SiO2 ... - ACS Publications

Subscriber access provided by the University of Exeter

Article

Performance Characteristics of Eggshell Co/ SiO2 Fischer-Tropsch Catalysts: A Modeling Study Syed Ali Gardezi, and Babu Joseph Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.5b01288 • Publication Date (Web): 30 Jul 2015 Downloaded from http://pubs.acs.org on August 11, 2015

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

2

Performance Characteristics of Eggshell Co/SiO2 Fischer-Tropsch Catalysts: A Modeling Study

3

Syed Ali Gardezi a,b*, Babu Josephb

1

4

a

5 6 7 8 9

Department of Chemical and Bio-molecular Engineering, University of Houston, Houston, TX, 77004 USA

b

Department of Chemical and Biomedical Engineering, University of South Florida, Tampa, FL, 33620 USA

* Email: [email protected]

10 11

Abstract

12

In this paper, we investigate the performance characteristics of cobalt catalyst (Co/SiO2) pellets with

13

eggshell morphology used in Fischer-Tropsch synthesis (FTS). An analytical strategy is developed first, to

14

estimate the optimum thickness of active shell. When the system is strongly mass transfer limited (Thiele

15

Modulus≫1) it can be represented by slab geometry in order to determine the shell thickness. Otherwise,

16

the analysis is done on a spherical geometry. In the latter case, the continuity equation simplifies into

17

Emden-Fowler equation which was solved via numerical iterative technique based on prior investigations.

18

A more accurate detailed numerical solution was also developed for this intra-pellet model. This numerical

19

result is compared with the analytical solution as well as with a previously published model. The sensitivity

20

of the catalyst performance to temperature and pressure variations was also examined. Performance

21

analysis was carried out considering both rate of reaction and α chain growth probability as yardsticks.

22

These performance parameters were examined using contour maps with Thiele modulus and H2/CO ratio

23

as their coordinates. On this map there exists an ideal combination of gas ratio and Thiele modulus that

24

fulfills the design requirements of product selectivity and high catalyst activity. This study confirms the

25

need for, and the advantage of, using egg-shell morphology for Fischer-Tropsch synthesis applications.

26 27 28 29 30 31 32 33

ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 45

1

1.

Introduction

2 3 4 5

Fischer-Tropsch synthesis (FTS) is a heterogeneous catalytic process that emerged out of technological advances during and after the First World War. This process is capable of transforming syngas (CO+H2) into petroleum based hydrocarbons. Generally speaking, Fischer-Tropsch synthesis includes the following two reactions

6 7

(2n + 1) H2 + n CO

Cn H2n+2 + n H2 O

(1.1)

8

2n H2 + n CO

Cn H2n + n H2 O

(1.2)

9 10 11 12 13 14 15

Syngas can be produced from a variety of raw materials including natural gas, coal-bed gas, landfill gas, coal and biomass 3. FTS converts syngas to liquid fuels on a solid catalytic surface. Surface adsorption models and subsequent analytical studies by Norskov et al. 4 indicated that there exists a volcano-curve type correlation between dissociative chemisorption energy of CO and the turnover frequency (TOF). The maxima of this curve correspond to ruthenium while lower cost metals iron and cobalt also show significantly high rates thus making them suitable industrial catalysts.

16 17 18 19 20 21 22 23

Heterogeneous catalysis presents practical flexibility due to strong interplay between chemical kinetics and mass transfer which can be manipulated by changing the catalyst size, shape and texture 5. The degrees of freedom involved in optimizing a heterogeneous catalytic process include geometry of the catalyst particles, topology of the active phase and the reactor configuration 5. Intra-pellet diffusion limitations (either due to size or process parameter) can render the interior of a pellet inactive. This utilization is characterized by the effectiveness factor (η), a function of Thiele Modulus (φ), which in turn is a function of shape of catalyst and the reaction kinetics. The morphology of the catalyst can be designed depending on mass and energy balances if reaction kinetics and diffusion rates are known.

24 25 26 27 28 29 30 31 32 33 34 35 36

For catalyst design, surface to volume ratio of the pellet is a main design parameter 5. This ratio relates the flux density of reactants (to catalyst surface) to the reactions inside the catalyst volume. Inverse of this ratio (called the characteristic length) represents the diffusional distance that needs to be penetrated by the reactants 5. A Diffusional limitation of reactants also affects the selectivity of hydrocarbon products especially if the reaction proceeds via a step-wise polymerization type mechanism. In FTS, re-adsorption is not always desired, especially if there are intermediate products such as jet fuel and gasoline that are preferred products. Another way to control selectivity is by using zeolite cages to control the product diffusion through the pellet. In case of zeolites the size of ring opening and super-cage limit the size of diffusing reactant molecules and the length of polymer chain that will form in the super-cage. Another factor of interest is the management of thermal energy within the catalyst bed. Radial heat transport, an essential component in fixed bed FTS reactors, is dependent on external diameter of catalyst 5 and bed void fraction. FTS is a gas-liquid reaction in which the bed voids are filled with condensed product after startup and drastically alters the heat transfer characteristics 6.


Page 3 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

These unique characteristics of FTS catalysts have led to many unique concepts for the design of the catalyst morphology. Generally, in the absence of diffusional limitations, it is customary to uniformly load the catalyst with metal as it increases catalyst activity per unit volume of catalyst bed 2, 5, 7. However, if η is less than unity, or the reaction causes pore mouth/ uniform poisoning, or in case of pore mouth blockage (e.g. wax deposition), the design requirement departs from uniform morphology giving rise to core-shell, egg-shell, egg-yolk and egg-white type designs5, 8. In line with these recommendations, Becker and Wei 8c developed a criterion for catalyst design based on the Thiele modulus for the main and the undesirable side reaction. According to them the required profile will either be egg-yolk, for low values of Thiele modulus or egg-white when the intra-pellet diffusion limitations increase significantly. For bimolecular reactions, such as FTS, where one reactant is both stoichiometric and diffusion limited, eggshell morphology works well 9. This design reduces the formation of soft and hard wax to an appreciable extent and decreases the risk of catalyst pore blockage. Similarly, if a reaction follows Langmuir kinetics where one reactant is rapidly adsorbed (e.g. CO oxidation) diffusion resistance can enhance its rate of reaction. For such situations the use of egg-yolk catalyst is more advantageous 5. Recently, core-shell and encapsulated bi-functional catalyst has also been employed e.g. Tsubaki et al. 10 synthesized the core-shell catalyst by coating zeolite membrane over Co/SiO2 pellet. The core produces long chain hydrocarbons while the zeolite in shell cracks these hydrocarbons into gasoline fractions. Likewise, hybrid catalysts have also been synthesized by physically mixing zeolite and Co/SiO2 during the synthesis step.

20 21 22 23 24 25

The objectives of this modeling study is to elucidate the interplay between catalyst design and operational performance taking into account mass and heat transfer effects within the catalyst particle and in the catalyst bed. The significance of catalyst morphology (metal loading profile) on product selectivity and overall effectiveness is investigated using a macroscopic model that couples heat and mass transfer effects. The purpose is to establish that catalyst design as an effective means of managing diffusional limitations and enhancing the desired product selectivity.

26 27 28 29 30 31 32

In this work, Fischer-Tropsch synthesis process is modeled over a spherical catalyst pellet. Table 1 lists different power law and Langmuir Hinshelwood type kinetics developed for Fischer-Tropsch synthesis. Kinetic model developed by Yates and Satterfield was used in this study 11. The size of catalyst pellet was based on previous recommendations i.e. it has to be large enough to avoid excessive pressure drop while allowing significant mass transfer to the interior of the pellet 12. Previous researchers have shown that an egg-shell catalyst of around 2mm diameter provides this design flexibility by decoupling the characteristic diffusion distance (within the catalyst pellet) from pressure drop and other reactor constraints 13.

33 34 35 36

Very limited modeling work has been done on how the FTS catalyst morphology affects the process performance characteristics 14. Two most important steady state variables i.e. temperature and pressure are explored in this paper, additionally the effect of pore filling is also investigated but its discussion is limited to specific conditions provided in section 5.1.

