Fischer-Tropsch Reactor: Particle Size & Tube Diameter Influence

Fischer-Tropsch synthesis
Andreas Jess
Christoph Kern
University Bayreuth,
Department of Chemical
Engineering, Bayreuth,
Germany.
369
Research Article
Influence of Particle Size and Single-Tube
Diameter on Thermal Behavior
of Fischer-Tropsch Reactors
Part I: Particle Size Variation for Constant Tube Size and Vice Versa
Simulation of a single tube of a wall-cooled multitubular Fischer-Tropsch (FT)
reactor with a cobalt catalyst indicates that the reactor performance is improved
by enlarging the catalyst particle diameter. This aspect is studied for variation of
the particle size for a fixed tube diameter and vice versa. For a syngas conversion
per pass of about 30 % as target and a typical industrially used single-tube
diameter of 40 mm, a particle size of > 3 mm is appropriate with regard to a
high production rate of higher hydrocarbons. For a particle diameter of < 3 mm,
a temperature runaway can only be avoided by rather low cooling temperatures,
and the target conversion cannot be reached. In addition, the pressure drop then
gets rather high. The reasons for this behavior are: (i) the heat transfer to the
cooled tube wall for a given tube size is considerably enhanced by increasing
the particle size; (ii) the influence of pore diffusion on the effective rate gets
stronger with rising particle size which decreases the danger of temperature runaway.
Keywords: Cobalt catalyst, Fischer-Tropsch synthesis, Fixed-bed reactor, Reactor modeling
Received: August 18, 2011; revised: November 04, 2011; accepted: November 18, 2011
DOI: 10.1002/ceat.201100615
1
CO + 3 H2 → CH4 + H2O
Introduction
The Fischer-Tropsch synthesis (FTS) has the potential to produce fuels like gasoline and diesel oil as well as petrochemicals
from fossil and renewable sources. In recent years, the availability of cheap natural gas and raw materials like coal and biomass has given momentum to FTS. The capacities will increase
in the near future with natural gas favored as feedstock, and
around 2015 the global annual production rate of fuels via FT
will be about 30 million tons, mostly produced in countries
like South Africa, Malaysia, and Qatar. FTS is described by:
CO + 2 H2 → (–CH2–) + H2O
0 = –152 kJ mol–1 (1)
DR H298
where the term (–CH2–) represents a methylene group of a
normal paraffin. For kinetic synthesis description, methane
formation is often considered as a separate reaction:
–
Correspondence: Prof. Dr.-Ing. A. Jess ([email protected]), University Bayreuth, Department of Chemical Engineering, Universitätsstraße 12, 95440 Bayreuth, Germany.
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
0 = –206 kJ mol–1
DR H298
(2)
The third reaction that plays an important role, at least if
iron-based catalysts are used, is the mostly unwanted formation of carbon dioxide by the water gas shift reaction:
CO + H2O → CO2 + H2
0 = –41 kJ mol–1
DR H298
(3)
Today, cobalt and iron are considered to be the most attractive FT catalysts. Iron is cheaper, but has a considerable water
gas shift activity (Eq. (3)). This is a drawback compared to
cobalt, as CO2 is an unwanted byproduct and less valuable
hydrocarbons are formed.
Two reactor types are currently favored for low-temperature
FT synthesis (< 250 °C): the multitubular fixed-bed and the
slurry bubble column. Here, only the multitubular reactor
with cobalt as catalyst is considered, although the analysis of
the reactor behavior done in this work is also valid for other
catalysts. Details on FT reactors and general aspects of the synthesis are found elsewhere [1–16].
One advantage of the multitubular reactor is that a modular
scale-up is possible from data of one representative single tube.
Nevertheless, reactor modeling is still needed to optimize the
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370
A. Jess, C. Kern
reactor with regard to productivity and safety aspects (temperature runaway). Such a modeling has to take into account the
intrinsic kinetics, pore diffusion limitations, and the radial
heat transfer from the fixed bed to the wall and finally to the
cooling medium, usually boiling water.
The most difficult problem to solve in the design of FT reactors is the high exothermicity combined with temperature
sensitivity of the product selectivity. Hence, an efficient heat
removal is needed. For the common tube length of about
10 m, the syngas conversion per pass is typically 30 %, but a
gas recycle, usually with a recycle ratio of 2, enables a high
overall conversion of about 90 %. Both the recycle and the limited conversion per pass help to remove reaction heat and
avoid a runaway.
