Assessment of 1-Dimensional Catalytic Reactors Using Constrained

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Assessment of 1-dimensional catalytic reactors using constrained Gibbs free energy minimization method: water-gas-shift and carbon monoxide methanation case. Eduardo Paiva, Risto Pajarre, Petteri Kangas, and Pertti Koukkari Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b01176 • Publication Date (Web): 28 Jun 2017 Downloaded from http://pubs.acs.org on July 2, 2017

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Industrial & Engineering Chemistry Research

Assessment of 1-dimensional catalytic reactors using constrained Gibbs free energy minimization method: water-gas-shift and carbon monoxide methanation case. Eduardo J. M. Paiva, Risto Pajarre, Petteri Kangas, Pertti Koukkari* VTT Technical Research Centre, Espoo, PO BOX 1000, FI-02044, Finland ABSTRACT: Catalytic 1-dimensional reactor models were developed using the Gibbs energy minimization approach in order to describe gas composition, molar fractions, and conversions of both water-gas-shift and carbon monoxide methanation at low partial pressure ratios of CO/H2. The extent of reaction in terms of CO amount in the system was used as an additional constraint on the chemical system while solving the local thermodynamic equilibrium. The validity of the model was checked against experimental data gathered from the literature. The known theory about heterogeneous catalysis was incorporated in the Gibbsian multiphase analysis by means of the advancements of a diffusional limited water-gas-shift reaction and the catalyzed methanation of CO using virtual phases in the conservation matrix. The advantages of the use of this technique to describe a 1D catalyzed reaction, namely qualitative data regarding the chemical system and reduced ordinary differential equations (ODE) input, among others, are outlined in this report.

KEYWORDS: Gibbs energy minimization; 1D reactor; water-gas-shift; methanation; heterogeneous catalysis; virtual phases; constrained thermodynamic equilibrium; carbon monoxide; extent of reaction.

1. Introduction Modeling and simulation of chemical reactors are advantageous as they enable the easy study of the influence of the operation conditions on the reactor performance, while the experimental work is limited to validation experiments1. Robust models are the first step toward optimization of reactors and, consequently, the chemical process. In addition, as modern society aims for controlling the emission levels of greenhouse gases, thus, technologies aiming to improve the existing catalytic process become highly desirable from both a scientific and an industrial point of view. In this context, the water-gas-shift reaction (WGS) is an alternative route to converting CO into useful hydrogen gas. Moreover, this process takes place in a lot of reforming reactions, whenever water and mono-dioxide carbon are present. Extensive work has been done in order to improve these reactions. Among them, Swickrath and Anderson1 explored potential Sabatier reactions to convert CO-CO2 into water and methane, aiming for carbon dioxide recycling and water reuse in space technology improvement; Kopyscinski, et al.2 performed an extensive study of methanation in a nickel-alumina catalytic plate reactor, using a sophisticated apparatus to collect data over several points of the bed. In both works, WGS took place to some extent. Salmi and Wärnå3 studied the transport phenomena involving iron-based catalyst to WGS conversion; the methodology proposed by Aris4 to treat diffusional phenomena was employed. A renewable natural gas substitute concept requires that it can be stored, distributed, and reconverted on demand. In

this novel approach, renewable energies are converted via reversible solid oxide cells into CO and hydrogen. In this context, the Syngas (H2 and CO) production itself is a valuable conversion route. Although, from an energetic perspective, one of the main conversion steps is methanation5. Industrially, CO methanation over nickel is currently used in large ammonia plants to remove the last fraction of a percentage of carbon oxides before synthesis, because iron-based ammonia catalysts are very sensitive to poisoning by oxygen6. The activation of carbon monoxide on nickel also plays a key role in metal-dusting corrosion on surfaces of equipment exposed to CO-rich gases7. Studies of reaction kinetics comprise a large share of the models of methanation processes, as the reaction rate has been the major issue to be improved. Often, Langmuir-Hinshelwood kinetics is applied to these studies1-2,8,9,. Rönsch, et al.10 performed a comprehensive review of the different modeling approaches regarding the methanation process. In order to improve the kinetics of the catalytic reactions, efforts have been made and several experimental techniques have been developed through the years. In many cases, sophisticated equipment cannot be used in studying a particular kinetic system because of problems in online analysis or in temperature and pressure control11. This is the practical situation involving some industrial processes, ionic and speciation reactions in geothermal processes, and reaction-diffusion phenomena inside catalytic pellets. Thus, a modeling approach that easily provides qualitative data regarding the chemical nature of the system and, in the meantime, allows changes without the need for new

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experimental data is an asset to tasks involving modeling and optimization. De facto, this is a gap that Gibbs Energy Minimization (GEM) and the Constrained Gibbs Free Energy (CFE) methodology may fill. The theory and application of Constrained Free Energy minimisation has been earlier discussed by the authors12. The CFE method is based on earlier formulations on inclusion of additional stoichiometric restrictions presented by Alberty13,13b and Smith and Missen and co-workers14, 15. Related approaches for calculating constrained equilibria have been introduced by Keck and co-workers16 and Pope and co-workers17,18 for ideal gas mixtures. The conventional Gibbs energy minimization methods apply elemental amounts of the system components as conservation constraints in the form of a stoichiometric conservation matrix. The resulting calculation then gives a global equilibrium condition at a given temperature and pressure. However, the Gibbs energy of a multiphase system is affected by conditions due to immaterial properties, due to force or relaxation effects, and new constraints must be adjusted by the respective entities. The constrained equilibrium state is then reached as a solution to the non-linear optimization problem of finding the free energy subject to these constraints. The CFE minimization method includes such conditions and incorporates every immaterial constraint accompanied with its conjugate potential11. Furthermore, the CFE approach is fully consistent with thermodynamics. It is worth emphasizing that the introduction of reaction rate constraint into the Gibbsian thermochemical calculation, for instance, does not indicate any deduction of kinetic data from thermodynamic principles. The method just makes it possible to calculate such dynamic chemical states for which it is possible to distinguish rate-controlling reactions while the system may otherwise reach partial equilibrium19. In this work, reaction rate constraints in GEM have been applied to the water gas shift and methanation reactions to demonstrate the feasibility of CFE in the resolution of catalytic 1-dimensional reactors (1D), since most industrial processes rely on packed bed reactors. In this novel approach, water-gas-shift and methanation, both catalyzed and key reactions in carbon capture and reuse technology, were selected as study cases. Some advantages over the mechanistic-stoichiometric method were pointed out, and the preliminary results pave the way for a more complex treatment of catalytic particles, which means taking into account intra-particle diffusion phenomena and the multiple steady states, which are allowed to form due to temperature and concentration gradients inside the pellets20. 2. Methods 2.1. CFE fundamentals and heterogeneous catalysis Gibbs energy minimization requires optimization of the nonlinear G-function with linear constraints, which can be performed by the Lagrange method of undetermined multipliers. In the conventional method, the molar amounts (mass balances) appear as necessary constraints. To incorporate the additional phenomena, a method with analogous immaterial constraints is needed. Thus, the question reverts to one of the fundamental problems in computational thermodynamics, namely the convex minimization

