Mathematical Modelling of Trickle-Bed Reactors forFischer-Tropsch Synthesis

Diogo M. Mosteiasa, Henrique A. Matosa, Stepan Spatenkab, Vasco Manacasb

aChemical Engineering Department, Instituto Superior Tecnico, Lisbon, PortugalbProcess Systems Enterterprise, Ltd., London, United Kingdom

October 2018

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 depletedhas 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, therefore threedifferent 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 reactor bedaxially and radially distributed and the pellet model is also distributed; the second one has the reactor bedaxially distributed as well as the pellet; the last and simplest model has the reactor bed axially distributedbut 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 errorstypically below 0.1%, the case study was based on one of the ARGE reactors from the Sasolburg FTplant [1]. However, when the axially and pellet distributed model was compared to the lumped pellet modelit was 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 order reduction

1. Introduction

Due to the growing energy demand and environ-mental regulations there has been a chase forclean synthetic fuel production. Because of that,the interest in FTS has been rekindled during thefirst decade of the 21st century. The FT technologyis quite old with coal utilisation roots. It was dis-covered in 1923 by Franz Fischer and Hans Trop-sch at the Kaiser Wilhelm Institute for Coal Re-search in Mullheim and first applied in Germanyin the 1930’s during the WWII. Being a coal-richcountry, Germany used FT process in order to pro-duce ersatz (replacement) fuels, it accounted for9% of German’s war fuels production and 25%of the auto mobile fuel production. However, thistechnology was expensive and very inefficient and

could not compete against cheap and readily avail-able crude oil. Regardless, the technology was sofascinating that research and technology develop-ment were carried on even when commercial ap-plications seemed unlikely.[2, 1, 3]

The most interesting fact about FTS is that agaseous mixture of H2 and CO, also known as syn-gas, enters the reactor and a hydrocarbon liquidexits it. Thermodynamically, the preferred prod-uct would be methane, however the predominantproducts are heavier hydrocarbons. The FTS hassyngas as feedstock that nowadays comes mostlyfrom natural gas through the steam methane re-forming (SMR) process1.[2, 1]

1Syngas can also come from coal burning but the steammethane reforming is most common pathway.

1

The conversion of natural gas to hydrocarbons(Gas-to-Liquids (GTL) route) is currently one of themost promising topics in the energy industry dueto the possibility of using remote natural gas else-where, to produce clean fuels, specialty chemi-cals and/or waxes. On the other hand, coal andheavy residues can be used on sites where theseare available at reduced costs. The products fromthe FTS can then be used to produce value addedproducts and/or fuels.[2, 1, 4]

Nonetheless, the decision on whether to build aFT plant or not lies in the assessment of the riskperceived from the future price and availability ofpetroleum crude oil as well as on local politics.FTS complexes need a heavy capital investmentspecially the section of production of purified syn-gas because its composition has to match the ratioof H2/CO used for this process, which depends onproduct selectivity.[5]

2. Fischer-Tropsch SynthesisThe FTS is extremely dependent on the syn-gas composition, hence the hard requirement ofpreparing it carefully. The syngas is preparedfrom carbonaceous feedstock, the only essentialrequirement is that the feed contains carbon and,preferably hydrogen. Otherwise the latter has tobe obtain from the scission of the water moleculeswhich requires large amounts of energy in order tobe achieved.[1]

CO, CO2 and H2 are produced through SMR,where the CO2 must be removed by its total or par-tial recycling back to the reformer. The CO2 formedduring FTS can also be recycled back to the re-former in order to reduce the fresh feed of naturalgas to the whole process [1]. Technologies such asautothermal reforming, pressure swing adsorption(PSA), methanation, water gas-shift (WGS) can beused to obtain the FTS desired H2/CO ratio, thatcannot be obtain through SMR alone.

