Optimizing fin design for a PCM-based thermal storage device using dynamic Kriging

https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.143Get rights and content

Highlights

  • A strategy for the optimization of latent heat thermal storage device is proposed.

  • Dynamic Kriging is used to create a response surface for the objective function.

  • Elementary effect variable screening and dynamic mesh refinement help reduce cost.

  • Horizontal fins reduce the significance of convective heat transfer.

  • Device shape and composition prove more important than internal fin configuration.

Abstract

A key challenge in the development of a practical thermal storage device (TSD) is the low thermal conductivity of common phase change materials (PCM). This low conductivity impedes both heat input and extraction. The most common solution is to use conductive metal fins to spread heat through the device. However, optimizing the effectiveness of the container and the fin arrangement is difficult due to the large number of potential design parameters. This paper develops a strategy to make simulation-based optimization process affordable and accurate. First, numerical techniques are designed to accurately and efficiently compute heat and mass transport in a variety of geometries without generating grids to conform to each geometry. This facilitates rapid prototyping and mitigates the expense of individual simulations. Second, a pre-screening process identifies the independent variables with the largest and most nonlinear effect on the objective function in the optimization process, thus narrowing the parameter space. Finally, a dynamic Kriging-based optimization approach constructs a multidimensional response surface using sparse input datasets; the response surface is then used to identify an optimal design. The combination of the above three strategies is shown to result in an approach that can aid in the design of optimal thermal storage devices that rely on a mixture of PCM and metal fins.

Introduction

The high energy density and stable temperature fields of latent heat TSDs make them promising in a range of applications, including solar energy storage [1], solar cooking [2], home heating and cooling [3], [4], and thermal buffering [5], [6]. The chief engineering challenge in building an effective latent heat TSD is to find a way to complement the storage capabilities provided by the low-conductivity PCM with a suitable enhanced heat transfer mechanism.

Researchers have investigated a wide range of approaches to conduction enhancement within latent heat TSDs. These approaches include both attempts to enhance the thermal conductivity of the PCM itself as well as the inclusion of additional geometric objects within the PCM [7], [8], [9], [10], [11]. The former tactic [12] includes doping the PCM with highly conductive nanoparticles [13], impregnation of porous materials such as expanded graphite or metal foam into the PCM [14], [15], [16], [17], and the inclusion of low-density materials like carbon fibers within the PCM [18]. The latter approach has produced innovations as well. Encapsulation of the PCM in a protective polymer coating can lead not only to increased thermal conductivity with the TSD but also a more stable PCM [19], [20], [21]. Heat pipes use evaporation and condensation in order to carry thermal energy between a heat transfer fluid and the PCM, and have been shown to significantly increase heat transfer into the PCM [22], [23].

However, the most common strategy to enhance heat transfer into and out of a PCM thermal storage unit is the use of high conductivity fins [11], [24], [25]. This paper aims to use numerical simulation to optimize the design for a bottom-heated TSD that uses a metal fin arrangement to spread heat to a low conductivity PCM. Previous computational and experimental work on finned heat spreaders demonstrates that they consistently enhance heat transfer within a variety of types of latent heat TSDs [26], including containers with flat plate fins [27], [28] and pipes with axial [29], [30], and most relevant to this work, containers with circular fins [31], [32], [33]. However, identifying the optimal design for these devices is complicated. The main challenge is the vast number of potential designs for such a device: even the simplest designs have multiple independent parameters, such as the number of fins, fin thickness, fin height, distance between fins and the angle between fins. Each independent variable exponentially increases the number of design points needed to cover the parameter space. On the one hand, in the past decade, some researchers have developed correlation equations using non-dimensional parameters to predict the behavior of TSDs at design points other than the tested design points [34], [35]. But for the most part, much of the work done with thermal storage devices has been restricted to the study of the effect of one or two fin parameters on the efficiency of heat storage or discharge [36], [37], [38], [39].

In recent years, however, researchers using numerical simulations have employed more sophisticated optimization tools to explore larger parameter spaces for devices that rely on enhanced heat transfer using fins. Sciacovelli et al. [40] optimized tree-shaped fins for a double pipe heat exchanger by using numerical simulations to create a response surface in a 4-dimensional parameter space. Pizzolato et al. [41] identified an optimal fin design for an annular heat exchanger with a topological optimization approach. More recently, Alayil et al. [42] used an artificial neural network to find an optimal melting solution for a finned rectangular TSD with four independent parameters. All of these studies employ state-of-the-art optimization tools and a moderate number (30–100) of simulations in order to estimate the effectiveness of heat transfer at untested design points; in doing so, they are able to pinpoint an optimal design over a much wider parameter space than would be possible without such tools.

The present work develops a novel approach to optimization in order to contribute to this emerging work on optimization of PCM-based, finned TSDs. To increase the efficiency of the optimization procedure, we streamline three different parts of the optimization procedure as follows:

  • 1.

