Parallel finite element modelling of multi-physical processes in thermochemical energy storage devices☆
Introduction
The current ambitions towards a transformation of the energy supply system to renewable resources have increased the need for technologies to decouple energy supply and demand as well as for means to increase energy efficiency. Building climatisation, hot water supply and industrial process heat generation consume a significant share of the primary energy supply and offer considerable potentials for efficiency improvements [1], [2], [3], [4]. Intense research effort is invested into numerous kinds of thermal energy storage systems in order to improve our technological capabilities in this regard [1], [2], [5], [6], [7]. Among the technologies considered for heat storage are various chemical reactions as well as adsorption and absorption processes [8], [9], [10].
The numerical simulation of such systems requires software specialised to capture the strongly coupled nonlinear processes characteristic for their operation. Computational models can be used to design and simulate heat storage devices based on a physical–chemical characterisation of the storage material in the laboratory [11], [12], [13] and aid in the screening of suitable reactive systems as performed in reference [14]. The model domains become large when practical applications are considered instead of laboratory examples. Heat exchangers and structures that manage flow patterns can have complex geometries, while irregularly shaped porous structures can be used as integrated thermochemical heat exchangers and reactors [15]. The reactive bed itself can develop inner structures and heterogeneities due to, e.g., adverse flow conditions. A detailed investigation of large scale thermochemical energy storage systems and their interaction with geometrically complex system components like heat exchangers and flow diverters requires simulations based on sufficiently refined grids.
Most current numerical simulations of thermochemical heat storage focus on the geometrically simple reaction bed and use either 1D [16], [17] or 2D [18], [19] mesh topologies. Only very few 3D numerical simulations using the finite element method have been published, see Nam et al. [20] for hydrogen storage in metal hydrides, Lele et al. [21] for salt hydrates and Bilke et al. [22] for an example of thermochemical heat storage. One obstacle for more complex analyses certainly originates from computational limitations. Other authors relied on different numerical methods, e.g. finite volume methods [13], [23], [24] or—especially for adsorption based systems—finite difference methods [25], [26]. In general, computational analyses of strongly coupled processes in thermochemical heat storage often remain academic in nature due to a lack of efficient computational schemes. With very few exceptions, detailed analyses of complex geometries, large problem sizes and detailed multi-physical aspects are lacking. In order to facilitate the transition of modelling from academic to applied research and practical design, computational performance is a key requirement.
The complexity of the mathematical description of strongly coupled and highly nonlinear processes characteristic for thermochemical heat storage render a computational solution numerically challenging and, above all, time-consuming. These difficulties are exacerbated when complex or large geometries are being considered. To facilitate the solution of such problems using the finite element method (FEM), the numerical algorithms have to be computationally efficient. Currently, parallisation is the method of choice for significantly improving the computational efficiency of FE computations.
In the past decades, much effort has been devoted to the parallelisation of the FEM, and many advances have been published. For example, a domain decomposition-based parallel FEM was introduced by Farhat and Roux [27]. The parallel FEM has been developed and applied in the context of a very broad range of physical problems, e.g. fluid–structure interaction [28], geotechnology [29], fatigue crack growth [30], Navier–Stokes flow [31], incompressible flow [32], as well as coupled thermal, hydraulic and mechanical processes in porous media [33], [34], [35]. The parallelised FEM method has been implemented into both open source software, e.g., deal.II [36] and OOFEM [37], and popular commercial FE software packages like COMSOL® and ANSYS®.
