R-132. The

outlet pressure was assigned an initial guess value of 1.23 bar.

7.2.8. Product receiver

Pure methanol is withdrawn during the finite reflux simulations to product receiver F-140. For this reason, a

tank model identical to the sump model was added. In this case, the following specifications were given:

8 L cylindrical receiver with flat heads and 21.7 cm diameter;

Constant liquid surface pressure of 0.9999 bar;

Initial temperature of 66.3 ºC;

Initial negligible holdup of 3,12 × 10−4 and 5.55 × 10−4 moles of methanol and water, respectively.

Note that S-142 is the product receiver sink which is representing a dummy stream with zero flowrate. As was

the case with the reboiler outlet, the product receiver sink was added to comply with the mandatory material

outlet required by the tank model.

7.3. Total reflux

As previously stated, a first total reflux simulation is scheduled in ModelBuilder to achieve the same steady-

state conditions given by Bonsfills. Indeed, the experimental column operates under a constant heat duty of

933.3 W. After steady-state is achieved the system operates with a constant finite reflux ratio of 𝐿

𝐷= 3.

Figure 7.3 shows the temperature profile for column stages 1, 10, 12, 13, 14 and 15 obtained in the total reflux

ModelBuilder simulation for a heat duty of 933.3 W. It is interesting to note that the steady-state temperature

does not vary significantly in the upper half of the column, that is, the temperature only increases from 65.5°C

to 65.7 °C for trays 1 and 10, respectively. On the contrary, on the bottom portion of the column the steady-

state temperature varies significantly from stage to stage. Indeed, between trays 10 and 15 a temperature

difference of 7.3°C is observed. At steady-state the first nine stages of the column contain mostly methanol

which is the lightest component of the mixture. Hence, the temperature of these stages is close to the boiling

point of this component (64.7 °C).

63

Figure 7.3 – Total reflux temperature profiles for column stages 1, 10, 12, 13, 14 and 15 and Q = 933.3 W.

From Figure 7.3 it can be noted that at the beginning of the total reflux simulation a sharp decrease in

temperature is predicted for all column stages. For example, for tray 14 the temperature decreases from

75.3°C at 𝑡 = 0 to 69.3°C at 𝑡 = 1.67 min. The reason for this sharp variation can be explained by analysing

the reboiler behavior at the beginning of the simulation.

Figure 7.4 illustrates the reboiler pressure profile for the same total reflux simulation of 933.3 W heat duty. In

this case, a sharp decrease from the initial specified pressure of 1.221 bar is shown. The initial specified

pressure is significantly higher than the steady-state reboiler pressure (1.05 bar). Consequently, when the

pressure decreases sharply in the reboiler in the beginning of the simulation a significant amount of vapour is

formed, i.e., this pressure decrease is associated with a vapour fraction increase in the reboiler. During this

period of time, the reboiler vapour outlet flowrate is also markedly high (0.15 kg/s vs. 6.3 × 10−4 kg/s at

steady-state). Because a significant initial amount of vapour is sent to the column at the very beginning of the

simulation, the pressure in the column stages for times close to 𝑡 = 0 is very high, even more so than the

initial specified pressure profile listed in section 7.2.1. Consequently, the equilibrium temperature in the

column stages is also high in the beginning of the simulation. As the system stabilizes, the temperature

decrease in the column stages shown in Figure 7.3 can be noticed.

64

66

68

70

72

74

76

78

0 5 10 15 20 25 30 35 40

T(°

C)

Time (min)

Tray 1Tray 10Tray 12Tray 13Tray 14Tray 15

64

Figure 7.4 – Reboiler pressure profile for Q=933.3 W in total reflux simulation.

In the Bonsfills experiment a heat duty of 933.3 W was supplied to the reboiler. To validate the ModelBuilder

results against the experimental data, the same average distillate flowrate must be obtained for a given reflux

ratio.

Figure 7.5 shows the distillate flowrate profile obtained in the second ModelBuilder simulation set (using a

constant specified reflux ratio of 3) as well as the average distillate flowrate provided for the experiment. Note

that this second simulation starts at steady-state using the saved variables obtained at the end of the previous

total reflux simulation. In the Bonsfills experiment a heat duty of 933.3 W leads to an approximately 0.28

mol/min distillate flowrate, which is considerably lower than the 0.35 mol/min average value given by the

ModelBuilder simulation. The fact that the experimental Oldershaw column is not adiabatic may explain this

discrepancy. This means that an actually heat loss to the exterior may occur in the experiment conducted by

Bonsfills. Hence, the next step to adequately validate the ModelBuilder dynamic column model is to

determine the theoretical heat duty that generates an identical 0.28 mol/min flowrate. From Figure 7.5 it can

also be noticed that a theoretical heat duty of 736.4 W generates a 0.28 mol/min average distillate flowrate,

matching the experimental value. Hence, if a 21.1% heat loss equivalent to a 196.9 W heat loss is assumed for

the Bonsfills experiment, the distillate flowrate profiles become identical and the validity of the ModelBuilder

column model may be assessed. For this reason a heat duty of 736.4 W was used in ModelBuilder to validate

all the simulation results.

1

1.05

1.1

1.15

1.2

1.25

0 5 10 15 20 25 30 35 40

P (

bar)

Time (min)

65

Figure 7.5 – Distillate flowrate profiles for RR=3 and Q=933 W or Q=736 W.