37 38 39

As the process parameters become more stringent, the thickness of accessible active region reduces due to an increase in Thiele modulus, i.e. active material needs to be deposited close to the surface. Prior work by the authors has shown that a slab model is adequate for characterizing thin egg-shell coatings



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3 4 5 6 7 8

whereas the sphere model is better at for FTS catalysts for a thicker shell 1. Under a typical fixed bed reactor operation, using cobalt catalyst, the preference towards eggshell design becomes debatable since the operational envelope is very narrow e.g. in a system where Tgas-in= Tcoolant, reaction runaway happens at much lower temperature i.e. at approximately 493 K (ΔT ≈ 20K) 6, 14. Likewise, there are conflicting arguments on whether an eggshell catalyst design improves both the selectivity and activity, however, it is generally accepted that selectivity does improves with egg-shell morphology (as discussed in Section 2) 1, 9 . Here, we develop a strategy to identify the ideal eggshell thickness for a given set of process parameters based on Langmuir-Hinshelwood kinetics.

9 10 11 12 13 14

To achieve the above mentioned objectives, an analytical isothermal reaction-diffusion model, in conjunction with an eggshell modulus model, is developed. Further, a detailed numerical model of a pellet is developed to carry out sensitivity analysis (for this diffusion limited regime) with respect to temperature and pressure. The activity and selectivity data are collected for the selected set of process parameters. The optimum eggshell thickness is estimated based on the desired selectivity while maintaining high activity given the process conditions.

15 16

2.

Prior Research Work:

17 18 19 20 21 22

Some of the prior modeling effort has been on eggshell catalyst synthesis. We start by reviewing modeling work on eggshell FTS catalysts. Work by Wang et al. 7 takes into account particle size effect; they showed that for an industrial catalyst (i.e. 2-4 mm size) with uniform loading, its effectiveness factor was in the range of 0.14-0.28, depicting diffusion limitations. They stated that per unit volume activity of eggshell catalyst is smaller than uniformly loaded pellet. Contrary to the activity, the effect of eggshell morphology on the selectivity of desired product was favorable.

23 24 25 26 27 28 29 30 31 32

A more detailed analysis has been presented by Vervloet et al. 2. They showed that at a constant temperature and H2/CO ratio, Thiele modulus “φ” affects both the rate of reaction and product selectivity. In their model, selectivity towards heavier hydrocarbons drops when φ >1 as well as H2/CO ratio is greater than 1 (due to secondary hydrogenation). Similar effects were observed for space time yield of C5+ for φ >1. They stated that in diffusion hindered systems, one method of increasing catalyst activity is by using eggshell morphology, however, a very thin active layer results in reduced production per unit volume of reactor 2. Generally, the product selectivity equations are based on Anderson-Schluz-Flory (ASF) distribution which does not take into account olefin re-adsorption. For an eggshell catalyst this assumption may hold true, however, as indicated by Wang et al. 7 when a uniformly loaded catalyst is used, overall chain growth follows ASF distribution adjusted for olefin re-adsorption.

33 34 35 36 37 38

There have been a number of studies on the effects of non-uniform intra-pellet activity for other types of reactions. Luss et al. 15 investigated non-uniform profile by using a local rate constant term which was a function of intra-pellet radial position. They correlated this local constant with volume average rate constant of a spherical pellet via rate constant density function. By using this correlation, they have mathematically proven that effectiveness factor of a catalyst with higher surface activity exceeds that of the catalyst with lower surface activity. Thus, it is favorable to concentrate the active material close to


Page 4 of 45

Page 5 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

the pellet surface (i.e. eggshell profile). For a first order reaction network, asymptotic selectivity is independent of Thiele modulus as the reaction is already confined to the outer surface. Thus, the rate constant density function affects catalyst product selectivity only for intermediate values of Thiele modulus. The effect of radial non-uniform distribution was also studied by Johnson et al. 16 to evaluate the performance of spherical pellet under typical ethylene oxidation conditions. Similar to Luss et al. 15 they defined a distribution function which correlated reaction rate constant with radially varying catalytic activity. They also developed an expression for local selectivity of ethylene oxide in terms of temperature and concentration which was similar to the mathematical expression used for calculating α chain growth probability in case of an FTS reaction 2. They also found that partially impregnated catalysts exhibit higher effectiveness factor than uniformly activated ones for a wide range of Thiele modulus except for φ ≃ 0, where the opposite holds true. For an exothermic reaction, they argued that larger temperature gradients in partially impregnated catalyst results in enhancement of reaction rates. A peculiar feature of their process was multiplicity in activity and selectivity. Minhas et al. 17 also studied the merits of partial impregnation for SO2 oxidation over platinum catalyst. In their mathematical treatment, they updated Thiele modulus by multiplying it with fractional occupancy of active material. However, contrary to their claim, for a spherical catalyst this occupancy factor has to be cubic in nature, as shown by some later research 1. The effect of loading was introduced by the updating pre-exponential factor for various loadings along the characteristic length. In their model they observed an increase in the effectiveness factor due to partial impregnation; likewise they observed definitive increase in the space time yield of partially impregnated catalyst. Yazdi and Peterson 18 studied the change in catalyst performance by varying the distribution of active catalyst within porous supports. Their activity distribution was similar to that of Johnson et al. 16 i.e. radially varying and the steepness of variation were determined by a factor α which in itself is dependent on active material distribution. They analytically studied a series of first order reactions. They presented the analytical solution of continuity equation using Bessel functions. According to their analysis the selectivity of an intermediate product, in a series reaction increases if the activity is non-uniformly distributed in such a way that it decreases (due to decrease in catalyst concentration) towards the center of pellet. This result is encouraging in a sense that product formation in FTS is a polymerization process and intermediate chain termination is often desired due to market demand 1, 9, 19 . Tsamalis and Szepe also investigated the diffusional effects of a partially impregnated cylindrical pore 20. They showed that for high Thiele modulus the “observed reaction rate” was completely independent of the intrinsic surface rate and it was equal to the rate of diffusion in the un-impregnated portion of the pore. The morphology of the pore resembled that of a typical egg-white catalyst in which the active zone was surrounded from both sides by un-impregnated zone. An important result of this study was the correlation between effective diffusion coefficient and the observed reaction rate. They also showed that the observed rate constant was independent of true rate constant.

36 37 38 39 40 41

These studies confirm that eggshell catalyst has the highest effectiveness factor for positive order reactions. However, maximum dispersion and resistance to sintering normally occurs at the minimum surface concentration for a given loading of active material. As a result, a uniform catalyst may have better activity and stability than an egg-shell catalyst, if mass transfer limitation is not an important factor 21. Egg-white and egg-yolk catalyst are beneficial in applications where severe attrition is a problem since only the inert support is worn off and the active material can still be preserved.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 45

1 2 3 4 5 6 7 8 9 10

Previous researchers have also modeled the synthesis process of an eggshell catalyst. Their focus was on the joint effect of the adsorption rate and the diffusivity of precursor during the catalyst synthesis 21-22. For this purpose again the concept of Thiele modulus was used to interpret the resulting catalyst distribution. An impregnation system of large Thiele modulus corresponds to fast reaction between the precursor and the support or slow diffusion of an impregnant in the support thus producing the eggshell profile. On the other hand, a smaller Thiele modulus leads to thicker or more diffused eggshell. Another factor of choice is the surface area of the catalyst, smaller surface area gives smaller Thiele modulus and thus favors uniform distribution of the catalyst and vice versa 23. The time of impregnation is also an important factor, despite favorable Thiele modulus; an extended contact with the bulk impregnant allows the solute to diffuse in the support and expands the eggshell.

11 12 13 14 15 16

For FTS reaction, the work done by Iglesia et al. 9 is of fundamental nature. They developed an eggshell profile by using both cobalt nitrate melt and an aqueous solution of high viscosity. The penetration of precursor with the support pores was modeled by Washburn analysis and then compared with experimental measurements. An eggshell design strategy based on a reaction transport model to predict hydrocarbon selectivity as a function of shell thickness and other reaction parameter was developed. A Dimensionless parameter chi (χ) was defined to characterize and optimize shell thickness.

17 18 19 20

A simple eggshell modulus model for measuring the required eggshell thickness to avoid reactant diffusion limitation was developed by the authors in prior work 1. This model was based on geometric transition of shell to slab geometry with the reduction in shell thickness (as discussed in section 1 and 3). The result of this model was in close agreement with Iglesia et al. 9.

21 22 23

This overview of the related research work shows that there is a need for a comprehensive model that takes into account the experimental work and is also predictive so that it can contribute in overall process improvement.

24 25

3.

