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Computers and Chemical Engineering 39 (2012) 75– 83

Contents lists available at SciVerse ScienceDirect

Computers and Chemical Engineering

jo u rn al hom epa ge : www.elsev ier .com/ locate /compchemeng

taging of the Fischer–Tropsch reactor with an iron based catalyst

hmad Rafiee, Magne Hillestad ∗

epartment of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Sem Sælandsvei 4, N-7491 Trondheim, Norway

r t i c l e i n f o

rticle history:eceived 29 August 2011eceived in revised form1 November 2011ccepted 18 November 2011vailable online 26 November 2011

a b s t r a c t

The Fischer–Tropsch reactor is sectioned into stages based on the systematic method given by Hillestad(2010). The design functions are optimized to maximize the concentration of C11+ at the end of reac-tor path. The decision variables are fluid mixing, hydrogen distribution, heat transfer area distribution,coolant temperature, and catalyst concentration. With the path temperature constrained by 250 ◦C, stag-ing of the reactor will increase the concentration of C11+. For a three-stage reactor, the concentration isincreased by 2.50% compared to a single-stage reactor. The optimal mixing structure is plug flow to have

eywords:eactor pathtagingischer–Tropsch synthesisesign functionslug flow

the maximum possible conversion. A case study is conducted to separate and distribute hydrogen alongthe reactor path. This will reduce H2/CO at the beginning of the path and increase chain growth prob-ability. The results show that for a three-stage reactor, the concentration of C11+ is increased by 15.93%compared to single-stage reactor.

© 2011 Elsevier Ltd. All rights reserved.

. Introduction

Interests in FT synthesis is increasing rapidly due to the recentmprovements to the technology, clean-burning fuels (low sulphur,ow aromatics) derived from the FT process and the realization thathe process can be used to monetize stranded natural gas resources,ry and Steynberg (2004).

The synthesis gas (mainly hydrogen and carbon monoxide) pro-uced in syngas production unit is converted to a wide range ofydrocarbons in Fischer–Tropsch reactor. Four types of reactorsave been utilized for FT synthesis: The fixed bed tubular reac-or known as the ARGE reactor, circulating fluidized bed reactorsnown as Synthol reactors, Sasol Advanced Synthol reactor andlurry bubble column reactor, Dry and Steynberg (2004), Davis2002), Spath and Dayton (2003). The catalysts mainly used for FTynthesis are iron and cobalt. Iron is very active catalyst and wateras shift (WGS) reaction is active on this catalyst. Cobalt catalystsre not suitable for CTL applications due to the risk of catalyst poi-oning from various impurities present in coal which are difficult toemove to the required low levels. The syngas obtained from coalasification are generally considered to be better suited for pro-

essing by iron catalysts. The Sasol plant in Secunda (South Africa)s an industrial example of iron based Fischer–Tropsch reactor. Theapacity of this plant is 124,000 bbl/day.

∗ Corresponding author.E-mail address: magne.hillestad@chemeng.ntnu.no (M. Hillestad).

098-1354/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.oi:10.1016/j.compchemeng.2011.11.009

In chemical engineering, sectioning of the process into stagesis not a new idea and has been used for many cases. Androulakisand Reyes (1999) studied oxidative coupling of methane (OCM)on a staged plug flow reactor. Maretto and Krishna (2001) haveoptimized the conversion of staged slurry bubble column reac-tor for Fischer–Tropsch synthesis. Miracca and Capone (2001)designed a staged fluidized bed reactor based on theory and exper-iments. Waku, Yu, and Iglesia (2003) have had studies on oxygenintroduction and selective hydrogen combustion during propanedehydrocyclodimerization reactions. Stuber and Delmas (2003)studied the influence of the main operating variables such as tem-perature, hydrogen pressure, and liquid and hydrogen flow rates onhydrogen conversion and selectivity of cyclododecene in consec-utive hydrogenation of 1,5,9-cyclododecatriene to cyclododecenein a multistage operation. Hwang and Smith (2004) investigatedthe effect of feed distribution and catalyst dilution of two casestudies involving nitrobenzene hydrogenation and ethylene oxi-dation. Diakov and Varma (2003) have investigated the effect offeed distribution in a packed-bed membrane reactor for methanoloxidative dehydrogenation. Guillou, Paul, and Courtis (2008) havestudied the effect of distributed hydrogen between stages in amicro-channel FT reactor. Ngwenya, Glasser, Hildebrandt, Coville,and Mukoma (2005) have investigated the effect of operatingparameters on the performance of the cobalt catalyst based FT

reactor for different objective functions; i.e. converting a givenamount of feed in the smallest possible reactor volume, mini-mizing the production of methane, and maximizing hydrocarbonselectivity.

