Mathematical Modelling of Trickle-Bed Reactors for Fischer-Tropsch Synthesis Diogo M. Mosteias Thesis to obtain the Master of Science Degree in Chemical Engineering Advisor(s)/Supervisor(s): Dr. ˇ Stˇ ep´ an ˇ Spatenka Prof. Dr. Henrique An´ ıbal Santos de Matos Examination Committee Chairperson: Advisor: Members of the Committee: Prof. a Dr. a Maria Teresa Duarte Prof. Dr. Henrique An´ ıbal Santos de Matos Prof. Dr. Filipe Gama Freire October 2018
Transcript
Mathematical Modelling of Trickle-Bed Reactors forFischer-Tropsch Synthesis
Diogo M. Mosteias
Thesis to obtain the Master of Science Degree in
Chemical Engineering
Advisor(s)/Supervisor(s): Dr. Stepan SpatenkaProf. Dr. Henrique Anıbal Santos de Matos
Examination CommitteeChairperson:
Advisor:Members of the Committee:
Prof.a Dr.a Maria Teresa DuarteProf. Dr. Henrique Anıbal Santos de MatosProf. Dr. Filipe Gama Freire
October 2018
I find the great thing in this world is not so muchwhere we stand, as in what direction we are moving...
OLIVER WENDELL HOLMES
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Acknowledgments
I would first like to thank Prof. Costas Pantelides for the opportunity to develop this project at ProcessSystems Enterprise, Ltd. I would also like to thank Dr. Stepan Spatenka and Eng. Vasco Manacas forbeing available to answer my questions and clear my doubts as well as pointing me in the right direction.
I want to thank to Prof. Henrique Matos and Prof. Carla Pinheiro for letting me know of this oppor-tunity and for taking care of everything that was necessary to put me in contact with the company.
To Renato, Andre, Artur and Tomasz a special thanks for giving us a roof over our heads when wefirst arrive to London without whom it would have been very difficult, if not impossible. To Francisco,Mariana, Joao, Cristian, Lylia, Ali, Sugandha and Piero for making me feel at home during these sixmonths abroad.
A special thanks to my London family, Patrıcia, Alexandre and Mauro with whom I shared our man-sion, gossips, bills and moments that I will always hold dear in my heart.
My greatest gratefulness goes to my loving family, girlfriend Rita and friends for being at my sidethroughout my life and, specially these last five years, where moments of happiness were made happierand moments of sadness and stress were lessen because you were there cheering for me. Speakingof the last five years I want to thank to my friends (and colleagues) with whom I shared the difficultiesand challenges of Tecnico, thank you Ana Jorge, Rita Ribeiro, Sara Sousa, Joao Gil, Ana Carvalho andClaudia Medeiros.
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Abstract
The present work comprises the mathematical modelling of a trickle bed reactor (TBR) for Fischer-Tropsch synthesis (FTS), more specifically the kinetic model, the thermodynamic model, transport prop-erties and heat and mass transfer were developed.
Environmental awareness, as well as the realisation that the natural oil reservoirs are being depleted,has led to an increased interested in the FTS, even though this process is known since the Second WorldWar (WWII). Due to the complexity of the Fischer-Tropsch (FT) process, there has been an increase indemand for mathematical models capable of accurately predicting this process behaviour, thereforethree different models were developed in gPROMS with different levels of detail.
The main development of this work, regarding the FT process, is the kinetic model implementedthat accounts for the different hydrocarbon chain growth probability as opposing to the more commonlyimplemented kinetic models that assume a constant probability regardless of the chain’s carbon number.
The models developed in this work are organised by the level of detail, the first one has the reactorbed axially and radially distributed and the pellet model is also distributed; the second one has thereactor bed axially distributed as well as the pellet; the last and simplest model has the reactor bedaxially distributed but uses a lumped pellet model.
The key performance indicators (KPIs) for the three models were compared for the different modelsin order to assess the level of detail required while still achieving accurate and meaningful results. Bycomparing the axially, radially and pellet distributed model against the axially and pellet distributed modelit was found that this simplification was acceptable for the case study in question with relative errors typ-ically below 0.1%, the case study was based on one of the ARGE reactors from the Sasolburg FT plant[1]. However, when the axially and pellet distributed model was compared to the lumped pellet model itwas found that this major simplification would have consequences in the prediction of the hydrocarbonmolar selectivities as well as in the reactants conversion, with relative errors as high as 59%.
Keywords: Trickle Bed Reactor, Fischer-Tropsch Synthesis, gPROMS, modelling, model orderreduction
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Resumo
O presente trabalho compreende a modelacao matematica de um reator trickle bed para a sıntesede Fischer-Tropsch, mais especificamente o desenvolvimento do modelo cinetico, termodinamico, aspropriedades de transporte e transferencia de massa e calor.
A preocupacao ambiental assim como a tomada de consciencia de que as reservas naturais depetroleo estao a esgotar-se tem levado a um aumento no interesse da sıntese de Fischer-Tropsch (FT),apesar desta ser conhecida desde a segunda guerra mundial. Dada a complexidade do processo deFT tem havido um aumento na procura de modelos matematicos capazes de prever com precisao ocomportamento deste processo. Desta forma, foram desenvolvidos tres modelos em gPROMS comdiferentes nıveis de detalhe.
O principal desenvolvimento deste trabalho, relativamente ao processo de FT, foi o modelo cineticoimplementado que tem em conta as diferentes probabilidades para o crescimento da cadeia de hidrocar-bonetos; em contraste com os modelos normalmente implementados que assumem um valor constantepara esta probabilidade independentemente do numero de carbono da cadeia.
Os modelos desenvolvidos neste trabalho estao organizados por nıvel de detalhe, o primeiro contacom o leito do reator distribuıdo axial e radialmente e o modelo da partıcula de catalisador encontra-setambem distribuıdo radialmente; o segundo modelo possui o leito do reator apenas axialmente dis-tribuıdo e o modelo da partıcula manteve-se distribuıdo; o ultimo e mais simples modelo tem o leitoaxialmente distribuıdo mas usa um modelo agregado para descrever a partıcula de catalisador.
Os indicadores chave de performance dos tres modelos foram comparados entre eles com o intuitode avaliar o nıvel de detalhe necessario mantendo a precisao e significancia dos resultados. Compa-rando o modelo distribuıdo nas tres dimensoes (axial, radial e a escala da partıcula) com o modeloque nao possui distribuicao radial foi observado que esta simplificacao era aceitavel sendo que os des-vios do segundo, relativamente ao primeiro, eram tipicamente abaixo de 0.1%. No entanto, quando osegundo modelo foi comparado com o modelo mais simples foi observado que esta simplificacao temconsequencias graves na previsao das seletividades molares dos produtos assim como nas conversoesdos reagentes, com desvios relativos ate 59%.
Keywords: Reator Trickle Bed, Sıntese de Fischer-Tropsch, gPROMS, modelacao, reducao da ordemdo modelo
D.1 Fitting of the (αβ)g term against catalyst diameter (dp) . . . . . . . . . . . . . . . . . . . . 81D.2 Fitting of the a term against dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82D.3 Fitting of the b term against dp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
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Nomenclature
Acronyms
0D Nildimensional
1D One dimensional
2D Two dimensional
ABC Asymptotic behaviour correlation
AML:FT-FBCR Advanced Model Library for Fischer-Tropsch and Fixed-bed catalytic reactor
AML:TBR Advanced Model Library for Trickle-Bed Reactors
ASF Anderson-Schulz-Flory
CAER Center for Applied Energy Research
CFB Circulating fluidised bed
CSTR Continuous stirred tank reactor
DOF Degrees of freedom
EoS Equation-of-state
EROEI Energy Returned on Energy Invested
FBR Fixed-bed reactor
FFB Fixed fluidized bed
FT Fischer-Tropsch
FTS Fischer-Tropsch synthesis
GHSV Gas hourly space velocity
gPROMS general Process Modelling System
GSA Global System Analysis
GTL Gas-to-Liquids
HTFTS High temperature Fischer-Tropsch synthesis
IP Initialisation procedure
KPI Key performance indicator
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LHHW Langmuir-Hinshelwood-Hougen-Watson
LNG Liquefied natural gas
LPG Liquefied petroleum gas
LTFTS Low temperature Fischer-Tropsch synthesis
Mo1DB0DP 1D bed and 0D pellet (lumped pellet) model
Mo1DB1DP 1D bed (axially distributed) and 1D pellet model
Mo2DB1DP 2D bed (axially and radially distributed) and 1D pellet (radially distributed) model
MOR Model order reduction
OLS Ordinary Least Squares
PSE Process Systems Enterprise, Ltd
PVT Pressure, volume and temperature
RTD Residence time distribution
SA Sensitivity analysis
SMR Steam methane reforming
TBR Trickle bed reactor
VdW Van der Waals
VLE Vapour-liquid equilibria
WGSR Water-gas shift reaction
WWII Second World War
Greek Letters
αn Chain growth probability
β1,...,β5 Temperature-dependent ABC parameters
∆Ei Change in 1-olefin desorption activation energy caused by weak force interactions forevery C-atom
∆Hi Henry constant differential between the ith component and the wax average carbonnumber
δl Thickness of the liquid film
η Effectiveness factor
ηw Wetting efficiency
λg Gas thermal conductivity
λl Liquid thermal conductivity
λer Effective radial thermal bed conductivity
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λger Gas dynamic contribution
λler Liquid dynamic contribution
λser Static contribution
µ Dynamic viscosity
φ Thiele’s module
ε Bed porosity
εg Gas hold-up
εl Total liquid hold-up
ϕLi Fugacity coefficient of the ith component in the liquid phase
Symbols
Ai Pre-exponential factor of the rate constant for the ith step
Bo Bond number
cL Concentration of the liquid phase
dp Particle diameter
DAB Diffusion coefficient
Dref Diffusion coefficient at reference temperature
Ei Activation energy of the ith reaction step
Ei,ads Activation energy of the ith equilibrium step
Fi Molar flowrate of the ith component
Hi Infinite-Dilution Henry’s constant
Hi,0 Henry constant at carbon number 0
htc Heat transfer coefficient
Ki Adsorption rate constant
ki Rate constant
mtc Mass transfer coefficient
Nu Nusselt number
Pi Partial pressure of the ith component
Pr Prandtl number
Rp Particle radius
Re Reynolds number
Sh Sherwood number
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Xi Conversion of the ith component
xi Molar fraction of the ith component in the liquid phase
yi Molar fraction of the ith component in the gas phase
[S] Fraction of vacant sites
c Constant determining chain length dependence
Cmax Maximum number of carbons
n Carbon number
P Pressure
R Ideal gas constant
T Temperature
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Chapter 1
Introduction
Due to the growing energy demand and environmental regulations there has been a chase for cleansynthetic fuel production. Because of that, the interest in Fischer-Tropsch synthesis (FTS) has beenrekindled during the first decade of the 21st century. The Fischer-Tropsch (FT) technology is quite oldwith coal utilisation roots. It was discovered in 1923 by Franz Fischer and Hans Tropsch at the KaiserWilhelm Institute for Coal Research in Mullheim and first applied in Germany in the 1930’s during theSecond World War (WWII). Being a coal-rich country, Germany used FT process in order to produceersatz (replacement) fuels, it accounted for 9% of German’s war fuels production and 25% of the automobile fuel production. However, this technology was expensive and very inefficient and could notcompete against cheap and readily available crude oil. Regardless, the technology was so fascinatingthat research and technology development were carried on even when commercial applications seemedunlikely.[2, 1, 3]
The most interesting fact about FTS is that a gaseous mixture of H2 and CO, also known as syngas,enters the reactor and a hydrocarbon liquid exits it. Thermodynamically, the preferred product wouldbe methane, however the predominant products are heavier hydrocarbons. The FTS has syngas asfeedstock that nowadays comes mostly from natural gas through the steam methane reforming (SMR)process1.[2, 1]
The conversion of natural gas to hydrocarbons (Gas-to-Liquids (GTL) route) is currently one of themost promising topics in the energy industry due to the possibility of using remote natural gas elsewhere,to produce clean fuels, specialty chemicals and/or waxes. On the other hand, coal and heavy residuescan be used on sites where these are available at reduced costs. The products from the FTS can thenbe used to produce value added products and/or fuels.[2, 1, 4]
Nonetheless, the decision on whether to build a FT plant or not lies in the assessment of the riskperceived from the future price and availability of petroleum crude oil as well as on local politics. FTScomplexes need a heavy capital investment specially the section of production of purified syngas be-cause its composition has to match the ratio of H2/CO used for this process, which depends on productselectivity.[5]
The FT process has suffered ups and downs throughout the years, in 1938 there were nine plants inoperation in Germany that had a combined capacity of about 660 thousand tons per year which ceasedoperation due to the WWII; In 1950s another plant was built in Brownsville, Texas based on syngasfrom methane with a capacity of 360 thousand tons per year that had to be shutdown due to a steepincrease in the methane price; Another example is the construction of a plant in Sasolburg, South Africahowever, before it was even finished, the huge oil fields of the Middle East were discovered which ledto a decrease in the oil price and a, temporary, disinterest in FTS. In any case, some of these plants
1Syngas can also come from coal burning but the steam methane reforming is most common pathway.
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continue to operate nowadays, as referred in 3.2.6.[5]
The quality of FT products is excellent and their environmental properties are being recognized asvery valuable in the ongoing drive towards cleaner fuels and engines.[1]
The developments of GTL technology can be group into the three E’s categories (Energy security,Ecology and Economy):[6]
1. Energy Security
• It consists in an effective use of untapped natural gas resources and diversification of fuelresources assuring substitutes for crude oil;
• It will reduce the world’s dependence on the Middle East oil fields;• It will reduce the GTL monopolization and cost controls by major oil companies overseas.
2. Ecology
• It provides clean resources of flue gases from sulphur and aromatic sweet-free fuels;
• It promotes the use of highly efficient diesel-powered vehicles (with low carbon dioxide gasdischarge) linked to GTL light oil;
• It reduces and effectively uses the associated gases by flaring in oil and gas producing coun-tries.
3. Economy
• It helps to develop projects and contributes to technology through independent and superiordomestic technologies;
• It promotes the development of domestic companies’ gas fields.
1.1 Motivation and Purpose
The performance of the FTS is intimately related to the feed gas composition, catalyst formulationand operating temperature. Moreover, the catalyst used experiences chemical and/or physical changeswhich complicates the reactor design and optimisation. For example for an increase in the operatingtemperature: 1. Methane formation is favoured 2. Deactivation of the catalyst and coke formation is alsofavoured 3. The average chain length of the hydrocarbon products is reduced.
On the other hand, with the raising of the temperature, the rate of reaction is improved as well asthe quality of the steam produced by the reactor’s cooling system, which are both desirable. For thisreason, the application of FT technology involves a great number of constraints and variables prone tooptimisation.[1]
Due to the recent improvements of the technology and the realisation that it can be used to obtainvalue from remotely produced natural gas i.e. stranded natural gas can be converted to liquid hydrocar-bon products to be sold in worldwide markets, this is a different approach to the conventional process ofliquefaction of natural gas that produces liquefied natural gas (LNG).[1]
The modelling of trickle bed reactors (TBRs) is a key step for the optimisation and control of existingplants as well as for the successful scale-up of this technology. The complexity of these models liesin the fact that it’s a three-phase system, with coupled phenomena of catalytic reaction, vapour-liquidequilibria, heat and multi-component mass transfer phenomena and very complex hydrodynamics.
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1.2 Outline
This thesis is arranged as follows: in Chapter 2 a background regarding the work in some countrieson the FTS, the possible reactor’s configuration used as well as alternative routes for the production offuels and chemicals; in Chapter 3 the literature review is presented which is divided into two differentmain sections: 1. Fischer-Tropsch synthesis 2. trickle bed reactors which cover the relevant key pointsof each section; in Chapter 4 the software, the libraries and the flowsheeting models used to set up themodel for this work. Also in this chapter, it is explained, step by step, what was developed in this workregarding each building block of the TBR model for the FTS and their respective results; in Chapter 5a key point is analysed and that is the model order reduction (MOR) and in what extend it is viable toreduce the models’ order; in Chapter 6 is present a sensitivity analysis on some parameters that areprone to changes whether it be while modelling the process or during operation on the site; Finally,Chapter 7 presents the conclusion of this work as well as some future work.
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Chapter 2
Background
Fischer-Tropsch (FT) reactional system is known since the early 20th century thus much researchand development have been done towards a more efficient and economical way to carry this reactiondue to its importance in the energy market.
