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This journal is c The Royal Society of Chemistry 2012 Catal. Sci. Technol., 2012, 2, 1221–1233 1221

Cite this: Catal. Sci. Technol., 2012, 2, 1221–1233

Fischer–Tropsch reaction–diffusion in a cobalt catalyst particle: aspects

of activity and selectivity for a variable chain growth probability

David Vervloet,*aFreek Kapteijn,

bJohn Nijenhuis

aand J. Ruud van Ommen

a

Received 2nd February 2012, Accepted 27th February 2012

DOI: 10.1039/c2cy20060k

The reaction–diffusion performance for the Fischer–Tropsch reaction in a single cobalt catalyst

particle is analysed, comprising the Langmuir–Hinshelwood rate expression proposed by Yates

and Satterfield and a variable chain growth parameter a, dependent on temperature and syngas

composition (H2/CO ratio). The goal is to explore regions of favourable operating conditions for

maximized C5+ productivity from the perspective of intra-particle diffusion limitations, which

strongly affect the selectivity and activity. The results demonstrate the deteriorating effect of an

increasing H2/CO ratio profile towards the centre of the catalyst particle on the local chain

growth probability, arising from intrinsically unbalanced diffusivities and consumption ratios of

H2 and CO. The C5+ space time yield, a combination of catalyst activity and selectivity, can be

increased with a factor 3 (small catalyst particle, dcat = 50 mm) to 10 (large catalyst particle,

dcat = 2.0 mm) by lowering the bulk H2/CO ratio from 2 to 1, and increasing temperature from

500 K to 530 K. For further maximization of the C5+ space time yield under these conditions

(H2/CO = 1, T = 530 K) it seems more effective to focus catalyst development on improving the

activity rather than selectivity. Furthermore, directions for optimal reactor operation conditions

are indicated.

Introduction

Selecting an appropriate catalyst dimension for heterogeneous

catalyzed reactions is crucial for realizing optimum catalyst

utilization and selectivity, as expressed by the catalyst effective-

ness factor (Z). The Thiele modulus (f) is the key parameter that

defines the interplay between reaction rate(s) (Ri) in a porous

catalyst (with characteristic length lcat) and mass transport by

effective diffusion (Di,eff). The derivations and expressions for fare well-known for numerous types of kinetics.1

The heterogeneously catalyzed Fischer–Tropsch (FT) synthesis,

in which syngas is converted into hydrocarbons and water, may be

strongly affected by diffusion limitations.2,3 Therefore, an analysis

of the Thiele moduli for the reactants (H2 and CO) is crucial for

catalyst and reactor design purposes, irrespective of the reactor

type in which the catalyst is applied. Furthermore, the selectivity

towards desired hydrocarbon chain-lengths, typically C5+ in

low-temperature Fischer–Tropsch synthesis (FTS), is a key

factor. This is generally expressed by the chain growth

probability parameter a, which depends on the local tempera-

ture (T) and reactant concentrations (ci).4

An analysis of the diffusivities of the reactants reveals that the

ratio of diffusivities of H2 over CO in a typical liquid hydro-

carbon product (e.g. C28 n-paraffin, following the relations by

Wang et al.5) at typical low temperature FT temperatures

(e.g. 500 K) is approximately 2.7; this is similar to values

reported by other authors, e.g. ref. 6. Not only is hydrogen

diffusion faster than that of CO, but its concentration in the

liquid phase is typically also higher. Although the CO solubility

is approximately 1.3 times higher than that of H2 in a typical

liquid product medium at 500 K (following the relations and

parameter values for the Henry coefficients by Marano and

Holder7), bulk syngas feed ratios of 2 (or slightly lower) are

typically chosen for stoichiometric reasons, resulting in a liquid

H2/CO concentration ratio of approximately 1.6.

The consumption ratio of H2 over CO on the other hand is a

value between 2 (for production of infinitely long hydrocarbon

chains) and 3 (for production of methane), so depending on a.It can be shown mathematically, analogous to,8 that the

consumption ratio of H2 over CO follows the remarkably

linear result (3 � a), given the assumption that a is independent

of the chain length. For typical desired a values, between 0.9 and

0.95, the conclusion is that the diffusivity and concentration ratios

do not match the consumption ratio of H2 and CO. Therefore,

under typical reaction conditions, a syngas ratio (H2/CO) gradient

is expected inside the catalyst particle for diffusion limited systems,

having an impact on the catalyst performance in terms of reaction

rate and selectivity. To avoid limitation in one of the reactants,

a Product & Process Engineering, Delft University of Technology,Faculty of Applied Sciences, Julianalaan 136, 2628 BL Delft,The Netherlands. E-mail: [email protected]

b Catalysis Engineering, Delft University of Technology,Faculty of Applied Sciences, Julianalaan 136, 2628 BL Delft,The Netherlands

CatalysisScience & Technology

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1222 Catal. Sci. Technol., 2012, 2, 1221–1233 This journal is c The Royal Society of Chemistry 2012

the H2/CO consumption ratio (3 � a) inside a particle should

preferably match its molar diffusion ratio, which is determined

by the ratio of the diffusivities and the concentration gradients

of H2 and CO. In a simple approximation with full conversion

the latter suggests syngas compositions with H2/CO ratios

below 1.0 as indicative feed composition, whereas generally

values around 2.0 are used for stoichiometric reasons.

Objective and relevance

The key objective of this paper is to map the performance of a

cobalt based FT catalyst particle as a function of a range of

operating conditions (fCO, H2/CO, T, and p) by a numerical

analysis. A numerical study on an FTS catalyst in itself is not

new. Multiple examples of models can be found in the

literature, many of which are directly coupled to some kind

of reactor model,6,9–20 although several studies also report the

performance of individual particles.3,5,21,22 However, all of the

models are typically based on either simplified kinetics (mostly

first order in hydrogen) and/or a limited parameter space

(mostly at a single operating point for pressure, temperature,

and/or syngas ratio). Furthermore, none of the previous

studies takes into account that the selectivity inside the

catalyst particle may change locally as a consequence of

changing syngas ratio. These approaches have their limitations

for several reasons:

� Assuming a constant chain growth probability a may be

too rudimentary.

� Investigation of a limited parameter space provides no

insight into possible optimal conditions.

� Simplified FT kinetics based on H2 are not valid for cases

that are CO diffusion limited (i.e. CO conversions above

0.6),23 which can easily occur in a single catalyst particle under

typical conditions.

� Reporting only dimensionalized parameters, e.g. reaction

rate in a catalyst particle of a certain size, does not give much

generic insight in the catalyst performance.