According to a recent review from Steynberg et al. [17], the
number of serious modeling studies on fixed-bed FT reactors
reported in the open literature is limited [1, 18–22]. Since the
publication of Steynberg‘s review, to our knowledge only two
more studies on the simulation of fixed-bed FT reactors have
been published: Guettel and Turek [2] compared different reactors by a 1D approach (fixed bed, monolithic reactor, and
microreactor) demonstrating the potential of new concepts to
decrease mass transfer resistances. In an own work [16], results
of the simulation of multitubular reactors were presented for
iron and cobalt catalysts. Comparison of the 2D with a simpler
1D model indicated that the calculated syngas conversion is
similar, but the runaway behavior predicted by the 1D model
leads to a critical, too high cooling temperature. Hence, if all
heat transfer data is available, the more accurate 2D model
should be preferred.
During modeling of a wall-cooled FT reactor we realized by
chance that the particle size has a strong impact on the thermal
behavior of a multitubular reactor. Surprisingly, at least at first
sight, the increase of the particle size showed a positive effect
and motivated to have a closer look. For example, a higher
conversion can be obtained for larger particles without
temperature runaway, although the FT reaction is strongly
influenced by pore diffusion and the effective rate for a
given temperature decreases with increasing particle size. One
reason is the influence of the particle size on heat transfer. As
already stated by Steynberg et al., the heat transfer to the tube
wall can be considerably enhanced by increasing the catalyst
particle diameter [17]. Unfortunately, no details have been
given.
The objective of this paper is, therefore, the detailed simulation of the influence of the catalyst particle size on the behavior of a multitubular FT reactor with regard to syngas conversion and production rate of hydrocarbons per tube which has,
to the best of our knowledge, not been performed until now.
The kinetic data of a commercial cobalt catalyst were taken as
example, and the attempt was made to cover all main aspects,
including particularly intrinsic kinetics, mass transfer limitations, all heat transfer parameters (involving the influence of
particle size), and runaway behavior. Finally, general conclusions beyond FT were derived to determine the general influence of the particle size on the behavior of wall-cooled fixedbed reactors.
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2
Intrinsic and Effective Reaction Rate of
FTS on Cobalt
Since the discovery of the synthesis in 1923 by Franz Fischer
and Hans Tropsch, the kinetics have been studied extensively.
The reaction rate is mostly described by using power law or
Langmuir-Hinshelwood equations. An overview of rate equations for cobalt is given in [23]. In the initial phase of the synthesis, the catalyst pores are filled with liquid higher hydrocarbons (wax), which leads to a decrease of the effective reaction
rate by pore diffusion for particle diameters of more than
∼1 mm (see below).
Although numerous publications on FTS kinetics exist, the
number of reports with quantitative kinetic equations for
commercial catalysts is very limited. The intrinsic rate constant
of H2 consumption on a commercial cobalt catalyst was determined by Yates and Satterfield [23] and Maretto and Krishna
[24]. According to [17], the intrinsic activities of new-generation industrial cobalt catalysts are threefold higher than those
reported by these authors. Based on the data given in the
above-mentioned literature, the equation of the rate of H2
consumption was estimated (using the threefold values) by:
rm;H2 ˆ
cCO;g
dn_ H2
ˆ km;H2 ;LH cH2 ;g
2
dmcat
1 ‡ KCO cCO;g
(4)
The rate constant of this Langmuir-Hinshelwood equation
and the adsorption coefficient are:
37 400 J mol 1
km;H2 ;LH ˆ 0:8 m6 mol 1 kg 1 s
1
e
RT
(5)
68 500 J mol 1
KCO ˆ 5 10
9
m3 mol
1
e
RT
(6)
Based on these equations, a pseudo-first-order rate constant
can be determined:
cCO;g
rm;H2 ˆ km;H2 cH2 ;g with km;H2 ˆ km;H2 ;LH
2 (7)
1 ‡ KCO cCO;g
The influence of pore diffusion is considered by the effectiveness factor gpore:
gpore ˆ
rm;H2 ;eff
tanh f
1
≈ (for f ≥ 2)
ˆ
f
f
km;H2 cH2 ;g
and in the case of FTS the Thiele modulus is given by:
s
km;H2 qp cH2 ;g
Vp
fˆ
Ap;ext Deff;H2 ;l cH2 ;l
(8)
(9)
Vp/Ap,ex is the ratio of particle volume to external surface
area, and cH2,l is the hydrogen concentration in liquid wax, calculated by the Henry coefficient (HH2,c ≈ 20 000 Pa m3mol–1):
cH2 ;l ˆ
pH2 ;g
RT
ˆ
c
HH2 ;c HH2 ;c H2 ;g
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Chem. Eng. Technol. 2012, 35, No. 2, 369–378
Fischer-Tropsch synthesis
Combination of Eqs. (9) and (10) yields the following equation for the Thiele modulus:
v
Vp u
u km;H2 qp
fˆ
(11)
t
Ap;ex Deff;H ;l RT
2
HH2 ;c
The effective diffusion of dissolved hydrogen in the liquidfilled porous catalyst is described by an effective diffusion coefficient Deff,H2,l, which considers that only a portion of the particle is permeable and that the path through the particle is random and tortuous. Both aspects are taken into account by
porosity eP and tortuosity sP:
Deff;H2 ;l ˆ
ep
D
sP mol;H2 ;l
(12)
The molecular diffusivity of H2 in liquid FT products is calculated by the Wilke-Chang equation [25]. For temperatures
of 200–260 °C, Dmol,H2,l is 4 · 10–8 m2s–1 [26], and eP/sP is 0.3
[26, 27].