This chapter is aimed at giving an overview on the work done in some countries in Europe regardingFischer-Tropsch synthesis (FTS), a brief explanation on how certain types of FT reactors operate as wellas some alternative routes for fuels and chemicals production.
2.1 FTS work in Germany
Soon after the discovery of the FT reaction, conversions in the liquid phase were initiated. Thedevelopments progressed from the initial fixed-bed at atmospheric pressure to a bubble column reactorat atmospheric pressure and then to a large medium-pressure pilot plant that was operated during 1940-1950 period. [1]
The six majors reactors before the Second World War (WWII) were as follows:[1]
1. Fixed-bed reactor (FBR) with internal cooling operating at high conversion. The catalyst waspacked in a rectangular box and water-cooled tubes fitted with cooling plates at short distanceswere installed in the bed to remove the reaction heat - Normaldruck Synthese;
2. Multi-tubular reactor with sets of double concentric tubes in which the catalyst occupies the annularspace, surrounded by boiling water. This type of reactor was operated at medium pressure -Mittedruck Synthese;
3. Adiabatic FBR with a single bed, large recycle of hot gas which was cooled externally - IG-Farben/Michael Verfahren;
4. FBR with multiple adiabatic beds, inter-bed quenching with cold feed gas, recycle of hot gas andexternal cooling - Lurgi Stufenoven;
5. Adiabatic FBR with large recycle of heavy condensate passing in upflow through the bed. Theliquid recycle stream was cooled externally - BASF/Duftschmid Verfahren;
6. Slurry reactor with entrained solid catalyst, large recycle of hot oil and external cooling - BASFSchaumverfahren.
The developments prior and during the WWII led to reactors with increased potential for large scaleapplications.[1]
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2.2 FTS work in The Netherlands
International Hydrocarbon Synthesis Co (N. V. Internationale Koolwaterstoffen Syntheses Maatschap-pij, The Hague, The Netherlands) was reported to employ a form of recycle technique that used vapour-phase synthesis in the first stage followed by a liquid-phase operation that utilised a catalyst suspendedin oil, the process had a product condensation step in between the two reactors. This company heldmany FTS patents during 1930s but only a few referred to liquid-phase synthesis processes.[1]
2.3 FTS work in United Kingdom
The Fuel Research Station at Greenwich first used 50 litres fixed-bed reactors. In 1947 beganworking on the fluidized bed reactor and in 1949 work began on the liquid-phase technique. The threetypes of reactor were compared and the decision was to build a pilot plant in Greenwich to produce 385L of product per day at 20 atm and 300°C, this plant was later moved to Warren Spring Laboratory in1958. A few problems were found during the operation of this plant and it was finally terminated uponthe discovery of plentiful supply of petroleum on the Middle East.[1]
2.4 Fischer-Tropsch Reactors
2.4.1 Slurry reactors
During the 50’s and 60’s decades of the twentieth century various sizes of slurry reactors weretested in Germany, England and USA, however the space velocities used were very low and thus theperformance for commercial situations could not be judged.[1]
The slurry reactor system was considered to be suitable for the production of wax at low temperature.The wax is the continuous phase inside which the finely divided catalyst are suspended. Nonetheless,an efficient way of separating the product from the catalyst was only developed much later and firstdemonstrated by Sasol in 1990.[1]
This implementation was first tested in a 1 m internal diameter demonstration unit and, in 1993, in a5 m internal diameter one and, finally, a 22 m high commercial unit was brought on-line and has beenworking ever since, with a capacity of 100 thousand tonnes per year.[1]
Figure 2.1: Slurry phase reactor.[1]
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2.4.2 Fixed bed reactors
The use of vertical spaced packed bed and radial flow reactors with cooling between the beds isnot satisfactory due to negative effect of the increased temperature within each adiabatic bed. Thealternative is using a multi-tubular reactor with the catalyst placed inside the tubes and the coolingmedium (usually water) on the shell side, very similar to a Shell&Tube heat exchanger configuration. Thisreactor set-up ensures an improved heat transfer rate from the catalyst particles to the cooling medium,as long as the reactor is operating at high gas linear velocities, thus ensuring turbulent flow, and a shortdistance between the catalyst particles and the tube walls is used (narrow tubes diameters).[1]
The reactants conversion is improved with the reduction of the catalyst particle size, however this willlead to unacceptable high pressure drop throughout the reactor. Depending on the catalyst being used(either iron or cobalt) the temperature profiles will have a, more or less, pronounced effect on the reactorperformance. Figure 2.2 is a diagram of an ARGE reactor used in Sasolburg, they have 3 m of diameterand contain 2050 tubes with 5 cm of internal diameter and are 12 m long.[1]
Figure 2.2: ARGE reactor installed in Sasolburg.[1]
This reactor configuration is discussed in more detail in section 3.2.
2.4.3 Two phase fluidized bed reactors
In contrast to the previous reactor configurations, the two phase fluidised bed reactors can be oper-ated at high temperature FT processes. There are two types of fluidised bed reactors, the fixed fluidisedbed (FFB) - figure 2.3a - or the circulating fluidised bed (CFB) - figure 2.3b. The former has been calledSasol Advanced Synthol (SAS) reactors by its developers.[1]
Due to their turbulence, they can cope with the huge amounts of heat released from the reactions athigh conversions with high feed gas throughputs. However, the beds are isothermal and the temperaturedifferential between the top and bottom of the reactor are of only a few degrees.[1]
These reactors must be operated in such a way that ensures that the selectivity towards long chainhydrocarbons is limited to avoid excessive condensation of liquids in the pores, this would result inparticle’s outer surface to get wet resulting in the de-fluidisation of the bed ceasing the normal functioningof the unit.[1]
Figure 2.3: The two types of two phase fluidised bed reactors: a) SAS reactor and b) CFB reactor
2.5 Alternative routes for the production of fuels and chemicals
Nowadays the major source of the world’s fuels and commodity chemicals are of carbon containingraw materials, in other words, fossil resources. In 2004, the relative reserves of carbon to oil were asfollows:[1]
Table 2.1: World’s relative reserves of carbon to oil, in 2004.
Source Reserves (oil equivalent)
Crude oil 1.0Tar sands 0.7Shale oil 1.2Natural gas 1.5Coal 26
Crude’s reserves-to-production ratio are estimated to last for another 40-55 years. However, forpolitical and economical reasons, alternative sources of carbon containing materials should be used toa greater extent in a very close future. From table 2.1 one can note that coal has the largest reservesof all the carbon sources, thus in the long term, it might become the main source for production of liquidfuels and commodity chemicals. [1, 7]
Regarding shale oil, the majority are fine-grained sedimentary rocks containing relatively large amountsof organic matter (also known as kerogen) from which significant amounts of shale oil and combustiblegas can be extracted. The United States of America has the largest reserves of, technically recoverable,shale oil thought to have 1.5-2.6 trillion barrels (240-410 thousands of millions of cubic meters). Theprofitability of shale oil extraction lies with the energy costs of extraction (mining and processing) againstthe energy produced by the shale oil, this ratio is called Energy Returned on Energy Invested (EROEI)in 2006 the Royal Dutch Shell reported an EROEI of 3 to 4 on its in-situ development in the MahoganyResearch Project, which looks promising.[8, 9]
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Chapter 3
State of the Art
The analysis on the State of the Art was performed in two phases; the first one was focussed onthe Fischer-Tropsch (FT) technology whereas the second one was focussed on reactors capable ofconducting the Fischer-Tropsch synthesis (FTS) efficiently.
Regarding FT technology, this chapter goes through two possible mechanism for the FTS, the mostrecent kinetic models developed, the catalyst used and their advantages and disadvantages as well astheir deactivation. Regarding the trickle bed reactors (TBRs), it goes through different phenomena thatoccur such as hydrodynamics, heat and mass transfer, transport properties and vapour-liquid equilib-rium; it is also made a brief reference to commercial FT plants and recent reactor developments.
3.1 Fischer-Tropsch Synthesis
The FTS is extremely dependent on the syngas composition, hence the hard requirement of prepar-ing it carefully. The syngas is prepared from carbonaceous feedstock, the only essential requirement isthat the feed contains carbon and, preferably hydrogen. Otherwise the latter has to be obtain from thescission of the water molecules which requires large amounts of energy in order to be achieved.[1]
CO, CO2 and H2 are produced through steam methane reforming (SMR), where the CO2 must beremoved by its total or partial recycling back to the reformer. The CO2 formed during FTS can alsobe recycled back to the reformer in order to reduce the fresh feed of natural gas to the whole process[1]. Technologies such as autothermal reforming, pressure swing adsorption (PSA), methanation, watergas-shift (WGS) can be used to obtain the FTS desired H2/CO ratio, that cannot be obtain through SMRalone.
FTS is a catalytic reaction in which syngas is converted into a broad range of hydrocarbons.
nCO + 2 nH2 CnH2n + nH2O (3.1)
(2n+1)H2 + nCO CnH2n+2 + nH2O (3.2)
The equations (3.1) and (3.2) represent the reactions for the formation of olefins and paraffins, respect-ively [10]. The side reactions are as follows:
2nH2 + nCO CnH2n+2O + (n-1)H2O (3.3)
2 CO C + CO2 (3.4)
CO + H2O CO2 + H2 (3.5)
Where the equations (3.3), (3.4) and (3.5) represent the formation of alcohols, the Boudouard reaction
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for the complete oxidation of CO to CO2 and the water-gas shift reaction (WGSR), respectively.[10]
This process is carried out, mainly, at two temperature ranges:
• 300 to 350°C - High temperature FTS (HTFTS) in the presence of iron-based catalyst used forthe production of gasoline and linear low molecular mass olefins;
• 200 to 250°C - Low temperature FTS (LTFTS) in the presence of either iron or cobalt-basedcatalyst used for the production of high molecular mass linear waxes.
The advantages of the cobalt-based is that it has a high activity and selectivity towards C5+ com-pounds and is not very active towards the WGSR when compared to the iron-based catalyst.[11]
Over cobalt-based catalyst, the FTS produces mostly n-alkanes and 1-alkenes (equations (3.1) and(3.2)). As previously explained, this catalyst is not very active towards the WGSR, therefore only a smallfraction of the water produced is converted to CO2.[11]
FTS consists in a complex reactional system that involves a large number of product species andsurface intermediates. There’s also the difficulty of a three phase operation, possible mass and heatresistance, deactivation of the catalyst, etc. Additionally, there is no consensus regarding the exactmechanism of the reaction. There are two main proposed mechanism for the FTS, the carbide formationand the CO-insertion mechanism.[2]
The first one was proposed by Fischer and Tropsch and later modified by Brady and Petit. It consistson the formation of the hydrocarbon by hydrogenation of a metal carbide forming a C1 monomer, whichis then polymerized. However, there were several studies that disbelieved this mechanism leading to theproposal of a series of other mechanisms from which emerged the CO-insertion that could explain theformation of all typical FTS products.[2]
3.1.1 Carbide formation mechanism
The CO activation consists on a direct CO dissociation, meaning that the bond between carbon andoxygen is broken before the hydrogenation of the first. There is also an alternative carbide formationmechanism where the hydrogen assists in the CO bond scission. In figure 3.1 are represented bothpathways.
3.1.2 CO-insertion mechanism
The CO-insertion can be summarized in the following steps, the chemisorbed CO is the monomerand the chain initiator is thought to be a surface methyl species. The chain growth takes place in ametal-alkyl bond leading to a surface acyl species. The elimination of oxygen from the surface leads tothe formation of other alkyl species. The detailed initiation step is represented in figure 3.1 and the othersteps are more detailed in figure 3.2.[1]
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Figure 3.1: Possible CO activation pathways: a) Carbide mechanism; b) Carbide mechanism withH-assisted dissociation; c) CO-insertion mechanism.[2]
Figure 3.2: Detailed CO-insertion mechanism.[1]
Assuming that the hydrocarbon chain is formed by addition or insertion of C1 intermediates withconstant growth probability then the chain length distribution is given by the Anderson-Schulz-Flory(ASF) distribution. However, for all FT catalysts, deviations from the ideal distribution are observed.[12]
3.1.3 Anderson-Schulz-Flory distribution
The ASF distribution is a probability distribution that describes the relative ratios of polymers ofdifferent lengths that are formed in an ideal step-growth polymerization process. The probability massfunction can take the following form:
fa(k) = a2k(1− a)k−1 (3.6)
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Where k is a variable that characterises the chain length and a is an empirically determined constantrelated to the fraction of unreacted monomer. Equation 3.6 is a solution of equation 3.7 [13]
(a− 1)(k + 1)f(k) + kf(k + 1) = 0,
f(0) = 0,
f(1) = a2
(3.7)
Equation 3.6 implies that shorter polymers are favored over longer ones. This probability distributionis relevant to the FT process, since it is conceptually related in the way that lighter hydrocarbons areconverted to heavier hydrocarbons.[13]
However, the FT process has some deviations to the ideal ASF distribution namely C1, C2 and C10+
components. Typical deviations include higher C1 selectivity, lower C2 selectivity (and to a lower extentalso for C3) and higher selectivities for long chain hydrocarbons [14]. A more extensive analysis on thepossible causes for these deviations can be found in [14].
3.1.4 Reaction kinetics
Several kinetic models have been developed trying to accurately predict such a complex reactionalsystem. One of the most recent is the Stamenic et al. [4] model that combines the kinetic expressionsfrom three other authors, namely 1. Yates-Satterfield [11] for the CO consumption 2. Ma et al. [15] for theCH4 formation 3. Todic et al. [2] for the production of the other hydrocarbon species.
Yates-Satterfield kinetic model
The expression for CO consumption rate proposed by Yates et al. [11] is in the form of:
−rCO =kP aCOP
bH2(
1 +∑i
KiPciCOP
diH2
)2 (3.8)
Where k is the kinetic parameter, a and b are the reaction orders and Ki represents the adsorptionparameter of the ith term1.
All forms of kinetics presented in the article [11] have subtle differences in the functional dependenceon PH2
and PCO, so experiments were run over a broad range of H2/CO ratios in order to avoid acovariance between these two variables. It was found that the reaction rate has a first order dependenceon the H2 partial pressure and it is therefore eliminated from the denominator while fixing parameter b to1, as follows:
−rCO =kPCOPH2
(1 + aPCO)2(3.9)
Where the adjustable parameters are the surface rate constant and the adsorption coefficient, hencethe Yates-Satterfield kinetics takes the form of a Langmuir-Hinshelwood kinetics.
Ma kinetic model
Ma et al. [15] proved that the kinetic model from the Center for Applied Energy Research (CAER),which takes into account the inhibition effect of water on a cobalt-based catalyst, is adequate to describethe experimental data. It also shows that methane formation can follow different pathways including
1The denominator is squared due to the assumption that the reaction follows a bimolecular surface mechanism.
12
methanation sites on cobalt catalyst. The kinetic expression proposed is as follows:
rCH4=
kMPaMCOP
bMH2(
1 +mMPH2O
PH2
) (3.10)
Todic kinetic model
The model developed by Todic et al.[2] describes the formation of hydrocarbon species based on theCO-insertion mechanism, assuming that the chain grows by addition of monomers to it (as described in3.1.2). This addition is based on the probability of the next polymer being formed from the current chain2,figure 4.9. The elementary steps of the CO-insertion mechanism used in this article are in Appendix A.
This model is based on the Langmuir-Hinshelwood-Hougen-Watson (LHHW) methodology, this ledto a more complex form of kinetic model but an improved fit of experimental data when compared to thecarbide mechanism.
3.1.5 Catalyst
The typical commercial catalysts used for FTS are: 1. fused iron catalyst; 2. precipitated iron catalyst;3. supported cobalt catalyst. Iron catalyst are better suited to be used with coal derived syngas, howeversince the most common process to obtain synthesis gas is the SMR this work will focus on the cobalt-based catalyst.
Modern cobalt catalysts are prepared by its depositing on a pre-shaped oxide support, usually silica,alumina, titania, zinc oxide or combinations of these oxides.
The support is prepared to meet the desired particle size through spray drying and, but not neces-sarily, a refine step of size distribution to be used in slurry reactors. For the preparation of the catalystsfor fixed-bed reactor (FBR) are usually used extrusion techniques.
The mechanical strength is improved by thermal treatment and the pore size is also controlled in orderto maximise the deposition of the metal on the support enhancing the performance of the catalyst. Themetal is then impregnated onto the support along with promoter metals such as lanthanum, platinum,palladium, rhenium and ruthenium. The impregnated support is then dried and reduced with hydrogen.