The distillate flowrate profiles illustrated in Figure 7.5 also show that for a given heat duty the pure methanol

distillate flowrate in the first period of the simulation is higher than the pure water distillate flowrate in the

final simulation phase. This is explained by the fact that at normal pressure methanol has a lower latent heat

of vaporization than water (35.3 kJ/mol vs 40.68 kJ/mol) and consequently for the same heat duty a higher

flowrate is obtained for a pure methanol distillate, when compared to a pure water product.

In the Bonsfills methanol/water experiment an initial volume of 4L of a 20 mol% methanol and 80 mol%

water mixture was charged to a distillation still at atmospheric pressure and heated up under total reflux until

steady-state conditions were achieved. Because no product is withdrawn in the total reflux regime, the molar

holdups of methanol and water were maintained during this period at a constant value of 33.7 moles and

135.0 moles, respectively.

As stated previously, the specifications for each model in ModelBuilder are given in each total reflux

simulation for a particular reboiler heat duty 𝑄. To validate the batch dynamic model, the steady-state that is

obtained in the Bonsfills experiment must be reproduced. Hence, the specified initial conditions must

generate a steady-state with the same total holdups of methanol and water in the system, that is, 33.7 moles

and 135.0 moles, respectively.

The steady-state molar holdups obtained in the total reflux ModelBuilder simulation using a heat duty of 736.4

W are listed in Table 7.5. In the steady-state regime the holdups in the system are split between the reboiler,

the 15 column stages and the sump. The Bonsfills experiment yields an average steady-state molar holdup of

0.175 per column stage which is approximately the value obtained in the ModelBuilder simulation, within a

9.2% error. The required total steady-state molar holdups are obtained for the ModelBuilder simulation with a

relative error of 0.25%. The methanol and water holdup errors are presented in Table 7.5, translating the

relative deviations between the ModelBuilder steady-state holdups and the steady-state holdups of 33.7 moles

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5 6 7 8 9

Dis

till

ate

flo

wra

te (

mo

l/m

in)

Time (hr)

Experimental (average)

ModelBuilder Q=933.3 W

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7 8 9

Dis

till

ate

flo

wra

te (

mo

l/m

in)

Time (min)

Experimental (average)

ModelBuilder Q=736 W

66

of methanol and 135.0 moles of water given by Bonsfills. Because the required steady-state is reached in the

total reflux ModelBuilder simulation, the results may now be validated against the data given by Bonsfills.

Table 7.5 - Reboiler, column and sump steady-state holdups for Q=736.4 W.

7.4. Model validation

In this section the batch multi-staged ModelBuilder model is validated against temperature and composition

profiles provided by Bonsfills for the distillate, for plate 5 and for the reboiler. The Bonsfills experimental data

is compared with the ModelBuilder, Fortran and Batchsim theoretical results and the discrepancies are

explained by analysing the theoretical simplifications assumed for the models.

7.4.1. Column section

The left-hand side of Figure 7.6 shows the molar methanol fraction profile in the distillate for a ModelBuilder

simulation with a constant reflux ratio 𝑅𝑅 = 3 and a reboiler heat duty of 736.4 W. The same profile is also

illustrated for the Bonsfills experiment, Fortran model and the Batchsim model for a reflux ratio of 3 and for

the heat duty required to generate an average distillate flowrate of 0.28 mol/min. Note that at time 𝑡 = 0 the

system is operating under steady-state conditions.

Globally three distinct time periods can be noted: a first period where a plateau occurs for a molar methanol

fraction of 1; a second phase where the methanol molar fraction decreases and a final period where pure water

is withdrawn from the column.

During the first period pure methanol is withdrawn from the column and collected in the product receiver. In

this work, the first plateau duration is defined by the methanol purity, i.e., the first plateau ends when the

methanol purity falls below 0.99. The estimated duration of this first plateau is 94 min for the ModelBuilder

model, 105 min for the Fortran model and 112 min for the Batchsim model. Because Bonsfills obtained an

experimental plateau time of 80 min, ModelBuilder gives the closest estimation to this value, with an error of

17.2%. The methanol plateau duration is directly related to the distillate flowrate profile since higher distillate

Reboiler Column Sump Total Holdup error (%)

Methanol 30.2 moles 0.180 moles

per plate 1.017 moles 33.9 moles 0.59

Water 134.2 moles 0.0108 moles

per plate 0.807 moles 135.2 moles 0.15

Methanol+water holdup

164.4 moles 0.191 moles

per plate 9.2 % error

1.824 moles 169.1 moles 0.25

67

flowrates in this period mean that more pure methanol is being withdrawn per unit of time and consequently

shorter first plateau durations are obtained. This fact explains why the ModelBuilder estimation gives the

closest approximation to the experimental methanol plateau duration: both the experimental setting and the

ModelBuilder model use a constant heat duty; because methanol has a lower heat of vaporization than water,

to maintain an average distillate flowrate of 0.28 mol/min, the experimental and ModelBuilder distillate

flowrate profiles must reveal higher flowrates in the pure methanol withdrawal period (>0.28 mol/min) when

compared to the pure water withdrawal period (<0.28 mol/min) and thus shorter methanol periods are to be

expected in these cases. Both the Fortran and Batchsim models assume a constant distillate flowrate of 0.28

mol/min and thus the pure methanol withdrawal period lasts longer.