    First, the expense of individual simulations is minimized. The design investigated is a TSD which is heated from the bottom and releases its energy through a top outlet plate. It uses a finned aluminum heat spreader to transfer heat into and out of solar salts, a non-eutectic mixture (70%/30%) of NaNO3 and KNO3 [43]. The complicated geometry of the flow domain, phase changes of solar salts, contact conditions between the salts and Al surfaces, high temperature gradients, and rapid transients make the simulations within this space numerically challenging and computationally expensive. Several tools are used to reduce this cost. First, the flow solver [44], [45], [46] uses a strongly coupled implicit scheme for conjugate heat transfer and a sharp interface method to define the boundary between different materials: this combination of techniques produces a robust and accurate simulation of the phase change and heat transfer phenomena. Second, the combination of a Cartesian grid-based immersed boundary approach [47] and solution-adaptive locally refined meshes [48] eliminates the need to manually generate a mesh for each geometry. This allows for the resolution of fine flow features and interfacial transport while removing a time-consuming and user-intensive intermediate step in conducting flow simulations. When a simulation begins, the algorithm automatically creates a non-boundary conforming mesh based on the geometry input. A more refined Cartesian grid is established near boundaries between materials, while a coarse grid is used in areas removed from boundaries. As the flow develops, the flow solver measures the first and second derivatives of the temperature and velocity fields and refines/coarsens the mesh as necessary [48]; no computational expense is wasted on areas of weak transport, which is especially important in phase change simulations where large portions of the domain can be quiescent.

  • 2.

    The second measure that increases the efficiency of the optimization process is the use of an elementary effect-based variable screening procedures [49]. Each design of the TSD is defined by several (say d) independent design variables. For large d, the so-called “curse of dimensionality” makes the creation of a d-dimensional response surface prohibitively expensive [50]. The present work employs an elementary effect-based screening strategy that identifies independent variables that have a negligible effect on the objective function. In this approach, a small number of simulations are used to determine both the average magnitude and the standard deviation of the elementary effect of each variable; that is, the change in the objective function created by small perturbations in the variable. The variable screening leads to a reduction of the number of independent design parameters while facilitating the identification of the optimal design.

  • 3.

    The third strategy used to improve the efficiency of the optimization procedure is to create a response surface using a dynamic Kriging (DKG) method [51], [52]. Response surface construction, also called surrogate modeling or metamodeling [53], uses the results of simulations from a finite number of design points to create a response surface that represents the objective function in the spaces between the known (input) design points. The advantage of this approach is that it limits the number of simulations that must be run in order to find the optimal design. Most approaches to response surface optimization, however, require a priori assumptions about the nature of the function: the order of the polynomial basis functions, for instance, must be decided upon before the creation of the response surface [50]. The dynamic Kriging (DKG) approach improves upon these methods by using likelihood estimates to test the accuracy of a range of polynomial radial basis functions, so that the final response surface can be built from basis functions that are most appropriate to a given set of data [51], [52]. The result is a surrogate modeling process that is both efficient and accurate [54].

The chief aim of the present work is to optimize the design of a practical latent-heat thermal storage device. The numerical and optimization methodology presented in this paper are designed to achieve this efficiently and accurately in order to guide the design of TSDs that combine PCMs for latent storage and metal fin structures for heat spreading. Section 2 presents an introduction to the optimization problem, and explains the setup of the simulations. Section 3 provides the governing equations solved and a brief discussion of the computational methods used. Section 4 explains the variable screening process and its results. Finally, Section 5 covers the design process and the results of the optimization procedure, including an analysis of the response surface behavior.

Section snippets

TSD geometry and materials

The basic design for the TSD, shown in Fig. 1, is a PCM-filled chamber with metal core, fins, and container surrounded by a layer of insulation. The materials used in the container are aluminum, rice husks (as insulation), and a non-eutectic mixture of solar salts (NaNO3 (70%) and KNO3 (30%)). Non-eutectic solar salts were chosen as the PCM because they are readily available, inexpensive, and have a large melting range (see Table 1), which allows for a long period of relatively stable

Governing equations

The governing equations are the Navier-Stokes equations and the energy equation. These nonlinear equations are simplified by assuming incompressibility, and by using the Boussinesq approximation and the enthalpy-porosity formulation for phase change. Because the TSD design is cylindrical, because all heat fluxes and boundary conditions are axially symmetric, and because the flow is laminar, two-dimensional axisymmetric flow is assumed, viz. vθ=0 and θ=0 for all variables. Under these

Results

The Results section is divided into five sections. The first section explains the screening process used to reduce the number of independent variables from five to three. The following two sections explain the dynamic Kriging strategy used to create the objective function response surface and introduce the overall results of this optimization process. The remaining two sections explore in detail the objective function surface for charging and discharging.

Conclusions

This paper uses numerical simulations to optimize the design for an aluminum-fin and PCM thermal storage device. The main challenge to optimizing such systems is the vast number of potential designs for such a TSD. We use three strategies to significantly decrease the expense of this process while retaining its accuracy. First, the expense of individual simulations is reduced by the numerical scheme, which utilizes adaptively refined Cartesian grid meshes, a strongly coupled implicit scheme for

Competing interests

There are no conflicts of interest for the authors of this paper.

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