This work combines the reaction–diffusion problem of the

Langmuir–Hinshelwood FT kinetics as reported by Yates and

Sattefield24 with a temperature- and H2/CO ratio dependent

chain growth probability parameter (a), based on experimental

data from the literature. The model results are presented for a

broad range of several operating parameters to provide detailed

insight in the catalyst performance. The performance of the

catalyst is investigated for several criteria: the average chain

growth probability (aave), catalyst effectiveness (Z), total COconversion rate per unit mass catalyst (RCO,total) and the C5+

space time yield (STYC5+).

This work focuses on the evaluation of the catalyst performance

from the perspective of a local reaction–diffusion process

and selectivity in a reactor. External mass and heat transfer

limitations are not considered. Reactor design aspects, such as

pressure drop or cooling duty, are also left out of the analysis,

as these take place on a different scale, although these may

impose changing boundary conditions on the particle scale.

The results are used as research motivation for the improvement

of FT catalysts, and as explorative guidelines for optimum

conditions for FTS, which can be used as a basis for reactor

operating strategies.

Model derivation and approach

Reaction–diffusion equations

The dimensionless steady state reaction–diffusion mass balances in

a catalyst particle with geometry indicated by s (0 for a slab, 1 for a

cylinder, and 2 for a sphere) are captured by a second order

differential equation (eqn (1)), where yi is the dimensionless

concentration of species i (yi = ci/ci,0), z is the dimensionless length

of the catalyst (z = x/lcat, where x denotes the location in the

catalyst, and lcat represents the characteristic dimension of the

catalyst, defined by lcat = Vcat/Scat, where Vcat is the catalyst

particle volume and Scat is the external surface area),fi is the Thiele

modulus and Ci is the dimensionless reaction rate (Ci = Ri/Ri,0,

where Ri and Ri,0 are the local and surface reaction rates of

species i). It is assumed that the catalyst is fully saturated with a

liquid medium in which all reactants and products are dissolved.

The effect of product flow leaving the catalyst is neglected

(Appendix A) and therefore not included in the differential

equation (eqn (1)) for the steady state reaction–diffusion problem.

0 ¼ 1

zsd

dzzsdyi

dz

� �� f2

i Ci ð1Þ

The Thiele modulus is defined as:

fi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil2cat

Di;effci;0Ri;0

sð2Þ

where Di,eff is the effective diffusivity, with Di,eff = (ecat/tcat) �Di,bulk. Di,bulk is assumed to follow an Arrhenius type temperature

dependency5 with pre-exponential diffusivity Di,0 and diffusion

activation energy ED,i according to Di,bulk = Di,0exp(�ED,i/RT).

ecat and tcat are the catalyst porosity and pore tortuosity, respec-

tively. The boundary conditions are:

centre : dyidz

��z¼0¼ 0

surface : yijz¼sþ1¼ 1ð3Þ

External transport effects can be easily incorporated by replacing

the surface boundary conditions with:

dyi

dz

��z¼sþ1

¼ Bimð1� yi;0Þ

where Bim denotes the Biot number for mass transfer:

Bim ¼kLSðsþ 1Þlcat

Di;eff

where kLS is the external liquid–solid mass transfer coefficient.

Langmuir–Hinshelwood kinetics for the FT reaction24 defines

the reaction rate per catalyst volume:

Ri ¼ jvijrcatFapCOpH2

ð1þ bpCOÞ2ð4Þ

a ¼ a0 expEA;a

R

1

493:15� 1

T

� ��

b ¼ b0 expDHb

R

1

493:15� 1

T

� �� ð5Þ

where ni is the stoichiometric constant for species i, rcat is thecatalyst particle density, F is a catalyst activity multiplication

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factor that accounts for improvements in catalyst activity since

publication of the original parameter values,15 and a and b are the

reaction rate and adsorption coefficients (eqn (5)), as reported by

Maretto and Krishna,6 and pi is the partial pressure of reactant i.

The partial pressures, pi, and the liquid phase concentra-

tions, ci, of species i are coupled through Henry’s law by

pi = Hi � (ci/cl), where Hi represents temperature dependent

Henry’s constant and cl is the total molar liquid concentration.

The local dimensionless reaction rate for species i, defined as

Ci = Ri/Ri,0, can be derived from the rate equation (eqn (4)).

Conveniently, the temperature inside the catalyst particle can

be assumed constant as the internal Prater number is small

(Appendix B).25,26 This was numerically verified, and agrees

with conclusions by Wang et al.5 Additionally, we verified the

absence of external interphase (liquid–solid) heat transport

limitations by applying Mears’ criterion27 (Appendix C). The

absence of local temperature gradients inside the particle

drastically reduces the complexity of the system. The temperature

dependent reaction rate (a) and adsorption (b) parameters that

appear in rate expression (eqn (4)) can now be assumed constant

over the entire particle, as well as the Henry coefficients (Hi) for

H2 and CO. The expression for the local dimensionless reaction

rate Ci becomes as (eqn (6)):

Ci ¼vi

vi;0

�� 1þ bpCO;0yCO;0

1þ bpCO;0yCO

� �2

yCOyH2ð6Þ

Since the reaction rate is defined per mol CO, |nCO|= |nCO,0| = 1

applies. For H2 we derived, analogous to Stern et al.,8 with the

assumption that a is independent of the chain length: |nH2|= 3� a.

The mass balances are now fully defined. The energy balance

over the particle is omitted, since both internal- and external

heat transport processes are not limiting (Appendix B and

Appendix C), and isothermicity can be assumed.5

Selectivity as a function of temperature and syngas ratio

To arrive at a suitable expression for the catalyst selectivity,

the following assumptions are made:13,28

� a is independent of chain length.

� a is a function of syngas ratio and temperature

� The selectivity can be described by the ratio of propagation

and termination reactions at the active sites on the catalyst,

following a standard Arrhenius dependency with temperature.

� The ratio of termination and propagation reactions scales

with some power of the local syngas ratio.

The model for the chain growth probability that was derived

is (Appendix D):

a ¼ 1

1þ kaðcH2cCOÞb expðDEa

Rð 1493:15� 1

TÞÞ

ð7Þ

where ka denotes the ratio of rate constants for the propaga-

tion and termination reactions, b is the syngas ratio power

constant, and DEa is the difference in activation energies for

the propagation and termination reactions.

We note that other proposals exist for the description of the

product distribution, either based on a single chain growth

probability a,29 a chain length dependent growth probability

based on, for example, a1 and a2 for specific chain lengths,20,30,31

or more sophisticated models that rely on microkinetics to

describe the product distribution,32 whether or not further

extended by olefin readsorption models that are influenced by

diffusion limitations.33 The suitability of all of these approaches

is highly catalyst dependent and sometimes under debate, but

may be included taking specific effects into account. Eqn (7),

which has a sigmoidal shape, satisfies the general limiting

trends observed in the literature: a approaches 1 with decreasing

temperature and syngas ratio, and 0 with increasing tempera-

ture and syngas ratio.