With the better understanding of the importance of the geometry of the catalyst in its activity, moderncobalt-based catalysts have a selectivity towards methane of only about 5%.[1]
3.1.6 Catalyst deactivation
Catalyst deactivation is the loss over time of catalytic activity and/or selectivity. This is a problem thatis present in every catalysed process, thus it has been the focus of numerous studies. The time-framein which a catalyst deactivates can vary considerably, it can take seconds to deactivate (the case of thecatalytic cracking) or take years (the iron-based catalyst takes between 5-10 years to deactivate in theammonia synthesis process). The main objective of studying the deactivation of catalyst is to developmodels for the design of stable catalysts and optimise processes in order to avoid or slow down catalystdeactivation.[16]
The factors involved in the deactivation of the catalyst during FTS are as follows: [1, 16]
• The presence of high molecular weight waxes and/or aromatic coke precursors in the catalystpores, resulting in diffusion limitations;
• Fouling of catalyst surface by coke deposits;
2”Next” and ”current” polymer/chain is related to their carbon number, e.g. if the current chain has 25 carbons in its structurethe next polymer will have 26 carbons.
13
• Poisons in the feed gas such as H2S and organic sulphur compounds;• Hydrothermal sintering;• Oxidation of the active metal to the inactive oxide;• Boudouard carbon deposition.
Poisoning
Poisoning of a catalyst consists in a very strong chemisorption of reactants, products or impuritieson active sites.
Poisoning by sulphur has a series of problems [16]: 1. a single atom of adsorbed sulphur physicallyblocks up to four active sites and three or four topside sites on the metal surface ; 2. due to its strongchemical bond, it electronically modifies its neighbouring metal atoms, modifying their abilities of ad-sorbing and/or dissociating reactants/products; 3. by addition of a strongly adsorbed poison, structuralchanges in the catalyst may happen which would inhibit reactions that are dependent on the structureof the catalyst; 4. an adsorbed poison also blocks the access of two or more reactant to each other; 5. italso prevents or slows the diffusion of adsorbed species.
For the FTS the most common poisons are: H2S, COS, As, NH3 and metal carbonyls.[16]
Regarding cobalt-based catalyst is usually observed that, during the first hours, the decline in the FTSrate is relatively rapid and then declines much more slowly. Being the common operating temperaturesaround 220°C it’s unlikely that the deactivation comes from deposition of coke or fouling. One plausiblereason for the activity decline is the build up on the surface and in the catalyst pores of very long waxchain that inhibits adsorption and slow down diffusion rates.[1]
Contrary to what happens with iron-based catalyst, where its deactivation can be compensated bycontinuous removal of catalyst and addition of fresh one due to its cheap price, the same cannot be saidabout the cobalt-based catalyst, due to their relatively high price. In this case, it becomes imperativethe minimisation of loss of activity of the catalyst to ensure a long on-stream lifetime. Poisoning due tosulphur will cause permanent deactivation which makes the purification of the syngas a key point of theprocess.[1]
It has also been demonstrated that the presence of water in the syngas feed results in the surfaceoxidation of the catalyst3 this effect seems to be related with the type of support and also on the level ofwater vapour pressure inside the reactor[1]. Due to the high activity of cobalt catalysts, the ratio H2O/H2
will be higher and at 90% conversion the ratio will be, approximately, 4.2 and it is known that these highratios accelerate the deactivation rate due to enhanced sintering rate, oxidation of smaller cobalt crystaland the formation of inert compounds with the catalyst support material.[1]
It seems that, by removing poisons and by operating the reactor at desirable values of water vapourpressure, one can prolong the lifetime of the catalyst. The cobalt metal can be recovered from spentcatalyst and re-used to produce fresh catalyst after been treated with aqueous nitric acid.[1]
3.2 Trickle Bed Reactors
TBRs, sometimes referred to as tubular fixed bed reactors, are comprised of a packed bed of catalyston which the reactants, both gas and liquid, flow in a co-current or counter-current flow - figure 3.3.These reactors represent an important class of multiphase reactors capable of carrying out gas-liquidreactions in the presence of a solid catalyst. One of their main advantages is the easy separation of theproducts from the catalyst.
3Tested on laboratory scale using Co/Al2O3
14
This type of reactor has a wide range of applications, namely in the petroleum industry on processessuch as hydrodesulfurisation of heavy oils and gasoline, hydrodenitrogenation, hydrotreating, hydro-cracking and demetallisation. Trickle-beds are also used in chemical processes mainly in partial oxida-tions and hydrogenation of certain organics with or without the presence of a supported metal catalyst,waste water treatment is also another application of these kind of reactors and biochemical processesare also in the grasp of TBRs where immobilized cells or enzymes promote the reactions between dis-solved liquid substrates and oxygen, e.g. the trickling bed filter used in waste water treatment.[17, 18]
Despite of all the above mentioned applications, the only main usage focus is in the petroleumindustry. The main drawbacks of using commercial size TBRs lies on difficulty of scale-up and designfrom bench and pilot-scale as well as high pressure drop and heat removal, the latter being an importantmatter due to the high exothermicity regarding FTS.[4, 17]
Figure 3.3: Representation of several TBR designs: a) adiabatic co-current, b) adiabatic countercurrent, c) jacketed and d) internally cooled.[19]
3.2.1 Hydrodynamics
The flow inside randomly packed catalyst particles is subject to the path which offers the least resist-ance. The outcome of this is an inhomogeneous flow in velocity as well as phase distribution.
Due to the characteristics of flow inside packed bed reactors it’s very difficult to determine a rangefor laminar flow and turbulent flow. TBRs can be operated at four different flow regime - figure 3.4:
At low gas and liquid flow rates, the interaction between both phases is small and trickle flow regime(or low interaction regime) is observed. At moderate gas and liquid flow rates, the interaction betweenphases increases and the liquid-phase occupies the entire flow cross-section and pulse flow regime (orhigh interaction regime) is observed. The other two regimes may occur at higher gas and liquid flowrates but are less common industrially.[19]
Industrial reactors are, usually, operated in the transition condition between trickle and pulse flow inorder to maximise mass/heat transfer, catalyst utilisation and by doing so, enhancing production.[19]
15
With gas flow rate fixed one can notice that, with the increasing of liquid flow rate, the liquid hold-upalso increases to an extent that it can completely block the gas passage in which case disturbances inthe reactor will be felt and they might grow along it’s length. When these grow to a measurable scale apulse flow regime is achieved.[19]
The other way around also causes disturbances in the reator, the shear stress of the gas flow rateon the liquid also generates blockage which can also lead to a pulse flow regime.[19]
Figure 3.4: Possible flow regimes in trickle bed reactors.[19]
3.2.2 Heat transfer
Most reactions carried out in TBRs are exothermic, therefore the heat removal is essential to avert thecatalyst deactivation/sintering as well as guarantee a safe operation of the equipment. Besides, TBRsare also prone to temperature runaway conditions due to poor transfer rates to its surroundings.[19]
The overall heat transfer phenomena can be divided into four different levels:• Intraparticle heat transfer;• Particle-to-fluid heat transfer;• Radial heat transfer through the catalyst bed;• Heat transfer from the bed to the reactor’s wall.
Particle-to-fluid heat transfer coefficient
This is a function of the velocity of both liquid and gas and their thermal and transport properties aswell as the particle’s thermal properties. This coefficient is independent of the reactor diameter. Dueto the complex interaction between parameters in this class of equipment, this coefficient cannot bedetermined by first principle methods; instead one has to use empirical correlations, namely dimension-less analysis. Therefore, equation (3.11) is used to relate the heat transfer coefficient with the operating
16
parameters:
Nu = aRebPr1/3 (3.11)
Where the parameters a and b are to be estimated experimentally, Nu is the Nusselt number, Re is theReynolds number and Pr is the Prandtl number [19]. Bœlhouwer et al. [20] found that the heat transferrate increases with both liquid and gas flow rates but it’s far more noticeable for increases in the liquidvelocity.
The Effective Transport Concept
One dimensional models neglect the resistance to heat and mass transfer in radial direction andtherefore predict uniform temperatures and conversions in the cross section.[21]
This has severe consequences when reactions with a pronounced heat effect are involved. For suchcases, there is a need for a model that predicts the detailed temperature and conversion pattern in thereactor, thus avoiding eventual detrimental of hotspots in the radial axis.[21]
Bearing this concept in mind, when the effective radial thermal bed conductivity (λer), is determinedfrom heat transfer experiments in packed beds, it is observed that λer decreases strongly in the vicinityof the wall. It is as if a new resistance appeared near the wall, which might have to do with variations inthe packing density and flow velocity.[21]
Depending on the model’s level of detail one of two approaches can be used:1. An overall heattransfer coefficient for 1D bed models or 2. use the λ-α(lambda-alpha) approach for radial heat transferin 2D bed models, by calculating the λer and the heat transfer near the wall (htcws) separately. Thereis also another approach for the 2D bed model called the λ-r (lambda-r) approach, where the thermalconductivity of the bed is a function of the radial position (it decreases towards the wall).[21]
The second approach is preferable when it is important to predict temperature values with thegreatest possible accuracy. De Wasch and Froment published data that is believed to have a higherdegree of precision needed for these predictions. Correlations (3.12) are for air.[21]
λer = λoer +0.0025
1 + 46(dpdt
)2Re (3.12)
htcws = htcows +0.0115dt
dpRe (3.13)
Where dt is the diameter of the tube. λoer and htcows are static contribution dependent on the type andsize of the catalyst. Since these equations are for air, some parameters were estimated and changed inorder to use these correlation for FTS.[21]
The correlation for htcws is completely different from other previously published but confirms withtheoretical calculations.[21]
3.2.3 Mass Transfer
Trickle bed operation does not have rigorous mixing mechanism like other multiphase catalytic react-ors, therefore mass transfer rates are lower than said reactors and it can become the rate-limiting stepof the process.
There are three types of mass transfer rates that are relevant for TBRs: 1. inter-phase mass transfer(gas to liquid) 2. external mass transfer (liquids to solids or gas to solid in dry zones) 3. intra-particlemass transfer.
17
Gas to liquid are dependant of several system parameters such as particle diameter, liquid and gasflow rates, properties of fluid and the reactor operating conditions. The mass transfer rate increases withdecrease of particle diameter. Besides that, mass transfer rate also depends on the gas-liquid flow ratesbecause these enhance the interaction between both phases as well as spreading the liquid leading to ahigher interfacial area. Some correlations have been proposed to account for one or both of the previousdependencies.
Generally, liquid to solid mass transfer depends on the extent of liquid contacting the solid surface,with the increase of the liquid flow rate, the wetting efficiency is improved and the mass transfer rate isalso enhanced, due to this there are a few correlations that account for the wetting efficiency - equation(3.14).
Liquid to solid mass transfer is very sensitive to particle size and increases with its decrease. It ismore sensitive to liquid flow rate when compared to gas flow rate and increases with the flow rate, butthat effect is more noticeable at lower flow rates than higher ones.
ηwSh = aRebPr1/3 (3.14)
Where the ηw is the wetting efficiency and Sh the Sherwood number, and the a and b take valuesreported in the literature[19]. However, in the specific case of FTS since the liquid is produced inside thecatalyst pellet there is not much concerned with the wetting efficiency and it will not severely affect theresults.
Intra-particle mass transfer
The FTS is a complex catalytic process where a network of parallel and consecutive reactions takeplace within the pore filled with waxy liquid hydrocarbon product, consequently severe intrapellet diffu-sion limitations might cause CO and H2 concentration gradients and shift away the reaction equilibriumfrom the production of species that are formed under gradientless conditions as well as decrease thereaction rates.[22]
Some studies claim that FTS rates are proportional to H2 concentration and independent of COconcentration, these assumptions have led to simple models that conclude that H2 is the diffusion lim-ited reactant and so neglecting the effect of CO concentration gradients [22]. However, in reality theFTS is described by Langmuir-Hinshelwood kinetics, as described in 3.1.4, which has a negative orderfor carbon monoxide [15]. The kinetic order for CO tends to become increasingly negative for lighterproducts, as a result, any decrease in this reactant concentration selectively favours the formation oflighter components.[22]
Experimental data shows that the larger reactant (CO) exhibits the more severe intrapellet concen-tration gradients and is the diffusion limited reactant. The intrapellet diffusion limitation not only slowsdown the arrival of reactants at the catalytic sites, but it also influences the removal rate and second-ary reactions of reactive FTS products, such as α-olefins. The decreased rate at which α-olefins areremoved from the catalyst pores allows for longer residence times inside the particles which leads toheavier and more paraffinic products.[22]
Despite external mass transfer in FTS has been deeply studied, one needs reliable values of diffusioncoefficients of both reactants and products inside the FT wax at operating pressures and temperatures tobetter model what happens inside the reactor and inside the catalyst particles. Erkey et al. [23] measuredthe diffusion coefficients through the Taylor dispersion method at three different temperatures (202, 231and 263°C) and at 1400kPa and the experimental data fitted very well when compared with the predictedvalues.
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3.2.4 Vapour-liquid equilibria
Being the TBR a three phase reactor, vapour-liquid equilibria (VLE) plays an important role in thesynthesis of hydrocarbons via FTS. The reactants are supplied in the gas phase, however the reactionoccurs at the catalyst surface based on liquid concentrations. These liquid concentrations are given bythe VLE.
The reaction products are distributed between the vapour and liquid phase; the lighter componentsand the unreacted syngas are in the vapour phase whereas the heavier components are in the liquidphase.
Early models almost completely ignored the liquid phase-composition which has a major influence inthe reactor performance. Recent investigations have applied equation-of-state (EoS)-based methods tocalculate the VLE for FT systems, one of them is the Peng-Robinson EoS that was used to predict theflow rates and composition of both liquid and vapour leaving the reactor.
Marano et al. [24] developed a model that accurately predicts the phase separation and compositionpresented by other authors for three different FT reactor’s effluents. It has also shown to give betterpredictions than those based solely in EoS formulation presented throughout the literature.
3.2.5 Models for Fixed-bed reactors for Fischer-Tropsch
When addressing the modelling of a three phase reactor such as FBRs one must take into accounta series of constraints and limitations due to the presence of different phases inside the reactor.
Most of the studies towards FBR for FTS use pseudohomogeneous models, meaning that thermalequilibrium is negleted. To account for diffusion limitations an effectiveness factor (η) is employed,which is calculated through the Thiele’s module (φ), using simple kinetics for CO consumption. Suchstudies reveal that this effectiveness factor is relatively lower than 1 which shows a negative influenceof diffusion limitation. Whereas heterogeneous models accounts for concentration and temperaturedifferences between the bulk gas phase and the catalyst particle surface that results from inter-particlemass and heat transfer resistances. [1, 4]
The decision on whether to use a two dimensional (2D) approach or a one dimensional (1D) is basedon the necessity of accurately predict radial distribution of temperature and composition or if assuminga uniform or average profile will suffice the modelling requirements. When considering exothermic, orhighly exothermic reactions, one must take into account the radial temperature gradients that will happeninside the tubes and use the former approach or the latter, accordingly.
A recent model for the FTS in a FBR have been developed by Stamenic et al. [4] where is developeda new 1D multiscale and multiphase mathematical model at the reactor scale and the particle scale dueto their effect on the performance of a FBR for FTS.
