Multiscale Modeling. a. Ab initio quantum chemical calculations

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1 Multiscale Modeling EUGENIUSZ J. MOLGA, Warsaw University of Technology, Warsaw, Poland K. ROEL WESTERTERP, Roses, Spain Modeling of chemical engineering systems must be realized at several levels, as is summarized in Figure 1, in which an integrated approach to multiscale and multidisciplinary modeling systems on different time and length scales is depicted. It covers timescales from picoseconds for motions of atoms in a molecule during a chemical reaction to hours for the operation of industrial installations or even to days, weeks, and months for the influence on the environment. The length scale ranges from nanometers for molecules and active sites to meters and kilometers for the macroscale of plants and sites and even to hundreds or thousands of kilometers for the environment, i.e., the atmosphere, oceans, and soils. Thus process modeling must take into account the scales and complexity levels as well as all the relationships between all phenomena at all levels. More detailed information on the activities at any modeling level is given in the following: 1. Nanoscale: molecules and molecule agglomerates a. Ab initio quantum chemical calculations results used to provide fundamental information on chemical reaction parameters like bond lengths and angles, density distribution of electrons, etc. basis for calculations of chemical and physical properties: enthalpy, entropy, specific heat, viscosity, thermal conductivity, etc. first-principles theoretical calculations create a new tool to investigate reaction mechanisms and predict kinetic data at present the state of development does not enable the optimal catalyst to be predicted b. Interpreting and describing chemical reactions at the catalyst surface structural chemistry and surface science results: catalyst structure and nature of active sites modern experimental techniques like atomic emission spectroscopy, X-ray photoelectron spectroscopy, Raman spectroscopy, scanning and transmission electron microscopy, NMR spectroscopy, surface microscopy the development of an effective catalyst still depends on trial and error procedures c. Nanostructure tailoring molecular engineering in catalyst research controlling the size of pores and particle agglomerates improvement of the catalyst activity by achieving a very high surface-to-volume ratio still no practical methods to describe directly reaction rates in terms of the catalyst structure 2. Microscopic scale: particles, bubbles, droplets advanced materials, matrices, and carriers like polymers, ceramics, and composites # 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim /

2 2 Multiscale Modeling Figure 1. Hierarchical multilevels and multiscale modeling in chemical engineering [5] demand a designed structure on both the molecular and microscopic level complex media involved like non- Newtonian liquids, amorphous solids, multiphase dispersions, powders, and aerosols fundamentals of rheology and surface science are required closer cooperation between chemical engineering and physical chemistry is needed computational chemistry and noninvasive measurement techniques like NMR and tomography can be employed the scientific basis of surface and interfacial phenomena may contribute to progress in manufacturing the microstructure of advanced materials and help to study the mutual relationship between the molecular and macroscopic structures of these materials. 3. Mesoscale: reactor level, unit operation noninvasive measurement techniques for the quantitative description of phenomena in chemical reactors description of processes with the conservation equations of momentum, energy, and mass balances volume-averaged quantities representing the macroscale process are in use, but they are developed from an exact quantitative formulation of the transport phenomena and chemical reactions occurring on the scale of particles, bubbles and droplets transforming microscopic phenomena to macroscopic levels is crucial for the quantitative description of the entire process application of computational fluid dynamics (CFD) codes is very useful; spatial and time distributions of reactants, products, and catalyst as well as the yield and selectivity of chemical reactors are determined 4. Macroscale level: plant level computer-aided process engineering (CAPE) is employed to design an installation CAPE can also be used to model a single apparatus, but due to significant simplifications the results are less accurate than for CFD pressure to integrate CFD and CAPE methods is strong, but a generally approved approach still has not been elaborated.

