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2013, Chemical Engineering & Technology

Journal of Industrial and Engineering Chemistry

Modeling and operating conditions optimization of Fischer–Tropsch synthesis in a fixed-bed reactor2012 •

Reviews in Chemical Engineering

Recent advances in reactors for low-temperature Fischer-Tropsch synthesis: process intensification perspective2015 •

Computers & Chemical Engineering

Staging of the Fischer–Tropsch reactor with an iron based catalyst2012 •

2017 •

A comprehensive kinetic model of the Fischer-Tropsch synthesis (FTS) is developed in a fixed bed reactor under operating conditions (temperature, 230–235°C, pressure, 20–25 bar, gas hourly space velocity, 4000–5000 cm3(STP)/h/gcatalyst ,H2/CO feed molar ratio, 2.1) over a Co based catalyst. Reaction rate equations based on Eley-Rideal (ER) type model for initiation step and Langmuir-Hinshelwood-Hougen-Watson (LHHW) type model for propagation and termination steps of the FTS reactions have been considered and the readsorption of olefins were taken into account. The model that was reported in the literature was modified in order to explain many significant deviations from the ASF distribution. Optimum parameters have been obtained by Genetic Algorithms (GA). The calculated activation energies to produce n-paraffins and 1-olefins were in the range of 82.24 to 90.68 kJ/mol and 100.66 to 105.24 kJ/mol, respectively. The hydrocarbon distribution in FTS reactions was ...

Chemical Engineering & Technology

Optimal Design of a Gas‐to‐Liquids Process with a Staged Fischer‐Tropsch Reactor2016 •

The optimal design of a natural gas‐to‐liquid hydrocarbons (GTL) process with a multistage cobalt‐based Fischer‐Tropsch reactor and interstage product separation is considered. The objective function is to maximize the wax (C21+) production rate at the end of the reactor path. Sectioning of the Fischer‐Tropsch reactor increases the chain growth probability inside the reactor which results in a higher production of wax. The carbon efficiency of the two‐stage reactor is distinctly higher than that of the single‐stage reactor.

Chemical Engineering Journal

Intensifying the Fischer–Tropsch Synthesis by reactor structuring – A model study2012 •

The optimal design of a natural gas to liquid hydrocarbons (GTL) process with multi stage cobalt-based Fischer-Tropsch reactor and inter-stage product separation is considered. The objective function is to maximize the wax (C21+) production rate at the end of reactor path. Sectioning of the FT reactor increases the chain growth probability inside the reactors, which results in producing more wax. Carbon efficiency of the two stage reactor is 5.8% higher than single stage reactor.

Review of Scientific Instruments

Six-flow operations for catalyst development in Fischer-Tropsch synthesis: Bridging the gap between high-throughput experimentation and extensive product evaluation2013 •

Korean Journal of Chemical Engineering

Operating strategies for Fischer-Tropsch reactors: A model-directed study2004 •

A comprehensive parametric study for a Fischer-Tropsch (FT) synthesis process has been conducted to investigate the relation between process parameters and reactor characteristics such as conversion, selectivity, multiplicity, and stability. A flexible model was employed for this purpose, featuring the dependence of Anderson-Shultz-Flory (ASF) factor on composition and temperature. All variable process parameters in industrial FT reactors were subject to variation, including reaction temperature, reactor pressure, feed ratio, inlet mass flux, feed temperature, heat transfer coefficient, catalyst concentration, catalyst activity, etc. While typical trade-off was encountered in most cases, i.e., the change of a parameter in one direction enhances one aspect but deteriorating another, the change of feed conditions gave some promising results. It has been found that decreasing the feed rate (or increasing the residence time) and/or lowering the feed concentration can successfully enhance the conversion up to more than 90% for our specific case, without hurting the product selectivity as well as effectively condense the region of multiple steady states. The benefits and limitations accompanied with the variation of the parameters were discussed in detail and a rational start-up strategy was proposed based on the preceding results. It is shown that the decrease of inlet mass flux (say, 85% decrease of the feed rate or 60% decrease of the feed concentration from the nominal condition chosen here) or the decrease of H2/CO ratio (specifically, below about 0.25), or their combination can eliminate multiple steady states. The resulting unique relation between temperature and manipulated variable (i.e., coolant flow rate) appears to assure a safe arrival at the target condition at the start-up stage.