This model, when compared to others from the literature, has fewer simplifications and phenomenasuch as kinetics, mass/heat transfer and phase equilibria have been taken into account. Liquid hold-upand pressure drop are also calculated through momentum balance equations, instead of using empiricalcorrelations for this purpose.[4]
Regarding heat removal, different studies have employed a 2D pseudohomogeneous models to ac-count for radial gradients of concentration and temperature inside the reactor. These studies were aimedat providing a better representation of temperature profiles in order to predict hot spot formations. Gen-erally, 1D models are appropriate for narrower tubes, since the liquid-filled pores and particle sizes (≥1 mm) guarantee a relatively low volumetric reaction rate and inter-particle mass and heat transfer arenegligible. For typical dimensions of particle sizes (≥ 1 mm), tube diameters (2.5 - 5 cm), tube lengths (6- 12 m) and superficial velocities (≥ 0.2 m/s), the Bond number (Bo) is very large and plug-flow can beassumed (no axial mixing) whereas for tubes with a diameter above 5 cm, a 2D should be the choice to
19
account for the radial mixing [1, 4]. However, in the majority of the experiments, non-reacting conditionsare used and so the solid and fluid temperatures are almost the same and pseudohomogeneous modelsare favoured. However, for reacting conditions, the 2D heterogeneous models are necessary for the re-actor simulation due to its dynamics as well as its radial temperature gradient. These gradients are alsoinfluenced by the radial heat transfer near the wall, which depend on the 1. increase in the void fractionwhich decreases the bed conductivity 2. a laminar boundary layer 3. a decrease in the convective heattransfer due to a reduction of fluid displacement. [25]
3.2.6 Commercial Fischer-Tropsch Plants
Sasol (Sasolburg and Secunda, South Africa)
Up until 2004, the fresh syngas, for all three Sasol plants, was produced from coal. About 40 milliontons per year of low grade coal is consumed by Sasol. [1]
Since 2004, pure syngas is fed to the LTFTS reactors in Sasolburg coming from Mozambique. Thereare five multitubular reactors that operated at 2.7 MPa and 230°C, to maximise the production of linearalkanes/alkenes and waxes, and also a slurry reactor with the capacity of the five multitubular reactorscombined. Sasol is the world’s largest producer of linear paraffinic waxes, these products are thenconverted to alkanes through hydroprocessing, figure 3.5. [1, 4, 5]
At the Secunda plants, the syngas gas is fed to the HTFTS fluidized bed reactors, filled with iron-based catalyst, the operating conditions are aimed at producing 1-alkenes and gasoline. The outlet ofthe reactor is then put through a series of separation steps in order to recover and purify each product ofthe outlet spectrum. The operating condition are set to the production of 1-alkenes and gasoline. Sincethe FT reactions produce a wide range of oils with high molecular mass, the separation and refiningprocess is complex, figure 3.6. [1]
Currently Sasol produces a total amount of 7.5 million tons per year of various products and satisfies30% of South Africa’s liquid fuel demand. [1]
Figure 3.5: Sasolburg plant block diagram. [1]
20
Figure 3.6: Secunda plant block diagram. [1]
Shell SMDS (Bintulu, Malaysia)
The SMDS (Shell Middle Distillate Synthesis) plant is fed by off-shore methane. The syngas isproduced by non-catalytic partial oxidation, the resulting syngas has a H2/CO ratio of about 1.7 whichis below the 2.1 required for the cobalt-based catalyst used in FTS, thus some additional hydrogen-richgas is required and is obtain through catalytic reforming of the FT tail gas, this is a high cost operation.[1]
There are four multitubular reactors operating at 3 MPa and 200 to 230°C and conversions of 80%and C5+ selectivities of about 85% are claimed. The products in the outlet of the reactor are thenseparated and purified in order to obtain high wax production. [1, 4, 5]
Shell Pearl GTL in Qatar is the world’s largest GTL plant and Shell’s most recent project in FTS. Apartnership between Shell and Qatar Petroleum created one of the world’s largest, most complex andchallenging energy projects ever commissioned. [26]
The plant did its first shipment in 2011 and reached full capacity in 2012. It has 24 TBRs each with29 thousand tubes containing cobalt catalyst. [26]
This plant is upstream-downstream fully integrated and is capable of processing up to 45.5 millioncubic meters per day of wellhead gas from 22 offshore wells, which are then converted to Gas-to-Liquids(GTL) using Shell’s Middle Distillate Synthesis (SMDS) process.
21
Figure 3.8: Shell Pearl GTL plant in Qatar at dawn. [26]
3.2.7 Recent Developments
To account for specific requirements of different applications, some variants of the TBRs have emerged.One of the most successful is the monolith reactor, this reactor is a micro-channel trickle bed and
has been used extensively in the automotive industry in order to reduce the exhaust gas emissions.This kind of reactor has several advantages such as low pressure drop, high mass transfer rates, easyscale-up and the active catalytic material is coated externally to reduce diffusion limitations. [19]
The micro-reactors are also one of the modern types of TBRs with characteristic length scales thatcan range from a few microns to less than 1 mm, such characteristics provide this reactor with enhancedrates of transport that can be up to one order of magnitude higher when compared to the conventionalreactors. These reactors also offer better control over the residence time distribution (RTD), heat andmass transfer, better process flexibility, easier scale-up and enhanced process safety. [19]
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Chapter 4
Modelling
Process modelling is the new step towards more efficient chemical processes. There is a wide rangeof simulators accessible to model a variety of processes. In this work, gPROMS ProcessBuilder v1.3.0was used. It is commercialised by Process Systems Enterprise, Ltd (PSE), whose modelling tool allowsfor custom-modelling. [27]
The first step of this work on gPROMS ProcessBuilder was to set up a model for Fischer-Tropschsynthesis (FTS) process using the Advanced Model Library for Trickle-Bed Reactors (AML:TBR). Thereactor geometry, dimensions, catalyst properties as well as the operating conditions were based ontypical industrial values present in [5].
The aforementioned was followed by the implementation of two kinetic models from the literature: 1)the Todic’s kinetics [2]; 2) the Staminic’s kinetic model [4]. For the vapour-liquid equilibria (VLE), PSE’sPSRK-NRTL model and Marano’s Henry coefficients [24] were used, this could be complemented byusing gSAFT equations of state, however it was not attempted due to time limitations.
Regarding physical properties it was used Multiflash as well as some other methods from literaturenamely Marano’s papers for liquid viscosity and thermal conductivity [28] and Erkey’s paper for diffusion[23].
For heat and mass transfer some correlation were implemented from the Stamenic’s paper. [4]
These steps are discussed in deeper detail the subsequent sections.
4.1 Multiflash®
gPROMS can use Multiflash® as a ”ForeignObject” for physical properties1 it requires to effectivelyrun the models developed. This software allows for the phase behaviour of complex mixture’s mod-elling using powerful pressure, volume and temperature (PVT) and physical properties packages forapplications in oil and gas transport and processing. [29]
4.2 Modelling environment
This chapter presents an overview of the gML and AML:TBR pre-built models used during the flow-sheeting. The gML is a library that contains basic models for the construction of general flowsheets,whereas the AML:TBR provides rigorous models for trickle bed reactor (TBR).[27]
For all models present in this chapter, the blue ports represent material ports, the orange portsrepresent energy connections. The green ports are called the ”bus ports” which are optional and are
1Thermodynamic and transport properties.
23
used to share information (reporting and specification variables) between models.
From the following models, the Trickle bed section 1D - chapter 4.2.3 - and the Trickle bed section 2D- chapter 4.2.3 - were the ones modified during this work. These models include parts of the AML:TBRlibrary that were changed in order to implemented what was developed and discussed throughout thenext chapters of the this work. These model require custom modelling in order to successfully simulatea TBR process.
4.2.1 gML models
Source material gML
This model is where a material stream enters the flowsheet. It needs, as specification, the com-ponents present and what models are to be used to predict the thermophysical properties, transportproperties and phase equilibria. It also requires stream conditions such as temperature, pressure, com-position and flow rate.
Figure 4.1: Topology representation of the source material block.
Sink material gML
This model is the opposite to the previous one, it is used to define streams that are leaving theflowsheet. It needs no specifications.
Figure 4.2: Topology representation of the sink material block.
4.2.2 AML:TBR basic models
To accurately model these kind of reactors, the model is divided into several sub-models that describedifferent parts and phenomena present in this piece of equipment such as, heat exchange, kinetics, bedsections and pellet.
The TBR can be modelled with different kind of detail, for instance, for catalyst pellet, it can be eitherone dimensional (1D) pellet model (distributed model) where temperature and concentration gradientsare present or a nildimensional (0D) model (lumped model) where the pellet is modelled using a pseudo-homogeneous approach with an effectiveness factor. One can also choose to model the catalytic bed(or inert bed) axially distributed where the radial gradients are taken into account or axially and radiallydistributed or even any combination of the above situations.
For mass and heat transport phenomena inside the catalyst-filled bed, first principle methods cannotbe applied, instead an effective approach is considered using correlations for the effective bed con-ductivity and wall-to-bed heat transfer coefficient whereas, for 1D pellet model, rigorous mass and heattransfer is considered along the pellet diffusion path.
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Distributor gML
This model transforms lumped material streams into radially distributed in the tube, for that reasonthe outlet stream of figure 4.3 are gray whereas the inlet streams are blue, this is done to differentiatebetween lumped material streams and distributed ones. This model has two inlets and two outlets sinceit allows, for example, for a gas inlet and a liquid inlet.
This model is only used at the top of a two dimensional (2D) distributed bed section - tube radial andaxial dimensions - (figure 4.6) to connect upstream models with the 2D reactor model.
The specification are geometry and number of reactor tubes.
Figure 4.3: Topology representation of the distributor block.
Aggregator gML
This model is the opposite to the previous one, it aggregates the distributed material streams intolumped ones, thus it needs no specifications.
Figure 4.4: Topology representation of the aggregator block.
4.2.3 AML:TBR bed sections models
Trickle bed section 1D gML
This model represents a section of a reactor tube filled with solid pellets. Gas and liquid trickle downthe fixed-bed and the solid pellets are filled with liquid.
This model is axially distributed (1D) and uses heterogeneous bed model by applying separate con-servation equations for fluid and catalyst phases. The pellets can be represented either by a lumpedmodel or by a distributed one considering multi-component mass transfer inside the catalyst pores.
This model can be either adiabatic or consider heat transfer though the wall.
Figure 4.5: Topology representation of the trickle bed section 1D block.
Trickle bed section 2D gML
This model is a 2D version of the model described in 4.2.3, meaning that the fixed-bed model isaxially and radially distributed.
25
Figure 4.6: Topology representation of the trickle bed section 2D block.
4.2.4 AML:TBR heat exchange models
Fixed coolant gML
This model describes the heat exchange between a single reactor tube and a coolant. It uses asimplified equation with fixed heat transfer coefficient and fixed coolant temperature to predict the heatflux and temperature profiles.
It takes as specification the temperature of the coolant as well as the heat transfer coefficient.
Figure 4.7: Topology representation of the fixed coolant block.
There are other heat exchange models available in the model library. Since this work’s focus wasthe tube side modelling, the ”Fixed coolant” was sufficient representation of the shell side of the tubu-lar Fischer-Tropsch (FT) reactor, typically the coolant is boiling water that can be assumed to have aconstant temperature.
4.3 Kinetic modelling
The kinetic models implemented were divided into two different kinetic models: the first one followedTodic et. al model [2] where a degrees of freedom (DOF) analysis was performed2. Since the productsof the FTS are hydrocarbons with different carbon numbers, this has to be taken into account whenperforming such analysis. An analysis of the maximum carbon number was also performed, figure 6.1.
The second one was the hybrid model developed by Stamenic et. al [4]. This model couples thekinetics developed by Yates-Satterfield [11] for the CO-consumption, the kinetic developed by Ma et.al [15] for the production of methane and the aforementioned model for the production of every otherhydrocarbon. Due to use of different approaches and expressions, a normalisation of the reaction rateswas performed by the author in [4].
Due to the assumption that the olefins and paraffins were lumped into a single representative com-ponent and all the waxes were also lumped into a single representative component, the kinetic modelshave to be put through some minor modifications by summing both reactions rates in order to createpseudo-components of C2, C3, C4 and C28
3. These pseudo-components are the ones that the modelrecognises as reaction products.
2In Appendix A are shown the equations used by this model.3The reaction rate of C1 is equal to the reaction rate of methane because there’s no olefinic compound with only one carbon
atom.
26
4.3.1 Todic’s kinetic model
When implementing this model into gPROMS the results stated in the [2] were not being achieved,upon contacting the corresponding author of this article, a corrected version of equation for the fractionof vacant sites ([S]) was provided and after the implementation of such, the results fitted almost perfectlyto the ones reported in the literature [2], figures 4.10, A.1a and A.1b.
It should be noted that the inputs - such as composition, pressure and temperature - given to themodel in order to obtain the aforementioned figures were the same as the paper with the objective oftesting the fidelity of the implemented kinetic model.
The DOF analysis was performed before the implementation of the Todic model due to relative highnumber of variables and equations involved.
Table A.1 in appendix A shows how many and which variables had to be assigned in order to maintainthe well posedness of the model4.
In figure 4.10 is shown the product formation rate for paraffins and olefins (implemented model)which matches the model developed by Todic et. al, they successfully validated their model againstexperimental data. [2]
In appendix A is shown the plots for the total hydrocarbon formation (figures A.1a and A.1b).
The term ec·n in equations (4.3) and (4.4) means that, with the increase in the carbon number,the before mentioned term decreases, causing the chain growth probability (αn) to increase. Aftera certain carbon number is reached, around 15, the aforementioned contribution becomes, essentially,null resulting in a constant value of chain growth probability and a constant decrease in total hydrocarbonformation rate as can be viewed in figures 4.9 and 4.8.
RCH4= k7MK
0.52 P 0.5
H2α1 · [S]2 (4.1)
RCnH2n+2 = k7K0.52 P 0.5
H2α1α2
n∏i=3
αi · [S]2 n ≥ 2 (4.2)
RC2H4= k8Ee
c·2α1α2 · [S]2 (4.3)
RCnH2n = k8ec·nα1α2
n∏i=3
αi · [S] n ≥ 3 (4.4)
Where equations (4.2) and (4.4) are the rate of production of all paraffins and olefins but methane(equation (4.1)) and ethylene (equation (4.3)), respectively.
Total hydrocarbon formation rate for each carbon number as a Anderson-Schulz-Flory (ASF) distri-bution.
4Regarding the maximum number of carbons (Cmax) it is both a variable name and a number.
27
0 5 10 15 20 25 30 35 40 45 50
10−5
10−4
10−3
10−2
Carbon number
Tota
lhyd
roca
rbon
form
atio
nra
te,m
ol/g c
at/h
Figure 4.8: Total hydrocarbon production as a ASF distribution plot obtained in gPROMS.
Chain growth probability plotted against the carbon number as well as the indication from whichcarbon number the probability can be considered constant, figure 4.9.
1 2 4 6 8 10 12 14 16 18 200.4
0.6
0.8
Carbon number
Cha
ingr
owth
prob
abili
ty(αn)
Chain growth probabilityConstant probability
Figure 4.9: Chain growth probability.
The lower C2 formation rate is explained due to the ethylene being more strongly adsorbed to thesurface than the other 1-olefins, which results in its higher desorption activation energy. [2]
Figure 4.10: Paraffinic and olefinic product formation rate. [2]
Figure 4.10 was obtained using the same operating conditions as the ones present in [2]; 1. Temper-ature = 478 K 2. Pressure = 15 bar and 3. H2/CO = 2.1.
The differences between the model from the literature and the one developed in this work are neg-ligible, thus the latter is a faithful representation of the Todic et. al kinetic model. However, one of thereasons that could have led to the differences is the value of Cmax, since it affects the value of the frac-tion of vacant sites, which will slightly change the model’s profiles. Another reason is the fact that the
28
literature model data was not available in a table form, the WebPlotDigitizer® software was used. Thedata points were recovered by hand which explains some of the differences between the calculated andthe literature values.
A sensitivity analysis on the Cmax is presented in chapter 6.1.1.
4.3.2 Stamenic’s kinetic model
As it was said before, Stamenic et. al [4] developed a model that combines three different kineticmodels:
• Yates-Satterfield for the CO consumption• Wenping Ma et. al for the CH4 production• Todic et. al for the production of the other hydrocarbon
Due to the fact that this model is made up of three different models and different approaches wereused in each one of them when defining the reaction rates, these have to be normalised. According toproduct formation kinetic model (superscript prod), the CO consumption rate, excluding methane, is asfollows [4]:
(−RCO)prodC2+
=
Cmax∑n=2
n ·(RprodCnH2n+2
+RprodCnH2n
)(4.5)
(−RCO)Y SC2+
= (−RCO)Y S −RMaCH4
(4.6)
Normalised rates for every other species of hydrocarbon, except methane, are defined as follows [4]:
RCnH2n= RprodCnH2n
·(−RCO)Y SC2+
(−RCO)prodC2+
n ≥ 2 (4.7)
RCnH2n+2= RprodCnH2n+2
·(−RCO)Y SC2+
(−RCO)prodC2+
n ≥ 2 (4.8)
The consumption of hydrogen is calculated as the sum of all the hydrocarbon in relation to theircarbon number [4]:
(−RH2) = 3 ·RMa
CH4+
Cmax∑n=2
[(2n+ 1)RCnH2n+2
+ 2n ·RCnH2n
](4.9)
The rate of water formation is equal to the consumption rate of CO, assuming that CO2 formation isnegligible [4]:
RH2O = (−RCO)Y S (4.10)
Figure 4.11 shows a comparison between the results of the implemented Todic and Stamenic models,where the deviations in C1 compounds have to do with different kinetic models and the differences inthe other pseudo-components have to with the normalisation procedure.
29
C1 pseudo-C2 pseudo-C3 pseudo-C4 C5+0
5 · 10−3
1 · 10−2
Pseudo-component
Tota
lhyd
roca
rbon
prod
uctio
n(m
ol/g c
at/h)
Todic modelStamenic model
Figure 4.11: Model comparison between the Todic and Stamenic models
The paraffinic and olefinic productions predicted by the implemented model can be found in AppendixA.2, figure A.2.
