Modeling and Dynamic Analysis of a Fluid Catalytic Cracking Unit...

Modeling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU)

Juan David Reyes Fernández

Universidad Nacional de Colombia

Facultad de Ingeniería

Departamento de Ingeniería Química y Ambiental

Bogotá D.C., Colombia

2017

Modeling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU)

Juan David Reyes Fernández

Tesis presentada como requisito parcial para optar al título de:

Magister en Ingeniería Química

Director:

Ph.D. Carlos Arturo Martinez Riascos

Línea de Investigación:

Modelado, simulación y análisis de procesos

Grupo de Investigación:

Ingeniería de sistemas de proceso

Universidad Nacional de Colombia

Facultad de Ingeniería

Departamento de Ingeniería Química

Bogotá D.C., Colombia

2017

Acknowledgments

I would like to thank to all the people that took part of this work, in particular to my advisor

Carlos Martinez for taking the time to explain me topics which at first sight I did not

understand and make them clear for me, to Adriana Rodriguez as an incredible project

partner, her time and knowledge, to Galo Carrillo Le Roux for his advices and opportunities,

to the program “Jóvenes Investigadores de Colciencias” for the economic support in the

first part of the project, to the postgraduate students in Chemical Engineering.

I give my sincerest gratitude to my beloved family and my closest friends, because without

their support I could not have reached this goal. Finally, I would like to express my gratitude

to the Universidad Nacional de Colombia, this has been a place where I have experienced

some of the most important moments of my life so far.

Resumen y Abstract IX

Resumen

Este proyecto presenta el desarrollo de un modelo dinámico para la unidad Orthoflow F de

craqueo catalítico en lecho fluidizado (FCCU por sus siglas en inglés) y el uso de diferentes

herramientas de ingeniería de sistemas de proceso (PSE por sus siglas en inglés) para

realizar un análisis basado en modelo y proponer usos futuros para el mismo. El modelo

considera el reactor, la sección de despojamiento y el regenerador del catalizador. El

modelo dinámico es capaz de predecir el rendimiento de gasolina y otros productos que

resultan del proceso. Se analizó la sensibilidad paramétrica del sistema con la ayuda de

la matriz de ganancia relativa (RGA por sus siglas en inglés). Se propuso la estructura de

una red neuronal dinámica utilizando el modelo no-lineal exógeno autorregresivo (NARX

por sus siglas en inglés) cerrado. Se determinó la estabilidad del sistema en estado

estacionario utilizando la integración del modelo por continuación y la definición de

estabilidad de estados estacionarios con valores propios complejos, los cuales dieron

indicios de regiones estables e inestables del sistema. Finalmente se propuso una

estrategia de control basada en el análisis previo de la unidad.

Palabras clave: FCCU, DAE, Matriz RGA, Red Neuronal, Análisis de estabilidad,

control de procesos.

X Modelling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU)

Abstract

This work presents several PSE techniques to analyze the behavior of a FCCU. In the

second chapter, the construction of the dynamic model for an Orthoflow F type FCCU is

presented. This model only considers the riser, reactor, stripping section, regenerator, slide

and interconnection valves. The model can predict, with good accuracy, the yield of

gasoline and other products, at the most frequent operating conditions. Parametric

sensibility of the model was developed by the Relative Gain Array analysis. A dynamic

neural network for the FCCU was proposed using the NARX model in closed loop scheme.

The stability of the system was determined using the continuation integration strategy and

the stability definition is based in the eigenvalues of the different steady states presented

by the unit. Finally, a control strategy was proposed based on the previous analysis made

to the unit.

Keywords: FCCU, DAE, RGA, Neural Network, Stability analysis, process control.

Contents XI

Contents

Page

Chapter 1 Introduction .................................................................................................. 19 1.1 Introduction ........................................................................................................... 19 1.2 Objectives ............................................................................................................. 26 1.3 Thesis outline ........................................................................................................ 27 References ................................................................................................................. 28

Chapter 2 FCCU dynamic model .................................................................................. 31 2.1 Introduction ........................................................................................................... 31 2.2 Model description .................................................................................................. 36

2.2.1 Mixing point and riser model ............................................................................ 38 2.2.2 Strippinig-Disengaging Model .......................................................................... 42 2.2.3 Regenerator dense phase model ..................................................................... 43 2.2.4 Regenerator bed characterization model ......................................................... 46 2.2.5 Freeboard model ............................................................................................. 46

2.3 Solution algorithm ................................................................................................. 46 2.3.1 Riser discretization .......................................................................................... 47 2.3.2 Numerical solution strategy ............................................................................. 47

2.4 Parameter estimation ............................................................................................ 48 2.5 Noise addition ....................................................................................................... 51 2.6 Results .................................................................................................................. 51

2.6.1 Riser solution independence ........................................................................... 51 2.6.2 Parameter estimation results ........................................................................... 54 2.6.3 Steady-state results ......................................................................................... 58 2.6.4 Dynamic-state results ...................................................................................... 64 2.6.5 Noise addition results ...................................................................................... 71

2.7 Conclusions .......................................................................................................... 73 References ................................................................................................................. 74

Chapter 3 RGA analysis of the FCCU........................................................................... 79 3.1 Introduction ........................................................................................................... 79 3.2 Methodology ......................................................................................................... 81

3.2.1 Output selection .............................................................................................. 82 3.2.2 Input selection ................................................................................................. 83 3.2.3 Sensitivity analysis .......................................................................................... 83 3.2.4 Variable pairing ............................................................................................... 84

3.3 Results .................................................................................................................. 85

XII Modelling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU) Título de la tesis o trabajo de investigación

3.4 Conclusions ........................................................................................................... 86 References .................................................................................................................. 87

Chapter 4 DNN construction for the FCCU ................................................................. 89 4.1 Introduction ............................................................................................................ 89 4.2 Methodology .......................................................................................................... 91

4.2.1 Data generation .............................................................................................. 91 4.2.2 DNN structure ................................................................................................. 92 4.2.3 Training algorithm ........................................................................................... 92

4.3 Results .................................................................................................................. 93 4.4 Conclusions ........................................................................................................... 97 References .................................................................................................................. 97

Chapter 5 Stability analysis of the FCCU .................................................................. 100 5.1 Introduction .......................................................................................................... 100 5.2 Methodology ........................................................................................................ 102

5.2.1 System transformation .................................................................................. 102 5.2.2 Continuation software ................................................................................... 103

5.3 Results ................................................................................................................ 103 5.4 Conclusions ......................................................................................................... 107 References ................................................................................................................ 108

Chapter 6 Control of the FCCU .................................................................................. 109 6.1 Introduction .......................................................................................................... 109 6.2 Methodology ........................................................................................................ 110

6.2.1 PID Controller ............................................................................................... 110 6.2.2 PID tuning ..................................................................................................... 110

6.3 Results ................................................................................................................ 111 6.4 Conclusions ......................................................................................................... 113 References ................................................................................................................ 113

Chapter 7 Conclusions and recommendations ........................................................ 114

Contents XIII

List of Figures

Page

Figure 1-1. FCCU location in the refinery. Adapted from (Gary et al., 2007). .................. 21

Figure 1-2. Schematic representation of the FCC process. Adapted from (Zanin, 2001). 23

Figure 1-3. Plant decision hierarchy. Adapted from (Darby, Nikolaou, Jones, & Nicholson,

2011). ............................................................................................................................. 25

Figure 1-4. RTO layer description. Adapted from (Reyes, Rodríguez, & Riascos, 2015) 26

Figure 2-1. Schematic representation of the Kellogg Orthoflow F unit, adapted from Zanin

(2001). ............................................................................................................................ 37

Figure 2-2. Mixing point subsystem diagram. ................................................................. 38

Figure 2-3. Riser subsystem diagram. ............................................................................ 39

Figure 2-4. Schematic representation of the 6-lump cracking kinetic scheme, the lumps are:

Gas oil (GO), light cycle oil (LCO), gasoline (G), light gases (LG), liquefied petroleum gas

(LPG) and coke (C). Adapted from Araujo-Monroy & López-Isunza (2006). ................... 40

Figure 2-5. Catalytic cracking reaction mechanism, a) Reactions of the PONA components

of the GO major lump (P1, O1, N1), b) Reactions of the PONA components of the LCO

major lump (P2, O2, N2, A2), c) Reactions of the PONA components of the G major lump

(P3, O3, N3, A3) and d) Reactions of the PONA components of the LPG major lump (P4,

O4). Adapted from (Araujo-Monroy and López-Isunza, 2006). ........................................ 41

Figure 2-6. Stripping-Disengaging subsystem diagram. .................................................. 42

Figure 2-7. Regenerator subsystem diagram. ................................................................. 44

Figure 2-8. a) Representation of the regenerator bubbling fluidization regime b) Detail of

the gas mass transference through the interface of the bubble-emulsion phases. .......... 46

Figure 2-9. Initialization algorithm for the DAE FCCU model, equations and variables are

presented in the appendix A. .......................................................................................... 50

Figure 2-10. Temperature profile of the riser for different number of divisions (N). ......... 53

Figure 2-11. Gasoline lump profile of the riser for different number of divisions (N). ....... 54

XI

V

Modelling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU) Título de la tesis o trabajo de investigación

Figure 2-12. Comparison of the base model riser outlet temperature prediction against

steady state data. ............................................................................................................ 55

Figure 2-13. Comparison of the base model second regenerator stage temperature

prediction against steady state data. ............................................................................... 56

Figure 2-14. Comparison of the final model riser outlet temperature prediction against

steady state data. ............................................................................................................ 57

Figure 2-15. Comparison of the final model second regenerator stage temperature

prediction against steady state data. ............................................................................... 57

Figure 2-16. Comparison of the final model gasoline yield prediction against steady state

data. ................................................................................................................................ 58

Figure 2-17. Freeboard outlet gas concentration as a function of the total air flow to

regenerator for a catalyst circulation rate of 901.6 kg/s, a gas oil feed rate of 90,8 kg/s and

an air flow division ratio of 0.11. ...................................................................................... 60

Figure 2-18. Regeneration stage temperatures and riser outlet temperature as a function of

the total air flow to regenerator for a catalyst circulation rate of 901.6 kg/s, gas oil feed rate

of 90,8 kg/s and an air flow division ratio of 0.11. ............................................................ 61

Figure 2-19. Gasoline yield and coke on catalyst concentration in the second regeneration

stage as a function of the total air flow to regenerator for a catalyst circulation rate of 901.6

kg/s, gas oil feed rate of 90,8 kg/s and air flow division ratio of 0.11. .............................. 61

Figure 2-20. Freeboard outlet gas concentration as a function of the catalyst circulation rate

for a total air flow to regenerator of 80,1 kg/s, a gas oil feed rate of 90,8 kg/s and an air flow

division ratio of 0.11. ....................................................................................................... 62

Figure 2-21. Regeneration stage temperatures and riser outlet temperature as a function of

the catalyst circulation rate for a total air flow to regenerator of 80,1 kg/s, a gas oil feed rate

of 90,8 kg/s and an air flow division ratio of 0.11. ............................................................ 63

Figure 2-22. Gasoline yield and coke on catalyst concentration in the second regeneration

stage as a function of the catalyst circulation rate for a total air flow to regenerator of 80,1

kg/s, a gas oil feed rate of 90,8 kg/s and an air flow division ratio of 0.11. ....................... 63

Figure 2-23. Gasoline yield and second regenerator stage temperature as a function of the

air flow to the second regenerator stage for an air flow to the first regenerator stage of 71,9

kg/s, a gas oil feed rate of 90,8 kg/s and a catalyst circulation rate of 901,6 kg/s. ........... 64

Figure 2-24. Step change experiment description of the control valve opening for the

dynamic state results. ..................................................................................................... 65

Contents XV

Figure 2-25. Dynamic response of the regenerator second stage temperature, riser outlet

temperature and flue gas composition for a step change in the air flow to the first

regeneration stage. ........................................................................................................ 67

Figure 2-26. Dynamic response of the first and second regenerator stages level, SD reactor

level and regenerator pressure for a step change in the air flow to the first regeneration

stage. ............................................................................................................................. 68

Figure 2-27. Dynamic response of the second regenerator stages coke on catalyst and

gasoline yield for a step change in the air flow to the first regeneration stage. ................ 69

Figure 2-28. Dynamic response of the regenerator second stage temperature, carbon

monoxide and oxygen composition in flue gas for a step change in the reactor outlet slide

valve. .............................................................................................................................. 70

Figure 2-29. Dynamic response of the regenerator first and second stage level, riser outlet

temperature and oxygen composition in flue gas for a step change in the reactor outlet slide

valve. .............................................................................................................................. 71

Figure 2-30. Noise addition to the regenerator second stage temperature signal for a SNR

of 40 and a 20% of data with gross error addition in different time scales, a) 0 to 4000

seconds; b) 0 to 500 seconds. ........................................................................................ 73

Figure 4-1. Data generation scheme for the input variables step test. ............................ 92

Figure 4-2. DNN structure for the FCCU ......................................................................... 92

Figure 4-3. DNN scheme for the FCCU. ......................................................................... 94

Figure 4-4. DNN prediction performance with two training algorithms for open loop training.

....................................................................................................................................... 94

Figure 4-5. DNN prediction performance with two training algorithms after the closure

without re-training. .......................................................................................................... 95

Figure 4-6. Error histogram for the DNN with Bayesian Regularization ........................... 96

Figure 4-7. Performance plot as a function of the Epochs. ............................................. 97

Figure 5-1. Steady-state multiplicity of the FCCU for a change in the air to the first

regenerator stage ..........................................................................................................105

Figure 5-2. Heat generation and consumed lines for the FCCU .....................................106

Figure 5-3. Stability region for the FCCU .......................................................................107

Figure 6-1. Tuning graphical user interface for the SIMULINK PID tuning toolbox. ........111

Figure 6-2. Set point change for the regenerator first stage temperature in the stable

operative region. ............................................................................................................112

X

VI

Modelling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU) Título de la tesis o trabajo de investigación

Figure 6-3. Step change in the riser outlet temperature in the steady state multiplicity region.

..................................................................................................................................... 112

Figure 7-1. Schematic representation of the the Orthoflow F FCCU. ............................. 118

Figure 7-2. Regenerated catalyst and gas oil feed mixing point subsystem. .................. 120

Figure 7-3. Riser subsystem. ........................................................................................ 122

Figure 7-4. Schematic representation of the 6 lump cracking kinetic scheme, the lumps

considered are: Gas oil (GO), light cycle oil (LCO), gasoline (G), light gases (LG), liquefied

petroleum gas (LPG) and coke (C), adapted from (Araujo-Monroy & López-Isunza, 2006).

..................................................................................................................................... 123

Figure 7-5. Catalytic cracking reaction mechanism, a) Reactions of the PONA components

of the GO major lump (P1, O1, N1), b) Reactions of the PONA components of the LCO major

lump (P2, O2, N2, A2), c) Reactions of the PONA components of the G major lump (P3, O3,

N3, A3) and d) Reactions of the PONA components of the LPG major lump (P4, O4) adapted

from (Araujo-Monroy & López-Isunza, 2006). ................................................................ 124

Figure 7-6. Stripping-disengaging (SD) subsystem ....................................................... 128

Contents XVII

List of Tables

Page

Table 2-1. Literature review of the lump kinetic scheme for catalytic cracking used in FCCU

modeling. ........................................................................................................................ 32

Table 2-2. Equations summary for each subsystem ....................................................... 47

Table 3-1. Global sensitivity analysis results for the primary output and manipulated

variables ......................................................................................................................... 84

Table 3-2. Input-output variable pairing .......................................................................... 86

Table 6-1. PID controller type for the variable pairing. ...................................................110

Contents XVIII

Chapter 1 Introduction

1.1 Introduction

Worldwide energy demand has had a remarkable increase in recent years. This behavior

is related to the population growth and the increasing per capita energy consumption

(British Petroleum, 2015). The energy market is under stress, in part, due to the volatility

and uncertainty of the oil sell price and its direct impact on the fuels obtained from it, such

as gasoline and diesel (U.S. Energy Information Administration, 2015).

New energy sources such as shale oil, offshore deep-water oil, biofuels, solar and wind

power and others, require an important level of development to be able to compete with the

conventional energy sources. The implementation of these new energy technologies also

requires a careful analysis for the energy based economies (UNEP Frankfurt School of

Finance and Management and Bloomberg New Energy, 2016).

Crude oil is still going to be considered as one of the main energy sources in the near future,

hence, the fuel oriented refineries need to be profitable even with disturbances in the raw

materials price, quality, and environmental requirements. Gasoline still plays a key role in

the transportation sector. Some promising technologies are to be expected to cover the

corporate average fuel economy (CAFE) as well as the requirements for green-house

gases (GHG) emissions (Magaril & Magaril, 2016; Walton & Rousseau, 2014; J. Wang,

2008). The projections of the energy consumption by fuel indicate an increase in the

gasoline demand. Therefore, the refineries will increase the production of this type of fuel.

A fuel oriented refinery has the main purpose of produce the maximum amount of useful

fuels from each barrel of crude oil, efficiently and with the maximum profitability. The main

type of fuels produced by this kind of refinery are: gasoline, diesel, jet fuel, kerosene and

gas oil (Gary, Handwerk, & Kaiser, 2007). Gasoline is a complex hydrocarbon mixture with

20 Modeling and Dynamic Analysis of a Fluid Catalytic Cracking Unit (FCCU)

a large boiling point range, that goes from 38°C up to 205°C, depending on the quality

requirements (Meyers & Meyers, 2004).

The typical gasoline used for transportation is a mixture of effluents from several process

units of the refinery, such as: The fluid catalytic cracking unit (FCCU), catalytic reformer

unit, isomerization unit, alkylation unit. Along with other additives, the mixture depends on

the quality or availability of refining process to cover the market requirement (Fahim,

Alsahhaf, & Elkilani, 2010). The gasoline from FCCU makes up the largest fraction gasoline

used for transportation, being this unit fundamental in the refinery profitability (Sadeghbeigi,

2012).

In that way, the FCCU is an important part of a fuel oriented refinery. This unit converts high

boiling point hydrocarbon cuts into light and high value products. The main products of the

FCCU are: fuel gas (C3 and C4), olefins, gasoline, light cycle oil (LCO) and heavy cycle oil

(HCO) (Sadeghbeigi, 2012). The figure 1.1 presents the location of the fluid catalytic

cracking unit (FCCU) in a conventional fuel oriented refinery; The FCCU is fed by heavy

cuts from the atmospheric and vacuum distillation units and it produces several lighter and

high-value products.

In the figure 1.2 a schematic representation of a stacked type of FCCU is presented. Due

to the high process complexity of the FCCU, a sub-system description is convenient to have

a broader understanding of the implications of every section in the performance of the full

system. In the sub-system description, typically are six main sections to consider: feed

preheating, air supply to the regenerator, riser, reactor-stripping, catalyst regenerator and

the product fractionator.

The FCCU requires a gaseous hydrocarbon feed stream, therefore, most refineries produce

the sufficient amount of gas oil to cover the heating and evaporation requirements

(Sadeghbeigi, 2012). Feed preheating typically has external heat integration and a furnace

or fired-heater to reach the temperature in the range of 200-450°C (Gary et al., 2007).

Chapter 1 Introduction 21

Figure 1-1. FCCU location in the refinery. Adapted from (Gary et al., 2007).

The riser section is where the catalytic cracking reactions occur, in this section the gaseous

hydrocarbon feed gets in contact with the regenerated catalyst. Modern catalysts achieve

the maximum conversion for cracking reactions in less than three seconds (Sadeghbeigi,

2012). The hot regenerated catalyst provides the energy for the cracking reactions with a

reduction in the outlet temperature due its endothermic nature.

In most FCCU the vapor volume expansion is the principal driving force to carry the catalyst

up to the riser, in some designs steam is used as a lift media (Oliveira, Cerqueira, & Ram,

2012). The final steps of the cracking reaction mechanism produce coke. The coke is

deposited over the catalyst surface creating a deactivation layer, at this stage the catalyst

Atmospheric distillationCrude oil

feed

Gas

recovery

Isomerization

HDTCatalytic

reforming

Fluid catalytic

cracking (FCC)

Vacuum

distillationHydrocracking

Vacuum

gas oil

Atmosferic gas oil

Naphta

Straight rungasoline

LPG and fuel gas

Delayed coker

Alkylation

Vacuum

gas oil

Residue

Coke

Light olefins

and fuel gas

HCO

Reformate

Gasoline

LCO

i-C4=

C3=,C4

=

Gasoline, naphta and middle distillates

Gasoline, naphta and

middle distillates

Ble

nd

ing


is considered spent (Arbel, Huang, & Rinard, 1996).

After the riser, the mixture of cracked hydrocarbons and spent catalyst gets into the reactor.

The reactor vessel is composed by the catalyst recovery cyclones and the hydrocarbon

disengaging system. Most of the FCCU designs have a separation scheme that separates

nearly 99% of the catalyst from the cracked products (Sadeghbeigi, 2012).

The stripping section uses steam to release the hydrocarbons adsorbed over the catalyst

surface and pores. To improve the contact between the steam and the catalyst several

types of internal designs have been developed, such as the shed trays, disk baffles and

structural packing (Meyers & Meyers, 2004).

The spent catalyst goes into the regenerator, this system has several essential functions; It

restores the catalyst activity by the combustion of the coke layer. The heat of combustion

is used in the recirculation catalyst stream to supply the energy for the cracking reactions.

The system is designed to generate a fluidization regime of the catalyst particles, and

delivers the regenerated catalyst to the riser inlet through a slide valve (Han & Chung,

2001).

The average catalyst particle size is about 5 micrometers. Inside the regenerator can be

identified two main regions: a region above the air distribution system which has a high

catalyst concentration, it is typically called dense phase, and a low catalyst concentration

region, just above the dense phase and below the catalyst recovery cyclones, called dilute

phase or freeboard (Chuachuensuk, Paengjuntuek, Kheawhom, & Arpornwichanop, 2013).

The FCCU products consist in a mixture of light hydrocarbons, the main purpose of the

fractionator is to recover and separate the high value products. The FCCU product stream

enters the fractionator at the base with a stream of steam. The reactor vapor must be cooled

down before enters to the fractionator, and the large amount of produced light gases will

carry the gasoline that must be recovered (Sadeghbeigi, 2012).

Due its high complexity, the modelling of the FCCU has been an interesting challenge for

several research groups (Ali, Rohani, & Corriou, 1997; Dasila et al., 2014; Fernandes,

Pinheiro, Oliveira, & Ramo, 2008; Ramachandran, Rangaiah, & Lakshminarayanan, 2007;


L. Wang, Shen, & Li, 2006; X. Wang, Jin, Zhong, & Xiao, 2010).

Figure 1-2. Schematic representation of the FCC process. Adapted from (Zanin, 2001).

Interactions between reactor and regenerator, the existence of phenomena with multi-scale

dynamics, the composition and variability of the feed are some characteristics of the system

that increase model complexity are ones of the considerable number of questions that

several authors have been looking for answers.

Gas oil

feed

Flue gas

Fractionator

Reactor

Stripping

steam

Regenerator

Riser

Spent

catalyst

Regenerated

catalyst

FCC

products

Air

Air blower

Fuel gas and olefins

Gasoline

LCO

HCO

Furnace


The process systems engineering (PSE) is a branch of engineering which provides the

industry with methodologies, tools and techniques in order to be effective and efficient in

the decision-making process (Grossmann & Westerberg, 2000). Considering the

importance of the FCCU in the refinery economic balance and profitability, the use of the

PSE techniques for improving its performance is a straightforward and attractive strategy.

In that way, precision and accuracy of the mathematical models are required to develop

adequate analysis tools, such as control and optimization.

Changes in the quality, demand or environmental requirements of the products of the FCCU

makes this process an interesting case of study for real time optimization (RTO) techniques

(Oliveira et al., 2012). In the figure 1.3 the typical plant decision hierarchy for a process

plant is presented.

Planning is an activity based in market and economic decisions, it addresses the questions

that structure the plant operation like: the procurement process, the type and quality of

products that the plant need to produce, the amount of each product accordingly to the

market demand.

Scheduling deal with the activities timing. The actions performed in this stage are a result

from the decisions made in the planning layer. The supply chain structure is developed in

this stage to address the inventory problem and the delivery of feeds and products. Several

mathematical models have been proposed to predict and optimize these two layers with

successful industrial application (Biegler, 2018).

The RTO layer implements the scheduling and planning decisions to the operation in real

time. It is based on a detailed model of the plant in steady-state. The RTO search for an

optimal operative condition to maximize the plant profit. The RTO results are passed directly

to the Model Predictive Control (MPC) layer. The MPC uses the results from the RTO as

set points. In the figure 1-4, a detailed description of the RTO layer is presented.

Each element of the RTO layer has a defined purpose. The output signals from the plant

have gross and gaussian error, this occur due instrumentation bias, mechanical stress,

climate alterations etc. Therefore, a gross detection module has to be placed in the

structure. A steady-state detection identifies the moment in which the plant output variables


have a stable magnitude with respect to time, in this moment the data reconciliation and

parameter estimation calibrate the economic optimization module and deliver the adequate

parameters to the MPC layer.

The MPC can perform control over the plant directly and get optimization over it inside its

algorithm. The lower layer that interacts with the plant is the distributed control system

(DCS) that is responsible for the regulatory control of the plant.

This work is part of a broader project, which aims to develop a RTO framework to the FCCU.

In each chapter of this document will be pointed out the application of each part to the RTO

framework.

Figure 1-3. Plant decision hierarchy. Adapted from (Darby, Nikolaou, Jones, & Nicholson, 2011).

Planning

Scheduling

RTO

MPC

Planty(k)

u(k)

d(k)

yset(k)

DCS


Figure 1-4. RTO layer description. Adapted from (Reyes, Rodríguez, & Riascos, 2015)

1.2 Objectives

The main objective of this work is to model and analyze the dynamics of a FCCU using

computational tools. To accomplish this objective some specific objectives should be

covered.

The first specific objective is to develop a dynamic model for the FCCU based on first

principle balances of mass, energy and momentum. The model should be able to represent

the most important operative variables of the unit.

The second specific objective corresponds to the parametric sensibility analysis to the

dynamic model. This analysis will provide much more knowledge about the relation

between inputs and outputs of the model.

The third specific objective requires the construction of a dynamic neural network (DNN)

that can be able to mimic the dynamic behavior described by the dynamic model of the

FCCU. It should consider the more sensitive input and output variables defined in the

sensitivity analysis.

The fourth specific objective is related with the main project in which this work is embedded,

Planty(k)

u(k)

d(k)

MPC

Steady State

Detection

Data

Reconciliation

Gross Error

Detection

Parameter

Estimation

Economic

Optimization

RTO


as it was mentioned earlier the RTO project for the FCCU. In this task the output variables

of the dynamic model are contaminated with gross and gaussian noise to resemble the

signals obtained in a real FCCU.

The fifth specific objective is to define the stability and controllability regions of the dynamic

model of the FCCU. Finally, the sixth task is to establish an adequate control structure for

the FCCU.

1.3 Thesis outline

This thesis is divided in six chapters. The first chapter presents an initial overview of the

scope and motivation of the project. The second chapter presents the construction of a

dynamic model for an Orthoflow F type FCCU. This model considers: riser, reactor, stripping

section, regenerator and interconnection valves. It can predict with good accuracy, the yield

of gasoline and other products, and the most relevant operative variables.

In the third chapter, the relative gain array (RGA) methodology will be used to analyze the

effect of certain input variables on the performance of the FCCU. The fourth chapter deals

with the construction of a dynamic neural network (DNN) for the FCCU.

In the fifth chapter, a stability analysis is performed. The stability region in terms of control

variables is presented. The multiplicity of steady states and eigenvalues analysis are also

part of the stability analysis.

Finally, the sixth chapter integrates the previous developments, it presents the evaluation

of the control strategy over the FCCU using the dynamic model developed in the chapter 2

as a virtual plant and the variable pairing obtained by the RGA results from the chapter 3.