Reactor design
Ahmad Rafiee
Magne Hillestad
Norwegian University of Science
and Technology (NTNU),
Department of Chemical
Engineering, Trondheim,
Norway.
1
Research Article
Staging of the Fischer-Tropsch Reactor
with a Cobalt-Based Catalyst
A method for systematic reactor design, described by Hillestad [1], is applied to
the Fischer-Tropsch synthesis. The reactor path is sectioned into stages and design
functions are optimized to maximize an objective function. Two different objective functions are considered: the yield of wax and a measure of the profitability.
With the chosen kinetic model [2] and the path temperature constrained by
240 °C, staging of the Fischer-Tropsch synthesis based on the first criteria will
increase the yield of wax. By introducing the cost of heat transfer area in the
objective function, the total heat transfer area requirement of a two-stage reactor
is significantly less than of a single-stage reactor.
Keywords: Fischer-Tropsch synthesis, Path optimization, Plug flow, Reactor design, Staging
Received: December 14, 2012; revised: May 29, 2013; accepted: June 07, 2013
DOI: 10.1002/ceat.201200700
1
Introduction
In the last decades, the conversion of natural gas through a
gas-to-liquids (GTL) process proved to be an alternative for
the use of remote natural gas reserves to produce liquid transportation fuels. An increasing world-wide demand for cleanburning fuels has sparked a renewed interest in the study of
the Fischer-Tropsch (FT) synthesis.
A GTL plant consists of three main sections: (i) synthesis gas
(syngas) production: conversion of natural gas to a mixture
consisting mainly of H2 and CO, (ii) FT synthesis: conversion
of H2 and CO to a wide range of hydrocarbons, (iii) upgrading
of products.
There are different routes for syngas production: auto-thermal reforming (ATR), steam methane reforming, combined
reforming, and heat exchange reforming involving series and
parallel arrangements [3]. Over the years, four types of reactors have been utilized for FT synthesis: the fixed-bed tubular
reactor known as the ARGE reactor, circulating fluidized-bed
reactors known as Synthol reactors, the Sasol Advanced
Synthol reactor, and the slurry-bubble column reactor [4, 5].
The upgrading unit involves mainly separation and hydroprocessing. The primary upgrading starts with the removal of light
ends and dissolved gases. The same basic technologies used in
crude oil refineries have been adapted for FT product refining
[3].
–
Correspondence: Prof. M. Hillestad (magne.hillestad@chemeng.ntnu.
no), Norwegian University of Science and Technology (NTNU),
Department of Chemical Engineering, Sem Sælandsvei 4, 7491 Trondheim, Norway.
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
The earliest catalysts used for FT synthesis were iron and
cobalt. Iron is a highly active catalyst and exhibits water-gas
shift (WGS) activity whereas cobalt catalysts do not have this
activity, leading to improved hydrocarbon yield. Cobalt catalysts yield mainly straight-chain hydrocarbon products and no
oxygenates as with iron catalysts. However, cobalt catalysts are
230 times more expensive than iron, but still a very good alternative to iron catalysts. The reason is that cobalt catalysts demonstrate activity at lower operating pressures [5].
Staging in chemical engineering is not a new idea and has
been applied on many processes. Androulakis and Reyes [6]
studied oxidative coupling of methane (OCM) and the role of
oxygen distribution and product removal on a staged plugflow reactor. Maretto and Krishna [7] introduced staging of an
FT slurry-bubble column reactor and optimized the reactor
conversion. The advantages of this reactor configuration compared to a single-stage reactor are increased syngas conversion
and reactor productivity. In their study, the multi-stage reactor
requires additional cooling tubes. Waku et al. [8] performed
studies on staged oxygen introduction and selective hydrogen
combustion during propane dehydrocyclodimerization reactions on cation-exchanged zeolite. Hwang and Smith [9] investigated the effect of catalyst dilution and distribution of feed to
control the reactor temperature of two case studies involving
nitrobenzene hydrogenation and ethylene oxidation. Diakov
and Varma [10] investigated the effect of feed distribution in a
membrane reactor for methanol oxidative dehydrogenation.
Guillou et al. [11] examined the influence of hydrogen distribution between stages in a micro-channel FT reactor. Rafiee
and Hillestad [12] evaluated staging of an FT reactor with an
iron-based catalyst. The design functions, i.e., fluid mixing,
