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Abstract
A new method to control heat flux, called thermal surface transformation (TST), is introduced from transformation thermodynamics. Compared with transformation thermodynamics, TST has many advantage. First, there is no mathematical calculation during the whole process in TST (novel thermal devices can be designed graphically in a surface-to-surface way). Second, all thermal devices of various functions, shapes and sizes designed by TST only require one homogenous anisotropic thermal medium, i.e., thermal-null medium (TNM). With the help of the effective medium theory, TNM can be realized by layered copper and expanded polystyrene, whose performance on controlling heat flux by TST is verified by numerical simulations. Many examples are given, including thermal imaging devices, thermal unidirectional cloak, concentrator, rotator and thermal focusing devices.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Controlling electromagnetic waves by coordinate transformation method, i.e. transformation optics (TO), has been successfully applied to design novel optical devices and explore new electromagnetic phenomena [1–3]. However, many TO-based devices require complex materials (often anisotropic and inhomogeneous), which have to be simplified before experimental demonstration with metamaterials [4–6]. Optical surface transformation (OST) is a new theoretical branch derived from TO, which can design optical devices by choosing suitable geometrical surfaces without involving any mathematical calculations [2,7]. All devices designed by OST only require one homogenous anisotropic medium (optic-null medium), which has been widely realized at the microwave frequency range [8,9]. Many novel optical devices have been designed by OST, such as invisibility cloaks [10], waveguide bends [11], novel lenses [7], etc.
In recent years, based on the invariant form of thermal conduction equation under coordinate transformations, transformation thermodynamics (TT) has been proposed [12–15]. Many novel thermal devices have been designed by transformation thermodynamics including thermal cloaks [16–18], concentrators [19–21], thermal illusion devices [22–24], thermal diodes [25], thermal buffering [26], thermal hose [27], thermal encoding [28], thermal printing [29], heat-source transformation [30], etc. However, TT also has two main drawbacks. First, it requires coordinate transformation and tensor calculations during the design process, which is not convenient to thermal engineers who are not good at TT. Secondly, if the shapes, sizes or functions of thermal devices designed by TT need to change, the required materials of the devices have to change accordingly. We cannot find one homogenous special material to realize all thermal devices designed by TT. In this study, we extend a traditional TT to novel thermal surface transformation (TST), which can greatly simplify both designing process and material requirements. In our TST, we use a geometric graphic method to make all designs without involving any mathematical calculations. All TST-based devices can be realized by one homogeneous anisotropic material, namely, thermal-null medium (TNM), which has been experimentally realized by layered copper and expanded polystyrene [26].
Some other reported methods can also be used to design thermal devices using isotropic coatings [31–33]. However, the coating method can only give “thermal scattering illusions” (i.e., thermal cloaking) when a thermal scatter is illuminated by a heat flux which has an initially uniform density. The proposed TST can not only give some thermal scattering illusions, but also some other kinds of thermal control (e.g., remote cooling and thermal imaging). If the thermal scatter changes, the required thermal conductivity of the coating will change accordingly for the coating method [31], while only one kind of material (TNM) is needed to realize all novel thermal devices designed by our TST regardless of the sizes, shapes and functions of these thermal devices.
2. Thermal surface transformation (TST) and thermal null medium (TNM)
In TT there are two spaces that are linked by coordinate transformations: one is the reference space (a virtual space) and the other one is the real space. In this paper, quantities with and without primes are in the real and reference spaces, respectively. Thermal conduction equation without heat sources in the real space can be written by:
From the perspective of TT, the function of the TNM is to project the temperature distribution from one arbitrarily shaped surface S1 to another arbitrarily shaped surface S2 along x’ direction when these two surfaces are connected by TNM with its main axis along x’ direction (see Fig. 1(a)). Next, we will use TT to derive the required thermal conductivity of TNM. We can compress the red region in the real space (see Fig. 1(a)) into a thin slab of thickness Δ in the reference space (see Fig. 1(b)) by the following coordinate transformations:
Fig. 1. The coordinate transformation from the real space (a) to the reference space (b) in Eq. (5).