Performance parameters

The performance of the full catalyst particle is evaluated on

several criteria listed in Table 1. Eqn (8) is the classical

definition of the effectiveness factor of a sphere based on the

volume integral of the reaction rate divided by the total

volume of the catalyst particle. Eqn (9) defines the average

chain growth probability as volume integral over the catalyst

particle, weighted with the local dimensionless reaction rate

and corrected for effectiveness. Eqn (10) is the expression for

the total CO conversion rate based on catalyst weight.

Eqn (11) is the definition of C4� selectivity by weight34

(ch. 6, p. 403), which is used to calculate the C5+ selectivity

by weight (eqn (12)), since both selectivities add up to unity.

Eqn (13) expresses the C5+ space time yield on hourly basis

as the product of total molar CO conversion rate, C5+

selectivity and molar weight of a CH2 building block in the

hydrocarbon chains.

Table 1 Performance parameters of the full catalyst particle and equations for calculation

Performance parameter Symbol (unit) Equation

Catalyst effectiveness, sphere (s = 2) Z (—)

Z ¼

RVcat0

CCOðVÞdV

CCO;0Vcat¼ 3

33

R30

CCOðzÞz2dz (8)

Average chain growth probability, sphere (s = 2) aave (—)aave ¼ 3

33Z

R30

aðzÞCCOðzÞz2dz (9)

Total CO consumption rate per unit mass catalyst RCO,total (mol kgcat�1 s�1) RCO;total ¼ Z

rcatRCO;0 (10)

C4� selectivity SC4�(—)

SC4� ¼P4n¼1

nð1� aaveÞ2an�1ave (11)

C5+ selectivity SC5+(—) SC5+

= 1 � SC4�(12)

C5+ space time yielda STYC5+(g gcat

�1 h�1) STYC5+= 3.6RCO,totalSC5+

MCH2(13)

a Using the molar mass of building block MCH2= 14 g mol�1.

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Numerical approach

The differential equation (eqn (1)) was solved numerically on a

grid using a second order central difference scheme implemen-

ted in Fortran 95. For small fi eqn (1) is solved over the entire

domain (z = [0, s + 1]). For large fi (fCO 4 B1, depending

on H2/CO surface conditions) the grid was scaled to the

reactive (outer) shell of the catalyst particle (z = [zleft, s + 1]) to

increase the numerical accuracy of the solution and the stability of

the code, as well as speed to avoid wasting computational

resources on the calculation of values in the centre of the catalyst

particle that are essentially zero. In a coordinate scaling

problem of a reaction–diffusion system the reactive shell

thickness is typically inversely proportional to fi.35 However,

due to the non-linear nature of the rate equation (eqn (4)) the

conventional scaling approach35 of estimating the left (inner)

boundary condition for a single-component first order

reaction, zleft = max[0, (s + 1) � (1 � 1/fi)], was found

inadequate. Instead, a satisfactory estimate of the location of

the dimensionless left boundary condition (for the investigated

range of parameters) was found, by trial and error, to be:

zleft ¼ max 0; ðsþ 1Þ 1� 2

fCO

ffiffiffiffiffiffiffifficH2 ;0

cCO;0

q0B@

1CA

264

375

For an appropriate range of analysis (fCO o 40), a number

of 50 equidistant nodes were found sufficient to satisfy a

relative and absolute tolerance of 10�3.

Results and discussion

The selectivity model

Numerous results are presented in the literature on the selectivity

of different Co catalysts (promoted/unpromoted, various sizes

in Co active sites, and several different support structures)

under different process conditions (syngas ratio, temperature,

pressure). Although the general trends in the data seem

consistent, i.e. a decreases with increasing syngas ratio and

temperature, the amount of scatter in the data points is too

large to draw satisfactory conclusions about the trends. More

than 220 reported chain growth parameters from various

sources,2,11,36–61 mostly from the last decade, are given in

Fig. 1A as a function of temperature and H2/CO ratio. The fit

on the model equation (eqn (7)) is rather poor, judging from the

R2 value. Remarkably, the reported a values are generally well

below the commonly conjectured industrial range of 0.9–0.95.

A different approach to estimate a is to use the values for

methane selectivity (a = 1 � Smethane). This approach leads to

a conservative (lower) estimate for a, because it is generally

observed that the methane selectivity is somewhat larger than

(1 � a)34 (ch. 6). Using values for the methane selectivity reported

by De Deugd et al. (ch. 4)46,61 for thin layer (30 mm) coated

monolithic catalysts, which are assumed to be little affected by

transport limitations,46 the fitting parameters for the model

equation (eqn (7)) are obtained with a reasonably good fit

(Fig. 1B). The a-values derived from methane selectivity follow

the expected decreasing trends with increasing syngas ratio and

temperature and, furthermore, the a-value for typical conditions isbetween 0.9–0.95, as conjectured for industrial application.

Model results of the reaction–diffusion equation: reference

conditions

The parameter values that are used in the model and conditions

explored are presented in Table 2. The results of the multi-

dimensional reaction–diffusion problem are presented in the

following paragraphs with an increasing number of variables,

starting with a single experimental condition (Fig. 2).

Fig. 2 compares the results of two single calculations: one with

a fixed a value of 0.86, which is the predicted value based on the

bulk composition conditions (H2/CO = 2 and T = 490 K), and

one with the variable a model. The profiles of the dimensionless

concentrations, the dimensionless conversion rate of CO and the

local chain growth probability parameter inside a spherical

catalyst particle are given (conditions given in the caption).

Table 3 contains a comparative overview of the performance

parameters (fCO, Z, aave, RCO,total, SC5+, STYC5+

).

Both cases in Fig. 2 (fixed and variable a) are primarily CO

diffusion limited. The dimensionless concentration of CO

Fig. 1 Fit of data on the chain growth parameter a on the model equation (eqn (7)). The grey intensity of the symbols represents the H2/CO ratio,

as indicated by the grey scale bar. (A) Data from various sources. (B) Data from De Deugd et al.61 a-Values obtained from methane selectivity:

a = 1 � Smethane. Fitted model parameters (dependent), 95% confidence intervals and R2 values given in plot.

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drops much faster than that of H2, as a result of the lower

diffusion to consumption ratios. This is also visible in the top

left inset in Fig. 2A, where the concentration ratio of H2 over

CO varies over orders of magnitude towards the centre of the

catalyst. The dimensionless CO conversion rate first increases

towards the centre of the catalyst particle due to its negative

reaction order, as a consequence of the higher order of the

adsorption term in the rate expression, and then sharply drops

to zero upon further decrease in CO concentration, whereby the

reaction order of CO changes from negative to positive. Due to

this phenomenon the catalyst effectiveness exceeds unity.