76 A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83

Nomenclature

Variablesa heat transfer area density, m2/m3

A heat transfer area, m2

B constantcp,i specific heat capacity of component i, kJ/kg KE diagonal matrix, E = diag(0, 0, . . . , (cp,ref /cp))J partial derivative of component reac-

tions with respect to x, J = ∂R0(x)/∂x +diag(0, 0, . . . , 1)((˛(1 − cp,F /cp,0)) − (ˇcp,ref /cp,0))

K diagonal matrix, K = diag(1, 1, . . . , (cp,F /cp))n carbon numbernS number of stagesRi component reaction rate on molar basis,

kmol/(m3reactor s)

Ri component reaction rate on mass basis,kg/(m3

reactor s)T temperature along the path, KTw temperature of the cooling medium along the path,

KuA catalyst dilutionuF feed distribution design function, uF = �˛uH heat transfer area distribution design function,

uH = �ˇuM mixing design functionuT temperature profile of cooling mediumu vector of decision variablesU overall heat transfer coefficient, kW/(m2 K)V residence volume along the path, m3

VR total residence volume along the path, m3

W total mass flow rate, kg/sW0 total mass flow rate at the inlet, kg/sWn weight fraction of hydrocarbons heavier than

methanex state vector of mass fractions and temperature, x =

[ω1, . . . , ωn, �]T

xF state vector of extra feedz state vector augmented with mass flow rate, z = [xT,

�]T

Greek letters˛ feed distribution, kg/(m3 s)

heat transfer area design function, =Ua/cp,ref , kg/(m3 s)

� mass flow rate relative to the inlet, (W/W0)�i volume fraction of stage i, Vi/VR

� dimensionless temperature, � = (T − Tref )/Tref

� dimensionless volume of the path (independentvariable), (V/VR)

� space time or residence time (VR/W0), m3s/kg

Hid

2

ia

Table 1Kinetic and adsorption parameters in Eqs. (5) and (6).

k0,i (mol/(s kg Pa)) Ei (kJ/mol) ai bi

FT reaction 8.58 86 4.8 0.33WGS reaction 9.33 × 106 132 21 0

Table 2Model for calculation of ˛ASF where Ki = Ki(T0) e−Ei/R((1/T)−(1/T0)) and T0 = 573 K. Theunit of pressure in Eq. (7) is bar.

Parameter Ki(T0) Ei (kJ/mol)

K 1.22×10−6 0

ωi mass fraction of component i

Here, the method of systematic reactor design given byillestad (2010) is applied on the Fischer–Tropsch synthesis. This

s the conceptual design of the FT reactor and not the detailedesign.

. Fischer–Tropsch kinetic model

The Fischer–Tropsch synthesis over iron catalyst can be approx-mated as simultaneous series–parallel reactions for the FT reactionnd the water gas shift (WGS):

1

K2 1.05×10−6 94.5K3 2.36×10−6 132.3

CO + UH2 H2rFT−→p1C1 + p2C2 + · · · + piCi + · · ·

+ H2O (−rH)FT = 172.00 kJ/mol (1)

CO + H2OrWGS� CO2 + H2 (−rH)WGS = 39.89 kJ/mol (2)

where UH2 is the usage ratio of H2, and pi is the stoichiometriccoefficient of the hydrocarbon of chain length i. The stoichiomet-ric coefficients will follow the ideal Anderson–Schulz–Flory (ASF)distribution and

pi = (1 − ˛ASF )2˛i−1ASF∞∑

i=1

pi = 1 − ˛ASF(3)

The hydrogen usage (UH2 ) is calculated based on the olefin toparaffin distribution ratio �FT which is here assumed to be 0.35. Thehydrogen usage for paraffin production can be shown to be 3 − ˛ASF

and for olefin production it is 2 + (1 − ˛ASF )2. The hydrogen usagefor combination is

UH2 = (3 − ˛ASF )1

1 + �FT+ (2 + (1 − ˛ASF )2)

�FT

1 + �FT(4)

Here we assume that ˛ASF is the same for paraffins and olefins.For some catalysts we know that this is not the case.