If the thermal material in the reference space is homogeneous background medium with κ= κ0, the required thermal conductivity between S1 and S2 in the real space can be calculated by Eqs. (2) and (5):
If Δ approaches zero in the reference space, the thermal medium between S1 and S2 in the real space becomes an extremely anisotropic medium (thermal conductivity is extremely large along x’ direction and close to zero in other orthogonal directions), which is a TNM with the main axis along x’ direction. From the perspective of TT, the thin slab of thickness Δ in the reference space will reduce to a surface (no longer a volume) when Δ→0. That is the reason why we call its corresponding region between S1 and S2 in the real space thermal-null medium (TNM). “Null” means its corresponding volume of space is zero in the reference space.
Two surfaces of arbitrary shapes S1 and S2 linked by the TNM with the main axis along x’ direction in the real space corresponds to the same surface in the reference. It means S1 and S2 are equivalent surfaces, i.e. each point on S1 has a corresponding point on S2 (the temperature distribution on S1 is ideally projected onto S2 along x’ direction). Although the above calculation is for the case when the TNM’s main axis is along x’ direction, the conclusion also holds when the TNM’s main axis is along any other direction, as we can always set up a local Cartesian coordinate with its x’ axis along the TNM’s main axis.
Now we get the key conclusion in TST: the function of TNM is to project the temperature distribution from its input surface to its output surface along its main axis. All surfaces vertical to the TNM’s main axis are equivalent surfaces, which correspond to the same surface in the reference space. In TST, we only need to design geometrical shapes of the thermal device’s input and output surfaces, and then fill TNMs with appropriate main axes that can project the temperature distribution from the input surface to the output surface.
3. Novel thermal devices and numerical simulations
Figures 2(a) and 2(b) verify the TNM’s projecting property, which can be used as a thermal imaging device. In Fig. 2(a), if we set two thermal sources on one plane, we will get their thermal images on the other plane of the same size, which is a thermal imaging device without magnification. In Fig. 2(b), the input plane is smaller than the output plane, and both are also connected by TNMs (the main axes are different in three regions). In this case, it is a thermal imaging device with magnification (the magnification factor is determined by the ratio of the input surface’s area to the output surface’s area). Thermal focusing device can be designed when the main axes of TNMs piecewise changes (see Fig. 2(c)). If the heat fluxes incident from left onto the thermal focusing device (see Fig. 2(d)), TNMs can guide the heat flux directionally, concentrate the heat flux gradually, and obtain a focused hot spot on the output surface of the device.
Fig. 2. (a) and (b) Temperature distributions: (a) Two planes of the same size are linked by the TNM with the main axis along the x’ direction. (b) TNMs are filled in three regions whose main axes are along x’, radical and x’ directions in each region. The circular boundary of the calculation region is set by a fixed low temperature 293K. Two hot sources with temperature 375K are set on the left input surface of the TNMs. The TNM blocks are bounded by the black lines and the main axis’ directions are indicated by the black arrows. (c) The structure of the thermal focusing device. The red and blue regions are TNMs with main axes along x’ and r’ directions, respectively. (d) The temperature distribution of the thermal focusing device when left and right boundaries are set by fixed temperatures 393K and 293K, respectively. The upper and lower boundaries are thermal insulation.
In a room with a hot source, one may wish to cool down the room, preferably making the temperature distribution in most area of the room nearly uniform and comfortable to live, by putting a cooling source (like a refrigerator) near the hot source. However, the temperature distributions of the whole region around the two sources are quite non-uniform even if the cooling source is close to the hot source. As shown in Fig. 3(a), the temperature of the region around the hot source is much higher than the temperature of the region around the cooling source. Figure 3(b) shows a different situation when we utilize the TNM to wrap around two sources to achieve a nearly uniform temperature distribution. Figures 3(c) and 3(d) are numerical simulated results corresponding to Figs. 3(a) and 3(b), respectively. In Fig. 3(d), the main axis of the TNM around the hot and cooling sources are along the line that links the centers of two sources (i.e., the x’ direction here). From the perspective of TT, the hot source and cooling source on the same line parallel to the x’ axis are equivalent points, which correspond to the same point in the reference space. In other words, with the help of the TNM, the hot source and cooling source are overlapped with each other. Therefore, the uniformity of temperature outside the TNM (the white region) in Fig. 3(d) is greatly improved compared with the case if the TNM is removed in Fig. 3(c) (the temperature outside the white region is obviously non-uniform). Figures 3(e) and 3(f) show the temperature distribution on a circle and a line parallel to the x axis outside two sources, which quantitatively shows that TNM can achieve remote cooling with high uniformity of temperature (around 293K outside the device).