26 27 28 29 30 31 32 33 34

First, we analyze a simplified isothermal reaction-diffusion model for the FTS reactions in a 2mm diameter spherical pellet. A diffusion limited regime is assumed if the Thiele modulus (i.e. φ) is greater than unity. Our prior work has shown that a Thiele modulus value of approximately 4 can often be encountered for a 2mm pellet under typical conditions 1. As discussed earlier, the active shell deposited on a spherical pellet can be characterized by slab geometry if it is significantly small 1, 24. In the present work, the required thickness of active zone is calculated by assuming the shell as a sphere or an equivalent slab.

35 36 37

Theoretical Background:

The general reaction diffusion continuity equation is given as. 1 d q dui (ξ ) = ϕi 2 . φi ξq dξ dξ

( i = CO, H2 )

(3.1)

In case of sphere, q= 2. The solution of this equation, for first order kinetics and i = CO, is available in literature 25. Likewise, for power-law FTS kinetics, the solution has been developed in the supporting


Page 7 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 2 3 4 5 6 7 8 9 10 11

information of this manuscript. The power law kinetics is selected for comparison with our previous published work. In case of slab approximation, an analytical solution was found. The thickness of active region, for a diffusion limited system, characterized by Thiele modulus (ϕCO ) value of 4, was approximately 0.27 mm (see supporting information). Without slab approximation, the reaction diffusion equation was first reduced to a special form of Emden-Fowler equation and then numerically solved. The resultant profile (in case of ϕCO = 4) suggested a required thickness of 0.1 mm (see supporting information), interestingly, this result is in close agreement with our prior work in relation to eggshell modulus development 1. For comparison, previous published results are also shown in Figure 1. Following our prior argument 1 and modeling work done by other research groups 9, a comparison of resultant Thiele modulus and the resultant thickness, as predicted by eggshell modulus approach, is shown in Table 2. Sample E-DH corresponds to this analytical work.

12 13 14 15 16

These results indicate that it is possible to predict an eggshell thickness for a given set of structural and process conditions for a spherical catalyst by augmenting the results of sphere and slab approximations. However, one limitation is that this strategy does not take into account product selectivity. In-fact it is only defining a bound based on diffusion limitations. In the next section, a detailed reaction-diffusion model that takes into account activity and selectivity variations is developed.

17 18

4

Intra-pellet Reaction- Diffusion with multiple reactions

19

4.1

Mathematical Development of Model

20 21 22 23 24 25 26 27 28

The basic assumption employed while developing a mathematical model for Fischer Tropsch reaction in a spherical pellet is that of steady state. Even for a partially filled pellet, discussed in section 5.1, the assumption of pseudo steady state is valid6. Following prior mathematical treatment, a single cylindrical pore is considered to be the representative of unimodal porosity 26. This cylindrical pore approximation is an idealization of the actual pore filling process. The result should therefore be considered an approximation of the actual pore filling situation. It is generally accepted that the intra-porous mass transport happens via concentration gradient26, thus convective term is neglected during the development of this radial model in spherical coordinates. The resultant material and energy balances are expressed by the following steady state differential equations.

29 30

1 ∂ 2 ∂ci (r D ) − Σ υij R j = 0 ieff r 2 ∂r ∂r

(4.1)

1 ∂ 2 ∂T (r λeff ) − Σ ∆Hj R j = 0 2 r ∂r ∂r

(4.2)

31 32 33 34

In their dimensionless form, they can be expressed by the generic equations, Eq. 4.3 & 4.4 (where q=0 for slab, 1 for cylinder and 2 for sphere). In these equations, “u” is the dimensionless concentration while “y”



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3

Page 8 of 45

is the dimensionless temperature. The water gas shift reaction has been neglected as cobalt catalyst is being investigated, thus summation is removed from the second term on the left hand side of equation. Similarly, the subscript “j” is also removed from rate and heat of reaction.

4 5

1 ∂ q ∂ui (ξ ) − ϕi 2 φi = 0 q ξ ∂ξ ∂ξ

(4.3)

1 ∂ q ∂y (ξ ) + β φi = 0 𝜉 q ∂ξ ∂ξ

(4.4)

6 7 8 9 10 11

The dimensionless term phi (ϕi ) in the Eq. 4.3 represents the strength of internal mass transfer resistance towards a gaseous component i. As discussed before (section 3.0) for ϕi 1, there is significant internal heating (y(r) > ys) while in case of β < −1 there is internal cooling of the catalyst. Mathematically, these dimensionless

13 14

numbers can be expressed as follows

15

ϕi = √

16

2 rpellet

Di,eff ci,s

Rs

(4.5)

17

β=

18

2 (−∆H) rpellet Rs

𝜆eff Ts

(4.6)

19 20 21 22 23 24

For this reaction-diffusion model, Dirichlet boundary condition is applied at the surface of the pore as given by Eq. 4.7 & 4.9. Likewise, fugacity correlation, as defined by Wang et al. 7, is applied at the intrapellet material boundary between the empty and filled fractions. This correlation is given in Eq. 4.8. However, at the center of pellet von-Neumann boundary condition applies as described by Eq. 4.10.

25

Ci = Ci,s

26

𝑦𝑖 =

Boundary conditions at the surface are:

∅𝐿𝑖 𝑥 ∅𝑣𝑖 𝑖


(4.7) (4.8)

Page 9 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


T = Ts

1

(4.9)

2 3 4

It is important to note that 𝑦𝑖 is related to concentration via equation of state, while xi via Henry’s law correlates with liquid concentration.

5 6

Likewise at the center of pellet

7

∂ci =0 ∂r

8

∂T =0 ∂r

&

(4.10)

9 10 11 12

The effective diffusivity, Deff , term appearing in the Eq. 4.1 varies with extent of pore filling (partially filled pore considered as special case in section 5). For an empty pore (or unfilled fraction) Deff can be calculated from the following formula 27.

13

Di eff =

14 15 16

𝜀p Dvi τ

(4.11)

Whereas 1 Dvi

=

1 DMi

+

1 DKi

(4.12)

17 18 19 20

The molecular component of the diffusivity DMi can be calculated from Fuller expression 28. The Knudsen diffusion coefficient DKi for a straight pore can be calculated by Eq. 4.13, which is corrected to account for tortuosity via τ. 1

21

DKi

T 2 = 97 rp ( ) Mi

(4.13)

22 23 24 25

Alternatively, for a completely filled pore or in case of filled fraction (of partially filled pore) Deff can be calculated by an expression as explained by Wang et al. 7 for infinite dilution 29 (concentration of gaseous species within hydrocarbon solvent is approximately 0 mole/m3) as given by Eq. 4.14

26



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Dieff =

1 2 3

εp DBi τ

Page 10 of 45

(4.14)

Whereas

DBi = DBio exp (

−EDi T

)

(4.15)

4 5 6

The kinetics for intra-pellet FTS reaction is taken from Yates and Satterfield 11. They developed Langmuir Hinshelwood type kinetics.

7 8

R i = |υi |

ρcat a Pco PH2 (1 + b PCO )2

(4.16)

9 10 11 12 13

The rate constant and adsorption constant are defined in terms of their pre-exponential factor and activation energy as shown in Eq. 4.17 and 4.18, values of ao and EA/R are provided by Maretto and Krishna, based on non-linear fitting of experimental data 30. Details of different model parameters are provided in the Table 3.

14 15

a = ao exp (

EA 1 1 ( − )) R g 493.15 T

(4.17)

∆Hb 1 1 ( − )) R g 493.15 T

(4.18)

16 17 18 19 20 21 22 23 24 25 26 27 28

b = bo exp (

In the above rate equation the partial pressure are related to concentration via ideal gas law when considering the empty fraction of the pore or by Henry’s law in case of pore filling 2. While dealing with different catalyst distributions, pre-exponential factor a0 is adjusted to account for the superior (or inferior) performance of same active material. In the further development of this model some assumptions are also taken into account, (i) single pore model is the representative of entire pellet (ii) hydrocarbon product distribution is approximated by alpha chain growth probability (α) (iii) for the special case of partially filled pellet (with liquid hydrocarbon), discussed in section 5.1, the hydro-thermal properties of a pellet is calculated by summing the product of fraction filled and its corresponding properties with the product of fraction unfilled and its associated properties (iv) the model is a spherical model, no slab approximation has been taken into account thus there is a certain degree of error inherently associated with it when ϕi >>1 (v) pore collapse effect is not the subject of this model.


Page 11 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3


For an empty pellet fraction, alpha chain growth probability was estimated by a correlation developed by Yermakova et al. 31 while for filled pellet the expression of Vervloet et al. 2 was used. These equations are given by Eq. 4.19 & 4.20.

