Simulation of Catalytic Reactions in Open-Cell Foam Structures

Received: October 9, 2020 Revised: Accepted:

ORIG I NAL AR TI CLE Journal Section

Simulation of catalytic reactions in open-cell foam structures

Sebastian Mühlbauer1 | Severin Strobl1 | Matthew Coleman2* | Thorsten Pöschel1

1Institute for Multiscale Simulation, Friedrich-Alexander-Universität We describe a technique for particle-based simulations Erlangen-Nürnberg, Cauerstraße 3, 91058 of heterogeneous catalysis in open-cell foam structures, Erlangen, Germany which is based on isotropic Stochastic Rotation Dynamics 2Department of Chemical and Biomolecular Engineering, University of Notre Dame, 182 (iSRD) together with Constructive Solid Geometry (CSG). Fitzpatrick Hall, Notre Dame, IN The approach is validated by means of experimental re- 46556-5637 sults for the low temperature water-gas shift reaction in Correspondence an open-cell foam structure modeled as inverse sphere Sebastian Mühlbauer, Institute for Multiscale Simulation, packing. Considering the relation between Sherwood and Friedrich-Alexander-Universität Reynolds number, we ﬁnd two distinct regimes meeting ap- Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany proximately at the strut size Reynolds number 10. For typi- Email: [email protected] cal parameters from the literature, we ﬁnd that the catalyst

Present address density in the washcoat can be reduced considerably with- * Institute for Multiscale Simulation, out a notable loss of conversion eﬃciency. We vary the Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 porosity to determine optimum open-cell foam structures, Erlangen, Germany which combine low ﬂow resistance with high conversion ef-

Funding information ﬁciency and ﬁnd large porosity values to be favorable not DFG, SFB, ZISC, FPN, Haus Grant/Award only in the mass transfer limited regime but also in the in- Number: 123456 arXiv:2010.03904v1 [cond-mat.soft] 8 Oct 2020 termediate regime.

KEYWORDS heterogeneous catalysis, particle based ﬂuid mechanics, isotropic Stochastic Rotation Dynamics (iSRD), open-cell foam structure, constructive solid geometry

;: © Wiley Periodicals, Inc. 1 2 Sebastian Mühlbauer et al.

1 | INTRODUCTION ously not adequate for the simulation of reactive flows. Therefore, here we use semi-periodic boundary condi- Open-cell foam structures are extremely promising sub- tions [13] which allow for a discontinuity of the concen- strates for heterogeneous catalysis [1]. As a result of tration field, while the other flow fields, namely density, their high porosity, specific surface, and tortuosity, these temperature, and flow velocity, are periodic. The flow is structures provide excellent mass transport properties driven by an external acceleration. at moderate pressure drop [2]. Therefore, the performance of open-cell foam structure for catalytic reactions was subject of several studies concerning both regular [3, For the verification of our simulation method we 4] and irregular structures [5,2]. In these studies, the have chosen the low temperature water-gas shift as pro- foam structure was modeled as Kelvin cells, and contin- totype reaction due to its great industrial relevance. We uum approaches such as finite volume methods were assume the reaction to follow the Langmuir-Hinshelwood used to solve for the fields of concentration and flow reaction mechanism [14, 15]. The foam structure serves velocity. Following this approach, the foam, which is of as substrate and is assumed to be coated with CuO/ZnO/Al2O3 complicated geometrical shape, can be described by a washcoat. We wish to point out that this study is not correspondingly shaped mesh or by immersed bound- restricted to the mass transfer limited regime, but also aries [6]. In both cases, the precision of the boundary the reaction rate limited regime is addressed. Therefore, description depends essentially on the spatial discretiza- it is eminently important to model the reaction mecha- tion of the simulation domain. Smorygo et al. [7] fol- nism in detail. The effective reaction rate in the wash- lowed a different approach and described open-cell foam coat layer is quantified by means of precomputed look- structures by an inverse sphere packing, that is, the do- up tables for the effectiveness factor [16]. Among other main is given by the union of overlapping spheres and parameters, the effective reaction rate depends on the the characteristics of the foam are represented by the partial surface pressures of the reactants, which can be positions of the centers and the radii of the overlapping obtained from the collision fluxes on the surface. Hence, spheres. They could show that inverse sphere packing the relevant quantities are obtained directly at the cat- models provide a good description for the specific sur- alytic boundary. face and hydraulic permeability of foams. The great ad- vantage of the inverse sphere packing model is that the simulation domain is defined analytically, thus the representation of the boundaries and consequently of the In the following section, we describe our simulation loci of the reactions is mathematically exact disregard- method and the model for the chemical reaction. In Sec.3, ing the resolution of the spatial discretization (lattice) of we validate the individual parts of our method with ex- the domain. This way, the foam can be described as a perimental data taken from the literature and in Sec.4, hierarchy of spheres of significantly different size at rea- we study the properties of the low temperature water- sonable computational effort. gas shift reaction in porous foam for a range of param- In the current paper, we employ isotropic Stochastic eters. Here, we vary both the active site density within Rotation Dynamics (iSRD) [8] to simulate the flow of a the washcoat and the porosity of the unit foam cell. For reactive gas in a porous media. The foam structure is de- the mass transfer controlled regime, Lucci et al. find that scribed using Constructive Solid Geometry (CSG) [9, 10]. the dimensionless performance index increases monotonously Special care is required in regard to the boundary condi- with the porosity [5]. We will reproduce this result for tions, since pressure boundary conditions suffer from in- the low temperature water-gas shift and explore the va- stabilities for particle-based fluid dynamics [11, 12]. On lidity of this relation towards the reaction rate limited the other hand, periodic boundary conditions are obvi- regime. Sebastian Mühlbauer et al. 3

2 | NUMERICALMODEL can be found in [10, 17].