Fig. 1 presents the effective (formally first-order) rate constant of H2 consumption (gporekm,H2) for spherical particles
with diameters ranging from 0.5 to 6 mm. Pore diffusion
strongly affects the effective rate constant within the typical
temperature regime of FT synthesis of 200–250 °C for particle
diameters above 0.5 mm. The reason is the slow diffusion of
dissolved H2 in the liquid-filled pores of the catalyst. External
diffusion limitations are negligible for temperatures below
400 °C, which is an unrealistically high temperature for FT
synthesis.
The apparent (effective) activation energy at a certain temperature is an important parameter characterizing the thermal
sensitivity of a catalyst and a reactor and is given by:
EA;eff ˆ
d ln km;H2 ;eff
1
≈R
R
T2
d 1=T †
1
T1
1
km;H2 ;eff T1 †
for DT→0
ln
km;H2 ;eff T2 †
(13)
If a reaction is limited by pore diffusion (like FTS), EA,eff
may vary from the maximum intrinsic value to about half of
this value, if the range of relevant particle sizes and reaction
temperatures induces a transition of the effective rate constant
from the intrinsic kinetic regime to the regime of strong pore
diffusion limitation. Then Eq. (13) has to be used to calculate
EA,eff. For FT on cobalt, Eq. (13) is also needed even in the intrinsic regime, because both the rate constant km,H2,LH
(Eq. (5)) and the adsorption coefficient KCO (Eq. (6)) are functions of the temperature, and the apparent activation energy is
the result of the interplay of both terms (Eq. (7)).
3
Pressure Drop of an FT Fixed-Bed
Reactor
For technical fixed-bed reactors, the particle size is limited by
the pressure loss:
Dpb ˆ fb
LR qmol Mg u2s
dp
2
(14)
where us is the superficial velocity (empty tube) and fb is the
friction factor of a packed bed which is given by the following
equation based on the particle Reynolds number, Rep = usdp/m
[28]:
1 e†
300
(15)
3:5
‡
1
e
†
fb ˆ
e3
Rep
For a packed bed of spheres (equal diameter, porosity
e = 0.4), Eq. (15) simplifies to:
fb ≈ 33 ‡
1700
Rep
(16)
Fig. 2 indicates that for typical conditions of the FTS (see
Tab. 1) the particle size should be higher than 1 mm to avoid
an excessively high pressure drop. The dotted lines in Fig. 2
represent the extrapolation to pressure drops > 5 bar,
where Eq. (14) leads to inaccurate values, because then the
change of the gas density has to be considered and the
pressure drop gets very high compared to the assumed
total pressure of 24 bar.
4
Figure 1. Intrinsic and effective rate constant of H2 consumption
(Eq. (7)) for a cobalt catalyst for different particle diameters (spheres)
ranging from 0.5 to 6 mm (reactor entrance, i.e., cCO = cCO,in; other reaction conditions in Tab. 1).