hydrogen distribution, heat transfer area distribution, coolant
temperature, and catalyst concentration are optimized to max-
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
2
A. Rafiee, M. Hillestad
liquid phase and the other species are in the gas phase. There
are small traces of these components in the gas phase which
are neglected here. The average heat capacities of lump 1–5
are: 2.86, 2.56, 2.52, 3.15, and 3.00 kJ kg–1K–1, respectively. The
price of products (lump 3–5) are 0.8, 0.9, and 1.1 kg–1, respectively [16].
The mass production rates of lump 2–5 are according to the
ASF distribution model:
imize the concentration of C11+ at the end of the reactor path.
Jess et al. [13, 14] described the effect of particle size and single-tube diameter on the thermal behavior of FT reactors.
The potential of sectioning the FT synthesis reactor into
stages based on the method proposed by Hillestad [1] is demonstrated. Each reactor stage is represented by design functions. Staging of the reactor path will provide more degrees of
freedom for optimization. This study focuses on the FT unit
only and investigates the possibilities of cost reduction of the
FT reactor.
2
~ 2 2aASF 1
R
~ 3 3a2ASF 1
R
Fischer-Tropsch Kinetic Model
~ 5 11a10
R
ASF 1
(1)
k1 PH2 PCO 0:05
1 K1 PCO
(2)
rCO
k2 PH2 0:6 PCO 0:65
1 K1 PCO
(3)
aASF 2 aASF n
1
n 2; 3; :::; ∞
k1
K1
k2
37 326
8:8 × 10 6 exp
RT
68 401:5
1:096 × 10 12 exp
RT
37
326
1:6 × 10 5 exp
RT
3
Problem Formulation
3.1
Staging of the Reactor Path
cI
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dx
~ x uF K xF
uM r~J ruA R
xn
x
uH E x
xw
(10)
The state vector x is the vector of mass fractions
augmented with the temperature (Rafiee and Hillestad [12]).
Unit
1:05
3
mreactor
s
1
x x 1 ; x2 ; x3 ; x4 ; x 5 ; xH2 ; xCO ; xH2 O ; hT
h T
Pa
(9)
A path is a line of production on which basic operations take
place which are represented by design functions. The path is
sectioned into a number of stages and the design functions
(decision variables) are optimized so as to maximize an objective function. The flow model given by Eq. (10) is a concise
formulation, representing the change of state variables (mass
fractions and temperature) along the path. The derivation of
Eq. (10) is described in detail by Hillestad [1].
(4)
kmolCH4 Pa
(8)
The first five reaction rates in Eq. (9) are the rates of lump
1–5 on mass basis. The value of B is determined from Eq. (9).
The consumption rate of H2 is calculated according to the stoichiometry of the reactions and the distribution of paraffins
and olefins.
Table 1. Kinetic and adsorption parameters in Eqs. (2) and (3) [2].
Arrhenius expression
aASF 2 . . . B
~ CO R
~2 R
~3 R
~4 R
~5 R
~H R
~H O 0
~1 R
R
2
2
In order to reduce the number of components, a lumping
technique is used for hydrocarbon species: (1) CH4, (2) C2, (3)
LPG (C3 and C4), (4) middle distillate (C5–C10), and (5) wax
(C11+).
At 20 bar and a temperature range of 210–240 °C, vaporliquid equilibrium calculations using the UNISIM DESIGN
process simulator indicate that lump 4 and 5 are mainly in the
Parameter
(6)
The conservation of mass requires that the sum of all component reaction rates add up to zero.
The kinetic and adsorption parameters are given in Tab. 1.
The process considered in this study is a once-through FT
reactor (s) and the pressure of the reactor is 20 bar.
The weight fractions of FT products heavier than methane
(Wn) are assumed to follow the ideal Anderson-Schulz-Flory
(ASF) distribution.
Wn n 1
aASF B
(7)
The kinetic model applied here is the one given by Iglesia
et al. [2] for a cobalt catalyst, where methane production and
CO consumption are defined by Eqs. (2) and (3), respectively.
This kinetic model is chosen because it takes into account a
higher selectivity of methane. However, other available kinetic
models could be applied [15].
rCH4
aASF 2 4a2ASF 1
(5)
2
~ 4 5a4ASF 1 aASF 2 6a5ASF 1 aASF 2 . . . 10a9ASF 1 aASF 2 B
R
In an FT reactor, the synthesis gas (syngas) reacts to form a
mixture of hydrocarbons.
CO + UH2H2 → (–CH2–) + H2O
aASF 2 B
Tref Tref1
11
1
kmolCO Pa
1:25
3
mreactor
s
1
The first five mass fractions in Eq. (11) are the
mass fractions of lump 1–5, respectively.
~ x
The dimension of the reaction rate vector R
–3 –1
is kg m s .
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
Reactor design
~ x R
~1; R
~2; R
~3; R
~4; R
~5; R
~H ; R
~ CO ; R
~H O; R
~h T
R
2
2
~ h D; H cp 1 Tref1 rCO
R
4.2
(13)
The total mass flow rate W is independent of pressure
change, temperature change, phase change, or mole number
change along the reactor path [1].