2.1 | Open-cell foam 2.2 | Gas flow Smorygo et al. have show that inverse sphere packings 2.2.1 | Fluid model are an excellent model for open-cell porous foam structures [7], therefore, we use constructive solid geome- The gas flow is simulated by means of a variant of Stochas- try (CSG) [9, 10] to model the geometrical shape of the tic Rotation Dynamics [18, 19]. In SRD, the fluid is mod- foam. This shape is then used as boundary of the simu- eled by point-like quasi-particles which do not directly lation for the particle based fluid dynamics described in correspond to actual particles, but represent a discretiza- Sec. 2.2. tion of the phase space. Each time step comprises one Figure1 illustrates the construction of our unit cell: streaming step, propagating the particles according to the foam geometry results from the geometric subtrac- their current velocity, and one collision step, allowing tion of the sphere packing from the hexagonal prism. the particles to exchange momentum. SRD has been This approach allows us to fully parametrize the geom- shown to be a reliable to model for fluid flow through etry by the prism width, w, and the sphere diameter, complex geometries [20, 21]. Here, we use isotropic d . The porosity of the foam is controlled by the ratio Stochastic Rotation Dynamics (iSRD) [8], a variant of stan- k w d w ≡ / . Thus, the prism width, , represents the pore dard SRD. While the streaming step is identical in SRD size of the foam. and iSRD, for the collision step, in iSRD the particles are grouped into randomly distributed spheres, and not into the cubic cells of a Cartesian simulation grid as in SRD. Hence, iSRD unlike SRD does not suffer from anisotropy = at complex shaped domain boundaries [8]. \

2.2.2 | Boundaries

Inlet and outlet. For the simulation of the gas flow we F I G U R E 1 The open-cell foam structure is modeled as an inverse sphere packing, resulting from the assume semi-periodic boundary conditions [13] in flow geometric subtraction of a sphere packing from a direction. This means, we apply periodic boundary con- hexagonal prism. The porosity depends on the ratio ditions at the inlet and outlet for the fields of density, between the sphere diameter and the width of the velocity, temperature, and pressure while for the field of prism. the composition of the simulated gas mixture, we allow for a discontinuity at the (otherwise) periodic boundary. The CSG concept works particularly well for parti- This type of boundary model was designed particularly cle based simulation methods, where the boundary con- to describe reactive gas flows. It assures temporally conditions can be enforced directly on every particle hitting stant reactant and product concentrations at the inlet. It the wall. Moreover, this approach allows us to evalu- has been shown that this boundary condition is suitable ate the partial pressure corresponding to each individual to mimic reactive flows with pressure boundary condi- particle species exactly at the surface via the collision tions at much lower computational effort [13]. fluxes, that is, simply by bookkeeping over the particles Solid wall. In order to simulate no-slip conditions having hit the wall. at the solid walls formed by the foam described in Sec. The numerically robust implementation of CSG bound- 2.1, the velocity of each fluid particle hitting the wall is aries to represent complex shapes is nontrivial; details inverted. This is, however, not sufficient to assure no- 4 Sebastian Mühlbauer et al. slip conditions if the collision bins (in SRD) or the colli- The back reaction is not taken into account. This reac- sion spheres (in iSRD) overlap with the domain bound- tion proceeds along the Langmuir-Hinshelwood reaction ary [20]. Therefore, we generate ghost particles within path [15] which is one of the most important prototype the solid walls. Their velocities are randomly chosen reaction paths for heterogeneous catalysis. Therefore, from a Maxwell-Boltzmann distribution. The number the results and conclusions attained for this example density of the ghost particles is chosen identical to the are applicable similarly also to other catalytic reactions. number density of particles in the fluid domain [22]. To In industrial applications the carrier structure is often demonstrate that this procedure delivers indeed reliable coated by a porous washcoat layer. This washcoat pro- results, in Fig.2 we show the numerically obtained flow vides a large effective surface containing catalytic sites velocity profile for plane Poiseuille flow. The gray shaded needed for the water-gas shift reaction to take place. We assume the foam structure to be coated evenly and 1.4 L simulation denote the washcoat layer thickness by . Since, the 1.2 exact solution reactant concentrations decrease with increasing depth 1.0 in the washcoat, the local reaction rate in the washcoat,

[-] deﬁned as the number of reactions per unit time and 0.8 n t max washcoat volume, d r d , varies as well. Note that due

u /

/ 0.6

u to the Langmuir-Hinshelwood mechanism, the reaction 0.4 rate is not necessarily a monotonous function of the re-

0.2 actant concentrations [16].