Fig. 3. (a) The temperature distribution is quite non-uniform around two sources even if the cooling source is very close to the hot source. (b) When TNM is put around two sources, the temperature distribution outside can be nearly uniform. (c) and (d) are simulated temperature distributions without and with the TNM, respectively. The hot source (393K) and cool source (193K) are indicated by the red and blue dots, respectively. The outer circular boundary is set as room temperature (293K). The white regions are TNM in (d) and background medium in (c). (e) The temperature distribution on a circle (the center is the middle point of the two sources and the radius is 0.7m) with (blue line) and without (red line) the TNM, respectively. (f) The temperature distribution on a line parallel to the x axis (5cm above the white region) with (blue line) and without (red line) the TNM, respectively.
Thermal unidirectional cloaks, concentrators and rotators can be designed by TT with suitable coordinate transformations. We can directly design these devices without any mathematical calculations by TST and realize all these devices of various shapes and sizes by one material (i.e., TNM). TNM can guide the heat flux directionally along its main axis, which can be treated as a special thermal hose/waveguide [26,27]. Properly arranged TNMs can guide the detecting heat flux around the concealed region and then guide them back to their incident direction, which perform as a thermal unidirectional cloak. Figure 4(a) shows one simple design to achieve a thermal guiding effect by TNMs, whose performance is shown in Figs. 4(b) and 4(c). Some heat flux incident onto the upper and lower edges of our unidirectional thermal cloak would influence its performance, which leads to the non-uniformed outer temperature distribution. Similarly, some thermal concentrator and rotator can also be designed by choosing TNMs with suitable orientations (see Figs. 4(d)–4(i)). For the rotator and concentrator, they are circularly symmetrical. Therefore, the rotator and concentrator can also work when the incident thermal flux is of oblique incidence. The performance of the unidirectional thermal cloak will be influenced when the background thermal flux is of oblique incidence.
Fig. 4. (a) Thermal unidirectional cloak by TST. The yellow region is the concealed region. The red and green regions are TNMs with main axes along + 45 degree and −45 degree (indicated by the black arrows), respectively. The heat flux is guided around the concealed region and smoothly redirected back to the original direction by TNMs (we have designed a similar structure for cloaking in microwave frequency by optic-null medium [10]). (b) and (c) are temperature distributions and isothermals for the thermal unidirectional cloak, respectively. (d) Thermal concentrator by TST. The purple region is TNM with main axis along the radical direction (indicated by the black arrows). (e) and (f) are temperature distributions and isothermals for the thermal concentrator, respectively. (g) 45 degree rotator designed by TST. The gray regions are TNMs whose main axes are indicated by the black arrows (consistent with the heat flux directions inside the rotator). (h) and (i) temperature distributions and isothermals for the thermal rotator, respectively. In the above simulations, we set left and right boundaries of the simulation area by fixed temperatures of 393K and 293K, respectively. The upper and lower boundaries in the simulations are set as thermal insulation. Thermal conductivity of the TNM is set as 1000 W·m−1·K−1 along the main axis and 0.001 W·m−1·K−1 in other orthogonal directions.
One important function of a thermal cloak is to achieve a zero gradient of the temperature distribution in the central concealed region. We plot the temperature distribution along x direction when heat fluxes of uniform density incident from various angles with and without our unidirectional thermal cloak in Fig. 5(a). Compared with the case when the cloak is removed, our unidirectional thermal cloak can still give a zero gradient of the temperature field along x direction in the central concealed region even if the incident angle of the background thermal flux changes. In some other applications, high gradient of the temperature field is desired. The thermal concentrator in Fig. 4(d) can help to enhance the gradient of the temperature field. Figure 5(b) shows the temperature distribution along the x direction when heat fluxes of uniform density normally incident onto the thermal concentrator in Fig. 4(d). For various sizes of the thermal concentrators (b/a are different; b and a are outer and inner radii of the concentrator, respectively), the gradients of the temperature field in the central region of the concentrator are different: the larger b/a the larger gradient along the x direction. The slopes of the curves in Fig. 5(b) give the gradient of the temperature field along the x direction, which means very large gradient of the temperature field can be achieved by choosing very large b/a.