Todic’s kinetic model, and as a consequence Stamenic’s one, both calculate the αn for each chainaccording to its carbon number whereas the kinetic model implemented on a previous example of theAML:TBR took this value as constant. This is the case for high molecular weight chains as can beobserved in figure 4.9, however for smaller values of carbon number the probability should not be con-sidered constant.
After the full implementation of the Stamenic’s kinetic model in the process, a problem was en-countered when it was applied to the models that use a radially distributed 1D pellet (these are explainedin detail in sections 5.1 and 5.2). Ma’s kinetics [15] has a negative power law towards the partial pressureof CO - equation (4.11) - and these models account for temperature and concentration gradients insidethe pellet, which are severely different from the conditions outside of it. Because of this, the conditionsinside the the pellet may fall outside the kinetics validity range, therefore some predicted production ratesare nonsense (e.g. the production of C1 being higher than the consumption of CO). Figure 4.12 showshow C2 is being consumed and the steep rate at which C1 is being produced which leads the model tofail prematurely.
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8−4
−2
0
2
4
6
·10−4
Rate factor
Rea
ctio
nR
ate,
mol/(kg
cats)
C1
C2
C3
C4
C +5
Figure 4.12: Reaction rates for each product using the Stamenic’s kinetic model
The x axis in figure 4.12 is related to the Initialisation Procedure on gPROMS where it graduallyincreases the ”Rate factor”5 until it reaches its final value of 1.
From equation (4.6) one can deduct that, if the RMaCH4
production rate is higher than the consumption ofCO predicted by Yates-Satterfield kinetic model, the normalisation procedure (given by equations (4.7)and (4.8)) will be compromised resulting in the behaviour mentioned in the previous paragraph.
5Rate factor is a constant by which the reaction rate is multiplied in order to make a smooth model initialisation.
30
From the kinetics point of view, CO is needed to produce methane through the FTS, therefore thenegative power law could be related with the adsorption of this reactant to the catalyst which, in greatquantities, could indeed inhibit the reaction rate. However, the conditions at which the reactor operateswere observed to, sometimes, be outside of this kinetic model’s application range.
RMaCH4
= kMaP−0.99CO P 1.28
H2
1 + 0.58PH2O
PH2
(4.11)
Equation (4.11)’s parameters are available in [4]6.For the reasons explained above, the kinetic model used throughout the rest of this work was the
one developed by Todic in [2].
4.4 Vapour-liquid equilibria (VLE) calculations
VLE determines how the chemical species are distributed between the vapour and liquid phase. Thisdistribution has a strong thermal dependency, meaning that, for a certain temperature value, the con-centration (or partial pressure) of a component in the vapour phase has a corresponding concentrationin the liquid phase and the same for the other way around.
These concentrations can be obtained experimentally or approximately calculated by Raoult’s, Dalton’sand/or Henry’s laws. For real gases, fugacity coefficients are usually employed to calculate VLE partialpressures, this is closely related to the thermodynamic activity.
The VLE calculation in the model can be done by using the thermodynamic model defined by PSEto use the PRSK-NRTL equation-of-state (EoS) to calculate the fugacity coefficients in order to predictthe solubility of the gases in the liquid phase. This model also allows for the calculation of the Henrycoefficients through equation (B.1).
Alternatively, the correlation developed by Marano et. al [24] (equation (4.12)) was implemented aspart of this work. The correlation calculates the Henry coefficients, thus predicting the solubility of thegases in the liquid phase. The Henry coefficient values, obtained by both methods, are presented intable 4.1 for the entrance and centre of the reactor. The reactor conditions are present in chapter 5.
lnHi = Hi,0 − n ·∆Hi
Hi,0 = β1 + β2
T + β3 · lnT + β4 · T 2 + β5
T 2
(4.12)
The temperature-dependent ABC parameters (β1,...,β5) and the Henry constant differential betweenthe ith component and the wax average carbon number (∆Hi) presented cover the hydrocarbons, aswell as the reactants, used in the case study with exception of the n-butane7. The parameters forthe n-butane were estimated through linear interpolation between propane and hexane. The resultingInfinite-Dilution Henry’s constant (Hi) is presented in figure B.1 along with the other compounds’ Hi -in order to obtain this plot the wax average carbon number (n) was set to 28 in accordance with [23].Henry’s constant for n-butane follows, mainly, the behaviour of the propane due to the interpolationprocedure.
Table 4.1 contains the Henry coefficients calculated by gPROMS using correlation (4.12) as well asMultiflash’s PSRK-NRTL EoS predictions at the centre and entrance of the reactor.
6In the Supporting Information.7The β1,...,β5 and ∆Hi for ethylene, propylene and n-hexane are also presented but these were ignored due to the way the
model is built.
31
Table 4.1: Henry coefficients predicted through Marano’s correlation and through PSRK-NRTL EoS.
From table 4.1 one can notice that the Multiflash’s predictions for H2 and CO are fairly close to theones predicted by the Marano’s correlation, this is because the binary interaction parameters for thesetwo components with the others present were fitted to experimental data whereas for CO2, N2 and fromC1 to C4 are one order of magnitude higher than the ones predicted from Marano’s correlation sincethese lack the experimental validation.
For that reason, the VLE will be predicted through Marano’s correlation throughout the model. Therelative error was calculated based on the Marano’s predictions since it’s based on experimental data.
4.5 Thermophysical Properties
Thermophysical properties are a key aspect in modelling because if they are not calculated theright way or are not obtained from the right source, the results might be severely affected with minordifferences in the properties.
4.5.1 Viscosity
The viscosity of the bulk component, in this case assumed as C28, will heavily affect the diffusionrate.
Marano et al. [28] developed a generalised asymptotic behaviour correlation (ABC) - equation (4.13)- to predict the liquid viscosity of n-paraffin and n-olefin.
ln(µ) = µ∞,0 + ∆µ∞(n− n0)−∆µ0e−β(n−n0)
γ
∆µ = β1 + β2
T + β3 · lnT + β4 · T 2 + β5
T 2
(4.13)
The values predicted by equations (4.13) are in cP. The parameters in equations (4.13) are presentin [28]. In the second equation of (4.13), the ∆µ parameter represents, both, ∆µ∞ and ∆µ0.
The analysis performed in this section will only focus in the liquid viscosity for almost pure C28 par-affin. The values predicted by ABC developed by Marano were compared to the values predicted byMultiflash software at five different temperatures - 446, 475, 504, 536 and 565 K - the pressure doesn’thave any noticeable effect on the liquid viscosity. The comparison is shown in table 4.2.
32
Table 4.2: Marano’s and Multiflash’s viscosity predictions for almost pure C28 at 446, 475, 504, 536 and565 K and the respective relative errors (µ× 104).
Temperature (K) Literature† (Pa.s) MF‡ (Pa.s) Relative Error
‡MF = Multiflash software using mixing rules method
It was also performed an analysis on the predictions from Multiflash and from Marano’s correlationusing gPROMS by running the implemented values. Since it had to be performed in different simulations,table 4.3 only contains viscosity values at the entrance of the reactor, where the conditions are the sameregardless of the method used.
Table 4.3: Comparison between Marano’s correlation, Multiflash mixing rules and MultiflashSuperTRAPP method (µ× 104).
Despite table 4.2 showing that the Mixing Rules method accurately predicts the viscosity for almostpure C28 wax, when it is used the full composition of the FT wax this method fails to predict the viscositywith accuracy. However, if the SuperTRAPP method is to be used, its predictions match the ones in theMarano’s paper, table 4.3.
Marano’s ABC method was the preferable one since it’s less computationally expensive when com-pared with Multiflash’s SuperTRAPP, guaranteeing a quicker simulation.
4.5.2 Diffusion
Diffusion is one of the properties with a major impact in FT process performance, this is due todifferent two-phase interfaces that the reactants must diffuse through to the inside of the pellet and thereaction products have to diffuse in the opposite direction, assuming that the pellet is covered in FT wax(C28), being this the bulk compound.
Erkey et. al [23] measured the diffusivities of H2, CO, CO2, C8H18, C12H26 and C16H34 at 14 bar andat three different temperatures (475, 504 and 536 K), being the liquid bulk component C28. However, themodel being developed in this work uses methane, ethane, propane and n-butane which are not presentin the aforementioned paper.
To overcome this limitation the Multiflash software was used to obtain the diffusivities of the missingcomponents as well as the hydrocarbons studied in the paper in C28, using the Hayduk-Minhas method.These results, when compared to the ones in the paper, have deviations that must be taken into account.
33
The ratios between the Multiflash diffusivities and Erkey’s results for each carbon number were cal-culated and plotted for 504 K, assumed as reference temperature, and 14 bar; then a second degreepolynomial trend-line was fitted to the ratios and the missing diffusivities were calculated by dividing theMultiflash diffusivities by the predicted ratio hence obtaining an estimation of Erkey’s diffusivities for themissing pseudo-components, figure 4.14.
8 9 10 11 12 13 14 15 16
0.8
0.85
0.9
0.95
Carbon number
Rat
io
RatioPolynomial trend-line
Figure 4.13: Ratio between Multiflash’s and Erkey’s diffusivity coefficients.
Where the polynomial trend-line is given by equation (4.14) and n represents the carbon number.
Ratio = 0.0033n2 − 0.0548n+ 1.0258 (4.14)
From figure 4.13 one can notice that the fit contains only three points, of course this not nearlyenough to come to a conclusion, however it was the only way to obtain values for the diffusivities otherthan the ones calculated through Multiflash software.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
0.5
1
·10−8
Carbon number
Diff
usio
nco
effic
ient
,m/s2
MultiflashErkey
Predicted values
Figure 4.14: Multiflash’s, Erkey’s and predicted diffusivity coefficients.
Temperature and viscosity dependence
It is known that the diffusion coefficient has a dependence on the temperature and viscosity. For 475and 536 K the missing coefficients were also predicted with the aforementioned method.
In order to implement it in gPROMS, an equation containing both dependences had to be developed.Equation (4.15) shows the generic form of such equation.
DAB = DAB,ref
(T
Tref
)a(µ
µref
)b(4.15)
The a and b are correlating parameters to be estimated. DAB,ref was calculated assuming 504 K asreference temperature.
34
To predict the dependence, a and b were set to 1 and an hypothetical DAB was calculated at 475 Kand 536 K for each compound.
The Ordinary Least Squares (OLS) method was then used for both temperatures at the same time.Equation (4.16) show that for this temperature range its dependence is linear, meaning that parametera is 1 and the dependence on the viscosity has an order of -0.782.
DAB = DAB,ref
(T
Tref
)(µ
µref
)−0.782(4.16)
In table C.1 are presented the values predicted for the three temperatures as well as the valuesobtain from Multiflash.
It should be noted that this parameter estimation was performed with diffusion coefficient for threecompounds covering a temperature range of 61 K. Besides not being a very broad range, the estimationis based on the predicted values performed in 4.5.2.
4.5.3 Liquid thermal conductivity
The liquid thermal conductivity plays a role in the heat transfer due to the liquid film layer around thecatalyst where the main heat transfer mechanism is conduction.
Marano et. al [28] developed another asymptotic behaviour correlation to predict this liquid thermo-physical property. Equations (4.17) were used to predict this property, their parameters are present in[28].
λL = λL,∞,0 + ∆λL,∞(n− n0)−∆λL,0e−β(n−n0)
γ
∆λL or λL = β1 + β2
T + β4 · T 2(4.17)
This correlation predicts the liquid thermal conductivity for pure hydrocarbon components, in thiscase for pure n-octacosane (C28).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
0.12
0.13
0.13
Axial position
Liqu
idth
erm
alco
nduc
tivity
,W/(mK
)
MultiflashMarano
Figure 4.15: Multiflash’s and Marano’s liquid thermal conductivity.
In figure 4.15 is plotted the different values for the liquid thermal conductivity. The relative error foreach axial position is around 9%, which is above of the acceptable 5%8. Despite the wax being almostpure C28, it still has some traces of other components that might change the overall liquid conductivity,thus the Multiflash’s mixing rules method is better suited to predicted this property.
8A relative error of 5% was considered as acceptable as a rule of thumb.
35
4.6 Heat and Mass Transfer
Due to the high exothermicity of the FTS, heat transfer is a major concern when modelling these kindof processes.
Besides, the heat and mass transfer between the different two-phases interfaces, the model mustalso take into account the conductivity of the bed as well as the heat duty that is transferred to the wallof the reactor.
For this section, no comparison is done between what was already part of the library and whatwas newly implemented, because the following correlations are an addition to the model and not animprovement to it. These will allow the user to choose the correlation that best suits their needs.
4.6.1 Gas-Liquid Heat Transfer
One can define the heat transfer in the interface either on the side of the gas or on the side of theliquid. Assuming that the heat transfer on the gas side is faster, correlation (4.18) was applied on theliquid phase.[4]
htcg−l =cp,gρgugεg
εPr−2/3g
(0.765
Re0.82g
+0.365
Re0.386g
)(4.18)
Reg =ρgdpugεg
µg(4.19)
Prg =cp,gµgλg
(4.20)
Where Reg and Prg are the Reynolds number and Prandtl number for the gas phase, respectively.
4.6.2 Liquid-Solid Heat and Mass Transfer
Both heat and mass transfer coefficients between the liquid-solid interface are assumed to be trans-ferred through a liquid film surrounding the catalyst where conduction and diffusion are considered asthe dominant transport (film theory) for the heat and mass transfer, respectively.
Heat transfer
Correlation (4.21) predicts the heat transfer coefficient between the liquid-solid interface.[4]
htcl−s =λl
δl(δl/2+Rp)2
Rp(δl+Rp)
(4.21)
Where htcl−s stands for the heat transfer coefficient between the liquid-solid interface.
(dp + 2δl)3 − d3p
d3p=
εl1− ε
(4.22)
The thickness of the liquid film (δl) is calculated based on the total liquid hold-up (εl) that is equallydistributed around the particle.
36
Mass transfer
The mass transfer coefficient for each component between the liquid-solid interface is calculatedthrough correlation (4.23).[4]
mtci,l−s =Di,j
δl(δl/2+Rp)2
Rp(δl+Rp)
(4.23)
Where the mtci,l−s is the mass transfer coefficient for the ith component between liquid-solid interface.
4.6.3 Effective Bed Conductivity
The effective radial bed conductivity accounts for the temperature profile from the reactor’s centre tothe wall, exclusively.
The model only has one variable for the effective bed conductivity whereas the correlation presentin Stamenic’s paper [4] has two separate variables for the gas and liquid phase, however the Stamenicpaper uses the correlation present in Brunner’s paper [30]. In the latter paper, the effective bed conduct-ivity has only one parameter, thus being this expression - equation (4.24) - the one implemented in thegPROMS model.
The terms λser, λler and λger have expressions of their own that are explicit in the appendix D.1. Thefull equation for the effective bed conductivity is (4.25).
This correlation is another improvement to the model since it grants the users the ability to bettermodel their processes, in case the previous correlations couldn’t represent, accurately, their process.
4.6.4 Bed-Wall Heat Transfer
This coefficient accounts for the temperature profile in the thin layer next to the wall.As it happened in 4.6.3 the model only takes one variable for this parameter, however in [4] it is
segregated into both phases. To account for this, equation (4.26) was derived.
htcws =εghtcw,g + εlhtcw,l
ε(4.26)
The contributions of both phases are stated in appendix D.2.
37
38
Chapter 5
Model Order Reduction
Model order reduction (MOR) is a technique of reducing the computational complexity of mathemat-ical models in order to make the simulations faster while assuring the model fidelity. For instance for amodel that is suppose to be used to continuously optimise a process, a computing time of around 3 to 4hours is impractical, thus the necessity of trying to simplify the models while maintaining it’s accuracy.
The trickle bed reactor (TBR) model with one dimensional (1D) pellet and two dimensional (2D) bedwas the first model to be created, it uses a radially distributed pellet model as well as a axially andradially distributed reactor model.
After successfully implementing this model, it was developed two more models with a sequentialMOR: the reactor model was reduced to axially distributed while the pellet model was kept distributedand then the latter was also reduced to a lumped pellet model (nildimensional (0D) model).
• Mo2DB1DP - 2D bed (axially and radially distributed) and 1D pellet (radially distributed) model;• Mo1DB1DP - 1D bed (axially distributed) and 1D pellet model;• Mo1DB0DP - 1D bed and 0D pellet (lumped pellet) model.