4 α = (A

5

uCO + B) (1 − 0.0039 (T − 533)) uCO + uH2

(4.19)

6

α=

7

1 C β ∆E 1 1 1 + k α (CH2 ) exp ( R α ( − T)) 493.15 CO g

(4.20)

8 9 10 11

In this expression k α denotes the ratio of rate constants for the propagation and termination reaction, β is syngas ratio power constant and ∆Eα is the difference in activation energies of propagation and termination reaction.

12 13

4.2

Solution Method

14 15 16 17 18 19 20 21 22

The single cylindrical radial pore covering the domain from surface to the center of the pore [i.e. 0, s+1] was divided in grid points. This grid was developed in such a way that it was finer near the surface, due to high rate especially when ϕi ≫ 1, however, a coarser grid was applied at the interior and near the center of pellet. The spacing between grid points was adaptive depending on the sensitivity of concentration profile. This increased numerical accuracy and stability of the code. Differential equations were converted to non-linear equations using central difference approximation. Degree of freedom was fulfilled by solving 2 material balance equations (for CO and H2) and 1 energy balance equation on each grid point. Local rate was calculated on each grid point from local concentration and temperature values. Overall rate of a spherical pellet was calculated using the volume integral. rpellet

23

Rate i,total = ∫

R i ( 4πr 2 ) dr

(4.21)

0

24 25 26

This integral was solved using 14-point Gauss Quadrature. Catalyst efficiency was then estimated in terms of effectiveness factor as given below

27 28



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 45

r

∫0 pellet R i ( 4πr 2 ) dr ηi = R i o Vcat

1

(4.22)

2 3 4 5

Similarly the overall chain growth probability of the product can be calculated from the correlation provided by Vervloet et al. 2 as given in Eq. 4.23. Individual “α” values are calculated from the reactants concentration gradients by solving radial non-linear equations.

6

αave =

7

3 1 ∫ α( ξ ) φCO ( ξ ) 𝜉 2 dξ η 0

(4.23)

8 9 10 11

Since carbon monoxide is both diffusion and stoichiometric limited reactant [11], so from here onwards we are going to restrict our discussion to CO only. Thus, ϕCO will be represented by simply φ, φCO by φ and likewise.

12 13

5.

Results and Discussion

14

5.1

Comparison with Analytical Results

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Theoretical estimations suggested a shell thickness of 0.1 mm for a pellet of 2mm diameter using sphere geometry. In that calculation the value of Thiele modulus was assumed to be 4.0 (as shown in supporting information). The numerical solution matched the analytical result as shown in Figure 2a. Model parameters used were: (i) catalyst density of 2500 kg/m3 (ii) assumption of complete pore filling by liquid hydrocarbons (iii) total pressure of 20 bar ≈ 300 psi and (iv) catalyst surface temperature of 493K. Preexponential factor was adjusted accordingly. The adjoining plot (Figure 2b) shows the variation in reaction rate with the change in radial diffusion length for different values of fractional filling. In case of empty pellet (f=0), the rate is constant at 0.0030 moles/kgcat.s. At approximately 20% filling system becomes diffusion limited with respect to CO (фCO =1) and reaction rate shows a maxima. This is due to reduction in rate inhibition by carbon monoxide (Eq. 4.16). Further increase in filling results in drop of rate as both carbon monoxide and hydrogen further depletes (value reduction in numerator overtakes the reduction in the denominator of rate equation). Fractional filling, however, also increases the overall intra-porous retention time, thus increasing the rate in empty fraction. With an increase in filling (i.e. f= 0.3, 0.4 and 0.6) the peak value of rate is reduced due to enhanced resistance in the flow path of hydrogen (carbon monoxide is already diffusion limited). Additionally, the radial rate reduces to zero at some distance from the center due to complete depletion of CO although the hydrogen was still available for reaction.

31 32

For a completely filled case, the active zone is limited to a thin shell of 0.1mm due to significantly high mass transfer resistance. The same result is shown in the contour of Figure 2a. However, within this shell


Page 13 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3 4


hydrogen is sufficiently available as it is near the surface of pellet thus increasing the peak height (maxima) once again. This confirms our earlier assertion that during steady state operation, it is best to use an eggshell catalyst as the inner core does not actively contribute to the reaction. However, this hypothesis is limited to its respective conditions.

5 6

5.2

Comparison with Previous Published Results

7 8 9 10 11

The results were compared with the model of Vervloet et al. 2. For this purpose, a catalyst having skeletal density of 2500 kg/m3 was considered under the assumption of complete filling (f=1). The process conditions were (i) T = 490K & (ii) P=30 bar, identical to that used by Vervloet et al. 2. Figure 3 shows the profile of dimensionless concentrations, reaction rate and hydrocarbon product selectivity along the intraporous radial length. This trend is generally similar to that obtained by Vervloet et al. 2.

12 13 14 15 16

Performance parameters are shown in the table insert in Figure 3. When compared with Vervloet et al. higher value of Thiele modulus was obtained from this model than that predicted by Vervloet et al. 2 due to larger pellet size (i.e. 2 mm vs. 1.5 mm). Effectiveness factor value and the resultant rate of reaction are in close agreement with their model 2 (η is 1.68 vs. 1.28 and RCO is 4.30 mmol/ (Kgcat.s) vs. 3.13 mmol/ (Kgcat.s)).

17 18 19 20 21 22 23 24 25 26

One major difference is in product selectivity. In the present work, resultant hydrocarbon product is in the range of middle distillate (i.e. Jet fuel and Diesel) which is in agreement with previously published results for eggshell catalysts deposited on 2mm pellets 1, 9. However, Vervloet et al. 2 predicted much lighter product (α= 0.57). It seems that the resultant mass transfer resistance (towards carbon monoxide) was much stronger for their case despite relatively small particle size. It is important to note that both models used the same diffusivity values, however, Vervloet et al.2 used a temperature dependent Henry’s coefficient while in the present work a constant value of HCO was used. This difference altered the intraporous CO solubility for the two cases; consequently the active shell was smaller for the case of Vervloet et al.2, due to shorter intra-porous residence time of carbon monoxide. This promoted secondary hydrogenation reaction resulting in chain termination at lower carbon values.

27 28

5.3

Sensitivity Analysis

29 30 31 32 33

In order to investigate the significance of catalyst morphology for an FTS process using cobalt catalyst, its sensitivity is tested against steady state temperature and pressure conditions. Table 4 shows a fractional factorial design, developed for this sensitivity analysis. Modeling results are represented in the form of contour plots. The effect of temperature and pressure on the catalyst morphology differs significantly. Thus, it would be prudent to discuss these effects separately.

34 35

The solubility of diffusing gases in liquid hydrocarbon depends on their partial pressure as characterized by Henry’s law 2.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 45

1 Ci Pi = Hi × ( ) Cl

2

(5.1)

3 4 5 6 7 8 9 10 11

In the above correlation, pi is the partial pressure of component “i", Hi is its Henry’s law constant, Ci is its liquid phase concentration and Cl is the total molar liquid concentration. The higher is the partial pressure, the more is the solubility of carbon monoxide molecules in liquid hydrocarbons thus making them accessible to the catalyst sites present at the interior of the pellet. Consequently, this reduces the thickness of inaccessible core (or inert core). For a catalyst with an intrinsic density of 2000 kg/m3, this effect is clearly visible up to 478K (comparing the values at the left column to those at the right columns of Table 4). Above 483 K, the solubility effect is not dominant; rather system behaves as if it is independent of pressure.

12 13 14 15

Temperature on the other hand affects the intra-pellet gas diffusivity as shown by Eq. 4.13-4.15. Increase in temperature increases mass transfer resistance towards diffusing gases especially carbon monoxide and consequently increases the boundary of gas de-void region. This is a general trend irrespective of system pressure as observed when the values at top are compared with that of bottom in the Table 4.

16 17 18 19 20 21 22 23

Thus, for a 2mm spherical pellet having a catalyst density of 2000 kg/m3, eggshell catalyst morphology is certainly required, even at low temperatures, to overcome the intra-porous mass transport limitation of carbon monoxide. Since the analysis in Table 4 is based on the mass transport of gases within the pellet, the inert core described in the table is in fact the gas de-void region with respect to carbon monoxide. Based on 2mm diameter (i.e. 1mm radius) pellet, the fraction of catalyst which comes in contact with the diffusing carbon monoxide (i.e. region where reaction takes place) is given by Eq. 5.2. Additionally, this fraction also represents the desired eggshell thickness of the pellet from the standpoint of mass transfer limitation alone.