This work has a confidentiality agreement with a third partner, therefore some of the

information and/or themes and/or results have been removed for public distribution.


References

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Chapter 2 FCCU dynamic model

In this chapter, a dynamic model for the Kellogg Orthoflow F model of FCCU is presented.

It considers the following subsystems: riser, stripper and disengaging unit, slide and control

valves and the two-stage regenerator. The riser is modeled as a one-dimensional plug flow

reactor employing a modified 6 lump kinetic model. The regenerator is modeled considering

the two regions, dense and freeboard, and the two-phase theory for the dense region. The

slide valves are coupled with pressure equations in both the stripper and regenerator

stages, to describe the overall dynamic behavior of the unit.

2.1 Introduction

Rigorous mathematical modeling of industrial processes is a challenging task regarding its

high complexity and scale. Additionally, the wide diversity of process disturbances and the

lack of validation measurements increase the challenge.

The initial attempts for the modeling of the fluid catalytic cracking unit (FCCU) were focused

on the kinetics of catalytic cracking. The lump kinetic strategy was developed for the

catalytic cracking process due to the chemical complexity of the FCCU feed and products

and the analytical requirements for the online experimental data acquisition which difficult

the complete characterization. This strategy considers groups of different molecules, called

lumps, according to its boiling point and molecular structure. The most characteristic lumps

are paraffins, olefins, naphtenes and aromatics (Oliveira, Cerqueira and Ram, 2012). This

kinetic approach has the advantage of reducing the computational effort in the parameter

estimation and reactor modeling. The accuracy and precision of the results which rely on

the number and type of lumps taken into consideration (Xiong et al., 2015b).

The three-lump kinetic model proposed by Weekman and Nace (Weekman and Nace,

1970) was one of the first widely used models in commercial FCCU. This model considers


feedstock, gasoline (C5-221°C) and remaining gases plus coke as lumps. For the modeling

and simulation of the entire FCCU, it is necessary that the kinetic cracking model consider

the coke on catalyst as a lump, as it was presented by Lee and coworkers in a four-lump

model (Lee, Chen, et al., 1989); this kinetic model has been the base of several studies for

the entire riser-regenerator system in FCCU (Gupta and Subba Rao, 2001; Han and Chung,

2001; Nayak, Joshi and Ranade, 2005).

Several other authors have extended the lump approach to cover a larger spectrum of

feedstocks and products of the FCCU.

The table 2-1 presents a brief review of these works. In a general way, there are two types

of kinetic model orientations in FCCU: Feedstock and Product. The feedstock kinetic model

orientation describes with high accuracy the hydrocarbon feed, meanwhile the product

kinetic perspective is used to correlate accurately the products conversion.

Table 2-1. Literature review of the lump kinetic scheme for catalytic cracking used in FCCU modeling.

Number

of lumps Description Reference

3

Product oriented kinetic model.

Lumps: Gas oil feedstock, gasoline and gases+coke.

The basic lumping strategy is related to the boiling point.

(Weekman &

Nace, 1970)

4

Product type oriented kinetic model, it considers the

following Lumps: Gas oil feedstock, gasoline, C1-C4 gas

and coke.

It considers the coke as an independent component.

(Lee, Chen,

Huang, & Pan,

1989)

5

Product oriented kinetic model.

Lumps: Gas oil feedstock, gasoline, dry gas (Hydrogen,

methane, ethane and ethylene), LPG (propane,

propylene, n-butane, isobutene and butene) and coke.

Useful to design and simulate compressors for the FCC

units considering the separation of the gas components.

(Ancheyta-

Juarez, Lopez-

Isunza, Aguilar-

Rodriguez, &

Moreno-Mayorga,

1997)

6 Product oriented kinetic model. (Xiong, Lu, Wang,

Chapter 2 FCCU dynamic model 33

Number

of lumps Description Reference

Lumps: Unconverted feedstock (340°C+), diesel (200-

340°C), gasoline (C5-200°C), LPG, dry gas and coke.

The study presents an analysis of apparent activation

energies indicating that low reaction temperature

increases gasoline and diesel production.

& Gao, 2015)

8

Feedstock oriented kinetic model.

Lumps: Paraffins, naphthenes and aromatics in light (220-

343°C) and heavy fractions (343°C+), gasoline (C5-

220°C) and gases+coke (C1-C4+coke).

It was used effectively for modeling a short reaction time

riser reactor at similar industrial operating conditions.

(Kraemer,

Sedran, & de

Lasa, 1990)

10


Lumps: Paraffins, naphthenes, aromatics and carbons

among aromatic rings in light (220-343°C) and heavy

fractions (343°C+), gasoline (C5-220°C) and gases+coke

(Cs-C4+coke). This model requires accurate feed

characterization and has been used for several authors

regarding its good fit with plant and experimental data

(Secchi, Santos, Neumann, & Trierweiler, 2001).

(Gross, Jacob,

Nace, & Voltz,

1976)

21


Based on the 10 lump mechanism (Gross et al., 1976),

but expanded to 21 lumps and changing the definitions of

several key lumps. The lumping strategy stands into two

classification approaches: Boiling point, being the light

(220-343°C), heavy (343-510°C) and residue fraction

(510°C+); and the chemical type: Paraffins, naphthenes,

aromatic carbon atoms and substituents, light gases (C1-

C5), coke and gasoline.

(Aspentech,

2011)

Another kinetic modeling approach is the single event model which takes into consideration

the transition state theory for the calculation of the kinetic parameters (Feng, Vynckier and


Froment, 1993). With this approach, it is necessary to calculate the Gibbs free energy of

the activated complex in the transition state, even if it is theoretically possible for a large

and complex reaction network, the high computational demand makes it unfeasible now,

as it requires the use of a quantum chemical software package (Lee et al., 2011).

The catalytic model nonlinearity increases with the consideration of the catalyst deactivation

process (Corma, Melo and Sauvanaud, 2005). The main cause of catalyst deactivation is

the coke layer deposition over the catalyst surface. One of the first attempts on the catalyst

deactivation model was made by Voorhies (Voorhies, 1945), in his work the amount of coke

on catalyst depends only on the catalyst residence time in the riser.

Several empirical equations have been used to describe the deactivation effect in the

catalytic cracking process with a good correlation with plant data (Weekman and Nace,

1970; Corella et al., 1985; Gianetto et al., 1994; Patel et al., 2013).

The main cracking reactions and the feed vaporization occur inside the riser, along with the

catalyst deactivation and the momentum transfer between the gaseous and solid phases.

These simultaneous physicochemical processes need to be considered for a

comprehensive mathematical abstraction into a successful model.

The feed vaporization has a key role in the FCCU performance, because the presence of

liquid inside the riser increases coke formation and reduces gasoline conversion (Gupta

and Subba Rao, 2001).

Several authors have considered the one-dimensional assumption to the riser reactor

modeling (Arbel, Huang and Rinard, 1996; Han, Chung and Riggs, 2000; Fernandes,

Verstraete, et al., 2007). In this riser modeling approach, the behavior of both catalyst and

gas phases are modeled as a one-dimension plug flow reactor (PFR), exchanging mass,

heat and momentum (Oliveira, Cerqueira and Ram, 2012).

Momentum balances have been developed to take into consideration the contact time

between the catalyst particles and the gaseous hydrocarbons, and the effect on the

products conversion and kinetics. Different ideas to cope with this issue have been

proposed, Malay et al. (Malay, Rohani and Milne, 1999) assumed that the ratio between


gas a catalyst phases is constant, this assumption results in a good agreement with plant

data and reduces the complexity of the momentum balance equations.

For the stripper and disengaging systems, models are often presented in a simplified way,

to the application of dynamic models of the FCCU (Secchi et al., 2001). The usual

approximation is to consider the system as a continuous stirred tank without thermal

cracking reactions involved (Oliveira, Cerqueira and Ram, 2012). Although, some authors

included thermal cracking reactions (Araujo-Monroy and López-Isunza, 2006).

Other important modeling consideration is related to the stripping efficiency: the fraction of

hydrocarbons that remain adsorbed in the catalyst pores after the stripping process. The

coke deposited over the catalyst surface during the catalytic cracking process has a lower

hydrogen to carbon (H/C) molar ratio in comparison to paraffin, naphthenic or aromatic

compounds.

The adsorbed hydrocarbon fraction after the stripping process increases the H/C ratio and

consequently the heat produced in the regeneration stage (Koon et al., 2000). Some

authors have proposed several empirical functions to estimate this adsorbed fraction as a

function of the steam, catalyst and hydrocarbon feed flowrates (Arbel, Huang and Rinard,

1996; Bollas et al., 2007).

As well as the other sub-systems of the FCCU, the regenerator has been subject of several

research studies, regarding its dynamic behavior, hydrodynamics and multiphase

characteristics. The first idea to accomplish the mathematical representation of this unit

was the consideration of the existence of two main regions inside the regenerator (Arbel,

Huang and Rinard, 1996). In this theory, the regenerator is composed by a dense region

and a dilute region. The dense region is where the bulk of the catalyst remains and

homogeneous and heterogeneous combustion reactions take place.

The dilute phase or freeboard, the catalyst particles are in low concentration, in comparison

to the combustion gases. In the dilute phase, mainly homogeneous combustion reactions

take place (Fernandes, Pinheiro, et al., 2007).


Particularly for the dense phase region, some authors have considered the two-phase

theory to model the bubble effect generated in the high gas speed fluidization process

(Faltsi-Saravelou and Vasalos, 1991). In this theory, the gas components are divided into

two different environments or phases, named emulsion and bubble phase (Kunii and

Levenspiel, 1991).

The emulsion phase is composed by catalyst particles dispersed in a gas mixture. The

bubble phase results from the higher gas velocity in comparison to the minimum fluidization

velocity, creating catalyst free bubbles.

In the bubble phase solely reactions of homogeneous gas phase combustion take place.

This theory has been successfully applied to industrial plant data, with good agreement in

comparison with other fluidization models (Lee, Yu, et al., 1989). In other approach, the

dense region has been modeled as a continuous stirred tank reactor (CSTR) by several

authors, with no significant differences with plant data. This approach allows to predict the

complex fluidization behavior of the FCCU (Secchi et al., 2001).

In other way, the freeboard region has been subject of several modeling strategies, such

as considering a one-dimensional plug flow reactor (PFR) (Hernández-Barajas, Vázquez-

Román and Salazar-Sotelo, 2006).

2.2 Model description

The figure 2-1 shows a schematic representation of the Kellogg Orthoflow F unit. In this

section, the structure and modeling assumptions of the dynamic model are presented. The

detailed equation development and input parameters of the model are presented in the

appendix A and the full subsystem representation of the model is presented in the appendix

B.


Figure 2-1. Schematic representation of the Kellogg Orthoflow F unit, adapted from Zanin (2001).

Gas oil

feed

Flue gas

Reactor

Stripping

steam

Regenerator

Riser

Spent

catalyst

Regenerated

catalyst

FCC

products to

fractionator

Air

1st

Regeneration

bed

2nd

Regeneration

bed


2.2.1 Mixing point and riser model

Figure 2-2. Mixing point subsystem diagram.

The regenerated catalyst-feed mixing point subsystem, presented in the figure 2-2, is

modeled as a quasi-steady state system; due to the consideration of the average contact

time for full vaporization takes about 0.1 seconds, which is about 3% of the total residence

time of the components in the riser (Ali, Rohani and Corriou, 1997).

For this subsystem is assumed that components reach the thermal equilibrium, the outlet

temperature for the gas and the catalyst are equal. No cracking reactions and coke

deposition over the catalyst surface are expected in this subsystem, therefore, the mass

flow of the catalyst and feed does not change.

The pressure in the mixing point is calculated based on the pressure drop of the gas phase

in the riser and the gas pressure exerted in the stripping-disengaging section.

Regenerated

catalyst-feed

mixing point

Gas oil

feed

Regenerated Catalyst

Gas oil

vaporized

Catalyst

Regenerated catalyst slide

valve


Figure 2-3. Riser subsystem diagram.

The figure 2-3 presents the riser subsystem diagram, in which the gas oil vaporized in the

mixing point subsystem reacts with the help of the catalyst. The physical riser is divided in

two zones based on the phase involved: Gas and solid phases. The phase interaction is

presented between the two distinct phases. The interaction is primarily of mass, heat and

momentum interchange between the phases.

The mass transference is constituted by the coke, which is the last step in the cracking

reaction mechanism, and it is deposited over the catalyst surface and changes in gas

composition due to the cracking reaction process.

The heat transference in the riser subsystem occurs due to the endothermic nature of the

cracking reactions and the heat gained by the catalyst in the regeneration process. The

momentum is transferred mainly between the gas and the catalyst solid particles and the

both phases with the riser internal components and walls.

The riser is modeled as a one-dimension tubular reactor in quasi-steady state. Solution

algorithm. The mass balance is performed by component using the lump kinetic scheme

presented by Araujo-Monroy & López-Isunza (2006) and adapted to this unit with a

parameter estimation strategy presented.

This lumping methodology is based on the Paraffinic, Olefinic, Naphthenic and Aromatic

(PONA) contents into the feed gas oil and cracking products. A schematic representation

of the major lump kinetic model for the large species is presented in the figure 2-4.

Riser gas

phase

Riser solid phase

Gas oil vaporized

Catalyst

Phase interaction

Cracked products

Deactivated catalyst


Figure 2-4. Schematic representation of the 6-lump cracking kinetic scheme, the lumps are: Gas oil (GO), light cycle oil (LCO), gasoline (G), light gases (LG), liquefied petroleum

gas (LPG) and coke (C). Adapted from Araujo-Monroy & López-Isunza (2006).

The reaction mechanism is based on the following steps:

1. Reversible adsorption of the PONA components of each lump over the catalyst

surface.

2. Formation of an adsorption reaction intermediate 𝜃𝑖.

3. Superficial cracking reaction and irreversible desorption.

Every major lump (GO, LCO, G, LPG, LG and C) has a PONA composition in terms of mass

fraction. The cracking reaction network generated for the six lumps with four components

is presented in the figure 2-5. All the reactions have a first-order reaction rate expression.

Gas oil

Coke

LCO Gasoline

LPG

Light gas


Figure 2-5. Catalytic cracking reaction mechanism, a) Reactions of the PONA components of the GO major lump (P1, O1, N1), b) Reactions of the PONA components of the LCO major lump (P2, O2, N2, A2), c) Reactions of the PONA components of the G

major lump (P3, O3, N3, A3) and d) Reactions of the PONA components of the LPG major lump (P4, O4). Adapted from (Araujo-Monroy and López-Isunza, 2006).

The riser pressure profile is developed by considering the approximation that the pressure

drop is caused by hydrostatic pressure of the solid catalyst (Fernandes, Verstraete, et al.,

2007).

P1 θP1K-1

K1

K2K3

K4

K5

K6

P2

O2

N2

A2

C

N1 θN1K-7

K7

K8

K9

K10

O2

A2

C

A1 θA1K-11

K11K12

K13

A2

C

a)

P2 θP2K-14

K14

K15K16

K17

K18

K19

P3

O3

N3

A3

C

O2 θO2K-20

K20

K21

K22

K23

P3

A3

C

b)

N2 θN2K-24

K24

K25

K26

K27

P3

A3

LG

A2 θA2K-28

K28

K29

K30

K31

A3

LG

C

P3 θP3K-32

K32K33

K34

P4

c)

O4

O3 θO3K-35

K35K36

K37

P4

C

N3 θN3K-38

K38K39

K40

O4

LG

A3 θA3K-41

K41K42

K43

LG

C

P4 θP4K-44

K44 K45 LG

b)

O4 θO4K-46

K46 K47 LG


The feed stream characterization uses the n-d-M correlation method (Sadeghbeigi, 2012).

It is used to predict the weight fraction of paraffinic, naphthenic and aromatic compounds

in the feed gas oil. This method requires only viscosity, boiling point and specific gravity

data, which is available from an experimental cut assay.

2.2.2 Strippinig-Disengaging Model

Figure 2-6. Stripping-Disengaging subsystem diagram.

In the Stripping-Disengaging (SD) section the catalyst level is an important variable to

control, therefore, a catalyst inventory mass balance is presented as a continuous stirred

tank dynamic model.

The catalyst flow from the SD section is controlled by a slide valve, as it is presented in the

figure 2-6, and the flow is driven by the pressure difference between the bottom of the

reactor and the bottom of the first regeneration bed.

The coke concentration in the stripper is calculated using the coke on catalyst concentration

at the riser outlet and an empirical correlation proposed by Fernandes, Verstraete, et al.

(2007) which calculates the amount of hydrocarbon adsorbed at the catalyst surface, this

Reactor/

Stripper

Stripping

steam

FCC

vapour

products

Spent

catalyst

slide valveGas

Catalyst

Spent

catalyst


concentration is usually called cat-to-oil coke. The correlation estimates this quantity as a

function of the SD temperature.

The gas phase pressure in the SD section is calculated with the ideal gas model. The

gaseous products mass flow from the SD section is determined by the opening of the control

valve between the SD section and the main fractionator.

The energy balance in the SD section is performed assuming that exist thermal equilibrium

between the catalyst and gas phases; it implies that the SD outlet streams are also in

thermal equilibrium in an adiabatic operation. The heat of desorption is neglected and the

specific heat for the gas and the catalyst are assumed constant with the temperature.

2.2.3 Regenerator dense phase model

The regenerator consists of two stages, in which the superficial coke layer is removed by

combustion and thus the catalyst surface is regenerated for the cracking reactions. A

schematic representation of the regenerator system for the Orthoflow F FCCU is presented

in the figure 2-7.


Figure 2-7. Regenerator subsystem diagram.

The two-phase fluidization theory is used to describe the combustion, particularly in the

dense phase of the regenerator (Kunii & Levenspiel, 1991). There is an emulsion phase, in

which a high concentration of catalyst reacts with the combustion air producing as products

carbon monoxide (CO), carbon dioxide (CO2), water (H2O), unreacted oxygen (O2) and

nitrogen (N2).

The combustion kinetics are diffusion controlled, taking into account the porosity of the

catalyst particles (Goodwin, R.D., Weisz, 1963). The intrinsic CO2/CO molar ratio is used

as the base for the reaction mechanism and the assumption that the coke is composed

solely of hydrocarbons. This is reasonable due the low composition of sulfuric and

nitrogenous compounds (Sadeghbeigi, 2012).

The base coke formula is 𝐶𝐻𝑞, where 𝑞 is the atomic ratio of hydrogen to carbon in the

catalytic coke.

1st regenerator

bed emulsion

phase

(gas+catalyst)

1st regenerator

bed emulsion

phase

(gas+catalyst)

1st regenerator

bed bubble

phase (gas)

1st regenerator

bed bubble

phase (gas)

Air blowerAir blower Air flow

division

Air flow

division

Air

Regenerator freeboard gas phaseRegenerator freeboard gas phase

2nd

regenerator

bed emulsion

phase

(gas+catalyst)

2nd

regenerator

bed emulsion

phase

(gas+catalyst)

2nd

regenerator

bed bubble

phase (gas)

2nd

regenerator

bed bubble

phase (gas)

Flue gas

to stack

Spent

catalyst

Phase

interaction

Phase

interaction

1st regeneration

stage catalyst

flow

Combustion

gases

Combustion

gases

Regenerated

Catalyst

1st regenerator

air flow

2nd

regenerator

air flow


𝐶𝐻𝑞 + (0.5 + 0.25𝑞)𝑂21. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂 + 0.5𝑞𝐻2𝑂

𝐶𝐻𝑞 + (1 + 0.25𝑞)𝑂22. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂2 + 0.5𝑞𝐻2𝑂

𝐶𝑂 +1

2𝑂2

3. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂2

𝐶𝑂 +1

2𝑂2

4. 𝑔𝑎𝑠→ 𝐶𝑂2

(2-1)

The carbon combustion reactions produce simultaneously CO2 and CO. However, the CO

produced undergoes further oxidation to CO2 through the so-called after-burning reactions.

The oxidation of CO to CO2 can be of two different natures: heterogeneous (catalytic) or

homogeneous combustion (Ali, Rohani, & Corriou, 1997).

The catalyst flow from the first bed to the second is determined by the weir height that

separates the two stages, as it is shown in figure 2-1. On the other hand, the catalyst flow

from the second regeneration bed section is determined by a slide valve to the regenerated

catalyst-feed mixing point.

The average regenerator pressure (𝑃𝑅𝐺𝑁𝑔𝑎𝑠) is calculated at the freeboard conditions of

composition, temperature and density, and it is assumed that the ideal gas equation of state

is adequate at the operating conditions of the regenerator.

The mass and energy balances in the regenerator are modelled as a continuous stirred

tank reactor (CSTR) dynamic model. The model is proposed for the coke and gases in both

emulsion and bubble phase. In the figure 2-8 a schematic representation of the fluidization

process inside the reactor.

The gases are transferred through the interface, but the catalyst particles are only placed

in the emulsion phase. The bubbles do not contain catalyst particles, hence only

homogeneous combustion kinetics are developed in this region. The mass transference

rate is calculated using an overall mass transference coefficient for each gas. In the

emulsion phase the catalyst and the gases are in thermal equilibrium.

Additionally, the heat loss to the environment is calculated using a convective heat transfer

coefficient from the regenerator bulk temperature and the inner regenerator wall, the


conductive heat transference inside the regenerator wall and the conductive heat

transference of the outer regenerator wall and the atmosphere.

Figure 2-8. a) Representation of the regenerator bubbling fluidization regime b) Detail of the gas mass transference through the interface of the bubble-emulsion phases.

2.2.4 Regenerator bed characterization model

The bed characterization is made considering that the dense phase is constituted by the

catalyst volume, the gas volume in the emulsion phase and the gas volume in the bubble

phase. The volume fractions of each phase were calculated using validated empirical

equations (Han & Chung, 2001).

2.2.5 Freeboard model

The freeboard region is modeled as a dynamic CSTR reactor for the mass and energy of

the gaseous species and it is assumed that coke is not present in the freeboard, just

homogeneous combustion occurs in that region and the catalyst lost in the flue gas is

neglected, this consideration is the high efficiency of the cyclone catalyst recovery system

(Sadeghbeigi, 2012). The catalyst particles ejected to the freeboard does not constitute an

important fraction of the total volume.

2.3 Solution algorithm

The dynamic FCC model consists of an equation system that mixes ordinary differential

(ODE) and algebraic ones (AE). It constitutes a differential algebraic equation system

(DAE). The total of 47+30*(N) equations are summarized in the table 2-2, where N is the

Regenerator dense phase

Regenerator

dilute phase

or freeboard

Air

Emulsion

phase

Bubble

phase

O2, CO, CO2,

H2O, N2

O2, CO, CO2,

H2O, N2, CHq,

Catalyst

Interface

a) b)


number of sections in which de riser is discretized.

Table 2-2. Equations summary for each subsystem

Subsystem Equations

AE ODE Total

Feed-catalyst mixing point 2 0 2

Riser 30*N 0 30*N

SD 0 4 4

Regenerator 1 36 37

FCCU 3+30*N 40 43+30*N

The quasi-steady consideration for the riser implies that, for the construction of the DAE

system, a discretization in the axial direction was developed to generate a set of N algebraic

equations.

2.3.1 Riser discretization

As an example of the riser discretization strategy, the mass balance for each PONA lump,

is considered:

𝜖𝑔𝑢𝑔𝜌𝑔

𝑑𝑋𝑖,𝑗

𝑑𝑧= 𝜌𝑐𝑅𝑖,𝑗

𝑔Ψ (2-2)

A progressive differential approximation for the first derivative is going to be used:

𝑑𝑋𝑖,𝑗

𝑑𝑧=𝑋𝑖,𝑗(𝑘) − 𝑋𝑖,𝑗(𝑘−ℎ)

ℎ+ 𝑂(ℎ) ∀ 𝑘 ∈ {ℎ, 2ℎ,…𝑁ℎ}

𝑑𝑋𝑖,𝑗

𝑑𝑧=𝑋𝑖,𝑗(𝑘) − 𝑋𝑖,𝑗(𝑘−ℎ)

ℎ=𝜌𝑐𝑅𝑖,𝑗

𝑔

(𝑘)Ψ(𝑘)

𝜖𝑔𝑢𝑔𝜌𝑔

ℎ =𝐻𝑟𝑖𝑠𝑁

(2-3)

The number of discretization sections are defined by a solution independence criteria

analysis.

2.3.2 Numerical solution strategy

Two numerical calculation tools are going to be used to obtain the dynamic solution of the

FCC model:

1. ODE15s: This ordinary equation solver based on MATLAB® coding platform, it can

be extended for the solution of index 1 DAE systems.

2. DASSLC: This solver does the multirate integration of DAE systems. The integration


algorithm used in DASSLC is an extension of the DASSL code of Linda R. Petzold

(Petzold, 1982). This solver is used in the EMSO® modeling, simulation and

optimization environment developed by R.P Soares and A. R. Secchi (Soares &

Secchi, 2003).

The Mass matrix must be developed and it enters as an input of the function, the mass

matrix (𝑀(𝒚,𝑡)) is the implicit representation of a DAE system:

𝑀(𝒚,𝑡)

𝑑𝒚

𝑑𝑡= 𝒇(𝒚,𝒖,𝑡) (2-4)

The semi-explicit representation of the DAE system is:

𝑑𝒚

𝑑𝑡= 𝒇′(𝒚,𝒖,𝑡)

𝟎 = 𝒈(𝒚,𝒖,𝑡)

(2-5)

The initialization strategy presented in the figure 2-9 is required for an adequate estimation

of the initial values for the DAE algorithm, particularly for the algebraic equations of the riser

discretization which are particularly difficult to estimate. This scheme is quite useful to

overcome the high nonlinearity of the model and a smooth transition to the dynamic state.

Another variation of the solution algorithm involves the full solution of the system in steady-

state with a nonlinear algebraic solution solver. This approach has the complications related

to the multiplicity of steady states usually presented in this process (Hernández-Barajas,

Vázquez-Román, & Salazar-Sotelo, 2006; Maya-yescas, Bogle, & Lo, 1998), but the

solution is attainable with a similar initialization algorithm. The full steady-state solution is

particularly useful for the step test analysis.

2.4 Parameter estimation

The model validation was performed through parameter estimation using steady state data

obtained from an industrial facility. The data set used in the parameter estimation includes

a large set of operational conditions obtained from different feedstocks and process quality

requirements. For this task, a nonlinear steady-state parameter estimation problem was

proposed, as a multivariable square difference problem:

min𝝁(𝑿𝑷𝒍𝒂𝒏𝒕 −𝑿𝑴𝒐𝒅𝒆𝒍(𝝁))

𝑇𝛌(𝑿𝑷𝒍𝒂𝒏𝒕 − 𝑿𝑴𝒐𝒅𝒆𝒍(𝝁))

𝝁 ∈ {𝐴𝑐𝑐𝑖, 𝐴𝑐𝑗, 𝑎} (2-6)


0 < 𝐴𝑐𝑐𝑖 < 1

0 < 𝐴𝑐𝑗 < 1

0 < 𝑎 < 0.5

Where 𝜇 is the set of model parameters to estimate to minimize the difference between the

plant data (𝑋𝑃𝑙𝑎𝑛𝑡) and the model state variables (𝑋𝑀𝑜𝑑𝑒𝑙). 𝜆 is the weighting matrix. The

available plant measurements are focused on the riser and the regenerator. For the riser,

the outlet temperature, the fraction of the products (LCO, LPG and gasoline) and total coke

on catalyst are available; and for the regenerator, the dilute and dense phase temperature,

flue gas composition and coke concentration on the catalyst at the regenerator flue gas

outlet.

The parameters estimated for the riser are the catalyst activities for the six lumps

considered in the reaction path of the catalytic cracking (𝐴𝑐𝑐𝑖), and the deactivation constant

(𝑎). For the regenerator, the combustion catalytic activity (𝐴𝑐𝑗).


Figure 2-9. Initialization algorithm for the DAE FCCU model, equations and variables are presented in the appendix A.

Start

Estimation of Stripper-Disengager system.

