Work packages

I) WP1: Numerical Solver for Thermal Energy Storage (TES)

(Scientific coordinator: Mejdi Azaiez, Teams : I2M and LMAC)

The main objective of this work package is to develop efficient, easy-to-implement, energy stable numerical schemes to accurately capture the dynamics of interface as well as the behavior of the PCM when is melting from solid state to liquid State.The WP1 is organized in three subwork packages as follow:

• WP 1.1: Numerical Solver ( Participants: M. Azaiez, F. Ben Belgacem and PostDoc)

  Following the work made by the I2M team on the subject (see [Li22], [Wan22], [Hou19], [Zho19]) we aim to construct a linear, fully decoupled, energy stable numerical scheme for solving the coupled nonlinear system (Allen-Cahn equation for the phase field, Navier-Stokes and energy equations for temperature, pressure and velocity). The proposed method is based on an auxiliary variable approach for the time discretization (see [She10], [She18]) and finite element method for the spatial discretization. The main contributions for this part will be :

  • to provide an unconditionally stable method for the time discretization of the Navier-Stokes, heat and phase-field coupled equations modeling the melting convective phase change;

  • to implement and valid a fast and stable 2D and 3D Finite element solver.

• WP 1.2: Numerical Analysis ( Participants: M.Azaiez, F. Ben Belgacem and J. Shen)

  This subwork package is dedicated to the analysis of the stability and the accuracy of the global solver and the convergence of the temperature, phasfield, velocity and pressure. This work will be done in collaboration with Prof. Jie Shen from Purdue University. He is one of the world's leading experts in phasefield methods. The risk arises from the difficulty of carrying out all the numerical analysis and convergence studies.

• WP 1.3: Validation ( Participants: M. Azaiez, F. Ben Belgacem and C. Lebot)

  We aim to validate the numerical model on some synthetic cases and benchmarks. We limit ourselves in this subwork package to simple cases. More complex cases will be faced in WP4. For some PCM, we will consider the question of how the Navier-Stokes equation can be reduced to linear Stokes one. We also focus on the effect of the densities of the two PCM on the trajectories of the spheres and the impact of these trajectories on storage.

II) WP 2: Optimization algorithms for Thermal Energy Storage (TES)

(Scientific coordinator: Jelassi Faten, Teams: LMAC and IMB)

TES systems with PCMs are expected to have high energy storage capacities. However, their poor thermal conductivities are an important drawback. The main effect is a slow heat flowing across the medium; unwanted contrasting heat storage rates may show up. Encapsulate a high conductive material in the TES support is an opportunity for a smart hybrid system to significantly improve the heat supply/storage/removal cycle. The challenging question is hence: how to disseminate these enhancer capsules to increase the effective conductivity while preserving the high storage heat capacity of the PCMs? This is a shape optimization problem.

The strategy starts by the identification of the pertinent parameters, those having the most important influence on the responses of the supply/removal process of the thermal energy within/from the heat composite support (heat capacity, conductivity, contact resistance, ..., and the geometry). Some parameters are fixed inputs, others are outputs of the optimization process, in particular those describing the geometrical structure of performing hybrid media (radius, thickness, position of capsules, minimum distances between them, ...). We restrict the study to the transient heat model, letting aside the dynamics generated by natural convection. In fact, during the solid/liquid phase change, heat transfer by conduction is dominant and natural convection is less efficient.

This Work Package follows a gradual methodology: the increasing difficulties are addressed sequentially. We start by Sensible Heat Storage problems and move toward Latent Heat Storage models. We study mathematically and numerically the direct equations in composite media before tackling the optimum design of the TES system.