Mixing: The function uM(n) is a design function representing the fluid mixing. When uM is zero, Eq. (10) represents a
plug-flow model and as the
functionincreases proportionally
duM
1 , Eq. (10) becomes the
with the length of path
dn
completely mixed
volume. For an intermediate slope
duM
0<
< 1 , Eq. (10) represents the plug-flow model with
dn
recycle [1].
Distributed feed: The design function uM = ar defines extra
feed distribution along the path volume. The distribution of
extra feed can be continuous or pointwise. In this study, a
pointwise distributed feed is applied [1].
Heat transfer: The design function uH = br represents the
heat transfer area distribution along the path. Heat transfer
area distribution is parameterized by a piecewise constant
function [1].
Chemical reactions: The design function uA is the catalyst
activity which can vary between 0 and 1. A value of 1 is considered to be the maximum catalyst concentration. Catalyst activity is parameterized by a piecewise constant function.
Parameterization of design functions leads to a set of inequality constraints [1]. The model applied here is a pseudoheterogeneous model. The gas and liquid phases flow in the
same direction and have the same degree of dispersion.
4
Optimization
4.1
Optimal Control Analogy
Objective Functions
(12)
Heat of reaction (–DrH) at 25 C is 172 kJ per mole of CO
consumed in the FT reactor. Simultaneously, the total mass is
conserved.
dc
uF
dn
The concentration of wax at the end of the path is a possible
objective function, J1 = xwax [1]. The design functions represent costs, such as heat transfer area, extra feed points, and
utility streams. A measure of the profit is another possible
objective function (J2).
Here, it is focused only on the FT synthesis, where the syngas
is imported. With a syngas capacity of around 67.45 kg s–1, the
price of syngas is 0.23 kg–1. The pressure of the reactor is
20 bar.
The costs associated with this study are categorized as:
1. Capital cost: Cost of FT reactor including reactor shell
[19], tubes [20], and the price of cobalt catalyst which for
the initial load amounts to ∼ 54 kg–1. The investment
required for the power production unit is assumed to be
300 kW–1. It is presupposed that process water is free.
The selling price of the produced electricity in the power
plant is 0.06 kW–1h–1 [19].
2. Equivalent Annual Operating Cost (EAOC): This cost is
the annual operating cost plus the annual capital charge
[19]. The annual capital charge is the amortized capital
cost over the operating life of the plant to establish an
annual cost. For an operating life time of 20 years and an
interest rate of 8 %, the annual capital charge is 10 % of
the capital cost [19].
The annual profit is equal to the income minus the EAOC.
The amount of steam produced in the FT reactor is calculated as the total heat transfer divided by the latent heat of vaporization (assuming 100 % efficient heat transfer):
R
UA T TW dn
_ steam
m
(16)
k
The latent heat of vaporization (k) is 1889 kJ kg–1.
Finally, J2 is formulated as below (tax is neglected):
J2 Income
EAOC
EAOC annual operating cost 0:1 × capital cost
nS
nS
X
X
uA;i Di 1:027 × 104
b 2i Di C2;i
4:792 × 108 8:660 × 105
i1
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
C2,i is the cost of tubes per meter ($ m–1) [20].
C2,i = 2.5 + 98.958(di – 0.02)
(14)
The objective function (J) is formulated as a function of
state variables z xT ; cT at the end of the path. Here, it is required that the temperature along the path does not exceed
240 °C, i.e., T(n) ≤ 240 °C [7, 17, 18].
The path constraints are represented by nonlinear inequality
constraints [1].
h(z,u) ≤ 0
i1
(17)
The optimal path configuration can be found by solving a
problem similar to that of an optimal control problem.
max J
r; u ∈ U
dz
s:t:
f z; u; z 0 z0
dn
3
(15)
(18)
All performed approximations have the same effect on all
the cases.
4.3
Mathematical Programming
The decision variables (design functions) and state vector
z = [xT, c]T are discretized. The design functions are discretized
by piecewise linear or piecewise constant functions [1]. The
system of ordinary differential equations given by Eq. (14) is
discretized by orthogonal collocation. On each reactor stage
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
4
A. Rafiee, M. Hillestad
optimizing the path, no extra feed is beneficial, because it will
dilute the wax concentration. The mixing structure for this
case is a PFR and there is no catalyst dilution. The reactor temperature hits the maximum to obtain the maximum converIn this study, the optimization algorithm based on SQP, fminsion and then declines. In this case, the concentration of water