FTS is a catalytic reaction in which syngas isconverted into a broad range of hydrocarbons.

nCO + 2 nH2 CnH2n + nH2O (1)(2n+1)H2 + nCO CnH2n+2 + nH2O (2)

The equations (1) and (2) represent the reactionsfor the formation of olefins and paraffins, respec-tively [6]. The side reactions are as follows:

2nH2 + nCO CnH2n+2O + (n-1)H2O (3)2 CO C + CO2 (4)

CO + H2O CO2 + H2 (5)

Where the equations (3), (4) and (5) represent theformation of alcohols, the Boudouard reaction for

Figure 1: Detailed CO-insertion mechanism.[1]

the complete oxidation of CO to CO2 and the water-gas shift reaction (WGSR), respectively.[6]

This process is carried out, mainly, at two tem-perature ranges:

• 300 to 350°C - High temperature FTS(HTFTS) in the presence of iron-based cata-lyst used for the production of gasoline andlinear low molecular mass olefins;

• 200 to 250°C - Low temperature FTS(LTFTS) in the presence of either iron orcobalt-based catalyst used for the productionof high molecular mass linear waxes.

Over cobalt-based catalyst, the FTS producesmostly n-alkanes and 1-alkenes (equations (1) and(2)). As previously explained, this catalyst is notvery active towards the WGSR, therefore only asmall fraction of the water produced is convertedto CO2.[7]

2.1. CO-insertion mechanismThe CO-insertion can be summarized in the follow-ing steps, the chemisorbed CO is the monomerand the chain initiator is thought to be a surfacemethyl 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 detailedinitiation step is represented in figure 1.[1]

3. Reaction kineticsThe model developed by Todic et al.[2] describesthe formation of hydrocarbon species based on theCO-insertion mechanism, assuming that the chaingrows by addition of monomers to it (as describedin 2.1). This addition is based on the probabilityof the next polymer being formed from the currentchain2.

2”Next” and ”current” polymer/chain is related to their carbonnumber, e.g. if the current chain has 25 carbons in its structure

2

Figure 2: Chain growth probability.

The Todic’s model is described by the followingequations:

α1 =k3K1PCO

k3K1PCO + k7M√K2PH2

(6)

α2 =k3K1PCO[S]

k3K1PCO[S] + k7√K2PH2

[S] + k8Eec·2(7)

αn =k3K1PCO[S]

k3K1PCO[S] + k7√K2PH2

[S] + k8ec·n(8)

Where ki is the rate constant and Ki is the ad-sorption rate constant. The constants k7M , k7, k8Eand k8 represent the reaction rate for the methane,every other paraffins, ethylene and every otherolefins, respectively.

[S] represents the fraction of vacant sites on thecatalyst surface, c is related to the weak Van derWaals (VdW) interactions through equation (9):

c = −∆E

RT(9)

Where ∆Ei is the change in 1-olefin desorptionactivation energy caused by weak force interac-tions, R and T are the ideal gas constant and tem-perature, respectively.

The fraction of vacant sites is calculated by solv-ing the site balance through equation (10):

[S] = 1/

[1 +K1PCO +

√K2PH2

+(1

K22K4K5K6

PH2O

P 2H2

+√K2PH2

)α1 + α1α2 + α1α2

n∑i=3

i∏j=3

αj

](10)

The rates of production can be calculated by the

the next polymer will have 26 carbons.

following expressions:

RCH4= k7MK

0.52 P 0.5

H2α1 · [S]2 (11)

RCnH2n+2= k7K

0.52 P 0.5

H2α1α2

n∏i=3

αi · [S]2 n ≥ 2

(12)

RC2H4= k8Ee

c·2α1α2 · [S]2 (13)

RCnH2n= k8e

c·nα1α2

n∏i=3

αi · [S] n ≥ 3

(14)

Where equations (12) and (14) are the rate of pro-duction of all paraffins and olefins but methane(equation (11)) and ethylene (equation (13)), re-spectively.

The reaction rate constants (ki) and equilib-rium and adsorption constants (Ki) are calculatedthrough equations (15) and (16), respectively [8]:

ki = Aie− EiRT (15)

Ki = Aie−Ei,adsRT (16)

The parameters needed for equations (15) and(16) can be found in [2].