Solve equations (13) to (20) in Steady StateK<10

K=K+1

FCCU Parameters and valve opening

fractions (xv) definition

WSDcat, CSD, WSD

g,

TSD

K=0

Variables estimated: Regenerator Pressure (PRGN),

Mixing Point Pressure (PMP), Riser Outlet

Temperature (Tris(hris)), Riser Outlet Coke

Concentration (Cris,out), Coke concentration in the

second regenration stage outlet (Crgn2).

Estimation of Regenerator system. Solve

equations (25) to (42) in Steady State

Yes

Feed Characterization n-d-M Correlation

Method Xi,GO Ɐ i ε {P,N,A}

Wc reg1,Wc

reg2 , Creg1, Treg1, Twall1,

Cireg1,E, Ci

reg1,B, Creg2, Treg2, Twall2,

Cireg2,E, Ci

reg2,B

Estimation of Freeboard system. Solve

equations (47) to (50) in Steady State

CiFB, TFB, TwallFB, Prgn

Estimation of Riser system. Solve equations

(3) to (12) in Steady State.The ODE15s

algorithm is used to solve ODE system in the

riser length dimension (z)

Xij, θij, Tris

Solve the full DAE System with the estimated

initial values for the Differential equations and

the Algebraic Equations. The DASSLC or

ODE15s algoritmh is used to solve the

system.

Save the output time series and plot the

required variables as a function of time.

No

End

Feed experimental properties:

Refractive Index (RI20°C),

Density (D20°C), Average Molecular

Weight (MW), Sulphur Content (S)

Define the number of divisions

of the riser to match the

DAE model (N)

Define the simulation time


2.5 Noise addition

To simulate the behavior of the real FCCU, error was added to the time series resulting

from the DAE model with the Signal Noise Ratio function presented in the MATLAB® coding

platform. The signal to noise ratio characterizes the quality of the signal detection of a

measuring system. It quantifies how much a signal has been corrupted by noise, particularly

this type of noise is known as gaussian error. It compares the level of a desired signal to

the level of background noise.

Additionally, the output signal from the gaussian error addition was contaminated with gross

error: a set of 20% of the overall data was randomly selected and errors up to 5% of the

maxima values were added.

In the RTO structure, it is required that the output signal from the plant have noise so that

the gross detection module can filter this type of error. It is a module of the application when

the dynamic model takes the place of the real plant.

2.6 Results

Three types of results are presented, the parameter estimation results are presented in the

first place to show the importance of this task in the modeling of the FCCU and the

divergence that exists if is only considered reference parameters into a real plant model.

Afterwards, the steady state results are shown in which the main relations of the model

variables are encountered. Finally, the dynamic open loop response of the model is

presented.

2.6.1 Riser solution independence

The riser was divided in N equally spaced sections for the construction of the DAE system.

The equation 2-3 can be written in terms of the limit where the length of the riser division

tends to zero, which is the same of N tends to infinity.

𝑑𝑋𝑖,𝑗

𝑑𝑧= limℎ=0

𝑋𝑖,𝑗(𝑘) − 𝑋𝑖,𝑗(𝑘−ℎ)

ℎ= lim𝑁=∞

𝑋𝑖,𝑗(𝑘) − 𝑋𝑖,𝑗(𝑘−ℎ)𝐻𝑟𝑖𝑠𝑁

∀ 𝑘 ∈ {ℎ, 2ℎ,…𝑁ℎ} (2-7)


ℎ =𝐻𝑟𝑖𝑠𝑁

Numerically an infinite number of divisions is unfeasible, but taking a restricted number of

divisions implies a large error, hence an adequate number of divisions is required for the

computation of the full FCCU model. The strategy proposed is the solution independence

with the following scenarios of riser divisions: N=5, 10, 20 and 50. The riser temperature

and gasoline lump concentration profiles are presented in the figure 2-10 and 2-11

respectively.

The number of divisions has an important impact on the accuracy and magnitude of the

variables of the riser, and its interdependence in the overall FCCU model. It has a tendency

of a unique solution for values of N larger than 50. To consider the different operational

cases that can lead to different solutions a value of N of 100 was used in the complete DAE

system description. With this parameter defined, the total number of equations solved at

each time step is 3043. Several computational issues were faced with the size of the

system. Particularly, the computational time required to solve the DAE dynamic system.

A numerical improvement for the model solution is to use an adaptive step size solver for

the riser system. This is a clear advantage, because the larger gradients that need to be

precisely calculated are essentially ant the first third of the riser axial direction. After this

region, the catalyst deactivation takes place and the catalytic cracking reactions are

neglected, therefore, the temperature and lump concentration practically has no change.


Figure 2-10. Temperature profile of the riser for different number of divisions (N).


Figure 2-11. Gasoline lump profile of the riser for different number of divisions (N).

2.6.2 Parameter estimation results

The figures 2-12 to 2-14 present the initial comparison of the steady-state solution of the

model with the reference parameters and the plant data. The model with the initial

parameters does not correlate adequately with the plant data considering the low

magnitude of the coefficient of determination (R2) in all cases.

This is the fundamental reason for the use of a parameter estimation strategy. The

substantial number of kinetic and adaptation parameters of the FCCU model, increases the

complexity of the optimization problem presented in the equation (2-6). Several

decomposition strategies have been used to speed the optimization process, and some

particularly to the FCCU modeling problem (Ancheyta-Juarez et al., 1997; Maronna &

Arcas, 2009; Özyurt & Pike, 2004).

The parameter estimation strategy assumes that the relations between the kinetic

coefficients of each PONA component in each major lump. The problem was solved after


45 iterations and a possible local minimum was found with the first-order optimality criteria

of 1x10-6 and the constraints over the optimization variables were satisfied.

Figure 2-12. Comparison of the base model riser outlet temperature prediction against steady state data.

Riser Outlet Temperature (K)

750 800 850 900 950

Mo

del

Ris

er

Ou

tlet

Tem

pera

ture

(K

)

750

800

850

900

950

R2=0.2632


Figure 2-13. Comparison of the base model second regenerator stage temperature prediction against steady state data.

The optimization problem was solved used the function fmincon in the MATLAB® coding

platform. This function search for the minimum of a constrained multivariable objective

function.

The figures 2-14 and 2-15 present the correlation plots for the regenerator second stage

temperature and riser outlet temperature, there was an improvement in the coefficient of

determination (R2) in comparison with the base case results. Although, the values are still

far from the unit, they still have representative values to continue with the study and the

magnitude is similar to other parameter estimation studies on FCC (Sildir et al., 2015).

Second stage regenerator temperature (K)

920 940 960 980 1000

Mo

del

seco

nd

sta

ge r

eg

en

era

tor

tem

pera

ture

(K

)

920

940

960

980

1000

R2=0.0227


Figure 2-14. Comparison of the final model riser outlet temperature prediction against steady state data.

Figure 2-15. Comparison of the final model second regenerator stage temperature prediction against steady state data.

Riser Outlet Temperature (K)

800 820 840 860 880 900

Mo

del

Ris

er

Ou

tlet

Tem

pera

ture

(K

)

800

820

840

860

880

900

R2=0.8420

Second stage regenerator temperature (K)

940 950 960 970 980 990

Mo

de

l s

ec

on

d s

tag

e r

eg

en

era

tor

tem

pe

ratu

re (

K)

940

950

960

970

980

990

R2=0.9302


Figure 2-16. Comparison of the final model gasoline yield prediction against steady state data.

2.6.3 Steady-state results

The figure 2-17 shows the freeboard outlet concentration of the main gaseous species as

a function of the total air flow inlet. It can be observed that the figure can be divided in two

regions. In the low air flow region, the CO concentration is higher than the CO2. This region

is sometimes referred as partial combustion operation (Arbel, Huang, & Rinard, 1996), in

which the coke regeneration is carried to the production of CO and lower regeneration

temperatures, as it can be seen in parallel in the figure 2-18.

The increase in the air flow produces an increase in the coke combustion, and considering

that the reaction heat for the CO2 production has a magnitude three times larger than the

CO combustion. This thermal effect is considerable higher than the cooling effect of the

nitrogen inert addition effect. The CO leaving the regenerator should be further combusted

in a boiler for additional heat recovery. One of the main advantages of the partial

combustion is that it allows the system to process heavier feeds, which has an increase

amount of Conradson carbon and a higher tendency to coke formation.

Gasoline Yield

0,48 0,50 0,52 0,54 0,56 0,58 0,60

Mo

de

l G

as

olin

e Y

ield

0,48

0,50

0,52

0,54

0,56

0,58

0,60

R2=0.9059


The regeneration elevated temperature has an important negative impact in the catalyst

activity, because it could change the structure and its catalytic properties (Sadeghbeigi,

2012). Other effect considered is the catalytic effect of the combustion of coke, because

the FCC catalyst has in its structure combustion promotors, which enhances the CO2

conversion in the catalyst surface.

The increase in the total air flow at a constant catalyst circulation rate and feed flowrate

generates an increase in the gasoline yield, as it is presented in the figure 2-19. This is a

consequence of the reduction of the coke on catalyst, having an increase effect in the

catalytic cracking reactions rates. The increase of the dense phase temperature of the

second regenerator stage also increase the temperature of the riser leading to an increase

in the cracking reactions.

There is a temperature tipping point in the figure 2-18, this is the maximum point in which

the heat released from the coke total combustion is equal to the heat removed by the

nitrogen in the air flow, after this point the regenerator temperature decreases and the

oxygen concentration in the flue gas increases, there is not enough coke to sustain the

increase in temperature.

In the figure 2-20 the freeboard outlet gas concentration is presented as a function of the

catalyst circulation rate at a fixed air flow and gas oil feed rate. At a low recirculation rate,

the amount of coke that enters to the regenerator is low, therefore, the regenerator has an

air excess and all the coke is combusted with an excess of oxygen in the flue gas.

With the increase of the circulation catalyst rate, more coke enters into the regenerator

system, therefore more carbon dioxide is produced and the excess oxygen is reduced. The

temperature of both regeneration stages and riser outlet are presented in the figure 2-21.

The trends that follow the regenerator temperature are associated to the amount of coke in

the regenerator and the coke to air ratio, which leads to total or partial combustion. The

increase in the gasoline yield is related mainly to the increase in the riser temperature and

catalyst fraction in the riser, as it is presented in the figure 2-21. This behavior is analogous

to the work done by Ali and Rohani (Ali et al., 1997).


Figure 2-17. Freeboard outlet gas concentration as a function of the total air flow to regenerator for a catalyst circulation rate of 901.6 kg/s, a gas oil feed rate of 90,8 kg/s and

an air flow division ratio of 0.11.

Total air flow to regenerator (kg/s)

60 70 80 90 100

Co

ncen

trati

on

(%

mo

l)

0

1

2

3

4

5

6

COCO2O2


Figure 2-18. Regeneration stage temperatures and riser outlet temperature as a function of the total air flow to regenerator for a catalyst circulation rate of 901.6 kg/s, gas oil feed

rate of 90,8 kg/s and an air flow division ratio of 0.11.

Figure 2-19. Gasoline yield and coke on catalyst concentration in the second regeneration stage as a function of the total air flow to regenerator for a catalyst

Total air flow to regenerator kg/s)

60 70 80 90 100

Te

mp

era

ture

(K

)

600

700

800

900

1000

1100

1200

First regenerator stage temperatureSecond regeneration stage temperatureRiser outlet temperature

Total air flow to regenerator (kg/s)

60 70 80 90 100

Gaso

lin

e y

ield

0,485

0,490

0,495

0,500

0,505

0,510

0,515

Co

ke o

n c

ata

lyst

(kg

co

ke/k

g c

ata

lyst)

0,000

0,001

0,002

0,003

0,004

0,005

0,006

Gasoline yieldCoke on catalyst second regeneration stage


circulation rate of 901.6 kg/s, gas oil feed rate of 90,8 kg/s and air flow division ratio of 0.11.

Figure 2-20. Freeboard outlet gas concentration as a function of the catalyst circulation rate for a total air flow to regenerator of 80,1 kg/s, a gas oil feed rate of 90,8 kg/s and an

air flow division ratio of 0.11.

Catalyst circulation rate (kg/s)

700 800 900 1000 1100

Ca

rbo

n D

iox

yd

e a

nd

C

arb

on

Mo

no

xyd

e c

on

ce

ntr

ati

on

(%

mo

l)

0,0

0,5

1,0

1,5

2,0

2,5

3,0

Ox

yg

en

co

nc

en

tra

tio

n (

%m

ol)

0

2

4

6

8

10

COCO

2

O2


Figure 2-21. Regeneration stage temperatures and riser outlet temperature as a function of the catalyst circulation rate for a total air flow to regenerator of 80,1 kg/s, a gas oil feed

rate of 90,8 kg/s and an air flow division ratio of 0.11.

Figure 2-22. Gasoline yield and coke on catalyst concentration in the second regeneration stage as a function of the catalyst circulation rate for a total air flow to


700 800 900 1000 1100

Te

mp

era

ture

(K

)

700

800

900

1000

1100

1200

1300

First regenerator stage temperatureSecond regenerator stage regenerator temperatureRiser outlet temperature


700 800 900 1000 1100

Ga

so

lin

e y

ield

0,2

0,3

0,4

0,5

0,6

Co

ke

on

ca

taly

st

(kg

co

ke

/kg

ca

taly

st)

0,0000

0,0005

0,0010

0,0015

0,0020

0,0025

Gasoline yield Coke on catalyst second regenerator stage


regenerator of 80,1 kg/s, a gas oil feed rate of 90,8 kg/s and an air flow division ratio of 0.11.

An important characteristic of the Orthoflow F FCCU type is the two-regenerator stage

system. The amount of air that enters to the regenerator stages can be calculated with the

measurement of the total air flow and the air flow division, and with the direct measurement

of the two flows.

The main advantage of the two-stage regenerator is that it can work at lower temperature

in comparison to the single stage regenerator. In the first stage, the main coke combustion

occurs, meanwhile in the second stage, the objective is to control the catalytic activity with

an excess of air maintaining a low temperature and assuring the maximum gasoline yield

possible. This behavior can be observed in the figure 2-23.

Figure 2-23. Gasoline yield and second regenerator stage temperature as a function of the air flow to the second regenerator stage for an air flow to the first regenerator stage of

71,9 kg/s, a gas oil feed rate of 90,8 kg/s and a catalyst circulation rate of 901,6 kg/s.

2.6.4 Dynamic-state results

The dynamic response of the FCCU model are presented according to the figure 2-24. In

which the step test profile for each of the valves opening fraction is presented. The main

control valves of the FCCU are presented in the figure 2-1.

Air flow to the second regenerator stage (kg/s)

4 6 8 10 12

Se

co

nd

re

ge

ne

rato

r s

tag

e t

em

pe

ratu

re (

K)

1010

1020

1030

1040

1050

1060

Ga

so

lin

e Y

ield

0,48

0,49

0,50

0,51

0,52

0,53

0,54

Second regenerator stage temperatureGasoline Yield


Figure 2-24. Step change experiment description of the control valve opening for the dynamic state results.

An increase in the air flow to the first regeneration stage increases the temperature in the

regenerator and riser temperature, as it is shown in the figure 2-25. This increase in

temperature is caused for an incremented coke and CO combustion. The changes in the

flue gas composition, presented in the figure 2-25, indicates the CO concentration reduction

and the increase of unreacted oxygen.

In the figure 2-26 can be observed that the increase in the air flow to the regenerator first

stage increases the pressure of the regenerator and to maintain the hydraulic balance of

the system, the level of the first and the second stage catalyst decreases. The increase in

the regenerator pressure acts as a counter pressure for the catalyst flow from the SD

section to the regenerator, therefore, the SD section in the catalyst level increases.

The reduction in the coke on catalyst concentration of for the increase in the air flow to the

second regeneration stage, illustrated in the figure 2-27, increases the gasoline yield with

a more active surface for the cracking reactions and the additional factor of the higher riser

temperature. The dynamic of the FCCU presented here have a long-time response in

general, this behavior is attributed by the large reactor and regenerator catalyst hold up.

Operational

base case

Steady-State

5% Increase 5% Decrease

Time (s)

Valve opening fraction

0 s 1200 s 10200 s 19200 s


In general, the decrease in the air flow to the base operational stage shows hysteresis, this

is typical for the high nonlinearities presented in the model and is related to other FCCU

models presented in the literature (Bollas et al., 2007; Sildir et al., 2015).


Figure 2-25. Dynamic response of the regenerator second stage temperature, riser outlet temperature and flue gas composition for a step change in the air flow to the first

regeneration stage.

Time (s)

0 5000 10000 15000 20000

Tem

pera

ture

(K

)

860

880

900

920

940

960

980

1000

1020

Co

ncen

trati

on

(%

mo

l)

0,0

1,0

2,0

3,0

4,0

5,0

Riser outlet temperatureRegenerator second stage temperatureO2CO2CO


Figure 2-26. Dynamic response of the first and second regenerator stages level, SD reactor level and regenerator pressure for a step change in the air flow to the first

regeneration stage.

Time (s)

0 5000 10000 15000 20000

Le

ve

l (m

)

0

10

20

30

40

50

Pre

ss

ure

(k

Pa

)

0

100

200

300

400

SD Reactor level First regenerator stage levelSecond regenereator stage levelRegenerator pressure


Figure 2-27. Dynamic response of the second regenerator stages coke on catalyst and gasoline yield for a step change in the air flow to the first regeneration stage.

The dynamic response of the FCCU to a change in the outlet catalyst slide valve are

presented in the figures 2-28 and 2-29. The main changes are in the regenerator

temperature, flue gas composition and hydraulics in the reactor and regenerator. For an

increase in the valve opening a reduction in the regenerator temperature is presented; this

behavior is explained by considering the increase in the catalyst flow to the regenerator and

the energy and the oxygen to coke ratio in the pre-combustion zone. This effect can be

seen for the increase in the CO composition in the flue gas.

In the figure 2-29 are presented the dynamic responses for the step change in the reactor

catalyst outlet slide valve opening for the catalysts levels in the unit and the regenerator

pressure. In particular, to reach a new steady state, the system changes the catalyst level

in both reactor and regenerator to match the catalyst circulation rate in both slide valves.

The pressure changes slightly but it also has direct effects over the catalyst level.

Time (s)

0 5000 10000 15000 20000

Co

ke o

n c

ata

lys

t (k

g c

ok

e/k

g c

ata

lys

t)

0,0000

0,0005

0,0010

0,0015

0,0020

Ga

so

lin

e Y

ield

0,46

0,48

0,50

0,52

0,54

Coke on catalyst second regeneration stageGasoline yield


Figure 2-28. Dynamic response of the regenerator second stage temperature, carbon monoxide and oxygen composition in flue gas for a step change in the reactor outlet slide

valve.

Time (s)

0 5000 10000 15000 20000

Te

mp

era

ture

(K

)

920

940

960

980

1000

Co

nc

en

tra

tio

n (

%m

ol)

0,2

0,4

0,6

0,8

1,0

Second stage regenerator temperatureO2 composition in flue gas

CO composition in flue gas


Figure 2-29. Dynamic response of the regenerator first and second stage level, riser outlet temperature and oxygen composition in flue gas for a step change in the reactor

outlet slide valve.

The changes in valve opening for the flue gas has an important effect on the reactor and

regenerator pressure, and additionally for the hydraulic driven operation of the FCCU, the

catalyst hold up is also changed which in part have effects directly on the regenerator state

variables and the cracking kinetics in the riser.

2.6.5 Noise addition results

The signal noise can be attributed to several factors, such as: the fluctuation in the

transmission network, distortions of the signal conversion, environmental changes, etc. The

distinction between Gaussian and gross error are related to the frequency of appearance

in the output signal. The gross error is associated to random events in the signal

Time (s)

0 5000 10000 15000 20000

Level

(m)

0

5

10

15

20

25

30

Pre

ssu

re (

kP

a)

260

280

300

320

First stage regenerator levelSecond stage regenerator level SD reactor level Regenerator pressure


transmission that produces a change in its intensity. Meanwhile, the Gaussian noise is a

constant source of noise which has a normalized intensity distribution.

In the figure 2-32 the signal from the second regenerator stage temperature has been

contaminated with Gaussian and gross error. The sample time of this analysis is 5 seconds.

The amplitude of the noise addition is qualitatively similar to the one presented by

Ramachandran et al. (Ramachandran, Rangaiah, & Lakshminarayanan, 2007). The

different time scales presented in the figure 2-32 allows to identify the effect of the gross

error in the final output signal with some random high intensity points which are interesting

for a further study in signal processing and filtering.

a)


b)

Figure 2-30. Noise addition to the regenerator second stage temperature signal for a SNR of 40 and a 20% of data with gross error addition in different time scales, a) 0 to

4000 seconds; b) 0 to 500 seconds.

2.7 Conclusions

A detailed model of a Kellogg Orthoflow F type of FCCU unit based on the following

subsystems: Mixing point, riser, stripping-disengaging and regenerator was developed. The

parameter estimation strategy improved the model accuracy, although, it could improve

extensively with the use of fast data reconciliation and parameter estimation strategies in a

framework that evaluates the most sensitive parameters subject to optimization.

The dense phase regenerator model is an improvement to previous FCCU models of a

Kellogg Orthoflow F, considering the discrepancies that the previous models had over the

coke layer concentration which require correction parameters to adjust the amount of coke

remaining in the process (Lautenschlager Moro & Odloak, 1995). The detailed description

and parameters allow the data reconciliation based in phenomenological considerations.


The model presented for the FCCU have capabilities to work as a virtual plant for control

and optimization studies. Particularly this model can take the place of the plant in the RTO

scheme presented in the figure 1-5, therefore the RTO structure can be used and tested

with the noise module as a simulation experiment.

The subsystem modeling approach can be customized by subsections for a different type

of FCCU. The increase use of heavy oil in refineries impact the performance of the units,

particularly the FCCU takes the heavy cuts from the fractionation process. The naphthenic

and aromatic compounds tend to increase the coke formation in the catalytic cracking

process. Therefore, the regeneration step generates more heat and the temperature of the

flue gas will increase. This effect is in part handled with catalyst coolers (Sadeghbeigi, 2012)

which are devices what refrigerate part of the catalyst streams to reduce the heat removal

requirements in the process. This is modeled straight forward with the subsystem model

approach and easily merged into the full FCCU model.

The dynamics of the FCCU model indicates that there is a dynamic response of several

output variables with the change of only one manipulated variable. Also, there is changes

in the same output variable for changes in different manipulated variables. This indicates

that the control system of the plant should cope with this mixed effect and manipulate the

variables accordingly.

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Chapter 3 RGA analysis of the FCCU

In this chapter, the results from the dynamic model for the FCCU are going to be analyzed

using the Relative Array Analysis (RGA) to observe the possible pairing and the relationship

between state variables and manipulated variables in the system.

3.1 Introduction

One of the key factors in the design of control systems is to achieve a good pairing between

control and manipulated variables. In large scale industries, such as the petrochemical

industry, the lowest level in the control system usually is the regulatory control (Hovd and

Skogestad, 1993).

The Relative Gain Array (RGA) method has been a useful tool for the design of proper

control structure. The original work presented by Bristol (Bristol, 1966) proposes the method

of RGA as a tool for the pairing selection in a system with several SISO (Single Input Single

Output) loops in a decentralized control system scheme. Each element in the RGA matrix

is defined as the ratio between the open-loop gain of the control variable and the closed-

loop gain of the same control variable.

𝜆𝑖𝑗 =

(𝜕𝑦𝑖𝜕𝑢𝑗)𝑢𝑘≠𝑗

(𝜕𝑦𝑖𝜕𝑢𝑗)𝑦𝑘≠𝑖

(3-1)

A linear system can be described in a simplified way as:

𝑌(𝑠) = 𝐺(𝑠)𝑈(𝑠) (3-2)


Where 𝑈(𝑠) is the vector of manipulated variables or inputs, 𝑌(𝑠) is the vector of control

variables or outputs and 𝐺(𝑠) is the matrix of transfer functions. In steady state, the matrix

𝐺 can be defined as the gain matrix and the steady-state RGA matrix can be defined as:

Λ(𝐺) = 𝐺 ∘ (𝐺−1)𝑇 (3-3)

In the equation (2), the symbol ∘ denotes the element-by-element multiplication or Schur

product. The transfer matrix is non-singular. Although, this equation is in steady-state

(𝑠 = 0), the same can be computed in terms of the frequency. The pairing rules regarding

the RGA matrix results are:

• Avoid input and output pairs which have negative relative gains.

• Avoid input and output pairs which have large relative gains.

• Select input and output pairs which have the relative gain close to 1.

The first rule is presented in the work of Grosdidier et al. (Grosdidier, Morari and Holt, 1985)

in which for several systems and the pairings with negative components in the RGA matrix

should be avoided if possible.

The large values should be avoided as pointed out by Hovd and Skogestad (Hovd and

Skogestad, 1993) because it implies a high control difficulty. The unity rule is based in the

definition of the equation (1), because it follows that the open loop gain is related to the

closed loop gain by a factor of 𝜆𝑖𝑗−1, and therefore a value near of the unity will indicate that

the pair 𝑦𝑖 , 𝑢𝑗 will indeed be controlled as a SISO feedback control (Haggblom, 1997).

Before conducting the RGA analysis the input and output variables need to be defined

beforehand. A sensitivity analysis is required to determine how much each of the output

variables change with a variation on the input variables. The Sobol method (Sobol′, 2001)

is a global sensitivity analysis that takes into account the direct input output sensitivity and

also the cross effect of the variation of every parameter simultaneously.

The model is represented in its analysis of variance (ANOVA) representation:

Chapter 3 RGA analysis of the FCCU 81

𝑦 = 𝑓(𝑢) = 𝑓0 +∑ ∑ 𝑓𝑖1…𝑖𝑗 (𝑢𝑖1 , … , 𝑢𝑖𝑗)

𝑛

𝑖1<⋯<𝑖𝑗

𝑛

𝑗=1

𝑦 = 𝑓(𝑢) = 𝑓0 +∑𝑓𝑖(𝑢𝑖)

𝑛

𝑖=1

+∑𝑓𝑖𝑗(𝑢𝑖, 𝑢𝑗)

𝑛

𝑖<𝑗

+⋯+ 𝑓123…𝑛 (𝑢1, 𝑢2, … , 𝑢𝑛)

(3-4)

Where 𝑦 are the output variables of the model and 𝑢 corresponds to the input variables.

The number of summands in (3-4) is 2𝑛. Numerically a Monte-Carlo Approach to calculate

the sensitivity index of each variable can be performed defining a number of samples N to

the Monte Carlo Sampling algorithm (Saltelli, 2005). The sensitivity index for the input

variable 𝑢𝑖 with respect to the output variable 𝑦 can be expressed as:

𝑆𝑖 =

𝐷𝑖𝐷

𝑓0 =1

𝑁∑𝑓(�̂�𝑗)

𝑁

𝑗=1

𝐷 =1

𝑁∑𝑓2(�̂�𝑗)

𝑁

𝑗=1

− 𝑓02

𝐷𝑖 =1

𝑁∑𝑓(�̂�𝑗)𝑓(�̂�𝑖)

𝑁

𝑗=1

− 𝑓02

(3-5)

Where �̂�𝑗 are the random numbers generated for the manipulated variables of the system

in the operating ranges of the model.

In this chapter, the results from the dynamic model for the FCCU are going to be analyzed

using the Relative Array Analysis (RGA) to observe the possible pairing and the relationship

between state variables and manipulated variables in the system, defined from the global

sensitivity analysis.

3.2 Methodology

The fundamental objective of the process control is to maintain the FCCU at a safety and

profitable operation regime. The regulatory control is in the lowest part of the hierarchy of

the control system of the plant and is in charge of keeping the plant in a defined series of

set points.


An intermediate level in the control system structure corresponds to the supervisory control.

In which the system can change the plant’ set points to optimize some objective. In this

control layer, the system should consider the constraints related to the product quality,

utilities availability, mechanical integrity limits etc.