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
371
FT Fixed-Bed Reactor Model
(Multitubular Reactor)
For simulation of a multitubular FT reactor, typical reaction conditions as listed in Tab. 1 were used. One aspect
was neglected for simplifying: The syngas entering the reactor may not only consist of CO and H2, but may contain
methane and also CO2, which are either already present in
the fresh syngas or in the recycle gas. If mass dispersion is
neglected (which is reasonable as proved by respective
calculations), the mass balance for hydrogen yields:
us
dcH2
ˆ rm;H2 qb
dz
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372
A. Jess, C. Kern
If axial dispersion of heat is neglected, the heat balance
of the 2D model is:
2
dT
d T dT
qmol cp us
ˆ krad
rm;H2 ;eff qb DR HH2
‡
dz
dr 2 r dr
(18)
Boundary conditions are:
T = Tin
(for z = 0)
(19)
dT
ˆ0
dr
(for r = 0 and all z)
(20)
krad
Figure 2. Pressure drop of fixed bed with a length of 12 m for different
particle diameters and two gas velocities (related to empty reactor; other
conditions in Tab. 1).
dT
ˆ aw;int Tw;int;1 Tw;int;2
dr
(for r = 0.5dR,int at the wall)
(21)
Eq. (18) considers the bed and the fluid as a pseudohomogeneous medium, i.e., the heat transfer in the bed up
For cobalt, the water gas shift reaction (Eq. (3)) does not
to the wall is represented by the effective radial thermal conplay a role. In this work, methane formation was also not conductivity krad. By Eq. (21), the model assumes a jump in temsidered as a parallel reaction to the formation of higher hydroperature at the wall from Tw,int,1 to Tw,int,2, which considers
carbons (C2+). Thus, Eq. (4) representing the overall H2 conthat krad decreases strongly in the vicinity of the wall, and the
sumption rate was applied.
introduction of aw,int accounts for the heat transfer near the
wall. The fourth boundary condition is
Table 1. Data on chemical media and reaction conditions of Fischer-Tropsch synthesis
related to the heat transfer from the exterused to model a multitubular fixed-bed reactor. All values at 24 bar and 240 °C, assumed
nal tube side to the heat transfer medium
standard internal tube diameter of 4 cm, and standard particle diameter of 3 mm
(here boiling water):
Parameter
Value
Superficial gas velocity us (empty reactor)a
0.55 m s–1
Total molar gas concentration (feed) qmol
563 mol m–3
Hydrogen concentration (feed) cH2,in
375 mol m–3
Total pressure p
24 bar
Diameter of spherical cobalt-catalyst particle dp
3 mm
Length of tubes LR
12 m
(Internal) diameter of single tube dR
4 cm
Bulk density of catalyst bed qb
700 kg m–3
Inlet concentration of hydrogen
66.6 vol.-% (rest CO)
Adiabatic rise in temperature DTad
1690 K
Kinematic viscosity m of gas mixture (feed)
4 · 10–6 m2s–1
Thermal conductivity of gas mixture (feed) kg
0.16 Wm–1K–1
Effective radial thermal conductivity krad
3.6 Wm–1K–1
Heat capacity of gas mixture (feed) cp
30 J mol–1K–1
Heat transfer coefficient (bed to internal tube wall) aw,int
1040 Wm–2K–1
Heat transfer coefficient (external tube wall to boiling water, 25 bar) aw,ex 1600 Wm–2K–1
Overall thermal transmittance Uoverall
336 Wm–2K–1
a) For a CO conversion of 27 % per pass and an assumed recycle rate of 2.5, the conversion related to the fresh syngas would be 95 %. For a selectivity of 90 % to C5+-hydrocarbons, which is given in the literature for modern cobalt catalysts [29], the production
rate would be 38 kg C5+ per day and tube.
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© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
q_ ˆ aw;int Tw;int;1 Tw;int;2 ˆ
1
dw
1
‡
Tw;int;2 Tcool
kwall aW;ex
(22)
The term (dw/kwall + 1/aw,ex)–1 summarizes the thermal resistance of heat
conduction in the wall with thickness dw
and of the heat transfer from the tube to
the cooling medium (here boiling water).
The resistance of heat conduction through
the wall (dw/kwall) is negligible: For 0.5 cm
wall thickness and a thermal conductivity
(steel) of 50 W m–1K–1, dw/kwall is only
0.0001 K W–1 compared to the (also rather
low) external heat transfer resistance
(1/aw,ex) to the boiling water of
0.0006 K W–1.
The coefficient aw,int for the heat transfer
from the bed to the internal wall was calculated by the following correlation for the
Nusselt number Nu [30, 31]:
aw;int dp
5 dp kbed
Nu ˆ
ˆ 1:3 ‡
kg
dR
kg
(23)
1=3
‡ 0:19 Re3=4
p Pr
The coefficient aw,ex for the external heat
transfer from the tube to the boiling water
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
Fischer-Tropsch synthesis
373
with Rep = usdp/m and Pr = m/a = mcpqmol/kg. Eqs. (23) and (24)
are not valid (or only to a limited extent) for low values of dR/
dp < 5.
kbed is the effective thermal conductivity of the bed without
flow, and kbed/kg is in the range of 2–10, if the thermal conductivity of the particles is high compared to the fluid (kp/kg >10).