4. Vapour-liquid equilibrium calculationsVapour-liquid equilibria (VLE) determines how thechemical species are distributed between thevapour and liquid phase. This distribution has astrong thermal dependency, meaning that, for acertain temperature value, the concentration (orpartial pressure) of a component in the vapourphase has a corresponding concentration in the liq-uid phase and the same for the other way around.

The VLE calculation in the model can be doneby using the thermodynamic model defined by PSEto use the PRSK-NRTL equation-of-state (EoS) tocalculate the fugacity coefficients in order to predictthe solubility of the gases in the liquid phase.

Alternatively, the correlation developed byMarano et. al [9] was implemented as part of this

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work. The correlation calculates the Henry coeffi-cients, thus predicting the solubility of the gases inthe liquid phase.

The Henry coefficient values, obtained by bothmethods, are presented in table 1 for the entranceand centre of the reactor.{

lnHi = Hi,0 − n ·∆Hi

Hi,0 = β1 + β2

T + β3 · lnT + β4 · T 2 + β5

T 2

(17)

The temperature-dependent ABC parameters(β1,...,β5) and the Henry constant differential be-tween the ith component and the wax averagecarbon number (∆Hi) presented cover the hy-drocarbons, as well as the reactants, used inthe Advanced Model Library for Trickle-Bed Reac-tors (AML:TBR) with exception of the n-butane3.The parameters for the n-butane were estimatedthrough linear interpolation between propane andhexane. The resulting Infinite-Dilution Henry’s con-stant (Hi) is presented in figure ?? along withthe other compounds’ Hi - in order to obtain thisplot the wax average carbon number (n) was setto 28 in accordance with [10]. Henry’s constantfor n-butane follows, mainly, the behaviour of thepropane due to the interpolation procedure.

the Multiflash’s predictions for H2 and CO arefairly close to the ones predicted by the Marano’scorrelation, this is because the binary interactionparameters for these two components with theothers present were fitted to experimental datawhereas for CO2, N2 and from C1 to C4 are oneorder of magnitude higher than the ones predictedfrom Marano’s correlation since these lack the ex-perimental validation.

From table 1 one can notice that the Multiflash’spredictions for H2 and CO are fairly close to theones predicted by the Marano’s correlation, thisis because the binary interaction parameters forthese two components with the others presentwere predicted from experimental data whereas forCO2, N2 and from C1 to C4 are one order of magni-tude higher than the ones predicted from Marano’scorrelation since these lack the experimental vali-dation.

5. DiffusionDiffusion is one of the properties with a major im-pact in FT process performance, this is due to dif-ferent two-phase interfaces that the reactants mustdiffuse through to the inside of the pellet and thereaction products have to diffuse in the oppositedirection, assuming that the pellet is covered in FTwax (C28), being this the bulk compound.

3The β1,...,β5 and ∆Hi for ethylene, propylene and n-hexane are also presented but these were ignored due to theway the model is built.

Erkey et. al [10] measured the diffusivities of H2,CO, CO2, C8H18, C12H26 and C16H34 at 14 bar andat three different temperatures (475, 504 and 536K), being the liquid bulk component C28. How-ever, the model being developed in this work usesmethane, ethane, propane and n-butane which arenot present in the aforementioned paper.

To overcome this limitation the Multiflash soft-ware was used to obtain the diffusivities of themissing components as well as the hydrocarbonsstudied in the paper in C28, using the Hayduk-Minhas method. These results, when comparedto the ones in the paper, have deviations that mustbe taken into account.

The ratios between the Multiflash diffusivitiesand Erkey’s results for each carbon number werecalculated and plotted for 504 K, assumed as ref-erence temperature, and 14 bar; then a second de-gree polynomial trend-line was fitted to the ratiosand the missing diffusivities were calculated by di-viding the Multiflash diffusivities by the predictedratio hence obtaining an estimation of Erkey’s dif-fusivities for the missing pseudo-components.