24 Eggshell Thickness(𝑚𝑎𝑠𝑠−𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟) = {1 − Inert Core}

25

(5.2)

26 27 28 29

However, as discussed earlier, any morphological adjustment based on the reactant mass transfer alone will not produce the desired activity and selectivity. For that purpose, product chain growth within the pellet also needs to be investigated. In the next section we have discussed this factor in detail.

30

5.4

31 32 33 34

After estimating the parametric sensitivity, this model was used to analyze catalyst performance under typical FTS conditions. Catalyst activity and selectivity was taken as the performance measures. As mentioned by previous researchers 2, 7, one design parameter for an eggshell catalyst is to find the optimal shell thickness (for a given size of pellet) which maximizes selectivity while keeping the activity

Performance Analysis


Page 15 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 2

comparable to that of uniform pellet. Both activity and selectivity has been represented in this work via contour maps.

3 4 5 6

The resultant Thiele modulus (φ) was used as the abscissa of the performance contour map, while H2/CO ratio at the pellet inlet was taken as ordinates, shown in Figure 4a and 4b. It may seem inappropriate to use Thiele modulus for this purpose as it is a model output variable. However, its mathematical expression (Eq. 5.3) provides the answer. Thiele Modulus “φ” can be expressed by the following function.

7 8

ϕ = f(rpellet, H2,conc, COconc, Deff , T)

(5.3)

9 10 11 12

Effective diffusivity (Deff) is the function of extent of filling. If the system temperature, catalyst filling, and the concentrations of gaseous species are kept constant, Thiele modulus can simply be expressed as following.

13 14

ϕ = f(rpellet )

(5.4)

15 16 17 18 19 20 21

As per the above condition, different Thiele Modulus values on the contour map, at a fixed H2/CO ratio, represents various pellet radius values. Alternatively, rpellet can also be considered as the intra-pellet diffusion distance having the corresponding H2/CO ratio. Likewise, if on contour, H2/CO ratio is changed, for a fixed modulus value, it can be taken as if radial length is changing. Additionally, this correlation can also be extended to an eggshell type catalyst. In case of an eggshell profile, Thiele modulus is a function of shell thickness, as shown by our previous work 6.

22 23

ϕeggshell = f(Xshell )

(5.5)

24 25 26 27 28

So, for an eggshell catalyst the modulus in the contour map of Figure 4 a & b can be considered as eggshell modulus, function of a certain shell thickness i.e. X shell . From this eggshell modulus value, optimum shell thickness can easily be calculated using our previously developed combined slab-sphere geometry approach, as discussed later.

29 30 31 32

The abscissa values on the map were based on analytical experience and previous discussion. As Thiele modulus (φ) of 4 provided an active thickness of 0.1 mm over 2mm pellet (size required to decouple mass transfer limitation from pressure drop). Beyond the value of 4, whole pellet will be rendered useless and the entire discussion will be meaningless. Thus φ was varied from 0 to 3. Additionally, as shown later in



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 45

1 2

this discussion, Thiele modulus above 3 will negatively impact the rate and selectivity. The stoichiometric H2/CO ratio of 2 for FTS operation is taken as the maxima of the ordinate.

3 4 5 6 7 8 9 10 11

On the contour plot of Figure 4a, at pellet-inlet H2/CO ratio of 2 and φ of 3, corresponding rate of reaction is approximately 3.3 mmol/ (kgcat.s). If the inlet H2/CO ratio is kept constant and the value of modulus is reduced, the rate of reaction starts to increase for all values of H2/CO ratio till the transitional φ of 1.6. This is due to reduction in characteristic diffusion length as φ is a function of rpellet. At φ value of approximately 1.6, maximum rate is observed i.e. 5 mmol/ (kgcat.s); for the inlet H2/CO ratio of 2. It seems that at this modulus value, system shifts from diffusion to kinetic limited regime. Interestingly, this change happens at the same φ value for all inlet ratios. However, the corresponding radial length will be different. For φ < 1.6, the relative drop in conversion is due to the fact that not enough catalyst is available for reaction to proceed further.

12 13 14 15

For all values of φ, the rate of reaction increases by increasing the inlet H2/CO ratio. This is due to a reduction in CO inhibition. However this change is more significant near φ = 1.6. On the further right corner i.e. extremely diffusion limited region, this variation is the least as most of the pellet interior is devoid of reactant gases.

16 17 18 19 20

Another advantage of this study is the possible reduction in hydrogen usage while keeping the rate at higher levels. In Figure 4a, this hypothesis can be proven by following the iso-line of 3.3mmol/kgcat.s. For an inlet H2/CO ratio of 2, this rate is attained at φ value of 3.0. However, if φ ≈1.6, a lower H2/CO ratio (i.e. ≈ 1.4) will result in the same rate of reaction. This lower value of φ does not necessarily require a smaller sized pellet; it can rather be achieved by having an eggshell morphology while keeping pellet size constant.

21 22 23 24 25 26 27

A similar analysis has been performed for products selectivity as shown in Figure 4b. Analogous to rate of reaction, the αave value for ASF product distribution is approximately 0.79 at φ value of 3.0 and the inlet H2/CO ratio of 2. Based on previous analytical results; this value give peak fractions in the region of middle distillate fuels along with low fractions of soft wax. Generally, the value of αave shows a drop in the region where φ > 1.6 for a given value of inlet syngas ratio. This is due to the drop in CO concentration in diffusion limited regime giving rise to secondary hydrogenation reaction. Thus, higher fractions of lighter hydrocarbon will be produced as Thiele modulus keeps on increasing.

28 29 30 31 32 33 34 35 36 37 38

Following the arguments presented during rate discussion, if H2 saving is implemented by reducing the inlet H2/CO ratio to 1.4 at φ ≈ 1.6, the corresponding αave value increases to 0.85, producing a significant fraction of soft wax. This is due to a decrease in hydrogenation reaction. However, at the same φ value, if the inlet H2/CO ratio is raised to 2, the αave value drops to 0.79-0.8, significantly increasing the productivity of middle distillates. Additional benefit of this rise is the significant increase of rate of reaction i.e. 5 mmol/kgcat.s as can be seen in Figure 4a. For φ values of less than 1.6, catalysts produce heavier hydrocarbons for all H2/CO ratios. This is due to more CO availability leading to longer hydrocarbon chains. Based on this discussion it is clear that Thiele Modulus (φ) of approximately 1.6 gives higher rate as well as the desired selectivity of middle distillate. For further calculation, a constant H2/CO ratio of 2, pressure of 20bar and temperature of 498 K is selected. Under these constant conditions, the overall Thiele modulus for 2 mm pellet, having skeletal density of 2500 kg/m3 is approximately 2.6. Thus either a smaller


Page 17 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 2

pellet will have to be used or if the size is to be kept constant then eggshell morphology is the second option.

3 4 5 6 7 8 9 10 11

The following formulae have been developed (based on shell and slab geometry) to correlate eggshell modulus to the Thiele modulus of entire pellet. These expressions are based on the spherical volume of an active zone that has a volume fraction “ρ” within a sphere of radius “R”. The active volume will then be 4/3 π R3 (1 - ρ3). The derivation details are provided in our prior work 1. Based on the sphere geometry alone (i.e. Eq. 5.6), for a ф value of 2.6, eggshell modulus value of 1.6 is obtained at an inert core fraction (ρ) of 0.75 (eggshell thickness of 0.25 mm). It is important to note that the corresponding pellet radius is 1 mm. However, the combined geometry result provides an eggshell thickness of approximately 0.38 mm as shown in Figure 5. This profile is based on the combination of Eq. 5.6, for sphere geometry, and Eq. 5.7, for slab geometry.