The second phase of the distillation process generates off-cuts, impure methanol/water mixtures that in an

industrial set context would be collected separately and recycled to a next batch. The off-cut duration refers to

the period where impure methanol/water products are obtained, i.e., where 0.01 < 𝑦𝑚𝑒𝑡ℎ𝑎𝑛𝑜𝑙 < 0.99. To

satisfactorily compare the off-cut periods, each one of the four profiles on the left-hand side of Figure 7.6 was

shifted in the xx axis by a time equal to the first plateau duration, thereby eliminating this first plateau (see

Figure 7.6- right-hand side). The shifted transformations show that the ModelBuilder and the Batchsim slopes

are close and almost parallel in this second time period, predicting very sharp decreases in the methanol

composition, i.e., efficient separations linked to short off-cut periods. The models used to determine the VLE

data might explain the off-cut slopes: both the ModelBuilder and the Batchsim models use the NRTL

equations to determine the activity coefficients of the liquid phase. A simpler assumption of constant relative

volatility is given in the Fortran model, explaining why the off-cut slope is shallower than the remaining

theoretical models.

The Bonsfills experiment gives a significantly longer off-cut period (>174 min). Unlike the models, in an

experimental setting the liquid and vapour phases leaving each stage are not in thermal, mechanical and

thermodynamic equilibrium primarily due to insufficient contact time between the two phases and deficient

mixing [23]. The assumption of equilibrium stages for all the models may have affected the distillate profiles

generating optimistic (short) off-cut periods and efficient separations. Indeed, the three models predict a

higher methanol recovery when compared to the experimental value of 79.7%: 85.4% for ModelBuilder,

86.7% for Fortran and 92.7% for Batchsim (see Table 7.6). Hence, a future model improvement would be to

possibly implement rate-based equations thereby considering mass and heat transfer limitations.

68

Figure 7.6 - Distillate methanol composition profiles with RR=3 and �̅� = 0.28 mol/min. The right-hand side figure represents the same profiles with an xx axis shift for each of the 3 functions to G(X) = F(X+T1

stplateau).

Finally, in the last phase the distillate is in the form of pure water. This is shown in the profiles illustrated in

Figure 7.6, where the methanol molar fraction decreases to approximately 0 during the last period.

Table 7.6 summarizes the information described above by listing the 1st plateau and off-cut period durations,

as well as the experimental, ModelBuilder, Fortran and Batchsim methanol recoveries.

Table 7.6 – 1st plateau and off-cut durations and methanol recoveries. The first plateau ends when the methanol purity

falls below 0.99. The off-cut duration refers to the period where 0.01 < 𝑦𝑚𝑒𝑡ℎ𝑎𝑛𝑜𝑙 < 0.99.

Figure 7.7 compares the distillate temperature profile obtained in ModelBuilder with the Fortran model for

the same simulation (RR=3 and D̅= 0.28 mol/min). In this case, the same three time periods can be noted. In

the first plateau pure methanol is distilled and the temperature given by both simulations is 65.3 °C, which is

close to the component’s boiling point (64.7 °C). The temperature increases and the second phase ends when

the distillate temperature reaches 100 °C, when the process of withdrawing pure water from the column starts.

A sharper increase in temperature during the second phase is given by ModelBuilder, more so than the

shallower slope obtained in Fortran. This sharp increase in temperature is in agreement with the shorter off-

cut time estimated in Figure 7.6 for the ModelBuilder simulation.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

y m

eth

an

ol

(mo

l fr

ac)

Time (hr)

Fortran

ModelBuilder

Batchsim

Experimental

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

y m

eth

an

ol

(mo

l fr

ac)

Time (hr)

Fortran

ModelBuilder

Batchsim

Experimental

Experimental

ModelBuilder

Fortran Batchsim

1st plateau duration (min)

80 94 105 112

Off-cut duration (min)

>174 78 174 84

Methanol recovery (%)

79.7 85.4 86.7 92.7

69

Figure 7.7 - Distillate temperature with RR=3 and �̅� = 0.28 mol/min.

Now that the behavior of the ModelBuilder simulations has been analysed and validated for the distillate

stream, let us consider an intermediate column plate 5. Figure 7.8 shows the ModelBuilder, Fortran and

Batchsim liquid outlet stream composition profiles for plate 5. It can be noticed that the three time periods

analysed for the distillate profile are still very clearly defined for plate 5. Pure methanol (xmethanol ≥ 0.99) is

obtained on plate 5 during 81 min for the ModelBuilder simulation, 86 min for Fortran and 103 min

according to Batchsim. The liquid composition profiles simulated by the three models are very similar for this

plate, with the only significant difference being in fact the duration of this first plateau.

As expected, the period where pure methanol is obtained in the distillate is longer than for plate 5. However,

for plate 5 the duration of the period where pure water is obtained increases. Evidently, as the stage column

number increases the methanol is held during shorter times while the water is retained for longer periods. This

fact is clearly illustrated in Figure 7.9 where the methanol holdup profiles (i.e., the sum of the vapour and

liquid tray holdups) are shown for column stages 1, 5 and 10-15 for the ModelBuilder simulation. Not only

does the first plateau last longer for the upper column stages, but the total molar methanol holdup is also

higher. For example, tray 1 starts with a 0.186 moles methanol holdup while on stage 15 the methanol molar

holdup is only 0.127 at time 𝑡 = 0. From stages 12 to 15 the first plateau ceases to exist and the methanol

holdup decreases immediately at the start of the simulation.

40

50

60

70

80

90

100

110

0 1 2 3 4 5 6 7 8 9 10 11

Tem

pera

ture

(°)

Time (hr)

FortranModelBuilder

70

Figure 7.8 - Methanol composition for plate 5 with RR=3 and �̅� = 0.28 mol/min.

Figure 7.9 – Methanol holdup profiles for column trays with RR=3 and �̅� = 0.28 mol/min (ModelBuilder).