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of the nonlinear objective function with its linear constraints12d. When temperature and pressure are assumed known, and in most industrial cases they are, Gibbs energy is naturally the appropriate function to be minimized. In terms of the chemical potentials of the constituents, the Gibbs energy of a chemical system, G, is defined in Eq. 1 subject to Eqs. 2 and 3, as presented in the scheme 1: Scheme 1. Formulation of the Gibbs Energy problem.

G= ∑ ∑ nαk μαk α

(1)

k

nαk ≥0 ∀ k

(2)

An − b = 0

(3)

where α refers to all phases in the chemical system and k to all constituents (species) in each of the phases. The variable n is the molar amount and μ is the chemical potential of each species. A refers to the stoichiometric matrix, n is a vector of molar amounts of species, and b is a vector of the total input amount of a system component (most commonly the elements). The minimization of Gibbs energy is often conducted with the Lagrange method of undetermined multipliers, Eq. 4:

L = G − πT (An − b)

(4)

where L is the Lagrange function, and π is the Lagrange multiplier vector. The physical meaning of the Lagrange multipliers is that they represent how much potential energy a component contributes to the molar Gibbs energy of a constituent or species15. Different computational tools can be applied to solve the chemical equilibrium by minimizing the non-linear constrained Gibbs free energy. The ChemSheet21 and ChemApp22 software packages were employed in our simulations, as they also enable the calculation of constrained thermodynamic equilibria. In a system controlled by reaction rates or, alternatively, extents of reaction can be included in the CFE calculation by extending the conservation matrix by a particular immaterial component connected with the advancement of a given reaction. The chemical reactions are then calculated by the minimization algorithm, and thus the respective changes in the thermodynamic properties are inherently taken into account within the thermodynamics procedure12d. For both catalyzed WGS and CO methanation, a system with three components (C, H, O) and their species is represented by Eq. 5 as the stoichiometric matrix, A. By using the dimension theorem, also known as the phase rule, the number of reactions is calculated as the number of species less the number of linearly independent entities (components)23. According to the theorem, the number of linear independent reactions, to the proposed reactive system, must be 3 as a direct consequence of the fact that matrix A is a full rank one.

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AT =


C

H

O

CO(g)

1

0

1

CO2(g)

1

0

2

H2(g)

0

2

0

H2O(g)

0

2

1

CH4(g)

1

4

0

C(s)

1

0

0

(5)

The proposed linear independent reactions to this system are described by the set of equations described in the scheme 2 (Eq. 6-8): Scheme 2. Main reactions involved in the description of the reactive system.

R1 (WGS)

CO + H2 O ⇔ CO2 + H2

(6)

R2 (CO meth.)

CO +3H2 ⇔ CH4 + H2 O

(7)

them to the system will not change the system energy or entropy12d,24. In our simulations, we assumed that the Boudouard reaction is fast enough to reach equilibrium at the proposed conditions, thus, its respective row and column were erased from the conservation entity matrix. Besides, its extent is negligible in such conditions as will be presented in the results session. Similarly, for the catalyzed WGS reaction, considering that mainly CO is undergoing catalytic interactions, CH4 and CO2 (rows 2 and 5, respectively) can also be supressed, while R2 (CO methanation) is assumed to be free to equilibrate. A similar procedure was employed with CO methanation, meaning that just the amounts of CO are constrained due to catalytic interactions, while R1 (WGS) is now free to equilibrate. The tableaux (symbol T) of the conserved matrices for both cases, after applying the simplifications described above, are presented in the scheme 3 by Eq. 10-a (WGS) and 10-b (CO methanation). Scheme 3. Reduced conservation matrices applied to WGS and CO methanation reaction. C

2H

O

𝑣1

1

0

1

1

0

1

1

0

0

1

0

0

R1

0

0

0

1

T

C

2H

O

𝑣2

1

0

1

0

0

1

1

0

0

1

0

0

1

2

0

1

0

0

0

1

T CO(g)

R3 (Boudouard)

2CO ⇔ C(s) + CO2

(8)

H2(g)

It is worth mentioning that CO2 methanation is just a linear combination of R1 and R2. Following the algorithm proposed by Blomberg and Koukkari23, an extended conservation matrix A'T is obtained (See Eq. 9). A systematic description of the algorithm is given in the supporting information. In Eq. 9, the vx entities with their respective subscripts that emerge naturally during the matrix base change process can be interpreted, for instance, as constraints related to the catalyst surface area, or simply as a set of elements in an immaterial (or virtual) phase acting as a buffer to the reactive system. The molar mass of an immaterial component is set to zero in order to avoid errors in mass balance.

