Towards computationally-efficient modeling of transport phenomena in three-dimensional monolithic channels

Abstract

In general, three-dimensional (3D) non-isothermal models for monolithic channels that seek to capture the local transport phenomena are computationally expensive. In this regard, we present a reduced model for a monolithic channel that reduces the computational cost, whilst preserving the 3D geometry and all of the essential physics – this is accomplished by exploiting the inherent slenderness of the monolith channel, coupled with scaling arguments, leading-order asymptotics and a fast space-marcher. The model takes into account conservation of mass, momentum, species and energy coupled with chemical kinetics, and is demonstrated for a three-way reaction mechanism for treatment of automotive exhaust. The results of the reduced model are verified against those of the full model and validated with axial temperature distributions for an experimental square channel. Overall, memory requirements and computing time are reduced by around 2–3 orders of magnitude as compared to the full set of equations. Finally, we discuss the suitability of the reduced model for reactor-scale modeling and extensions for transient simulations and other slender chemical engineering systems.

Introduction

Monolithic reactors are widely used for reducing emissions through conversion of toxic by-products of combustion into less harmful substances in gasoline-powered vehicles. In addition to their well-established role as automotive converters, they represent potential alternatives in many other applications: e.g., catalytic combustion of methane, catalytic oxidation, hydrogenation or dehydrogenation of aromatic compounds, hydrogen generation for fuel cells, steam reforming of light hydrocarbons and methanol, water gas shift reactions, etc. [1], [2]. Typically, a monolithic reactor – ceramic or metallic – comprises several hundreds of slender, parallel, and straight channels as shown in Fig. 1. The channels are in turn coated with a porous material that serves as a support for the catalytic materials: usually a noble metal such as platinum (Pt), rhodium (Rh) or palladium (Pd). The resulting structure provides a high surface area and gives rise to a low pressure drop in comparison with, for example, packed bed reactors.

Mathematical modeling and simulation have found widespread use in research and development of monolithic reactors: numerous mathematical models have been reported with varying degrees of complexity in terms of model dimensionality, washcoat modeling, chemical kinetics and number of species, physics and dynamics [1], [3], [4], [5], [6], [7], [8]. These models, as a complement to experimental design and materials research, help to gain a fundamental understanding of the complex interactions between reactions, mass and heat transfer, and fluid-flow phenomena. They are also useful in elucidating the effect of design parameters, material properties, and operating conditions – such studies would otherwise be time-consuming and expensive to carry out experimentally. Furthermore, they can be employed in optimization, control and system studies.

Detailed mechanistic models consider conservation of mass, momentum, species and heat transfer and thus result in a computationally intensive system of highly-coupled partial differential equations (PDEs). Monolith channels are manufactured with various cross-sectional shapes, including circles, squares, hexagon, triangles and sinusoids. All these geometries require three-dimensional (3D) models in order to capture the physical phenomena – except for the circle-shaped cross-section. As such, simulating the mechanistic models for 3D geometries requires significant compute resources and computing time, where the latter can range from hours to days [1], [4], [9]. This limits their applicability, because they cannot easily be exploited in wide-ranging parametric studies, optimization, and in simulations of more than one channel in the monolith [10], [11], [12], [13], [14].

To lower the computational cost, simplifications are commonly invoked to reduce the number of spatial dimensions: numerous one- (1D), and two-dimensional (2D) models have been reported [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], and there have also been attempts to derive reduced three-dimensional models [27], [28], [29], [30], [31], [32], [33] that can predict the effect of geometry on the performance of a monolithic channel. However, the nature of the physical phenomena is usually simplified in the reduction process; the simplifications usually take the form of model assumptions: e.g., the channel is treated as isothermal; the flow is assumed to be fully developed; zero thickness washcoat; or the flow is incompressible, i.e., there is no change in density despite exothermic reactions. While these models require fewer computing resources as compared to the full 3D models, they do not preserve geometrical resolution and/or physics. Moreover, these model reductions have been limited to modeling of single channels; their scalability to reactor-scale simulations has not been established.

In this regard, the aim of this paper is twofold: first, to develop a reduced 3D model that lowers the computational cost and preserves the 3D nature of a monolithic channel and the essential physics; and second, to explore how well the reduced 3D model scales numerically for simulation of an entire monolith reactor comprising hundreds or thousands of channels. For this purpose, we employ an asymptotic reduction methodology, whereby the steady-state governing equations are systematically simplified by exploiting the slenderness of monolith channels and the impermeable nature of the washcoat as compared to the flow channel. Thus, the full set of conservation equations that consist of a system of 3D elliptical PDEs are reduced to a system of 3D parabolic PDEs and 2D elliptic PDEs, as illustrated schematically in Fig. 2(b). As a result, a 3D elliptic steady-state model can be reduced to a transient-like propagation counterpart by marching along the streamwise direction $({\hat{\mathbf{e}}}_z)$; that is, the 3D $({\hat{\mathbf{e}}}_x\text{,}{\hat{\mathbf{e}}}_y\text{,}{\hat{\mathbf{e}}}_z)$ steady-state model is reformulated as a 2D $({\hat{\mathbf{e}}}_x\text{,}{\hat{\mathbf{e}}}_y)$ model for the cross-section and integrated in the streamwise direction $({\hat{\mathbf{e}}}_z)$. Here, we demonstrate the procedure on a monolith channel with a square-shaped cross-section. The resulting reduced model and numerical implementation are then verified with the full set of equations and validated with experiments. Further, we explore the scalability of the reduced formulation for full monolith reactors comprising multiple number of channels, and discuss the limitations of the proposed reactor model. We finish with conclusions and a discussion of extensions of the current methodology for transient simulations, complex surface chemistry, and other chemical engineering systems.