In general, the values of the static contribution kbed/kg given in
the literature vary. Hofmann and Chao found that under reaction conditions kbed/kg is smaller than without reaction
[34, 35]. Apparently, there is an interaction between this static
thermal conductivity and the kinetics [36]. According to Westerterp, a value for kbed/kg of 4 should be used (as done in this
work) for modeling of wall-cooled reactors [36].
All heat transfer parameters for typical reaction conditions
are listed in Tab. 1. It is important to note that the particle size
has a strong influence both on aw,int and krad, and thus, on the
overall thermal resistance (1/Uoverall):
conditions in Tab. 1; resistance of wall (dw/kwall) was neglected).
With increasing particle diameter, the heat transfer resistance of the bed (Rheat,bed = dR/(8kbed) decreases, whereas the
thermal resistance at the internal reactor wall (Rheat,wall,int =
1/aw,int) increases. The external resistance to the boiling water
(Rheat,wall,ex = 1/aw,ex) is in general small and has a very low influence on the overall heat transfer. In total, the overall heat
transfer resistance decreases until a particle diameter of about
4 mm is reached (Fig. 3).
The influence of the particle size on the overall thermal
transmittance (Uoverall) of a cooled tubular reactor is more
pronounced for a large tube diameter (Fig. 4). For example, an
increase of the particle diameter from 1 to 6 mm leads to an
increase of Uoverall by about 14 % and 80 % for a single tube of
2 and 7 cm diameter, respectively. For particle Reynolds numbers above 100, which in this case is already reached at a small
particle size of 1 mm, kbed is proportional to dp, whereas aw,int
is almost independent of dp (∼dp–0.25).
The influence of dp on Uoverall vanishes for high values of dp
and Rep (Fig. 4 for dp > 6 mm), as the heat transfer resistance
at the internal wall gets high and determines the overall heat
transfer. For dR/dp values < 5, the applicability of Eqs. (22) and
(23) is also limited, as undesirable wall effects (bypassing, slippage) may occur (dashed line in Fig. 4).
1
d
1
d
1
ˆ R ‡
‡ w ‡
Uoverall 8 krad aw;int kwall aw;ex
5
Simulation of a Multitubular FT Reactor
5.1
Reactor Behavior for Constant Tube Diameter
and Particle Size
was determined by well-known correlations [32]. The coefficient krad for the radial effective heat conduction in the bed is
given by [30, 33]:
krad kbed
ˆ
‡ h
kg
kg
7 2
Rep Pr
1
2 i with
2dp dR
kbed
≈4
kg
(24)
(25)
The overall thermal transmittance Uoverall is not needed in
this work for reactor modeling (in contrast to a 1D model
where only axial temperature gradients are considered), but
nevertheless Eq. (25) is useful to illustrate the effect of the
particle size on the overall heat transfer. Fig. 3 depicts for a
typical tube diameter of 4 cm that the particle size has a strong
influence on the overall heat transfer resistance (Rheat,overall =
1/Uoverall) as well as on the respective individual contributions
of the heat transport within the bed (Rheat,bed = dR/8kbed), at
the internal reactor wall (Rheat,wall,int = 1/aw,int), and of the
external heat transfer to the boiling water (Rheat,wall,ex = 1/aw,ex;
For simulation of reactor behavior, the differential equations
(Eqs. (17)–(22)) were solved numerically by the commercial
program Presto Kinetics. The reactor behavior with regard to
the axial temperature profile and syngas conversion is illustrated in Fig. 5 for an assumed standard tube diameter of 4 cm
and a particle diameter of 3 mm.
Obviously, cooling temperatures of more than 206 °C should
be avoided to exclude a temperature runaway. Here, the following definitions are used:
– The ignition temperature (Tig) characterizes the
cooling temperature, where a clear temperature
runaway occurs (e.g., 211 °C in Fig. 5).