The control structure strategy proposed by (Hovd and Skogestad, 1993) describes first the

outputs selection. In which are included the primary and secondary controlled variables.

The primary variables are the output variables that have a fast control and are important to

accomplish the main control objective, such as levels, temperatures, pressures that must

be in certain limits. The secondary variables are variables that are in themselves have a

low priority to control in the low hierarchy of the control system, but it has importance in the

upper layers.

The inputs are usually related to degrees of freedom of the system which have values

according to the operation philosophy of design. In this type of process could be the opening

fraction of control valves, variable speed motors or drivers, mechanical devices inside the

equipment, etc.

After the input and output definition the variable pairing is performed. This step is

particularly important because it is the fundamental for ability of the control system to

overcome disturbances and the performance of the change in the set point for the upper

layers of the control hierarchy.

3.2.1 Output selection

There are several nonlinearities in the FCCU model presented in the chapter 2, this gives

the system a high interrelation between the manipulated variables and the output variables.

The primary variables that need to be controlled are:

• Regenerator first stage dense phase temperature 𝑇𝑟𝑒𝑔1

• Regenerator second stage dense phase temperature 𝑇𝑟𝑒𝑔2

• Riser outlet temperature 𝑇(𝑧=𝐻𝑟𝑖𝑠)𝑟𝑖𝑠

• Reactor-Regenerator pressure difference 𝑃𝑅𝐺𝑁𝑔𝑎𝑠− 𝑃𝑆𝐷

𝑔𝑎𝑠

• Reactor catalyst level 𝐿𝑆𝐷


• Regenerator first stage dense phase catalyst level 𝐿𝑟𝑒𝑔1

• Regenerator second stage dense phase catalyst level 𝐿𝑟𝑒𝑔2

The secondary output variables are:

• Gasoline yield 𝑌𝐺

• LPG Yield 𝑌𝐿𝑃𝐺

• LCO Yield 𝑌𝐿𝐶𝑂

• Gas oil conversion 𝜒𝐺𝑂

• Oxygen composition in the flue gas 𝐶𝑂2𝐹𝐵

• Coke on catalyst riser outlet 𝐶(𝑧=𝐻𝑟𝑖𝑠)𝑟𝑖𝑠

• Coke on catalyst regenerator first stage 𝐶𝑟𝑔𝑛1

• Coke on catalyst regenerator second stage 𝐶𝑟𝑔𝑛2

• Feed flowrate 𝐹𝑓𝑒𝑒𝑑

3.2.2 Input selection

The manipulated variables of the FCCU are related to the valve opening fractions of the

control valves presented in the unit.

• SD section catalyst slide valve opening fraction 𝑥𝑣𝑆𝐷

• Second regeneration stage catalyst slide valve opening fraction 𝑥𝑣𝑟𝑒𝑔2

• Flue gas valve opening fraction 𝑥𝑣𝐹𝐺.

• Air flow to the first regenerator stage valve opening fraction 𝑥𝑣𝐴𝑖𝑟1

• Air flow to the second regenerator stage valve opening fraction 𝑥𝑣𝐴𝑖𝑟2

3.2.3 Sensitivity analysis

Considering all the input and output variables, the sensitivity analysis will rank the input

variables for each output variable of the model. The algorithm for the Monte Carlo

simulations is to first generate random numbers for the manipulated variables In its

operation range.

The computation of the variances defined in the equation (3-5) uses the steady state

representation of the model and its results are placed in a chart in which every input and

output variable are ranked accordingly to the magnitude of its sensitivity indexes. Ans

considering:


∑𝑆𝑖

𝑛

𝑖=1

= 1 ∧ 0 ≤ 𝑆𝑖 ≤ 1

The table 3.1 indicates the relationship among the input variables and the output variables

for the primary output variables:

Table 3-1. Global sensitivity analysis results for the primary output and manipulated variables

Input/Output 𝑇𝑟𝑒𝑔1 𝑇𝑟𝑒𝑔2 𝑇(𝐻𝑟𝑖𝑠)𝑟𝑖𝑠 𝑃𝑅𝐺𝑁

𝑔𝑎𝑠− 𝑃𝑆𝐷

𝑔𝑎𝑠 𝐿𝑆𝐷 𝐿𝑟𝑒𝑔1 𝐿𝑟𝑒𝑔2

𝑥𝑣𝑆𝐷 0,020 0,016 0,012 0,002 0,098 0,039 0,018

𝑥𝑣𝑟𝑒𝑔2 0,008 0,022 0,081 0,004 0,008 0,021 0,075

𝑥𝑣𝐹𝐺 0,010 0,007 0,005 0,098 0,010 0,018 0,018

𝑥𝑣𝐴𝑖𝑟1 0,067 0,049 0,039 0,028 0,018 0,014 0,010

𝑥𝑣𝐴𝑖𝑟2 0,020 0,069 0,047 0,016 0,014 0,010 0,012

In the table 3-1 can be noted that the relative sensibility of every primary output for the

manipulated variable are mixed and a defined group of variables cannot be selected

straightforward from the obtained data. The regenerator catalyst levels are a hard variable

to measure or estimate considering the turbulent regime in the unit. This makes an

important factor in for the control purposes.

3.2.4 Variable pairing

In practice, the output variables considered follows the control structure similar to the one

proposed by Lautenschlager and Odloak (Lautenschlager and Odloak, 1995) for a Kellog

Orthoflow F FCCU type in which the output variables are the riser outlet temperature, gas

oil conversion, the temperatures of the regenerator dense phase at the first and second

stages, the pressure difference of the regenerator and reactor and the reactor level.

Considering the results from the sensitivity analysis, the most significant output variables

regarding the manipulated variables are similar to the ones found in practice, therefore this

are the variables used in the variable pairing.

The input or manipulated variables are the air flow rate to the regenerator stages, the

second regeneration stage catalyst slide valve opening fraction, the feed flowrate, the SD

section catalyst slide valve opening fraction and the flue gas valve opening fraction.


The riser temperature is important because it is an indicator of the cracking reaction

conversion and total product yields. The gas oil conversion can be measured in field with

the data from the fractionator bottoms and the feed flowrate and it is a crucial measurement

in the reactor performance and the main control objective. The temperatures of the

regenerator dense phase at the first and second stages are important for the stability of the

combustion process in the FCCU and with this control, the temperature of the freeboard is

kept below the upper limit for the mechanical integrity of the regenerator and the catalyst

activity.

The air flowrate to the regenerator is limited by the maximum capacity of the air blower and

this constrained makes the use of the total air flowrate useful with the ratio to specify the

air flow to the two regenerator stages. The feed flowrate is usually determined by the

refinery gas oil production or the market requirements for the products of the FCCU, it is

also limited by the maximum design capacity of the FCCU. The problem is therefore

constituted by 5 x 5 inlet- outlet variables. The equation (3-2) for this system is:

(

𝑇(𝑧=𝐻𝑟𝑖𝑠)𝑟𝑖𝑠

𝑇𝑟𝑒𝑔1𝑇𝑟𝑒𝑔2

𝑃𝑆𝐷𝑔𝑎𝑠− 𝑃𝑅𝐺𝑁

𝑔𝑎𝑠

𝐿𝑆𝐷 )

= (

𝑔11 ⋯ 𝑔16⋮ ⋱ ⋮𝑔61 ⋯ 𝑔55

)

(

𝑥𝑣𝐴𝑖𝑟1

𝑥𝑣𝐴𝑖𝑟2

𝑥𝑣𝑟𝑒𝑔2

𝑥𝑣𝑆𝐷

𝑥𝑣𝐹𝐺 )

(3-4)

3.3 Results

The process limits for the outlet variables let to define the outlet signal fraction to have a

dimensionless similar scale between variables.

�̂� =

𝑌𝑚𝑎𝑥 − 𝑌

𝑌𝑚𝑎𝑥 − 𝑌𝑚𝑖𝑛 (3-5)

The equation (3-5) is applied to the selected state variables except the gas oil conversion

which it is already dimensionless. The transfer function matrix (𝐺) for the base case

operational steady-state is:


𝐺 =

(

0.32154 0.21458 0.45121 −0.14521 −0.058640.684630.47895−0.324580.01423

0.154780.54769−0.078510.004785

−0.01467−0.154780.012970.14852

−0.09842−0.054780.004120.00012

0.098420.078430.748570.10482)

The RGA matrix is:

Λ(𝐺) =

(

−1.4136 −0.0787 2.0038 4.5131 0.09941.1021−0.7126−0.31450.0024

−0.27251.0199−0.02210.0012

0.00860.31620.02400.1254

0.10760.38640.03311.0003

0.0521−0.02941.27890.0076 )

According to the RGA matrix and the pairing rules, the input-output pairing proposed is:

Table 3-2. Input-output variable pairing

Output variable Input variable

𝑇(𝑧=𝐻𝑟𝑖𝑠)𝑟𝑖𝑠 𝑥𝑣

𝑟𝑒𝑔2

𝑇𝑟𝑒𝑔1 𝑥𝑣𝐴𝑖𝑟1

𝑇𝑟𝑒𝑔2 𝑥𝑣𝐴𝑖𝑟2

𝑃𝑆𝐷𝑔𝑎𝑠

− 𝑃𝑅𝐺𝑁𝑔𝑎𝑠

𝑥𝑣𝐹𝐺

𝐿𝑆𝐷 𝑥𝑣𝑆𝐷

The pairing results are similar to the relations made by Moro and Odloak (Lautenschlager

Moro and Odloak, 1995) with the difference in the riser outlet temperature which is

controlled with the second regeneration stage catalyst slide valve opening fraction. The

result is plausible accordingly with the qualitative analysis of the dynamic responses made

in the chapter 2.

3.4 Conclusions

The steady-state RGA method was applied for the FCCU model and the pairing results

agree with the qualitative analysis previously made. The RGA matrix also indicates that

there is another configuration plausible, considering the multiple effect of some manipulated


variables over the control variables. This pairing scheme is going to be evaluated in the

chapter 6 of this work.

The RGA analysis is a valuable tool, because it gives a quantitative idea of the most

responsive variable match for a multiple SISO system. In this case, the steady-state gain

matrix indicates that there is an important effect on several control variables, this could be

an indication of the possibility of a MIMO arrangement, and is going to be concluded with

the results from the chapter 6.

Other interesting fact about this analysis is that a system with N possible inputs and M

possible outputs, the total number of alternative control schemes is (Cao and Rossiter,

1997):

∑∑(

𝑀𝑚)(𝑁𝑛)

𝑁

𝑛=1

𝑀

𝑚=1

(𝑀𝑚) =

𝑀!

𝑚! (𝑀 − 𝑛)!

(3-6)

For our case, the number of possible control schemes is 961.

References

Bristol, E. (1966) ‘On a new measure of interaction for multivariable process control’, IEEE

Transactions on Automatic Control, 11(1), pp. 133–134. doi: 10.1109/TAC.1966.1098266.

Cao, Y. and Rossiter, D. (1997) ‘An input pre-screening technique for control structure

selection’, Computers & Chemical Engineering, 21(6), pp. 563–569. doi: 0098-1354/97.

Grosdidier, P., Morari, M. and Holt, B. R. (1985) ‘Closed-loop properties from steady-state

gain information’, Industrial & Engineering Chemistry Fundamentals. American Chemical

Society, 24(2), pp. 221–235. doi: 10.1021/i100018a015.

Haggblom, K. E. (1997) ‘Partial Relative Gain : A New Tool for Control Structure Selection’,

(0), pp. 1–6.

Hovd, M. and Skogestad, S. (1993) ‘Procedure for regulatory control structure selection

with application to the FCC process’, AIChE Journal. American Institute of Chemical

Engineers, 39(12), pp. 1938–1953. doi: 10.1002/aic.690391205.


Lautenschlager Moro, L. F. and Odloak, D. (1995) ‘Constrained multivariable control of fluid

catalytic cracking converters’, Journal of Process Control, 5(1), pp. 29–39. doi:

https://doi.org/10.1016/0959-1524(95)95943-8.

Chapter 4 DNN construction for the FCCU

This chapter presents the methodology used for the construction of a DNN for the FCCU.

The input data is gathered from the dynamic model presented in the chapter 2 arranged

as a time series with fixed step size. The input and output variables are the same that the

presented in the chapter 3.

4.1 Introduction

The artificial neural network (ANN) is a widely used method for modeling a large and diverse

type of systems. It has been useful in systems where the deep phenomena is not well

understood (Khalil, 2011).

The ANN design is an analogy to the human brain function. For instance, the biological

neurons are composed of dendrites, axon and the synaptic terminal. In the human brain,

the neurons transfer information through electrical impulses. The impulse travels through

the axon up to the dendrites and the information transference occurs through the dendrite

and the synaptic terminal of other cell based on a potential difference between the two cells

(Rutecki, 1992).

The ANN are composed of many processing units called neurons, just as the human brain.

Other similitude is the learning ability of the ANN, which is related to the training stage. The

ANN training is a procedure in which the parameters and internal structure of the network

adapts to a series of historic data from the system and replicate its behavior.

The neurons in the ANN are grouped into layers with a high degree of connectivity. The

training algorithm adjusts its architecture and parameters in a minimization problem


framework. The neuron is constituted of a series of inputs, a function and a series of output

signals (Hussain, 1999).

The design of a ANN follows 6 essential steps:

• Data collection: In this step, the data from the system is collected and organized,

the input data is placed along with the output responses.

• Network creation and configuration: The basic structure of the network is

established. The number of layers, the number of neurons on each layer, the input

and output neurons, which in many cases corresponds to the number of input and

output variables selected. The type of transference functions used in each neuron.

• Weights and delays initialization: For each input, a series of weights and delays are

placed. It depends on the type of data and its impact on the system performance.

• Network training: The training algorithm is selected and a random part of the data

is selected as training material.

• Network validation: After the training algorithm, the network uses the part of the data

that was not used in the previous step and performs a comparison of the prediction

values, determining the performance of the network and the ability of predicting data

away from the training spectra.

• Network usage: The last step consists in the use of the network, in this point the

network can be used to predict new states of the system or use it as an experimental

subject.

The neural networks can be classified as static or dynamic. The static neural networks

(SNN) does not require feedback data or delays, since its independent from time and the

output is calculated in the feedforward scheme. In the case of the dynamic neural networks

(DNN) the output depends also from the previous inputs and the actual state of the network.

(Adebiyi and Corripio, 2003).

The Nonlinear Autoregressive Exogenous (NARX) model is a recurrent nonlinear model

based on the linear model ARX and can model dynamic systems as time series. The

predicted value in the time series is generated using previous values from the series and

Chapter 4 DNN construction for the FCCU 91

an exogenous signal. The inputs and outputs can be multidimensional, and it is quite useful

in nonlinear dynamic system modeling (Diaconescu, 2008).

There has been extensive use of ANN in chemical and petrochemical applications, such as

dynamic modeling of chemical processes (Lennox et al., 1998), identification and control of

a FCCU (Vieira et al., 2005), fault diagnosis of FCCU (Sengupta and Khurana, 1995),

process optimization (Nascimento, Giudici and Guardani, 2000). Particularly for the FCC

process, the highly nonlinearities of the system and the capability of the ANN to work with

noise due instrumentation make it a suitable modeling and analysis approach (Bollas et al.,

2003).

4.2 Methodology

4.2.1 Data generation

Considering the development of a DNN for the FCCU, a great quantity of data is necessary

for training, validation and testing stages; this data was generated using the model

presented in the chapter 2. The input and output variables corresponds to the definition

presented in the chapter 3.

The results from the FCCU model solution were presented as a time series with 20 seconds

of time step. The six input variables were submitted to a step test like the one presented for

the dynamic response of the chapter 2 but with a larger time span to be able to fit correctly

the dynamic and steady state for each input change, as presented in the figure 4-1.


Figure 4-1. Data generation scheme for the input variables step test.

4.2.2 DNN structure

The main objective of the DNN for the FCCU is to generate a model that can replicate the

plant in for the RTO structure, therefore, it at least should be able to predict the output

variables for the control of the FCCU in terms to couple it with the control layer. In the figure

4-2 the base structure of the DNN is presented.

Figure 4-2. DNN structure for the FCCU

The figure 4-2 indicates that the number of internal neurons and the output regression order

are the discrete variables that need to be adjusted for the test and validation of the network.

A factorial experimental design was developed. The internal neurons are going to vary from

1 to 25 and the output order from 1 to 6. The ranges were selected considering the limited

computing power for training and testing the DNN.

4.2.3 Training algorithm

In the MATLAB® platform for the NARX dynamic neural network approach are two different

Operational

base case

Steady-State

5% Increase 5% Decrease

Time (s)

Valve opening

fraction

0 s 2000 s 12000 s 22000 s

5% Decrease 5% Increase

32000 s 42000 s

DNN

NARX model

N internal neurons

Nd Output order

Tris

Treg1

Treg2

PSD-PRGN

LSD

xvreg2

xvAir1

xvAir2

xvFG

xvSD


training algorithms, the Levenberg-Marquardt and the Bayesian Regularization. The two

methods were used and compared.

The Levenberg-Marquardt method is used to solve nonlinear least squares problems. This

method adapts the parameter update between the gradient descend method and the

Gauss-Newton. This method can cause overfitting, which is an inconvenient scenario in

which the training solution is stiff in a hyper dimensional point and reduces the learning

capabilities of the DNN. This is particularly notorious when new data is presented to the

DNN, large errors are encountered (Mahapatra and Sood, 2012).

The Bayesian regularization maximizing the evidence to train the parameters and weights

in the DNN structure. In this method, the maximization of the evidence is inversely related

to the model complexity which reduces the probability of overfitting and improves model

generalization (Chan, Ngan and Rad, 2003).

For the train of the DNN, the 70% of the data collected was used for training, the 15% for

validation and the other 15% for testing.

4.3 Results

The DNN using the NARX model in the MATLAB® platform has a limitation regarding the

hidden layers are constrained to one only, therefore the limitation of the number of neurons

per layer can be a limitation for highly non-linear dynamic model.

The final structure of the DNN that fits the final tolerances is presented in the figure 4-2. In

this figure, in the hidden layer are 18 hidden neurons, there are 5 input neurons and 5

output neurons. In closed loop for the straight time series prediction.


Figure 4-3. DNN scheme for the FCCU.

Even if the both training algorithms react the required tolerances with a finite number of

iterations. In closed loop, there is always a reduction in the performance. The MATLAB®

documentation argues that this numerical issue is related to the accumulation of error of

the output variables and the targets and the change of the DNN architecture changes the

expected response. The figures 4-3 and 4-4 indicates the effect on the prediction

performance of the DNN with the closed loop arrangement.

Figure 4-4. DNN prediction performance with two training algorithms for open loop training.

DNN

NARX model

18 internal neurons

2 Output order

Tris

Treg1

Treg2

PSD-PRGN

LSD

xvreg2

xvAir1

xvAir2

xvFG

xvSD

Time (s)

0 5000 10000 15000 20000

Re

ge

ne

rato

r s

ec

on

d s

tag

e t

em

pe

ratu

re (

K)

985

990

995

1000

1005

1010

1015

1020

FCCU model responseBayesian regularizationLevenberg-Marquardt


Figure 4-5. DNN prediction performance with two training algorithms after the closure without re-training.

It is strongly recommended to retrain the DNN after the closure to avoid large prediction

errors, in particular for applications where the precision plays an important role, such as

optimization or inventory control. In the figure 4-3 it can be observed a slightly better

prediction for the DNN trained with the Bayesian regularization algorithm and It is not as

sensitive in the performance detriment after the closure, therefore for this application the

Bayesian regularization is the most appropriate algorithm.

The error histogram for the Bayesian Regularization training algorithm is presented in the

figure 4-5. In this graph can be observed that most of the data have an adequate tendency

to the near cero relative error. In the training stage, some of the data is placed before and

after the zero-error line, but for the variable magnitude it is still in the tolerance considering

the variables magnitude.

Time (s)

0 5000 10000 15000 20000

Reg

en

era

tor

seco

nd

sta

ge t

em

pera

ture

(K

)

985

990

995

1000

1005

1010

1015

1020

FCCU model responseBayesian regularizationLevenberg-Marquardt


Figure 4-6. Error histogram for the DNN with Bayesian Regularization

The performance curve as a function of the epoch is presented in the figure 4-6, there for

an epoch of 5, it is encountered the minimum mean square difference. It is in this point

where the DNN takes the final parameters after the training.


Figure 4-7. Performance plot as a function of the Epochs.

4.4 Conclusions

A DNN was constructed for the FCCU model presented in the chapter 2. A strategy for the

data generation as a time series finally have successful results and a 17-hidden layer

structure was capable of predict the nonlinear behavior of the unit near to the base

operational case. It is important to consider that the DNN is trained to correlate adequately

in near to the operational base case. There is an elevated risk of inaccuracy for

extrapolation. For applications such as model predictive control, it is recommended to use

dynamic training to cope with the extrapolation and increase the area of prediction.

References

Adebiyi, O. A. and Corripio, A. B. (2003) ‘Dynamic neural networks partial least squares (

DNNPLS ) identification of multi v ariable processes’, 27.


Bollas, G. . et al. (2003) ‘Using hybrid neural networks in scaling up an FCC model from a

pilot plant to an industrial unit’, Chemical Engineering and Processing: Process

Intensification, 42(8–9), pp. 697–713. doi: 10.1016/S0255-2701(02)00206-4.

Chan, Z. S. H., Ngan, H. W. and Rad, A. B. (2003) ‘Improving Bayesian Regularization of

ANN via Pre-training with Early-Stopping’, pp. 29–34.

Diaconescu, E. (2008) ‘The use of NARX Neural Networks to predict Chaotic Time Series’,

3(3).

Hussain, M. A. (1999) ‘Review of the applications of neural networks in chemical process

control Ð simulation and online implementation’, 13.

Khalil, R. A. (2011) ‘Comparison of Four Neural Network Learning Methods Based on

Genetic Algorithm for Non-linear Dynamic Systems Identification صخلم ال ةيطخ ’, pp. 8–12.

Lennox, B. et al. (1998) ‘Case study investigating the application of neural networks for

process modelling and condition monitoring’, Computers & Chemical Engineering, 22(11),

pp. 1573–1579. doi: https://doi.org/10.1016/S0098-1354(98)00234-8.

Mahapatra, S. S. and Sood, A. K. (2012) ‘Bayesian regularization-based Levenberg–

Marquardt neural model combined with BFOA for improving surface finish of FDM

processed part’, The International Journal of Advanced Manufacturing Technology, 60(9),

pp. 1223–1235. doi: 10.1007/s00170-011-3675-x.

Nascimento, C. A. O., Giudici, R. and Guardani, R. (2000) ‘Neural network based approach

for optimization of industrial chemical processes’, Computers & Chemical Engineering,

24(9–10), pp. 2303–2314. doi: 10.1016/S0098-1354(00)00587-1.

Rutecki, P. A. (1992) ‘Neuronal excitability: voltage-dependent currents and synaptic

transmission.’, Journal of clinical neurophysiology : official publication of the American

Electroencephalographic Society. United States, 9(2), pp. 195–211.

Sengupta, S. and Khurana, H. (1995) ‘Neural network application for fault diagnosis in

FCCU’, Proceedings of IEEE/IAS International Conference on Industrial Automation and

Control. Ieee, pp. 445–450. doi: 10.1109/IACC.1995.465799.


Vieira, W. G. et al. (2005) ‘Identification and predictive control of a FCC unit using a MIMO

neural model’, Chemical Engineering and Processing: Process Intensification, 44(8), pp.

855–868. doi: 10.1016/j.cep.2004.08.008.

Chapter 5 Stability analysis of the FCCU

In this chapter, the stability analysis for the FCCU is presented. The continuation theory is

used with the differential-algebraic equation (DAE) model presented in the chapter 2. The

FCCU model is turned into an ordinary differential equation (ODE) model by analytical

transformation. This analysis is performed using the manipulated variables as the

continuation parameters to search the multiplicity of steady states of the FCCU.

5.1 Introduction

The different approaches to the mathematical modelling of the FCCU are based on the

interactions between the riser and the regenerator, in addition to the nonlinear cracking and

combustion kinetics, the modeling strategy is directly related to the application objective,

for example the mechanistic models based in mass and energy balances in the unit

subsystems have had important success in control and online optimization (Kasat et al.,

2002).

The control of the FCCU must deal with the complex behavior of the state variables with

respect to the changes in the control variables. Arbel et al. (Arbel et al., 1995) presented a

FCCU type IV model which presented multiplicity of steady states and defined its stability

through eigenvalues analysis, several other authors have made this type of analysis with

simplifications in the base model to perform bifurcation analysis on ODE type of models

(Maya-yescas, Bogle and Lo, 1998).

A system with multiplicities has low controllability, because exist more than one set of

steady state for a given manipulated variables definition and the situation of reaching a

point different from the originally was desired could be attained. (Fernandes et al., 2007).

Some authors argue that all FCCU should present steady state multiplicity, due the

Chapter 5 Stability analysis for the FCCU 101

autothermic characteristic of the process (Arbel et al., 1995), in which the heat released

from the coke layer combustion in the regenerator is used as the reaction heat in the

catalytic cracking reactions in the riser.

The multiplicity in steady states have other point to be discussed, that is the stability of each

steady state, which is sometimes defined graphically by the gradient of the curves of

removed and generated heat with the regenerator temperature (Fernandes et al., 2007),

the definition implies that if the gradient of the heat removed against the regenerator

temperature is greater. then the generated heat gradient, therefore the steady state is

stable.

This definition is based in the work of Levenspiel (Levenspiel, 1999), in which if there are

three steady states in a system, the intermediate is called the ignition point, The upper

steady state is the ignited state and the lower steady state is the extinguished state. Other

alternative to determine the stability of the steady state multiplicity is the numerical

continuation algorithm. The numerical continuation is an algorithm to compute a

consecutive sequence of points which approximate a desired branch.

Consider a smooth function 𝐹: ℝ𝑛+1 → ℝ𝑛. The algorithm will try to compute the solution for

the function 𝐹(𝑥) = 0. The main idea is to generate the sequence of points 𝑥 = {𝑥1, 𝑥2, … , 𝑥𝑛}

that numerically satisfy the tolerance imposed over the function ‖𝐹(𝑥𝑖)‖ ≤ 𝛿. The algorithm

initiates in an equilibrium point and the new point prediction is commonly made using the

tangent prediction or the Moore-Penrose Continuation. In this case the 𝐹 function is

constituted by the FCCU dynamic model and the eigenvalues of the system are computed

each point to verify the stability of the equilibrium curve.

In general, the continuation is widely spread in the ODE systems but it is not the same case

for DAE systems (Clausbruch, Biscaia and Melo, 2006). A possibility is an index reduction

of the DAE system via explicit differentiation of the algebraic equations. Once the system

is presented in an explicit ODE form, a specialized software for continuation analysis can

be used straightforward.


5.2 Methodology

5.2.1 System transformation

Considering the FCCU base model which consists in differential and algebraic equations,

we can structure this DAE model in its general or explicit formulation with the use of the

mass matrix M, as follows:

𝑴(𝒚,𝑡)

𝑑𝒚

𝑑𝑡= 𝑭(𝒚,𝒖,𝑡) (5-1)

In the equation (5-1) the mass matrix is a function of the state variables (y), manipulated

variables (u) and time. To transform the DAE model into an ODE model its necessary to

derivate the algebraic equations and reconstruct the mass matrix. This process is easier to

understand if we consider the semi-explicit form of the DAE model:

𝑑𝒚

𝑑𝑡= 𝒇(𝒚,𝒖,𝑡)

0 = 𝒈(𝒚,𝒖,𝑡)

(5-2)

The mass matrix that interchanges the explicit and semi-explicit DAE forms can be written

as:

𝑴 =

(

𝑘1 0 ⋯ 0 0

0⋮0

⋱ 0 00 𝑘𝑖 00 0 ⋱

0⋮0

0 0 ⋯ 0 𝑘𝑁)

𝑁𝑥𝑁

𝑘𝑖 = 1 𝑖𝑓 𝑖 ≤ 𝑁𝑂𝐷𝐸

𝑘𝑖 = 0 𝑖𝑓 𝑁𝑂𝐷𝐸 < 𝑖 ≤ 𝑁

𝑁 = 𝑁𝐴𝐸 +𝑁𝑂𝐷𝐸

(5-3)

Where 𝑁𝑂𝐷𝐸 is the number of ordinary differential equations (ODE) and 𝑁𝐴𝐸 is the number

of algebraic equations (AE) in the differential-algebraic equation (DAE) model.