5.1. ViscosityThe viscosity of the bulk component, in this caseassumed as C28, will heavily affect the diffusionrate.

Marano et al. [11] developed a generalisedasymptotic behaviour correlation (ABC) - equation(18) - to predict the liquid viscosity of n-paraffin andn-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

(18)

The values predicted by equations (18) are in cP.The parameters in equations (18) are present in[11]. In the second equation of (18), the ∆µ pa-rameter represents, both, ∆µ∞ and ∆µ0.

5.2. Diffusion’s temperature and viscosity depen-dence

It is known that the diffusion coefficient has a de-pendence on the temperature and viscosity. For475 and 536 K the missing coefficients were alsopredicted with the aforementioned method.

In order to implement it in gPROMS, an equa-tion containing both dependences had to be de-veloped. Equation (19) shows the generic form ofsuch equation.

DAB = DAB,ref

(T

Tref

)a(µ

µref

)b(19)

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Table 1: Henry coefficients predicted (bar) through Marano’s correlation and through PSRK-NRTL EoS.

Comp. Literature† MF‡ Relative errorH2 503.00 503.66 0%CO 385.04 395.33 3%CO2 152.77 1322.31 766%N2 455.66 4135.67 808%

H2O 49.23 26.19 47%C1 207.78 2434.13 1071%C2 99.05 752.14 659%C3 53.57 317.34 492%C4 32.45 143.94 344%

†Literature = Marano et al.[9]

‡MF = Multiflash software using PSRK-NRTL EoS

Figure 3: Ratio between Multiflash’s and Erkey’s diffusivity coefficients.

The a and b are correlating parameters to be es-timated. DAB,ref was calculated assuming 504 Kas reference temperature.

To predict the dependence, a and b were set to 1and an hypothetical DAB was calculated at 475 Kand 536 K for each compound.

The Ordinary Least Squares (OLS) method wasthen used for both temperatures at the same time.Equation (20) show that for this temperature rangeits dependence is linear, meaning that parametera is 1 and the dependence on the viscosity has anorder of -0.782.

DAB = DAB,ref

(T

Tref

)(µ

µref

)−0.782(20)

It should be noted that this parameter estimationwas performed with diffusion coefficient for threecompounds covering a temperature range of 61 K.Besides not being a very broad range, the estima-tion is based on the predicted values performed in5.

6. Heat and Mass transferDue to the high exothermicity of the FTS, heattransfer is a major concern when modelling thesekind of processes.

Besides, the heat and mass transfer betweenthe different two-phases interfaces, the model mustalso take into account the conductivity of the bedas well as the heat duty that is transferred to thewall of the reactor.

The correlations implemented for the Gas-Liquidheat transfer and Liquid-Solid heat and mass trans-fer coefficients are the ones present in the Sta-menic paper[4], with exception for the effective bedconductivity and the bed-wall heat transfer, as fol-lows.

6.1. Effective Bed Conductivity

The effective radial bed conductivity accounts forthe temperature profile from the reactor’s centre tothe wall, exclusively.

The model only has one variable for the effectivebed conductivity whereas the correlation present inStamenic’s paper [4] has two separate variables forthe gas and liquid phase, however the Stamenicpaper uses the correlation present in Brunner’s pa-per [12]. In the latter paper, the effective bed con-ductivity has only one parameter, thus being thisexpression - equation (21) - the one implemented

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Figure 4: Multiflash’s, Erkey’s and predicted diffusivity coefficients.

in the gPROMS model.

λer = λser + λler + λger (21)λer = 1.5λl + (αβ)gλgRegPrg + (αβ)lλlRelPrl

(22)

The terms λser, λler and λger have expressions oftheir own that are explicit in [12]. The full equationfor the effective bed conductivity is (22).

This correlation is another improvement to themodel since it grants the users the ability to bettermodel their processes, in case the previous cor-relations couldn’t represent, accurately, their pro-cess.