Comparing the two cases for the selectivity (dashed lines:

a = 0.86, solid lines: a = variable) it is clear that the

dimensionless concentration profiles for CO are not very

different and that of H2 is somewhat lower for the variable acase. The H2/CO concentration ratio profile is lower for a

variable a case (top left inset, Fig. 2A) due to the a-dependentconsumption ratio. The catalyst effectiveness and overall CO

consumption rates do not differ much (Table 3).

The major difference between the two cases is visible in the

selectivity. In the variable a case the chain growth probability

deteriorates as the H2/CO ratio increases in the region where

the reaction rate is highest (z between 1.5 and 2.5). This causes

significant differences in the overall selectivity of the catalyst

particle, which results in a threefold reduction of the C5+

space time yield.

Model results for variable /CO and H2/CO ratio

The calculations with variable a have been performed for a

wide range of conditions. In Fig. 3 aave, Z, RCO,total, SC5+, and

Table 2 Parameter values and condition ranges used in the model

Description Symbol Value Motivation/reference

Temperature T 470–530 K Varied rangePressure p 12–36 bar Varied rangeSyngas ratio in the bulk — 0.1–3.0 Varied rangeCatalyst particle diameter dcat 10–5000 mm Varied rangeCatalyst intrinsic (skeleton) density rcat 2500 kg m�3 TypicalCatalyst porosity ecat 0.5 TypicalCatalyst pore tortuosity tcat 1.5 TypicalYates and Satterfield reaction rate constant a(T) T-dependent relation/mol s�1 kgcat

�1 bar�2 24Yates and Satterfield adsorption constant b(T) T dependent relation/bar�1 24Catalyst activity multiplication factor F 1–10 Estimated catalyst activity improvement15

CO diffusion constant in product medium D0,CO 5.584 � 10�7 m2 s�1 5CO diffusion activation energy ED,CO 14.85 � 103 J mol�1 5H2 diffusion constant in product medium D0,H2

1.085 � 10�6 m2 s�1 5H2 diffusion activation energy ED,H2

13.51 � 103 J mol�1 5Henry coefficient Hi(T) T dependent relation/bar 7Chain growth probability a Model This work (eqn (7))Selectivity constant ka 56.7 � 10�3 This work (eqn (7))Selectivity exponential parameter b 1.76 This work (eqn (7))Selectivity activation energy difference DEa 120.4 � 103 J mol�1 This work (eqn (7))

Fig. 2 (A) Dimensionless concentration yi, reaction rateCi and a profiles in a spherical catalyst particle for a fixed a (dashed lines, a=0.86) and for the

variable amodel (solid lines). Conditions: T=490 K, p=30 bar, syngas ratio at the catalyst surface = 2, dcat = 1.5 mm, F=1; other parameters as in

Table 2. z=0 at the centre and z=3 at the surface of the particle. Inset: the syngas (H2/CO) ratio profile in the catalyst. (B) Graphical representation of

the dimensionless CO reaction rate profile in the catalyst sphere (variable a). An overview of other performance parameters is given in Table 3.

Table 3 Comparative overview of performance parameters for a fullcatalyst particle for the fixed a model versus variable a model.Conditions: T = 490 K, p = 30 bar, syngas ratio at catalystsurface = 2, dcat = 1.5 mm, F = 1; other conditions as in Table 2

Fixed a model Variable a model

fCO (—) 0.88 0.88Z (—) 1.37 1.28aave (—) 0.86 0.57RCO,total (mmol kgcat

�1 s�1) 3.36 3.13SC5+

(—) 0.86 0.29STYC5+

(g gcat�1 h�1) 0.15 0.046

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STYC5+are given as a function of fCO and the bulk syngas

ratio at constant temperature and pressure using the variable

a model. The corresponding particle diameters, dcat, are

also given.

From the results follows that aave drops significantly in the

region where fCO 4 0.6 and bulk syngas ratio 41 (Fig. 3A).

The catalyst effectiveness Z shows a local maximum in the

region where aave drops sharply, and decreases with increasing

fCO for fCO 4 1 (Fig. 3B). The total reaction rate in the

catalyst particle, RCO,total, increases with increasing H2/CO

ratio due to less CO inhibition (Fig. 3C). Furthermore,

RCO,total (Fig. 3C) and the C5+ selectivity (Fig. 3D) are

coupled to Z and aave, respectively, and follow the same trends.

The STYC5+contour plot (Fig. 3E) shows that under these

conditions (p = 30 bar, T = 490 K, F = 1 � Yates and

Satterfield activity) the maximum C5+ productivity is

obtained at H2/CO 4 1.8 and fCO o 0.5. The contour lines

of the corresponding particle diameters of the model results

are given in Fig. 3F.

The strength of this analysis is that (un-)favourable regions

in the contour plots can be clearly distinguished, and directly

compared to other (performance) parameters. Nevertheless,

Fig. 3 is still only part of the full picture, since temperature,

pressure, and catalyst activity are fixed, whereas in a reactor a

broad range of conditions is covered.

Isolines at varying temperature, pressure and catalyst activity

Investigation of the same maps for other temperatures

(T = 470–530 K), pressures (p = 12–36 bar) and catalyst

activity (F = 1–10) shows that the general features of the

contour plots remain the same, but that the values become

different. Fig. 4 shows the changing position of some relevant

contour lines depending on several performance parameters

(column 1: aave = 0.9, column 2: Z = 1, column 3: RCO,total =

3 mmol kgcat�1 s�1, column 4: STYC5+

= 0.1 or 0.2 g gcat�1 h�1)

at varying temperature (top row), pressure (middle row) and

catalyst activity factor F (bottom row).

The most important observation in Fig. 4 is that the contour

plots remain qualitatively similar to those of Fig. 3. Several

isolines in the contour plots show even hardly any appreciable

sensitivity to the varied parameters (Fig. 4B, E–G, I and J).

The discussion is focused on the contour plots that do show

sensitivity, i.e. the plots in which the isolines shift.

In Fig. 4A the influence of temperature on aave = 0.9 is

shown for various fCO and H2/CO ratios at the catalyst

surface. The negative impact of increased temperature on

a can be compensated by decreasing the H2/CO ratio. For

kinetically controlled conditions, fCO o 0.5, the unmatched

diffusivity ratio of H2/CO in the catalyst particle does not play

an important role, and a gradual shift of the isolines with

temperature to smaller ratios is observed.

For diffusion controlled particles, fCO 4 0.6, the contour

lines are much closer to each other. In this situation the

sensitivity between temperature and syngas ratio to maintain

aave = 0.9 has changed tremendously, because of the unmatched

diffusion and consumption ratios of H2 and CO. Increasing the

H2/CO ratio slightly above 1 has a detrimental effect on the

average selectivity of the catalyst.