If the DAE system is of index 1, there is only necessary only one step of differentiation of

therefore it can be written as:

𝑑𝑔𝑖𝑑𝑡= 𝛼1𝑖(𝒚,𝒖,𝑡)

𝑑𝑦1𝑑𝑡+ 𝛼2𝑖(𝒚,𝒖,𝑡)

𝑑𝑦2𝑑𝑡+⋯+ 𝛼𝑁𝑖(𝒚,𝒖,𝑡)

𝑑𝑦𝑁𝑑𝑡

= 0 (5-4)

With the definition of the equation (5-4), the mass matrix can be modified as follows:


�̂� =

(

1 0 … 0 … 0 0 1 … 0 … 0 ⋮𝛼11⋮𝛼1𝑁

⋮𝛼21⋮𝛼2𝑁

⋱…⋮…

00⋮…

… 0 … 𝛼𝑁1⋱ ⋮ … 0 )

𝑁𝑥𝑁

𝑁 = 𝑁𝐴𝐸 +𝑁𝑂𝐷𝐸

(5-5)

The explicit form of the DAE with the modified mass matrix (�̂�) can be expressed as:

�̂�(𝒚,𝒖,𝑡)

𝑑𝒚

𝑑𝑡= (𝒇(𝒚,𝒖,𝑡)𝟎

)

𝑑𝒚

𝑑𝑡= �̂�(𝒚,𝒖,𝑡)

−1(𝒇(𝒚,𝒖,𝑡)𝟎

)

(5-6)

The equation (5-6) is the explicit ODE formulation after the differentiation of the algebraic

equations of the DAE model.

5.2.2 Continuation software

The continuation analysis of ODE models has been widely performed and studied by

several authors (Elnashaie, Mohamed and Kamal, 2004). The continuation software used

in this chapter is MatCont®. This software is developed over the MATLAB® platform, and

it is used for the numerical continuation study of continuous and discrete parameterized

dynamical systems. To use MatCont® for the purposes of analysis of the FCCU, the system

must be an explicit ODE to make use of the software capabilities.

MatCont® takes the advantage of the MATLAB® ODE solver to perform the continuation

analysis, and therefore, the ODE15s solver is used regarding its capabilities with the stiff

and nonlinear FCCU model.

5.3 Results

The ODE model for the FCCU have some important remarks, considering that the solution

algorithms for the DAE model use the initial point as a guess point in which the solver itself

reduces the value of the algebraic equations to a certain tolerance before starting the

calculations. In this case a well-defined initial steady state must be defined to avoid


The figure 5-1 presents the multiplicity of steady states of the FCCU in two different

operational conditions, the first is the change of the air flow to the first regenerator stage.

The multiplicity occurs at a high catalyst circulation rate, this a similar result with the found

by Hernandez et al. (Hernández-Barajas, Vázquez-Román and Salazar-Sotelo, 2006), in

which the operating conditions affects the multiplicity regions in the steady state.

Considering the figure 5-1, the middle zone in the bifurcation diagram presented in the has

all the real part of the eigenvalues positive, it is an indication an unstable behavior, in

comparison with the upper and lower zones, presenting linear stability. The diagram also

presents two bifurcation points, associated as limit points which differentiate the stability

regions.

There were not found multiplicities with the air flow to the second regeneration stage in the

valve opening range. Considering that the main coke combustion occurs in the first

regenerator stage and therefore the main energetic coupling supposed to occur in that

stage.


Figure 5-1. Steady-state multiplicity of the FCCU for a change in the air to the first regenerator stage

The energetic coupling between the reactor and the regenerator and the autothermic

behavior were analyzed in the figure 5-2 in which for a defined steady state multiplicity

region, the two components of the heat generated and consumed by the regenerator and

reactor were put together. The cross points between these two lines corresponds to steady

state points. The middle point was determined to be unstable by the continuation solution

and considering the graphical relationship presented by Fernandes et al (Fernandes et al.,

2006):

𝑑𝐻𝑟𝑒𝑎𝑐𝑡𝑜𝑟𝑑𝑇𝑟𝑔𝑛

>𝑑𝐻𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑

𝑑𝑇𝑟𝑔𝑛 𝑓𝑜𝑟 𝑎 𝑠𝑡𝑒𝑎𝑑𝑦 𝑠𝑡𝑎𝑡𝑒 𝑡𝑜 𝑏𝑒 𝑠𝑡𝑎𝑏𝑙𝑒

The graphical relationship and the eigenvalue method of stability agree in this case,

therefore the two methods are good to determine the stability of the steady states in the

FCCU.

Air flow first regenerator stage (kg/s)

60 70 80 90 100

Reg

en

era

tor

firs

t sta

ge t

em

pera

ture

(K

)

800

850

900

950

1000

1050

1100

Catalyst recirculation rate 901.6 kg/sCatalyst circulation rate 1050.2 kg/sUnsteady steady state


Figure 5-2. Heat generation and consumed lines for the FCCU

Finally, the stability criteria to determine the stability regions is the presence or not of

multiplicity of steady states. A search in the operation region generates the figure 5-3. In

which three zones were identified to have multiplicity of steady states and should be

avoided in normal operative conditions.

Some authors (Arbel et al., 1995) deals with the possibility for the existence of a region of

5 multiple steady states, in this search only the three multiple steady states were identified.

Regenerator first stage temperature (K)

860 880 900 920 940 960 980 1000

Heat

(kJ

/kg

)

0

500

1000

1500

2000

2500

3000

3500

Heat required at the reactorHeat generated combustion


Figure 5-3. Stability region for the FCCU. The blue dots are the regions where only one single steady state are present. The red dots are the region where exist multiplicity of

steady states.

5.4 Conclusions

A strategy of identification and stability analysis was performed for a FCCU model using

the continuation algorithm. The stability method was put in comparison with a graphical

method with agreement in the stability of the steady state. A stability region was determined

and it also limit the controllability region of the FCCU considering the operative difficulties

of the multiple steady states.

Air flow first regenerator stage (kg/s)

60 65 70 75 80 85 90 95 100

Cata

lyst

cir

cu

lati

on

rate

(kg

/s)

700

750

800

850

900

950

1000

1050

1100

Steady-stateMultiplicity

Steady-stateMultiplicity

Steady-state Multiplicity

Single steady state


References

Arbel, A. et al. (1995) ‘Dynamics and Control of Fluidized Catalytic Crackers. 2. Multiple

Steady States and Instabilities’, Industrial & Engineering Chemistry Research, 34(9), pp.

3014–3026. doi: 10.1021/ie00048a013.

Clausbruch, B. C. Von, Biscaia, E. C. and Melo, P. A. (2006) ‘Stability Analysis of

Differential-Algebraic Equations in AUTO _ DAE’, pp. 297–302.

Elnashaie, S. S. E. H., Mohamed, N. F. and Kamal, M. A. I. (2004) ‘Simulation and Static

Bifurcation Behavior of Industrial FCC Units’, Chemical Engineering Communications.

Taylor & Francis, 191(6), pp. 813–831. doi: 10.1080/00986440490275859.

Fernandes, J. L. et al. (2006) ‘Multiplicity of steady states in an UOP FCC unit with high

efficiency regenerator’, pp. 1575–1580.

Fernandes, J. L. et al. (2007) ‘Steady state multiplicity in an UOP FCC unit with high-

efficiency regenerator’, Chemical Engineering Science, 62(22), pp. 6308–6322. doi:

10.1016/j.ces.2007.07.033.

Hernández-Barajas, J. R., Vázquez-Román, R. and Salazar-Sotelo, D. (2006) ‘Multiplicity

of steady states in FCC units: effect of operating conditions’, Fuel, 85(5–6), pp. 849–859.

doi: http://dx.doi.org/10.1016/j.fuel.2005.08.007.

Kasat, R. B. et al. (2002) ‘Multiobjective Optimization of Industrial FCC Units Using Elitist

Nondominated Sorting Genetic Algorithm’, Industrial & Engineering Chemistry Research,

41(19), pp. 4765–4776. doi: 10.1021/ie020087s.

Levenspiel, O. (1999) Chemical reaction engineering. Wiley.

Maya-yescas, R., Bogle, D. and Lo, F. (1998) ‘Approach to the analysis of the dynamics of

industrial FCC units’, 8(2).

Chapter 6 Control of the FCCU

In this chapter, the evaluation of the proposed control structure in the chapter 3 is

performed. A PID type of controller is placed for each of the 5 SISO arrangement

considered. The key factor to evaluate is the performance in the stable operating zone and

in the steady state multiplicity region.

6.1 Introduction

The control of the FCCU have been studied for several authors due its necessity to operate

near process constraints and the strong interaction between the control loops, as it was

shown in the chapter 3. The nonlinearities presented in the FCCU present also a challenge

in on the implementation even of the basic regulatory system (Zanin, Tvrzská de Gouvêa

and Odloak, 2002).

The open loop response of the FCCU also presents interesting features, like the different

time constants for the different state variables, which is important in the control structure

design and evaluation. The variable pairing presents an important role in the control

efficiency and have been studied by several research works (Arbel, Rinard and Shinnar,

1997; Vieira et al., 2005; Pandimadevi, Indumathi and Selvakumar, 2010; Oliveira,

Cerqueira and Ram, 2012).

In the literature, the most important control variables are the SD catalyst level which gives

stability and plays a key role in the catalyst circulation inside the unit. The riser temperature

is an important variable for the catalytic cracking kinetics. Flue gas oxygen concentration

which is a measure of the regeneration performance. The regenerator temperature

indicates the activity of the regenerated catalyst and the regenerator-reactor pressure

difference for the catalyst circulation rate (Oliveira, Cerqueira and Ram, 2012).


6.2 Methodology

6.2.1 PID Controller

PID feedback control are proposed for the variable pairing presented in the RGA analysis

in the chapter 3. In the table 6-1 are described the type of PID controller for each loop

considering the open loop response presented in the chapter 2.

Table 6-1. PID controller type for the variable pairing.

Input variable Output variable PID controller type

𝑇(𝑧=𝐻𝑟𝑖𝑠)𝑟𝑖𝑠 𝑥𝑣

𝑟𝑒𝑔2 PID

𝑇𝑟𝑒𝑔1 𝑥𝑣𝐴𝑖𝑟1 PID

𝑇𝑟𝑒𝑔2 𝑥𝑣𝐴𝑖𝑟2 PID


− 𝑃𝑅𝐺𝑁𝑔𝑎𝑠

𝑥𝑣𝐹𝐺 PI

𝐿𝑆𝐷 𝑥𝑣𝑆𝐷 PI

6.2.2 PID tuning

The PID tuning and performance of blocked response is done with the SIMULINK PID

Tuning toolbox®. In which the Plant linearization and tuning parameters are coupled with

the following compensator formula:

𝐶 = 𝑃 + 𝐼𝑇𝑠

1

𝑧 − 1+ 𝐷

𝑁

1 + 𝑁 𝑇𝑠1

𝑧 − 1

(6-1)

The tuning algorithm in the Matlab® platform has three main objectives. The first one is the

closed-loop stability, in which the closed loop system output remains bounded for a

bounded input. The second objective is the adequate performance of the controller, the

performance is enhanced by suppressing the disturbances. It can be achieved by taking

the loop bandwidth based on the plant model to get a fast response to changes or

disturbances. Finally, the las objective is the adequate robustness. In which the software

calculate the gain and phase margin to cope with the system dynamics, it is achieve by

minimizing the phase margin.

Chapter 6 Control of the FCCU 111

6.3 Results

The tuning of the PID was performed with the SIMULINK PID tuning toolbox® as it is

presented in the figure 6-1. In which the controller parameters appear in the right inferior

part of the interface. This tool allows to change the control response time, between a slower

scale and a faster scale in seconds, this tool was used to cope with the different time

responses of the FCCU. It also allows to change the transient behavior from robust to

aggressive, which affects the identification coefficient of the tuning algorithm

The control performance for a set point change is presented in the figures 6-2 and 6-3. The

control structure works well in the stable zone, as it is presented in the figure 6-2. The

controlled variables reach the new set point with a stable response. In comparison, the

steady state multiplicity affects the control structure performance and it cannot drive the

system through the set point transition.

Figure 6-1. Tuning graphical user interface for the SIMULINK PID tuning toolbox.


Figure 6-2. Set point change for the regenerator first stage temperature in the stable operative region.

Figure 6-3. Step change in the riser outlet temperature in the steady state multiplicity region.

Time(s)

0 1000 2000 3000 4000

Re

ge

ne

rato

r te

mp

era

ture

0,18

0,20

0,22

0,24

0,26

0,28

0,30

0,32

O2

ma

ss

fra

cti

on

in

flu

e g

as

0,300

0,305

0,310

0,315

0,320

0,325

0,330

0,335

0,340

Set point

Regenerator temperature

O2 mass fraction flue gas

Time (s)

0 500 1000 1500 2000

Re

ge

ne

rato

r te

mp

era

ture

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

Ris

er

ou

tle

t te

mp

era

ture

0,06

0,08

0,10

0,12

0,14

0,16

Regenerator temperature

Regenerator temperature set point

Riser outlet temperature


6.4 Conclusions

The control structure proposed have a good qualitative performance in the stable operability

region. Instead it cannot control the system in the steady state multiplicity region. It is

important to consider the change in the process gains within the two defined operation

regimes: the pre-combustion and post-combustion operation types. In this case all the

control structure has to change in order to cope the new relations which in part can lead to

different pairing schemes and control performance.

References

Arbel, A., Rinard, I. H. and Shinnar, R. (1997) ‘Dynamics and Control of Fluidized Catalytic

Crackers . 4 . The Impact of Design on Partial Control’, (1996), pp. 747–759.

Oliveira, N. M. C., Cerqueira, H. S. and Ram, F. (2012) ‘Fluid Catalytic Cracking ( FCC )

Process Modeling , Simulation, and Control’, Industrial & Engineering Chemistry Research,

51(1), pp. 1–29. doi: 10.1021/ie200743c.

Pandimadevi, G., Indumathi, P. and Selvakumar, V. (2010) ‘Design of Controllers for a

Fluidized Catalytic Cracking Process’, Chemical Engineering Research and Design.

Institution of Chemical Engineers, 88(7), pp. 875–880. doi: 10.1016/j.cherd.2009.11.021.

Vieira, W. G. et al. (2005) ‘Identification and predictive control of a FCC unit using a MIMO

neural model’, Chemical Engineering and Processing: Process Intensification, 44(8), pp.

855–868. doi: 10.1016/j.cep.2004.08.008.

Zanin, A. C., Tvrzská de Gouvêa, M. and Odloak, D. (2002) ‘Integrating real-time

optimization into the model predictive controller of the FCC system’, Control Engineering

Practice, 10(8), pp. 819–831. doi: http://dx.doi.org/10.1016/S0967-0661(02)00033-3.

Chapter 7 Conclusions and recommendations

A detailed model of a Kellogg Orthoflow F type of FCCU unit was developed. The feed

characterization enhances the model usage by the possibility of use distinct types of cuts

into It and the effect on the unit performance. It is quite convenient considering the

variations on the feed quality and availability.

The use of a PONA characterization on each lump in the catalytic cracking mechanism, can

be used further to define quality correlations and perform different optimization strategies.

The RTO structure usually uses a profit objective function, but due to the quality and

environmental requirements of the FCCU products a multiobjective optimization strategy

can be implemented and the model can be subject to further study for the best simultaneous

environmental and economic performance (Sankararao & Gupta, 2007).

The detailed model of the bubble phase behavior of the regenerator is able to predict the

coke on catalyst concentration after the regeneration process accurately. This is a further

step in this kind of units where this variable and the ones related are usually far from the

results encountered in the real plant.

The parameter estimation strategy improved the model accuracy, although, it could improve

extensively with the use of fast data reconciliation and parameter estimation strategies in a

framework that evaluates the most sensitive parameters subject to optimization. these

modules are placed in the RTO layer to achieve a better match between the steady-state

model and the real plant.

The steady-state results show relations similar of those found in the literature, among others

related to the second regeneration bed of the FCCU analyzed. The partial and total


operation can be represented using this model. This flexibility can be used to evaluate

different operative scenarios and to define new operational procedures.

The catalyst inventory control can be achieved with the dynamic model. The importance of

the catalyst inventory lies in the necessity of the addition of catalyst make up to the FCCU

and the quality control of the catalyst activity and product yield.

The sensitivity analysis of the primary output variables of the FCCU indicates that all are

sensitive to several input variables, this is remarkable considering its effects on SISO

control loops strategy. Some of the

In the open literature bifurcation studies for FCCU models have been focused on

simplifications and construction of ODE models based in spatial average variables. The

DAE transformation to ODE is performed as a methodology for the possibility of the study

of the bifurcation analysis in this type of mathematical models. The methodology and the

structure of the explicit formulation is one of the fundamental contributions to the

understanding of the multiplicity of the FCCU of this work. The fact that the empirical

relationships and the bifurcation analysis reached to the same conclusions indicates that

the methodology is predicting adequately the behavior of the unit and it can be used in other

types of FCCU and refinery plants. The stability regions identified are useful to restrict the

RTO and the control layer to avoid regions where the unit has an unstable behavior.

The DNN developed for the unit was able to predict the behavior of the unit for the structure

identified by the sensitivity analysis. This model has a lower computational requirement to

calculate the plant dynamics in comparison to the phenomenological model. It can be used

as a virtual plant in the RTO layer among with the noise addition module.

This work has accomplished the following academic contributions:

• Oral presentation on the joint PSE2015/ESCAPE25 – an event uniting the two

conferences: Process Systems Engineering (PSE) and European Symposium on

Computer Aided Process Engineering (ESCAPE) at the Bella Center in

Copenhagen, Denmark, during 31 May to 4 June 2015 with the presentation


entitled: “Data analysis and modelling of a Fluid Catalytic Cracking Unit (FCCU) for

an implementation of Real Time Optimization”

• Juan D. Reyes, Adriana L. Rodríguez, Carlos A.M. Riascos, Data Analysis and

Modelling of a Fluid Catalytic Cracking Unit (FCCU) for an Implementation of Real

Time Optimization, Editor(s): Krist V. Gernaey, Jakob K. Huusom, Rafiqul Gani,

Computer Aided Chemical Engineering, Elsevier, Volume 37, 2015, Pages 611-616.

In this paper the dynamic model, DNN and gross error detection evaluation has

been based in the results from this thesis.

• Oral presentation on the XXI Brazilian Congress of Chemical Engineering COBEQ

2016 held in Fortaleza/Ceara, during 25 to 29 of September of 2016, with the

presentation entitled: “Análise de estabilidade de uma unidade de craquamento

catalitico em leito fluidizado”.

• J. D. Reyes, G.A. Carrillo Le Roux, C. A. M. Riascos, Steady state multiplicity and

stability analisys of a fluidized catalytic cracking unit (FCCU), congress article of the

XXI Brazilian Congress of Chemical Engineering COBEQ 2016. In this work

bifurcation analysis of the FCCU are based on the results from this work.

A. Detailed Dynamic model description

In this appendix, the dynamic model development for the Orthoflow F Fluid Catalytic

cracking unit (FCCU) is presented. In the figure 1, a schematic representation of the FCCU

is presented.


Figure 7-1. Schematic representation of the the Orthoflow F FCCU.

Model development

Each section of the FCCU presented in the figure 1 are included into a sub-system

structure, except for the air blower, the furnace and the fractionator. With this consideration,

Gas oil

feed

Flue gas

Reactor

Stripping

steam

Regenerator

Riser

Spent

catalyst

Regenerated

catalyst

FCC

products to

fractionator

Air

1st

Regeneration

bed

2nd

Regeneration

bed

Appendix A Detailed model description 119

the FCCU is decomposed into five sub-systems:

• Regenerated catalyst and gas oil feed mixing point

• Riser.

• Stripping-disengaging

• Regenerator dense phase

• Regenerator Freeboard

Feed characterization

The feed stream characterization uses the n-d-M correlation method (Sadeghbeigi, 2012)

used to predict the weight fraction of paraffinic (P), naphthenic (N) and aromatic (A)

compounds in a highly complex hydrocarbon mixture, as the gas oil. To use this method, it

is necessary to have the following information:

• Mass density of the hydrocarbon mixture at 20°C, g/mL.

• Molecular weight. Use ASTM D2502 for estimation.

• Refractive index at 20°C. If it is not available, the TOTAL correlation method could

be used for an accurate estimation.

The first step for the n-d-M correlation method is to calculate the correlation factors 𝜈 and

𝜔.

𝜈 = 2.51(𝑅𝐼(20°𝐶) − 1.4750) − 𝐷20 + 0.8510

𝜔 = 𝐷20 − 0.8510 − 1.11(𝑅𝐼(20°𝐶) − 1.4750)

There are 4 cases for the calculation of the aromatic weight fraction (%𝐶𝐴) and the weight

fraction of the total ring-type compounds (%𝐶𝑅), depending on the sign of the n-d-M factors:

• If 𝜈 is positive: %𝐶𝐴 = 430𝜈 + 3660/𝑀𝑊.

• If 𝜈 is negative: %𝐶𝐴 = 670𝜈 + 3600/𝑀𝑊.

• If 𝜔 is positive: %𝐶𝑅 = 820𝜔 − 3𝑆 + 10000/𝑀𝑊.

• If 𝜔 is negative: %𝐶𝑅 = 1440𝜔 − 3𝑆 + 10600/𝑀𝑊.


Finally, it is possible to calculate the weight percentage of naphthenic (%𝐶𝑁) and paraffinic

(%𝐶𝑝) compounds in the mixture as follows:

%𝐶𝑁 = %𝐶𝑅 −%𝐶𝐴

%𝐶𝑃 = 100 −%𝐶𝑅

Where:

• %𝐶𝑁, Naphthenic rings weight percentage.

• %𝐶𝑅, Total ring type compounds weight percentage.

• %𝐶𝐴, Aromatic compounds weight percentage.

• %𝐶𝑃, Paraffinic compounds weight percentage.

• 𝑅𝐼(20°𝐶), refractive index at 20°C.

• 𝐷(20°𝐶), mass density at 20°C, g/mL.

• 𝑆, sulfur mass percentage.

Regenerated catalyst and gas oil feed mixing point model

Figure 7-2. Regenerated catalyst and gas oil feed mixing point subsystem.

This subsystem is composed of the regenerated catalyst-feed mixing point and the gas and

solid phases that enter the riser subsystem. The regenerated catalyst-feed mixing point is

modeled as a steady state heat transfer mixing point, considering that the average contact

Regenerated

catalyst-feed

mixing point

Regenerated

catalyst-feed

mixing point

Gas oil

feed

Regenerated

Catalyst

Gas

Catalyst

Regenerated

catalyst slide

valve

Regenerated

catalyst slide

valve


time for full vaporization takes about 0.1 second.

The energy balance for the mixing point is:

𝐹𝑐𝑎𝑡𝑀𝑃,𝑖𝑛ℎ𝑐𝑎𝑡

𝑀𝑃,𝑖𝑛 + 𝐹𝑓𝑒𝑒𝑑𝑀𝑃,𝑖𝑛ℎ𝑓𝑒𝑒𝑑

𝑀𝑃,𝑖𝑛 − 𝐹𝑐𝑎𝑡𝑀𝑃,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑀𝑃,𝑜𝑢𝑡 − 𝐹𝑔𝑎𝑠𝑀𝑃,𝑜𝑢𝑡ℎ𝑔𝑎𝑠

𝑀𝑃,𝑜𝑢𝑡 = 0

Where:

• 𝐹𝑖𝑗, mass flow of the component 𝑖 at the position 𝑗, kg/s.

• ℎ𝑖𝑗, mass specific enthalpy of the component 𝑖 at the position 𝑗, J/kg.

The following assumptions are made over the energy balance of the regenerated catalyst-

feed mixing point, in order to solve for the outlet temperature:

1. The outlet temperature for the gaseous stream and the catalyst are equal

(𝑇𝑐𝑎𝑡𝑀𝑃,𝑜𝑢𝑡 = 𝑇𝑔𝑎𝑠

𝑀𝑃,𝑜𝑢𝑡 = 𝑇𝑀𝑃,𝑜𝑢𝑡).

2. The reference temperature for the energy balance is the vaporization temperature

of the gas oil feed mixture (𝑇𝑣𝑎𝑝 = 𝑇𝑟𝑒𝑓).

3. The specific heat for the gas and the catalyst are constant with the temperature.

4. Adiabatic system.

5. Given that the residence time for the mixing point is about 0.1 seconds, and the

average residence time for the riser is about 2-3 seconds. No cracking reactions

and coke deposition over the catalyst surface are expected, therefore, the mass

flow of the catalyst and feed are unchanged during the operation (𝐹𝑐𝑎𝑡𝑀𝑃,𝑖𝑛 =

𝐹𝑐𝑎𝑡𝑀𝑃,𝑜𝑢𝑡 = 𝐹𝑐𝑎𝑡

𝑟𝑔𝑛,2 𝑎𝑛𝑑 𝐹𝑓𝑒𝑒𝑑

𝑀𝑃,𝑖𝑛 = 𝐹𝑔𝑎𝑠𝑀𝑃,𝑜𝑢𝑡 = 𝐹𝑓𝑒𝑒𝑑).

6. There is no pressure drop in the mixing point.

𝑇𝑀𝑃,𝑜𝑢𝑡 =𝐹𝑐𝑎𝑡𝑟𝑔𝑛2,𝑜𝑢𝑡�̅�𝑝

𝑐𝑎𝑡𝑇𝑐𝑎𝑡𝑟𝑔𝑛2,𝑜𝑢𝑡 + 𝐹𝑓𝑒𝑒𝑑�̅�𝑝

𝑙𝑖𝑞𝑇𝑓𝑒𝑒𝑑𝑖𝑛 − 𝐹𝑓𝑒𝑒𝑑Δ𝐻𝑣𝑎𝑝

𝐹𝑐𝑎𝑡𝑟𝑔𝑛2,𝑜𝑢𝑡�̅�𝑝

𝑐𝑎𝑡+ 𝐹𝑓𝑒𝑒𝑑�̅�𝑝

𝑔𝑎𝑠 (1)

Where:

• 𝑇𝑀𝑃,𝑜𝑢𝑡, outlet temperature of the mixing point, K.

• 𝐶�̅�𝑖 , mass specific heat of the component 𝑖, J/kg-K.

• 𝑇𝑖𝑗, temperature of the component 𝑖 at the position 𝑗, K.


• Δ𝐻𝑣𝑎𝑝, vaporization heat of the hydrocarbon feed mixture at the operation pressure,

J/kg.

The pressure in the mixing point is calculated based on the pressure drop of the gas phase

in the riser and the pressure exerted in the stripping-disengaging section.

𝑃𝑀𝑃 = 𝑃𝑆𝐷 + Δ𝑃𝑟𝑖𝑠 (2)

Where:

• 𝑃𝑀𝑃, pressure in the catalyst-feed mixing point, Pa.

• 𝑃𝑆𝐷, pressure in the stripping-disengaging section, Pa.

• 𝑃𝑟𝑖𝑠(𝑖), pressure in the 𝑖 position of the riser section, Pa.

Riser model

Figure 7-3. Riser subsystem.

The riser is modelled as a one-dimension tubular reactor in steady-state. The mass balance

is performed by component using the lump kinetic scheme presented by (Araujo-Monroy &

López-Isunza, 2006). This lumping methodology is based on the paraffinic, olefinic,

naphthenic and aromatic contents (PONA) of the feed gas oil and the cracking products. A

schematic representation of the major lump kinetic model for the large species is presented

in the figure 3.