12 13

ϕeggshell = ϕ (1 − ρ3 )

(5.6)

(1 − ρ3 ) = ϕ 3

(5.7)

14 ϕeggshell

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

6.0

Conclusion

Fischer Tropsch synthesis is a heterogeneous catalytic reaction. For a 2mm porous spherical pellet, at 490 K and 20 bar pressure, deposited with highly dispersed catalyst material (resultant ф = 4) an analytical reaction-diffusion model predicts an intra-porous active reaction zone of 0.1 mm (desired eggshell thickness) within which there is no diffusion limitation for the inward motion of gas mixture. However, if slab approximation is applied the resultant thickness comes out to be 0.27 mm. Based on our previously published results 1 it is more prudent to use a combined slab-sphere geometric approach to converge to a final thickness value. However, this analysis is incomplete as it does not take into account product selectivity The fundamental design aspect of a catalyst is to enhance both the activity and the desired selectivity. For this purpose, analytical approach was further improved via numerical modeling that took into account pore filling, rate of reaction, hydrocarbon product chain growth and thermodynamic phase equilibrium equations. At a fixed temperature and pressure of 490 K and 20 bars and at an inlet H2/CO ratio of 2, system exhibits maximum rate of 5 mmol/kgcat.s and gives product in the range of middle distillate for a Thiele modulus value of approximately 1.6. For a 2mm pellet with a catalyst density of 2500 kg/m3 this value converged to an eggshell thickness of 0.38 mm. Apart from thickness estimation, sensitivity analysis was also performed to check the morphology requirement under typical conditions. Although these results are specific to 2 mm pellet and the range of temperature and pressure values given in Table 4, these conditions are frequently encountered in previously published FTS research work. During the numerical modeling, while comparing with analytical result, pre-exponential factor of Yates and Satterfield 11 equation was adjusted in order to match the observed activity of an experimentally synthesized catalyst 1. While, for the rest of the modeling, pre-exponential factor was kept same as



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1 2 3 4 5 6

Page 18 of 45

suggested by Yates and Satterfield 11. Thus the two results are not comparable and are specific to their respective conditions. One advantage of using Thiele modulus as a performance parameter is its dependence on several structural and process parameters as shown in Eq. 4.26. If rest of the values is kept constant, Thiele modulus becomes the function of a single process or structural variable. In this manner, Thiele modulus values provided in Figure 4 can either represent a range of pellet size or a range of filling or temperature values.


Page 19 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 1: Fischer-Tropsch kinetics developed for Co based catalyst by previous researchers along with the type of reactors employed. In the current work, model developed by Yates and Satterfield has been used.

Reference

Catalyst

Reactor Type

Intrinsic Kinetics

Yang et al. 32

Co/ CuO /Al2O3

Fixed Bed

- RCO+H2 = a PH2 PCO - 0.5

Rautavouma et al.33

Co/ Al2O3

Fixed Bed

Co/ Kisselguhr

Berty

-RCO= (a PH21/2 PCO1/2)/ (1+b PCO1/2+ c PH21/2+ d PCO)2

Wang et al. 35

Co/ B/ Al2O3

Fixed Bed

- RCO = a PH20.68 PCO - 0.5

Yates et al. 11

Co/ MgO /SiO2

Slurry

-RCO = (a PCO PH2)/(1+b PCO)2

Iglesia et al. 9

Co/ SiO2

Fixed Bed

-RCO = (a PH20.6 PCO 0.65)/(1+b PCO)

Withers et al. 36

Co (CO)8/ Zr(O Pr)4 / SiO2

CSTR

-RFT = (a PH2)/(1+(b PH2O/(PCO PH2)))

Davis et al. 37

Co/ Al2O3

CSTR

-RFT = (a PCO -0.31 PH2 0.88)/(1+(b PH2O/ PH2))

Sarup et al.

34


-RCO= (a PH2 PCO1/2)/(1+b PCO1/2)3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Table 2: Required eggshell thickness for a 2mm spherical catalyst employed in an FTS process, characterized by Thiele Modulus φ. Thickness value calculated using eggshell modulus concept. For details on nomenclature refer to Table 3 of our prior work 1

Sample

Thiele Modulus (ф)

W-SA E-SA E-DH W-DH

1.00 3.02 4.00 1.90

Thickness (Slab) mm 1.00 0.8 0.35 1.00

Sphere

Page 20 of 45

Slab

Thickness (Sphere) mm 1.00 0.12 0.10 0.22

Mass Transfer Resistance

Rate Limiting

Figure 1: Variation in the eggshell modulus with the change in the fraction (i.e. thickness) of inner core “ρ” based on combined sphere and slab geometry, ACS Paragon Plus Environment

Page 21 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 3: Modeling parameter values used in the model described in section 4.1

Symbol

Value

dpellet

2mm

Catalyst Pore Radius

dp

5nm

Catalyst Intrinsic Density Catalyst Porosity

ρcat εp

Reference

Description Catalyst Specific Catalyst Diameter

Catalyst Tortuosity

Τ

2000-2500 kg m-3 0.7

Sensitivity- Comparison 2

2.0

Diffusivity Specific Molecular Diffusivity

DMi

Knudsen Diffusivity

DKi

Fuller Correlation 28a Equation 4.13 -7

2 -1

CO Diffusion Constant in Hydrocarbon

DB−COo

5.584 10 m s

CO Diffusion Activation Energy

ED−CO

14.85 103 J mol-1

H2 Diffusion Constant in Hydrocarbon

DB−H2o

1.085 10-6 m2 s-1

H2 Diffusion Activation Energy

ED−H2

13.51 103 J mol-1

Henry’s Coefficient for CO

HCO

1.3 104 Pa. m3 mol-1

Henry’s Coefficient for H2

HH2

2.0 104 Pa. m3 mol-1

Solubility Specific

Fugacity Coefficient

FROM SRK EOS 38

𝜙𝑖

Performance Specific Yates and Satterfield Rate Constant Preexponential Factor Yates and Satterfield Adsorption Constant Pre-Exponential Factor Activation Energy Enthalpy of Adsorption Selectivity Constant

ao

8.852 10-3 mol s-1 Kgcat-1

11

bo

2.226 bar-2

11

EA /R ΔH/R kα

4494.1 K -8236.15 K 56.7 10-3

30, 39


30 2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Selectivity Exponential Parameter Selectivity Activation Energy Difference Empirical Constant in Yermakova’s Product Selectivity Expression Empirical Constant in Yermakova’s Product Selectivity Expression Thermal Properties Heat of Reaction Effective Thermal Conductivity

Page 22 of 45

β

1.76

2

ΔEα

120.4 103 J mol-1

2

A

0.2332

B

0.6330

ΔH

-170 KJ mol-1

40

𝜆𝑒𝑓𝑓

0.23 W m-1 k-1

6


Page 23 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(a)



0.0050

ɸ (1.0) =3.9

ɸ (0.2) =1.04 ɸ (0.8) =2.1

ɸ (0.4) =1.25 0.0040

ɸ (0.6) =1.46

ɸ (0.3) =1.1

Rate (mol/ kgcat .s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 45

ɸ (0.1) =0.89

0.0030

0.0020

0.0010

0.0000 0

0.2

0.4

0.6

0.8

1

Dimensionless Radius ( ξ )

(b)

Figure 2: (a) Contour plot showing the profile of CO partial pressure. Pre-exponential factor of Satterfield equation adjusted to give Thiele modulus of (ф) ≈ 4 in order to reproduce the result from analytical section (filling =1) (b) Rate of reaction profile at various fillings for the same system (fractional filling “f” given in parenthesis). Process conditions provided in section 5.1.


Page 25 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ϕCO

α

yH2 yCO

Figure 3: Dimensionless concentration yi, reaction rate ϕco and α profile in spherical catalyst. Conditions were kept same as used by Vervloet et al. 2 i.e. T=490 K, P = 30 bar, H2/CO ratio = 2 and filling = 1. 10 bar 15 bar 20 bar 25 bar 30 bar



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 45

Table 4: Numerically calculated fraction of radial inert core for a 2 mm dia. catalyst. The values of temperature and pressure are those frequently encountered in a catalytic FTS process using cobalt catalyst. The point of investigation is the requirement of eggshell morphology.

473 K

Radial Inert Core =0.40 Φ= Inert Core =

Radial Inert Core = 0.38


Φ= Inert Core =

Radial Inert Core =0 .30 Φ= Inert Core =


478 K

Radial Inert Core = 0.47 Φ= Inert Core =



Φ= Inert Core =


Radial Inert Core = 0.42 00.24

Φ= Inert Core =

483 K

Radial Inert Core =0.52



42

Radial Inert Core = 0.54 40

Radial Inert Core = 0.52 0.00.44 Φ= Inert Core =

488 K






493 K




Φ= Inert Core =




Page 27 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(a)

(b)

Figure 4 (a) The contour map of rate of reaction. (b) The contour map of chain growth probability. Modeling conditions are P = 20 bar, T = 490 K & f =1. Different Thiele modulus (ϕ) values were obtained by varying catalyst radius



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 45

Figure 5: Variation in eggshell modulus with a change in the fraction of inert core. The value of Thiele modulus at ρ = 0 corresponding to fully loaded catalyst. The geometric transition zone is based on previously published results.