H2O(g)

CO(g) H2(g) H2O(g) CH4(g) R2

𝐴′𝑇 =

C

2H

O

𝑣1

𝑣2

𝑣3

1

0

1

1

0

0

CO(g)

1

0

2

2

0

0

CO2(g)

0

1

1

0

0

0

H2(g)

0

1

0

0

0

0

H2O(g)

1

2

0

1

1

0

CH4(g)

1

0

0

0

0

1

C(s)

0

0

0

1

0

0

R1

0

0

0

0

1

0

R2

0

0

0

0

0

1

R3

(9)

The standard chemical potential of virtual phases (reactions) is likewise set to zero at all temperatures, so adding

(10-a)

(10-b)

In addition to the description of the chemical system, the thermodynamic data, as well as the input composition and conditions (for example, temperature and pressure), need to be defined before solving the equilibrium. In this study, the thermodynamic properties of species were obtained from HSC software25. Alternatively, density and functional theory calculations (DFT) can be used to provide thermodynamic properties26. In order to model catalytic reactors, it is often necessary to include the effects of internal or external diffusion resistances. Mathematically, the simplest case is the one of a first-order reaction occurring isothermally in pellet geometry such as a sphere, infinite cylinder, or slab particle, assuming one-dimensional dispersion model. In our model, the diffusion interactions with the species are directly connected to Eq. 10A (row 1, column 4) and Eq. 10B (row 4, column 4). For practical purposes, they can be understood



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as a function between the immaterial species and their respective reactions, vx (Rx). From the result, one can establish a connection between reaction advancements and diffusion resistances using Eqs. 11-14. Furthermore, if Young and Finlayson's27 criterion is satisfied, the diffusion resistance arising from the interface between the bulk and the pellet surface can also be neglected; this is often the condition achieved with a plug flow reactor with high interstitial velocities. When the adsorption phenomenon is noticeable in catalytic reactions, the mathematical treatment proposed by Hayes, et al.28, which is briefly presented here, may be adopted. These authors considered a base case involving dimensionless parameters, intra-particle diffusion, and LHHW adsorption as a fast approach to compute effectiveness factors in catalytic reactions. Assuming that a first-order reaction can be measured through a key species A in the bulk, and the axial diffusion is negligible, the set of equations presented in scheme 4 can be written: Scheme 4. Set of equations describing the mass transfer resistance in heterogeneous catalysts bind to CFE routines.

𝑣𝑥 (𝑅𝑥 ) ≅ 𝑑𝜉 = −

𝑑𝛹 =

𝑑𝐶𝐴𝑠 𝑣

𝑑𝐶𝐴𝑠 𝑑𝐶𝐴𝑏

(11)

(12)

𝑑𝛹 2 𝛹𝑝 2 −𝜙 =0 (1 + 𝜏𝛹)𝑚 𝑑𝜆2

(13)

𝑝

𝜙 2 = 𝑅2

𝑘𝑝′′ 𝜌𝑐𝑎𝑡 𝑆𝑎 𝐶𝐴𝑠 𝐷𝑒

(14)

where 𝑑𝛹 is the dimensionless concentration taking into account the concentration of an arbitrary species A, at the surface of a spherical catalyst d𝐶As with the concentration at the bulk d𝐶Ab ; 𝜆 is the dimensionless particle radius; τ is the dimensionless adsorption coefficient, and 𝜙 is the Thiele Modulus, which in turns incorporates: 𝑅 the characteristic particle diameter, 𝑘 ′′ the intrinsic kinetics, Sa the surface area, ρcat the catalyst density, and the effective diffusion 𝐷𝑒 . In Eq. 13, m is the magnitude order of the adsorption phenomena and p the assumed order to an arbitrary reaction. Both of these can be estimated using the algorithm proposed by Hayes, et al.28. Thus, the Gibbsian matrix incorporates diffusion resistance as a modified constraint. It must be emphasized that, in our simulations, such diffusive effects were addressed to the virtual phases vx. In the WGS case, the adsorption phenomena were lumped into kinetic parameters as proposed by Keiski et al.11. The methanation case was assumed to be in a pseudo-

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heterogeneous condition, meaning that both intra- and inter-particle diffusion were considered negligible under the experimental conditions. 2.2. Chemical systems studied 2.2.1. Water gas shift Eq. 15 describes the chemistry ongoing in a WGS reaction catalysed by iron oxides. The WGS reaction involves CO, H2O, CO2 and H2 (N2 used as an inert gas). A classical numeric approach requires the activation energy of reaction 15 (Scheme 5), the equilibrium constant and the pre-exponential factor values to be solved. Scheme 5. The chemistry involved with the WGS reaction catalysed by iron oxides. Fe2 O3

CO+H2 O ⇔

CO2 +H2

∆H(298 K) = -41 kJ.mol-1

(15)

∆S(298 K) = -42 J.K-1 . mol-1 The experimental survey performed by Keiski et al.11 was primarily chosen due to the setup and conditions employed, which was an attempt to mimic an industrial process, meaning non-isothermal continuous reactors with typical feed rates, as compiled in Table 1. Due to the large amount of data available in this work, including catalyst performance and stability analysis, one can draw important conclusions concerning the macrokinetics and stability of the models predicting bulk concentrations from catalytic particles. The mathematical description of the catalytic phenomena was performed using the classical and elegant theory on heterogeneous catalysis4, 29. The base comparison data used in this modeling was obtained using a ferrochrome catalyst that contained typically 89% Fe2O3 and 9% Cr2O3, with the rest being Al2O3. Further details are given in the original publication. In our simulations, the kinetic models from Keiski et al.11 and Salmi and Wärnå3 were solved using the Matlab ODE45 algorithm, and CFE just used a discretized form of Eq. 15 to increment the constraints settled for CO specie in Matrix A. The approach described in scheme 6 was employed in both of them: Scheme 6. Mass balance equation applied to the catalysed WGS reaction under mass transfer resistance.