Fig. 5. (a) The temperature distribution in the concealed central region (yellow region in Fig. 3(a)) along the x direction with the unidirectional thermal cloak (blue line series) and without the cloak (red line series) for various incident angles of the uniform background thermal flux. Here the concealed object is a material of high thermal conductivity (к=1000 W·m−1·K−1). (b) The temperature distribution in the central circular region in Fig. 4(e) along the x direction with the concentrators of two different sizes (the ratio of the outer radius to the inner radius b/a varies) and without the concentrator. If the ratio of the outer radius to the inner radius b/a becomes larger, the slope of the line increases (see the blue line), which indicates high gradient of the temperature field along the x direction in this region. Other parameters are the same as those in Fig. 4.
4. Realization design and discussion
Ideal TNM’s thermal conductivity is infinitely large along its main axis and nearly zero in other orthogonal directions. The reduced TNMs, whose thermal conductivity is very large along its main axis and close to zero in other orthogonal directions, also keep good performance (i.e. guiding the heat flux directionally). The reduced TNM can be realized by layered copper (кc = 394 W·m−1·K−1) and expanded polystyrene (кe=0.03 W·m−1·K−1) on the thermal epoxy background (кb=3.4 W·m−1·K−1) [26,27]. Based on thermal effective medium theory [34,35], the effective thermal conductivity of layered copper and expanded polystyrene is 197.015 W·m−1·K−1 along the boundary of two materials and 0.015 W·m−1·K−1 in the direction vertical to the boundary of two materials, which can be treated as a reduced TNM. If we replace the ideal TNM in Fig. 4 by the layered natural materials (layered copper and expanded polystyrene), these thermal devices can also give good performance (see Fig. 6).
Fig. 6. (a)–(c) Simulated temperature distributions for the unidirectional cloak, concentrator and rotator when the TNMs are realized by layered copper and expanded polystyrene on the thermal epoxy background.
As many applications of the heat conduction are on the surface, novel thermal devices designed in this study are in 2D case. TST is derived from the 3D coordinate transformation formula and 3D transformation thermodynamics, which can be directly applied to design novel 3D thermal devices. 3D expression should be used for the thermal conductivity of TNM in Eq. (6), which can be realized by inserting copper wires inside an expanded polystyrene plate.
5. Summary
TT is a powerful tool to design novel thermal devices of amazing functions. However, its applications are limited due to the complex mathematics (i.e., coordinate transformations and tensor calculations) and complex material requirements (which often require inhomogeneous anisotropic thermal conductivity, and the required materials are different once the shapes, sizes or functions of thermal devices change). TST proposed in this study can overcome these two main restrictions in TT (i.e., greatly simplify both the design process and material requirement). All devices in TST can be designed graphically in a surface-to-surface way without involving any mathematical calculations, and all the designed devices of various functions, shapes and sizes only require one material (TNM). TNMs can project temperature distribution from one surface to another surface along its main axis, which can be treated as a special thermal waveguide/hose that can guide the heat flux directionally. Due to the unique properties of TNMs, novel thermal devices designed by TST is in a graphical way without any mathematical calculations. In TST, the whole designing process only takes two steps: design the geometrical shapes of the surfaces for a thermal device and fill TNMs with proper main axes between these surfaces. All thermal devices designed by TST can be realized by reduced TNMs, which only require two types of natural materials (staggered copper and expanded polystyrene). TST will provide a new theoretical method in controlling temperature fields and designing novel thermal devices.
Funding
National Natural Science Foundation of China (61971300, 11604292, 11621101, 11674239, 61905208, 91833303, 60990322, 91233208); Scientific and Technological Innovation Programs (STIP) of Higher Education Institutions in Shanxi (2019L0146, 2019L0159); China Postdoctoral Science Foundation (2017T100430, 2018M632455); National Key Research and Development Program of China (2017YFA0205700); Program of Zhejiang Leading Team of Science and Technology Innovation.
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