Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Tiancheng Han^{a},
Jiajun Zhao^{a},
Tao Yuan^{a},
Dang Yuan Lei^{b},
Baowen Li^{cd} and
Cheng-Wei Qiu*^{a}
^{a}Department of Electrical and Computer Engineering, National University of Singapore, 119620, Republic of Singapore. E-mail: chengwei.qiu@nus.edu.sg; Fax: +65-6779-1103; Tel: +65-65162559
^{b}Department of Applied Physics, The Hong Kong Polytechnic University, Hong Kong, China
^{c}Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, 117546, Republic of Singapore
^{d}Center for Phononics and Thermal Energy Science, School of Physical Science and Engineering, Tongji University, 200092, Shanghai, China

Received
2nd May 2013
, Accepted 8th July 2013

First published on 8th July 2013

Three-dimensional devices capable of efficiently harvesting light energy or microwave radiation from arbitrary directions are still challenging to make due to the stringent requirement of inhomogeneous and extreme material parameters. This usually requires the use of metamaterials and results in time-consuming and complicated fabrication, narrow bandwidth performance and huge losses, which prevent these devices from being extended to large-scale energy-related applications. In this paper, we demonstrate that thermodynamic cells harvesting heat energy in three dimensions can be achieved by employing naturally available materials with constant thermal conductivity. Particularly, the thermal-energy harvesting efficiency of the proposed devices is independent of geometrical size and may achieve nearly 100% with tunable anisotropy, much superior to the concentrating devices reported so far. Theoretical analysis and numerical experiments validate the excellent performance of the advanced thermal cells. We further show that such thermal cells can be practically realized by using two naturally occurring conductive materials in a simplified planar geometry, which may open a new avenue for potential applications in solar thermal panels and thermal-electric devices.

## Broader contextOver the last few decades, much more attention has been paid to renewable energy. In addition to solar energy, another important energy source is thermal energy, which can be collected from the environment. Three-dimensional devices capable of efficiently harvesting thermal energy from arbitrary directions are still challenging to make due to the stringent requirement of inhomogeneous and extreme material parameters. This usually invokes the use of metamaterials and results in time-consuming and complicated fabrication, and huge losses, which prevent these devices from being extended to large-scale energy-related applications. In this paper, we demonstrate that thermodynamic cells harvesting heat energy in three dimensions can be achieved by employing naturally available materials with constant thermal conductivity. Particularly, the thermal-energy harvesting efficiency of the proposed devices is independent of geometrical size and may achieve nearly 100% with tunable anisotropy, much superior to the concentrating devices reported so far. We further show that such thermal cells can be practically realized by using two naturally occurring conductive materials in a simplified planar geometry, which may open a new avenue for potential applications in solar thermal panels and thermal-electric devices. |

On the other hand, light-harvesting devices based on transformation optics,^{13,14} capable of efficiently harvesting and focusing the incident light energy without severe reflection or absorption, are believed to play an important role in improving the energy-conversion efficiency of current solar cell devices in which high field-intensities are usually preferable.^{15–17} In addition to solar energy, another important energy source is thermal energy, which can be collected from the environment. Transformation-optic method has also been extended to manipulate heat current.^{18,19} However, practical realization of concentrating devices, particularly in three dimensions, often relies on the spatially varying material parameters comprising tensor components,^{15,16} though the ideal parameters can be reduced partially.^{17} Decoupling electric and magnetic effects, two-dimensional static electric concentrators based on resistor networks^{20} and static magnetic concentrators based on superconductor-ferromagnetic metamaterials^{21} have been demonstrated recently. Analogous to wavedynamics,^{15–17} thermal concentrating devices reported so far are also facing similar serious bottlenecks, such as inhomogeneity,^{19} which is challenging for the practical applications of three-dimensional thermal cells. Although a two-dimensional cylindrical concentrator made of latex rubber and processed silicone has recently been demonstrated in experiment and shown the fascinating ability to concentrate heat current, its concentrating efficiency is severely limited by the two-dimensional configuration, with an optimal value of ∼12%.^{22} Thus, designing three-dimensional thermal concentrators with naturally available materials is of particular importance in enhancing the heat-harvesting efficiency in many energy-related applications.

In this paper, we establish the theoretical account and a general design road map for creating a realizable three-dimensional thermal cell made of natural conduction materials. Different from the previously studied concentrating devices in wave dynamics^{15–17} and thermodynamics^{18,19} obtained from rigorous transformation optics,^{13} the proposed novel thermal cell is homogeneous in materials composition, and its performance is independent of the geometrical size but mainly dominated by the thermal-conduction anisotropy. By judiciously selecting the natural materials with strong thermal-conduction anisotropy or constructing composite materials of highly anisotropic thermal-conduction, the heat-concentrating efficiency of the studied cell can reach unity in an ideal situation. Due to the homogeneous response of the materials' conductivity in each individual direction, we further demonstrate that such advanced thermal cells can be realized in experiment through sophisticated spatial-arrangement of two naturally occurring conductive materials.