Riser gas

phase

Riser gas

phase

Riser solid

phase

Riser solid

phase

Gas oil

vaporized

Catalyst

Phase

interaction

Cracked

products

Deactivated

catalyst


Figure 7-4. Schematic representation of the 6 lump cracking kinetic scheme, the lumps considered are: Gas oil (GO), light cycle oil (LCO), gasoline (G), light gases (LG),

liquefied petroleum gas (LPG) and coke (C), adapted from (Araujo-Monroy & López-Isunza, 2006).

The reaction mechanism proposed is based in the following steps:

1. Reversible adsorption of the PONA component of each lump over the catalyst

surface.

2. Formation of an adsorption reaction intermediate 𝜃𝑖 for each PONA component.

3. Superficial cracking reaction and irreversible desorption.

Every major lump (GO, LCO, G, LPG, LG and C) has a PONA mass composition. The

cracking reaction network generated for the six loops with four components is presented in

the figure 4, all the reactions are considered of first order.

GO

C

LCO G

LPG

LG


Figure 7-5. Catalytic cracking reaction mechanism, a) Reactions of the PONA components of the GO major lump (P1, O1, N1), b) Reactions of the PONA

components of the LCO major lump (P2, O2, N2, A2), c) Reactions of the PONA components of the G major lump (P3, O3, N3, A3) and d) Reactions of the PONA

components of the LPG major lump (P4, O4) adapted from (Araujo-Monroy & López-Isunza, 2006).

The mass balance equation system for the gaseous species inside the riser, according to

the figure 4, for the 𝑖 PONA component of the lump 𝑗 is:

𝜖𝑔𝑟𝑖𝑠𝑢𝑔

𝑟𝑖𝑠𝜌𝑔𝑟𝑖𝑠𝑑𝑋𝑖,𝑗

𝑑𝑧= 𝜌𝑏𝑐

𝑟𝑖𝑠𝑅𝑖,𝑗𝑔Ψ ∀ 𝑖 ∈ {𝑃𝑗, 𝑂𝑗, 𝑁𝑗 , 𝐴𝑗} ∧ 𝑗

∈ {𝐺𝑂, 𝐿𝐶𝑂, 𝐺, 𝐿𝑃𝐺, 𝐿𝐺}

(3)

P1 θP1

K-1

K1

K2

K3

K4

K5

K6

P2

O2

N2

A2

C

N1 θN1

K-7

K7

K8

K9

K10

O2

A2

C

A1 θA1

K-11

K11K12

K13

A2

C

a)

P2 θP2

K-14

K14

K15

K16

K17

K18

K19

P3

O3

N3

A3

C

O2 θO2

K-20

K20

K21

K22

K23

P3

A3

C

b)

N2 θN2

K-24

K24

K25

K26

K27

P3

A3

LG

A2 θA2

K-28

K28

K29

K30

K31

A3

LG

C

P3 θP3

K-32

K32K33

K34

P4

c)

O4

O3 θO3

K-35

K35K36

K37

P4

C

N3 θN3

K-38

K38K39

K40

O4

LG

A3 θA3

K-41

K41K42

K43

LG

C

P4 θP4

K-44

K44 K45 LG

d)

O4 θO4

K-46

K46 K47 LG


Where:

• 𝜖𝑔𝑟𝑖𝑠, riser gas fraction, m3

gas/m3riser.

• 𝑢𝑔𝑟𝑖𝑠, riser gas velocity, m/s.

• 𝜌𝑔𝑟𝑖𝑠, riser gas mass density, kggas/m3

gas.

• 𝑋𝑖,𝑗, weight fraction of every gaseous 𝑖 PONA component in the major lump 𝑗,

kgi,j/kggas.

• 𝑧, riser axial direction, m.

• 𝜌𝑏𝑐𝑟𝑖𝑠, bulk catalyst density inside the riser, kgcatalyst/m3

riser.

• 𝑅𝑖,𝑗𝑔

, reaction rate in the gas phase of every 𝑖 PONA component in the major lump 𝑗,

kgi,j/kgcat.s.

• Ψ, catalyst deactivation function.

As an example, the reaction rate for the paraffinic composition of the gas oil lump (GO) 𝑃1

accordingly to the reaction mechanism presented in the figure 5 is:

𝑅𝑃1,𝐺𝑂𝑔

= −𝐾1𝑋𝑃1,𝐺𝑂 + 𝐾−1𝜃𝑃1,𝐺𝑂

The reaction rate for the light gas lump (LG) is:

𝑅𝐿𝐺𝑔= 𝐾27𝜃𝑁2,𝐿𝐶𝑂 + 𝐾30𝜃𝐴2,𝐿𝐶𝑂 + 𝐾40𝜃𝑁3,𝐺 + 𝐾42𝜃𝐴3,𝐺 + 𝐾45𝜃𝑃4,𝐿𝑃𝐺 +𝐾47𝜃𝑂4,𝐿𝑃𝐺

The mass balance for the catalytic cracking reaction intermediate inside the riser, according

to the figure 4, for the 𝑖 PONA component of the lump 𝑗 is:

(1 − 𝜖𝑔

𝑟𝑖𝑠)𝑢𝑐𝑟𝑖𝑠𝜌𝑐

𝑑𝜃𝑖,𝑗

𝑑𝑧= 𝜌𝑏𝑐

𝑟𝑖𝑠𝑅𝑖,𝑗𝑠 Ψ ∀ 𝑖 ∈ {𝜃𝑃𝑗 , 𝜃𝑂𝑗 , 𝜃𝑁𝑗 , 𝜃𝐴𝑗} ∧ 𝑗

∈ {𝐺𝑂, 𝐿𝐶𝑂, 𝐺, 𝐿𝑃𝐺, 𝐿𝐺, 𝐶}

(4)

Where:

• 𝜃𝑖,𝑗, weight fraction of every catalytic cracking reaction intermediate 𝑖 PONA

component in the major lump 𝑗, kgi,j/kgcat.

• 𝑅𝑖,𝑗𝑠 , reaction rate of every catalytic cracking reaction intermediate 𝑖 PONA

component in the major lump 𝑗, kgi,j/kgcat.s.

• 𝜌𝑐, catalyst mass density, kgcat/m3catalyst.

• 𝑢𝑐𝑟𝑖𝑠, riser catalyst velocity, m/s.

The catalyst deactivation function corresponds to the following exponential expression:

Ψ = exp(−𝛼 ∙ 𝐶) (5)


Where:

• 𝛼, deactivation parameter, kgcat/kgcoke.

• 𝐶, coke mass fraction kgcoke/kgcat.

The reaction rate expressions have an Arrhenius type of equation:

𝐾𝑖 = 𝐴0𝑖 exp(−

𝐸𝑎𝑖𝑅𝑇𝑔

) (6)

𝐴0𝑖 = 𝛼0𝑖 [(

𝐴

𝑁)−0.42

] (7)

Where:

• 𝛼0𝑖, frequency factor parameter for every component in the riser, 1/s.

• 𝐴

𝑁, aromatic-naphthenic ratio of the feed gas oil.

• 𝐸𝑎𝑖, activation energy of the 𝑖 component, J/mol.

• 𝑇𝑔, riser gas phase temperature, K.

• 𝑅, universal gas constant, J/mol-K.

The energy balance is developed by considering an adiabatic operation:

𝑑𝑇𝑟𝑖𝑠

𝑑𝑧=

𝜌𝑏𝑐𝑟𝑖𝑠

𝜌𝑔𝑟𝑖𝑠𝑢𝑔𝐶�̅�

𝑔𝑎𝑠∑∑Δ𝐻𝑖𝑅𝑖,𝑗𝑔

𝑁𝑗

𝑗=1

4

𝑖=1

(8)

Where:

• 𝑇𝑟𝑖𝑠, riser temperature, K.

• Δ𝐻𝑖, Reaction heat for each major lump, J/kg.

The gas and catalyst velocity in the riser is assumed to be constant. Using the continuity

equation for each phase:

𝑢𝑔𝑟𝑖𝑠 =

𝐹𝑓𝑒𝑒𝑑

𝜌𝑔𝑟𝑖𝑠𝜖𝑔

𝑟𝑖𝑠𝐴𝑟𝑖𝑠 (9)

𝑢𝑐𝑟𝑖𝑠 =

𝐹𝑐𝑎𝑡𝑟𝑔𝑛,2

𝜌𝑐(1 − 𝜖𝑔𝑟𝑖𝑠)𝐴𝑟𝑖𝑠

(10)

𝜖𝑔𝑟𝑖𝑠 =

𝐹𝑓𝑒𝑒𝑑𝜌𝑔

𝐹𝑓𝑒𝑒𝑑𝜌𝑔

+𝐹𝑐𝑎𝑡𝑟𝑔𝑛,2

𝜌𝑐

𝜖𝑐𝑟𝑖𝑠 = 1 − 𝜖𝑔

𝑟𝑖𝑠


Where:

• 𝐴𝑟𝑖𝑠, riser cross section, m2.

The riser bulk catalyst mass density is calculated as follows:

𝜌𝑏𝑐𝑟𝑖𝑠 = 𝜖𝑔𝜌𝑔

𝑟𝑖𝑠 + 𝜖𝑐𝜌𝑐 (11)

The riser pressure profile is developed by considering the approximation that the pressure

drop is caused by hydrostatic head of solids, and the acceleration effects are only important

at the riser base (Fernandes, Verstraete, Pinheiro, Oliveira, & Ramôa Ribeiro, 2007).

𝑑𝑃𝑟𝑖𝑠

𝑑𝑧= −𝜌𝑏𝑐

𝑟𝑖𝑠𝑔 (12)

Where:

• 𝑔, gravity constant, m/s2.

The boundary conditions for the riser equations at the inlet (𝑧 = 0) are:

{

𝑋𝑖,𝐺𝑂 =

%𝐶𝑖100

∀ 𝑖 ∈ {𝑃, 𝑁, 𝐴}

𝑋𝑖,𝑗 = 0 ∀ 𝑖 ∈ {𝑃𝑗, 𝑂𝑗, 𝑁𝑗 , 𝐴𝑗} ∧ 𝑗 ∈ {𝐿𝐶𝑂, 𝐺, 𝐿𝑃𝐺, 𝐿𝐺}

𝜃𝑖,𝑗 = 0 ∀ 𝑖 ∈ {𝜃𝑃𝑗 , 𝜃𝑂𝑗 , 𝜃𝑁𝑗 , 𝜃𝐴𝑗} ∧ 𝑗 ∈ {𝐺𝑂, 𝐿𝐶𝑂, 𝐺, 𝐿𝑃𝐺, 𝐿𝐺}

𝐶 = 𝐶𝑟𝑔𝑛2

𝑇𝑟𝑖𝑠 = 𝑇𝑀𝑃

𝑃𝑟𝑖𝑠 = 𝑃𝑀𝑃

Stripping-disengaging model


Figure 7-6. Stripping-disengaging (SD) subsystem

In the stripping-disengaging (SD) section, the catalyst level is an important variable to

control; hence, a catalyst inventory mass balance is done considering a continuous stirred

tank model in unsteady state.

𝑑𝑊𝑐𝑎𝑡𝑆𝐷

𝑑𝑡= 𝐹𝑐𝑎𝑡

𝑟𝑖𝑠,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡 (13)

𝑊𝑐𝑎𝑡𝑆𝐷 = 𝜌𝑐𝜖𝑐𝐴𝑆𝐷𝐿𝑆𝐷

𝐿𝑆𝐷 =

𝑊𝑐𝑎𝑡𝑆𝐷

𝜌𝑐𝜖𝑐𝐴𝑆𝐷

Where:

• 𝐹𝑐𝑎𝑡𝑗,𝑜𝑢𝑡

, catalyst mass flow coming out from the section 𝑗, kg/s.

• 𝐿𝑆𝐷, stripping-disengaging section catalyst level, m.

• 𝐴𝑆𝐷, stripping-disengaging cross section, m2.

• 𝑊𝑐𝑎𝑡𝑆𝐷, catalyst inventory in the stripping-disengaging section, kg.

The catalyst flow from the SD section (𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡) is determined by a slide valve, as it is

presented in the figure 6, and the flow is driven by the pressure difference between the

bottom of the reactor and the bottom of the first regeneration bed:

Reactor/

Stripper

Reactor/

Stripper

Stripping

steam

FCC

vapour

products

Spent

catalyst

Spent

catalyst

slide valve

Spent

catalyst

slide valve

Cracked

products

Deactivated

catalyst


𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡 = 𝑘𝑣

𝑆𝐷

√

Δ𝑃𝑆𝐷𝑣

1𝜌𝑐[(1𝐴𝑥𝑣)2

− (1𝐴0)2

]

(14)

Δ𝑃𝑆𝐷𝑣 = 𝑃𝑆𝐷𝑔𝑎𝑠+ 𝜌𝑐𝜖𝑐𝑔𝐿𝑆𝐷 − 𝑃𝑅𝐺𝑁

𝑔𝑎𝑠− 𝜌𝑐𝜖𝑐,𝑟𝑔𝑛1𝑔𝐿𝑟𝑔𝑛1

𝐴𝑥𝑣 = 𝑎1 + 𝑏1𝑥𝑣2

Where:

• 𝑘𝑣𝑆𝐷, stripper disengaging slide valve rating factor, kg/(s.Pa)0.5.

• 𝑥𝑣, valve opening fraction.

• 𝑃𝑆𝐷𝑔𝑎𝑠

, SD gas phase pressure, Pa.

• 𝑃𝑅𝐺𝑁𝑔𝑎𝑠

, regenerator gas pressure, Pa.

• 𝐿𝑟𝑔𝑛1, catalyst level of the dense phase in the first regenerator stage.

• 𝐴0, full open valve cross section passage, m2.

• 𝐴𝑥𝑣, valve cross section passage at a specific valve opening fraction, m2.

• 𝑎1, 𝑏1, valve passage cross section and valve opening fraction correlation factors,

m2.

The coke concentration in the stripper is calculated using the coke on catalyst concentration

coming out from the riser and the following empirical correlation proposed by (Fernandes

et al., 2007) which calculates the amount of feed that contributes to cat-to-oil coke at the

stripper temperature:

𝛾 = exp(5.2113 − 0.0144𝑇𝑆𝐷)

The coke mass balance is:

𝑑𝐶𝑆𝐷


𝑟𝑖𝑠,𝑜𝑢𝑡(𝐶𝑟𝑖𝑠,𝑜𝑢𝑡 + 𝛾) − 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡𝐶𝑆𝐷 (16)

The gas phase pressure in the SD section is calculated considering the ideal gas

assumption, therefore:


= 𝑊𝑆𝐷𝑔 𝑅𝑇𝑆𝐷

𝑔

𝑀𝑊𝑆𝐷 𝑉𝑆𝐷𝑔 (17)

𝑉𝑆𝐷𝑔= 𝑉𝑆𝐷 − 𝜖𝑐𝐴𝑆𝐷𝐿𝑆𝐷

Where:

• 𝑊𝑆𝐷𝑔

, SD gas mass, kg.

• 𝑉𝑆𝐷𝑔

, gas phase SD section volume, m3.


• 𝑇𝑆𝐷𝑔

, gas phase SD section temperature, K.

• 𝑉𝑆𝐷, SD section volume, m3.

• 𝑀𝑊𝑆𝐷, average molecular weight in the SD section, kg/kmol.

The gas phase inventory in the SD section is calculated as:

𝑑𝑊𝑆𝐷𝑔

𝑑𝑡= 𝐹𝑔

𝑟𝑖𝑠,𝑜𝑢𝑡 + 𝐹𝑠𝑡𝑒𝑎𝑚 − 𝐹𝑔𝑆𝐷,𝑜𝑢𝑡 (18)

Where:

• 𝐹𝑔𝑗,𝑜𝑢𝑡

, gas mass flow coming out from the section 𝑗, kg/s.

• 𝐹𝑠𝑡𝑒𝑎𝑚, stripper disengaging steam mass flow, kg/s.

The gaseous products mass flow from the SD section (𝐹𝑔𝑆𝐷,𝑜𝑢𝑡) is determined by the

opening the control valve between the SD section and the main fractionator. It is calculated

using the following equation:

𝐹𝑔𝑆𝐷,𝑜𝑢𝑡 = 𝑘𝑣

𝑀𝐹𝑥𝑣√𝑃𝑆𝐷𝑔𝑎𝑠− 𝑃𝑀𝐹 (19)

The energy balance in the SD section is performed by considering the following

assumptions:

1. Exist thermal equilibrium between the catalyst and gas phases, it implies that the

SD outlet streams are also in thermal equilibrium.

2. The heat of desorption is neglected.

3. The specific heat for the gas and the catalyst are constant within the temperature

range considered.

4. Adiabatic operation.

5. The reference temperature for the enthalpy calculation corresponds to the SD

temperature (𝑇𝑆𝐷).

𝑑𝐻𝑆𝐷𝑑𝑡

= 𝐹𝑠𝑡𝑒𝑎𝑚ℎ𝑠𝑡𝑒𝑎𝑚 + 𝐹𝑔𝑟𝑖𝑠,𝑜𝑢𝑡ℎ𝑔

𝑟𝑖𝑠,𝑜𝑢𝑡 + 𝐹𝑐𝑎𝑡𝑟𝑖𝑠,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑟𝑖𝑠,𝑜𝑢𝑡 − 𝐹𝑔𝑆𝐷,𝑜𝑢𝑡ℎ𝑔

𝑆𝐷,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑆𝐷,𝑜𝑢𝑡

𝑑

𝑑𝑡(𝑊𝑆𝐷

𝑐𝑎𝑡𝐶�̅�𝑐𝑎𝑡𝑇𝑆𝐷 +𝑊𝑆𝐷

𝑔𝐶�̅�𝑔𝑇𝑆𝐷)

= 𝐹𝑠𝑡𝑒𝑎𝑚𝐶�̅�𝑠𝑡𝑒𝑎𝑚(𝑇𝑠𝑡𝑒𝑎𝑚 − 𝑇𝑆𝐷) + 𝐹𝑔

𝑟𝑖𝑠,𝑜𝑢𝑡𝐶�̅�𝑔(𝑇𝑟𝑖𝑠(ℎ𝑟𝑖𝑠) − 𝑇𝑆𝐷)

+ 𝐹𝑐𝑎𝑡𝑟𝑖𝑠,𝑜𝑢𝑡𝐶�̅�

𝑐𝑎𝑡(𝑇𝑟𝑖𝑠(ℎ𝑟𝑖𝑠) − 𝑇𝑆𝐷)


𝑑𝑇𝑆𝐷𝑑𝑡

=1

𝐶�̅�𝑐𝑎𝑡𝑊𝑆𝐷

𝑐𝑎𝑡 + 𝐶�̅�𝑔𝑊𝑆𝐷𝑔 [𝐹𝑠𝑡𝑒𝑎𝑚�̅�𝑝

𝑠𝑡𝑒𝑎𝑚(𝑇𝑠𝑡𝑒𝑎𝑚 − 𝑇𝑆𝐷)

+ 𝐹𝑔𝑟𝑖𝑠,𝑜𝑢𝑡𝐶�̅�

𝑔(𝑇𝑟𝑖𝑠(ℎ𝑟𝑖𝑠) − 𝑇𝑆𝐷) + 𝐹𝑐𝑎𝑡

𝑟𝑖𝑠,𝑜𝑢𝑡𝐶�̅�𝑐𝑎𝑡(𝑇𝑟𝑖𝑠(ℎ𝑟𝑖𝑠) − 𝑇𝑆𝐷)

− 𝐶�̅�𝑐𝑎𝑡𝑇𝑆𝐷(𝐹𝑐𝑎𝑡

𝑟𝑖𝑠,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡)

− 𝐶�̅�𝑔𝑇𝑆𝐷(𝐹𝑔

𝑟𝑖𝑠,𝑜𝑢𝑡 + 𝐹𝑠𝑡𝑒𝑎𝑚 − 𝐹𝑔𝑆𝐷,𝑜𝑢𝑡)]

(20)

Regenerator dense phase model

The regenerator consists into two combustion stages, in which the superficial coke layer is

removed by combustion and the catalyst surface is regenerated for the cracking reactions.

A subsystem representation of the FCC regenerator is presented in the figure 4. The two-

phase fluidization theory is used to describe the combustion, particularly in the dense phase

of the regenerator (Kunii & Levenspiel, 1991). There is an emulsion phase, in which a high

concentration of catalyst reacts with the combustion air producing as products carbon

monoxide (CO), carbon dioxide (CO2), water (H2O), unreacted oxygen (O2) and nitrogen

(N2).

The combustion kinetics are considered diffusion controlled, considering the porosity of the

catalyst particles (Goodwin, R.D., Weisz, 1963). The intrinsic CO2/CO molar ratio is used

as the base for the reaction mechanism and the consideration that the coke is composed

solely of hydrocarbons. This is reasonable due the low composition of sulfuric and

nitrogenous compounds (Sadeghbeigi, 2012).

The base coke formula is 𝐶𝐻𝑞, where 𝑞 is the atomic ratio of hydrogen to carbon in the

catalytic coke.

𝐶𝐻𝑞 + (0.5 + 0.25𝑞)𝑂21. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂 + 0.5𝑞𝐻2𝑂

𝑟1 = 𝑘1𝐶𝑟𝑒𝑔𝑖𝐶𝑂2 (21)

𝐶𝐻𝑞 + (1 + 0.25𝑞)𝑂22. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂2 + 0.5𝑞𝐻2𝑂

𝑟2 = 𝑘2𝐶𝑟𝑒𝑔𝑖𝐶𝑂2 (22)

𝜎 = 𝜎1 exp (𝜎2𝑇𝑟𝑒𝑔𝑖

) 𝑘1 =𝑘𝑐1 + 𝜎

𝑘2 =𝑘𝑐𝜎

1 + 𝜎


Figure 4. Interaction among dense phases and freeboard in a two-regeneration bed

system.

Where:

• 𝑟𝑗, reaction rate expression for the 𝑖 reaction.

• 𝑘𝑖, pre-exponential factor for the 𝑖 reaction.

• 𝜎, CO/CO2 molar ratio.

• 𝑇, temperature.

The carbon combustion reactions produce simultaneously CO2 and CO. However, the CO

produced undergoes further oxidation to CO2 through the so-called after-burning reactions.

The oxidation of CO to CO2 can be of two different natures: heterogeneous (catalytic) or

homogeneous combustion (Ali et al., 1997).

𝐶𝑂 +1

2𝑂2

3. 𝑔𝑎𝑠−𝑠𝑜𝑙𝑖𝑑→ 𝐶𝑂2

1st regenerator

bed emulsion

phase

(gas+catalyst)

1st regenerator

bed emulsion

phase

(gas+catalyst)

1st regenerator

bed bubble

phase (gas)

1st regenerator

bed bubble

phase (gas)

Air blowerAir blower Air flow

division

Air flow

division

Air

Regenerator freeboard gas phaseRegenerator freeboard gas phase

2nd

regenerator

bed emulsion

phase

(gas+catalyst)

2nd

regenerator

bed emulsion

phase

(gas+catalyst)

2nd

regenerator

bed bubble

phase (gas)

2nd

regenerator

bed bubble

phase (gas)

Flue gas

to stack

Spent

catalyst

Phase

interaction

Phase

interaction

1st regeneration

stage catalyst

flow

Combustion

gases

Combustion

gases

Regenerated

Catalyst

1st regenerator

air flow

2nd

regenerator

air flow


𝑟3 = 𝑘3 exp (−

𝐸3

𝑅𝑇)𝐶𝑂2

0.5𝐶𝐶𝑂 (23)

𝐶𝑂 +1

2𝑂2

4. 𝑔𝑎𝑠→ 𝐶𝑂2

𝑟4 = 𝑘4 exp (−

𝐸4

𝑅𝑇)𝐶𝑂2

0.5𝐶𝐶𝑂𝐶𝐻2𝑂0.5 (24)

For the first regeneration bed, an overall mass balance for the catalyst is performed as

follows:

𝑑𝑊𝑐𝑟𝑒𝑔1


𝑆𝐷,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑅𝑒𝑔1,𝑜𝑢𝑡 (25)

𝑊𝑐𝑟𝑒𝑔1 = 𝜌𝑐𝜖𝑐

𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1𝐿𝑟𝑒𝑔1

𝐿𝑟𝑒𝑔1 =𝑊𝑐𝑟𝑒𝑔1

𝜌𝑐𝜖𝑐𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1

For the second regeneration bed, the catalyst mass balance is:

𝑑𝑊𝑐𝑟𝑒𝑔2


𝑅𝑒𝑔1,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑅𝑒𝑔2,𝑜𝑢𝑡 (26)

𝑊𝑐𝑟𝑒𝑔2 = 𝜌𝑐𝜖𝑐

𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2𝐿𝑟𝑒𝑔2

𝐿𝑟𝑒𝑔2 =𝑊𝑐𝑟𝑒𝑔2

𝜌𝑐𝜖𝑐𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2

Where:

• 𝑊𝑐𝑟𝑒𝑔𝑖, catalyst mass in the 𝑖th regenerator bed.

• 𝐿𝑟𝑒𝑔𝑖, catalyst level in the 𝑖th regenerator bed.

The catalyst flow from the first regeneration bed section (𝐹𝑐𝑎𝑡𝑅𝑒𝑔1,𝑜𝑢𝑡) to the second

regeneration bed is determined by the weir height that separates the two stages:

𝐹𝑐𝑎𝑡𝑅𝑒𝑔1,𝑜𝑢𝑡 = 𝐾𝑤√𝐿𝑟𝑒𝑔1 −𝐻𝑤 (27)

Where:

• 𝐾𝑤, weir flow constant, kg/s.m0.5.

• 𝐻𝑤, weir height, m.

The catalyst flow from the second regeneration bed section (𝐹𝑐𝑎𝑡𝑅𝑒𝑔2,𝑜𝑢𝑡) is determined by a

slide valve, as it is presented in the figure 4, and the flow is driven by the pressure difference

between the regenerator pressure and the regenerated catalyst-feed mixing point:


𝐹𝑐𝑎𝑡𝑅𝑒𝑔2,𝑜𝑢𝑡 = 𝑘𝑣

𝑅𝑒𝑔2

√

Δ𝑃𝑅𝑒𝑔2𝑣

1𝜌𝑐[(1𝐴𝑥𝑣)2

− (1𝐴0)2

]

(28)

Δ𝑃𝑅𝑒𝑔2𝑣 = 𝑃𝑅𝐺𝑁𝑔𝑎𝑠

+ 𝜌𝑐𝑔𝐿𝑟𝑔𝑛2 − 𝑃𝑀𝑃

𝐴𝑥𝑣 = 𝑎2 + 𝑏2𝑥𝑣2

Where:

• 𝑘𝑣𝑅𝑒𝑔2, second regeneration bed slide valve rating factor, kg/(s.Pa)0.5.

The gas volume (𝑉𝑟𝑒𝑔𝑔) is calculated as follows:

𝑉𝑟𝑒𝑔𝑔= 𝑉𝑟𝑒𝑔 − 𝐴𝑟𝑒𝑔1

𝐿𝑟𝑒𝑔1𝜖𝑐𝑟𝑒𝑔1− 𝐴𝑟𝑒𝑔2

𝐿𝑟𝑒𝑔2𝜖𝑐𝑟𝑒𝑔2

(29)

The average regenerator pressure (𝑃𝑅𝐺𝑁𝑔𝑎𝑠) is calculated at the freeboard conditions of

composition, temperature and density, and it is assumed that the ideal gas equation of state

is adequate, considering the operating conditions of the regenerator.

𝑃𝑅𝐺𝑁𝑔𝑎𝑠

=𝑁𝑇𝑅𝑇𝐹𝐵

𝑉𝑟𝑒𝑔𝑔 (30)

Where:

• 𝑁𝑇, total gas moles in the regenerator, mol.

• 𝑇𝐹𝐵, freeboard temperature, K.

• 𝐹𝐹𝐺, flue gas mass flow, kg/s.

• 𝑃𝑎𝑡𝑚, atmospheric pressure, Pa.