6.2. Bed-Wall Heat TransferThis coefficient accounts for the temperature pro-file in the thin layer next to the wall.

As it happened in 6.1 the model only takes onevariable for this parameter, however in [4] it is seg-regated into both phases. To account for this,equation (23) was derived.

htcws =εghtcw,g + εlhtcw,l

ε(23)

The contributions of both phases are stated in[4].

7. Model Order ReductionModel order reduction (MOR) is a technique of re-ducing the computational complexity of mathemat-ical models in order to make the simulations fasterwhile assuring the model fidelity. For instance for amodel that is suppose to be used to continuouslyoptimise a process, a computing time of around 3to 4 hours is impractical, thus the necessity of try-ing to simplify the models while maintaining it’s ac-curacy.

The TBR model with one dimensional (1D) pelletand two dimensional (2D) bed was the first modelto be created, it uses a radially distributed pelletmodel as well as a axially and radially distributedreactor model.

After successfully implementing this model, itwas developed two more models with a sequentialMOR: the reactor model was reduced to axially dis-tributed while the pellet model was kept distributedand then the latter was also reduced to a lumpedpellet model (nildimensional (0D) model).

• Mo2DB1DP - 2D bed (axially and radially dis-tributed) and 1D pellet (radially distributed)model;

• Mo1DB1DP - 1D bed (axially distributed) and1D pellet model;

• Mo1DB0DP - 1D bed and 0D pellet (lumpedpellet) model.

The characteristics of the multi-tubular reactorbase case from which the simplified models werecreated are as follows and are based on the litera-ture [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 met-

ric tonnes per year.

7.1. Mo2DB1DPThe fact that the pellet is distributed implies thatthe temperature and concentration profiles insidethe pellet will not be uniform, this is due to the re-action and diffusion steps occurring either in seriesor in parallel, meaning that a certain molecule ofreactant can 1. keep diffusing towards the centreof the pellet or 2. react at the internal surface of thepellet. [13]

The KPIs for this model are presented in table 2,these stand as reference when studying the fidelityof the following order reduced models.

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Table 2: Mo2DB1DP model KPIs.

KPIs Mo2DB1DP

CO conversion (%) 64.1H2 conversion (%) 77.8C1 molar selectivity (%) 13.7C2 molar selectivity (%) 1.4C3 molar selectivity (%) 3.4C4 molar selectivity (%) 3.0C5+ molar selectivity (%) 78.5Cat. Productivity (kg/kgcat h)) 0.083

7.2. Mo1DB1DPComparing the previous model to this one, itwas performed a model order reduction of one-dimension by neglecting the radial distribution inthe reactor, this reduction is acceptable for narrowtubes where the radial temperature and concentra-tion gradients are not very pronounced.

Figure 5 shows the axial profiles of theMo2DB1DP and Mo1DB1DP models. Regardingthe former at the bed’s centre, near the wall andalso the average temperature of the bed.

From figure 5 one can conclude that the aver-age bed temperature of the Mo2DB1DP matchesthe Mo1DB1DP model temperature profile. How-ever, when looking at the temperatures at the cen-tre of the reactor or near the wall, the differencesbetween them are noticeable.

Taking the point of maximum temperature differ-ence between both models (around the axial po-sition of 0.09) and plotting the temperature radialprofile in this point, figure 6 is obtained.

Figure 6 shows the radial profiles of theMo2DB1DP and Mo1DB1DP models. On the latterthe value is constant since there’s no radial gradi-ents in this model.

From figure 6 it can be noted that there’s a dif-ference of about 3°C spread throughout the radiusof the reactor’s tubes (2.5 cm). This differencehas consequences in the reaction rates, especiallynear the surface of the catalyst, since these areextremely dependent on the temperature.

This model order reduction showed a goodconcordance between the Mo2DB1DP and theMo1DB1DP which means that the latter can beused to accurately reproduce the Mo2DB1DP casestudy. Therefore, for this tube diameter (5 cm) theradial distribution can be neglected.