As shown in Figure 7.10, the temperature profiles for plate 5 generated by the theoretical models

(ModelBuilder, Fortran and Batchsim) are quite similar. However, the experimental temperatures given by

Bonsfills for plate 5 in the first plateau are higher than predicted by all theoretical models. For example, an

average experimental temperature of 71.3 °C is given for the first plateau, which is significantly higher than

the average temperature predicted by ModelBuilder.

It is important to analyse the differences between the theoretical models and the pilot plant experiment. In an

experimental setting, systematic or random experimental errors may affect the results. In the Bonsfils

experimet, for example, the distillate flowrate is measured by weighing the distillate weight in a scale with time,

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

x m

eth

an

ol

(mo

l fr

ac)

Time (hr)

FortranModelBuilderBatchsim

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Meth

an

ol

ho

ldu

p (

mo

l)

Time (hr)

Tray 1

Tray 5

Tray 10

Tray 11

Tray 12

Tray 13

Tray 14

Tray 15

71

liquid compositions are determined off-line by gas chromatography and the temperatures are measured on-

line using temperature transducers Pt-100. All these measurements represent scenarios where human errors

may possibly occur. For the theoretical models, the possibility of compromising the results by making

simplified assumptions must be considered. For instance, all the theoretical models analysed in this work

assume equilibrium trays, a total condenser and an adiabatic column. Additionally, for both the Fortran and

Batchsim models, constant liquid holdup and negligible vapour holdup on the trays are assumed.

Figure 7.10 - Plate 5 temperature with RR=3 and �̅� = 0.28 mol/min.

Figure 7.11 shows the temperature profile for column stages 1, 5, 10 and 13-15 for the ModelBuilder

simulation. The behavior of the first 10 column stages is quite similar, with a well defined pure methanol

plateau with an average duration of 79 min and temperature of approximately 65.5°C. As previously stated,

the plateau ceases to exist for the bottom column stages and the temperature slope is less steep during the first

period. For example, on stage 15 a pure methanol holdup is never obtained and consequently the minimum

temperature is 75.4 °C, which is almost 10°C higher than the boiling point of pure methanol. A final

temperature of 100°C is reached in all column stages, when pure water holdups are obtained.

50

60

70

80

90

100

110

0 1 2 3 4 5 6 7 8 9 10 11

Tem

pera

ture

(°C

)

Time (hr)

FortranModelBuilderBatchsimExperimental

72

Figure 7.11 - Temperature profiles for column trays with RR=3 and �̅� = 0.28 mol/min (ModelBuilder).

7.4.2. Reboiler

Figure 7.12 compares the reboiler temperature profile given by ModelBuilder with the results generated by the

Fortran and Batchsim models, for a reflux ratio of 𝑅𝑅 = 3 and average distillate flowrate of �̅� = 0.28

mol/min.

A good agreement is observed between the ModelBuilder and the Batchsim models regarding the reboiler

temperature profile predictions. In fact, both profiles start at the exact same temperature of 83.8°C. Note also

that the initial temperature determined by Fortran is significantly higher (88.7 °C). Additionally, the

temperature increase predicted in ModelBuilder and Batchsim before the 100°C plateau is attained faster than

in the Fortran model, where a shallower steep is estimated. Again, the steepness of the temperature slopes can

be explained considering the VLE assumptions used in each model. The ModelBuilder and Batchsim models

both use the NRTL activity coefficient equations for the liquid phase, fact that might explain the observed

similarities. For the Fortran model a constant relative volatility is assumed in the VLE calculations.

40

50

60

70

80

90

100

110

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Tem

pera

ture

(°C

)

Time (hr)

Tray 1

Tray 5

Tray 10

Tray 13

Tray 14

Tray 15

73

Figure 7.12 – Reboiler temperature profile with RR=3 and �̅� = 0.28 mol/min.

The reboiler molar holdup and vapour fraction profiles are illustrated in Figure 7.13 for the ModelBuilder

simulation. As expected, the reboiler holdup profiles clearly indicate two distinct phases: a methanol

withdrawal period and a water removal phase. Indeed, the methanol holdup decreases from 30 to

approximately 0 moles during the first two hours whereas the water holdup which is 4.5 times higher is

removed during the following 7.7 hours. For almost the entire simulation the reboiler contains a saturated

liquid mixture. However, at time 𝑡 = 9 hours a dramatic rise in the vapour fraction is observed as the reboiler

is run dry.

Figure 7.13 – Reboiler holdup and vapour fraction profiles for RR=3 and �̅�=0.28 mol/min (ModelBuilder).

60

65

70

75

80

85

90

95

100

105

0 25 50 75 100 125 150 175 200 225 250

Tem

pera

ture

(°C

)

Time (min)

BatchsimFortranModelBuilder

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9 10 11

Ho

ldu

p (

mo

l)

Time (hr)

Water

Methanol

0

0.04

0.08

0.12

0.16

0.2

0 1 2 3 4 5 6 7 8 9 10 11

Vap

ou

r fr

acti

on

(m

ol/

mo

l)

Time (hr)

74

7.5. Sensitivity analysis

The relative importance of key operating conditions is now assessed by performing a sensitivity analysis. The

effect of the reflux ratio and the heat duty on the total separation time and on the methanol recovery was

analysed in a total of 9 simulations with ±0.5 reflux ratio differences and ±10% heat duty relative variations

starting from the base case scenario with a reflux ratio of 3 and 736 W heat duty. Hence, the simulations were

carried out combining reflux ratios of 2.5, 3 and 3.5 with heat duties of 663 W, 736 W and 810 W.