0.0 The gas ﬂow through the mesoporous foam struc- 0.6 0.4 0.2 0 0.2 0.4 0.6 − − − ture is simulated as described in Sec. 2.2.1. Albeit both x channel width [-] / foam and washcoat have a similar geometric structure,

F I G U R E 2 Test of the described boundary it is not feasible to model the open-cell foam and the conditions for the case of plane Poiseuille flow. washcoat simultaneously, since the associated processes Obviously, the desired no-slip boundary conditions at occur on very different length and time scales. Hence, the solid wall are obtained and at the same time the in the simulation the reaction-diffusion process related analytical result is reproduced up to very good to the washcoat is decoupled from the particle simula- precision. tion by introducing an effective model for the reactions. Again, we neglect the back reaction and obtain for the areas indicate the regions populated by ghost particles. average reaction rate in the washcoat layer We see that the simulation data agrees up to a high pre- n cision with the analytical solution for plane Poiseuille d r = ζ R t (2) flow. d

with

2.2.3 | Reaction model k P P R = CO H2O 2 1+K P +K P +K P +K P For demonstration of the model and the algorithm, we CO CO H2O H2O CO2 CO2 H2 H2 have chosen the low temperature water-gas shift reac- (3) tion for its great importance in chemical engineering, that where k is the reaction rate constant and Ki and Pi with i , , , is, the production of carbon dioxide and hydrogen from ∈ {CO H2O CO2 H2 } are the adsorption constant and carbon monoxide and water steam, the partial pressure of the component i at the outer surface of the washcoat, respectively. Thus, the average re- + + . 3 CO H2O → CO2 H2 (1) action rate, Eq. (2), of unit mol/(m s) results from the Sebastian Mühlbauer et al. 5

2.06 3 local reaction rate at the outer surface of the washcoat, g/cm including the pores. The characteristic pore R, given by Eq. (3), and the effectiveness factor, ζ. diameter in the catalyst is in the range of 10 ... 20 nm. We follow the procedure proposed in [16] to com- The most accurate model considered in [15] is based pute the effectiveness factor for the relevant range of on the Langmuir-Hinshelwood reaction mechanism. For the partial reactant pressures and the model parameters, the temperature, T = 453 K, the kinetic constant for this k = 1.217 104 3 2 which vary with temperature. Note that for the compu- model is × mol/(m s Pa ), where the vol- tation of the effectiveness factor for a given parameter ume refers to catalyst volume. The kinetic constant is set, the one-dimensional reaction-diffusion equation for assumed proportional to the active site density. Thus, Langmuir-Hinshelwood kinetics in the washcoat has to lower active site density is achieved by reducing k . The be solved [16], while the reaction rate directly at the sur- adsorption constants referring to the different species face can be evaluated very easily. are:

The domain boundary described by the inverse sphere 3 1 K = 8.562 10− Pa− packing corresponds to the outer surface of the wash- CO × 4 1 K = 8.898 10− − coat. The catalytic boundary is discretized into individ- H2O × Pa 4 1 (5) ual boundary cells. The partial surface pressures are com- K = 4.581 10− − CO2 × Pa puted from the given wall temperature and the tempo- K = 9.594 10 4 1 . H2 − Pa− Z i , , , × rally averaged collision fluxes, i , with ∈ {CO H2O CO2 H2 }. Subsequently, the effectiveness factor is read from the The geometric washcoat properties are taken from sam- precomputed table, and the average reaction rate fol- ple A15 examined by Novak et al. in [23]. Here the wash- lows from Eq. (2). To transform the simulated quasi- coat thickness is L = 15 µm. Below we will also need particles according to the chemical reaction, we need the macro- and mesoporosity given as εmacro = 0.26 the reaction probability for each reactant species, which and εmeso = 0.43, respectively. The pore diameter, re- is obtained via quired to compute the Knudsen diffusion coefficients in 13 n ν the mesopores, is nm. The effective diffusion coef- p = d r LA i , i , , i t Z for ∈ {CO H2O} (4) ficients in the washcoat are estimated based on what d i Novak et al. call the “standard model” [23]: where L and A denote the washcoat thickness and the 6 2 D wc = 5.068 10− surface area in the concerned boundary cell, respectively. CO × m /s 6 2 Thus, whenever a particle of type i hits the catalytic D wc = 7.342 10− H2O × m /s boundary, this particles undergoes a reaction with prob- 6 2 (6) D wc = 4.425 10− CO2 m s ability pi , which has to be computed beforehand in the × / 5 2 D wc = 1.553 10− . corresponding boundary cell. For the reaction consid- H2 × m /s ered here, the stoichiometric coefficient, νi = 1, for both reactants. This approach, on average, recovers the expected number of conversions for each reactant species. 3 | VALIDATION