In Fig. 4C and K the effect of varying temperature and

catalyst activity factor F on the isolines of RCO,total = 3 mmol

kgcat�1 s�1 can be observed. The trends in the two plots are

quite comparable. The shift of the contour lines indicates that

an increase in temperature or catalyst activity can be compensated

by a reduction in the H2/CO ratio. The trend is approximately

inversely proportional to the catalyst activity, which means that a

doubling of the catalyst activity can be compensated by reducing

the H2/CO ratio with 50%.

This effect can be explained by simplifying the denominator

of the kinetic expression (eqn (4)), obtaining a reaction rate

that shows these proportionalities:

RCO ¼ rcatFa

b2

� � pH2

pCO

� �

Under this simplification the rate scales linearly with F and the

syngas ratio.

The combined influences of FT catalyst activity and selectivity

are captured in Fig. 4D, H and L, where the effects of varying

temperature, pressure and catalyst activity on the STYC5+

isolines are shown as a function of Thiele modulus and H2/CO

ratio. In Fig. 4D the trade-off between activity, selectivity and

catalyst effectiveness is clearly visible. The regions where

STYC5+4 0.2 for several temperatures are bound by the

respective isolines at three sides. The north-boundaries of the

regions exist as a consequence of selectivity loss with increasing

Fig. 3 Catalyst performance for the variable amodel as a function of

fCO and the syngas ratio at the catalyst surface. Conditions: T= 490 K,

p=30 bar, F=1�Yates and Satterfield; other conditions as in Table 2.

A: aave, B: Z, C: RCO,total, D: SC5+and E: STYC5+

and F: corresponding

catalyst particle diameter dcat.

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H2/CO ratio. The east-boundaries are marked by a decreasing

effectiveness with increasing fCO. And finally, south-boundaries

exist, because of the decreasing reaction rate with decreasing

H2/CO ratio. With increasing temperature, the optimum

STYC5+region moves south, and expands to the east. While

the onset of selectivity loss occurs at lower H2/CO ratios with

increasing temperature (the north-boundary), a decrease in

H2/CO ratio also results in less CO diffusion limitation,

allowing larger fCO (the east-boundary). The influence of

pressure (Fig. 4H) is relatively small under the investigated

conditions. When the catalyst activity factor F is increased

(Fig. 4L) the region where STYC5+4 0.2 expands to larger

fCO and lower H2/CO ratios.

The presented contour plots in Fig. 4 are useful, both for

giving direction to catalyst development studies as well as for

explorative purposes for optimum conditions under which the

FT catalyst may be operated. Both topics are addressed below.

Research directions for FT catalyst development

From Fig. 1 it is clear that a decreases with temperature. To better

exploit the activity increase of catalysts at higher temperatures,

improving the catalyst selectivity at higher temperatures is

definitely a required objective. Fig. 2 shows that the increasing

syngas H2/CO ratio profile in a CO diffusion limited system

can have a detrimental effect on the overall selectivity of the

catalyst, due to its coupled effects on the local a.Improving the chain growth parameter with the syngas ratio

is therefore useful in systems where CO is the limiting species.

The additional advantage of being able to operate at higher

syngas ratios is that the reaction rate also increases

(viz. Fig. 3C). Obviously, the benefits are smaller for systems

that do not suffer from CO diffusion limitations, i.e. at low

bulk syngas ratios. At bulk syngas ratios of approximately 1,

and depending on the temperature, H2 becomes the limiting

reactant. Obviously, under these conditions the advantage of

improved a at higher syngas ratio has vanished.

Improving the catalyst activity seems, as always, an attractive

objective to increase the space time yield. For reaction controlled

systems this simply means higher mass-based production rates.

In diffusion hindered systems the increased activity has to be

exploited differently. One approach to tackle this problem is with

geometrical solutions. Egg-shell catalysts33 or structured reactor

internals that serve as catalyst support structure,62 such as

monoliths,15,19,63,64 are used if the minimum particle diameter

is constrained by the pressure drop in packed bed reactors.

Egg-shell catalysts with thin active layers can be applied to

reduce the diffusion length in the used catalyst, but at the same

time increasing the inert catalyst core, thereby reducing the

effective production per reactor volume.

However, additional opportunities for improved performance

of diffusion controlled systems exist by optimizing the operating

conditions. Lowering the syngas ratio in the bulk and/or the

temperature leads to reduced CO diffusion limitations and

improved selectivity, at the expense of a reduced total reaction

rate. The combined effect can lead to improvements in C5+

productivity—for example, in Fig. 3E at fCO = 1, where a

reduction in the bulk syngas ratio from 2 to 1 leads to an almost

Fig. 4 Matrix of selected isolines with conditions as in Table 2, unless specified. Top row (A–D): varying temperature from 470 to 530 K. Middle

row (E–H): varying total pressure from 12–36 bar. Bottom row (I–L): varying catalyst activity multiplication factor from 1 to 10. The effects of

the varied input parameters on the isolines of several performance indicators are shown. Column 1 (A, I and L): average chain growth

parameter isolines of 0.9. Column 2 (B, F and J): catalyst effectiveness isolines of 1. Column 3 (C, G and K): total CO conversion rate isolines of

3 mmol kgcat�1 s�1. Column 4 (D, H and L): C5+ space time yield isolines of 0.1 (varying P) or 0.2 (varying T and F) g gcat

�1 h�1.

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doubled STYC5+. Therefore, improving the catalyst activity is

also an important objective, even for systems where diffusion

limitations are present.

The question arises whether it is more interesting to improve

the chain growth probability or the activity. We address this

question by comparing improvements of Yates and Satterfield

activity multiplication parameter F and the chain growth

selectivity constant ka. The base case is represented by

F = 1 and 1 � ka. Both parameters were independently

improved 2, 5, and 10 times by multiplication or division,

respectively. The latter implies a weaker dependency of the

chain growth probability on the temperature and syngas ratio,

viz. eqn (7). The effects of the improvements on the C5+ space

time yield of a small particle (dcat = 50 mm) and a large

particle (dcat = 1.5 mm, respectively) under typical conditions

(p = 30 bar, bulk syngas ratio = 2.0 and T = 470–530 K) are

shown in Fig. 5.

The C5+ space time yield of small particles benefits much

more from improvements in catalyst activity (parameter F,

dashed lines) than in selectivity at temperatures below

approximately 520–530 K, depending on the improvement

factor (Fig. 5A). Only at the high end of the investigated

temperature range, where a is sufficiently negatively impacted

by temperature, the improvements in ka (solid lines) are

approximately equally effective compared to improvements

in F. The data in Fig. 5A are also presented on a normalized

scale with respect to the base case (F = 1 and 1 � ka). These

normalized values are shown in Fig. 5C and D and represent

the STYC5+sensitivity with F and ka.