7.5.1. Total separation time

The distillate flowrate profiles obtained for reflux ratios of 2.5, 3 and 3.5 with a fixed 736 W heat duty are

represented in Figure 7.14 – a). As can be seen, the behavior of these profiles is similar. A pure methanol

withdrawal period where the distillate flowrate increases up to a maximum can be noticed, followed by an off-

cut removal phase with an associated distillate flowrate decrease and a final pure water withdrawal period

where the distillate flowrate remains approximately at a constant relative minimum. As previously stated,

because at normal pressure methanol has a lower latent heat of vaporization than water (35.3 kJ/mol vs 40.68

kJ/mol), for the same heat duty the pure methanol distillate flowrate in the first period of the simulation is

necessarily higher than the pure water distillate flowrate in the final simulation phase. Figure 7.14 – a) also

shows that for all simulation times the distillate flowrate increases with a reflux ratio decrease, which should

evidently be expected since less reflux is returned to the top of the column and more product is withdrawn

per unit of time. From Figure 7.14 – b) it can also be noticed that for a fixed reflux ratio of 3, a heat duty

increase generates higher distillate flowrates throughout the entire simulation runs.

Figure 7.14 – a) Distillate flowrate profiles for a fixed Q = 736 W and RR = 2.5, 3 and 3.5. b) Distillate flowrate profiles for a fixed RR = 3 and Q = 663 W, 736 W and 810 W.

Figure 7.15 shows the average distillate flowrate (ADF) and the total separation time (TST) for all simulations.

The total separation time starts in the steady-state regime and ends when the reboiler is run dry, i.e., it is the

0.20

0.24

0.28

0.32

0.36

0.40

0 1 2 3 4 5 6

Dis

till

ate

flo

wra

te (

mo

l/m

in)

Time (hr) a)

RR=3

RR=2.5

RR=3.5

0.20

0.24

0.28

0.32

0.36

0.40

0 1 2 3 4 5 6

Dis

till

ate

flo

wra

te (

mo

l/m

in)

Time (hr) b)

Q=736 W

Q=810 W

Q=663 W

75

period of time where a constant finite reflux separation is established. As can be seen from Figure 7.15, the

highest average distillate flowrate of 0.35 mol/min is obtained combining the lowest reflux ratio of 2.5 with

the highest heat duty of 810 W. On the contrary, the lowest average distillate flowrate of 0.22 mol/min is

given for the highest reflux ratio and the lowest heat duty combination (3.5 and 663 W, respectively).

Evidently, higher distillate flowrates generate a shorter total separation time since more product is withdrawn

per unit of time. Consequently, the shortest total separation time of 7.8 hours is given by the 2.5 RR – 810 W

simulation whereas the longest total separation time of 12.1 hours is predicted by the 3.5 RR – 663 W

simulation.

Figure 7.15 – ADF and TST for Q = 663 W, 736 W and 810 W and RR = 2.5, 3 and 3.5.

The average distillate flowrate (ADF) and the total separation time (TST) variations based on the base case 3

RR – 736 W simulation are listed in Table 7.7. As expected, for a particular reflux ratio and heat duty the ADF

and the TST relative variations present similar symmetrical values. For instance, a 1.8% increase in the average

distillate flowrate generates a 1.8% reduction in the total separation time for the 2.5 RR – 663 W simulation.

Another conclusion that may be drawn from Table 7.7 is that ±10% heat duty variations with fixed reflux

ratios and ±0.5 reflux ratio differences with fixed heat duties affect the total experiment time approximately

12%. For example, when maintaining a constant heat duty of 736 W the total separation time varies -12.3%

for a reflux ratio of 2.5 and +12.8% for a reflux ratio of 3.5. Similarly, when fixing a reflux ratio of 3 the total

separation time varies 12% and -8.4% for the 663 W and the 810 W simulations, respectively.

Table 7.7 – ADF and TST relative variations. Base simulation: 736 W with RR = 3.

0.20 0.25 0.30 0.35 0.40

Q=663 W

Q=736 W

Q=810 W

ADF (mol/min)

RR=3.5

RR=3

RR=2.5

6 8 10 12 14

Q=663 W

Q=736 W

Q=810 W

TST (hr)

RR=3.5

RR=3

RR=2.5

ADF variation (%) 663 W 736 W 810 W TST variation (%) 663 W 736 W 810 W

RR = 2.5 1.8 14.0 24.2 RR = 2.5 -1.8 -12.3 -19.7

RR = 3 -11.2 0.0 8.6 RR = 3 12.0 0 -8.4

RR = 3.5 -20.2 -11.6 -2.9 RR = 3.5 25.3 12.8 2.7

76

7.5.2. Methanol recovery

The distillate methanol composition profiles are represented in Figure 7.16 for a constant heat duty of 736 W

and reflux ratios of 2.5, 3 and 3.5. Here, two main aspects should be noted: firstly, the pure methanol

withdrawal duration is in agreement with the distillate flowrate profiles given in the section above, i.e., the

duration of the first plateau increases for higher reflux ratios due to an associated decrease in the distillate

flowrate (see Table 7.8); secondly, during the off-cut phases the profiles are relatively dissimilar directly

affecting the separation efficiency and therefore the product recovery. To satisfactorily compare the off-cut

periods, each one of the three profiles was shifted in the xx axis by a time equal to the first plateau duration,

thereby eliminating this first plateau (see Figure 7.16 - right-hand side). The shifted transformations show that

the separation is not as sharp for lower reflux ratios, where the off-cut slope is shallower and the off-cut

duration is longer.

Figure 7.16 – Distillate methanol composition profiles for a fixed Q = 736 W and RR = 2.5, 3 and 3.5. The right-hand side figure represents the same profiles with an xx axis shift for each of the 3 functions to G(X) = F(X+T1

stplateau).