The WP2 is organized in three subwork packages as follow:

• WP2.1: Optimization of Sensible and Latent Heat Storage (Participants: F. Jelassi, S. Ervedoza, M. Tucsnak, Postdoc IMB and Post Doc LMAC)

  The Sensible Hybrid TES devices are often composed of a high thermal mass fluid (or solid) that encompasses conductive full metal capsules (enhancers). No phase change occurs during the heat up or cool down (heat loading/unloading) processes. As stated above, the capsules are there to enhance the effective conductivity of the whole device so to facilitate the heat flow, when injected in (or extracted from) the fluid medium. We are here interested in the loading step. The target is therefore to reach a more or less uniform storage density in the TES domain, regardless the heat inlet location. The lever is to optimize the dissemination of the capsules within the composite support. The preliminary mathematical model we work with is a coupled transient heat equation, with disparate parameters according to the region where they are set: high conductance for metallic capsules and high thermal capacitance of the fluid. The idealistic interface is thermally perfect: are prescribed there the continuity of the temperature and the fluxes conservation. A more realistic modeling is when, at the defected interface (capsules/media), the heat transfer rate is reduced due to a contact resistance; the temperature is hence discontinuous and has jumps. The pursued challenge is to achieve a numerical maximization of the stored heat in the device after a fixed time; the variables being the positions and the sizes of the spherical capsules. Obtaining the gradient formula of the cost function is relying on the adjoint state and requires the differentiation of the temperature field, solution to the coupled heat equations, with respect to the capsules (location and size). For spherical capsules, simulation may be sped-up, by implementing a domain decomposition algorithm with an analytic solver, for the inside heat equation matched with finite element method outside.

  The alternative to Sensible Heat Storage is Latent Heat Storage in PCMs materials: a powerful means for increasing the quantity of heat storage. The melting process absorbs a substantial heat amount (loading stage) while the solidification releases ‘an equivalent heat energy’ (unloading stage). In a first step, the linear heat equation is replaced by the non-linear Phase Change Enthalpy Stefan model to track the moving interface at the melting temperature. Depending on the research advances and the computing progress, we ambition later on to call for a rich phase-field equation of the mushy area. Either, the global model is composed of two Enthalpy Stefan equations in PCMs coupled with a linear heat equation in the capsules shells, or the shell can be accounted for as a thin membrane and interface transmission formulas on the temperature are mathematically derived. The following step is to review the optimization developments and run the algorithm after adapting it. In particular, the mathematical derivation of the temperature gradient, with respect to the capsules should be revisited to take into account the rough non-linearity which has to be regularized, beforehand. The stopping criteria should be re-investigated.

• WP 2.2: Shape design: Parametric and Topological Optimization of the Capsules( Participants: F. Jelassi, S. Ervedoza, M. Tucsnak, Postdoc IMB and Postdoc LMAC)

  Once the location and the size of the spherical conductive enhancers are optimized, one may wonder whether one can engineer a better feasible shape of the capsules that result in the twofold aim: higher effective conductivity and higher amount of heat storage. To help answer these points, we will implement a parametric shape optimization method where the starting configuration is the one obtained from the optimum spherical capsules. A mathematical sensitivity analysis is underway in the case of the linear heat equation, used for the sensible storage problem. We need to extend this analysis to handle the non linearity of the Enthalpy equation for PCM, in case of the Latent Heat Storage. In the numerical chapter, we need to customize the computing methods for the optimum design of interfaces (capsules/PCMs). To ensure the success of the approach, so to obtain at least a local maximizer, we will test and assess some regularization tools for the algorithm combined with suitable penalization procedures so to impose realistic constraints on the total surface of the interfaces, for example (see [All21b], [Hen07]). During the prediction/correction iterations of the geometry, one of the difficulties we expect has to do with updating the mesh near the interfaces. This can be achieved by applying the level set based mesh evolution method (see [All18] ). So far, the number of capsules to scatter in the TES device needs to be known in advance. However, it may be profitable to add or to suppress capsule(s). Parametric optimization fails to do so, and a remedy is to resort to the topological shape design alternative. Considering the number of enhancers as an optimization variable becomes possible. An asymptotic expansion at the vicinity of a virtual capsule produces the topological gradient. That sensitivity helps to decide whether to add that virtual capsule or not, and even to retrieve an existing one, so to increase the cost function (full heat stored in the media). The power of this approach resides in its simplicity, since it has been showed in [Gar01] how to use skilfully duality for an accessible formula to the topological gradient (or sensitivity).