con in Matlab is applied. The objective functions are maximiis 41.87 wt %.
zation of wax production at the end of the path and a measure
Case 2: Two stages are selected. The concentration of wax
of profitability. The inequality constraints, h(z,u), are path
and
the average heat transfer area density are higher than in
constraints and bounds on decision variables. The equality
case 1. There is no cost on heat transfer area. All the design
constraints are the discretized state variables at each collocafunctions are free and there are 13 degrees of freedom, i.e., two
tion point on each reactor stage.
design parameters for each of heat transfer area density, coolant temperature, catalyst activity, and extra feed and five for
mixing structure. The results are indicated in Fig. 2.
5 Results
Removing liquid wax from the outlet stream of the first
stage has no effect on the objective function, because the reacIn all cases (if otherwise stated), the residence time (s) and
tion rates are calculated based on the partial pressure of H2
chain growth probability are kept constant at 2.97 m3s kg–1 and
and CO in the gas phase. On the other hand, removing water
0.9 [7, 17, 18], respectively. The maximum operating temperafrom the first stage and reoptimizing the path will increase the
ture of the reactor is limited by catalyst deactivation. Catalyst
concentration of wax to 23.98 %. In this case, CO conversion
deactivation is not considered here.
and average heat transfer area are 81.90 % and 163.43 m2m–3,
Tab. 2 presents the optimization results. In each case, the
respectively.
input parameters are: objective function criteria, number of
Case 2a: In this case, the heat transfer density is required to
stages, number of design parameters for coolant temperature,
be uniform along the path. The reason may be that similar
and number of design parameters for heat transfer area denreactors are wanted. The mass fraction of wax and CO conversity. In Tab. 2, the value of 1 for nuT (or nuH) means that all the
sion are increased compared to case 1. The first stage is plug
stages have the same coolant temperature or heat transfer area
flow with recycle (rP) due to less degrees of freedom compared
density. The optimization outputs are: value of objective functo case 2, and rP is there to level out the temperature peak. In
tion, mixing structure (PFR, CSTR, or PFR with recycle), averthis case, there are 12 degrees of freedom.
age heat transfer area density, coolant temperature, catalyst
Case 2b: The same criteria as above are applied and both
dilution, and mass fraction of each lump of components. The
the coolant temperature and the heat transfer density are refresh syngas and extra feed contain only H2 and CO and H2/
quired to be the same along the path. The reason of having
CO to the reactor is 2.1.
equal coolant temperatures is that it is intended to use the
Case 1: One stage is selected and the target is to maximize
same pressure level of boiling water as coolant. The mixing
the mass fraction of wax. The profile is illustrated in Fig. 1. By
structure for this case is PFR with recycle for the first stage
and PFR for the second stage.
The number of degrees of freeTable 2. Cases with different optimization criteria, number of stages (ns), number of design padom for this case is 11.
rameters for coolant temperature (nuT), and heat transfer area density (nuH).
Case 2c: Chain growth probability is varying along the path as a
Input
Results
function of temperature and parnuT a
nuH b J
uM c
ad
xWax
CO
Criteria Case
ns
tial pressures of H2 and CO based
conversion [%]
on the correlation proposed by
Song et al. [21]. The chain growth
1
1
1
1
21.92
P
123.5
21.92
74.91
probability at the end of the path is
2
2
2
2
22.81
P-P
156.5
22.81
77.97
0.78 and the concentration of wax
2a
2
2
1
22.66
rP-P
151.7
22.66
77.5
is 8.05 wt %. The concentration of
J1
lump 4 is increased compared to
2b
2
1
1
22.64
rP-P
151.0
22.64
77.43
case 2 due to a low chain growth
2c
2
2
2
8.05
rP-P
157.2
8.05
75.55
probability.
Case 3: Three stages are applied
3
3
3
3
23.02
rP-P-P
155.1
23.02
78.72
here. The optimal mixing structure
4
1
1
1
100.00 % P
117.2
21.79
74.47
is rP-P-P (Fig. 3) and the concenJ2
tration of wax is increased com4a
2
2
2
122.00 % C-P
97.6
22.21
75.97
pared to case 1. All design func4b
2
2
2
125.00 % C-P
96.1
22.31
76.28
tions are free and 20 degrees of
a
freedom are given.
Number of design parameters for coolant temperature; 1 means the same coolant temperature.
b
Case 4: One stage is selected and