The kinetic model applied in this study is the one given byZimmerman and Bukur (1990) for iron catalyst:

rFT = kFTPCOPH2

PCO + aFT PH2O + bFT PCO2

(5)

rWGS = kWGSPCOPH2O − PH2 PCO2 /Kp

PCO + aWGSPH2O + bWGSPCO2

, Kp = e(4578/T−4.33) (6)

The kinetic parameters are given in Table 1.The process considered in this study is a once through FT reactor

(s) and the pressure of the reactor is 30 bar.The chain growth probability ˛ASF is calculated based on the cor-

relation developed by Lox and Froment (1993a,1993b). The modelparameters are given in Table 2.

˛ASF = K1PCO

K1PCO + K2PH2 + K3(7)

In order to reduce the number of components, a lumping tech-nique is used for hydrocarbon species.

1. CH4

2. C23. LPG (C3 and C4)4. Middle distillate (C5–C10)5. Wax (C11+)

d Che

(

n

a

n

c

C

p

M

b

3

3

dtv2p�

�

[

J

i

s

A. Rafiee, M. Hillestad / Computers an

The average carbon number of each lump is, Marano and Holder1997):

¯ [i,j] = i + ˛ASF

1 − ˛ASF− (j − i + 1)˛j−i+1

ASF

1 − ˛j−i+1ASF

(8)

nd

¯ [i,∞) = i + ˛ASF

1 − ˛ASF(9)

The reaction 1 may also be formulated based on the lumpedomponents. For the chosen lumps, the reaction can be written as:

O + UH2 H2rFT−→p1C1 + p2C2 + p[3,4]C[3,4] + p[5,10]C[5,10]

+ p[11,∞)C[11,∞) + H2O (10)

The stoichiometric coefficients of the lumps are

[i,j] = (1 − ˛ASF )(˛i−1ASF − ˛j

ASF ) (11)

The mean molar mass of each lump is:

¯ [i,j] = 14n[i,j] + 21 + �FT

(12)

The rates of production/consumption of components on massasis are

R1 = rFT M1p1

R2 = rFT M2p2

R3 = rFT M[3,4]p[3,4]

R4 = rFT M[5,10]p[5,10]

R5 = rFT M[11,∞)p[11,∞)

RCO = (−rFT − rWGS)MCO

RH2 = (−rFT UH2 + rWGS)MH2

RH2O = (rFT − rWGS)MH2O

RCO2 = rWGSMCO2

(13)

The sum of all reactions on mass basis will add up to zero.

. Problem formulation

.1. Staging of the reactor path

The reactor path is divided into a number of stages and theesign functions are optimized so as to maximize an objective func-ion. The dimensionless model of plug flow and completely mixedolume are given by Eqs. (14) and (15), respectively (Hillestad,010). The independent variable � is dimensionless volume of theath � = V/VR and � is the mass flow rate relative to the inlet

= W/W0.

dxd�

= �uAR(x) + uF K · (xF − x) − uHE · (x − xw) (14)

�I − �� J]dxd�

= �uAR0(x) + uF K0 · (xF − x) − uHE0 · (x − xw) (15)

= ∂R0(x)∂x

+ diag(0, 0, . . . , 1)

(˛

(1 − cp,F

cp,0

)− ˇ

cp,ref

cp,0

)(16)

The subscript zero on R, K and E indicates that the heat capacitys taken at inlet condition.

The models of plug flow and completely mixed volume have theame structure and may be comprised into one to represent the

mical Engineering 39 (2012) 75– 83 77

change of mass fractions and temperature (state variables) alongthe path, Eq. (17).

[�I − uM� J]dxd�

= �uAR(x) + uF K · (xF − x) − uHE · (x − xw) (17)

x = [ω1, ω2, ω3, ω4, ω5, ωH2 , ωCO, ωH2O, ωCO2 , �]T,

� = T − Tref

Tref(18)

The first 5 mass fractions in Eq. (18) are the mass fractions oflump 1–5, respectively.

The dimension of the reaction rate vector R(x) is (kg m−3 s−1).

R(x) = [R1, R2, R3, R4, R5, RH2 , RCO, RH2O, RCO2 , R�]T,

R� = (−rH)FT

cpTrefrFT + (−rH)WGS

cpTrefrWGS (19)

The parameter � is defined as the total volume divided by massflow rate of syngas.

Simultaneously, the total mass is conserved.

d�

d�= uF (20)

Mixing: The function uM(�) represents the fluid mixing. WhenuM is zero, Eq. (17) is a plug flow model (PFR) and as the functionis proportional to path length (duM/d� = 1), Eq. (17) becomes thecompletely mixed volume (CSTR). For an intermediate slope (0 <(duM/d�) < 1), Eq. (17) represents the plug flow model with recycle(rPFR). The design function uM(�) is parameterized by a piecewiselinear function and consists of 3nS − 1 parameters.

