This example is taken from the M.Sc. Thesis of Sandro Goisis and Alessandra Osio (Chapter 4). For further details and comments, we suggest to download the Thesis here!
The test case selected for these numerical tests is taken from (Maestri, Beretta et al. 2008). In Chapter 5 the simulations results are compared to the experimental data provided in the article. Maestri and co‐workers developed a dynamic two‐dimensional model of a catalytic annular reactor and validated it comparing model and experimental results. A UBI based kinetic scheme for the combustion of hydrogen on rhodium was used, as reported in Appendix C. The simulations were performed under isothermal and isobaric conditions. A schematic representation of the catalytic reactor is given in Figure 1. As shown in Figure 1, the Rh/α‐Al2O3 catalyst is deposited over the surface of the inner wall of the reactor (colored in orange).
|Inner radius||2.35 mm|
|Outer radius||4.50 mm|
|Reactor length||15 mm|
|O2 mole fraction||0.04|
|H2 mole fraction||0.01|
|N2 mole fraction||0.95|
The operating conditions and the geometric parameters of the reactor are presented in Table 4.1. The ratio between catalytic area and effective reactor volume (Vreact) is known and is called acat. From this parameter the αcat is obtained by the following equation:
where Areact is the geometric area of the catalytic surface. The spatial discretization of the geometrical domain was simplified in order to reach the steady state conditions with a smaller computational effort. Specifically, thanks to the symmetry of the annular reactor, it is possible to consider only one slice of this reactor. This is very convenient because this allows one to consider a 2D mesh instead of a 3D one. The 2D mesh is obtained considering the slice of a cylinder with a width of 5°. The number of required cells is thus 72 times lower than the one required for a 3D grid. A schematic view of the mesh is presented in the Figure 4.2.
As shown in Figure 4.2, the geometrical domain is divided in a series of cells by partitioning the axial and radial coordinates. The simulations are performed over an optimized mesh composed by 250 units in axial direction and 24 units in the radial one for an overall amount of 6000 cells. Further details about the optimized number of cells are provided in the following sections. In addition the grid is refined with a specific grading, i.e. the length of each cell of the mesh is not constant. The expansion ratio of the cells is calculated as the ratio among the length of the first and the last cell along one edge of a block. This enables the mesh to be graded, or refined, in specified directions for a specific factor. The introduction of a non‐constant step grid allows one to describe certain areas of the system in a more detailed way. In our case, since the zone close to the catalyst is interested by strong normal gradients, the mesh is highly refined near the catalytic wall in the radial direction. A further grading was introduced in the axial direction to provide a proper description of the rapid consumption of reactants at the reactor inlet. For the temporal discretization a Courant number of 0.1 is adopted. In order to provide an exhaustive analysis on the performance of catalyticFOAM, the simulations are performed at two different temperatures. In the first case (423.15 K) the system is under chemical regime, in the second one (773.15 K) it is controlled by mass transfer phenomena. The convergence tests were performed on an Intel core i7 950 CPU @ 3.07 GHz with 8 GB of available RAM memory. In the following the main results of these numerical tests are presented.
In Figure 4.3 the velocity profiles for the two temperatures investigated are reported. Since the inlet flow is constant, the velocity is higher in the second case (773.15 K). It can be noticed that the flow is laminar and thus the radial profile of velocity is parabolic. In Figure 4.4 and 4.5 the O2 and H2O molar fractions are reported.
The simulation results show the amount of O2 increase in the radial direction. This is an evidence of the reactions occurring at the catalytic surface. At 773.15 K the conversion of O2 is fairly complete. It noticeable that near the catalyst, in the first 0.3 mm more than the 90% of the available O2 is consumed. The trend of the water molar fraction is opposite to that of O2. The production of H2O is considerably lower in case of low temperature. In Figure 4.6 the axial O2 profiles sampled at different radial positions are presented, the first at 423.15 K and the second at 773.15 K. From Figure 4.6 it is observable that in the first part of the reactor there is an abrupt consumption of O2.
In Figure 4.7 a rapid production of H2O is shown along the axial coordinate. The nearer to the catalytic surface the profile is taken, the higher the conversion of the reactant is. It can be noticed that at 423.15 K the slope of the oxygen and water profiles in the axial direction is smaller than in the case of higher temperature. This highlights the limited reactivity at low temperatures.
In Figures 4.8 and 4.9 the radial profiles of the oxygen and water mole fractions are represented. The profiles are sampled along the radial coordinate at different axial length, at the temperatures of 423.15 K and 773.15 K.
The slope of the curves decreases going from the inlet to the outlet of the reactor. This trend underlines the high reactivity of the system in the very first millimeters of the reactor. According to the heterogeneous kinetic model, the H* is the most abundant surface species, as shown in Figure 4.10, in which are depicted the profiles of the adsorbed species on the catalytic surface, at steady‐state conditions. Water desorbs as soon as it is produced. The decrease of O* at 773,15 K is caused by the elevate conversion of the gaseous oxygen.
The achievement of these results has been made possible after solving a series of problems. These regards the numerical nature of the problem and are discussed in the following.