𝑑𝐶𝐶𝑂 𝜂𝑒 𝜌𝑏𝑢𝑙𝑘 (−𝑘𝐶𝐶𝑂 ) ⁄𝑣 = 𝑑𝑧

(16)

𝑘 = 𝜌𝑐𝑎𝑡 𝑘 ′ = 1.1 × 108 𝑒𝑥𝑝(−11455 /𝑇)[𝑠 −1 ]

(17)


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Table 1. Typical industrial feeding rates and mol fractions in water gas shift reactor (adapted from Keiski et al.11 and Salmi and Wärnå3). Setup 1

Setup 2

Setup 3

mol flow

Fraction

mol flow

Fraction

mol flow

Fraction

[mol/s]

%

[mol/s]

%

[mol/s]

%

CO

3.1E-04

9.4

3.1E-04

7.4

3.1E-04

7.4

H2O

1.0E-03

31.1

1.9E-03

45.3

1.9E-03

45.2

CO2

2.1E-04

6.3

2.0E-04

5.0

2.0E-04

5.0

H2

1.2E-03

37.4

1.6E-03

39.4

1.2E-03

29.6

N2

5.2E-04

15.7

1.2E-04

2.9

5.3E-04

12.9

Total Dry-gas

-

-

2.2E-03

54.7

2.3E-03

54.8

Total gas

3.0E-03

-

4.1E-03

-

4.1E-03

-

*Inlet temperature and pressure: 600-675 [K] and 101-133 [kPa]. Reactor dimensions: volume of 93.66 cm³ and length of 16 cm.

where 𝜂𝑒 is the effectiveness factor assumed for a first-order reaction, 𝜙 is the respective Thiele Modulus solved for a cylindrical pellet with a characteristic dimension of 3.9 mm and 𝜌𝑐𝑎𝑡 3.74 g.cm-3. 𝜈 is the interstitial velocity of the fluid in cm.s-1, and 𝜌𝑏𝑢𝑙𝑘 is the catalyst bulk density in g.cm-3 and z is the length coordinate. Further details about the catalyst and experimental setup can be found in the original manuscript. Eqs. 18 and 19 are employed with first order kinetics, showing the inter-relationship of Thiele modulus, the respective effectiveness factor and the diffusion resistance (De).

𝑛𝑒 =

1 1 1 [ − ] 𝜙 𝑡𝑎𝑛ℎ(𝜙) 𝜙

(18)

0.4𝑘 2𝐷𝑒

(19)

𝜙2 =

2.2.2. Methanation over Ni-alumina catalyst The Gibbs Energy minimization was performed considering the kinetic survey by Hayes30. This reaction involves the following species: CO, H2, CH4, H2O. An inert gas is often employed (Nitrogen or Argon). The set of equations and required parameters is described in scheme 7 by Eqs. 21-23: Scheme 7. Mass balance equation applied to the catalysed CO methanation reaction. Ni/Pt

CO + 3H2 ⇔

(20)

A relatively complex formulation was used by Keiski et al.11 and Salmi and Wärnå3. These authors employed modified power law equations including a reversible factor. The CFE

r = k × (pCO )1.27 × (pH2 ) k = 8.8 × exp (

(21)

CH4 + H2 O −0.87

As the pellets were assumed to be in isothermal conditions, the diffusion taking place within and on the particles can be considered constant throughout the bed. The effective diffusion was calculated using the Knudsen (DK) and Damköhler (DA) numbers for the gaseous species. The adjusted parameter of tortuosity is the only uncertainty about the catalyst, with values ranging from 2 to 9. In these simulations, we assumed a base value of 2.25. The porosity (with a value of 0.55) was taken from Keiski et al.11. manuscript. 1 𝑝 1 1 = ( − ) 𝐷𝑒 𝜏 𝐷𝑘 𝐷𝐴

model inherently takes into account the possible equilibrium state12d, but the effect of reverse reactions was not otherwise considered.

−9270 ) T

× (pH2O )

−0.13

[mol. g −1 . s −1 ]

(22) (23)

Briefly, these authors used both commercial and synthetized catalysts based on Ni-Al2O3, promoted by several metals like ruthenium and platinum. In order to avoid diffusional effects, the particles were crushed to sizes less than 0.7 mm, and the experiments were performed at temperatures lower than 598 K. In addition, the Mears criterion was checked for inter-phase diffusion, thus, under the experimental conditions applied, the kinetic data collected were assumed to be intrinsic to the catalyst. In our simulation, it was performed using their Nickel-Alumina-Platinum (9% Ni and 1.5% Pt) catalyst dataset, which, according to the authors, has stable activity for more than 60 hours. The set of conditions employed by these authors is summarized in Table 2. According to the authors, the power law kinetics proposed incorporates the adsorption phenomenon and the intrinsic kinetics. Based on their ex-



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perimental findings, the authors included a factor to account for the influence of the partial pressures of H2O and stated order zero on the effect of partial pressures of CH4 and CO2. Table 2. Experimental conditions for CO methanation over Ni-Pt-alumina (adapted from Hayes30). Conditions

Units

Values

Molar flow rate

mmol/s

0.86-1.27

Volumetric flow rate

cm3/s

410-600

Mass of catalyst

g

0.25-0.50

Temperature

K

533-595

Pressure

kPa

100-131

CO

%

0.22-6.0

H2

%

9.30-15.0

H2O

%

0.0-2.6

N2

%

85-89

Bed length

mm

19

Cross sec. area

mm²

9.5

Reactor volume

mm³

473

(STP)

Feeding

Figure 1. WGS equilibrium composition as a function of temperature (setup 1); (y) in logarithmic scale.

A simulated profile from both CFE and the kinetic model (KM) was plotted against the experimental results and is presented in Figure 2.

Reactor

The water gas shift reaction was neglected in their catalytic reaction due to the absence of detected CO2 at temperatures lower than 573 K. Then, the mass balance proposed by the authors was coupled to the reaction kinetics and matrix A’ was updated iteratively with the results calculated from Eq. 24:

𝑑𝐶𝐶𝑂 = 𝑟 × 𝜌𝑏𝑢𝑙𝑘 × 𝑣 −1 𝑑𝑧

(24)

where 𝜌𝑏𝑢𝑙𝑘 is the density of the bulk catalyst in g.mm-3, and 𝜈 is the flow velocity in mm.s-1, z is the length coordinate. 3. Results 3.1. Water gas shift The equilibrium composition of a gaseous mixture, under the same conditions employed in our survey, was previously performed in order to identify significant species in the range of temperatures investigated. Figure 1 presents the chemical equilibrium for WGS in a range of 300 to 1200 K. It reveals that considerable methane formation is expected at temperatures lower than approximately 750 K, then becomes negligible after 1200 K. In addition, good yields of H2 can be obtained at a temperature of approximately 800 K.

Figure 2. Comparisons between experimental data, CFE, and KM; (+) data taken from Keiski et al.11; (x) Salmi and Wärnå3; CH4-WH means simulations with methane phase and CH4WT without. Inlet conditions described in the setup 1 and 2 (See, Table 1).