Distributed feed: The design function uF = ˛� defines the distri-bution of extra feed along the reactor path. The distribution of extrafeed is parameterized by a piecewise constant function. The extrafeed distribution can be continuous or point-wise just prior to thestage.

Heat transfer: The design function uH = ˇ� represents the dis-tribution of heat transfer area along the path where = Ua/cp, U isthe overall heat transfer coefficient [kW/m2 K], a is the heat trans-fer area density [m2/m3] and cp is the average heat capacity of themixture [kJ/kg K]. The overall heat transfer coefficient depends onthe reactor geometry and fluid properties. Here, the cp values aretaken from UNISIM DESIGN process simulator. The average heatcapacities of lump 1–5 are: 2.86, 2.56, 2.52, 3.15 and 3.00 kJ/kg K,respectively. The distribution of heat transfer area is parameterizedby a piecewise constant function.

Chemical reactions: The design function uA is the activity of thecatalyst and is normalized between 0 and 1. A value of 1 is consid-ered to be the maximum catalyst concentration. Catalyst activity isparameterized by a piecewise constant function.

4. Optimization

4.1. Optimal control analogy

The optimal reactor configuration may be found by solving aproblem similar to that of an optimal control problem.

max J[�, u] ∈ U

s.t.dzd�

= f(z, u); z(0) = z0

(21)

where z = [xT, �]T.

78 A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83

Table 3Cases with different number of stages (nS), number of design parameters for mixing, catalyst activity, extra feed distribution, heat transfer area density and coolant temperature.For all the cases � = 13.34 m3 s/kg.

Input Results

Case nSa [M, A, F, H, T]b J (%) uM

c uAd ˛e (kg/m3 s) CO conversion (%)

I.1 1 [1, 1, 1, 1, 2] 13.701 P 0.946 0 75.23I.2 2 [2, 2, 2, 2, 5] 14.043 P-P 0.937 0-0 74.38I.3 3 [3, 3, 3, 3, 8] 14.043 P-P-P 0.937 0-0-0 74.38

a Number of stages.b [M, A, F, H, T] are number of design parameters for mixing, catalyst activity, extra feed distribution, heat transfer area density and coolant temperature, respectively.

For each stage, the number of design parameters for mixing structure is 2. The volume fraction of each stage, �i = Vi/VR , is also a degree of freedom and we have nS − 1independent degree of freedom for �i . The total number of design parameters for mixing structure is 2nS + (nS − 1) = 3nS − 1.

m

J

6

2ti

h

r

4

cltf

csiti(

fppsp

i

5

tfi

5

tp

tion of CO2 and water are 34.81 and 13.56%, respectively. H2/COratio at the outlet of the reactor is 5.33. Water is produced by theFT reaction and consumed by the water gas shift reaction. This willproduce CO2 and increase H2/CO which is not of interest.

c Mixing structure.d Average catalyst concentration.e Extra feed distribution.

The objective function (J) considered here is to maximize theass fraction of wax at the end of path.

= ωWax(1) (22)

With reactor path volume of 900 m3 and feed mass flow rate of7.45 kg/s, the space time � is 13.34 m3 s/kg.

Here, the temperature along the path is not allowed to exceed50 ◦C, T(�) ≤ 250 ◦C, and the total mass flow at the outlet is equal tohe inlet flow, �(1) = 1. These are represented by nonlinear inequal-ty constraints.

(z, u) ≤ 0 (23)

The reason for setting a maximum temperature of 250 ◦C iselated to the catalyst deactivation and product selectivity.

.2. Mathematical programming

The design functions and state vector z = [xT, �]T

are dis-retized. The design functions are parameterized by piecewiseinear (mixing structure) or piecewise constant (coolant tempera-ure, heat transfer area density, extra feed point or catalyst activity)unctions.

The system of ordinary differential equations, Eq. (21), is dis-retized by orthogonal collocation on finite elements. On eachtage, the number of internal collocation points is n. The solutions approximated by the polynomial of degree n + 1 which satisfieshe initial condition, z(0) = z0, and the differential equation at allnternal points. On each stage and for each collocation point, Eq.21) is formulated as nonlinear equality constraints.

Here we have applied an optimization algorithm based on SQP,mincon in Matlab. The objective function is maximization of waxroduction at the end of path. The inequality constraints, h(z, u), areath constraints and bounds on design functions. The equality con-traints are the parameterized state variables at each collocationoint on each stage.