Washcoat parameters 3.1 | Pressure drop – Hagen number For the current study, we refer to Ayastuy et al. [15]. who assessed various approaches to model the low tem- A typical simulation result is shown in Fig.3 where the perature water-gas shift reaction. In their experiments, concentration field and the field of flow velocity are rep- the catalyst is composed of 24.9 % CuO, 43.7 % ZnO and resented by color and stream lines, respectively. From 3 31.4 % Al2O3. Having a pore volume of 0.29 cm per the flow fields obtained from the simulations we deter- gram catalyst, this yields an effective catalyst density of mine the superficial velocity, U , and the pore size Reynolds 6 Sebastian Mühlbauer et al.

stream velocity [m/s] 1.7 1.6 1.0 1.5 2.0 2.5 3.0 1.5 [-]

i 1.4

ρ ρ max 1.1

/h ρ ≥ 1.3 h i max ρ 1.2 10 1.1 % density change

1.0 0 10 20 30 40 50 60 70 80 90 Re [-]

xCO [%] F I G U R E 4 Maximum compression within the 0.0 1.0 2.0 3.1 simulation domain as a function of Reynolds number. ρ ρ < 10 The dashed line separates regions with max/h i % F I G U R E 3 Typical simulation result. The figure from regions with larger deviation. This separation = 30 shows the fields of flow velocity (left) and corresponds to Re . concentration (right) in a periodic unit foam cell, represented by color and streamlines, respectively. In 1.0 this example the porosity is ε = 0.902. The remaining approach the ideal value of due to fluctuations in parameters are taken from Tab.1 explained in detail in density and velocity which are inherent to the statistical Sec. 4.1. nature of iSRD and similar particle-based models of CFD. Thus, part of the density variation shown in Fig.4 must be attributed to statistical fluctuations of the system. To number, obtain the results shown in this manuscript, we average wU 2 104 , over × time steps, after the steady state has been Re ν (7) ≡ reached. where w is the prism width and ν denotes the kinematic viscosity. Further, we compute the dimensionless pressure drop represented by the Hagen number,

w 3 P d , Hg ≡ − ρ ν2 z (8) h i d In Figure5, we show the relation between the Ha- P z where is the pressure, is the coordinate along the gen number, representing the pressure drop, and the flow direction, and ρ is the average mass density in = 30 h i Reynolds number, Hg Hg(Re). For Re . , the devia- the system. In mesoscopic simulations [24, 19,8], the tions between simulation results and experimental data, Reynolds number can be increased either by increasing taken from [25], remain below 6 %. These deviations are the spatial and temporal resolution (which leads to a de- due to the imperfect representation of the foam geom- crease of viscosity) or by increasing the flow velocity. etry by the inverse sphere packing. For Re & 30, the Here, we vary the superficial velocity, while the pore deviations grow with increasing Reynolds number. This size and the resolution and, consequently also the kine- growth is associated with the compressibility effects, de- 30 matic viscosity remain invariant. For Re & , this ap- picted in Fig.4. These effects play a minor role be- 10 proach leads to density variations of more than %, as low Re = 30. We conclude that our model and simula- shown in Fig.4. tion method reliably predicts the pressure drop for metal 0 ρ ρ Even for Re → , the value of max/h i would not foam structures in the low Reynolds number regime. Sebastian Mühlbauer et al. 7

104 2.0 · where s is the eﬀective strut size, simulation, ε = 0.914 ε = 0.915 4 1.5 experiment, s = w 1 ε . (12) r 3π ( − ) ρ max 1.1 1.0 ρ ≥ η h i In the deﬁnition of the Sherwood number, denotes the Hg [-] conversion rate, 0.5

dn t d CO, out 0.0 η = 1 . (13) 0 10 20 30 40 50 60 70 80 90 − dn t d CO, in Re [-]

within a system of length h. DCO and As are the diffu- F I G U R E 5 Dimensionless pressure drop quantified sion coefficient for carbon monoxide (the limiting educt by the Hagen number, Eq. (8), as a function of the Reynolds number. Symbols: simulation results; soid species) and the specific surface of the foam, respec- line: experimental data (sample 2 in [25], see Fig. 2(i) tively. there). Dashed line: see caption to Fig.4. 2.5 0.64 x 0.43 1.84 x 2 0.37 x 3.2 | Mass transport – Sherwood 2.0 − x 2 + 0.39 x 0.25 [-] − number 3 / 1 1.5 Sc / s Sh 1.0 simulation, ε = 0.902 Further, we wish to validate the model and algorithm by simulation, ε = 0.914 0.5 evaluating simulation data for the mass transport to the 0 2 4 6 8 10 12 14 16 18 catalytic surface. Following Giani et al. [26], we consider Res [-] the Sherwood number as a function of the Reynolds num- F I G U R E 6 ber in the mass transfer limited regime. Numerically, this Sherwood number as a function of the is achieved by using Eq. (4) with Reynolds number – comparison of simulation and experiment. Symbols: simulation results; lines: fit dnr formulae. For this plot, we adopt the definitions of = min Z , Z 2 . (9) dt CO H O Sherwood and Reynolds number provided in [26], that ε = 0.902 = 0.204 is, for porosity , Res × Re and for ε = 0.914 = 0.190 For a quantitative comparison of the simulation data with , Res × Re. In this scaling, the function Shs = Shs Res is independent of the porosity, for the experimental results from Giani et al., in Fig.6, we ( ) details see [26]. The Schmidt number, normalize the simulation data to the data provided in [26], = ν D = 0.77 Sc / CO is kept constant in this study. as detailed in the figure caption. Hence, we define a

Reynolds and Sherwood number based on the eﬀective = The function Shs Shs (Res) reveals two diﬀerent strut size [26], regimes: For small Reynolds numbers, the data follows a sU rational function while for large Reynolds number they Res ≡ ν (10) and

s 1 η = − ln( − ) , Shs D A h U (11) CO s / 8 Sebastian Mühlbauer et al. obey a power law [26]: tures.