Page 29 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Symbols r

= Radial distance (m)

υ

= The stoichiometric coefficient

i

= ith component

j

= jth reaction

R

= Rate of reaction of CO (mol. m-3.s-1)

λ

= Thermal conductivity (W. m-1. K-1)

∆Hj

= Heat of reaction (J.mol-1.K-1)

Rg

= Ideal gas constant (J.mol-1.K-1)

ξ

= Dimensionless Radius

D

= Diffusivity (m-2 .s-1)

eff

= Effective

f

= Fractional filling

β

= Measure of heat of reaction

φ

= Dimensionless Rate

c

= Concentration of ith component (mol. m-3)

T

= Temperature (K)

𝜑𝐿

= Fugacity coefficient of liquid hydrocarbons

𝜑𝑣

= Fugacity coefficient of vapor hydrocarbons

𝑥𝑖

= Liquid fraction

𝑦𝑖

= Vapor fraction

s

= Surface specific specie

K

= Knudsen component

M

= Molecular component



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

𝜀p

= Porosity

τ

= Tortuosity

rp

= Pore Radius (m)

ρcat

= Catalyst density (Kg. m-3)

Pi

= Partial pressure (bar)

DBi

= Diffusion constant for component i (m2.s-1)

EDi

= Activation energy for diffusion (J/mol)

ao

= Rate constant pre-exponential factor (mol. s-1 Kgcat-1)

bo

= Adsorption constant pre-exponential factor (bar-2)

EA

= Activation energy (J.mol-1)

∆Hb

= Heat adsorption (J.mol-1)

α

= Chain growth probability

kα

= Selectivity constant

∆Eα

= Selectivity activation energy difference

Vcat

= Catalyst volume (m3)

Hi

= Henry’s constant (Pa. m3.mol-1)

Cl

= Liquid concentration (m3. mol-1)

η

= Effectiveness factor

ϕ

= Thiele Modulus

ρ

= Fractional inert core


Page 30 of 45

Page 31 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Acknowledgement: This research is funded through a University grant by Florida Energy Systems Consortium (FESC).

Supporting Information Available: A complete development of analytical model for sphere and slab geometry is provided in the supporting document. This information is available free of charge via the internet at http://pubs.acs.org.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

References 1. Gardezi, S. A.; Wolan, J. T.; Joseph, B., Effect of catalyst preparation conditions on the performance of eggshell cobalt/SiO2 catalysts for Fischer-Tropsch synthesis. Appl Catal a-Gen 2012, 447, 151-163. 2. Vervloet, D.; Kapteijn, F.; Nijenhuis, J.; van Ommen, J. R., Fischer-Tropsch reaction-diffusion in a cobalt catalyst particle: aspects of activity and selectivity for a variable chain growth probability. Catal Sci Technol 2012, 2 (6), 1221-1233. 3. Chahbazpour, D.; LaRusso, A. Renewable Gas —Vision for a Sustainable Gas Network; National Grid: Waltham, Massachusetts, 2010. 4. Bligaard, T.; Norskov, J. K.; Dahl, S.; Matthiesen, J.; Christensen, C. H.; Sehested, J., The Bronsted-Evans-Polanyi relation and the volcano curve in heterogeneous catalysis. J Catal 2004, 224 (1), 206-217. 5. Sie, S. T., Design of Catalyst Morphology Tailored to Process Needs. Precision Process Technology 1993, 139-155. 6. Gardezi, S. A. Development of Catalytic Technology for Producing Sustainable Energy. Doctoral Dissertation University of South Florida Tampa, Florida, United States 2013. 7. Wang, Y. N.; Xu, Y. Y.; Xiang, H. W.; Li, Y. W.; Zhang, B. J., Modeling of catalyst pellets for FischerTropsch synthesis. Ind Eng Chem Res 2001, 40 (20), 4324-4335. 8. (a) Delancey, G. B., Optimal Catalyst Activation Policy for Poisoning Problems. Chem Eng Sci 1973, 28 (1), 105-118; (b) Dougherty, R. C.; Verykios, X. E., Nonuniformly Activated Catalysts. Catal Rev 1987, 29 (1), 101-150; (c) Becker, E. R.; Wei, J., Nonuniform Distribution of Catalysts on Supports .2. First-Order Reactions with Poisoning. J Catal 1977, 46 (3), 372-381. 9. Iglesia, E.; Soled, S. L.; Baumgartner, J. E.; Reyes, S. C., Synthesis and Catalytic Properties of Eggshell Cobalt Catalysts for the Fischer-Tropsch Synthesis. J Catal 1995, 153 (1), 108-122. 10. Jun B.; H., J.; Yi Z.; Yoshiharu Y.; Noritatsu T., A Core/Shell Catalyst Produces a Spatially Confined Effect and Shape Selectivity in a Consecutive Reaction. Angewandte Chemie 2008, 120, 359-362. 11. Yates, I. C.; Satterfield, C. N., Intrinsic Kinetics of the Fischer-Tropsch Synthesis on a Cobalt Catalyst. Energ Fuel 1991, 5 (1), 168-173. 12. Khodakov, A. Y.; Chu, W.; Fongarland, P., Advances in the development of novel cobalt FischerTropsch catalysts for synthesis of long-chain hydrocarbons and clean fuels. Chem Rev 2007, 107 (5), 1692-1744. 13. Post, M. F. M.; Vanthoog, A. C.; Minderhoud, J. K.; Sie, S. T., Diffusion Limitations in FischerTropsch Catalysts. Aiche J 1989, 35 (7), 1107-1114. 14. Jess, A.; Kern, C., Modeling of Multi-Tubular Reactors for Fischer-Tropsch Synthesis. Chem Eng Technol 2009, 32 (8), 1164-1175. 15. Corbett, W. E.; Luss, D., Influence of Nonuniform Catalytic Activity on Performance of a Single Spherical Pellet. Chem Eng Sci 1974, 29 (6), 1473-1483. 16. Johnson, D. L.; Verykios, X. E., Selectivity Enhancement in Ethylene Oxidation Employing Partially Impregnated Catalysts. J Catal 1983, 79 (1), 156-163. 17. Minhas, S.; Carberry, J. J., On Merits of Partially Impregnated Catalysts. J Catal 1969, 14 (3), 270&. 18. Shadmany.F; Petersen, E. E., Changing Catalyst Performance by Varying Distribution of Active Catalyst within Porous Supports. Chem Eng Sci 1972, 27 (2), 227-&. 19. Gardezi, S. A.; Joseph, B.; Prado, F.; Barbosa, A., Thermochemical biomass to liquid (BTL) process: Bench-scale experimental results and projected process economics of a commercial scale process. Biomass Bioenerg 2013, 59, 168-186.