Number of design parameters for heat transfer area density; 1 means the same heat transfer
area density. cP is PFR, C is CSTR, and rP is a PFR with recycle. dAverage heat transfer area dena measure of the annual profit is
sity along the path.
maximized. The mixing structure
and each collocation point, Eq. (14) is formulated as nonlinear
equality constraints [1].
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© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
Reactor design
5
Figure 1. Case 1 and the criteria defined by J1.
is a single PFR and the wax concentration of wax is
21.79 wt %.
Case 4a: Two stages are applied and there is no distributed
feed. The mixing structure is CSTR for the first stage and
plug flow for the second stage. The value of objective function is increased by 22 % which is due to the lower heat
transfer area requirements compared to the single-stage reactor (case 4). Comparing the heat transfer area density of this
case with case 2 demonstrates that by introducing the cost
of heat transfer area, the total area is reduced. The results
are presented in Fig. 4. If the CSTR in Fig. 4 is set to a PFR,
the peak temperature exceeds the maximum temperature.
The CSTR serves to level out the temperature peak with less
heat transfer area.
Case 4b: The same criteria as above are applied but the original feed is distributed between the stages. In this case, 97.48 %
of the original syngas is fed to the first reactor and the objective function increases by 25 % compared to case 4. The concentration of wax at the end of the path is 22.31 wt %.
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
6
Discussion
6.1
Model Verification
The model presented in this study is verified by: (i) UNISIM
DESIGN process simulator, and (ii) performing two-phase calculations based on the correlations proposed by Marano and
Holder [22] for FT systems. The values of b, coolant temperature, and mixing structure are the same as for case 1. The
results presented in Tab. 3 are close to the results given by
UNISIM DESIGN and two-phase calculations. Consequently,
Table 3. Verification of model. I: this study, II: UNISIM DESIGN,
III: two-phase calculations.
I
II
III
xWax [wt %]
21.92
21.70
21.63
CO conversion [%]
74.91
74.04
74.47
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
6
A. Rafiee, M. Hillestad
Figure 2. Case 2 and the criteria defined by J1.
assuming an ideal split between gas and liquid phases is a good
approximation.
6.2
Interpretation of the Results
This study focuses on the FT unit only and investigates the
possibilities of cost reduction of an FT reactor. The kinetic
model is the one given by Iglesia et al. [2] which takes into
account a higher selectivity of methane. The pressure drop is
neglected here. The model applied is a pseudo-heterogeneous
model.
With the chosen kinetic model, staging of the FT synthesis
based on the first criteria will increase the mass fraction of wax
and CO conversion. The concentration of wax for a single-,
two-, and three-stage FT reactor of the same total volume is
21.92, 22.81, and 23.02 wt %, respectively. By optimizing the
path, no extra feed is beneficial because it will dilute the wax
concentration. The reactor temperature hits the maximum to
obtain the maximum conversion. For a two-stage reactor
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where all the design functions are free, the number of degrees
of freedom is 13. In case 2a, the heat transfer area density is
forced to be the same for both stages, and the first stage is a
plug flow with recycle to level out the temperature peak.
By introducing the cost of heat transfer area in the objective
function, the total heat transfer area requirement of a twostage reactor is 16.7 % less than for a single-stage reactor. The
optimal mixing structure of a two-stage FT reactor is completely mixed (CSTR) for the first stage and plug flow (PFR)
for the second stage. The CSTR is to level out the temperature
peak with less heat transfer area requirement.
A case study is conducted to investigate the effect of distributing the original syngas feed between the stages. In this case,
97.48 % of the original syngas is fed to the first reactor and the
concentration of wax at the end of the path is 22.31 wt %.
For a single-stage FT reactor with iron-based catalyst and
syngas flow rate of 67.45 kg s–1, the conversion of CO and concentration of wax at the end of the path are 75.23 % and
13.70 %, respectively, [12]. In this case, the water-gas shift
reaction is active and part of CO is converted to CO2, i.e., the
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