1.84 2 0.37 Res − Res for ε . 10 Sh = 2 + 0.39 0.25 (14) s  Res Res −  0.43 3.3 | Timescales of diﬀusion and 0.64 Re for ε & 10 s reaction – Thiele modulus   This behavior can be explained as follows: An important dimensionless quantity for heterogeneous Steep increase for small Re: The Sherwodd number, catalysis is the Thiele modulus [27],

Shs, is directly proportional to the superficial velocity τ Φ = d , and increases monotonously with the conversion rate. τ (15) r r Further, in addition to the convective transport, there is another process providing an inflow of reactants, namely where τd and τr are the diffusion and reaction timescales, the diffusive transport along the main flow direction. For respectively. For large Φ, the mass transfer to the cat- small superficial velocity, the conversion is mainly fed by alytic surface is the limiting factor, while for small Thiele diffusion, and the conversion rate depends only weakly Φ the reaction rate limits the conversion [28]. For the on the velocity. Therefore, increasing Res, which is pro- considered reaction, the bottleneck in the diffusion proportional to superficial velocity, leads to a steep increase cess is CO as H2O is abundant in the considered system. of Shs. Therefore, we define the diffusion time scale, τd , as the inverse of the diffusion rate per volume at which CO is Plateau for intermediate Re: For larger Res, however, transported to the catalytic surface. Neglecting the du- the convective transport dominates, and Shs approaches ration of the reaction, that is, no CO accumulates at the a plateau. To explain this plateau, we consider the prob- surface, the diffusion rate per volume is ability for a particle to reach the catalytic surface, which 1 DCO n = CO, in , is proportional to its residence time and, therefore, in- 2 (16) τd R versely proportional to U . On the other hand, the in- U n flux of reactants is directly proportional to , if we ne- where CO, in is the number density of CO at the inlet, glect diffusion along the main flow direction. Conse- and R denotes the pore radius. Analogously, τr is the quently, the product of both, that is, the mass transfer inverse of the reaction rate per volume. Assuming the to the catalytic surface and, thus, the Sherwood number diffusion to the reactive surface being much faster than is nearly independent of the superficial velocity, leading the reaction, the reaction rate per volume is to a plateau in Fig.6. n 1 d r, in = As L , (17) τr dt Power law for large Re: The explanation above does 10 n t not hold true for Res & , and even before the men- where d r, in/d is the average reaction rate in the wash- tioned above plateau is reached, the function turns into coat, as given by Eq. (2) for inlet conditions. As and L a power law with the exponent reported in [26]. While are the specific surface and the washcoat thickness, re- our simulation data are in very good agreement with the spectively. fit to the experimental data for the exponent, we under- estimate the prefactor by Giani et al. [26] slightly. We We study the Thiele modulus as a function of the ρs ρ believe that the deviation should be attributed to the in- normalized density of active reaction sites, / s,ref. The ρ verse sphere packing model of the foam which is more reference value, s,ref used from normalization corresponds regular and less tortuous compared to real foam struc- to the reaction parameters defined in the end of Sec. Sebastian Mühlbauer et al. 9

2.2.3. We obtain heterocatalytic reactive ﬂows. Particularly good agree-

1 ment we found for ﬂows of low Reynolds number. There- 2 ρs ρs 4 Φ for . 3 10− fore, in the subsequent Secs. 4.2-4.3, we will exploit ∝ ρs, ρs, × ref ref our method’s predictive power to study the dependence 1 (18) 4 ρs ρs 4 of the system on the density of active sites in the cat- Φ 3 10− ∝ ρ for ρ & × s,ref s,ref alyst and on the foam porosity, in the regime of low Reynolds number. Both parameters are diﬃcult to vary as shown in Fig.7. and, thus, not easily accessible in experimental investi-

102 gations. Having in mind applications like microﬂuidics, where due to small system sizes, also the Thiele modu- [-] 1 Φ 10 lus is typically small, we extend the study of the mass 0.25 1.0 transfer limited regime also to the intermediate range 100 towards the reaction rate limited regime.