Page 32 of 45

Page 33 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20. Tsamalis, C. P.; Szepe, S., On Diffusional Effects in Partially Impregnated Catalysts. Chem Eng Sci 1988, 43 (7), 1705-1708. 21. Lee, S. Y.; Aris, R., The Distribution of Active Ingredients in Supported Catalysts Prepared by Impregnation. Catal Rev 1985, 27 (2), 207-340. 22. Summers, J. C.; Ausen, S. A., Catalyst Impregnation - Reactions of Noble-Metal Complexes with Alumina. J Catal 1978, 52 (3), 445-452. 23. Harriott, P., Diffusion Effects in Preparation of Impregnated Catalysts. J Catal 1969, 14 (1), 43-&. 24. van Houwelingen, A. J.; Kok, S.; Nicol, W., Effectiveness Factors for Partially Wetted Catalysts. Ind Eng Chem Res 2010, 49 (17), 8114-8124. 25. Aris, R., The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts: Vol. 1: The Theory of the Steady State Oxford University Press: England, 1975; p 460. 26. Bartholomew, C. H.; Farrauto, R. J., Fundamental of Industrial Catalytic Processes. WileyInterscience New Jersey 2005. 27. Hayes, R. E.; Kolaczkowski, S. T., Introduction to Catalytic Combustion. Gordon and Breach Science Publishers Netherlands 1997. 28. (a) Fuller, E. N.; Schettle.Pd; Giddings, J. C., A New Method for Prediction of Binary Gas-Phase Diffusion Coeffecients. Ind Eng Chem 1966, 58 (5), 19-&; (b) Fuller, E. N.; Ensley, K.; Giddings, J. C., Diffusion of Halogenated Hydrocarbons in Helium . Effect of Structure on Collision Cross Sections. J Phys Chem-Us 1969, 73 (11), 3679-&. 29. Reid, R. C.; Prausnitz, J. M.; Poling, B. E., The Properties of Gases and Liquids McGraw-Hill: New York 1987. 30. Maretto, C.; Krishna, R., Modelling of a bubble column slurry reactor for Fischer-Tropsch synthesis. Catal Today 1999, 52 (2-3), 279-289. 31. Yermakova, A.; Anikeev, V. I., Thermodynamic calculations in the modeling of multiphase processes and reactors. Ind Eng Chem Res 2000, 39 (5), 1453-1472. 32. Yang, C. H.; Massoth, F. E.; Oblad, A. G., Kinetics of carbon monoxide and hydrogen reaction over cobalt-copper-alumina oxide catalyst. Advances in Chemistry Series 1979, 178, 35-46. 33. Rautavuoma, A. O. I.; Vanderbaan, H. S., Kinetics and Mechanism of the Fischer-Tropsch Hydrocarbon Synthesis on a Cobalt on Alumina Catalyst. Appl Catal 1981, 1 (5), 247-272. 34. Sarup, B.; Wojciechowski, B. W., Studies of the Fischer-Tropsch Synthesis on a Cobalt Catalyst .3. Mechanistic Formulation of the Kinetics of Selectivity for Higher Hydrocarbon Formation. Can J Chem Eng 1989, 67 (4), 620-627. 35. Wang, J. Physical, Chemical and Catalytic Properties of Borided Cobalt Fischer-Tropsch Catalyst Ph.D. Thesis Bringham Young University Provo, Utah, 1987. 36. Withers, H. P.; Eliezer, K. F.; Mitchell, J. W., Slurry-Phase Fischer-Tropsch Synthesis and KineticStudies over Supported Cobalt Carbonyl Derived Catalysts. Ind Eng Chem Res 1990, 29 (9), 1807-1814. 37. Ma, W. P.; Jacobs, G.; Das, T. K.; Masuku, C. M.; Kang, J. S.; Pendyala, V. R. R.; Davis, B. H.; Klettlinger, J. L. S.; Yen, C. H., Fischer-Tropsch Synthesis: Kinetics and Water Effect on Methane Formation over 25%Co/gamma-Al2O3 Catalyst. Ind Eng Chem Res 2014, 53 (6), 2157-2166. 38. Karimi, Z.; Rahmani, M.; Moqadam, M. In A study on vapour-liquid equilibria in Fischer-Tropsch synthesis, 20th International Congress of Chemical and Process Engineering Prague, Czech Republic, Elsevier: Prague, Czech Republic, 2012. 39. (a) Gholami, F., M. Torabi and Z. Gholami, F. Gholami, M. Torabi and Z. Gholami, Modeling the Fischer Tropsch Reaction in a Slurry Bubble Column Reactort. World Academy of Science 2009, 3, 168171; (b) Guillean, D. P., Anatasia M. Gribik, Jonathan K. Shelley, Steven P. Antal, Incorporation of Reaction Kinetics into a Multiphase Model of a Fischer Tropsch Slurry Bubble Column Reactor In AiChE Annual Meeting, AIChE: Philadelphia, 2008. 40. Andrzej, C.; Moulijn, J. A., Structured Catalysts and Reactors CRC Press Boca Raton, Florida 2006.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

380x221mm (96 x 96 DPI)


Page 34 of 45

Page 35 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


380x221mm (96 x 96 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

380x221mm (96 x 96 DPI)


Page 36 of 45

Page 37 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


380x221mm (96 x 96 DPI)



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

380x221mm (96 x 96 DPI)


Page 38 of 45

Page 39 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 1: Fischer-Tropsch kinetics developed for Co based catalyst by previous researchers along with the type of reactors employed. In the current work, model developed by Yates and Satterfield has been used.

Reference

Catalyst

Reactor Type

Intrinsic Kinetics

Yang et al. 32

Co/ CuO /Al2O3

Fixed Bed

- RCO+H2 = a PH2 PCO - 0.5

Rautavouma et al.33

Co/ Al2O3

Fixed Bed

Co/ Kisselguhr

Berty

-RCO= (a PH21/2 PCO1/2)/ (1+b PCO1/2+ c PH21/2+ d PCO)2

Wang et al. 35

Co/ B/ Al2O3

Fixed Bed

- RCO = a PH20.68 PCO - 0.5

Yates et al. 11

Co/ MgO /SiO2

Slurry

-RCO = (a PCO PH2)/(1+b PCO)2

Iglesia et al. 9

Co/ SiO2

Fixed Bed

-RCO = (a PH20.6 PCO 0.65)/(1+b PCO)

Withers et al. 36

Co (CO)8/ Zr(O Pr)4 / SiO2

CSTR

-RFT = (a PH2)/(1+(b PH2O/(PCO PH2)))

Davis et al. 37

Co/ Al2O3

CSTR

-RFT = (a PCO -0.31 PH2 0.88)/(1+(b PH2O/ PH2))

Sarup et al.

34


-RCO= (a PH2 PCO1/2)/(1+b PCO1/2)3


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Table 2: Required eggshell thickness for a 2mm spherical catalyst employed in an FTS process, characterized by Thiele Modulus φ. Thickness value calculated using eggshell modulus concept. For details on nomenclature refer to Table 3 of our prior work [2]

Sample

Thiele Modulus (ф)

W-SA E-SA E-DH W-DH

1.00 3.02 4.00 1.90

Thickness (Slab) mm 1.00 0.8 0.35 1.00


Thickness (Sphere) mm 1.00 0.12 0.10 0.22

Page 40 of 45

Page 41 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48


Table 3: Modeling parameter values used in the model described in section 4.1

Symbol

Value

dpellet

2mm

Catalyst Pore Radius

dp

5nm

Catalyst Intrinsic Density Catalyst Porosity

ρcat εp

Reference

Description Catalyst Specific Catalyst Diameter

Catalyst Tortuosity

τ

2000-2500 kg m-3 0.7

Sensitivity- Comparison 2

2.0

Diffusivity Specific Molecular Diffusivity

DMi

Knudsen Diffusivity

DKi

Fuller Correlation 28a Equation 4.13 -7

2 -1

CO Diffusion Constant in Hydrocarbon

DB−COo

5.584 10 m s

CO Diffusion Activation Energy

ED−CO

14.85 103 J mol-1

H2 Diffusion Constant in Hydrocarbon

DB−H2o

1.085 10-6 m2 s-1

H2 Diffusion Activation Energy

ED−H2

13.51 103 J mol-1

Henry’s Coefficient for CO

HCO

1.3 104 Pa. m3 mol-1

Henry’s Coefficient for H2

HH2

2.0 104 Pa. m3 mol-1

Solubility Specific

Fugacity Coefficient

FROM SRK EOS 38

𝜙𝑖

Performance Specific Yates and Satterfield Rate Constant Preexponential Factor Yates and Satterfield Adsorption Constant Pre-Exponential Factor Activation Energy Enthalpy of Adsorption Selectivity Constant

ao

8.852 10-3 mol s-1 Kgcat-1

11

bo

2.226 bar-2

11

EA /R ΔH/R kα

4494.1 K -8236.15 K 56.7 10-3

30, 39


30 2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48

Page 42 of 45

β

1.76

2

ΔEα

120.4 103 J mol-1

2

A

0.2332

B

0.6330

Thermal Properties Heat of Reaction

ΔH

-170 KJ mol-1

40

Effective Thermal Conductivity

𝜆𝑒𝑓𝑓

0.23 W m-1 k-1

6

Selectivity Exponential Parameter Selectivity Activation Energy Difference Empirical Constant in Yermakova’s Product Selectivity Expression Empirical Constant in Yermakova’s Product Selectivity Expression


Page 43 of 45

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 4: Numerically calculated fraction of radial inert core for a 2 mm dia. catalyst. The values of temperature and pressure are those frequently encountered in a catalytic FTS process using cobalt catalyst. The point of investigation is the requirement of eggshell morphology. 10 bar

15 bar

20 bar




25 bar

30 bar

473 K

Φ= Inert Core =

Φ= Inert Core =

Φ= Inert Core =

Radial Inert Core =0 .30 Φ= Inert Core =


478 K




Φ= Inert Core =


Radial Inert Core = 0.42 00.24

Φ= Inert Core =

483 K




42

Radial Inert Core = 0.54 40

Radial Inert Core = 0.52 0.00.44 Φ= Inert Core =

488 K






493 K




Φ= Inert Core =





1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

380x221mm (96 x 96 DPI)


Page 44 of 45

Page 45 of 45


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ACS Paragon Plus Environment

SiO2 ... - ACS Publications

Recommend Documents