For each case, the optimization is conducted with different start-ng points.

. Results

Two groups of cases are considered. In the first group, H2/CO tohe first reactor is set to 2.1 and in the second group; H2/CO to therst reactor is optimized.

.1. I: Feed with H2 to CO ratio of 2.1

Table 3 shows the results of the optimization. For each case,he input parameters are; number of stages, number of designarameters for mixing, catalyst activity, extra feed distribution,

Fig. 1. Volume distribution of the cases given in Table 3.

heat transfer area density and coolant temperature. The optimiza-tion outputs are; the value of objective function, mixing structure,catalyst dilution, amount of extra feed and mass fraction of eachlump of components. The fresh syngas and extra feed contain H2and CO.

Figs. 1 and 2 summarize the optimization results given in Table 3.The volume distribution along the path is shown in Fig. 1. Fig. 2illustrates the production rate of wax for the cases given in Table 3.

Case I.1. One stage is selected and the target is to maximize thewax concentration. The profile is shown in Fig. 3. By optimizingthe path, no extra feed is beneficial, because it will dilute the waxconcentration. The mixing structure for this case is plug flow (PFR)to have maximum possible conversion. In this case, the concentra-

Fig. 2. Production rate of wax for the cases given in Table 3.

A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83 79

is to m

pbgirttTch

h(

h

f

a

c(

rtocm

ptpttwtz

t

Fig. 3. Case I.1 and the target

In this case, the catalyst activity is 0.946 and the reactor tem-erature does not hit the maximum temperature of 250 ◦C. This isecause of the fact that high temperatures will reduce the chainrowth probability. Heat transfer area density reaches the max-mum specified value. More heat transfer area will reduce theeactor temperature and will increase chain growth probability. Onhe other hand, lower heat transfer area will increase the reactoremperature and reduce the mass fraction of wax at the end of path.he area needed to remove the heat produced in the reactor can bealculated as:eat produced = heat removed

eat produced = (−r H)FT (amount of CO converted to FT products) +−r H)WGS(amount of CO converted to CO2) = 200.50 MW

eat removed = UA(T − Tw)avg = UaVR(T − Tw)avg

rom Fig. 1, (T − Tw)avg = 4.54 ◦C

nd Ua = 49.102 kW/m3 K ⇒ ˇ/40 = 0.44This means that the maximum specified value of area is suffi-

ient in order to remove the heat produced in the reactor. IncreasingT − Tw)avg will reduce the heat transfer area even more.

By setting the upper bound of to 40 and re-optimizing theeactor path, the mass fraction of wax becomes 14.97%. In this case,he optimal value of reaches the maximum value of 40. In casef unconstrained ˇ, the optimal value of becomes 859.6 and theoncentration of wax is 15.36%. More heat transfer area will costore.Lower H2/CO will increase chain growth probability along the

ath. A case study is conducted to reduce H2/CO inside the reac-or by adding some CO2 to the stage. Optimizing the reactorath with the extra feed containing CO2 shows that H2/CO athe end of path is not reduced. The reason is that the tempera-ure is low and adding CO2 cannot reduce H2/CO by the reverseater gas shift reaction. The optimal value of the objective func-

ion is the same as Case I.1 with the flow-rate of CO2 equal toero.

Case I.2. In this case, two stages are selected. All the design func-ions are free and there are 13 degrees of freedom, i.e. 2 design

aximize wax concentration.

parameters for each of heat transfer area density, coolant tempera-ture, catalyst activity and extra feed and 5 for mixing structure. Theresults are shown in Fig. 4. In this case, the mass fraction of wax andCO conversion at the end of path are increased compared to CaseI.1. Coolant temperature for both stages is the same and equal to220 ◦C. This means that we can use the same pressure level of boil-ing water as the coolant. The catalyst activity of the first stage is0.684 and for the second stage is equal to 1. The volume of the firststage is 20% of the total volume.

The main feed can be distributed between the stages; optimizingthe reactor path shows that all of the syngas must be fed to the firstreactor. The optimal value of the objective function is close to thatof Case I.2.

The difference between Case I.1 and Case II.2 is the catalystconcentration and the reason is due to the temperature.

Case I.3. Three stages are applied here. The optimal mixing struc-ture is PFR-PFR-PFR. There are 20 degrees of freedom, i.e. 3 designparameters for each of heat transfer area density, coolant tempera-ture, catalyst activity and extra feed and 8 for mixing structure. Themass fraction of wax is increased compared to Case I.1. The profileis similar to Fig. 4.