1 0.5 10− 1.0 Thiele modulus 4.1 | Simulation setup 10 2 − 8 7 6 5 4 3 2 1 0 10− 10− 10− 10− 10− 10− 10− 10− 10 4.1.1 | Bulk transport coefficients ρs ρ / s,ref Typical mole fractions of CO, CO2,H2,N2, and H2O F I G U R E 7 Thiele modulus as a function of the for the low temperature water-gas shift reaction at the normalized density of active sites. In agreement with inlet are [29], 3 %, 13 %, 30 %, 28 %, and 26 %, respec- literature [16], we find two characteristic regimes. tively. Characteristic temperature and pressure are T = 453 K and P = 1 atm, respectively. We obtain the bi- For low densities of active sites, the Thiele modulus nary diffusion coefficient for each pair of species using grows with the square root of the normalized density of Hirschfelder’s method [30] and subsequently the bulk active sites and, consequently, the average reaction rate diffusion coefficients for each species in the mixture are in the washcoat increases linearly with the relative ac- computed [31]. For the gas mixture at the inlet, we ob- tive site density. This indicates that the reactions take tain place homogeneously within the washcoat, that is, ζ = 1. 5 2 D = 5.842 10− For larger active site density, the exponent changes to CO × m /s 5 2 1/4, indicating that the effectiveness factor decreases D = 8.795 10− H2O × m /s with the exponent 0.5. The results of the simulation co- 5 2 (19) D = 5.225 10− CO2 × m /s incides with the findings by Roberts and Satterfield [16]. 4 2 D = 1.682 10− H2 × m /s

4 | SIMULATION-BASEDPREDICTIONSAs the reaction proceeds, the diffusion coefficients change FORREACTIVEFLOWINPOROUS with the composition of the gas mixture. Assuming that MEDIA CO is completely consumed due to the reaction, the diffusion coefficient for CO, H2O, CO2 and H2 would change 0.9 +1.2 4.2 1.9 Based on the excellent quantitative agreement of our by − %, %, − %, and %, respectively. We simulation results with experimental data from different neglect this effect here. sources of the literature, shown in Secs. 3.1-3.3 we feel For the computation of the viscosity of the gas mix- confident that our numerical method and the correspond- ture, according to [32], we need the mass density and ing models deliver a correct and reliable description of the viscosity of each involved component. The mass 10 Sebastian Mühlbauer et al. densities are computed from the ideal gas law. The vis- face is secondary. cosity of overheated steam is computed according to The ratio between diffusion coefficient and viscos- IAPWS-IF97 [33], while the dynamic viscosities of the ity can be tuned via the rotation angle and the ratio be- remaining components are computed as described in [34]. tween time step and collision diameter [8]. For the cho- D ν σ = 27.0 From the densities and the dynamic viscosities, sen discretization, CO,sim and sim are times larger than the values given in Tab.1. To compensate 3 5 ρ = 0.7535 µ = 2.412 10− CO kg/m CO × Pa s for that, the time scale of convective transport as well as 3 5 ρ = 0.4846 µ = 1.537 10− the reaction time scale in the unit cell can be chosen ap- H2O kg/m H2O × Pa s 3 5 U ρ = 1.184 µ = 2.188 10− propriately. To this end, both the superficial velocity, , CO2 kg/m CO2 × Pa s and the effective reaction rate, nr , must be increased ρ = 0.05423 3 µ = 1.169 10 5 h ¤ i H2 kg m H2 − Pa s / × by the same factor σ. This approach ensures that, while 3 5 ρ = 0.7536 µ = 2.406 10− N2 kg/m N2 × Pa s the dimensional parameters chosen in the simulations (20) may deviate from what is shown in Tab.1, all relevant and the mole fractions given above, we compute the av- dimensionless parameters – Reynolds number, Schmidt ρ = 0.5298 3 erage density, kg/m , and the average dy- 5 number and Thiele – match the actual application. Note namic viscosity, µ = 2.368 10− Pa s. × that for a given Reynolds and Schmidt number also the Péclet number is known. The temperature is assumed to be constant within the simulation domain. Therefore, 4.1.2 | Simulation parameters all dimensionless numbers that refer to heat transfer are In particle-based, mesoscopic simulation methods, the irrelevant for the considered setup. fluid properties are represented by quasi-particles. Fur- ther, the transport coefficients of the simulated fluid de- 4.1.3 | System geometry and boundary pend on the spatial and temporal discretization as well conditions as some additional parameters [35, 36, 24,8]. In practical cases, the parameters of the fluid model, such as, We are interested in foam structures, as shown in Fig- rotation angle (in SRD and iSRD), collision volume size, ure8. In the simulation, we consider representative foam or time step, must be chosen in such a way that the di- cells. Figure3 shows the stream lines and the field of mensionless characteristics of the real system, such as mole fraction, Reynolds number, Péclet number, and Schmidt number, ni match the corresponding characteristics in the real sys- xi = , (21) j nj tem. Here we describe one way to determine suitable Í simulation parameters. for a typical simulation setup. At the inlet, the concen- As for most particle-based methods, for iSRD the tration is chosen homogeneous. transport coefficients depend on the spatial and tempo- In order to vary the porosity, we adjust the diameter ral discretization. Moreover, for iSRD, the diffusion co- of the spheres used for the inverse sphere packing de- efficient of a particle is directly related to its mass: For scribed in Sec. 2.1 while keeping the width of the foam D ∆t kBT m fixed density and rotation angle, ∝ / [8]. cell constant. At the same time, the superficial velocity In order to avoid an additional thermostat to be neces- is kept constant. sary for the inlet-outlet boundary condition, we choose For practical applications one aims for foam substrates m = m m = m CO, sim CO2,sim and H2O,sim H2,sim, which im- for heterogeneous catalysis which combine a large con- D = D D = D η ∆P plies CO, sim CO2,sim and H2O,sim H2,sim. This version rate, , with a small pressure drop, . The con- simplification has only minor influence on the simulation version rate, η, describes the amount of educt which is results, since the transport away from the reactive sur- converted along the reactor. Naturally, this quantity de- Sebastian Mühlbauer et al. 11