– The appropriate cooling temperature (Tcool =
Tin = 206 °C in Fig. 5) characterizes Tcool for a
safe operation of the reactor. Thereby, it is assumed that Tcool should be 5 K below Tig. In
some cases discussed here, Tcool may be even
lower than 5 K below Tig, if a certain syngas conversion as target is already reached. A syngas
conversion of preferably 27 % (per pass) was
supposed, because the total syngas conversion
would then be 90 % for the assumed typical
recycle ratio of 2.5 used in FT reactors. In this
case, the cooling temperatures listed in the tables
are marked with a star as superscript.
Figure 3. Influence of the particle diameter on the thermal resistances of a
cooled tubular reactor for a constant tube diameter of 4 cm.
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
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374
A. Jess, C. Kern
short-chain hydrocarbons are formed more and
more, and the catalyst may also deactivate.
– To characterize the reaction temperature, Tmean
= (Tcool + Tmax)/2 is used.
5.2
Figure 4. Influence of the particle diameter on the overall thermal transmittance
Uoverall of a cooled tubular reactor. Dashed line: limitation of the applicability of
Eqs. (22) and (23) for dR/dp < 5 (other conditions in Tab. 1).
Influence of Single-Tube Diameter for
Constant Particle Size
The influence of the tube diameter (dR) on the H2
conversion per pass and on the corresponding production rate of C5+-hydrocarbons for a particle
diameter of 3 mm is depicted in Fig. 6 (additional
data in Tab. 2; NC,crit, Nad, and km,H2,eff at Tig are
explained in Part II of this paper).
For the cooling temperature, a safety distance of
5 K to the critical cooling temperature, where a
temperature runaway occurs, was chosen (Tcool =
Tcool,crit – 5 K). Values of Tcool, Tmax, and of Uoverall
etc. are listed in Tab. 2. Other parameters and assumptions are summarized in Tab. 1.
A conversion per pass of preferably 27 % was assumed as target, i.e., for a typical recycle ratio of
2.5 the total conversion related to fresh feed would
then be 90 %. For the given conditions, this target
is missed for a tube diameter of more than 39 mm,
because a temperature runaway is then only
avoided by a rather low cooling temperature,
which in turn leads to a low reaction temperature
and rate (Tab. 2). This tube diameter can be regarded as optimal for a particle size of 3 mm, as it
is the largest possible diameter to reach the target
of 27 % conversion. For slightly higher tube diameters of up to 45 mm, the production rate of C5+hydrocarbons per tube would be higher, but the
target of 27 % conversion is then more and more
missed, e.g., only 20 % conversion at the maximum of the production rate of 37 kg per day,
which is almost the same value as for dR = 39 mm
(35 kg per day).
5.3
Influence of Particle Size for Constant
Tube Diameter
The impact of the particle size (dp) on the performance of a cooled single-tube FT reactor for a tube
diameter of 40 mm and a conversion of preferably
27 % is illustrated in Fig. 7 and Tab. 3. For Tcool, a
safety distance of 5 K to the critical cooling temperature, where a runaway occurs, was chosen
(Tcool = Tig – 5 K). The respective values of Tcool
and of Tmax are listed in Tab. 3.
Figure 5. Influence of cooling temperature on the behavior of a wall-cooled FT reactor (single tube with 4 cm diameter; spherical particles with 3 mm diameter;
For dp < 3.5 mm, the target of 27 % conversion
Tcool = Tin; other conditions in Tab. 1).
cannot be reached with regard to Tig. The thermal
transmittance Uoverall is low (201 Wm–2K–1 for
0.5 mm compared to 355 Wm–2K–1 for 3 mm, Fig. 6). In addi– The temperature in the tube center at the position of the
tion, the influence of pore diffusion increases with increasing
axial temperature maximum is denoted as Tmax. The value is
particle size and hence the effective activation energy drops,
limited to 250 °C. At higher temperatures, methane and
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Chem. Eng. Technol. 2012, 35, No. 2, 369–378
Fischer-Tropsch synthesis
Figure 6. Influence of single-tube diameter (dR) on H2 conversion (XH2 = XCO)
per pass and production rate of C5+-hydrocarbons of a cooled single-tube FT
reactor for 3 mm particle diameter and a syngas conversion of preferably 27 %.
375
which helps to avoid a thermal runaway (details in
Section 6). The decrease of gpore (Fig. 7, right) and
with that of the effective rate is actually considered
as a drawback, but this is compensated by a higher
cooling temperature as long as the reaction temperature remains below the limit of 250 °C. This is
needed to avoid excessive formation of methane as
well as of short-chain hydrocarbons. For the given
tube size of 40 mm, a particle size of 4 mm is optimal. For higher dp values, the ratio dR/dp gets too
low and with that undesirable wall effects (bypassing) may occur (dR/dp < 5 for dp > 8 mm).