For modeling the component mass and energy balances in the regenerator, a continuous

stirred tank reactor (CSTR) dynamic model is proposed for the coke energy and mass

balances, and a CSTR dynamic reactor for the mass balances of the gaseous species in

both emulsion and bubble phase. The freeboard region is modelled as a dynamic CSTR

reactor for the mass and energy of the gaseous species and it is assumed that coke is not

present in the freeboard, just homogeneous combustion occurs in that region.

The following assumptions are made for the regenerator mass and energy balances:

1. The bubbles do not contain catalyst particles, hence only homogeneous combustion

kinetics are developed in this region.

2. The catalyst particles ejected to the freeboard does not constitute an important

fraction of the total volume.

3. The catalyst in the dense phase is in thermal equilibrium with the gaseous

components.

4. The gas phase in the regenerator is composed by oxygen, nitrogen, carbon dioxide,


carbon monoxide and water.

For the first regenerator bed, the coke balance is:

𝑑

𝑑𝑡(𝑊𝑐

𝑟𝑒𝑔1𝐶𝑟𝑒𝑔1) = 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡𝐶𝑆𝐷 − 𝐹𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1 +𝑀𝑊𝐶𝐴𝑟𝑔𝑛1𝜌𝑐𝜖𝑐𝑟𝑒𝑔1𝐿𝑟𝑒𝑔1(

−𝑟1𝐶𝑟𝑒𝑔1 − 𝑟2

𝐶𝑟𝑒𝑔1)

𝑊𝑐𝑟𝑒𝑔1

𝑑𝐶𝑟𝑒𝑔1


𝑆𝐷,𝑜𝑢𝑡𝐶𝑆𝐷 − 𝐹𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1 + 𝑀𝑊𝐶𝐴𝑟𝑔𝑛1𝜌𝑐𝜖𝑐𝑟𝑒𝑔1𝐿𝑟𝑒𝑔1

(−𝑟1𝐶𝑟𝑒𝑔1 − 𝑟2

𝐶𝑟𝑒𝑔1)

− 𝐶𝑟𝑒𝑔1(𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡

− 𝐹𝑐𝑎𝑡𝑅𝑒𝑔

1,𝑜𝑢𝑡)

(31)

Where:

• 𝜖𝑐𝑟𝑒𝑔1, catalyst fraction in the first regeneration bed.

• 𝐶𝑟𝑒𝑔1, coke mass concentration, kg coke/kg catalyst.

• 𝑀𝑊𝐶, carbon molecular weight, 0,012 kg/mol.

The gas inventory in the first regenerator bed is:

𝑊𝑔𝑟𝑒𝑔1 = 𝐿𝑟𝑒𝑔1𝐴𝑟𝑔𝑛1(1 − 𝜖𝑐

𝑟𝑒𝑔1)∑𝑀𝑊𝑖(𝐶𝑖𝑟𝑒𝑔1,𝐸 + 𝐶𝑖

𝑟𝑒𝑔1,𝐵)

5

𝑖=1

∀ 𝑖{𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(32)

𝐹𝑎𝑖𝑟𝑟𝑒𝑔1 = 𝑘𝑣

𝐴𝑖𝑟1𝑥𝑣√𝑃𝑎𝑖𝑟 − 𝑃𝑟𝑔𝑛

Where:

• 𝑊𝑔𝑟𝑒𝑔1, gas phase mass of the first regeneration bed, kg.

• 𝐹𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡, mass flow of the gas leaving the first regeneration bed, kg/s.

• 𝐹𝑎𝑖𝑟𝑟𝑒𝑔1, air mass flow for the first regeneration bed, kg/s.

• 𝑘𝑣𝐴𝑖𝑟1 , air valve coefficient for the first regeneration bed kg/s.Pa0.5

The energy balance for the first regeneration bed is:

𝑑𝐻𝑟𝑒𝑔1𝑑𝑡

= 𝐹𝑎𝑖𝑟𝑟𝑒𝑔1ℎ𝑎𝑖𝑟

𝑟𝑒𝑔1 + 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑆𝐷,𝑜𝑢𝑡 − 𝐹𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑔

𝑟𝑒𝑔1,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡 + �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛

+ �̇�𝑙𝑜𝑠𝑡

𝑑

𝑑𝑡(𝑊𝑐

𝑟𝑒𝑔1𝐶𝑝𝑐𝑇𝑟𝑒𝑔1 +𝑊𝑔𝑟𝑒𝑔1𝐶𝑝𝑔𝑇𝑟𝑒𝑔1)


𝑟𝑒𝑔1 + 𝐹𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑆𝐷,𝑜𝑢𝑡 − 𝐹𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑔


𝑟𝑒𝑔1,𝑜𝑢𝑡

+ �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛1 + �̇�𝑙𝑜𝑠𝑡1

𝑑

𝑑𝑡(𝑊𝑐


= 𝐶𝑝𝑐𝑇𝑟𝑒𝑔1𝑑𝑊𝑐

𝑟𝑒𝑔1

𝑑𝑡+𝑊𝑐

𝑟𝑒𝑔1𝐶𝑝𝑐𝑑𝑇𝑟𝑒𝑔1𝑑𝑡

+ 𝐶𝑝𝑔𝑇𝑟𝑒𝑔1𝑑𝑊𝑔

𝑟𝑒𝑔1

𝑑𝑡+𝑊𝑔

𝑟𝑒𝑔1𝐶𝑝𝑔𝑑𝑇𝑟𝑒𝑔1𝑑𝑡


𝑑𝑇𝑟𝑒𝑔1𝑑𝑡

= (1

𝑊𝑐𝑟𝑒𝑔1𝐶𝑝𝑐 +𝑊𝑔

𝑟𝑒𝑔1𝐶𝑝𝑔)(𝐹𝑎𝑖𝑟

𝑟𝑒𝑔1ℎ𝑎𝑖𝑟𝑟𝑒𝑔1 + 𝐹𝑐𝑎𝑡

𝑆𝐷,𝑜𝑢𝑡ℎ𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡 − 𝐹𝑔

𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡

− 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡 + �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 + �̇�𝑙𝑜𝑠𝑡 − 𝐶𝑝𝑐𝑇𝑟𝑒𝑔1𝑑𝑊𝑐

𝑟𝑒𝑔1

𝑑𝑡

− 𝐶𝑝𝑔𝑇𝑟𝑒𝑔1𝑑𝑊𝑔

𝑟𝑒𝑔1

𝑑𝑡)

(33)

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 0.232𝐹𝑎𝑖𝑟𝑟𝑒𝑔1𝐶�̅�𝑂2

(𝑇𝑟𝑒𝑓° − 𝑇𝑟𝑒𝑔1) + (𝐹𝑐𝑎𝑡

𝑆𝐷,𝑜𝑢𝑡𝐶𝑆𝐷 − 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1)𝐶�̅�𝑐𝑜𝑘𝑒(𝑇𝑟𝑒𝑓

° − 𝑇𝑟𝑒𝑔1)

+ 𝐹𝐻2𝑂𝑟𝑒𝑔1,𝑜𝑢𝑡Δ𝐻𝑓,𝐻2𝑂 + 𝐹𝐶𝑂2

𝑟𝑒𝑔1,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂2 + 𝐹𝐶𝑂𝑟𝑒𝑔1,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂

+ 𝐹𝐻2𝑂𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶�̅�,𝐻2𝑂(𝑇𝑟𝑒𝑔1 − 𝑇𝑟𝑒𝑓

° ) + 𝐹𝑂2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶�̅�,𝑂2(𝑇𝑟𝑒𝑔1 − 𝑇𝑟𝑒𝑓

° )

+ 𝐹𝐶𝑂2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶�̅�,𝐶𝑂2(𝑇𝑟𝑒𝑔1 − 𝑇𝑟𝑒𝑓

° ) + 𝐹𝐶𝑂𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶�̅�,𝐶𝑂(𝑇𝑟𝑒𝑔1 − 𝑇𝑟𝑒𝑓

° )

ℎ𝑎𝑖𝑟𝑟𝑒𝑔1 = �̅�𝑝𝑎𝑖𝑟(𝑇𝑎𝑖𝑟 −𝑇𝑟𝑒𝑓)

ℎ𝑐𝑎𝑡𝑆𝐷,𝑜𝑢𝑡 = 𝐶�̅�𝑐(𝑇𝑆𝐷 −𝑇𝑟𝑒𝑓)

ℎ𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡 = 𝐶�̅�𝑔 (𝑇𝑟𝑒𝑔1 −𝑇𝑟𝑒𝑓)

ℎ𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡 = 𝐶�̅�𝑐 (𝑇𝑟𝑒𝑔1 −𝑇𝑟𝑒𝑓)

Where:

• 𝑇𝑟𝑒𝑔1 , temperature of the first regeneration bed, K.

• �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛, reaction heat flow, J/s.

• �̇�𝑙𝑜𝑠𝑡, heat flow lost to the environment, J/s.

• 𝑇𝑟𝑒𝑓° , thermodynamic reference temperature, 298.15K.

• 𝑇𝑟𝑒𝑓, energy balance reference temperature, K.

To reduce the number of terms in the equation(34), the energy balance reference

temperature (𝑇𝑟𝑒𝑓) is going to be equal to the temperature of the first regeneration

bed(𝑇𝑟𝑒𝑔1). The heat loss to the environment is calculated considering the temperature

change in the regenerator wall.

𝑑𝑇𝑤𝑎𝑙𝑙1𝑑𝑡

=1

𝑑𝑤𝑎𝑙𝑙𝐶𝑝𝑤𝑎𝑙𝑙𝜌𝑤𝑎𝑙𝑙(�̇�𝑤𝑎𝑙𝑙1 − �̇�𝑙𝑜𝑠𝑡1) (34)

�̇�𝑤𝑎𝑙𝑙1 =𝐴𝑤𝑎𝑙𝑙1

1𝑈𝑖𝑛

+ 0.5𝑑𝑤𝑎𝑙𝑙𝑘𝑤𝑎𝑙𝑙

(𝑇𝑟𝑒𝑔1 − 𝑇𝑤𝑎𝑙𝑙1)

�̇�𝑙𝑜𝑠𝑡1 =𝐴𝑤𝑎𝑙𝑙1

1𝑈𝑜𝑢𝑡


(𝑇𝑤𝑎𝑙𝑙1 − 𝑇𝑎𝑡𝑚)


𝐴𝑤𝑎𝑙𝑙1 = 2𝐿𝑟𝑒𝑔1√𝜋𝐴𝑟𝑒𝑔1

Where:

• 𝑇𝑤𝑎𝑙𝑙1, wall temperature of the first regeneration bed, K.

• 𝑑𝑤𝑎𝑙𝑙, regenerator wall thickness, m.

• 𝜌𝑤𝑎𝑙𝑙, regenerator wall material density, kg/m3.

• 𝑘𝑤𝑎𝑙𝑙, regenerator wall material thermal conductivity, J/m2s.

• 𝐴𝑤𝑎𝑙𝑙1, heat transference area of the first regeneration bed, m2.

• 𝑇𝑎𝑡𝑚, atmospheric temperature, K.

The mass balances for the gaseous species depends on the phase in consideration. For

the emulsion phase:

𝑑

𝑑𝑡(𝑉𝑔

𝑟𝑒𝑔1,𝐸𝐶𝑖𝑟𝑒𝑔1,𝐸)

= 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐸 − 𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸𝐶𝑖𝑟𝑒𝑔1,𝐸 + 𝑉𝑔

𝑟𝑒𝑔1,𝐸𝐾𝑟𝑒𝑔1𝑖 (𝐶𝑖

𝑟𝑒𝑔1,𝐵 − 𝐶𝑖𝑟𝑒𝑔1,𝐸)

+ 𝑉𝑔𝑟𝑒𝑔1,𝐸 (

𝜖𝑐𝜌𝑐𝜖𝑔𝐸

∑𝑟𝑗𝜈𝑗𝑖

4

𝑗=1

+ 𝑟5𝜈𝑖)

𝑑

𝑑𝑡(𝑉𝑔𝑟𝑒𝑔1,𝐸𝐶𝑖

𝑟𝑒𝑔1,𝐸) =𝑑

𝑑𝑡(𝐿𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1𝜖𝑔𝐸𝐶𝑖

𝑟𝑒𝑔1,𝐸) =

= 𝐴𝑟𝑒𝑔1𝜖𝑔𝐸 (𝐶𝑖𝑟𝑒𝑔1,𝐸

𝑑𝐿𝑟𝑒𝑔1𝑑𝑡

+ 𝐿𝑟𝑒𝑔1𝑑𝐶𝑖

𝑟𝑒𝑔1,𝐸

𝑑𝑡)

𝐴𝑟𝑒𝑔1𝜖𝑔𝐸𝐿𝑟𝑒𝑔1

𝑑𝐶𝑖𝑟𝑒𝑔1,𝐸

𝑑𝑡

= 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐸 − 𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸𝐶𝑖𝑟𝑒𝑔1,𝐸

+ 𝑉𝑔𝑟𝑒𝑔1,𝐸𝐾𝑟𝑒𝑔1

𝑖 (𝐶𝑖𝑟𝑒𝑔1,𝐵 − 𝐶𝑖

𝑟𝑒𝑔1,𝐸)

+ 𝐿𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1𝜖𝑔𝐸 (𝜖𝑐𝜌𝑐𝜖𝑔𝐸


3

𝑗=1

+ 𝑟4𝜈𝑖)

− 𝐴𝑟𝑒𝑔1𝜖𝑔𝐸𝐶𝑖𝑟𝑒𝑔1,𝐸


∀ 𝑖 ∈ {𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(35)

𝑄𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸 = 𝑄𝑂2

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸 + 𝑄𝐶𝑂2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸 +𝑄𝐶𝑂

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸 + 𝑄𝑁2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸 + 𝑄𝐻2𝑂

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸

Where:


• 𝑉𝑔𝑟𝑒𝑔1,𝐸, gas volume of the emulsion in the first regeneration bed, m3.

• 𝐶𝑖𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸, molar concentration of the 𝑖 component in the emulsion phase, mol/m3.

• 𝐶𝑖𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵, molar concentration of the 𝑖 component in the bubble phase, mol/m3.

• 𝜖𝑔𝐸, gas fraction in the emulsion phase.

• 𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸, total outlet volumetric flow of the emulsion phase, m3/s.

• 𝑄𝑖

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸, outlet volumetric flow of the 𝑖 component of the emulsion phase, m3/s.

• 𝐾𝑟𝑒𝑔1𝑖 , mass transference coefficient of the 𝑖 component between the emulsion

phase and the dense phase, s-1.

• 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐸, molar flow of the component 𝑖 to the first regenerator stage in the

emulsion phase, mol/s.

For the bubble phase:

𝑑

𝑑𝑡(𝑉𝑔𝑟𝑒𝑔1,𝐵𝐶𝑖

𝑟𝑒𝑔1,𝐵)

= 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐵 −𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵𝐶𝑖𝑟𝑒𝑔1,𝐵 + 𝑉𝑔

𝑟𝑒𝑔1,𝐵𝐾𝑟𝑒𝑔1𝑖 (𝐶𝑖

𝑟𝑒𝑔1,𝐸 − 𝐶𝑖𝑟𝑒𝑔1,𝐵)

+ 𝑉𝑔𝑟𝑒𝑔1,𝐵𝑟4𝜈𝑖

𝑑

𝑑𝑡(𝐿𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1𝜖𝑔𝐵𝐶𝑖

𝑟𝑒𝑔1,𝐵) = 𝐴𝑟𝑒𝑔1𝜖𝑔𝐵 (𝐶𝑖𝑟𝑒𝑔1,𝐵



𝑟𝑒𝑔1,𝐵

𝑑𝑡)

𝐴𝑟𝑒𝑔1𝜖𝑔𝐵𝐿𝑟𝑒𝑔1

𝑑𝐶𝑖𝑟𝑒𝑔1,𝐵

𝑑𝑡

= 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐵 − 𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵𝐶𝑖𝑟𝑒𝑔1,𝐵

+ 𝑉𝑔𝑟𝑒𝑔1,𝐵𝐾𝑟𝑒𝑔1

𝑖 (𝐶𝑖𝑟𝑒𝑔1,𝐸 − 𝐶𝑖

𝑟𝑒𝑔1,𝐵) + 𝐿𝑟𝑒𝑔1𝐴𝑟𝑒𝑔1𝜖𝑔𝐵𝑟5𝜈𝑖

− 𝐴𝑟𝑒𝑔1𝜖𝑔𝐵𝐶𝑖𝑟𝑒𝑔1,𝐵


∀ 𝑖 ∈ {𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(36)

𝑄𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵 = 𝑄𝑂2

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵 +𝑄𝐶𝑂2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵 +𝑄𝐶𝑂

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵 + 𝑄𝑁2𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵 + 𝑄𝐻2𝑂

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵

Where

• 𝑄𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵, total outlet volumetric flow of the bubble phase, m3/s.

• 𝑄𝑖

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵, outlet volumetric flow of the 𝑖 component of the bubble phase, m3/s.

• 𝑁𝑖𝑟𝑒𝑔1,𝑖𝑛𝐵, molar flow of the component 𝑖 to the first regenerator stage in the bubble

phase, mol/s.


For the second regenerator bed, the coke balance is:

𝑑

𝑑𝑡(𝑊𝑐

𝑟𝑒𝑔2𝐶𝑟𝑒𝑔2)

= 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1 − 𝐹𝑐𝑎𝑡

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐶𝑟𝑒𝑔2

+𝑀𝑊𝐶𝐴𝑟𝑔𝑛2𝜌𝑐𝜖𝑐𝑟𝑒𝑔2𝐿𝑏𝑒𝑑2(−𝑟1

𝐶𝑟𝑒𝑔2 − 𝑟2𝐶𝑟𝑒𝑔2)

𝑊𝑐𝑟𝑒𝑔2

𝑑𝐶𝑟𝑒𝑔2


𝑆𝐷,𝑜𝑢𝑡𝐶𝑆𝐷 − 𝐹𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1

+ 𝑀𝑊𝐶𝐴𝑟𝑔𝑛1𝜌𝑐𝜖𝑐𝑟𝑒𝑔1𝐿𝑏𝑒𝑑1(−𝑟1

𝐶𝑟𝑒𝑔1 − 𝑟2𝐶𝑟𝑒𝑔1)

− 𝐶𝑟𝑒𝑔2 (𝐹𝑐𝑎𝑡𝑅𝑒𝑔1,𝑜𝑢𝑡 − 𝐹𝑐𝑎𝑡

𝑅𝑒𝑔2,𝑜𝑢𝑡)

(37)

Where:

• 𝜖𝑐𝑟𝑒𝑔2, catalyst fraction in the second regeneration bed.

• 𝐶𝑟𝑒𝑔2, coke mass concentration in the second regeneration bed, kg coke/kg

catalyst.

The gas inventory in the second regenerator bed is:

𝑊𝑔𝑟𝑒𝑔1 = 𝐿𝑟𝑒𝑔1𝐴𝑟𝑔𝑛1(1 − 𝜖𝑐

𝑟𝑒𝑔1)∑𝑀𝑊𝑖(𝐶𝑖𝑟𝑒𝑔1,𝐸 + 𝐶𝑖

𝑟𝑒𝑔1,𝐵)

5

𝑖=1

∀ 𝑖{𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(38)

𝐹𝑎𝑖𝑟𝑟𝑒𝑔2 = 𝑘𝑣

𝐴𝑖𝑟2𝑥𝑣√𝑃𝑎𝑖𝑟 − 𝑃𝑟𝑔𝑛

Where:

• 𝑊𝑔𝑟𝑒𝑔2, gas phase mass of the second regeneration bed, kg.

• 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡, mass flow of the gas leaving the second regeneration bed, kg/s.

• 𝐹𝑎𝑖𝑟𝑟𝑒𝑔2, air mass flow for the second regeneration bed, kg/s.

• 𝑘𝑣𝐴𝑖𝑟2 , air valve coefficient for the second regeneration bed kg/s.Pa0.5

The energy balance for the second regeneration bed is:



𝑟𝑒𝑔2 + 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡 − 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡ℎ𝑔





𝑑

𝑑𝑡(𝑊𝑐



𝑟𝑒𝑔2 + 𝐹𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡 − 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡ℎ𝑔




𝑑

𝑑𝑡(𝑊𝑐


= 𝐶𝑝𝑐𝑇𝑟𝑒𝑔2𝑑𝑊𝑐

𝑟𝑒𝑔2

𝑑𝑡+𝑊𝑐

𝑟𝑒𝑔2𝐶𝑝𝑐𝑑𝑇𝑟𝑒𝑔2𝑑𝑡

+ 𝐶𝑝𝑔𝑇𝑟𝑒𝑔2𝑑𝑊𝑔

𝑟𝑒𝑔2

𝑑𝑡+𝑊𝑔

𝑟𝑒𝑔2𝐶𝑝𝑔𝑑𝑇𝑟𝑒𝑔2𝑑𝑡

𝑑𝑇𝑟𝑒𝑔2𝑑𝑡

= (1

𝑊𝑐𝑟𝑒𝑔2𝐶𝑝𝑐 +𝑊𝑔

𝑟𝑒𝑔2𝐶𝑝𝑔)(𝐹𝑎𝑖𝑟

𝑟𝑒𝑔2ℎ𝑎𝑖𝑟𝑟𝑒𝑔2 + 𝐹𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑐𝑎𝑡𝑟𝑒𝑔1,𝑜𝑢𝑡

− 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡ℎ𝑔


𝑟𝑒𝑔2,𝑜𝑢𝑡 + �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛2 + �̇�𝑙𝑜𝑠𝑡2

− 𝐶𝑝𝑐𝑇𝑟𝑒𝑔2𝑑𝑊𝑐

𝑟𝑒𝑔2

𝑑𝑡− 𝐶𝑝𝑔𝑇𝑟𝑒𝑔2

𝑑𝑊𝑔𝑟𝑒𝑔2

𝑑𝑡)

(39)

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 = 0.21𝐹𝑎𝑖𝑟𝑟𝑒𝑔2𝐶�̅�𝑂2

(𝑇𝑟𝑒𝑓° − 𝑇𝑟𝑒𝑔1) + (𝐹𝑐𝑎𝑡

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐶𝑟𝑒𝑔1 −𝐹𝑐𝑎𝑡𝑟𝑒𝑔2,𝑜𝑢𝑡𝐶𝑟𝑒𝑔2) 𝐶�̅�𝑐𝑜𝑘𝑒(𝑇𝑟𝑒𝑓

° − 𝑇𝑟𝑒𝑔2)

+ 𝐹𝐻2𝑂𝑟𝑒𝑔2,𝑜𝑢𝑡Δ𝐻𝑓,𝐻2𝑂 +𝐹𝐶𝑂2

𝑟𝑒𝑔2,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂2 +𝐹𝐶𝑂𝑟𝑒𝑔2,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂

+𝐹𝐻2𝑂𝑟𝑒𝑔2,𝑜𝑢𝑡�̅�𝑝,𝐻2𝑂(𝑇𝑟𝑒𝑔2 − 𝑇𝑟𝑒𝑓

° )+𝐹𝑂2𝑟𝑒𝑔2,𝑜𝑢𝑡�̅�𝑝,𝑂2(𝑇𝑟𝑒𝑔2 − 𝑇𝑟𝑒𝑓

° )

+𝐹𝐶𝑂2𝑟𝑒𝑔2,𝑜𝑢𝑡�̅�𝑝,𝐶𝑂2(𝑇𝑟𝑒𝑔2 − 𝑇𝑟𝑒𝑓

° )+𝐹𝐶𝑂𝑟𝑒𝑔2,𝑜𝑢𝑡�̅�𝑝,𝐶𝑂(𝑇𝑟𝑒𝑔2 − 𝑇𝑟𝑒𝑓

° )

ℎ𝑎𝑖𝑟𝑟𝑒𝑔2 = �̅�𝑝𝑎𝑖𝑟(𝑇𝑎𝑖𝑟 −𝑇𝑟𝑒𝑓)




Where:

• 𝑇𝑟𝑒𝑔2 , temperature of the second regeneration bed, K.

• �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛2, reaction heat flow of the second regeneration bed, J/s.

• �̇�𝑙𝑜𝑠𝑡2, heat flow lost to the environment of the second regeneration bed, J/s.



To reduce the number of terms in the equation(39), the energy balance reference

temperature (𝑇𝑟𝑒𝑓) is going to be equal to the temperature of the second regeneration

bed(𝑇𝑟𝑒𝑔2). The heat loss to the environment is calculated considering the temperature

change in the regenerator wall.


𝑑𝑇𝑤𝑎𝑙𝑙2𝑑𝑡

=1

𝑑𝑤𝑎𝑙𝑙𝐶𝑝𝑤𝑎𝑙𝑙𝜌𝑤𝑎𝑙𝑙(�̇�𝑤𝑎𝑙𝑙2 − �̇�𝑙𝑜𝑠𝑡2) (40)

�̇�𝑤𝑎𝑙𝑙2 =𝐴𝑤𝑎𝑙𝑙2

1𝑈𝑖𝑛


(𝑇𝑟𝑒𝑔2 − 𝑇𝑤𝑎𝑙𝑙2)

�̇�𝑙𝑜𝑠𝑡2 =𝐴𝑤𝑎𝑙𝑙2

1𝑈𝑜𝑢𝑡


(𝑇𝑤𝑎𝑙𝑙2 − 𝑇𝑎𝑡𝑚)

𝐴𝑤𝑎𝑙𝑙2 = 2𝐿𝑟𝑒𝑔2√𝜋𝐴𝑟𝑒𝑔2

Where:

• 𝑇𝑤𝑎𝑙𝑙1, wall temperature of the first regeneration bed, K.

• 𝐴𝑤𝑎𝑙𝑙2, heat transference area of the second regeneration bed, m2.

The mass balances for the gaseous species depends on the phase in consideration. For

the emulsion phase:

𝑑

𝑑𝑡(𝑉𝑔𝑟𝑒𝑔2,𝐸𝐶𝑖

𝑟𝑒𝑔2,𝐸)

= 𝑁𝑖𝑟𝑒𝑔2,𝑖𝑛𝐸 −𝑁𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸𝐶𝑖𝑟𝑒𝑔2,𝐸 + 𝑉𝑔

𝑟𝑒𝑔2,𝐸𝐾𝑟𝑒𝑔2𝑖 (𝐶𝑖

𝑟𝑒𝑔2,𝐵 − 𝐶𝑖𝑟𝑒𝑔2,𝐸)

+ 𝑉𝑔𝑟𝑒𝑔2,𝐸 (

𝜖𝑐𝜌𝑐𝜖𝑔𝐸


4

𝑗=1

+ 𝑟5𝜈𝑖)

𝑑

𝑑𝑡(𝐿𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2𝜖𝑔𝐸𝐶𝑖

𝑟𝑒𝑔2,𝐸) = 𝐴𝑟𝑒𝑔2𝜖𝑔𝐸 (𝐶𝑖𝑟𝑒𝑔2,𝐸



𝑟𝑒𝑔2,𝐸

𝑑𝑡)

𝐴𝑟𝑒𝑔2𝜖𝑔𝐸𝐿𝑟𝑒𝑔2

𝑑𝐶𝑖𝑟𝑒𝑔2,𝐸

𝑑𝑡

= 𝑁𝑖𝑟𝑒𝑔2,𝑖𝑛𝐸 − 𝑁𝑡𝑜𝑡𝑎𝑙


+ 𝑉𝑔𝑟𝑒𝑔2,𝐸𝐾𝑟𝑒𝑔2

𝑖 (𝐶𝑖𝑟𝑒𝑔2,𝐵 − 𝐶𝑖

𝑟𝑒𝑔2,𝐸)

+ 𝐿𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2𝜖𝑔𝐸 (𝜖𝑐𝜌𝑐𝜖𝑔𝐸


4

𝑗=1

+ 𝑟5𝜈𝑖)

− 𝐴𝑟𝑒𝑔2𝜖𝑔𝐸𝐶𝑖𝑟𝑒𝑔2,𝐸


∀ 𝑖 ∈ {𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(41)

𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸 = 𝑁𝑂2

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸 +𝑁𝐶𝑂2𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸 +𝑁𝐶𝑂

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸 +𝑁𝑁2𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸 +𝑁𝐻2𝑂

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸

Where:


• 𝑉𝑔𝑟𝑒𝑔2,𝐸, gas volume of the emulsion in the second regeneration bed, m3.