For large particles an increase in activity has a larger effect

on the C5+ space time yield than improvements in selectivity

at temperatures below 480–490 K (Fig. 5B), depending on the

improvement factor. However, at higher temperatures, the

effect of improved selectivity is much larger, as CO becomes

more diffusion limited in the particle. The increase of STYC5+

with ka at high temperature is even more than proportional

(Fig. 5D, solid lines)—e.g. a five-fold improvement of ka(blue solid line) at 530 K increases the C5+ productivity with

a factor 14. Note that the kink in some of the curves in Fig. 5B

and D (around 482 K) represents the onset of CO depletion in

the centre of the particle.

Disregarding the research effort and complexity of improving

either F or ka, it is clear that both developments are interesting

for the improvement of the FT catalyst in general, but their

impact depends on the catalyst design, reactor design and

operating conditions. A priori, without additional design and

operating considerations, one cannot be preferred over the other.

In the next section we explore the effect of operating conditions

on the performance of both small and large particles.

Exploration of favourable operating conditions

The catalyst particle size typically does not vary in reactors. This is

exploited by converting the x-axes of Fig. 3 and 4 to dimensional

units. One example is given in Fig. 6, where the contour-lines for

aave = 0.9, which may be regarded as a boundary condition for

industrial processes, are given as a function of particle radius and

bulk syngas ratio. If we assume that the pressure effects are

negligible for this map, which seems reasonable from Fig. 4E, it

can be used to determine inlet and outlet conditions for reactor

operation. For example, operating a fixed bed reactor with

spherical catalyst particles of 1.5 mm diameter (particle radius

rcat = 0.75 mm) and a substoichiometric syngas ratio of 1.8 at the

inlet at a desired selectivity of aave = 0.9 shows that the tempera-

ture should be approximately 480K at the reactor inlet. The syngas

ratio will gradually decrease with reactor length, due to the chosen

substoichiometric starting feed conditions. As the syngas ratio

drops, the corresponding optimal operational temperature to

maintain the desired aave = 0.9 can be found from the map

(red arrow), in this case 505 K at a single-pass CO conversion of

73%. This information can be used to determine boundary

conditions for the temperature profile in the reactor.

Fig. 5 Comparative analysis of improvement of the Yates and Satterfield

activity parameter F (dashed lines) and the chain growth selectivity

constant ka (solid lines). Conditions: p = 30 bar and H2/CO = 2.

(A) STYC5+of a small particle (dcat = 50 mm, fCO o 0.35). (B) STYC5+

of a large particle (dcat = 1.5 mm, fCO= 0.35–9.9). (C) STYC5+sensitivity

of a small particle (dcat = 50 mm) and (D) STYC5+sensitivity of a large

particle (dcat = 1.5 mm) with F and ka relative to the reference case.

Fig. 6 Contour-lines of aave = 0.9 vs. catalyst radius (rcat) and bulk

syngas ratio at p = 30 bar and F = 1, other conditions as in Table 2.

The desired temperature conditions can be estimated as a consequence

of desired starting conditions and conversion levels.

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Another way of looking at the data is to plot the effect of

varying temperature and bulk syngas ratio at constant particle

diameter and pressure. In Fig. 7 the C5+ space time yield is given

as a function of temperature and bulk syngas ratio for four

particle diameters. Clearly, for all particle diameters, the opti-

mum operating point for maximized STYC5+is at a bulk syngas

ratio just below 1, where both H2 and CO are approximately

equally limiting, and the maximum presented temperature of

530 K. The figures also display the isocontour for aave = 0.9.

The most important observation from the contour plots in

Fig. 7 is that the bulk syngas ratio and the temperature are

strongly coupled. No single optimum value can be determined

for either process parameter without specifying the other.

The coupling between temperature and bulk syngas ratio

takes place through the processes of diffusion, reaction and

selectivity. Furthermore, this coupling is moderately dependent

on the particle diameter. Finally, we observe that the operating

conditions for maximum productivity are located in the regime

where aave o 0.9.

Similarly, these types of maps can be used to, for example,

estimate the productivity of various parts of a slurry bubble

column, where the temperature remains fairly constant and the

gas phase reactants can be considered to travel in plug flow.

Fig. 8 is an overview of the results obtained at four different

temperatures for a typical slurry catalyst particle diameter of

50 mm, effectively eliminating the influence of any internal

diffusion limitations, at varying pressure and bulk syngas

ratio. Again, the figures also display the isocontour for

aave = 0.9, which is in this case not influenced by diffusion

effects or pressure and is therefore horizontal.

Also the bulk syngas ratio and the pressure show some

coupled behaviour, although for this example less complex

than for larger particles (Fig. 7) due to the absence of diffusion

effects. At temperatures of 490 K (Fig. 8A) and 500 K

(Fig. 8B) the contour plots are rather flat (few contour

lines) and show relatively little influence of the pressure. At

temperatures of 510 K (Fig. 8C) and 520 K (Fig. 8D) the

effect on a becomes apparent again and the C5+ productivity

is optimal at high pressures (p 4 30 bar) and a bulk

syngas ratio of approximately 1. Similarly as for large

particles, the optimum STYC5+is found in the region where

aave o 0.9,

Interestingly, both cases (large particle, Fig. 7, and small

particle, Fig. 8) show significant potential to improve a single

particle STYC5+at high temperatures (4520 K), high pres-

sures (p 4 30 bar) and a bulk syngas ratio of approximately 1,

in contrast to what is typically considered (T E 500 K, and a

bulk syngas ratio of approximately 2, much closer to the

stoichiometric consumption ratio). Although the temperature

has a negative impact on a, the lowered bulk syngas ratio

and increased reaction rate with temperature more than

compensate this, leading to a significant increase in STYC5+

values by a factor of 3 (small particles) to 10 (large particles).

This modelling analysis corroborates the reasoning in the

introduction that substoichiometric H2/CO feed ratios may

be favourable in FTS.

This insight presents opportunities for reactor configura-

tions that, for example, make use of staged feeding of H2 along

the reactor coordinate to control the bulk syngas ratio around

a value of 1 as the CO conversion increases. This can also be

Fig. 7 C5+ space time yield (STYC5+in g gcat

�1 h�1) contour plots as a function of temperature and bulk syngas ratio at constant pressure

(p = 30 bar) and particle diameter. (A) dcat = 0.5 mm, (B) dcat = 1.0 mm, (C) dcat = 1.5 mm, (D) dcat = 2.0 mm. Calculations performed with a

catalyst activity multiplication factor F = 1. The dotted line indicates the isocontour for aave = 0.9.

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achieved by a catalytic water gas shift functionality in the

reactor, whereby the relatively increasing CO is converted with

the produced water to CO2 and H2, keeping up the proper

desired H2/CO ratio. The indication to operate at low syngas

ratios is especially interesting when the syngas is produced

from coal or biomass, where typically low H2/CO ratios

are found.65 Using a strict boundary condition for the chain

growth parameter, for example aave = 0.9 (dotted line, Fig. 7

and 8) for carbon efficiency reasons, the conclusion is that the

maximum STYC5+for a single catalyst particle is achieved at

even lower bulk syngas ratios (H2/CO = 0.5–0.8) and high

temperature (T 4 520 K).

