Parallel finite element modelling of multi-physical processes in thermochemical energy storage devices

Highlights

• 3D simulations of CaO/Ca(OH)₂ thermochemical heat storage.

• Multi-physical model parallelised using PETSc routines.

• Open-source finite element implementation.

• Significant reduction of computation times.

Abstract

Thermal energy storage technologies are of current interest in order to improve the integration of renewable energy sources as well as energy efficiency. Numerical simulations of thermochemical heat storage are especially challenging and time consuming due to the complexity of the mathematical description of the strongly coupled and highly nonlinear processes characteristics for such systems. These difficulties are exacerbated once practically relevant complex or large geometries are considered as they can occur around heat exchangers or due to internal heterogeneities of the reactive bed. To allow a computationally efficient simulation of such applications, an existing finite element implementation of a thermochemical heat storage model was parallelised using PETSc routines. Input/output, global assembly and the linear solver all work in a distributed fashion. The approach is implemented into the open source framework OpenGeoSys. The performance of the present parallelisation approach is tested by simulating the discharge of a heat store based on calcium oxide and water as a benchmark problem. The algorithm is tested on 2D as well as 3D meshes. The computational time required for the simulation could be reduced significantly. For example, a 3D model running almost 7 days on a single core could be solved in less than 1 h on 120 cores using the developed framework. The results strongly depend on linear solver and preconditioner settings.

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.