7.3. Mo1DB0DPThis is the simplest model developed, it not onlyneglects the radial distribution of the reactor butalso neglects the radial distribution on the pellet,turning it into a lumped model.

This simplification implies that the internal dif-fusion limitations are not accounted for, meaningthat the concentration, as well as the temperature,of the reactants in the catalyst layer is the sameas their concentration on the particle’s surface.For this reason an additional parameter should beadded to account for drawback of this simplifica-tion. The Thiele’s module (φ) is used to calculatethe effectiveness factor (η) of the catalyst. In [13] ispresented a general expression for an irreversiblereaction of order n for any particle geometry to cal-culate φ and an expression for the η for sphericalparticles.

From the Mo1DB1DP, the η was calculatedthroughout the axial domain of the reactor beingits average 0.438, which shows that the influenceof the internal diffusion limitation is noticeable.

Even though the Mo1DB0DP showed a fairlygood prediction of the temperature profile whencompared to the Mo1DB1DP, the prediction of theeffectiveness factor is very poor. This leads toroughly good prediction of the reactants’ conver-sion, however the molar selectivity is significantlydifferent from the Mo1DB1DP predictions.

8. Sensitivity AnalysisDue to the complexity of a TBR model, its sensi-tivity analysis (SA) was divided into three differentsections, as follows:

• Numerical parameters - where the most suit-able Cmax as well as discretisation parameterswere chosen among the ones studied in orderto reduce the computing time while aiming ataccurate results;

• Design parameters - this analysis fell uponsome of the reactor’s design parameters suchas length, radius, and gas hourly space veloc-ity (GHSV);

• Operating parameters - these contemplateoperational parameters such as H2/CO ratio,coolant temperature and gas inlet pressure.

To compare the different values studied foreach parameter of each section, some variableswere plotted to better understand the system’s be-haviour. In addition, some KPIs were put togetherin the form of table, these are the same as the onesused in chapter 7.

8.1. Parametric Analysis of Cmax

Regarding computer simulations, there is a trade-off between model detail and computational capa-bility, meaning that a model too detailed will takelonger to be simulated when compared to a lessdetailed one. Thus, a model should be detailedenough to guarantee accurate results within an ac-ceptable run-time.

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Figure 5: Mo2DB1DP and Mo1DB1DP reactor axial temperature profiles.

Figure 6: Radial temperature profiles of the Mo2DB1DP and Mo1DB1DP models at axial position 0.09.

Figures 7 and 8 show a comparison on the re-sults obtained using different values of Cmax4.

From figures 7 and 8 one can deduce that avalue of Cmax of 10 is not enough when comparedto Cmax = 100, besides C28 is considered to be theFT wax which would not be produced if the formervalue of Cmax was used. The decision was to useCmax = 35. That way, the formation of C28 is ac-counted for and the results are very similar to theobtained when using a Cmax of 100.

Regardind the discretisation of the bed and thepellet, several values were tested and the oneschosen were so based on the number of equationsand computing time.

8.2. Sensitivity Analysis on Design ParametersThis analysis was performed over the 1. tubelength 2. tube radius 3. GHSV.

This analysis showed that the C5+ molar selectiv-ity increased with the tube length mainly becauseof the lower average bed temperature. The in-crease in the tube radius increases the bed av-erage temperature which translates in higher con-version and catalyst productivity, however withhigher temperatures, the risk of thermal runawayincreases due to the decreased rate of heat trans-fer from the bed. Regrading the GHSV, its increasepromotes higher C1 molar selectivity, which means

4Values were obtain using the Mo1DB1DP on account for re-ducing the simulation time required for this sensitivity analysis.

that higher values of GHSV promote the formationof methane, since the molar selectivities for C2, C3,C4 and C5+ decrease with the increase of this vari-able.

8.3. Sensitivity Analysis on Operating ConditionsThis analysis was performed over the 1. H2/CO ra-tio 2. coolant temperature 3. gas inlet pressure.