Reactor design
7
Figure 3. Case 3 and the criteria defined by J1.
concentration of CO2 at the end of the path is 34.81 %. This
will increase H2/CO at the end of the reactor path to 5.33. The
optimal mixing structure is plug flow [12].
Maretto and Krishna [7] studied the staging of an FT slurrybubble column reactor and optimized the reactor conversion.
Staging of the reactor increases syngas conversion and reactor
productivity compared to a single-stage reactor. In their study,
the catalyst activity is equal to 1 and a temperature difference
of 10 °C is considered between the reactor and the coolant.
Furthermore, there is no cost on heat transfer area and the
multi-stage reactor requires additional cooling tubes. Here,
catalyst activity, reactor temperature, and heat transfer area are
optimized.
6.3
Realization of the Optimal Design
In an industrial GTL plant, the FT reactor will require large
volumes and sectioning of the reactor may be necessary in any
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
case. Systematic reactor staging will provide more degrees of
freedom for reactor optimization.
The mixing structure of a two-stage reactor based on the
profitability measure (second objective function) is CSTR for
the first stage and plug flow for the second stage. The geometry
of the slurry reactor, gas and liquid velocities, and the arrangement of the cooling tubes will affect the flow pattern inside the
reactor [7]. With the same value of b for fixed-bed and slurrybubble column reactors, more area is needed for a fixed-bed
reactor due to a lower heat transfer coefficient.
7
Conclusions
The method developed by Hillestad [1] was applied to FT synthesis by means of a kinetic model given by Iglesia et al. [2].
The design functions such as fluid mixing, heat transfer area
density, catalyst dilution, and distribution of extra feed are
optimized along the path to maximize an objective function.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
8
A. Rafiee, M. Hillestad
Figure 4. Case 4a and the
criteria defined by J2.
The objective function is here defined to be the yield of wax
(J1) and a measure of profitability (J2). Staging of the FT reactor based on J1 will increase the mass fraction of wax and CO
conversion. No distributed feed is beneficial because it will
only dilute the wax concentration.
By introducing the cost of heat transfer area in the objective
function (J2), the total heat transfer area requirement is reduced which will increase the annual profit. The optimal mixing structure for a two-stage FT is completely mixed (CSTR)
for the first stage and plug flow (PFR) for the second stage.
Having less heat transfer area, the purpose of the CSTR is to
level out the temperature peak. If the CSTR is set to a PFR, the
peak temperature will exceed the maximum temperature. The
syngas feed can be distributed between the stages. In this case,
97.48 % of the main syngas feed is fed to the first reactor and
the objective function will increase compared to a single-stage
reactor due to less heat transfer area requirements.
www.cet-journal.com
Staging with optimal distribution of heat transfer area, distribution of syngas, and mixing configuration will increase the
profit measure as demonstrated here. The results of this study
can serve as initial points for staging of the FT reactor in the
overall GTL process with different syngas production configurations [23].
Acknowledgment
The authors gratefully acknowledge financial support from the
Research Council of Norway through the GASSMAKS program.
The authors have declared no conflict of interest.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
Reactor design
Symbols used
a
A
B
[m2m–3]
[m2]
[–]
–1 –1
cp,i
cp,ref
C2,i
di
E
[kJ kg K ]
[kJ kg–1K–1]
[m–1]
[m]
[–]
h
I
J1
J2
~J
[–]
[–]
[wt %]
[–]
[–]
K
[–]
heat transfer area density
heat transfer area
scaling parameter to ensure a
consistent ASF distribution for all
components heavier than methane
specific heat capacity of component i
reference heat capacity
cost of tubes per meter
tube diameter
diagonal matrix,
E = diag (0,0,...., cp,ref cp–1)
path constraints
identity matrix
wax concentration
measure of profitability
partial derivative of component
reactions with respect to x,
~
~J ∂R0 x diag 0; 0; . . . ; 1 ×
∂x
a 1 cp;f cp;01
bcp;ref cp;01
diagonal matrix,
diag (1,1,...., cp,F cp–1)
[kmolCH4Pa–1.05mreactor–3s–1]
k1
kinetic parameter
k2
[kmolCOPa–1.25mreactor–3s–1]
kinetic parameter
K1
[Pa–1]
adsorption parameter
ṁsteam [kg s–1]
steam production rate
n
[–]
carbon number
nc
[–]
number of components
ns
[–]
number of stages
nuT [–]