Figure 7.17 shows the distillate methanol composition profiles for a fixed reflux ratio of 3 and heat duties of

663 W, 736 W and 810 W. Once again, the 1st plateau durations are in agreement with the distillate flowrate

profiles analysed in the previous section since higher heat duties generate higher distillate flowrates and thus a

shorter first plateau duration (Table 7.8). The off-cut periods can be conveniently compared using the xx axis

transformation mentioned above, generating the profiles shown in the right-hand side of Figure 7.17. In this

case, the off-cut slopes do not vary significantly for different heat duties and the expected separation

efficiency should be quite similar.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

y m

eth

an

ol (m

ol fr

ac)

Time (hr)

RR=2.5

RR=3

RR=3.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.0 0.2 0.4 0.6 0.8 1.0

y m

eth

an

ol (m

ol fr

ac)

Time (hr)

RR=2.5

RR=3

RR=3.5

77

Figure 7.17 - Distillate methanol composition profiles for a fixed RR=3 and Q = 663 W, 736 W and 810 W. The right-hand side figure represents the same profiles with an xx axis shift for each of the 3 functions to G(X) = F(X+T1

stplateau).

Table 7.8 – 1st plateau duration for Q = 663 W, 736 W and 810 W and RR = 2.5, 3 and 3.5. The first plateau duration is defined by the methanol purity, i.e., the first plateau ends when the methanol purity falls below 0.99.

1st plateau duration (min) 663 W 736 W 810 W

RR = 2.5 85.5 75.5 68.7

RR = 3 102.3 94 83.9

RR = 3.5 120.2 107 98.6

It is interesting to note that the off-cut profiles analysed above are in agreement with the methanol recoveries

listed in Table 7.9. The methanol recoveries are affected by ±0.5 differences in the reflux ratio, fact that

should be expected considering the above-mentioned differences in the off-cut profiles, namely the off-cut

slopes. For instance, a 6.8% relative decrease and a 2.6% relative increase in the methanol recovery is obtained

using the 2.5 RR – 736 W and the 3.5 RR – 736 W simulations, respectively, when compared to the base case

3.0 RR – 736 W simulation. Correspondingly, a ±10% heat duty variation practically does not modify the

methanol recovery since the off-cut period profiles are almost identical, i.e., the off-cut slopes are not altered

and hence the separation efficiency is maintained. This fact is demonstrated, for example, in the fixed RR = 3

simulations where a negligible methanol recovery relative variation of -0.7% is given for the 3.0 RR – 663 W

simulation and of -0.1% for the 3.0 RR – 810 W simulation. In conclusion, the methanol recovery varies

between 2.2 and 6.4% in absolute values when fixing the heat duties and modifying the reflux ratio in ±0.5

increments; on the contrary, a maximum absolute change of only 1.2% is observed in the methanol recovery

when the reflux ratio is maintained and the heat duties are varied ±10%.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.0 1.5 2.0 2.5 3.0 3.5 4.0

xm

eth

an

ol

(mo

l fr

ac)

Time (hr)

Q=736 W

Q=663 W

Q=810 W

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

xm

eth

an

ol

(mo

l fr

ac)

Time (hr)

Q=736 W

Q=663 W

Q=810 W

78

Table 7.9 – Methanol recovery and associated relative variation. Base simulation: 736 W with RR = 3.

7.6. Conclusion

In this chapter a multi-staged batch dynamic distillation model based on an experimental pilot plant assembled

by Bonsfills [45] was successfully implemented in ModelBuilder for a methanol/water mixture.

Model validation was also successfully carried out by comparing the ModelBuilder simulations with

experimental data and theoretical results given by Bonsfills for a Fortran model and for simulator Batchsim of

Pro/II for a reflux ratio of 3 and an average distillate flowrate of 0.28 mol/min. The assumption of constant

heat duty vs. constant distillate flowrate created differences in the duration of the pure methanol withdrawal

period. It was also found that the assumption of equilibrium stages generated for all models a non-negligible

overestimation of the separation efficiency and thus the methanol recovery.

The robustness of the dynamic ModelBuilder model was attested by performing a sensitivity analysis with

10% heat duty variations for the reflux ratios of 2.5, 3 and 3.5. As expected, shorter separation times were

obtained for lower reflux ratios and higher heat duties whereas higher reflux ratios and lower heat duties led to

longer separation times. The positive impact on the methanol recovery generated by using higher reflux ratios

was also analysed.

Methanol recovery (%)

663 W 736 W 810 W Methanol recovery

variation (%) 663 W 736 W 810 W

RR = 2.5 80.4 79.6 78.9 RR = 2.5 -5.9 -6.8 -7.6

RR = 3 84.8 85.4 85.4 RR = 3 -0.7 0.0 -0.1

RR = 3.5 88.7 87.6 88.8 RR = 3.5 3.9 2.6 4.0

79

8

General conclusions

and Future work

8.1. General conclusions

The background research presented in this work provided a broad overview on the field of batch distillation.

In particular, the literature review focused on introducing batch distillation with regard to operating policies

and modelling approaches, the latter comprising a general explanation of equilibrium and rate-based models as

well as tray hydraulic modelling issues. Because the demand for productivity in the fine chemical industry is

increasing, robust models that can predict separation performance indicators such as operation time and

component purity are required. Thence, the main contribution of this study has been the implementation of a

dynamic multi-staged distillation model in gPROMS ModelBuilder® based on equilibrium stages and

including the effects of non-negligible tray holdup and of tray pressure drop.