3 Multiscale Modeling 3 Figure 2. Schematic representation of the possible multiscale modeling integration [1] Expected integration steps between some modeling levels are shown in Figure 2 [1]. Integration of computational chemistry (CCh) and CFD to create a joint discipline called computational transport modeling CTM and, in parallel, integration of CFD and CAPE methods into computational process engineering (CPE) are postulated. Furthermore, CTM and CPE can be integrated giving computational product processing (CPP), which covers the whole length scale. However, a wide gap in integration of CCh and CFD still exists, while more pronounced integration of CFD and CAPE is noticed [1, 2]. Since the possibilities for multiscale modeling are unlimited, the discussion here is restricted to one example of multiscale modeling to illustrate the perspectives of this approach in reaction engineering. An example of a practical application of the multiscale approach is demonstrated by the results obtained for the multiscale modeling of reaction and transport in a porous catalyst [3]. The considered case covers only a relatively small section of the full length scale, but it is quite representative to illustrate the methodology itself as well as the characteristic problems met with during multiscale modeling. In the problem presented in [3] the hierarchical modeling of a catalytic reactor on three scales nano-, micro-, and macroscale is described. Following a general concept of multiscale modeling, data obtained at the nanoscale were transferred to the microscale model and further successively to the full-scale macromodel. The oxidation of CO in porous Pt/g- Al 2 O 3 catalyst washcoated on a monolith was used as the tested reaction. On the nanolevel, a spatially 3D model was defined which describes a single micro g-al 2 O 3 particle with a size of mm constructed by virtual agglomeration of primary Al 2 O 3 nanoparticles with a size of 1 3 nm and deposition of individual Pt crystallites thereon. Such a model takes into account size effects of the catalytic active sites and the diffusion in small mesopores with a size of a few nanometers. On the microlevel, a spatially 3D model of the porous catalytic washcoat layer with a thickness of 30 mm was formulated, in which the g-al 2 O 3 microparticles accurately described by the nanomodel are virtually packed. By using the micromodel, effectiveness factors and spatially averaged reaction rates were evaluated as a function of the temperature, reactant concentration, and macroporous structure, the last-named of which included the g-al 2 O 3 microparticle size due to sintering. On the macrolevel, a spatially 1D model of the catalytic monolith with a size on the order of magnitude of 10 cm was formulated which, assuming plug flow of the gaseous phase, also takes into account the mass and heat transport in the monolith channels. In the macromodel

4 4 Multiscale Modeling the results obtained with the nano- and micromodels are employed, i.e., the locally averaged reaction rates as estimated at the nano- and microlevels. Such a linking of the detailed description of a porous catalyst structure with the full-scale model of the reactor is essential for multiscale modeling and offers a significant improvement of modeling and optimization of chemical reactors. A schematic illustration of the hierarchical three-scale approach applied in [3] is shown in Figure 3. The considered process was modeled as follows: Modeling of the Porous Catalyst. To obtain reliable and accurate results, reconstruction of the catalyst particles should follow the real process of catalyst preparation. In the presented approach the properties of porous catalyst are represented by a 3D matrix containing the information about the phase function in each voxel, i.e., the volumetric element within a three-dimensional space. Data for such a representation have been obtained by scanning electron microscopy (SEM) and transmission electron microscopy (TEM) on a catalyst support with the same morphology, i.e., porosity and pore and particle size distributions. The description of the porous catalyst layer must be carried out at two levels of the nanoand microsize. This is because for a representation of single Pt nanoparticles and small mesopores a resolution on the order of 1 nm is required, while the Pt/g-Al2O3 microparticles have a size on the order of mm and the thickness of catalyst layer in the washcoat is on Figure 3. Schematic illustration of a hierarchical three-scale approach [3]