Operating at low H2/CO ratios may lead to additional

effects that are important for industrial application, such as

the changed olefin/paraffin ratio in the product distribution,4

or the catalyst deactivation rate.66 These elements are not

addressed in this modelling approach, and may be considered

for further analysis and deeper insight in economical viability.

Also, we note that the catalyst is an integral part of the reactor, in

which gradients (T, H2/CO, P) are to be expected and must be

taken into account. Reactor and overall process design—as other

units, such as syngas manufacturing and product upgrading,

roughly size with the amount of gas and liquid that needs to be

processed—are ultimately a decision based on capital invest-

ment, operating cost and total productivity. These results

aid in the exploration and selection of favourable operating

conditions, whether or not under additional constraints, for

maximum productivity.

As a final element, we readdress the earlier question on

whether to focus catalyst development on improving the activity

Fig. 8 C5+ space time yield (STYC5+in g gcat

�1 h�1) contour plots as a function of total pressure and bulk syngas ratio at constant catalyst

particle diameter (dcat = 50 mm) and various temperatures. (A) T = 490 K, (B) T = 500 K, (C) T = 510 K, (D) T = 520 K. Calculations

performed with a catalyst activity parameter F = 1. The dotted line indicates the isocontour for aave = 0.9.

Table 4 Performance analysis of a small catalyst particle (dcat = 50 mm) and a large catalyst particle (dcat = 1.5 mm) at a base case (F = 1 and1 � ka), improved catalyst activity (F = 10) and improved selectivity (0.1 � ka). Other conditions: T = 530 K, p = 36 bar and H2/CO = 1

dcat = 50 mm dcat = 1.5 mm

F (—) 1 10 1 10 1 10 1 10ka (—) 1 1 0.1 0.1 1 1 0.1 0.1fCO (—) 4.0 � 10�3 4.0 � 10�2 4.0 � 10�3 4.0 � 10�2 3.6 35.8 3.6 35.8Z (—) 1.00 1.00 1.00 1.00 0.53 0.19 0.56 0.20aave (—) 0.70 0.70 0.96 0.96 0.72 0.73 0.93 0.93RCO,total (mmol kgcat

�1 s�1) 29.1 289.6 29.1 290.4 15.9 57.9 16.8 61.1SC5+

(—) 0.52 0.52 0.98 0.98 0.58 0.60 0.97 0.96STYC5+

(g gcat�1 h�1) 0.76 7.60 1.44 14.39 0.47 1.75 0.82 2.95

Relative STYC5+(—) 1.00 10.00 1.89 18.93 1.00 3.72 1.74 6.28

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or the selectivity. The chosen operating conditions follow from our

previous analysis at maximum STYC5+(T = 530 K, p = 36 bar

and H2/CO = 1). In Table 4 the results are presented for a base

case calculation (F = 1 and 1 � ka) and a ten-fold improvement

of the parameters for activity and selectivity, both for a small

(dcat = 50 mm) and a large particle (dcat = 1.5 mm). Clearly,

under these conditions, the STYC5+improves more with

catalyst activity than with selectivity. Improving both para-

meters at the same time increases the STYC5+, but without

synergistic effects, as judged from the relative STYC5+for the

individual and combined parameter improvements.

Validity of the results

The kinetic rate expression (eqn (4)) and parameters in the

original paper by Yates and Satterfield24 were extensively

validated on several data-sets that covered a temperature

range of 454–523 K and a syngas ratio range of 0.2–8.3,

although not varied independently over the entire range.

Despite the broad range of conditions, diffusion limitations

have been shown in this paper to cause situations where the

H2/CO ratio approaches 0 or N, for which the validity of the

equation was not proven. However, under the conditions

outside the studied H2/CO range the reaction rate also rapidly

decreases as one of the reactants becomes depleted, attenuating

uncertainty issues.

The data range used for the selectivity relation (eqn (7)) is

narrower than that of the Yates and Satterfield rate expression

(H2/CO = 1–3 and T = 450–500 K). Therefore, the inter-

pretation of the exact numerical values of the analysis at some

of the limits of the investigated domain (T= 530 K) should be

taken with caution.

As a final consideration, we note that olefin formation is

expected especially at low H2/CO ratios. Diffusion limitations

of these molecules will result in further hydrogenation or

reinsertion in the chain growth, just leading to more paraffinic

products, but not essentially changing the chain length

product distribution.46 Therefore, we remain confident that

the displayed trends are a good indication of catalyst perfor-

mance and present an incentive for future experimental and

numerical studies.

Conclusions

The calculated H2 and CO concentration profiles inside a

cobalt based Fischer–Tropsch catalyst particle under typical

operating conditions (temperature = 490 K, pressure= 30 bar,

bulk syngas ratio = 2, catalyst sphere diameter = 1.5 mm)

demonstrate the severity of CO diffusion limitation that

can occur. Incorporating a variable chain growth probability

a shows the deteriorating effect of strong gradients in

the syngas ratio over the catalyst particle on the local

chain growth probability. These gradients are due to

intrinsically unbalanced diffusivities and consumption

ratios of H2 and CO, and cause significant reduction (a factor

3 in the presented example) of the desired C5+ space

time yield.

Analysis of the modelling results for a wide range of

conditions (CO Thiele modulus fCO from 0.01–5, bulk syngas

ratio from 0.1–3.0) at constant pressure (p = 30 bar) and

temperature (T = 490 K) emphasizes the highly non-linear

dynamics of the interplay between reaction, diffusion and

selectivity. A common characteristic of all results is the critical

conditions beyond which catalyst performance is impacted

negatively: fCO 4 0.6 and a bulk syngas ratio4 1. Analysis of

an expanded parameter space (T= 470–530 K, p= 12–36 bar,

and a catalyst activity multiplication factor F = 1–10) reveals

the strong change in selectivity dependence between tempera-

ture and syngas ratio for reaction controlled particles and

diffusion controlled particles.

The maximum space time yield of the desired C5+ products

was found at high temperatures (T = 530 K), high pressures

(p=36 bar) and relatively low bulk syngas ratios (H2/CO= 1).

Under these conditions the STYC5+can be improved by a

factor 3 (small particles, dcat = 50 mm) to 10 (large particles,

dcat = 2.0 mm) compared to typical conditions (T = 500 K,

p = 30 bar, and H2/CO = 2).

Under the proposed operating conditions for maximizing

STYC5+it is more effective—a factor 5 for a small catalyst

particle (dcat = 50 mm) and a factor 2 for a large catalyst

particle (dcat = 1.5 mm)—to focus catalyst research on

improving the activity rather than the selectivity.