Preliminary steps were undertaken prior to implementing the dynamic multi-staged model. A one equilibrium

stage separator ModelBuilder model was successfully validated against a base case benzene/toluene separation

supplied by Seader et al. [13]. The composition, temperature and holdup profiles simulated in ModelBuilder

were overall in agreement with the data given by Seader et al. Indeed, the average benzene molar fraction in

the distillate predicted by ModelBuilder after 8.9 hours was found to be only 1.8% lower, being coupled with a

1.9% lower still temperature estimation. The thermodynamic models used to predict equilibrium data in both

cases should explain the noted divergences: a constant relative volatility was assumed by Seader et al. whereas

in ModelBuilder the more accurate NRTL and Soave-Redlich-Kwong equations accounted for the liquid and

the vapour non-idealities, respectively.

Tray holdup and pressure drop equations should be incorporated when developing a rigorous dynamic multi-

staged model in which the flowrates are pressure-driven and where non-negligible tray holdups are considered.

Thus, in this work several tray holdup and pressure drop correlations were added to the multi-staged column

80

model. Furthermore, column diameter correlations based on flooding limit estimations were also included as

preliminary design options.

To ensure model convergence, supplying accurate tray geometrical specifications when applying these

correlations is of significant importance. In this work, a methanol/water continuous distillation case study

with suitable geometrical specifications was used to preliminarily test the implemented hydraulic correlations.

Here, the type of flow regime was found to significantly affect the liquid holdup, dry vapour head loss and

aerated liquid head loss profiles. This fact was clearly illustrated considering that in the rectifying section

operating under a spray regime, Jeronimo & Sawistowski was found to be applicable, giving a liquid holdup of

approximately 977 moles; this value increased to 2517 moles for the froth-type stripping section where the

Bennett holdup correlation was found to be applicable.

Another noteworthy aspect relates to the importance of pondering the accuracy gain with a non-desirable

complexity increase when applying a more rigorous hydraulic correlation. In the dry vapour head loss case the

more complex Hughmark & O’Connell and Liebson equations predicted maximum absolute differences

relative to the simpler standard orifice equation of only 4.0% and 2.1%, respectively. On the contrary, for the

case study in question using a more complex column diameter equation was compensated by the accuracy

gain: the estimated column diameter was found to be 18% higher for the simpler Lowenstein correlation

compared to the more complex Fair correlation. In essence, all the implemented hydraulic correlations were

successfully simulated in ModelBuilder generating appropriate values if taken into account the occurring flow

regime and the complexity of each correlation.

After testing the one-staged equilibrium separator model and the hydraulic relations for the methanol/water

continuous distillation case study, a multi-staged dynamic distillation model based on an experimental pilot

plant assembled by Bonsfills [45] was implemented in ModelBuilder for a methanol/water mixture. As

expected, the distillate profiles highlighted three distinct phases: two pure methanol and pure water

withdrawal periods and an intermediate phase where impure mixtures were obtained.

Model validation was carried out by comparing the ModelBuilder simulations with experimental data and

theoretical results given by Bonsfills for a Fortran model and for simulator Batchsim of Pro/II for a reflux

ratio of 3 and an average distillate flowrate of 0.28 mol/min. The distillate flowrate profile was found to

significantly affect the duration of the pure methanol withdrawal period. A constant heat duty was supplied

experimentally. The ModelBuilder model assumed a constant heat duty profile, unlike the Fortran and

Batchsim models where a constant distillate flowrate was given. Regarding the duration of the methanol

withdrawal period, ModelBuilder predicted a closer agreement to the experimental data (94 vs. 80 min) than

the Fortran and Batchsim simulators where longer first plateau durations of 105 and 112 min were estimated,

respectively. This fact illustrates the non-negligible effect generated when assuming a constant distillate

81

flowrate profile adapting the heat duty accordingly, rather than providing a constant heat duty obtaining the

corresponding distillate flowrate profile. It was also found that the assumption of equilibrium stages generated

for all models a non-negligible overestimation of the separation efficiency and thus the methanol recovery.

The experimental methanol recovery of 80% was thereby lower than the ModelBuilder prediction of 85%, the

87% Fortran estimation and the Batchsim 93% methanol recovery. Nevertheless, the dynamic ModelBuilder

model was successfully validated simulating the expected composition, temperature and flowrate profiles with

occasional divergences from the experimental data being explained by the factors described above.

The robustness of the dynamic ModelBuilder model was attested by performing a sensitivity analysis with

10% heat duty variations for the reflux ratios of 2.5, 3 and 3.5. It was shown that ±10% heat duty variations

with a fixed reflux ratio and ±0.5 reflux ratio differences with a fixed heat duty affect the total experiment

time in approximately 12%. Furthermore, ±0.5 variations in the reflux ratio were found to have an absolute

impact of 2.2 to 6.4% on the methanol recovery. On the contrary, this recovery was practically not affected by

the 10% heat duty variation.

The main objective of this work was accomplished: a dynamic multi-staged batch distillation model enhanced

with tray hydraulic relations was successfully implemented in ModelBuilder. This tool may be applied, for

example, to the scheduling of chemical plants thereby facilitating industrial management at the scheduling

level. Future model improvements are discussed in the section presented below.

8.2. Future work

Future work is suggested for further development of the dynamic model presented in this thesis. A first

direction for future work involves further model improvement in the following aspects: simulation time and

convergence. If the initial conditions specified in the total reflux simulations are very far from steady-state, the

simulation time is long. This issue could be improved by appropriately scaling the model equations,

maintaining all the terms between 10−3 and 103. In the finite reflux period there are two phases where the

model is more likely to fail: at the beginning of the simulation, when defining the reflux ratio in the splitter

and during the first moments of off-cut mixtures removal from the column. In most cases the issue is solved

by temporarily increasing the heat duty to keep the system in the vapour-liquid phase equilibria region, since a

bubble point limit is impeding model convergence.