As one can see from Figure 2, minor deviations can be found when comparing KM and CFE outputs. Both models were able to predict the carbon monoxide conversions. At this point we can speculate that KM codes tend to slightly over-predict conversions compared to CFE, especially at lower temperatures. These over-prediction patterns were also observed by Keiski et al.11 in their simulations, even when employing the most complex model (i.e. n=0.8, m=0.55, p=-0.18 and q=0; the respective exponents to CO, H2, CH4, and H2O in their power law). Figure 3 presents the results obtained with this routine along with calculations of the Gibbs energy of the catalytic system.


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Figure 3. WGS simultaneous concentration profiles obtained with CFE in the steady state condition (setup 1, 675 K and 1 bar): (A) Pellet characteristic dimension of 1.3 mm; (B) Pellet increased by a factor of 10.

It is worth highlighting that minor species like CH2O, C2H4, and C2H6 with respective molar quantities around 1x10-13, 1x10-14, and 1x10-11, and methane with a non-negligible (~1x10-4), were detected in our calculations. However, solid carbons were found to have practically zero concentration (< 1x10-40 mol/cm3). Thus, one can argue, without taking into consideration nucleation and particle-growing chemistry, that such a finding provides evidence that carbon deposit formations do not represent a real issue in these equilibrium conditions, thus endorsing our assumptions with the Boudouard reaction. However, the discussion about this topic related to carbon particles growing on catalytic surfaces is beyond the scope of our report. The model was also used as a tool to predict the qualitative aspects involved in the catalyst parameter changing (i.e. increasing the particle characteristic dimension by a factor of 10). As expected, CO conversion rates are faster with smaller particles, as also indicated by the respectively steeper descent of Gibbs energy. These figures also reveal that more methane is formed with smaller pellets, and water and nitrogen concentrations have minor differences. As can be seen in Figure 3, the Gibbs energy per pellet size does not change significantly, and at some point, it will reach the same plateau. One may propose that such a gap in Gibbs energy, considering the bed length investigated might be attributed to a natural withdrawal of the free energy by the adsorption followed by the pellet augmentation. The simulated results demonstrated in Figure 4 reveals that KM retains the same trend when the catalyst particle diameter is increased, while CFE exhibits minor differences that become more pronounced with the characteristic dimension increasing, which is consistent with mass transfer resistance. The influence of total pressure upon the reactive system was also investigated by changing the partial pressure of the inert gas, the results is presented in Figure 5. Simulations show that overall pressure has a significant influence on carbon monoxide conversion. It is reasonable to assume that CFE is sensitive to falloff effects, attributed to pressures changes in a reactive system, enhanced by the presence of an inert gas.

Figure 4. Comparisons between CFE (dotted lines) and KM (solid lines), performed at different pellet characteristic dimensions, increased by a factor of 10.

Figure 5. CO conversion as a function of bed length and inert gas partial pressure change (profiles recorded at total pressure of 203 kPa).

3.2. Methanation over Ni-Alumina (9% Ni and 1.5% Pt) As stated above, equilibrium calculations are a worthwhile step to perform before simulations. In our CFE simulations, the entire thermodynamic databank, comprising 62 species, was considered. Then, Figure 6 shows that the amount of NH3 formed cannot be neglected within a range



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of 300 to 700 K, even though, as will be shown further, its formation does not greatly affect the overall conversion. In any case, this evidence points out that ammonia synthesis may compete with the hydrogen available for the methanation reaction under certain conditions.

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studied, and a correlation of 0.97 was achieved. Lack of experimental data of CO2 prevented a proper comparison with this by-product. Controversially, Hayes reported that Ni-Al2O3 promoted by platinum (9% Ni and 1.5% Pt) had a conversion to carbon dioxide of up to 5% at 595 K. In contrast, our simulations were able to detect just 0.02% (131 kPa and 595 K). A possible explanation is that their measured values should have incorporated noise due the catalyst aging or, due the fact that the power law considered did not account for CO2 partial pressure influence. Within a universe of 62 species, excluding the reactants and products, only NH3, CO2, and C2H6 presented considerable mol fraction in the proposed conditions. Except for the ammonia presence, these predicted results are in good agreement with the literature2, 31, 32. Figure 8 reveals the simulated results for both main (A) and secondary (B) species.

Figure 6. Equilibrium composition of a gaseous mixture of 3.1% CO, 9.6% H2, and 87.6% N2 at 131 kPa (y-axis on a logarithmic scale).

The water and CH4 are in the same molar proportion and virtually do not have any significant change in the range studied. The amount of CO2 found corroborates the assumption made by some authors2, 30, 31, namely neglecting its influence at lower temperatures. From this preliminary analysis and from a thermodynamic point of view, an ideal catalyst should promote CH4 conversions and be selective enough to avoid CO2 or even NH3 formation. Temperatures higher than 600 K have a negative influence in the conversion and higher pressures (results not shown here) do not improve the yield of methanation. However, pressures higher than 101 kPa and up to 500 kPa decrease the yield of CO2 under equilibrium conditions. The validations of simulated and experimental points are presented in Figure 7.

Figure 7. Model validation; CFE compared to experimental data.

Figure 8. CFE molar fraction profiles with CO methanation over Ni-Pt.

From the secondary species identified (B), ammonia is the only one not covered in the literature. The reason could be the low hydrogen and nitrogen partial pressures adopted by many authors. However, in this study, nitrogen accounts for more than 85% of the feeding gas. Figure 9 shows the ammonia influence over the reactive system; the results were properly compared with some experimental points. In any case, the extent of its formation can be considered negligible and, taking into account the gas-drying procedure employed before the analysis by some authors30, 32, its presence as a by-product most probably remained disguised. It is stated in the literature that the water concentration decreases the yield of methanation. Hayes30 reported that 2.6% v/v of water in the gas feeding leads to a decrease of 4% in conversion to methane. Remarkably, our simulated results with CFE agreed with their experimental results. Figure 10 shows a decrease of 5% in the conversions to methane when water, at the same proportions, is added to the gas feeding.