The characteristics of the multi-tubular reactor base case from which the simplified models werecreated are as follows and are based on the literature [5]:
• Inert section length⇒ 0.2 m;• Reactor length⇒ 12 m;• Tube diameter⇒ 5 cm;• Number of tubes⇒ 2050 tubes;• Coolant temperature⇒ 230°C;• Pressure⇒ 27 bar;• Gas Flowrate⇒ 11.8 kg/s;• Bed porosity⇒ 0.5;• Catalyst productivity⇒ 0.083 kg/kgcat/h;• Annual total production⇒ 51.8 thousand metric tonnes per year.
To account for the differences in the activity of the catalyst used in the Todic’s paper [2], a parameterwas added to the kinetic model implemented. This was adjusted in order to obtain an acceptable processbehaviour and at the same time reach the catalyst productivity reported in [6] and [31]. The productivityreported in the literature was not achieved and this might have to do with the fact that in Wakamura’s pa-per [6] a constant chain growth probability is assumed, whereas, in this work, this parameter is calculatedand the gas inlet pressure used in the paper is also different (22 bar).
39
Every single one of the following models has the same value for the catalyst activity in order to allowfor their comparison under the same operating conditions and analysis of the model fidelity.
The key performance indicators (KPIs) are mostly self explanatory with the exception of the catalystproductivity and the total production. The former accounts for the mass of wax produced per mass ofcatalyst per hour while the latter is the mass of wax annually produced.
5.1 TBR model with 2D bed and 1D pellet
This model is the most detailed from the three developed, since it uses a radially distributed pelletand a radially and axially distributed reactor model. For this reason, it accounts for temperature andconcentrations gradients across the pellet and across and throughout the reactor, thus the results ob-tained by the simulation of such model will have the highest fidelity for the reasons stated in the previouschapters. This model will, henceforth, be referred to as Mo2DB1DP.
Figure 5.1 represents the flowsheet model created to simulate 2D bed reactor models.
Figure 5.1: Flowsheet topology for the trickle-bed reactor with 1D pellet and 2D bed from gPROMS.
The fact that the pellet is distributed implies that the temperature and concentration profiles insidethe pellet will not be uniform, this is due to the reaction and diffusion steps occurring either in series orin parallel, meaning that a certain molecule of reactant can 1. keep diffusing towards the centre of thepellet or 2. react at the internal surface of the pellet [32]. This phenomena will create a concentrationprofile similar to the one represented in figure 5.21.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.3
0.601
0.902
1.202
1.503
1.804
·10−2
Radial pellet position, rc
CO
mol
arfra
ctio
n
Figure 5.2: CO molar fraction across the catalyst particle. The 0 in the x axis represents the particle’score and 1 represents its surface
The values for the catalyst radius are normalised.The KPIs for this model are presented in table 5.1, these will stand as reference when studying the
fidelity of the following order reduced models.1This profile was obtained at an axial position around 0.09, since is the most active zone of the reactor.
Comparing the previous model to this one, it was performed a model order reduction of one-dimensionby neglecting the radial distribution in the reactor, this reduction is acceptable for narrow tubes wherethe radial temperature and concentration gradients are not very pronounced, as explained in 3.2.5. Thismodel will, henceforth, be referred to as Mo1DB1DP. Figure 5.3 represents this model’s flowsheet.
Figure 5.3: Flowsheet topology for the trickle-bed reactor with 1D pellet and 1D bed from gPROMS.
Figure 5.4 shows the axial profiles of the Mo2DB1DP and Mo1DB1DP models. Regarding the formerat the bed’s centre, near the wall and also the average temperature of the bed.
From figure 5.4 one can conclude that the average bed temperature of the Mo2DB1DP matches theMo1DB1DP model temperature profile. However, when looking at the temperatures at the centre of thereactor or near the wall, the differences between them are noticeable.
41
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1495
500
505
510
Axial position, z
Tem
pera
ture
,KMo2DB1DP at the centreMo2DB1DP at the wallMo2DB1DP average
Mo1DB1DP
Figure 5.4: Mo2DB1DP and Mo1DB1DP reactor axial temperature profiles.
Taking the point of maximum temperature difference between both models (around the axial positionof 0.09) and plotting the temperature radial profile in this point, figure 5.5 is obtained.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
509
510
511
512
Radial position, r
Tem
pera
ture
,K
Mo2DB1DPMo1DB1DP
Figure 5.5: Radial temperature profiles of the Mo2DB1DP and Mo1DB1DP models at axial position0.09.
Figure 5.5 shows the radial profiles of the Mo2DB1DP and Mo1DB1DP models. On the latter thevalue is constant since there’s no radial gradients in this model.
From figure 5.5 it can be noted that there’s a difference of about 3°C spread throughout the radiusof the reactor’s tubes (2.5 cm). This difference has consequences in the reaction rates, especially nearthe surface of the catalyst, since these are extremely dependent on the temperature - figure 5.6.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.4
3.6
3.8
4
4.2·10−4
C1 radial position, r
Rea
ctio
nR
ate,
mol/kg c
at/s
Mo2DB1DPMo1DB1DP
Figure 5.6: C1 radial reaction rate profile at the surface of the catalyst.
42
Table 5.2: KPIs comparison between Mo2DB1DP and Mo1DB1DP models developed.
Computing time (s) 175 59Number of equations 521450 136297
The relative error on table 5.2 is calculated using the following expression:
|KPI(Mo2DB1DP )−KPI(Mo1DB1DP )|KPI(Mo2DB1DP )
× 100
From table 5.2 one can notice that the relative errors are all below 0.1% which shows that theMo1DB1DP model can be used to accurately reproduce the Mo2DB1DP model. Therefore, for thistube diameter (5 cm) and the conditions considered in this work the radial distribution can be neglected.The values presented in table 5.2 are all the same, even though the relative error is not 0%, this has todo with the value being truncated to one decimal digit.
It is expected the conclusion to be the same for narrower tubes and the differences between themodels’ results even smaller, whereas for wider tube this analysis should be performed once again dueto the radial gradients being more pronounced in wider tubes.
The computing time on table 5.2 was obtained without the Initialisation Procedure, instead it used aSaved Variable Set that already contained some initial guesses for all model variables from previous sim-ulations. This Saved Variable Set had to be obtained by running the model with the Initialisation Proced-ure, in which case the Mo2DB1DP model with take around 40 minutes to run whereas the Mo1DB1DPmodel takes around 4 minutes.
5.3 TBR model with 1D bed and lumped pellet
This is the simplest model developed, it not only neglects the radial distribution of the reactor but alsoneglects the radial distribution on the pellet, turning it into a lumped model. The topology is the same asprevious model and is represented in figure 5.3.
This simplification implies that the internal diffusion limitations are not accounted for, meaning thatthe concentration, as well as the temperature, of the reactants in the catalyst layer is the same as theirconcentration on the particle’s surface. For this reason an additional parameter should be added toaccount for drawback of this simplification. The Thiele’s module (φ) is used to calculate the effectivenessfactor (η) of the catalyst. In [32] is presented a general expression for an irreversible reaction of order nfor any particle geometry to calculate φ and an expression for the η for spherical particles. This modelwill, henceforth, be referred to as Mo1DB0DP.
43
η =tanhφ
φ(5.1)
From the Mo1DB1DP, the η was calculated throughout the axial domain of the reactor being itsaverage 0.438, which shows that the influence of the internal diffusion limitation is noticeable.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
500
505
510
Axial position, z
Tem
pera
ture
,K
η = 0.438η using Thiele’s module
Mo1DB1DP
Figure 5.7: Reactor’s axial temperature profiles.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.35
0.4
0.45
0.5
Axial position, z
Effe
ctiv
enes
sfa
ctor
(η)
η = 0.438η using Thiele’s module
Mo1DB1DP
Figure 5.8: Effectiveness factor axial profile.
Even though figure 5.7 shows a fairly good prediction of the temperature profile between the Mo1DB1DPand the Mo1DB0DP, figure 5.8 shows a very poor prediction of the effectiveness factor which is reflectedin the KPI in table 5.3.
Table 5.3 shows the KPIs of the Mo1DB1DP model as well as the KPIs for the Mo1DB0DP modelusing the average η and using the η calculated through the Thiele’s module. It also contains the relativeerrors by comparing the different results from the Mo1DB0DP model with the Mo1DB1DP one.
44
Table 5.3: Comparison of the KPIs using different values of η in the Mo1DB0DP model against theMo1DB1DP model.
Total Production (ton/year) 51803 60754 17.28% 55638 7.40%
From table 5.3 one can conclude that the Mo1DB0DP might predict the reactants conversion withsome accuracy, however the molar selectivity is significantly different from the Mo1DB1DP predictions.
For the reasons stated above, the Mo1DB0DP should not be used to predicted such a complexreactional system such as the Fischer-Tropsch synthesis (FTS) at the conditions of the case study. TheKPI obtained cannot be compared with experimental ones due to their scarcity throughout the literature.
45
46
Chapter 6
Sensitivity Analysis
Due to the complexity of a trickle bed reactor (TBR) model, its sensitivity analysis (SA) was dividedinto three different sections, as follows:
• Numerical parameters - where the most suitable Cmax as well as discretisation parameters werechosen among the ones studied in order to reduce the computing time while aiming at accurateresults;
• Design parameters - this analysis fell upon some of the reactor’s design parameters such aslength, radius, and gas hourly space velocity (GHSV);
• Operating parameters - these contemplate operational parameters such as H2/CO ratio, coolanttemperature and gas inlet pressure.
To compare the different values studied for each parameter of each section, some variables wereplotted to better understand the system’s behaviour. In addition, some key performance indicators (KPIs)were put together in the form of table, these are the same as the ones used in chapter 5.
6.1 Sensitiviy Analysis on Numerical Parameters
6.1.1 Parametric Analysis of Cmax
Regarding computer simulations, there is a trade-off between model detail and computational capab-ility, meaning that a model too detailed will take longer to be simulated when compared to a less detailedone. Thus, a model should be detailed enough to guarantee accurate results within an acceptablerun-time.
Taking into account the aforementioned, an analysis on the Cmax parameter was performed where,for each value of Cmax, the total hydrocarbon production was analysed. The results from figure 6.1 wereobtained under the same operating conditions as figure 4.10 using just the kinetic model, meaning thatthe thermodynamic and transport phenomena present in the reactor were not being accounted for1.
From the results plotted in figure 6.1, one can deduce that for values of Cmax higher than 100 thereis no increase in the total production of hydrocarbons, therefore this can be considered, regarding thisparameter, the model’s required maximum level of detail for these operating conditions.
1This kinetic model is integrated into the reactor model developed in this work.
47
15 50 100 150 200 25013
14
15
16
Cmax
Tota
lhyd
roca
rbon
prod
uctio
n,mol/kg c
at/s
Figure 6.1: Total hydrocarbon production Vs Cmax.
However, when the Mo2DB1DP reactor model was tested using a Cmax value of 100 the simulationtime was too long - Around 3 to 4 hours - using the Initialisation Procedure. To avoid simulations thattook too long, the value of Cmax had to be lower yet, the simulation results still had to be accurate, withinan acceptable range.
Figures 6.2, 6.3 and 6.4 show a comparison on the results obtained using different values of Cmax2.
From figures 6.2, 6.3 and 6.4 one can deduce that a value of Cmax of 10 is not enough when com-pared to Cmax = 100, besides C28 is considered to be the Fischer-Tropsch (FT) wax which would notbe produced if the former value of Cmax was used. The decision was to use Cmax = 35. That way, theformation of C28 is accounted for and the KPIs are very similar to the optimal Cmax value case - table6.1 - with relative errors below 3%.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
2
3
4·10−4
Reactor’s length, m
C1
form
atio
nra
te,m
ol/kg
cat/s
Cmax = 10Cmax = 35Cmax = 50Cmax = 100
Figure 6.2: C1 formation rate throughout the reactor for different values of Cmax.
2These values were obtain using the Mo1DB1DP model on account for reducing the simulation time required for this sensitivityanalysis.
48
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
2
·10−4
Reactor’s length, m
C+ 5
form
atio
nra
te,m
ol/kg c
at/s
Cmax = 10Cmax = 35Cmax = 50Cmax = 100
Figure 6.3: C5+ formation rate throughout the reactor for different values of Cmax.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
500
505
510
Reactor’s length, m
Tem
pera
ture
,K
Cmax = 10Cmax = 35Cmax = 50Cmax = 100
Figure 6.4: Temperature profiles throughout the reactor for different values of Cmax.
Table 6.1: KPIs for the two different values of Cmax
Computing time without IP (s) 990 165Number of equations 168940 78850
Where IP stands for initialisation procedure, meaning that a Saved Variable Set file was used tospeed up the simulation.
6.1.2 Bed Axial Domain Discretisation Analysis
When using distributed models one of the parameters to take into account is the number of pointsused in the discretisation procedure. This procedure consists on transforming continuous functions,
49
equations, variables and models into discrete counterparts.The following analysis consists in assessing the minimum number of axial discretisations points
while maintaining accurate results. The discretisation scheme used was the BFDM 1st order and thegrid transformation used was logarithmic.
From figures 6.5, 6.6 and 6.7 it can be noted that the profiles using 20 and 30 points are very similarwhich allows for the conclusion that higher number of points will not provide better predictions.
A discretisation of 10 points also gives fairly good predictions, however in the beginning of the reactorthere are some noticeable differences and since this zone is the reactor’s most active zone, using only10 points introduces some inaccuracy on the results.
The KPIs are present in table 6.2.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1
2
3
4·10−4
Reactor’s length, m
C1
form
atio
nra
te,m
ol/kg
cat/s
Points = 5Points = 10Points = 20Points = 30
Figure 6.5: C1 formation rate throughout the reactor for different axial domain discretisation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5
1
1.5
2
·10−4
Reactor’s length, m
C5+
form
atio
nra
te,m
ol/kg c
at/s
Points = 5Points = 10Points = 20Points = 30
Figure 6.6: C5+ formation rate throughout the reactor for different axial domain discretisation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
500
505
510
Reactor’s length, m
Tem
pera
ture
,K
Points = 5Points = 10Points = 20Points = 30
Figure 6.7: Temperature profiles throughout the reactor for different axial domain discretisation.
From figures 6.5, 6.6 and 6.7 it can be noted that 10 and 20 points give good predictions when
50
compared to 30 points - table 6.2 - however, for 10 points, there’s a poor prediction around between z = 0
and z = 0.1 which is the most active zone in the reactor, therefore the bed axial domain discretisationwill use 20 points. This is also highlighted by the KPIs in table 6.2
Table 6.2: KPIs for different discretisations of the axial domain.
The radial pellet domain discretisation is also another parameter that should be taken into account,because it will model how the gradients are developed inside the pellet. Too many points will cause themodel to have too many equations which will lead to a slow simulation without granting an increase inthe accuracy of the results, however insufficient number of points might lead to big gaps between themwhich will compromise the results as well as the stability of the model developed.
Bearing in mind the previous paragraph and its consequences, an analysis on the number of pointsfor the radial pellet domain was performed. The discretisation scheme used was the CFDM 2nd orderand with uniform grid transformation.
Figure 6.9: C5+ formation rate throughout the reactor for different radial pellet domain discretisation.
From figures 6.8 and 6.9 one can note that using 5 points provide not a very good prediction. When10 points are used, the prediction is improved however is still has some deviations from the one using25 points. The decision on whether to use 15 or 20 points, is based on the KPIs, computing time andnumber of equations - table 6.3 - this leads to the decision of using 20 points for the radial pellet domaindiscretisation, since the C2 molar selectivity has a relative error above 5% when using 15 points.
Table 6.3: KPIs for different discretisations of the radial pellet domain.
Computing time (s) 67 59 43Number of equations 166432 136297 106162
6.1.4 Bed Radial Domain Discretisation Analysis
The bed radial domain discretisation will model the radial gradients throughout the tube radius. It willbe, more or less, important depending on the radius of tube, meaning that, for wider tubes the profileswill be more pronounced than for narrower tubes.
This analysis approach is very similar to the previous ones; less points will lead to a faster but lessaccurate simulation whereas higher number of points will lead to a more accurate simulation but thecomputing time might be too long. A trade-off must be made in order to achieve a quick simulation whilemaintaining accurate and faithful results. The discretisation scheme used was the CFDM 2nd order andwith uniform grid transformation.