• WP 2.3: Neural Networks and Deep Learning ( Participants: F. Ben Belgacem, N. Gmati and trainee)

  In the challenge we face, geometrical parameters are the variables that drive the scalar heat cost functions to optimize. That non-linear dependence is complex (necessity to solve computationally difficult PDE systems, see WP1 and WP3) and the resulting problem may be of high dimensionality (location and size of enhancers). Deep Neural Networks (DNN) appear as an affordable method for the (nonlinear) approximation of the cost correspondence: Geometrical Parameters maps to Stored Heat Amount. The obtained cost function, related to either the partial or full physical model (heat and motion), is afterwards maximized using classical optimization algorithms (descent gradient methods, Newton, BFGS, ...) (see [Luo21], [Viq21]). DNN needs efficient stochastic-Gradient method (size of mini-batch), a fitting network architecture (suitable hyper-parameters: hidden layers, ...) and suitable activation functions. The training or learning stage of the DNN enables the evaluation of the weights of the perceptrons connections and their biases. To do so, a significant amount of data should be collected, either numerically (synthetization) or experimentally (measurements). Theses datas will be obtained by FOM simulations. If additional data are needed, ROM computations might complete the data-set.

III) WP 3: Reduced Order Models for Thermal Energy Storage (TES)

(Scientific coordinator: Cyrille Allery, Teams: LaSIE, I2M)

This work package is dedicated to model reduction of the global system and its use in the context of optimization framework. Since the optimization process of the physical parameters or/and of the capsule's location and geometry, requires to solve numerically many times the problem, the methodology is very time consuming and a large amount of memory storage is needed. It is why it is necessary to build efficient and robust ROMs (Reduced Order Models). In this work package, first we will investigate the reduction of the solver, developed in WP1, by POD for a given geometry and a given set of physical parameters. The shortcomings of the POD-Galerkin method are generally due to the evaluation of the non-linear term. Therefore, here the difficulty concerns the evaluation of the cubic term of the Allen Cahn equation. In the second step, we will investigate a challenging and risky subwork package of how to adapt the optimization algorithms developed in WP2 in the ROM framework. The most used optimization techniques for optimal control are of gradient descent type such as developed in WP2. Although their practical effectiveness, gradient-based strategies are susceptible to generate spurious local minima, which may inhibit their capabilities in some control applications. To circumvent this limitation, it is possible to employ stochastic optimization strategies, such as genetic algorithms, but the number of the cost function evaluations required exceeds in general the number required by a gradient-based optimization. Whatever the chosen approach, the integration of ROM inside optimization algorithm is now an attractive and growing research. Since these two reduced optimization approaches are already used by the LaSIE team, in the context of fluid mechanics [Oul18] [Oul20], the both approaches can be used in the project. The aim is to extend these approaches to thermal energy storage.

• WP 3.1: Parametric POD-ROMs for a given geometry and a given set of physical parameters (Participants: C. Allery, M. Azaiez, A. Hamdouni, E. Liberge , and PostDoc)

  In this sub work package, we will investigate the construction of POD ROM associated to the solver developed in WP1 for a given geometry and a given set of physical parameters. As already mentioned a special attention should be done to treat the non-linear term of the Allen Cahn equation by using, for example, the DEIM approach. The I2M team [Zho19] has already considered this approach in previous works. Concerning the coupling of these equation with the heat transfer and Navier Stokes equations, we will endorse on the experience of the LaSIE Team on the development of ROMs for anisothermal flows [Tal16] and multiphasic problems to take into account the interaction between the spheres and the PCM [Lib10].