A good fit was obtained, and it must be emphasized that the fit was obtained throughout the range of temperatures


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Figure 9. Ammonia influence on CO methanation over Ni-Pt. (+, x) Refers to the experimental points recorded at the same temperature and pressure conditions.

Figure 10. Influence of water feeding in CO methanation over Ni-Pt (simulated results).

4. Discussion A 1D-reactor model of WGS and CO methanation based on the constrained thermodynamic equilibrium (CFE technique) provided good agreement with experimental results given in the literature. Kinetic mechanisms were compared with CFE in the WGS case. Based on the results shown in section 3, and from a purely mathematical and descriptive perspective, both the KM and CFE approaches were able to describe the steady state conditions accurately. However, despite mechanistic codes being powerful tools, they cannot accommodate the dynamic interactions arising from a chemical system without a proper reformulation of the ODE set. This can be a considerable drawback in industrial applications and often leads to a constant review of the set of ODEs adopted in such scripts, which is, indeed, an experimental and time-consuming task. A reasonable explanation of the relative differences spotted with CFE (e.g. Figs. 2 and 4) is that, during the minimization procedure, the molar amounts of reactant and product species of the constrained reaction will adjust according to the incremental changes as obtained from the reaction rate equation, while the rest of the system will reach

the constrained equilibrium12d. Thus, the composition with CFE is dynamic while KMs are rigid, meaning that once the routine starts, a purely mathematical solution is obtained. De facto, in our simulations with WGS, a considerable amount of CH4 was detected, which is in accordance with several authors’ findings33, 34, 35, 36, 37 and with our already discussed equilibrium calculations. If, on one hand, the KM set of ODEs did not include the methane formation equation, so that its influence on the system could not be evaluated, on the other hand, when using CFE, a very useful feature is available, the detected stable species, CH4 in this case, which can easily be put in a “dormant state” and calculations performed without its influence. In the absence of methane species, our simulated data are virtually the same as those obtained from KM (see Fig. 2). Thermodynamic calculation bears the major advantage of giving all the measurable thermodynamic quantities, including enthalpies, without additional elaboration in the kinetic model. At this point, the main advantages of using CFE, instead of other codes, is linked to the fact that all the species, at least the ones that present meaningful chemical potential, are automatically incorporated at every iterative step performed inside Gibbs energy minimization routines. From the research point of view, these data regarding partial equilibrium or stable phases are a valuable extract, promptly produced by CFE. Moreover, the simulated results presented in Figure 3 show consistency with mass transfer phenomena, meaning that pellet augmentation should lead to a decrease in the CO conversions in the WGS reaction as observed, which was properly addressed to the mass balance equation by means of diffusional coefficients. At this point, a new hypothesis can be formulated, considering other catalyst parameters constant and the mass exchange between the bulk, film, and catalyst inner surfaces can be artificially modified, a possible meta-stable phase will then be allowed to form due the gradients20, 28. Thus, the first derivative of Gibbs energy may present different slopes, as observed. However, this hypothesis must be validated experimentally, utilizing, for example, a catalyst that presents strong mass transfer resistance. Similarly, the results presented for CO methanation show that other species can be formed, like the predicted ammonia due to nitrogen gas feeding, which, in turn, is often considered to be an inert gas. In Figure 5, another interesting qualitative result is demonstrated regarding CFE simulations and pressure changes. The observed differences between KM and CFE are, most probably, due to the falloff effects. KM without a proper adjustment in the kinetic parameter was unable to reproduce the same trends captured by CFE (i.e. simulated results whether with or without N2 in the feeding gas). However, the discussion about this topic is beyond the scope of this report. Of course, these simulated results themselves cannot be used to prove this assumption, namely that CFE is sensitive to falloff effects38, 39, but they are not in conflict with such a theory. The CFE approach plus the matrix entity extending procedure showed an explicit connection between the virtual phases and catalyst pellets. These constraints, which naturally emerge from the extended conservation matrix base



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change process (see supporting information), can be interpreted, for instance, in both reactions, as a sink to carbon monoxide due to the presence of an active metallic surface. 5. Conclusions This work showed that catalytic reactions operating in a 1D reactor can be successfully simulated using the CFE technique. The approach presented, namely incorporating the mass transfer resistance to the extent of reactions, which, in turn, was used as a constraint to chemical equilibrium, seems to be a viable strategy for use in reactor design. From a mathematical and descriptive point of view, both KM and CFE produce similar results, as shown in WGS simulations. However, CFE calculations employ chemical potentials that make it more consistent with physical-chemical phenomena, as they are also versatile, meaning that they enable stable phases to be iteratively added or removed from the chemical system, whenever a stable specie is allowed or not to be formed. CFE has also been shown to be consistent with the decrease in conversions due the increase in diffusion resistance. These preliminary results point out that CFE has the potential to describe the complex chemistry ongoing inside catalyst particles. The strict connection of the extended matrix conservation, including it entities and virtual phases, and the Gibbs energy minimization technique, enables the use of a powerful tool to screen several industrial process with complex chemistry, which the mechanism is not fully understood, or whenever an experimental setup is unfeasible to assemble, as in geothermal chemistry for instance, due the high pressures, temperatures and ionic interactions between the species. From this perspective, one of the major advantages of using CFE toward KM is that its capacity to predict stable phases, like CH4, was predicted in a WGS reaction, even without using its stoichiometric mass balance, that is, setting a constraint based on R2 (Eq. 7). The primary outcomes from the procedure extending stoichiometric matrices are that virtual phases naturally emerge from the iterative process, hence providing qualitative data regarding the reactive system, which is indeed such a valuable tool in the assessment of chemical processes, optimization, and reactor design. 6. Author information *Corresponding author: Pertti Koukkari E-mail: [email protected] Address: Biologinkuja 7, Espoo, FI 02044, PO BOX 1000 Contact: +358 20 722 6366

Funding sources The first author is grateful by scholarship grant nr. 232259/2014-3 CNPq-Brazil. The authors also wish to tank Academy of Finland grant nr. 303 453 Author’s contribution The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. 7. Abbreviations 1D: 1 dimensional reactors, typically plug flow conditions and packed beds. CFE: Constrained Gibbs free energy technique.