52
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 13.4
3.6
3.8
4
4.2·10−4
Radial position, r
C1
form
atio
nra
te,m
ol/kg c
at/s
Points = 4Points = 6Points = 8
Figure 6.10: C1 formation rate throughout the reactor for different radial domain discretisation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12
2.1
2.2
2.3
2.4·10−4
Radial position, r
C5+
form
atio
nra
te,m
ol/kg c
at/s
Points = 4Points = 6Points = 8
Figure 6.11: C5+ formation rate throughout the reactor for different radial domain discretisation.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
509
510
511
512
Radial position, r
Tem
pera
ture
,K
Points = 4Points = 6Points = 8
Figure 6.12: Temperature profiles throughout the reactor for different radial domain discretisation.
Figures 6.10, 6.11 and 6.12 show how the model behaves, regarding the C1 and C5+ formation ratesand temperature profiles, for different number of points in the bed radial domain.
Table 6.4 shows the KPIs for the different number of discretisation points studied. It can be noted thatby using, either, 6 or 4 points the errors relative to 8 points are all below 5%. The number of discretisationpoints chosen is 4, to ensure a faster simulation time. The difference between using 4 or 8 discretisationpoints, regarding the computing time is 213 seconds, roughly three and half minutes; however, this isunder the conditions of these work, especially for a Cmax of 35, if the latter were to be increased thecomputing time difference would be much higher.
53
Table 6.4: KPIs for different discretisations of the radial domain.
Unlike section 6.1, where numerical parameters were analysed and choose from the tested range,this section will focus on how the model behaves when changes on design parameters are introduced.The parameters tested were 1. tube length 2. tube radius 3. GHSV.
To perform this analysis the ideal tool to use would be the Global System Analysis (GSA), this al-lows the comprehensive exploration of the behaviour of a system over domains of any user-selectedinput variables, and output variables [27]. However, some of these analysis would take up to 20 hours3.Therefore, this analysis was performed by manually changing the parameters and comparing the results.
6.2.1 Tube length
The reaction rate inside TBRs, as any steady-state reactor, depends on its volume or the mass ofcatalyst. By increasing, or decreasing, the tube length the catalyst mass will also increase or decrease,respectively; this will lead to different reactant’s conversions.
Figures 6.13, 6.14, 6.15 and 6.17 were obtained for a tube length of 15 m since it was the highesttube length tested and the reaction rate and temperature profiles are the same when using differentlengths.
The CO and H2 conversions were calculated using equations (6.1) and (6.2), these equations wereapplied by both using the inlet and outlet of the reactor or the inlet and outlet of the discretisation pointbeing calculated.
3Using a laptop with 8Gb of RAM and an Intel® Core™ i7-2640M CPU @ 2.80 GHz.
54
XCO =F inletCO − F outletCO
F inletCO
(6.1)
XH2 =F inletH2
− F outletH2
F inletH2
(6.2)
0 2 4 6 8 10 12 14
0
20
40
60
80
Tube length, m
Con
vers
ion,
%
COH2
Figure 6.13: CO and H2 conversion for different values of tube lengths.
0 2 4 6 8 10 12 1413
14
15
16
17
Tube length, m
Mol
arse
lect
ivity
,%
C1
Figure 6.14: C1 molar selectivities for different values of tube lengths.
0 2 4 6 8 10 12 14
1.5
2
2.5
3
3.5
Tube length, m
Mol
arse
lect
ivity
,%
C2
C3
C4
Figure 6.15: C2, C3 and C4 molar selectivities for different values of tube lengths.
55
0 2 4 6 8 10 12 14
76
77
78
79
Tube length, m
Mol
arse
lect
ivity
,%
C5+
Figure 6.16: C5+ molar selectivities for different values of tube lengths.
0 2 4 6 8 10 12 14
0.1
0.2
0.3
0.4
Tube length, m
Cat
alys
tpro
duct
ivity
,kg/kg
cath
Catalyst Productivity
Figure 6.17: Catalyst productivity for different values of tube lengths.
Figure 6.17 can be viewed as the C5+ catalysed reaction rate. The same is also applicable to figures6.22, 6.28 and 6.39.
Figures 6.13, 6.14, 6.15 and 6.17 show how the KPIs change along the reactors length for 15 mtubes.
Table 6.5 shows the relative variation, either increase or decrease, of the KPIs with the change onthe design parameters. The C5+ molar selectivity increases with the tube length, this has mainly to dowith the lower average bed temperature. The catalyst productivity (i.e. catalyst utilisation) decreaseswith the tube length, resulting in higher amount of catalyst per mass unit of wax produced.
This case study uses a multitubular reactor, thus an analysis on the radius of such tubes is imperative.To do so the tube radius was being changed from 0.015 to 0.030 m at steps of 0.001. Since by changingthe tube radius the total amount of catalyst is also changed, the number of tubes was calculated suchthat the GHSV was fixed to 1275 h-1 in order to keep the total amount of catalyst constant thus allowingthe results’ comparison between different tube sizes.
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03
65
70
75
80
Tube radius, m
Con
vers
ion,
%
COH2
Figure 6.18: CO and H2 conversion for different values of tube radius.
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03
13.5
13.6
13.7
13.8
Tube radius, m
Mol
arse
lect
ivity
,%
C1
Figure 6.19: C1 molar selectivities for different values of tube radius.
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03
1.5
2
2.5
3
3.5
Tube radius, m
Mol
arse
lect
ivity
,%
C2
C3
C4
Figure 6.20: C2, C3 and C4 molar selectivities for different values of tube radius.
57
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03
78.2
78.4
78.6
78.8
79
Tube radius, m
Mol
arse
lect
ivity
,%
C5+
Figure 6.21: C5+ molar selectivities for different values of tube radius.
Figure 6.22: Catalyst productivity for different values of tube radius.
Figures 6.19, 6.20 and 6.21 show that there is no significant external diffusion limitation, since areno major increases in these KPIs regarding the variation of the tube radius.
The increase in KPIs of figures 6.18 and 6.22 has mainly to do with the increased bed averagetemperature for wider tubes - figure 6.23. However, with higher temperatures, the risk of thermal runawayincreases due to the decreased rate of heat transfer from the bed.
Another important aspect to look at is the maximum temperature of the bed since this has to bedecided in order to prevent runaway behaviours, therefore in figure 6.23 the maximum temperature ofthe bed is plotted against the different tube radius.
0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03506
508
510
512
Tube radius, m
Max
imum
bed
tem
pera
ture
,K
Maximum temperature
Figure 6.23: Bed’s maximum temperature for different values of tube radius.
From figure 6.23 shows that for wider tubes the bed’s maximum temperature is higher, which isexpected since the cooling is done on the outside of the tubes, therefore the wider the tubes the morepronounced is the thermal gradient.
58
6.2.3 Gas hourly space velocity (GHSV)
The GHSV which is the volumetric flowrate divided by the volume of catalyst inside the reactor wasstudied by changing the number of tubes. The number of tubes is intimately related with the tube radius.
Bearing the previous paragraph in mind, this analysis was performed by fixing the tube radius to 2.5cm and manually changing the number of tubes from 1200 to 5300 with steps of 100 tubes while keepingthe feed flowrate and coolant temperature the same.
600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200
50
60
70
80
90
GHSV, h−1
Con
vers
ion,
%
COH2
Figure 6.24: CO and H2 conversion for the different values of GHSV.
600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200
13
13.5
14
14.5
15
15.5
GHSV, h−1
Mol
arse
lect
ivity
,%
C1
Figure 6.25: C1 molar selectivities for different values of GHSV.
600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200
1.5
2
2.5
3
3.5
4
4.5
GHSV, h−1
Mol
arse
lect
ivity
,%
C2
C3
C4
Figure 6.26: C2, C3 and C4 molar selectivities for different values of GHSV.
59
600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200
77
77.5
78
78.5
GHSV, h−1
Mol
arse
lect
ivity
,%
C5+
Figure 6.27: C5+ molar selectivities for different values of GHSV.
600 800 1,000 1,200 1,400 1,600 1,800 2,000 2,200
0.04
0.06
0.08
0.1
GHSV, h−1
Cat
alys
tpro
duct
ivity
,kg/kg
cath
Catalyst Productivity
Figure 6.28: Catalyst productivity for different values of GHSV.
By increasing the number of tubes inside the reactor, even though the mass of catalyst per tube doesnot increase since they have the same radius and length, the GHSV changes and therefore the velocityof the gas inside the tubes as well as the residence time. The higher the GHSV, the higher the gasvelocity which decreases the residence time leading to lower conversions in the reactor.
This leads to the same behaviour as the previous two analysis. Therefore, in the beginning of thereactor, the reactants will contact with more catalyst, relative to their flowrate, being their consumptionsrates higher - figure 6.24, 6.25, 6.27. The behaviour is inverted when the number of tubes is reduced.
Figure 6.25 shows an increase in C1 molar selectivity, which means that higher values of GHSVpromote the formation of methane, since the molar selectivities for C2, C3, C4 and C5+ decrease with theincrease of this variable - figures 6.25, 6.26 and 6.27.
The number of tubes could also influence the external diffusion limitations due to the variation of thetotal cross area, however as was explained in chapter 6.2.2 these limitations are very low and the majorreason for the different results come from the variation of the overall total mass of catalyst.
6.3 Sensitivity Analysis on Operating Conditions
Just like the previous chapter - chapter 6.2 - this chapter focus on understanding how the modelbehaves when changes in its operating conditions are made. The conditions studied are 1. H2/CO ratio2. Coolant temperature 3. Gas inlet pressure.
6.3.1 H2/CO Ratio
As it was explained in the chapter 3, the Fischer-Tropsch synthesis (FTS) is very dependent on theH2/CO ratio, therefore an analysis on this operating condition was performed.
60
0 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
4
·10−4
Reactor’s length, m
C1
form
atio
nra
te,m
ol/kg
cat/s
H2/CO=1.75H2/CO=1.8H2/CO=2
Figure 6.29: C1 formation rate throughout the reactor for different H2/CO ratios.
0 1 2 3 4 5 6 7 8 9 10 11 12
0.5
1
1.5
2
·10−4
Reactor’s length, m
C5+
form
atio
nra
te,m
ol/kg c
at/s
H2/CO=1.75H2/CO=1.8H2/CO=2
Figure 6.30: C5+ formation rate throughout the reactor for different H2/CO ratios.
0 1 2 3 4 5 6 7 8 9 10 11 12
500
505
510
Reactor’s length, m
Tem
pera
ture
,K
H2/CO=1.75H2/CO=1.8H2/CO=2
Figure 6.31: Temperature profiles throughout the reactor for different H2/CO ratios.
It is expected that a decrease in the H2/CO ratio should promote the product shifting towards heavyhydrocarbons, since a relatively high CO quantity promotes the formation of heavy hydrocarbon chains.
Figure 6.30 might suggest otherwise, however table 6.6 shows that the molar selectivity towardsthese compounds is indeed increased. The reason why figure 6.29 and 6.30 suggest otherwise isbecause they only reflect the rate of formation of C1 and C5+, respectively, at the surface of the catalystnot taking into account that a variation on the syngas composition will also imply different conversions.The reactants conversions are present in table 6.6 that shows that a decrease on the ratio results in adecrease on the conversion, which explains why the formation rate lowers with the decrease in H2/COratio.
For an increase in the H2/CO ratio there is a clear improvement in the molar selectivity for lightcompounds such as C1 and C2 - table 6.6.
61
Table 6.6: KPIs for different H2/CO ratios.
H2/CO ratio 1.8 1.75 Relative variation 2.0 Relative variation
The cooling of a reactor, where exothermic reactions are taking place, is of the utmost importanceand the FTS is no exception to this. Therefore, an analysis on the temperature of the coolant temperaturewas performed.
0 1 2 3 4 5 6 7 8 9 10 11 12
2
4
6·10−4
Reactor’s length, m
C1
form
atio
nra
te,m
ol/kg c
at/s
498 K503 K508 K
Figure 6.32: C1 formation rate throughout the reactor for different coolant temperatures.
0 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
·10−4
Reactor’s length, m
C5+
form
atio
nra
te,m
ol/kg c
at/s
498 K503 K508 K
Figure 6.33: C5+ formation rate throughout the reactor for different coolant temperatures.
62
0 1 2 3 4 5 6 7 8 9 10 11 12
500
510
Reactor’s length, m
Tem
pera
ture
,K
498 K503 K508 K
Figure 6.34: Temperature profiles throughout the reactor for different coolant temperatures.
From figures 6.32 and 6.33 it can be noted that for higher temperatures of the coolant both formationrates are improved, since the system is not at equilibrium an increase in the system’s temperature willresult in an increase in the reaction rates.
Nevertheless, increasing the temperature too much might lead to a thermal runaway process whichhas to be kept in mind when trying to optimise the reactor’s temperature.
Table 6.7: KPI for different coolant temperatures.
Coolant temperature (K) 503 498 Relative variation 508 Relative variation
Being the FTS reactants in the gas phase the inlet pressure will affect the rate of reaction by changingthe volumetric flowrate, thus changing the space velocity. Besides, the increased inlet pressure willalso increase the solubility of the reactants in the FT wax which will influence the reaction rates andselectivities.
63
15 20 25 30 35 40 45 50
50
60
70
80
90
Gas inlet pressure, bar
Con
vers
ion,
%
COH2
Figure 6.35: CO and H2 conversion for different values of gas inlet pressure.
15 20 25 30 35 40 45 50
10
12
14
16
18
20
Gas inlet pressure, bar
Mol
arse
lect
ivity
,%
C1
Figure 6.36: C1 molar selectivities for different values of gas inlet pressure.
15 20 25 30 35 40 45 50
1
1.5
2
2.5
3
3.5
4
4.5
Gas inlet pressure, bar
Mol
arse
lect
ivity
,%
C2
C3
C4
Figure 6.37: C2, C3 and C4 molar selectivities for different values of gas inlet pressure.
15 20 25 30 35 40 45 50
70
72.5
75
77.5
80
82.5
Gas inlet pressure, bar
Mol
arse
lect
ivity
,%
C5+
Figure 6.38: C5+ molar selectivities for different values of Gas inlet pressure.
64
15 20 25 30 35 40 45 500.05
0.06
0.07
0.08
0.09
0.1
Gas inlet pressure, bar
Cat
alys
tpro
duct
ivity
,kg/kg
cath
Catalyst Productivity
Figure 6.39: Catalyst productivity for different values of gas inlet pressure.
The increased CO and H2 solubility for higher pressures results in an decreased H2/CO ratio at thepellet core - figure 6.40 - which results in an increased molar selectivity towards C5+ compounds whichcan be observed in figures 6.37, 6.36 and 6.38.
15 20 25 30 35 40 45 50
200400600800
1,0001,2001,4001,600
Gas inlet pressure, bar
H2/
CO
mol
arra
tioin
the
cent
reof
the
pelle
t H2/CO
Figure 6.40: H2/CO molar ratio for different values of gas inlet pressure.
65
66
Chapter 7
Conclusion and Future Work
7.1 Conclusion
The literature review performed in this work shows that alternatives to the current paths to producefuels and other carbon derived products are needed to account for the rapid rate at which the fossilresources are being consumed. Being the Fischer-Tropsch synthesis (FTS) one of such alternatives,a lot of research has been directed to this topic since it was first discover in the second decade ofthe 90’s. One type of equipment capable of carrying out the FTS is the trickle bed reactor (TBR).Even though this kind of reactor has an improved mass and heat transfer coefficients, which are a majorconcern in the Fischer-Tropsch (FT) process, it is a piece of equipment with a complex hydrodynamic andthermodynamic interactions as well as other complex phenomena such as, for instance, mass transferwith multi-component diffusion happening inside the catalyst pores.
The modelling and optimisation of the TBR are of the utmost importance when improving the effi-ciency of such processes hence the increased demand on increasingly detailed models to accuratelypredict the outputs of this process. Process Systems Enterprise, Ltd (PSE) already had the libraryneeded to simulate the a TBR for the FTS, however some other levels of detail were required for someof the models present, either for increasing the accuracy of the predictions or to provide the user with abroader range of options when modelling their own process.
The kinetic model implemented in this work accounted for the different chain growth probability whichis a major improve to the models commonly used that assume a constant value for this parameter. Byaccounting for the change in the probability of said (n)-chain being formed from the previous (n-1)-chainthe results are significantly better since the probability for light hydrocarbon chains is not same as theheavier hydrocarbon chains, which after a certain point can be considered constant.
The vapour-liquid equilibria (VLE) is another important step in the modelling of this process sincethe reactants and some products are in the gas phase whereas the heavier products are in the liquidphase, therefore the solubility of each and every component must be predicted. The asymptotic beha-viour correlation (ABC) developed by Marano and implemented in this work showed good results whencompared to the PSRK-NRTL equation-of-state (EoS) fitted by PSE since it was based on experimentaldata and the latter lacks experimental validation on the binary interactions on some of the components.