Before the mid-1990s, the Parallel Virtual Machine (PVM) was de facto standard for distributed computing and used to realise the domain decomposition based parallel FEM (e.g. the study presented in [33], [34]). Eventually, the Parallel Virtual Machine (PVM) was largely replaced by the much more successful Message Passing Interface (MPI) standard. Base on MPI, the Argonne National Laboratory started to develop PETSc (Portable, Extensible Toolkit for Scientific Computation) [38] in the mid 1990s. PETSc is intended for use in large scale applications with high scalability in parallel computing and has by now become the most widely used parallel numerical software library for sparse matrix computation and partial differential equations. PETSc provides a large suit of parallel routines for sparse matrix/vector computations and linear/non-linear solvers, and has been integrated into several scientific software packages for parallel computing. Among the frameworks for the numerical analysis of physical processes in porous media, parallel computing with PETSc has been implemented in DUNE [39], [40], a numerical simulator for multi-component multi-phase flow processes; MOOSE (The Multiphysics Object-Oriented Simulation Environment) [41], a finite-element, multiphysics framework; PFLOTRAN [42], a numerical simulator for subsurface flow processes; and also in the commercial software COMSOL®. For additional references see, for example, [43]. With PETSc, these codes show excellent scalability in parallel computations on Linux clusters and supercomputers. Therefore, PETSc has also been selected to parallise the solution process in OpenGeoSys (OGS) [44], [45], an open source scientific FEM software.
To the authors’ knowledge, the study of parallel finite element modelling for the coupled reactive mass and heat transport in models of thermochemical heat storage is still lacking. Simulations of thermochemical heat storage devices are characterised by a very strong coupling between chemical reactions, flow processes and heat transport [17], [19], [46]. The active reaction zone is localised in different regions of the discretised reactor domain during different times of the process simulation. The question arises whether the algorithms developed for other multiphysics processes in porous media will also provide sufficient scaling behaviour in parallel computations of thermochemical energy storage. This question is addressed by this article. The objective of the present effort is to provide researchers with an open-source software solution with parallel computing features to enable more detailed simulation studies and thus aid in the design of materials and reactors for thermochemical heat storage.
Similar to previous work on the parallelisation of the FEM modelling of two-phase flow processes in porous media [45], the present scheme takes the domain decomposition approach [47] and employs PETSc [38] routines for global assembly and the solution of the linear equation system. Meanwhile, parallel input/output (I/O) is realised via MPI functions. The approach has been implemented into the scientific open source software package OGS.
The outline of this paper is as follows: first, the theoretical background of the computational model used to simulate thermochemical heat storage is introduced in Section 2, followed by an outline of the parallel FEM approach and its implementation; subsequently, several simulations are performed using the developed parallel algorithms and analysed with respect to their computational efficiency in Section 3. The article closes with concluding remarks in Section 4.
Section snippets
Methods
In this section we briefly describe the governing equations for thermochemical energy storage process simulations as well as the corresponding numerical methods applied for their solution.
Computer platform
We conducted a series of performance tests on a Linux based cluster, EVE, available at the authors’ institution. EVE has 85 nodes connected by a 40 GB Infiniband network interconnect, where each node has 12 computing cores with Intel(R) Xeon(R) CPUs (2.67 GHz). The source code of OGS together with PETSc has been compiled using the Intel compiler optimised for Intel architectures.
Model problem
A calcium hydroxide reaction system similar to the one considered in Shao et al. [17] and Nagel et al. [19] was used
Conclusions
A parallel finite element approach for the numerical modelling of thermochemical heat storage using OGS and PETSc was presented. To the authors’ knowledge this is the first detailed investigation of this kind in the context of multi-physical models of thermochemical heat stores. The numerical scheme has been implemented into scientific open-source software which allows the adjustment of any particular algorithmic component or model feature to specific research questions or engineering demands.
Acknowledgements
Funding was provided by the Helmholtz Initiating and Networking Fund through the NUMTHECHSTORE project. We very much appreciate the efforts by the PETSc development team. Finally, we would like to thank Thomas Schnicke and his team for continued EVE support.
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This paper was presented at the 7th International Conference on Applied Energy (ICAE2015), March 28-31, 2015, Abu Dhabi, UAE (Original paper title: “A parallel FEM scheme for the simulation of large scale thermochemical energy storage with complex geometries using PETSc routines” and Paper No.: 336).