• WP 3.2: Optimization in the ROM framework (Participants: C. Allery, M. Azaiez, A. Hamdouni, E. Liberge , and PostDoc)

  The aim of this sub work package is to adapt the optimization algorithms developed in WP2 in the ROM framework, in order to obtain an efficient reduced optimization solver able to optimize in quasi real time the Thermal Energy Storage according to physical properties of PCMs or theirs position or size. A difficulty is due to the validity of the POD bases according to the variation of physical or geometrical parameters. In this project, to avoid this difficulty, the POD subspaces associated at different values of parameters will be interpolated with specific and adapted interpolation approaches on the tangent space of the Grassmann manifold [Mos18] [Mos20]. In particular, we will focus to extend this interpolation methods, for directly interpolating the ROM coefficients (in particular the nonlinear term) that will enable to avoid the Galerkin projection at each iteration of the optimization algorithm and to have significant time speedups. The developed approach will be first tested on nonlinear academic problems.

IV) WP 4: Validation

(Scientific coordinator: Fouzia Achaqchaq, Team: I2M)

This work package aims at validating experimentally the numerical model developed on some synthetic cases and benchmarks described in WP1. There are several numerical works considering a single and/or multiple hollow sphere encapsulating a PCM and arranged in a packed bed system, which is supposed to behave as a porous media. Analysis are mainly directed to the description of fluid flow and heat transfer inside these packed beds [Rad09] [Bel15]. Besides, the recent few works are dedicated to the study of these physical phenomena inside a hollow sphere fulfilled with a PCM [Ken20] or to the heat flow of a PCM in and around a solid sphere [Azi18]. To the best of our knowledge, there is none work, numerical or experimental, dedicated to the description of fluid flow and heat transfer simultaneously through both configurations described in Figure 1 (section Project). These experiments are very rare due to the difficulties related, on the one hand, to the material’s behavior such as how to avoid the leakage phenomenon and/or how to exploit the latent heat energy of a material in the due time because of its too high viscosity, among others. On the other hand, the addition of the other elements such as the heat transfer fluid and the consideration of the container’s scale change impact directly on the final performance of the PCM but also of the whole thermal energy storage system. This is the reason why the experiments, but also the numerical works, are either mainly oriented towards the study of the impacts of the addition of micro- and/or nanoparticles inside a PCM at Laboratory scale or oriented towards the optimization of the fins of a container’s experimental prototype to enhance the low thermal conductivity of the PCMs [Mao21] [Agr20] and to check the amount of available thermal energy density.

The experimental part of numerical work validation is challenging. Even the simplest experimental case that can be considered involves complex physical phenomena when it concerns a fluid dynamic problem coupled with heat transfer one. Moreover, the available means is too low to consider a potential application at high temperature (≥ 200°C), if this application is successful. This is the reason why we will focus on to the case of the molten PCM + solid sphere system for low temperature applications.

The WP4 is organized in two subwork packages, as follow:

• WP 4.1: Specification, conception and design of the experimental device (Participants: F. Achchaq and C. Lebot)

  The aim of this sub work package is to find out the pure PCM to neglect any chemical and/or sedimentation effects encountered with PCM-based alloys and the macroscale spheres with the suitable physical parameters to set up the experiment device described previously. The difficulties will be mainly on finding an interesting PCM in terms of thermal energy density with the lowest viscosity as possible and the right densities allowing the manifestation of the Archimedes’ principle effect or, at least, minimizing the gravity effect on the spheres.

• WP 4.2: Experimental device implementation and comparison and optimization of the results (Participants: F. Achchaq, C. Lebot, M. Azaiez and Master internship)

  This sub work will be dedicated to the set up of the experiment device and to the different protocols to achieve the objectives described above. Innovative methods added to the use of the visible and infrared cameras will be explored to follow the spheres’ trajectories embedded inside the molten PCM, in the vicinity of their initial positions, and their impact on the final thermal energy storage system performance. The results will be compared with the numerical ones to optimize the working conditions’ parameters of the device as well the targeted numerical models.

References