number of design parameters for
coolant temperature
number of design parameters for
nuH [–]
heat transfer area density
PCO [Pa]
partial pressure of CO
PH2 [Pa]
partial pressure of H2
rCO [kmol mreactor–3s–1] consumption rate of CO
rCH4 [kmol mreactor–3s–1] production rate of CH4
[kmol mreactor–3s–1] component reaction rate on molar
Ri
basis
~i
R
[kg mreactor–3s–1] component reaction rate on mass
basis
T
[K]
temperature along the path
Tref
[K]
reference temperature
Tw
[K]
temperature of the cooling medium
along the path
uA
[–]
catalyst dilution
uF
[–]
feed distribution design function,
uF = ra
uH
[–]
heat transfer area distribution
design function, uH = rb
[–]
mixing design function
uM
u
[–]
vector of decision variables
U
[kWm–2K–1]
overall heat transfer coefficient
UH2 [–]
usage ratio of H2
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
V
VR
W
W0
Wn
[m3]
[m3]
[kg s–1]
[kg s–1]
[–]
x
[–]
xF
xW
z
[–]
[K]
[–]
z(0)
[–]
9
residence volume along the path
total residence volume of the path
total mass flow rate along the path
total mass flow rate at the inlet
weight fraction of hydrocarbons
heavier than methane
state vector of mass fractions and
temperature, x = [x1,...,xn,h]T
state vector of extra feed
coolant temperature
state vector augmented with mass
flow rate, z = [xT,c]
z at the inlet of the reactor path
Greek symbols
a
aASF
b
[kg m–3s–1]
[–]
[kg m–3s–1]
c
[–]
Di
k
h
[–]
[kJ kg–1]
[–]
n
[–]
r
[m3s kg–1]
xi
[–]
feed distribution
chain growth probability
heat transfer area design function,
b = Ua cp,ref–1
mass flow rate relative to the inlet,
W W0–1
volume fraction of stage i, Vi VR–1
latent heat of vaporization of water
dimensionless temperature,
h = (T–Tref )Tref–1
dimensionless volume of the path
(independent variable), V VR–1
space time or residence time,
VR W0–1
mass fraction of component i
References
[1] M. Hillestad, Chem. Eng. Sci. 2010, 65, 3301–3312.
[2] E. Iglesia, S. C. Reyes, S. L. Soled, in Computer-Aided Design
of Catalysts (Eds.: E. R. Becker, C. J. Pereira), Exxon Research
and Engineering Co., New York 1993, 199–257.
[3] M. E. Dry, A. P. Steynberg, in Fischer-Tropsch Technology
(Eds: M. E. Dry, A. P. Steynberg), Elsevier Science & Technology Books, Amsterdam 2004, 406–481.
[4] B. H. Davis, Catal. Today 2002, 71, 249–300.
[5] http://www.fischer-tropsch.org/DOE/DOE_reports/510/
510-34929/510-34929.pdf
[6] I. P. Androulakis, S. C. Reyes, AIChE J. 1999, 45, 860–868.
[7] C. Maretto, R. Krishna, Catal. Today 2001, 66, 241–248.
[8] T. Waku, S. Y. Yu, E. Iglesia, Ind. Eng. Chem. Res. 2003, 42,
3680–3689.
[9] S. Hwang, R. Smith, Chem. Eng. Sci. 2004, 59, 4229–4243.
[10] V. Diakov, A. Varma, Ind. Eng. Chem. Res. 2003, 43 (2), 309–
314.
[11] L. Guillou, S. Paul, V. L. Courtis, Chem. Eng. J. 2008, 136,
66–76.
[12] A. Rafiee, M. Hillestad, Comput. Chem. Eng. 2012, 39, 75–83.
[13] A. Jess, C. Kern, Chem. Eng. Technol. 2012, 35 (2), 369–378.
[14] A. Jess, C. Kern, Chem. Eng. Technol. 2012, 35 (2), 379–386.
[15] I. C. Yates, C. N. Satterfield, Energy Fuels 1991, 5, 168–173.
[16] http://www.opec.org/opec_web/en/publications/338.htm
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
10
A. Rafiee, M. Hillestad
[17] R. Krishna, S. T. Sie, Fuel Process. Technol. 2000, 64, 73–105.
[18] C. Maretto, R. Krishna, Catal. Today 1999, 52, 279–289.
[19] R. Turton, R. C. Bailie, W. B. Whiting, J. A. Shaeiwitz, Analysis, Synthesis, and Design of Chemical Processes, 3rd ed., Prentice Hall, New York 2008.
[20] M. S. Peters, K. Timmerhaus, R. West, Plant Design and Economics for Chemical Engineers, 3rd ed., McGraw-Hill, New
York 2003.
www.cet-journal.com
[21] H. S. Song, D. Ramkrishna, S. Trinh, H. Wright, Korean
J. Chem. Eng. 2004, 21 (2), 308–317.
[22] J. J. Marano, G. D. Holder, Fluid Phase Equilib. 1997, 138, 1–
21.
[23] A. Rafiee, M. Hillestad, Chem. Eng. Technol. 2012, 35 (5),
870–876.
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
Reactor design
Research Article: A cobalt-based catalyst
Fischer-Tropsch (FT) reactor is
sectioned into stages. Design functions
are optimized to maximize the two
selected objective functions, namely, the
wax concentration at the end of the
reactor path and a measure of
profitability. Staging of the FT reactor
increases the production rate of wax
based on the first objective function
criterion.
Chem. Eng. Technol. 2013, 36, No. 00, 1–11
11
Staging of the Fischer-Tropsch Reactor
with a Cobalt-Based Catalyst
A. Rafiee, M. Hillestad*
Chem. Eng. Technol. 2013, 36 (䊏),
XXX … XXX
DOI: 10.1002/ceat.201200700
© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com

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