TA B L E 1 Summary of the dimensional and dimensionless parameters relevant for the following studies.

dimensional value ν 4.469 10 5 2 kinematic viscosity × − m /s D 5.842 10 5 2 diffusion coefficient CO × − m /s w 0.635 10 3 prism width × − m h = 8 3w 1.037 10 3 prism height / × − m U 1.0 superficial velocity p m/s dimensionless value = w U ν 14.2 Reynolds number Re / = w U D 10.9 Péclet number Pe / CO = ν D 0.77 Schmidt number Sc / CO

pends on the system’s length. Assuming an exponential decay of the reactant concentration between inlet (position: z = 0) and outlet (position: z = h),

n n z d = d η , t t exp −h i h (22) d CO, z d CO, in

an eﬀective conversion rate,

η = 1 η , h i − ln( − ) (23)

can be deﬁned, which is independent of the channel length.

4.2 | Active site density

η First we study the eﬀective conversion rate, h i, as a function of the active site density in the washcoat. Chang- ing this parameter allows to control the mode of opera- tion of the system from the reaction rate limited regime

F I G U R E 8 Cylindrical cutout of a foam structure to the mass transfer limited regime. Figure9 shows the modeled as an inverse sphere packing. conversion rate over the density of active sites, normalized by the density of active sites in the reference system according to the reaction parameters proposed by Ayastuy et al. [15] and the geometric washcoat properties described by Novak et al. [23] (see Sec. 2.2.3). Clearly, ρs ρ = 1 the reference system, corresponding to / s,ref , op- erates in the mass transfer limited regime. Starting from the reference value, we gradually reduce the active site 12 Sebastian Mühlbauer et al.

4 3 ative site density of 3.07 10− . This corresponds to η = 2.60 × h i Φ 2 ≈ , which is also where the two regimes meet in Fig.

2 2.60 7.

[-] 4 1 + 3.07 10− x i × / η h 1 limited by 4.3 | Porosity study reaction rate limited by mass transfer 0 In this section, we will perform a porosity study for two

8 7 6 5 4 3 2 1 0 different values of the active site density: 10− 10− 10− 10− 10− 10− 10− 10− 10 ρs ρs, / ref Φ = 16.8 * reference case (based on [15, 23]) → , 0.01 Φ = 1.34 F I G U R E 9 For the fixed porosity value ε = 0.902, * active sites reduced to % → . we vary the relative active site density in the washcoat. The dashed line has been fitted to the data. The reference case represents the mass transfer limited regime, while the other configuration yields an intermediate regime, as can be seen in Figure9. density to drive the system towards the reaction rate limited regime. Table2 and Figure7 show the Thiele modulus as a function of the relative site density.

TA B L E 2 Thiele modulus, Φ, as function of the 20 ρs ρ active site density over reference site density, / s,ref. 18 [-]

) 16 2 ρs ρs, Φ 14 / ref ρ U 8 /( 12

10− 0.0145 P ∆ 10 7 10− 0.0458 8 6 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 10− 0.143 5 porosity [-] 10− 0.443 4 F I G U R E 1 0 10− 1.34 For fixed mass flow rate, we plot the dimensionless pressure drop as a function of porosity. 10 3 2.98 − The flow resistance decreases with increasing porosity. 2 10− 5.31 1 10− 9.44 As Figures 10 and 11 show, both dimensionless 100 16.8 pressure drop and conversion rate decline, as the porosity is increased. This is expected, since with increasing porosity, the strut size as well as the surface area decreases. For the performed porosity variation, in the Even in the considered case, where the Reynolds mass transfer limited regime, the effective conversion 8 , +5 and Péclet numbers are at the lower end of the industri- rate varies in the range [− % %] around the value ally relevant range, the active site density, and such, the for the median porosity, ε = 0.902. In the intermediate total amount of catalyst, may be reduced by the factor regime, the effective conversion rate varies in the range 100 15 , +14 compared to the reference configuration [15] with- [− % %]. The variation is more distinctive in the out notably reducing the conversion rate. Further, half intermediate regime, since there the reactants need more of the maximum effective conversion is reached at a rel- wall encounters on average, before the reaction finally Sebastian Mühlbauer et al. 13

with porosity. Note that – due to the stronger relative η decline in h i depicted in the lower half of Figure 11– 2.80 the increase of the performance index is less distinct in 2.60 the intermediate regime.