• 𝐶𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸, molar concentration of the 𝑖 component in the emulsion phase, mol/m3.

• 𝐶𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵, molar concentration of the 𝑖 component in the bubble phase, mol/m3.

• 𝜖𝑔𝐸, gas fraction in the emulsion phase.

• 𝑁𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸, molar flow of the 𝑖 component at the outer part of the emulsion phase,

mol/s.

• 𝐾𝑟𝑒𝑔2𝑖 , mass transference coefficient for the 𝑖 component between the emulsion

phase and the dense phase, s-1.

For the bubble phase:

𝑑

𝑑𝑡(𝑉𝑔𝑟𝑒𝑔2,𝐵𝐶𝑖

𝑟𝑒𝑔2,𝐵)

= 𝑁𝑖𝑟𝑒𝑔2,𝑖𝑛𝐵 −𝑁𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵𝐶𝑖𝑟𝑒𝑔2,𝐵 + 𝑉𝑔

𝑟𝑒𝑔2,𝐵𝐾𝑟𝑒𝑔2(𝐶𝑖𝑟𝑒𝑔2,𝐸 − 𝐶𝑖

𝑟𝑒𝑔2,𝐵)

+ 𝑉𝑔𝑟𝑒𝑔2,𝐵𝑟5𝜈𝑖

𝑑

𝑑𝑡(𝐿𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2𝜖𝑔𝐵𝐶𝑖

𝑟𝑒𝑔2,𝐵) = 𝐴𝑟𝑒𝑔2𝜖𝑔𝐵 (𝐶𝑖𝑟𝑒𝑔2,𝐵



𝑟𝑒𝑔2,𝐵

𝑑𝑡)

𝐴𝑟𝑒𝑔2𝜖𝑔𝐵𝐿𝑟𝑒𝑔2

𝑑𝐶𝑖𝑟𝑒𝑔2,𝐵

𝑑𝑡

= 𝑁𝑖𝑟𝑒𝑔2,𝑖𝑛𝐵 −𝑁𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵𝐶𝑖𝑟𝑒𝑔2,𝐵

+ 𝑉𝑔𝑟𝑒𝑔2,𝐵𝐾𝑟𝑒𝑔2

𝑖 (𝐶𝑖𝑟𝑒𝑔2,𝐸 − 𝐶𝑖

𝑟𝑒𝑔2,𝐵) + 𝐿𝑟𝑒𝑔2𝐴𝑟𝑒𝑔2𝜖𝑔𝐶𝑟5𝜈𝑖

− 𝐴𝑟𝑒𝑔2𝜖𝑔𝐵𝐶𝑖𝑟𝑒𝑔2,𝐵


∀ 𝑖 ∈ {𝑂2, 𝑁2, 𝐶𝑂, 𝐶𝑂2, 𝐻2𝑂}

(42)

𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵 = 𝑁𝑂2

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵 +𝑁𝐶𝑂2𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵 +𝑁𝐶𝑂

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵 +𝑁𝑁2𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵 +𝑁𝐻2𝑂

𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵

Regenerator bed characterization model

First, we describe the relationship between the catalyst volume fraction (𝜖𝑐), gas volume

fraction in the emulsion phase (𝜖𝑔𝐸) and the gas volume fraction of the bubble phase (𝜖𝑔𝐵):

𝜖𝑐𝑟𝑒𝑔𝑖 + 𝜖𝑔𝐸

𝑟𝑒𝑔𝑖 + 𝜖𝑔𝐵𝑟𝑒𝑔𝑖 = 1 (43)

The empirical correlation presented by (Han & Chung, 2001) for the calculation of the

catalyst volume fraction is used and presented in the equation (44).

𝜖𝑐𝑟𝑒𝑔𝑖 = 0.3418exp(−0.975𝑢𝑔

𝑟𝑒𝑔1) + 0.1592 (44)


𝑢𝑔𝑟𝑒𝑔𝑖 =

𝐹𝑎𝑖𝑟𝑟𝑒𝑔𝑖

𝜌𝑔𝐴𝑟𝑒𝑔𝑖

Where:

• 𝑢𝑔𝑟𝑒𝑔𝑖, average superficial gas velocity, m/s.

The volume fraction of the bubble phase is calculated according to (Kunii & Levenspiel,

1991):

𝜖𝑔𝐵𝑟𝑒𝑔𝑖 =

𝑢𝑔𝑟𝑒𝑔𝑖 − 𝑣𝑔𝑖

𝑣𝑔𝐵𝑖 − 𝑣𝑔𝑖

𝑣𝑔𝑖 =𝑑𝑐2(𝜌𝑐 − 𝜌𝑔)𝑔(1 − 𝜖𝑐

𝑟𝑒𝑔𝑖)3𝜙𝑐2

150𝜇𝑔𝜖𝑐𝑟𝑒𝑔𝑖

𝑣𝑔𝐵𝑖 = 𝑢𝑔𝑟𝑒𝑔𝑖 − 𝑣𝑔𝑖 + 0.711√𝑑𝑏𝑔

Where:

• 𝑑𝑐, catalyst diameter, m.

• 𝜌𝑐, catalyst density, kg/m3.

• 𝜌𝑔, average gas density, kg/m3.

• 𝑔, gravity, m/s2.

• 𝜙𝑐, catalyst average sphericity.

• 𝜇𝑔, average gas viscosity, Pa s.

• 𝑑𝑏, average bubble diameter, m.

The average bubble diameter (𝑑𝑏) is calculated as the mean value through the axial

direction of the regenerator bed:

𝑑𝑏𝑖 =1

𝐿𝑟𝑒𝑔𝑖∫ 𝑑𝑏𝑚 − (𝑑𝑏𝑚 − 𝑑𝑏0) exp(−

0.3𝑧

𝑑𝑟𝑒𝑔𝑖)𝑑𝑧

𝐿𝑟𝑒𝑔𝑖

0

𝑑𝑏𝑖 = 𝑑𝑏𝑚 +

𝑑𝑟𝑒𝑔𝑖0.3𝐿𝑟𝑒𝑔𝑖

(𝑑𝑏𝑚 − 𝑑𝑏0) exp(−1 −0.3𝐿𝑟𝑒𝑔𝑖𝑑𝑟𝑒𝑔𝑖

) (45)

𝑑𝑏0 =2.78

𝑔(𝑢𝑔𝑟𝑒𝑔𝑖 − 𝑣𝑔)

2

𝑑𝑏𝑚 = 0.59(𝑢𝑔𝑟𝑒𝑔𝑖 − 𝑣𝑔)

0.4𝑑𝑟𝑒𝑔𝑖0.8

Where:

• 𝑑𝑟𝑒𝑔𝑖, 𝑖 regenerator bed diameter, m.

The overall mass transference coefficient between the emulsion phase and the dense


phase for each regeneration bed (𝐾𝑟𝑒𝑔𝑖𝑗) can be calculated by using the interchange

coefficient between bubble and cloud phases (𝐾1𝑗) and between cloud and emulsion

phases (𝐾2𝑗). The difussion coefficient for the component 𝑖 in the mixture (𝐷𝑖) is calculated

with the methodology developed by Fairbanks and Wilke (Fairbanks & Wilke, 1950).

𝐾𝑟𝑒𝑔𝑗𝑖 =

𝜖𝑔𝐵𝑟𝑒𝑔𝑗

1𝐾1+1𝐾2

(46)

𝐾1𝑖 = 4.5

𝑣𝑔𝑗

𝑑𝑏𝑗+ 5.85(

𝐷𝑓𝑖1/2𝑔1/4

𝑑𝑏𝑗5/4

)

𝐾2𝑖 = 6.77(

𝐷𝑓𝑖𝜖𝑔𝐸𝑟𝑒𝑔𝑗𝑣𝑔𝑗

𝑑𝑏𝑗)

1/2

𝐷𝑓𝑖 = 0.35(𝑔𝑢𝑔𝑟𝑒𝑔𝑖)

1/3𝐷𝑖4/3

𝐷𝑖 =1 − 𝑦𝑖

∑𝑦𝑘𝐷𝑖𝑘𝑘≠𝑖

𝐷𝑖𝑘 = 𝐷𝑖𝑘0 (𝑇𝑟𝑒𝑔𝑗𝑇0)

2

(𝑃0

𝑃𝑅𝐺𝑁𝑔𝑎𝑠) ∀ 𝑖 ≠ 𝑘 ∈ {𝑂2, 𝑁2, 𝐶𝑂. 𝐶𝑂2, 𝐻2𝑂}

Where:

• 𝐷𝑖𝑘0 , Diffusion coefficient of the component 𝑗 in 𝑘 at standard conditions, m2/s.

• 𝑇0, Standard temperature, 298.15 K.

• 𝑃0, Standard pressure, 101325 Pa.

• 𝑦𝑖, molar fraction of the component 𝑖.

Freeboard model

An overall gas mass balance in the freeboard is:

𝑑𝑊𝑔𝐹𝐵

𝑑𝑡= 𝐹𝑔

𝑟𝑒𝑔1,𝑜𝑢𝑡 + 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡 − 𝐹𝐹𝐺 (47)

𝐹𝑔𝑟𝑒𝑔𝑖,𝑜𝑢𝑡 = 𝑀𝑊𝑂2𝑁𝑂2

𝑟𝑒𝑔𝑖,𝑜𝑢𝑡 +𝑀𝑊𝐶𝑂2𝑁𝐶𝑂2𝑟𝑒𝑔𝑖,𝑜𝑢𝑡 +𝑀𝑊𝐶𝑂𝑁𝐶𝑂

𝑟𝑒𝑔𝑖,𝑜𝑢𝑡 +𝑀𝑊𝑁2𝑁𝑁2𝑟𝑒𝑔𝑖,𝑜𝑢𝑡

+𝑀𝑊𝐻2𝑂𝑁𝐻2𝑂𝑟𝑒𝑔𝑖,𝑜𝑢𝑡 ∀ 𝑖{1,2}

𝐹𝐹𝐺 = 𝑘𝑣𝐹𝐺𝑥𝑣√𝑃𝑅𝐺𝑁

𝑔𝑎𝑠− 𝑃𝑎𝑡𝑚

Where:

• 𝑘𝑣𝐹𝐺, flue gas slide valve rating factor, kg/(s.Pa)0.5.


• 𝐹𝐹𝐺, flue gas mass flow, kg/s.

• 𝑃𝑎𝑡𝑚, atmospheric pressure, Pa.

The component mass balance in the freeboard is:

𝑑

𝑑𝑡(𝑉𝑔𝐹𝐵𝐶𝑖

𝐹𝐵) = 𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸𝐶𝑖

𝑟𝑒𝑔1,𝐸 +𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵𝐶𝑖

𝑟𝑒𝑔1,𝐵 +𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸𝐶𝑖

𝑟𝑒𝑔2,𝐸 + 𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵𝐶𝑖

𝑟𝑒𝑔2,𝐵

−𝑁𝑡𝑜𝑡𝑎𝑙𝐹𝐵 𝐶𝑖

𝐹𝐵 + 𝑉𝑔𝐹𝐵𝑟5𝜈𝑖

𝑑

𝑑𝑡(𝑉𝑔𝐹𝐵𝐶𝑖

𝐹𝐵) = 𝐶𝑖𝐹𝐵𝑑𝑉𝑔

𝐹𝐵

𝑑𝑡+ 𝑉𝑔

𝐹𝐵𝑑𝐶𝑖

𝐹𝐵

𝑑𝑡

𝑉𝑔𝐹𝐵𝑑𝐶𝑖

𝐹𝐵

𝑑𝑡= 𝑁𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸𝐶𝑖𝑟𝑒𝑔1,𝐸 +𝑁𝑡𝑜𝑡𝑎𝑙

𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵𝐶𝑖𝑟𝑒𝑔1,𝐵 +𝑁𝑡𝑜𝑡𝑎𝑙


+𝑁𝑡𝑜𝑡𝑎𝑙𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵𝐶𝑖

𝑟𝑒𝑔2,𝐵 − 𝑁𝑡𝑜𝑡𝑎𝑙𝐹𝐵 𝐶𝑖

𝐹𝐵 + 𝑉𝑔𝐹𝐵𝑟5𝜈𝑖 − 𝐶𝑖

𝐹𝐵𝑑𝑉𝑔

𝐹𝐵

𝑑𝑡

(48)

The energy balance for the freeboard section is:


= 𝐹𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡ℎ𝑔

𝑟𝑒𝑔1,𝑜𝑢𝑡 + 𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡ℎ𝑔

𝑟𝑒𝑔2,𝑜𝑢𝑡 − 𝐹𝐹𝐺ℎ𝑔𝐹𝐺,𝑜𝑢𝑡 + �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝐹𝐵 + �̇�𝑙𝑜𝑠𝑡𝐹𝐵

𝑑

𝑑𝑡(𝑊𝑔

𝐹𝐵𝐶𝑝𝑔𝑇𝐹𝐵)



𝑟𝑒𝑔2,𝑜𝑢𝑡 − 𝐹𝐹𝐺ℎ𝑔𝐹𝐺,𝑜𝑢𝑡 + �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝐹𝐵 + �̇�𝑙𝑜𝑠𝑡𝐹𝐵

𝑑

𝑑𝑡(𝑊𝑔

𝐹𝐵𝐶𝑝𝑔𝑇𝐹𝐵) = 𝐶𝑝𝑔𝑇𝐹𝐵𝑑𝑊𝑔

𝐹𝐵

𝑑𝑡+𝑊𝑔

𝐹𝐵𝐶𝑝𝑔𝑑𝑇𝐹𝐵𝑑𝑡

𝑊𝑔𝐹𝐵𝐶𝑝𝑔

𝑑𝑇𝐹𝐵𝑑𝑡



𝑟𝑒𝑔2,𝑜𝑢𝑡 − 𝐹𝐹𝐺ℎ𝑔𝐹𝐺,𝑜𝑢𝑡

+ �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝐹𝐵 + �̇�𝑙𝑜𝑠𝑡𝐹𝐵 − 𝐶𝑝𝑔𝑇𝐹𝐵𝑑𝑊𝑔

𝐹𝐵

𝑑𝑡

(49)

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛 =𝑀𝑊𝑂2 (𝑁𝑂2𝑟𝑒𝑔1,𝑜𝑢𝑡 +𝑁𝑂2

𝑟𝑒𝑔2,𝑜𝑢𝑡) 𝐶�̅�𝑂2(𝑇𝑟𝑒𝑓

° − 𝑇𝐹𝐵)+𝑀𝑊𝐶𝑂2𝑁𝐶𝑂2𝐹𝐵,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂2

+𝑀𝑊𝐶𝑂𝑁𝐶𝑂𝐹𝐵,𝑜𝑢𝑡Δ𝐻𝑓,𝐶𝑂 +𝐹𝑂2

𝐹𝐵,𝑜𝑢𝑡�̅�𝑝,𝑂2(𝑇𝐹𝐵 − 𝑇𝑟𝑒𝑓° )+𝐹𝐶𝑂2

𝐹𝐵,𝑜𝑢𝑡�̅�𝑝,𝐶𝑂2(𝑇𝐹𝐵 − 𝑇𝑟𝑒𝑓° )

+𝐹𝐶𝑂𝐹𝐵,𝑜𝑢𝑡�̅�𝑝,𝐶𝑂(𝑇𝐹𝐵 − 𝑇𝑟𝑒𝑓

° )



ℎ𝑔𝐹𝐺,𝑜𝑢𝑡 = 𝐶�̅�𝑔(𝑇𝐹𝐵 −𝑇𝑟𝑒𝑓)

Where:

• 𝑇𝐹𝐵, temperature of the freeboard, K.

• �̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝐹𝐵, reaction heat flow of the freeboard, J/s.

• �̇�𝑙𝑜𝑠𝑡𝐹𝐵, heat flow lost to the environment of the freeboard, J/s.




The heat loss to the environment is calculated considering the temperature change in the

regenerator wall of the freeboard section.

𝑑𝑇𝑤𝑎𝑙𝑙𝐹𝐵𝑑𝑡

=1

𝑑𝑤𝑎𝑙𝑙𝐶𝑝𝑤𝑎𝑙𝑙𝜌𝑤𝑎𝑙𝑙(�̇�𝑤𝑎𝑙𝑙𝐹𝐵 − �̇�𝑙𝑜𝑠𝑡𝐹𝐵) (50)

�̇�𝑤𝑎𝑙𝑙𝐹𝐵 =𝐴𝑤𝑎𝑙𝑙𝐹𝐵

1𝑈𝑖𝑛


(𝑇𝐹𝐵 − 𝑇𝑤𝑎𝑙𝑙𝐹𝐵)

�̇�𝑙𝑜𝑠𝑡𝐹𝐵 =𝐴𝑤𝑎𝑙𝑙𝐹𝐵

1𝑈𝑜𝑢𝑡


(𝑇𝑤𝑎𝑙𝑙𝐹𝐵 − 𝑇𝑎𝑡𝑚)

𝐴𝑤𝑎𝑙𝑙𝐹𝐵 = 2𝐿𝑟𝑒𝑔2√𝜋𝐴𝑟𝑒𝑔2

Nomenclature list

𝐹𝑖𝑗, mass flow of the component 𝑖 at the position 𝑗, kg/s.

ℎ𝑖𝑗, mass specific enthalpy of the component 𝑖 at the position 𝑗, J/kg.

𝑇𝑀𝑃𝑜𝑢𝑡, outlet temperature of the mixing point, K.

𝐶�̅�𝑖 , mass specific heat of the component 𝑖, J/kg.K.

𝑇𝑖𝑗, temperature of the component 𝑖 at the position 𝑗, K.

Δ𝐻𝑣𝑎𝑝, vaporization heat of the hydrocarbon feed mixture at the operation pressure, kJ/kg.

𝑃𝑀𝑃, pressure in the catalyst-feed mixing point, Pa.

𝑃𝑆𝐷, pressure in the stripping disengaging section, Pa.

𝑃𝑟𝑖𝑠(𝑖), pressure in the 𝑖 position of the riser section, Pa.

𝐻𝑟𝑖𝑠, riser length, m.

𝜖𝑔, riser gas fraction, m3gas/m3

riser.

𝑢𝑔, gas velocity, m/s.

𝜌𝑔𝑟𝑖𝑠, gas density in the riser, kggas/m3

gas.

𝑋𝑖,𝑗, weight fraction of every gaseous 𝑖 PONA component in the major lump 𝑗.

𝑧, riser axial direction, m.

𝜌𝑐, bulk catalyst density inside the riser, kg/m3.

𝑅𝑖,𝑗𝑔

, reaction rate in the gas phase of every 𝑖 PONA component in the major lump 𝑗.

Ψ, catalyst deactivation function.


𝜃𝑖,𝑗, weight fraction of every catalytic cracking reaction intermediate 𝑖 PONA component in

the major lump 𝑗.

𝑅𝑖,𝑗𝑠 , reaction rate in the gas phase of every catalytic cracking reaction intermediate 𝑖 PONA

component in the major lump 𝑗.

𝛼, deactivation parameter, kg catalyst/kg coke.

𝛼0𝑖, frequency factor parameter for every component in the riser.

𝐴

𝑁, aromatic-naphthenic ratio of the gas oil.

𝐸𝑎𝑖, activation energy of the 𝑖 component, kJ/mol.

𝑇𝑔, riser gas phase temperature, K.

𝑅, universal gas constant, kJ/mol K.

𝑇𝑟𝑖𝑠, riser temperature, K.

𝐴𝑟𝑖𝑠, riser cross section, m2.

𝑔, gravity constant, m/s2.

%𝐶𝑁, Naphthenic rings weight fraction.

%𝐶𝑅, Total ring type compounds weight fraction.

%𝐶𝐴, Aromatic compounds weight fraction.

𝑅𝐼(20°𝐶), refractive index at 20°C.

𝐷(20°𝐶), mass density at 20°C, g/mL.

𝑆, sulfur mass percentage.

𝐹𝑐𝑎𝑡𝑗,𝑜𝑢𝑡

, catalyst mass flow coming out from the section 𝑗, kg/s.

𝐿𝑆𝐷, stripping-disengaging section catalyst level, m.

𝐴𝑆𝐷, stripping-disengaging cross section, m2.

𝑘𝑣𝑆𝐷, stripper disengaging slide valve rating factor, kg/(s.Pa)0.5.

𝑥𝑣, valve opening fraction.


, SD gas phase pressure, Pa.

𝑃𝑅𝐺𝑁𝑔𝑎𝑠

, regenerator gas pressure, Pa.

𝐿𝑟𝑔𝑛1, catalyst level of the dense phase in the first regenerator stage.

𝑘𝑣𝑀𝐹, main fractionator slide valve rating factor, kg/(s.Pa)0.5.

𝑊𝑆𝐷𝑔

, SD gas mass, kg.

𝑉𝑆𝐷𝑔

, gas phase SD section volume, m3.

𝑇𝑆𝐷𝑔

, gas phase SD section temperature, K.


𝑉𝑆𝐷, SD section volume, m3.

𝑀𝑊𝑆𝐷, average molecular weight in the SD section, kg/kmol.

𝐹𝑔𝑗,𝑜𝑢𝑡

, gas mass flow coming out from the section 𝑗, kg/s.

𝐹𝑠𝑡𝑒𝑎𝑚, stripper disengaging steam mass flow, kg/s.

𝑟𝑖𝑗, reaction rate expression for the component 𝑗 in the 𝑖 reaction.

𝑑𝑟𝑦, combustion kinetics in a free-water conditions.

𝑤𝑒𝑡, combustion kinetics in a non-free-water conditions.

𝑘𝑖𝑗, pre-exponential factor for the 𝑖 reaction in the 𝑗 condition.

𝐸𝑖𝑗, activation energy for the 𝑖 reaction in the 𝑗 condition.

𝑃𝑖, partial pressure of the 𝑖 component.

𝜎, CO/CO2 ratio.

𝑋, molar carbon conversion.

𝑇, temperature.

𝑊𝑐𝑟𝑒𝑔𝑖, catalyst mass in the 𝑖th regenerator bed.

𝐿𝑟𝑒𝑔𝑖, catalyst level in the 𝑖th regenerator bed.

𝐾𝑤, weir flow constant, kg/s.m0.5.

𝐻𝑤, weir height, m.

𝑘𝑣𝑅𝑒𝑔2, second regeneration bed slide valve rating factor, kg/(s.Pa)0.5.

𝑁𝑇, total gas moles in the regenerator, mol.

𝑇𝐹𝐵, freeboard temperature, K.

𝐹𝐹𝐺, flue gas mass flow, kg/s.

𝑃𝑎𝑡𝑚, atmospheric pressure, Pa.

𝜖𝑐𝑟𝑒𝑔1, catalyst fraction in the first regeneration bed.

𝐶𝑟𝑒𝑔1, coke mass concentration, kg coke/kg catalyst.

𝑀𝑊𝐶, carbon molecular weight, 0,012 kg/mol.

𝑊𝑔𝑟𝑒𝑔1, gas phase mass of the first regeneration bed, kg.

𝐹𝑔𝑟𝑒𝑔1,𝑜𝑢𝑡, mass flow of the gas leaving the first regeneration bed, kg/s.

𝐹𝑎𝑖𝑟𝑟𝑒𝑔1, air mass flow for the first regeneration bed, kg/s.

𝑇𝑟𝑒𝑔1 , temperature of the first regeneration bed, K.

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛, reaction heat flow, J/s.

�̇�𝑙𝑜𝑠𝑡, heat flow lost to the environment, J/s.


𝑇𝑟𝑒𝑓° , thermodynamic reference temperature, 298.15K.

𝑇𝑟𝑒𝑓, energy balance reference temperature, K.

𝑇𝑤𝑎𝑙𝑙1, wall temperature of the first regeneration bed, K.

𝑑𝑤𝑎𝑙𝑙, regenerator wall thickness, m.

𝜌𝑤𝑎𝑙𝑙, regenerator wall material density, kg/m3.

𝑘𝑤𝑎𝑙𝑙, regenerator wall material thermal conductivity, J/m2s.

𝐴𝑤𝑎𝑙𝑙1, heat transference area of the first regeneration bed, m2.

𝑇𝑎𝑡𝑚, atmospheric temperature, K.

𝑉𝑔𝑟𝑒𝑔1,𝐸, gas volume of the emulsion in the first regeneration bed, m3.

𝐶𝑖𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸, molar concentration of the 𝑖 component in the emulsion phase, mol/m3.

𝐶𝑖𝑟𝑒𝑔1,𝑜𝑢𝑡𝐵, molar concentration of the 𝑖 component in the bubble phase, mol/m3.

𝜖𝑔𝐸, gas fraction in the emulsion phase.

𝑁𝑖𝑟𝑒𝑔1,𝑜𝑢𝑡𝐸, molar flow of the 𝑖 component at the outer part of the emulsion phase, mol/s.

𝐾, mass transference coefficient between the emulsion phase and the dense phase, s-1.

𝜖𝑐𝑟𝑒𝑔2, catalyst fraction in the second regeneration bed.

𝐶𝑟𝑒𝑔2, coke mass concentration in the second regeneration bed, kg coke/kg catalyst.

𝑊𝑔𝑟𝑒𝑔2, gas phase mass of the second regeneration bed, kg.

𝐹𝑔𝑟𝑒𝑔2,𝑜𝑢𝑡, mass flow of the gas leaving the second regeneration bed, kg/s.

𝐹𝑎𝑖𝑟𝑟𝑒𝑔2, air mass flow for the second regeneration bed, kg/s.

𝑇𝑟𝑒𝑔2 , temperature of the second regeneration bed, K.

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛2, reaction heat flow of the second regeneration bed, J/s.

�̇�𝑙𝑜𝑠𝑡2, heat flow lost to the environment of the second regeneration bed, J/s.



𝑇𝑤𝑎𝑙𝑙1, wall temperature of the first regeneration bed, K.

𝐴𝑤𝑎𝑙𝑙2, heat transference area of the second regeneration bed, m2.

𝑉𝑔𝑟𝑒𝑔2,𝐸, gas volume of the emulsion in the second regeneration bed, m3.

𝐶𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸, molar concentration of the 𝑖 component in the emulsion phase, mol/m3.

𝐶𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐵, molar concentration of the 𝑖 component in the bubble phase, mol/m3.

𝜖𝑔𝐸, gas fraction in the emulsion phase.


𝑁𝑖𝑟𝑒𝑔2,𝑜𝑢𝑡𝐸, molar flow of the 𝑖 component at the outer part of the emulsion phase, mol/s.

𝐾𝑟𝑒𝑔2, mass transference coefficient between the emulsion phase and the dense phase, s-

1.

𝑑𝑐, catalyst diameter, m.

𝜌𝑐, catalyst density, kg/m3.

𝜌𝑔, average gas density, kg/m3.

𝑔, gravity, m/s2.

𝜙𝑐, catalyst average sphericity.

𝜇𝑔, average gas viscosity, Pa s.

𝑑𝑏, average bubble diameter, m.

𝑑𝑟𝑒𝑔𝑖, regenerator 𝑖 diameter, m.

𝑘𝑣𝐹𝐺, flue gas slide valve rating factor, kg/(s.Pa)0.5.

𝐹𝐹𝐺, flue gas mass flow, kg/s.

𝑃𝑎𝑡𝑚, atmospheric pressure, Pa.

𝑇𝐹𝐵, temperature of the freeboard, K.

�̇�𝑟𝑒𝑎𝑐𝑡𝑖𝑜𝑛𝐹𝐵, reaction heat flow of the freeboard, J/s.

�̇�𝑙𝑜𝑠𝑡𝐹𝐵, heat flow lost to the environment of the freeboard, J/s.



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B. Subsystem representation of the FCCU dynamic model.

Modeling and Dynamic Analysis of a Fluid Catalytic Cracking Unit...

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