Regarding the thermophysical properties and the heat and mass transfer, some additional correla-tions were successfully implemented to either improve the predictions fidelity with experimental data oradd some additional options to custom modelling to the user.
A model order reduction (MOR) was performed and it was concluded that assuming a uniform pro-file in the radial direction was acceptable with relative errors on the key performance indicators (KPIs)below 0.1%, for the given conditions, this simplification reduces the number of equations by 385 153
67
equations. However, neglecting the gradients throughout the pellet showed significant deviations up to59%, therefore the model becomes too simplified preventing it to perform accurate predictions. Despitethe computing time of this model not being typically long - up to 3 to 4 hours - since it is supposed to beused to continuously optimise a process, the computing time must be minimised.
The sensitivity analysis performed on the numerical parameters showed that the model is very sens-itive to changes in the carbon number as well as in the bed and radial pellet discretisation domain. Theanalysis performed on the other two groups of parameters (design and operating parameters) showedthat there are small external diffusion limitations; the quantity of catalyst used and the H2/CO heavilyinfluence the performance of the process; the coolant temperature also has a significant influence inthe reactor since the FTS is a highly exothermic set of reactions. An increase in the gas inlet pressureshowed an improvement in the reactants’ conversion as well as an improvement on the molar selectivitytowards c5+ compounds.
7.2 Future Work
Given the intricacy of the TBRs in addition to the complexity of the FTS there’s still work to bedeveloped in order to improve the fidelity of this model.
In what concerns the kinetic model, a simplification could be performed by using a constant valueof the chain growth probability, without reducing the fidelity of the model, so the number of equations isreduced, thus reducing the computing time.
The binary interaction parameters in the equation-of-state fitted by PSE should be estimated basedon experimental data for the components whose this parameter is missing, in order to have a fullyfunctional EoS able to successfully predict the VLE for the whole set of components.
Finally, a correlation for the effectiveness factor (η) of the catalyst should be derived to accuratelypredict both the conversion of reactants and the selectivities of the hydrocarbon chains when the lumpedpellet model is used. A better prediction could be attempted in order to better match the Mo1DB1DPpredicted effectiveness factor, thus improving the lumped pellet model results.
On account of the pellet, a whole model for the deactivation of the catalyst should be developed sothat the loss of catalyst activity can also be accounted to better model the performance of the reactor.
68
Bibliography
[1] M. E. Dry and A. P. Steynberg, Eds., Fischer-Tropsch Technology, 1st ed. Elsevier, 2004.
[2] B. Todic, W. Ma, G. Jacobs, B. H. Davis, and D. B. Bukur, “CO-insertion mechanism based kineticmodel of the Fischer-Tropsch synthesis reaction over Re-promoted Co catalyst,” Catalysis Today,vol. 228, pp. 32–39, 2014. [Online]. Available: http://dx.doi.org/10.1016/j.cattod.2013.08.008
[3] D. Leckel, “Diesel Production from Fischer-Tropsch: The Past, the Present, and New Concepts,”Energy and Fuels, vol. 23, no. 6, pp. 2342–2358, 2009.
[4] M. Stamenic, V. Dikic, M. Mandic, B. Todic, D. B. Bukur, and N. M. Nikacevic, “Multiscale andMultiphase Model of Fixed Bed Reactors for Fischer-Tropsch Synthesis: Intensification PossibilitiesStudy,” Industrial and Engineering Chemistry Research, vol. 56, no. 36, pp. 9964–9979, 2017.
[5] M. E. Dry, “The Fischer-Tropsch process: 1950-2000,” Catalysis Today, vol. 71, no. 3-4, pp. 227–241, 2002.
[6] O. Wakamura, “Development of GTL (Gas to Liquid) Technology,” Nippon Steel Technical Report,no. 92, p. 6, 2005.
[7] J. Clemente, “How Much Oil Does the World Have Left?” 2015. [Online]. Available:https://www.forbes.com/sites/judeclemente/2015/06/25/how-much-oil-does-the-world-have-left
[8] World Energy Council, “2010 Survey of Energy Resources,” Tech. Rep., 2010.
[9] S. Reiss, “Tapping the Rock Field,” 2005. [Online]. Available: https://www.wired.com/2005/12/oilshale/
[10] I. Kroschwitz and M. Howe-Grant, “Encyclopedia of Chemical Technology,” in Journal ofthe American Chemical Society, 4th ed. Wiley & Sons, 1996, vol. 17. [Online]. Available:http://pubs.acs.org/doi/abs/10.1021/ja9656107
[11] I. C. Yates and C. N. Satterfield, “Intrinsic Kinetics of the Fischer-Tropsch Synthesis on a CobaltCatalyst,” Energy & Fuels, vol. 5, no. 1, pp. 168–173, 1991.
[12] J. Patzlaff, Y. Liu, C. Graffmann, and J. Gaube, “Interpretation and kinetic modeling of productdistributions of cobalt catalyzed Fischer-Tropsch synthesis,” Catalysis Today, vol. 71, no. 3-4, pp.381–394, 2002.
[13] IUPAC, Compendium of Chemical Terminology. Wiley, 2014. [Online]. Available: http://goldbook.iupac.org/PDF/goldbook.pdf
[14] D. Fortsch, K. Pabst, and E. Groß-Hardt, “The product distribution in Fischer-Tropsch synthesis:An extension of the ASF model to describe common deviations,” Chemical Engineering Science,vol. 138, pp. 333–346, 2015. [Online]. Available: http://dx.doi.org/10.1016/j.ces.2015.07.005
[15] W. Ma, G. Jacobs, T. K. Das, C. M. Masuku, J. Kang, V. R. R. Pendyala, B. H. Davis, J. L. Klettlinger,and C. H. Yen, “Fischer-tropsch synthesis: Kinetics and water effect on methane formation over25%Co/γ-Al2O3 catalyst,” Industrial and Engineering Chemistry Research, pp. 2157–2166, 2014.
[16] C. H. Bartholomew, “Mechanism of catalyst deactivation.” Applied Catalysis A: General, vol. 212,pp. 17–60, 2001.
[17] P. A. Ramachandran, M. P. Dudukovic, and P. L. Mills, “A new model for assessment of externalliquid-solid contacting in trickle-bed reactors from tracer response measurements,” Chemical En-gineering Science, 1986.
[18] P. Harriott, Chemical Reactor Design, 1st ed. CRC Press, 2003.
[19] V. V. Ranade, R. V. Chaudhari, and P. R. Gunjal, Trickle Bed Reactors; Reactor Engineering &Applications. Elsevier, 2011.
[20] J. G. Boelhouwer, H. W. Piepers, and A. A. Drinkenburg, “Particle-liquid heat transfer in trickle-bedreactors,” Chemical Engineering Science, vol. 56, no. 3, pp. 1181–1187, 2001.
[21] G. F. Froment and K. B. Bischoff, “Chemical Reactor Analysis and Design,” p. 664, 1990. [On-line]. Available: http://trove.nla.gov.au/work/11802998?selectedversion=NBD1251091%5Cnhttp://books.google.com/books?id=Y9tTAAAAMAAJ&pgis=1
[22] R. J. Madon and E. Iglesia, “Hydrogen and CO intrapellet diffusion effects in ruthenium-catalyzedhydrocarbon synthesis,” pp. 428–437, 1994.
[23] C. Erkey, J. B. Rodden, and A. Akgerman, “Diffusivities of Synthesis Gas and n-Alkanes in Fischer-Tropsch Wax,” Energy and Fuels, vol. 4, no. 3, pp. 275–276, 1990.
[24] J. J. Marano and G. D. Holder, “Characterization of Fischer-Tropsch liquids for vapor-liquid equilibriacalculations,” Fluid Phase Equilibria, vol. 138, no. 1-2, pp. 1–21, 1997.
[25] A. G. Dixon, “Fixed bed catalytic reactor modelling - the radial heat transfer problem,” CanadianJournal of Chemical Engineering, vol. 90, no. 3, pp. 507–527, 2012.
[28] J. J. Marano and G. D. Holder, “General Equation for Correlating the Thermophysical Propertiesof n -Paraffins, n -Olefins, and Other Homologous Series. 3. Asymptotic Behavior Correlationsfor PVT Properties,” Industrial & Engineering Chemistry Research, vol. 36, no. 5, pp. 1895–1907,1997. [Online]. Available: http://pubs.acs.org/doi/abs/10.1021/ie960512f
[30] K. M. Brunner, J. C. Duncan, L. D. Harrison, K. E. Pratt, R. P. S. Peguin, C. H. Bartholomew,and W. C. Hecker, “A Trickle Fixed-Bed Recycle Reactor Model for the Fischer-Tropsch Synthesis,”International Journal of Chemical Reactor Engineering, vol. 10, 2012.
[31] P. M. Maitlis and A. de Klerk, Greener Fischer-Tropsch Processes for Fuels and Feedstocks, 2013.
[32] F. Lemos, J. M. Lopes, and F. R. Ribeiro, Reactores Quımicos, 1st ed. Lisbon: IST Press, 2002.
[33] B. Todic, W. Ma, G. Jacobs, B. H. Davis, and D. B. Bukur, “Corrigendum: CO-insertion mechanism based kinetic model of the Fischer-Tropsch synthesis reaction overRe-promoted Co catalyst,” Catalysis Today, vol. 242, no. PB, p. 386, 2015. [Online]. Available:http://dx.doi.org/10.1016/j.cattod.2014.08.020
[34] B. Todic, T. Bhatelia, G. F. Froment, W. Ma, G. Jacobs, B. H. Davis, and D. B. Bukur, “KineticModel of Fischer–Tropsch Synthesis in a Slurry Reactor on Co–Re/Al2O3 Catalyst,” Industrial& Engineering Chemistry Research, vol. 52, no. 2, pp. 669–679, 2013. [Online]. Available:http://pubs.acs.org/doi/abs/10.1021/ie3028312
The mechanism steps and the model’s equation are from Todic et. al [2] with the exception of equation(A.6) which was mistyped in [2] and corrected in [33].
A.1.1 CO-insertion mechanism steps
The labels of the mechanism steps are the notation for rate constant (ki) and adsorption rate constant(Ki).
CO + S CO-S K1
H2 + 2 S 2H-S K2
CO S + H S CHO S + S
CO S + CH3 S CH3CO S + S k3
CO S + CnH2n+1 S CnH2n+1CO S + S n = 2, 3, ...
CHO S + H S CH2O S + S
CH3CO S + H S CH3CHO S + S K4
CnH2n+1CO S + H S CnH2n+1CHO S + S n = 2, 3, ...
CH2O S + 2H S CH3 S + OH S + S
CH3CHO S + 2H S CH3CH2 S + OH S + S K5
CnH2n+1CHO S + 2H S CnH2n+1CH2 S + OH S + S n = 2, 3, ...
OH S + H S H2O + 2S K6
CH3 S + H S CH4 + 2S k7M
73
CnH2n+1 S + H S CnH2n+1 + 2S n = 2, 3, ... k7
C2H5 S C2H4 + H S k8E
CnH2n+1 S CnH2n + H S k8,n
A.1.2 Kinetic model’s equations
The equation derived are based on the gas phase, thus the partial pressure of the ith component(Pi) must be calculated for each component according to equation (A.1):
Pi = yiP (A.1)
Where the yi is the molar fraction of the ith component in the gas phase and P the total pressure of thesystem.
Since the model is based on chain growth CO-insertion mechanism, the chain growth probability (αn)has to be calculated as follows:
α1 =k3K1PCO
k3K1PCO + k7M√K2PH2
(A.2)
α2 =k3K1PCO[S]
k3K1PCO[S] + k7√K2PH2[S] + k8Eec·2
(A.3)
αn =k3K1PCO[S]
k3K1PCO[S] + k7√K2PH2[S] + k8ec·n
(A.4)
Where ki is the rate constant and Ki is the adsorption rate constant. The constants k7M , k7, k8E andk8 represent the reaction rate for the methane, every other paraffins, ethylene and every other olefins,respectively.
[S] represents the fraction of vacant sites on the catalyst surface, c is related to the weak Van derWaals (VdW) interactions through equation (A.5):
c = −∆E
RT(A.5)
Where ∆Ei is the change in 1-olefin desorption activation energy caused by weak force interactions,R and T are the ideal gas constant and temperature, respectively.
The fraction of vacant sites is calculated by solving the site balance through equation (A.6):
[S] =1/
[1 +K1PCO +
√K2PH2 +
(1
K22K4K5K6
PH2O
P 2H2
+√K2PH2
)α1 + α1α2 + α1α2
n∑i=3
i∏j=3
αj
](A.6)
The reaction rate constants (ki) and equilibrium and adsorption constants (Ki) are calculated throughequations (A.7) and (A.8), respectively [34]:
ki = Aie− EiRT (A.7)
Ki = Aie−Ei,adsRT (A.8)
74
Table A.1: Number of equations present in model and variables assigned - degrees of freedom (DOF)analysis.
Equation Number of Equations Variable Number of Variables
(A.1) 3PH2, PCO, PH2O,xgH2
, xgCO, xgH2O,
P7
(A.2) 1k3, k7M ,K1, K2,α1
5
(A.3) 1[S],
k7, k8E ,α2, c
5
(A.4) maximum number of carbons (Cmax)-2 k8,αn, n Cmax
(A.5) 1 ∆E, T 2(A.6) 1 K4, K5, K6 3(4.1) 1 RCH4
1(4.2) Cmax-1 RCnH2n+2
Cmax-1(4.3) 1 RC2H4 1(4.4) Cmax-2 RCnH2n Cmax-2
(A.7)+(A.8) 10 AiEi, Ei,ads
20
Total 3Cmax+14 Total 3Cmax+41 DOF=27Assigns Number of Variables Assigned
Ai 10Ei 5
Ei,ads 5∆E 1
Total 21 DOF=6Inputs Number of InputsT 1P 1xgi 3
Cmax 1
Total 6 DOF=0
75
A.1.3 Degree of Freedom Analysis
A.1.4 Model comparison
2 4 6 8 10 12 1410−4
10−3
10−2
Carbon number
Tota
lhyd
roca
rbon
form
atio
nra
te,
mol/g c
at/h
gPROMS ModelTodic model
(a)
2 4 6 8 10 12 140
0.5
1
1.5
2
Carbon number
1-ol
efin/
n-pa
raffi
nra
tio
gPROMS ModelTodic model
(b)
Figure A.1: (a) Total hydrocarbon formation i.e. ASF plot; (b) 1-olefin to n-paraffin ratio
A.2 Stamenic’s model
Paraffin and olefin production using the Stamenic model, figure A.2.
1 5 10 15 20
10−4
10−3
10−2
Carbon number
Par
affin
prod
uctio
n,mol/g
cat/h
(a)
2 5 10 15 20
1 · 10−4
2 · 10−4
3 · 10−4
4 · 10−4
Carbon number
Ole
finpr
oduc
tion,
mol/g c
at/h
(b)
Figure A.2: (a) Paraffin production; (b) Olefin production
76
Appendix B
Vapour-liquid equilibria (VLE)
Equation (B.1) is used to calculate the fugacity coefficients through Henry coefficients.
ϕLi = xiHicL (B.1)
Figure B.1 shows the dependence of the Henry coefficient with the temperature.
Since the model must be robust enough to account for different catalyst diameter (dp), the parametersfrom table D.1 were plotted and fitted against dp.
0.001 0.002 0.002 0.003 0.003 0.004 0.004 0.005
0.3
0.35
0.4
particle diameter, dp (m)
(αβ
) g
Brunner’s parametersFitting
Figure D.1: Fitting of the (αβ)g term against dp
81
0.001 0.002 0.002 0.003 0.003 0.004 0.004 0.005
0.16
0.18
0.2
particle diameter, dp (m)
a
Brunner’s parametersFitting
Figure D.2: Fitting of the a term against dp
0.001 0.002 0.002 0.003 0.003 0.004 0.004 0.005
0.010
0.020
0.030
particle diameter, dp (m)
b
Brunner’s parametersFitting
Figure D.3: Fitting of the b term against dp
Where the trend-lines are as follows, respectively:
(αβ)g = −38.88dp+ 0.4503 (D.4)
a = −15.56dp+ 0.2153 (D.5)
b = −6.99dp+ 0.0349 (D.6)
Where a and b are parameters to calculate the (αβ)l, as follows:
(αβ)l = a(1 + bReg) (D.7)
D.2 Bed-Wall Heat Transfer
The gas phase contribution (htcw,g) is divided into static (equation (D.9)) and dynamic contributions(equation (D.10)). The liquid phase contribution is calculated through the Nusselt number (Nu), howeverequation (D.11) is rearranged for the htcw,l term to be explicit.