5.2. II: H2 to CO ratio to the first reactor is optimized

A case study is conducted to investigate the effect of hydro-gen distribution on the production rate of C11+. The total syngasflow-rate and reactor volume are the same as the cases givenin Table 3. The H2/CO to the first reactor is optimized andthe rest of H2 is distributed continuously along the reactorpath as shown in Fig. 5. The minimum value of H2/CO tothe first reactor is set to 0.5 to prevent numerical errors in

calculations.

The profiles of single-, two- and three-stage reactors are shownin Figs. 6–8. In all the cases, the optimal H2/CO to the first reactoris 0.5 (Table 4).

80 A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83

Fig. 4. Case I.2 and the target is to maximize wax concentration.

Fig. 5. Hydrogen distribution along the reactor path.

Fig. 6. Case II.1 and H2/CO at the inlet of reactor is 0.5.

A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83 81

at the

teIti

Fig. 7. Case II.2 and H2/CO

Case II.1.For a single-stage reactor, the concentration of wax athe end of path, and CO conversion are 18.159 and 90.30%. The conc-ntration of wax and CO conversion are increased compared to Case

.1 in Table 3. As it can be seen from Fig. 6, at 40% of the reactor pathhe concentration of wax is increased from 0 to about 15% whilen Case I.1 it is increased from 0 to about 10%. The average catalyst

Fig. 8. Case II.3 and H2/CO at the

inlet of first reactor is 0.5.

activity is higher than Case I.1. The peak temperature is higherthan Case I.1 but still does not hit the maximum temperature.

Case II.2. For a two-stage reactor configuration, the optimal value

of the wax concentration is 20.843%. Conversion of CO is increasedto 92.05%. In this case, there are 14 degrees of freedom. The profileis shown in Fig. 7.

inlet of first reactor is 0.5.

82 A. Rafiee, M. Hillestad / Computers and Chemical Engineering 39 (2012) 75– 83

Table 4Cases with different number of stages (nS) and the optimal H2/CO to the first reactor is 0.5.

Case nS J (%) uMa uA

b ˛c (kg/m3 s) CO conversion (%)

II.1 1 18.159 P 0.98 0.0075 90.30II.2 2 20.843 P-P 1.00 0.0012-0.1250 92.05III.3 3 21.052 P-P-P 1.00 0-0.0012-0.1991 93.47

a Mixing structure.b Average catalyst concentration.c Extra feed distribution (pure H2).

Fig. 9. H2/CO along the path.

tb

Ahwc

Case II.3. For three-stage reactor, the concentration of wax athe end of path becomes 21.052%. The concentration is increasedy 15.93% compared to Case II.1.

The change of H2/CO along the reactor path is shown in Fig. 9.s it can be seen, for all the cases the H2/CO at the end of path isigher than 2. This ratio is increasing with path length due to theater gas shift reaction. Low H2/CO at the inlet will increase the

oncentration of wax at the end of path, as shown in Fig. 10.

Fig. 10. Chain growth probability along the path.

6. Conclusions

In industrial GTL plants, the Fischer–Tropsch reactors requirelarge volume and sectioning of the reactor into stages will pro-vide more degrees of freedom. In this study, we focus on the FTunit only where the reactor volume is divided into one, two andthree stages. The design functions are optimized so as to maximizeconcentration of wax at the end of path. With the chosen kineticmodel, staging of the Fischer–Tropsch synthesis will increase themass fraction of wax and CO conversion. For a two-stage reactor,the production is increased by 2.50% compared to a single-stagereactor.

Separating the hydrogen from the main syngas feed (with H2/COof 2.1) and distributing the pure H2 continuously along the reactorpath will increase the concentration of wax and CO conversion. Inthis case, H2/CO to the first reactor is lower than the main syngasfeed. The H2/CO to the first reactor is 0.5 and the rest of hydrogen isdistributed continuously along the path. The concentration of waxat the end of path becomes 18.159 and 20.843, and 21.052% goingfrom one- to three-stage reactor, respectively. The H2/CO at thebeginning of the path is low which will increase the selectivity toheavier hydrocarbons.

In this study, the lumped parameter is optimized and for anoptimal value we have to realize an area based on a rigorous calcu-lation of U and cp. With the same value of for fixed bed or slurrybubble column reactor, the fixed bed reactor needs more area dueto the lower heat transfer coefficient.

Acknowledgment

This work has been supported by the Norwegian Research Coun-cil through the GASSMAKS program.

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