Fig. 8 depicts radial temperature profiles for
small and large particles (1 and 6 mm) at the position, where the axial temperature maximum is just
reached. For dp = 6 mm, the effective radial thermal conductivity (krad) is threefold higher compared to 1 mm, whereas the heat transfer coefficient at the internal wall side (aw,int) is lower for
Table 2. Influence of tube size dR on Uoverall, Tcool = Tin, Tmax, Tig, gpore at Tmean = (Tcool + Tmax)/2, km,H2,eff at Tig, NC,crit at Tig, Nad, and EA,eff
at Tig; Tcool should be 5 K below Tig; if Tcool < Tig – 5 K to reach a conversion of 27 % per pass, the values are marked with a star.
dR
[mm]
dp
[mm]
15
20
60
70
Tig
[°C]
gpore at Tmean
[%]
km,H2,eff at Tig
[m3kg–1s–1]
NC,crit
Nad
EA,eff at Tig
[kJ mol–1]
477
215*
219
> 250
45
–
360
40
55
Tmax
[°C]
381
35
50
Tcool
[°C]
437
30
45
Uoverall
[W m–2K–1]
3
214*
211*
209*
221
224
225
242
223
217
44
44
45
–
–
–
4.91 · 10
–5
150
75
84
2.27 · 10
–5
189
80
90
1.72 · 10
–5
202
86
97
–5
341
206
225
211
48
1.26 · 10
229
96
107
324
202
220
207
56
1.00 · 10–5
243
104
116
310
198
214
203
65
0.77 · 10–5
272
112
126
75
0.58 · 10
–5
314
121
136
0.46 · 10
–5
348
128
143
0.33 · 10
–5
385
135
151
296
284
263
194
191
187
206
201
196
199
196
192
81
87
Figure 7. Influence of particle diameter (dp) on H2
conversion per pass, production rate of C5+-hydrocarbons, and pore effectiveness factor (gpore at Tmean =
(Tcool + Tmax)/2, 40 mm tube
diameter, conversion per
pass of preferably 27 %).
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
www.cet-journal.com
376
A. Jess, C. Kern
Table 3. Influence of particle diameter dp on Uoverall, Tcool = Tin, Tmax, Tig, km,H2,eff, TC,crit at Tig, Nad, and EA,eff.
Other data is shown in Fig. 7; Tcool should be 5 K below Tig; if Tcool < Tig –5 K to reach the target conversion of
27 % per pass, the values are marked with a star as superscript.
dp
[mm]
dR
[mm]
0.5
1
Tcool
[°C]
Tmax
[°C]
Tig
[°C]
km,H2,eff at Tig
[m3kg–1s–1]
NC,crit
Nad
EA,eff at Tig
[kJ mol–1]
201
184
189
189
0.29 · 10–5
586
156
175
193
–5
513
153
172
–5
367
136
152
–5
229
96
109
–5
191
80
88
–5
255
2
313
3
4
Uoverall
[W m–2K–1]
341
40
355
188
195
206
214*
194
204
225
232
200
211
221
0.42 · 10
0.72 · 10
1.26 · 10
1.57 · 10
5
362
220*
237
229
1.76 · 10
174
79
86
6
365
225*
241
236
1.95 · 10–5
158
78
85
7
367
229*
245
242
2.11 · 10–5
147
76
84
248
–5
135
75
82
8
367
232*
247
6-mm particles (201 Wm–2K–1) compared to 1811 Wm–2K–1
for 1-mm particles. This diminishes the positive effect of dp
on krad. The overall thermal transmittance (Eq. (25)) is
365 Wm–2K–1 for dp = 6 mm, which is 30 % higher compared
to 1-mm particles (255 Wm–2K–1). This underlines the advantage of large particles for the overall radial heat transport.
Fig. 8 indicates that the overall heat transfer resistances in the
bed (8 K) and at the internal wall side (7 K) are almost equal
for dp = 6 mm, whereas for dp of 1 mm, the heat transfer resistance of the bed is dominant (DTbed >> DTwall).
6
Conclusions and Outlook
Simulation of a wall-cooled single tube of a multitubular FT
reactor with cobalt as catalyst demonstrates that the reactor
performance is improved by increasing the catalyst particle
2.30 · 10
size. The reasons for this at first sight unexpected behavior are:
(i) The radial heat transfer for a given tube diameter is considerably enhanced by increasing the particle size. (ii) The influence of pore diffusion on the effective reaction rate of syngas
gets stronger with rising particle diameter, which lowers the
temperature sensitivity of the reactor and decreases the danger
of a temperature runaway.