A second direction for future work would be to further validate both the Rayleigh separator and the batch

column model with mixtures comprising three or more components. The need for experimental data must

also be highlighted here since there is a lack of batch experiments available in the open literature, with only a

82

few research works being validated experimentally. In addition, an interesting possibility would be to describe

the vapour-liquid equilibrium differently by choosing other thermodynamic models for the Rayleigh

benzene/toluene separation presented in Chapter 4 and for the batch multi-staged methanol/water distillation

shown in Chapter 7 (e.g., Peng-Robinson equation of state for the vapour phase and the UNIQUAC model

for the liquid phase).

The extension of the dynamic equations to model mass and heat transfer processes could be another

consideration for future work. A rate-based model of this sort would be useful to simulate mixtures exhibiting

liquid non-idealities or to model reactive separations. In the methanol/water multi-staged distillation

presented in this work, the existence of a rate-based model would allow an accuracy reassessment in the

predicted methanol recoveries and off-cut durations. However, the increase in accuracy should be measured

against the additional complexity level and the availability of parameters such as mass and heat transfer

coefficients.

83

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88


89

Appendix A

This section shows how the material balance equations listed in section 5.2.1 were prepared for possible future

equilibrium reactive terms. Indeed, adding equilibrium reactive terms causes the typical high index

complications described below.

High Index DAE Systems

The behavior of dynamic systems can be modelled using a mixed system of differential and algebraic

equations (DAEs) of the form:

𝑓(𝑥, �̇�, 𝑦, 𝑢) = 0 (1)

𝑔(𝑥, 𝑦, 𝑢) = 0 (2)

where 𝑓 refers to the differential equations, 𝑔 to the algebraic equations and 𝑥, 𝑦 and 𝑢 are the differential,

algebraic and input variables, respectively.

DAE systems can be classified according to their index, defined as the minimum number of differentiations

with respect to time that are necessary to obtain a pure ordinary differential equations system (ODE). An

ODE system is a particular case of a DAE system, where there are no algebraic equations or variables. In this

case, �̇� and �̇� are obtained as explicit functions of 𝑥 and 𝑦:

�̇� = 𝐹(𝑥, 𝑦) (3)

�̇� = 𝐺(𝑥, 𝑦) (4)

Note that ODE systems are classified as index-0. DAE index-1 systems are similar to ODE systems regarding

the number of initial conditions that are specified arbitrarily and the numerical algorithm integration behavior.

In this case, the number of initial conditions specified arbitrarily is equal to the number of differential

variables in the system.

On the contrary, for DAE systems with an index higher than 1 the number of arbitrary initial conditions is

reduced according to the number of differentiations that lead to an ODE system. Each differentiation takes

into account additional relations that provide further information and that reduce the number of arbitrary

initial conditions. Moreover, the integration error is difficult to control when applying the typical numerical

algorithms to DAE systems of high index. To avoid these complications, the index of such systems should be

reduced to 1 via differentiation with respect to time.

90

When equilibrium reactions occur in a system, the component holdups are implicitly related to each other by

equilibrium constants. In this case, the fact that the differential variables (holdups) are not independent and

cannot be assigned initial arbitrary values causes the high index problem. For this reason, the index of the

system is reduced to 1 introducing the reaction invariant holdups 𝑀𝑖𝑛𝑣 𝑗,𝑘 instead of the molar holdups 𝑀𝑗,𝑖.

Note that the invariant terms are used in the material balance equations developed in this work (section 5.2.1).

The reaction invariant holdup terms 𝑀𝑖𝑛𝑣 𝑗,𝑘 represent the 𝑘 invariant quantities that remain unchanged by

the equilibrium reactions that occur on each stage 𝑗 (𝑑𝑀𝑖𝑛𝑣 𝑗,𝑘

𝑑𝑡= 0). These terms are determined using the

molar holdups 𝑀𝑗,𝑖 of species 𝑖 on stage 𝑗:

𝑀𝑖𝑛𝑣 𝑗,𝑘 =∑𝑀𝑗,𝑖𝑖

× 𝑃𝑖𝑛𝑣𝑗,𝑘,𝑖 (5)

∑𝑃𝑖𝑛𝑣𝑗,𝑘,𝑖 × 𝜈𝑗,𝑖,𝑟𝑒𝑞

𝑖

= 0 (6)

In the above equations, the terms 𝑃𝑖𝑛𝑣𝑗,𝑘,𝑖 are defined using 𝜈𝑗,𝑖,𝑟𝑒𝑞, that is, using the 𝑖 stoichiometric

coefficients for each of the 𝑟 reactions that occur on stage 𝑗.

Finally, to reduce the index of the system to 1, all the 𝑖 inlet and outlet compositions 𝑥𝑖,𝑗 and 𝑦𝑖,𝑗 of each

stage 𝑗 in the M equations must be transformed 𝑘 times (𝑘 = 𝑁𝑜. 𝑖𝑛𝑣𝑎𝑟𝑖𝑎𝑛𝑡𝑠), as follows:

𝑥𝑖,𝑗 →∑{𝑥𝑖,𝑗 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

(7)

𝑦𝑖,𝑗 →∑{𝑦𝑖,𝑗 × 𝑃𝑖𝑛𝑣,𝑗,𝑘,𝑖}

𝑖

(8)

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