Appendix A—Peclet mass number in the catalyst

particle

Differential eqn (1) does not include the effect of product

flow leaving the completely liquid filled catalyst particle.

This is justified by the result of estimating the Peclet number

for mass:

Pem ¼vllcat

Deff¼ 0:06� 1

where product velocity vl is estimated (vl = 8 � 10�7 m s�1) by

assuming a uniform (high) benchmark hydrocarbon productivity

of 1 g gcat�1 h�1 34 (ch. 6, p. 432) of a liquid product with a density

of 700 kg m�3 in a catalyst slab with thickness lcat = 1 mm, a

density of 1000 kg m�3, a porosity of 0.5 and a pore tortuosity

of 1.5, where the effective diffusion coefficient at a temperature

of 500 K is 1.4 � 10�8 m2 s�1.

Appendix B—absence of internal temperature

gradients

The temperature inside the catalyst particle can be assumed

constant if the following criterion, depending on dimensionless

activation energy g, the internal Prater number bi and the

Wheeler–Weisz modulus Zf2, is satisfied:

gbiðZf2Þ ¼ EA

RT

� �ð�DHrÞDCO;effcCO;0

lcat;effT

� �

� RCO;totalrcatl2cat

DCO;effcCO;0

!o0:05

The following values for the parameters are assumed or

estimated: apparent activation energy EA = 100 kJ mol�1,34

T = 500 K, a reaction enthalpy DHr = �165 kJ mol�1, a

uniform large reaction rate of 0.02 mol CO/(kgcat s) is

assumed, which is approximately 1 g hydrocarbon gcat�1 h�1 34

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(ch. 6, p. 432), catalyst particle density rcat = 1000 kg m�3, an

effective diffusion coefficient of CO DCO,eff = 5.2 � 10�9 m2 s�1

(at 500 K, ecat = 0.5 and tcat = 1.5), the bulk

concentration of CO cCO,0 = 300 mol m�3, the effective heat

conductivity of the particle, corrected for a catalyst

porosity ecat = 0.5, lcat,eff = 1 W m�1 K�1, and spherical

catalyst particle with a diameter of 2 mm (lcat = 0.33 mm).

This yields:

gbi(Zf2) = (24.1) � (0.51 � 10�3) � (1.4) = 0.017 o 0.05

Appendix C—Mears’ criterion for external

interphase (liquid–solid) heat transport limitations

According to Mears27 external interphase heat transport

limitations can be safely neglected if

EAð�DHrÞRCO;totalrcatlcathRT2

o0:05

which was found satisfied (0.026) under the assumptions below.

A spherical catalyst particle is assumed with a diameter of 2 mm

(lcat = 0.33 mm, rcat = 1000 kg m�3), large uniform

hydrocarbon production of 1 g gcat�1 h�1, RCO,total =

0.02 mol kgcat�1 s�1, with an estimated apparent activation

energy EA = 100 kJ mol�1,34 a reaction enthalpy DHr =

�165 kJ mol�1 at T = 500 K. A typical value for the film heat

transfer coefficient (h = 2 kW m�2 K�1) was found from the

Nusselt number (Nu = h � lcat/ll, with liquid conductivity

coefficient ll = 0.14 W m�1 K�1), where Nu was estimated with

a packed bed correlation based on bed porosity (ebed = 0.35),

Reynolds (Re = rlvldcat/ml, with liquid density rl = 700 kg m�3,

liquid velocity vl = 0.02 m s�1, and liquid viscosity ml = 2.5 mPa s)

and Prandtl (Pr = Cp,lml/ll, with liquid heat capacity Cp,l =

2.2 kJ kg�1 K�1) number through Nu = 1.31Re1/3Pr1/3/ebed.

Appendix D—selectivity model

The selectivity can be described by the ratio of propagation (p)

and termination (t) reactions at the active sites on the catalyst,

following a standard Arrhenius dependency with temperature,

and the ratio of termination and propagation reactions scales

with some power of the local syngas ratio.

a ¼ kp

kp þ ðcH2cCOÞbkt

And:

ki ¼ ki0 expEi

R

1

493:15� 1

T

� �� i ¼ p or t

Substitution and rearranging results in:

a ¼ 1

1þ kaðcH2cCOÞbexpðDEa

Rð 1493:15� 1

TÞÞ

where ka = kt0/kp0 and DEa = Et � Ep.

Appendix E—list of symbols

Table 5 List of symbols

Romana Yates and Satterfield reaction rate

constantmol s�1 kgcat

�1

bar�2

Bim Biot mass number —b Yates and Satterfield adsorption constant bar�1

Cp Specific heat J kg�1 K�1

ci Concentration of species i mol m�3

D0,i Diffusion constant in product medium forspecies i

m2 s�1

dcat Catalyst particle diameter mED,i Diffusion activation energy for species i J mol�1

F Catalyst activity multiplication factor —Hi Henry coefficient for species i barh Film heat transfer coefficient W m�2 K�1

ka Selectivity model fitting parameter —kLS External liquid–solid mass transfer

coefficientm s�1

lcat Characteristic catalyst length mNu Nusselt number —p Pressure barPem Peclet mass number —Pr Prandtl number —R Gas constant J mol�1 K�1

Ri Reaction rate of species i per unit masscatalyst

mol kgcat�1 s�1

Ri Reaction rate of species i per unit volumecatalyst

mol mcat�3 s�1

Re Reynolds numberrcat Radius of a catalyst sphere mSC4� C4� selectivity by weight g g�1

SC5+ C5+ selectivity by weight g g�1

Scat External catalyst surface area m2

STYC5+ Space time yield of C5+ g gcat�1 h�1

s Geometric parameterT Temperature KVcat Catalyst volume m3

v Velocity m s�1

x Location in the catalyst myi Dimensionless concentration of species i —z Dimensionless length of the catalyst —

Greeka Chain growth probability —b Selectivity exponential fitting parameter —DHb Heat of absorption J mol�1

DHr Heat of reaction J mol�1

DEa Selectivity activation energy difference J mol�1

e Porosity —f Thiele modulus —lcat Catalyst thermal conductivity W m�1 K�1

m Viscosity mPa sr Density kg m�3

ni Stoichiometric constant for species i —tcat Catalyst pore tortuosity —Ci Dimensionless reaction rate —

Subscripts0 At the catalyst surfaceave Averaged over the catalyst particlebed Packed bedC5+ Hydrocarbon chains longer than

4 carbon atomscat CatalystCO Carbon monoxideeff EffectiveH2 Hydrogeni Species il Liquidtotal Summation over the entire catalyst

particle

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Acknowledgements

This research is supported by the Dutch Technology Founda-

tion STW, which is the applied science division of NWO, and

the Technology Program of the Ministry of Economic Affairs,

Agriculture and Innovation.

References

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