5 Multiscale Modeling 5 the order of magnitude of 30 mm. Thus, to describe a catalyst layer of representative size, say mm, with the 1 nm resolution, as many as voxels would be required. Simulation of such a large system is beyond the abilities of existing computers. Therefore, multiscale modeling is the only way to maintain high accuracy of calculations and simultaneously solve the task effectively. Hence, to represent the entire catalyst layer section with a resolution of 100 nm, the microscale model is used, while the nanoscale model with a resolution of 1 nm is employed to describe the structure of each Pt/g-Al 2 O 3 microparticle. This strategy, depicted in Figure 3, is described in detail below. Catalyst Reconstruction at the Nanolevel. According to SEM and TEM images, the approximate size of the primary g-al 2 O 3 particles, shaped as ellipsoids or cylinders with spherical caps, can be considered as having a diameter d s of 4 nm and a length l s of 12 nm, while the radius r Pt of the Pt particles, shaped as hemispheres, is in the range of nm. As shown schematically in Figure 3, the Al 2 O 3 nanoparticles are agglomerated, forming a structure of size n m, where n and m are numbers of primary particles, while sintering is quantified by the fractional overlap factor v. Such agglomerates are randomly packed g-al 2 O 3 microparticles, each with a size of 0.1 to a few micrometers. On the surface of such reconstructed primary particles of the support, the required number of Pt particles N Pt ¼ with chosen size r Pt of 1 4 nm are randomly inserted. From the catalyst reconstruction, modeled as presented above at the nanolevel, the following information can be obtained as a function of the characteristic values d s, l s, v, n, m, r Pt, and N Pt : the microparticle diameter d p, the pore size distribution for a single g-al 2 O 3 microparticle, and the magnitude of the accessible surface of Pt particles, i.e., the number of active sites. Catalyst Reconstruction at the Microlevel. The Pt/g-Al 2 O 3 microparticles, as elaborated in the nanoscale modeling step, are used to construct the catalyst layer on the microscale. Further, for microparticles forming the washcoat layer the size distribution as well as the macropore size distribution within this layer can be obtained. Thus, within the entire catalyst layer a bimodal pore size distribution is found: the macropores due to the voids between the Pt/g-Al 2 O 3 microparticles, which are on the order of magnitude of a few micrometers, and the mesopores due to porosity of the individual g-al 2 O 3 microparticles, which are on the order of a few nanometers. The bimodal pore size distribution obtained from this modeling can be easily verified with experimental tests [3]. Modeling of the Reaction and Transport Phenomena. After determination of the carrier and catalyst properties, the reaction and transport phenomena were modeled after the following assumptions: low reactants and products concentrations, constant pressure, and a sufficient number of molecules to apply the continuity approach. The last-named assumption should be validated particularly for the nanoscale model instead of the alternative and more complex approach based on molecular dynamics simulations. On the nanoscale level, for each gaseous reactant present in the system, the 3D reaction diffusion model consists of the set of mass balance equations which contain the molar concentration of each component in the gas phase and the porosity for the mesopores e m. Spatial coordinates determine a location within the catalyst pores on the nanoscale. The surface deposition of each reacting component u k, i.e., the fractional coverage of the active sites, can be modeled with the concentrations of platinum active catalytic site c Pt.To estimate the reaction rates in the nanopores, a Langmuir Hinshelwood reaction mechanism was assumed with the appropriate kinetic data as published in [4]. The enthalpy balance for the considered catalyst section can be formulated on the basis of r eff, c p,eff, and l eff as the effective thermophysical properties of the catalyst. By solving the mentioned set of balance equations at the nanoscale, the spatial concentration and temperature profiles of CO in the reconstructed Pt/g-Al 2 O 3 microparticle of size mm are obtained. Significant mass- and heat-transfer resistances are detected as well as distinct temperature and concentration gradients in the considered Pt/g-Al 2 O 3 microparticle. By solving in turn the model equations at the microscale, i.e., for a catalyst section of a representative size of mm, and

6 6 Multiscale Modeling using the data evaluated with the nanoscale model, the spatial CO concentration and temperature profiles in the washcoat layer are obtained. These results can directly be transferred to the full-scale model of the catalytic monolith. Results obtained for the washcoat layer indicate that even at a relatively low temperature significant temperature and concentration gradients in the layer exist, which correspond to significant variations of the local reaction rates. With these local values of the reaction rate the effectiveness factor h and the spatially averaged reaction rate R j,avg can be evaluated as follows: h j ¼ 1 Z V s R j;o Vs R j dv R j; avg ¼ 1 Z R j dv V W VW where V s is the volume of the Pt/g-Al 2 O 3 microparticles, so that this volume excludes the macropores, in which the reaction rate is equal to zero, while V w is the total volume of the catalyst layer and R j,o the rate of the j-th reaction at the layer surface. Values of h and R j,avg can be evaluated at different temperatures and CO concentrations at the boundary of the washcoat layer as well as for a different morphology of the catalyst. After approximation, such dependencies can be directly transferred to the full-scale model of the entire monolith reactor and enable optimization of the process in terms of catalyst morphology and operating conditions. Modeling of the Monolith Reactor. By assuming plug flow of the gas phase, a spatially 1D model for the entire monolith reactor can be formulated at the macrolevel, in which values of R j,avg or h are employed. In this model a set of mass- and heat-balance equations must be formulated for the entire monolith reactor (gas phase, washcoat pores, and catalyst surface). This set of model equations can be easily solved by employing the values of R j,avg estimated as shown above. The presented approach makes possible accurate and efficient modeling of the monolith reactor, taking into account an influence of the catalyst morphology at the nanolevel. With such a model the entire process can be easily optimized. References 1 Z. Jaworski, B. Zakrzewska, Chem. Proc. Eng. 29 (2008) Dzwinel, D.A. Yuen, K. Boryczko, Chem. Eng. Sci. 61 (2006) P. Koci, V. Novak, F. Stepanek, M. Marek, M. Kubicek, Chem. Eng. Sci. 65 (2010) R.H. Nibbelke, A.J. Nievergeld, J.H.B.J. Hoebink, G.B. Marin, Appl. Catal. B 19 (1998) J. Li, W. Ge, J. Zhang, M. Kwauk, Chem. Eng. Res. Des. 83 (2005) 574.