(1) |

The schematic diagram for the design of a three-dimensional thermal cell is shown in Fig. 1(a), where the spherical region (0 ≤ r ≤ b − δ) and the shell region (b − δ ≤ r ≤ b) in virtual space are respectively mapped onto the inner core (0 ≤ r′ ≤ a) and coating (a ≤ r′ ≤ b) in real space. When δ → 0, the resulting thermal cell then consists of two parts:

Fig. 1 (a) Spatial coordinate transformation for the design of a homogeneous three-dimensional thermal cell. (b) The scheme for the realization of the homogeneous thermal cell in (a) with two naturally occurring materials. |

For 0 ≤ r′ ≤ a (inner core):

κ^{′}_{r} = κ^{′}_{θ} = κ^{′}_{φ} = b/a
| (2a) |

For a ≤ r′ ≤ b (coating):

κ^{′}_{r} → ∞, κ^{′}_{t} = κ^{′}_{θ} = κ^{′}_{φ} → 0
| (2b) |

Because the thermal conductivity of the inner core is homogeneous and isotropic, we then mainly focus on the conductivity of the coating shell. We may set κ^{′}_{r} = 2^{n} and κ^{′}_{t} = 2^{−n}, which exactly satisfy eqn (2b) if n is large enough. Assuming n = 4 (κ^{′}_{r} = 16 and κ^{′}_{t} = 1/16), Fig. 2(a) shows the temperature profile and the heat flux streamline of a homogeneous spherical concentrator with a = 0.1 m and b = 0.4 m. It is clear that nearly all of the heat flux in the region (0 ≤ r ≤ b) is focused into the inner core (0 ≤ r ≤ a) without any reflection and distortion. According to the above theoretical analysis, the thermal cell will function more and more perfectly with the increase of n. However, what will happen if one pushes n toward the other extreme, i.e., decreasing n? Fig. 2(b) shows the temperature profile and heat flux streamline of the thermal cell in Fig. 2(a) with n = 1 (κ^{′}_{r} = 2 and κ^{′}_{t} = 0.5). It is very surprising that the thermal cell is still able to focus the heat current into the inner core without any reflection and distortion, which has never been achieved in wave dynamics^{15–17} counterparts reported so far.

Fig. 2 Normalized temperature profile and heat flux streamline for a spherical thermal cell with a = 0.1 m and b = 0.4 m. (a) n = 4. (b) n = 1. Streamlines of thermal flux are also represented by white color in the panel. |

(3) |

(4) |

Considering the symmetry in the φ direction and antisymmetry in the z direction, the temperature field in the three regions can be respectively expressed as

(5a) |

(5b) |

(5c) |

(6) |

According to the boundary condition T|_{z=±z0} = ±T_{0}, it is found that T_{3} is the linear distribution (which means E_{2m−1} = 0) and one only needs to consider m = 1. Assuming κ_{1} = κ_{3} = 1, we can obtain

(7) |

Clearly, if n is large enough, T_{1}/T_{3} → b/a, which means that nearly 100% efficiency is achieved. To quantitatively examine the concentrating efficiency (CE) with variance of the heat conduction anisotropy (denoted by n), we define

CE^{3D}_{SR} = |T|_{z=a} − T|_{z=−a}|/|T|_{z=b} − T|_{z=−b}|
| (8) |

Fig. 3 Concentrating efficiency of the thermal cell with a = 0.1 m and b = 0.4 m as a function of n. |

Fig. 4 (a) The scheme for the realization of a thermal cluster with alternating naturally available materials, such as epoxy and rubber. (b) Top view of the temperature profile and heat flux streamline for the thermal cluster. Streamlines of thermal flux and isothermal are also represented by white and yellow colors in the panel, respectively. (c) Concentrating efficiency of the thermal cell in (b) as a function of n. |

Based on the theoretical analysis of the temperature distribution of the homogeneous cylindrical concentrator, we can obtain T_{1}/T_{3} = (b/a)^{1−l} with . Clearly, we can obtain T_{1}/T_{3} → b/a if l → 0, which means that nearly 100% efficiency is achieved. To quantitatively examine the concentrating efficiency of the homogeneous cylindrical thermal cell in Fig. 4(b), we calculate CE^{2D}_{SR} = |T|_{x=a} − T|_{x=−a}|/|T|_{x=b} − T|_{x=−b}| from simulation results and CE^{2D}_{TR} = T_{1}a/T_{3}b from theoretical results, as shown in Fig. 4(c). The theoretical results are in good agreement with the simulation results. As can be seen from Fig. 4(c), the concentrating efficiency up to 100% can be achieved when n = 8. This example shows that an ultra-efficient thermal concentrator could be created, if a natural material with strong conduction anisotropy can be found or an effective material of highly anisotropic conduction can be constructed artificially.

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