Three questions are still open: Is it possible to improve the
reactor performance by means of an eggshell catalyst, where
the influence of pore diffusion is negligible? Which is the optimal combination of particle size and tube diameter? Is it possible to derive general guidelines for the appropiate choice of
particle and tube size for wall-cooled fixed-bed reactors beyond the case of FTS? These questions will be addressed in
Part II of this series of two papers.
The authors have declared no conflict of interest.
Figure 8. Radial temperature profiles in a 40-mm single tube for two different particle sizes (in both cases,
Rheat,wall,ex = 1/aW,ex = 1/1600 m2K W–1; axial positions at Tmax: z = 4.2 m for dp = 1 mm and 2.1 m for dp = 6 mm;
other conditions in Fig. 6).
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© 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
Fischer-Tropsch synthesis
Symbols used
a
Ap,ex
c
cp
dp
dR
dw
Deff
Dmol
EA
fb
HH2,c
km
km,H2,LH
km,eff
KCO
LR
mcat
Mg
Nad
NC
Ntubes
Nu
p
pi
Pr
q_
r
rm
rm,eff
R
Rheat
Rep
T
us
Uoverall
Vp
X
z
[m2s–1]
[m2]
thermal diffusivity
external surface area of a catalyst
particle
[mol m–3]
concentration
[J mol–1K–1]
heat capacity of gas phase
[m]
particle diameter
[m]
diameter of a single tube
[m]
wall thickness of a single tube
[m2s–1]
effective diffusion coefficient
[m2s–1]
molecular diffusion coefficient
[J mol–1]
activation energy
[–]
friction factor of packed bed
[Pa m3mol–1]
Henry coefficient of hydrogen in
liquid wax
[m3kg–1s–1]
rate constant for (pseudo)
first-order equation
[m6mol–1kg–1s–1] rate constant for LangmuirHinshelwood equation for FT
on Co
[m3kg–1s–1]
effective reaction rate constant
[m3mol–1]
adsorption coefficient for CO
[m]
reactor length (single tubes)
[kg]
mass of catalyst
[kg mol–1]
molar mass of gas phase
[–]
dimensionless parameter of heat
generation
[–]
dimensionless parameter of
cooling capacity
[–]
number of single tubes
[–]
Nusselt number
[Pa]
total pressure
[Pa]
partial pressure of component i
[–]
Prandtl number
[W m–2]
heat flux
[m]
radial direction
[mol kg–1s–1]
reaction rate per unit of mass of
catalyst
[mol kg–1s–1]
effective rate per mass of catalyst
[J mol–1K–1]
ideal gas law constant (8.314)
[m2K W–1]
heat transfer resistance
[–]
Reynolds number related to
particle
[ °C, K]
temperature
[m s–1]
superficial gas velocity
[W m–2K–1]
overall thermal transmittance
[m3]
particle volume
[–]
conversion
[m]
axial coordinate
Greek symbols
aW,int
[W m–2K–1]
aW,ex
[W m–2K–1]
DRHH2
[J mol–1]
heat transfer coefficient from
bed to internal wall
heat transfer coefficient from
outer shell tube to boiling water
enthalpy of FT reaction related
to 1 mol H2 (76 kJ mol–1)
Chem. Eng. Technol. 2012, 35, No. 2, 369–378
DTad
e
eP
f
gpore
kg
[K, °C]
[–]
[–]
[–]
[–]
[W m–1K–1]
kbed
[W m–1K–1]
krad
[W m–1K–1]
kwall
[W m–1K–1]
m
qp
qb
qmol
sP
[m2s–1]
[kg m–3]
[kg m–3]
[mol m–3]
[–]
377
adiabatic rise in temperature
porosity of fixed bed
particle porosity
Thiele modulus
pore effectiveness factor
thermal conductivity of gas
phase
thermal conductivity of the fixed
bed without flow
effective radial thermal
conductivity in the fixed bed
thermal conductivity of reactor
wall material
kinematic viscosity
density of catalyst
bulk density of catalyst bed
molar density of gas phase
particle tortuosity
Subscripts
cool
crit
eff
ex
g
ig
in
int
l
max
w
cooling
critical
effective
external
gas phase
ignition
reactor inlet
internal
liquid
maximum value
wall
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