This last analysis showed that low H2/CO ra-tios results in higher selectivity towards C5+ com-pounds. The same conclusion was achieved re-garding the gas inlet pressure analysis where theH2 and CO solubilities increase with the pressurewhich also improved the selectivity towards c5+compounds.

The analysis on the coolant temperature showedthat, since the system is not at equilibrium, an in-crease in the system’s temperature leads to in-crease reaction rates, however the selectivity to-wards C5+ compounds is decreased and the selec-tivity towards light chain hydrocarbons increased.

9. ConclusionsThe trickle bed reactor (TBR) for the Fischer-Tropsch synthesis (FTS) was successfully imple-mented in gPROMS by modelling the reactor’smodel specifics such as kinetics, thermodynamicmodel, transport properties and heat and masstransfer. The reactor was modelled with three dif-ferent levels of detail.

8

Figure 7: C1 formation rate throughout the reactor for different values of Cmax.

Figure 8: C5+ formation rate throughout the reactor for different values of Cmax.

The kinetic model implemented in this work ac-counted for the different chain growth probabilitywhich is a major improve to the models commonlyused that assume a constant value for this parame-ter. By accounting for the change in the probabilityof said (n)-chain being formed from the previous(n-1)-chain the results are significantly better sincethe probability for light hydrocarbon chains is notsame as the heavier hydrocarbon chains, which af-ter a certain point can be considered constant.

The vapour-liquid equilibria (VLE) is another im-portant step in the modelling of this process sincethe reactants and some products are in the gasphase whereas the heavier products are in the liq-uid phase, therefore the solubility of each and ev-ery component must be predicted. The asymptoticbehaviour correlation (ABC) developed by Maranoand implemented in this work showed good resultswhen compared to the PSRK-NRTL equation-of-state (EoS) fitted by PSE since it was based onexperimental data and the latter lacks experimen-tal validation on the binary interactions on some ofthe components.

Regarding the thermophysical properties andthe heat and mass transfer, some additional corre-lations were successfully implemented to either im-prove the predictions fidelity with experimental dataor add some additional options to custom mod-elling to the user.

The models with different levels of detail werecompared against themselves and the conclusionwas that the Mo1DB1DP (1D bed (axially dis-tributed) and 1D pellet model) had an acceptablesimplification, for this case study, when comparedto the Mo2DB1DP (2D bed (axially and radially dis-tributed) and 1D pellet (radially distributed) model)with relative errors below 0.1%, this means thatthe radial profile can be considered negligible, atthe given conditions. Furthermore, it is expectedthe conclusion to be the same for narrower tubesand the differences between the models’ resultseven smaller, whereas for wider tube this analysisshould be performed once again due to the radialgradients being more pronounced in wider tubes.

On the other hand, the Mo1DB0DP (1D bed and0D pellet (lumped pellet) model) showed unsatis-factory results when compared with relative errorsas high as 59%. Therefore, the Mo1DB0DP shouldnot be used to predicted such a complex reactionalsystem such as the FTS at the conditions of thecase study. The key performance indicator (KPI)obtained could not be compared with experimentalones due to their scarcity throughout the literature.

The sensitivity analysis performed on the numer-ical parameters showed that the model is very sen-sitive to changes in the carbon number as well asin the bed and radial pellet discretisation domain.The analysis performed on the other two groups

9

Figure 9: H2/CO molar ratio for different values of gas inlet pressure.

of parameters (design and operating parameters)showed that there are small external diffusion limi-tations; the quantity of catalyst used and the H2/COheavily influence the performance of the process;the coolant temperature also has a significant in-fluence in the reactor since the FTS is a highlyexothermic set of reactions. An increase in the gasinlet pressure showed an improvement in the reac-tants’ conversion as well as an improvement on themolar selectivity towards c5+ compounds.

References[1] M. E. Dry and A. P. Steynberg, Eds., Fischer-

Tropsch Technology, 1st ed. Elsevier, 2004.

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