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DFT: Density Functional Theory. EQ: Chemical Equilibrium. GEM: Gibbs Energy Minimization. KM: mechanistic-stoichiometric kinetic models. LHHW: Langmuir-Hinshelwood-Hougen-Watson. ODE: Ordinary Differential Equations. WGS: water-gas-shift reaction. STP: Standard temperature and pressure conditions. Symbols μαk : The chemical potential of the species k at the respective phase 𝛼; n: The total molar amount of the species; 𝜋: Lagrange undefined multiplier; b: Vector of the total input amount of a system component; A: Stoichiometric matrix; 𝜗 : Reaction matrix; 𝐴′𝑇 : The modified and transposed conservation matrix; 𝜉: Extent of the reaction; 𝜆 : Dimensionless particle radius; 𝜏 : Dimensionless adsorption coefficient; 𝜙: Thiele Modulus; 𝑅: Characteristic particle dimension (m); 𝑣𝑥 : Virtual phases or constraints, acting like a sink or source of species during the reaction; 𝑘 ′′ : Intrinsic kinetics (m3.kg-1.s-1); 𝑆𝑎 : Catalyst surface area (BET) (m2); 𝜌𝑐𝑎𝑡 : Catalyst density (kg.m3); 𝐷𝑒 : Effective diffusion (m2.s-1); 8. Supporting information The Supporting Information is available free of charge on the ACS Publications website. The conservation matrix augmentation taken step by step following the algorithm proposed by Blomberg and Koukkari23 (file type, PDF) 9. References (1) Swickrath, M. J.; Anderson, M. The Development of Models for Carbon Dioxide Reduction Technologies for Spacecraft Air Revitalization. In 42nd International Conference on Environmental Systems; 15-19 Jul. 2012; San Diego, CA; United States (2) Kopyscinski, J.; Schildhauer, T. J.; Vogel, F.; Biollaz, S. M. A.; Wokaun, A. Applying Spatially Resolved Concentration and Temperature Measurements in a Catalytic Plate Reactor for the Kinetic Study of CO Methanation. J. Catal. 2010, 271, 262-279. (3) Salmi, T.; Wärmå, J. Modelling of Catalytic Packedbed Reactors—comparison of Different Diffusion Models. Comput. Chem. Eng. 1991, 15, 715-727. (4) Aris, R. The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press: Oxford UK, 1975. (5) Er-rbib, H.; Bouallou, C. Modelling and Simulation of Methanation Catalytic Reactor for Renewable Electricity Storage. Chemical Engineering Transactions, 35 , 541-546, 2013. (6) Sehested, J.; Dahl, S.; Jacobsen, J.; Rostrup-Nielsen, J. R. Methanation of CO Over Nickel: Mechanism and Kinetics at High H2/CO Ratios. J. Phys. Chem. B 2005, 109, 2432-2438.


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(7) Rostrup-Nielsen, J. R.; Pedersen, K. Sulfur Poisoning of Boudouard and Methanation Reactions on Nickel Catalysts. J. Catal. 1979, 59, 395-404. (8) Xu, J.; Froment, G. F. Methane Steam Reforming, Methanation and Water-gas-shift: I. Intrinsic Kinetics. AIChE J. 1989, 35, 88-96. (9) Kangas, P.; Vázquez, F. V.; Savolainen, J.; Pajarre, R.; Koukkari, P. Thermodynamics Modelling of the Mehtanation Process with Affinity Constraints. Fuel 2017, 197, 217-225. (10) Rönsch, S.; Schneider, J.; Matthischke, S.; Schlüter, M.; Götz, M.; Lefebvre, J.; Prabhakaran, P.; Bajohr, S. Review on Methanation – From Fundamentals to Current Projects. Fuel 2016, 166, 276-296. (11) Keiski, R. L.; Salmi, T.; Pohjola, V. J. Development and Verification of a Simulation Model for a Non-isothermal Water-gas-shift Reactor. The Chemical Engineering Journal 1992, 48, 17-29. (12) (a) Kangas, P. Modelling the Super-equilibria in Thermal Biomass Conversion - Applications and Limitations of the Constrained Free Energy Method. Dissertation Åbo Akademi University, Turku, Finland, 2015; (b) Kangas, P.; Hannula, I.; Koukkari, P.; Hupa, M. Modelling Superequilibrium in Biomass Gasification with the Constrained Gibbs Energy Method. Fuel 2014, 129, 86-94; (c) Koukkari, P.; Pajarre, R. Introducing Mechanistic Kinetics to the Lagrangian Gibbs Energy Calculation. Comput. Chem. Eng. 2006, 30, 11891196; (d) Koukkari, P.; Pajarre, R.; Blomberg, P. Reaction Rates as Virtual Constraints in Gibbs Energy Minimization. Pure Appl. Chem. 2011, 83, 1243-1254; (e) Pajarre, R. Modelling of Chemical Processes and Materials by Free Energy Minimization. Additional Constraints and Work Terms. Aalto University, Espoo, Finland, 2016. (13) (a) Alberty, R. A. Thermodynamics of the Formation of Benzene Series Polycyclic Aromatic Hydrocarbons in a Benzene Flame. J. Phys. Chem. 1989, 93, 3299-3304; (b) Alberty, R. A. Use of Legendre Transforms in Chemical Thermodynamics (IUPAC Technical Report). Pure Appl. Chem. 2001, 73, 1349-1380. (14) Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms. Krieger publishing company: Malabar - Florida US, 1991. (15) Cheluget, E. L.; Missen, R. W.; Smith, W. R. Computer Calculation of Ionic Equilibria Using Species-or reaction-related Thermodynamic Data. J. Phys. Chem. 1987, 91, 2428-2432. (16) Keck, J. C.; Beretta, G. P.; Ghoniem, A.; Hatsopoulos, G. In Rate‐Controlled Constrained‐Equilibrium Theory of Chemical Reactions, AIP Conf. Proc., AIP: 2008; 329-334. (17) Pope, S. B. Gibbs Function Continuation for the Stable Computation of Chemical Equilibrium. Comb. Flame 2004, 139, 222-226. (18) Hiremath, V.; Pope, S. B. A Study of the Ratecontrolled Constrained-equilibrium Dimension Reduction Method and its Different Implementations. Combust. Theory Modell. 2013, 17, 260-293. (19) Koukkari, P. Introduction to Constrained Gibbs Energy Methods in Process and Materials Research. VTT Technical Research Centre Of Finland LTD: Espoo, 2014. (20) Weisz, P. B.; Hicks, J. S. The Behaviour Of Porous Catalyst Particles In View Of Internal Mass And Heat Diffusion Effects. Chem.l Eng. Sci. 1962, 17, 265-275.