2.40 reference case 2.20 [-] i

η 4 h 0.90 10 7.00 · − 0.80 6.50 reference case 0.70 6.00 0.60 active sites reduced to 0.01 % 5.50 0.50 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 [m] 5.00 s

A 4 10−

porosity [-] i/ 1.80 · η h 1.75 F I G U R E 1 1 For fixed mass flow rate, we plot the 1.70 effective conversion rate as a function of porosity. The 1.65 active sites reduced to 0.01 % conversion decreases with increasing porosity. 1.60 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 porosity [-] takes place. F I G U R E 1 3 To evaluate the surface utilization, we consider the effective conversion rate over the specific surface. In this study, the unit cell size is fixed, i.e., the actual surface area is directly proportional to the 0.32 specific surface. 0.28 reference case 0.24 [-] 0.20 )) Further, the surface utilization rises with porosity, 2 0.16 0.12 as shown in Figure 13. For the specific surface, A , an an-

ρ U s 2 /( 10− alytical expression is given in [7]. For the same reason as P 8.00 · ∆ 7.00 active sites reduced to 0.01 % above, the rise in surface utilization is less pronounced i/(

η 6.00

h in the intermediate regime. 5.00 4.00 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 Note that in the reaction rate limited regime, the nr porosity [-] concentrations of the reactants, and also h ¤ i, are ap- proximately constant throughout the system. Thus, the F I G U R E 1 2 To evaluate the suitability for the use total reaction rate, as substrate in heterogeneous catalysis, we consider the performance index suggested by Giani et al. in [26]. L nr da = L nr da , h ¤ i h ¤ i (24) ¹ ¹ An optimum substrate for heterogeneous catalysis where da denotes the surface element, is directly pro- should provide maximum conversion rate, while causing portional to the available surface area. For given inlet minimum pressure drop. However, these two require- concentrations, from the particle balance one sees that ments cannot be fulﬁlled at the same time. Toﬁnd a com- also η is proportional to the surface area. Since η is typ- promise, the performance index as in [26] is evaluated η = 1 η η ically small in this regime, h i − ln( − ) ≈ . Hence, for varying porosities in Figure 12. In both considered to assess the performance index in the mass transfer lim- cases, the performance index monotonously increases ited regime, it is reasonable to regard the ratio between 14 Sebastian Mühlbauer et al.

ber in the simulations qualitatively agrees with experimental data [26]. Interestingly, for smaller Reynolds num- 13 bers, we ﬁnd a qualitatively diﬀerent behavior of the m] /

1 Sherwood number, indicating the transition between two [ 12 10 )) different flow regimes at Res . 2 11 ≈ With this simulation setup, we are able to assess re- ρ U 10 /( active flows depending on the reaction parameters in P ∆ 9 the washcoat. As a starting point, we have assumed /( s

A 8 typical parameters from literature [15, 23]. We have 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 found that, even in the small Reynolds number regime, porosity [-] the number of active sites may be reduced by a factor of 100 F I G U R E 1 4 To estimate the suitability for the use with respect to this reference configuration without as substrate in heterogeneous catalysis in the reaction a notable change in conversion efficiency. rate limited regime, we consider the ratio between Further, we have examined the performance index specific surface and dimensionless pressure drop. and surface utilization for different values of porosity and Thiele modulus. For the considered setup, the sur- surface area and pressure drop. Figure 14 indicates that face utilization improves as the porosity is increased, both in the reaction rate limited regime, lower porosity values in the mass transfer limited and in the intermediate Thiele are beneficial, while according to Figure 12 the oppo- modulus regime. In accordance with literature [5], we site is true in the intermediate and mass transfer limited have found that the performance index also rises with regime. increasing porosity in the mass transfer limited regime. The same is true in the intermediate Thiele modulus regime, however, as for the surface utilization, the effect is less 5 | CONCLUSION pronounced. Conversely, in the reaction rate limited regime, the surface utilization is expected to stay roughly con- In this work, we have developed a technique for particle- stant, while the performance index, consequently, de- based simulations of heterogeneous catalysis in open- creases as porosity is increased. cell foam structures, consisting of four main components. The foam geometry is modeled as an inverse sphere pack- Acknowledgements ing using CSG [9, 17]. To eliminate any anisotropy at the complex boundaries, an isotropic variant of SRD is em- We thank Michael Blank for sharing his expertise in chem- ployed to simulate the gas dynamics [8]. The inlet and ical reaction engineering. We also thank the German Re- outlet are simulated with semi-periodic boundary con- search Foundation (DFG) for funding through the Clus- ditions [13]. Finally, the heterocatalytic reaction in the ter of Excellence “Engineering of Advanced Materials”, washcoat is decoupled from the particle simulation by ZISC, and, FPS. introducing an effective reaction model. Assuming the low temperature water-gas shift reac- References tion, we have validated the developed simulation technique based on experimental data. We have found the [1] Twigg MV, Richardson JT. Theory and applications relation between Hagen and Reynolds number to agree of ceramic foam catalysts. Chem Eng Res Des. 2002; with experimental findings, especially at low Reynolds 80:183 – 189. numbers [25]. Moreover, within the Reynolds number range covered in the experiments, the Sherwood num- [2] Lucci F, Della Torre A, Montenegro G, Dimopou- Sebastian Mühlbauer et al. 15

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