(21) Koukkari, P.; Penttilä, K.; Hack, K.; Petersen, S., CHEMSHEET - An Efficient Worksheet Tool for Thermodynamic Process Simulation. 2005, 323-330. (22) Petersen, S.; Hack, K. The Thermochemistry Library Chemapp And Its Applications. Int. J. Mater. Res. 2007, 98, 935-945. (23) Blomberg, P. B. A.; Koukkari, P. S. A Systematic Method To Create Reaction Constraints For Stoichiometric Matrices. Comput. Chem. Eng. 2011, 35, 1238-1250. (24) Pajarre, R.; Koukkari, P.; Kangas, P. Constrained And Extended Free Energy Minimisation For Modelling Of Processes And Materials. Chem. Eng. Sci. 2016, 146, 244-258. (25) Roine, A. HSC Chemistry 7.0 User’s Guide. Report 09006-ORC-J; Outotec Research Oy: Finland, 2009 (26) West, R. H.; Celnik, M. S.; Inderwildi, O. R.; Kraft, M.; Beran, G. J.; Green, W. H. Toward A Comprehensive Model Of The Synthesis Of Tio2 Particles From Ticl4. Ind. Eng. Chem. Res. 2007, 46, 6147-6156. (27) Young, L. C.; Finlayson, B. A. Axial Dispersion In Nonisothermal Packed Bed Chemical Reactors. Ind. Eng. Chem. Fundam. 1973, 12, 412-422. (28) Hayes, R. E.; Mok, P. K.; Mmbaga, J.; Votsmeier, M. A Fast Approximation Method For Computing Effectiveness Factors With Non-Linear Kinetics. Chem. Eng. Sci. 2007, 62, 2209-2215. (29) Rawlings, J. B.; Ekerdt, J. G. Chemical Reactor Analysis And Design Fundamentals. Nob Hill Pub, LLC: Madison-WI US, 2002. (30) Hayes, R. A Study Of The Nickel-Catalyzed Methanation Reaction. J. Catal. 1985, 92, 312-326. (31) Kopyscinsky, J. Production Of Synthetic Natural Gas In A Fluidized Bed Reactor. Ph.D. Dissertation, Swiss Federal Institute of Technology, Zurich, Switzerland, 2010. (32) Klose, J.; Baerns, M. Kinetics Of The Methanation Of Carbon Monoxide On An Alumina-Supported Nickel Catalyst. J. Catal. 1984, 85, 105-116. (33) Storch, H. H.; Pinkel, I. I. Preparation of an Active Cobalt-copper Catalyst for The Water-Gas Shift Reaction. Ind. Eng. Chem. 1937, 29, 715-715. (34) Gottschalk, F. M.; Copperthwaite, R. G.; Van Der Riet, M.; Hutchings, G. J. Cobalt/Manganese Oxide Water Gas Shift Catalysts. Appl. Catal. 1988, 38, 103-108. (35) Mellor, J. R.; Copperthwaite, R. G.; Coville, N. J. The Selective Influence Of Sulfur On The Performance Of Novel Cobalt-Based Water-Gas Shift Catalysts. Appl. Catal. A: General 1997, 164, 69-79. (36) Park, J. N.; Kim, J. H.; Lee, H. I. A Study on the SulfurResistant Catalysts for Water Gas Shift Reaction IV. Modification of CoMo/γ-Al2O3 Catalyst with K. Bull. Korean Chem. Soc. 2000, 21, 1239-1244. (37) Gawade, P.; Mirkelamoglu, B.; Tan, B.; Ozkan, U. S. Cr-Free Fe-Based Water-Gas Shift Catalysts Prepared Through Propylene Oxide-Assisted Sol–Gel Technique. J. Mol. Catal. AChem. 2010, 321, 61-70. (38) Troe, J. Predictive Possibilities Of Unimolecular Rate Theory. J. Phys. Chem., 1979, 83. (39) (a) Gilbert, R. G.; Smith, S. C. Theory Of Unimolecular And Recombination Reactions. Publishers' Business Services [distributor]: 1990; (b) Zhang, P.; Law, C. K. A Fitting Formula For The Falloff Curves Of Unimolecular Reactions. Int. J. Chem. Kinet. 2009, 41, 727-734.



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Figure Captions Figure 1. WGS equilibrium composition as a function of temperature (setup 1); (y) in logarithmic scale. Figure 2. Comparisons between experimental data, CFE, and KM; (+) data taken from Keiski et al. 11; (x) Salmi and Wärnå3; CH4WH means simulations with methane phase and CH4-WT without. Inlet conditions described in the setup 1 and 2 (See, Table 1). Figure 3. WGS simultaneous concentration profiles obtained with CFE in the steady state condition (setup 1, 675 K and 1 bar): (A) Pellet characteristic dimension of 1.3 mm; (B) Pellet increased by a factor of 10. Figure 4. Comparisons between CFE (dotted lines) and KM (solid lines), performed at different pellet characteristic dimensions, increased by a factor of 10. Figure 5. CO conversion as a function of bed length and inert gas partial pressure change (profiles recorded at total pressure of 203 kPa). Figure 6. Equilibrium composition of a gaseous mixture of 3.1% CO, 9.6% H2, and 87.6% N2 at 131 kPa (y-axis on a logarithmic scale). Figure 7. Model validation; CFE compared to experimental data. Figure 8. CFE molar fraction profiles with CO methanation over Ni-Pt. Figure 9. Ammonia influence on CO methanation over Ni-Pt. (+, x) Refers to the experimental points recorded at the same temperature and pressure conditions. Figure 10. Influence of water feeding in CO methanation over Ni-Pt (simulated results).


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