Keywords

1 Opening Remarks

Thermal regulation has been an important issue since ancient times, and it is related to the survival and development of human beings. Therefore, thermal regulation has attracted countless researchers to continue to explore. Thermal metamaterials refer to artificially designed and prepared structural materials that can achieve novel thermal functions [1,2,3,4,5]. Since transformation thermotics was proposed by Fan et al. [6], thermal metamaterials have become a powerful tool to manipulate heat energy [1,2,3]. A lot of thermal metadevices based on transformation thermotics emerge endlessly, such as thermal cloaks [7,8,9,10,11,12], thermal transparency [13,14,15,16], thermal sensor [17,18,19,20], thermal concentrators [21, 22], thermal rotators [23] and thermal camouflages [23,24,25,26,27]. Although transformation theory has made breakthrough achievements, it requires that the parameters of thermal metamaterials are anisotropic and inhomogeneous, which makes the fabrication of devices quite troublesome. However, over the past ten years, the theory for designing thermal metamaterials has been continuously developed. In addition to the traditional transformation thermal theory, researchers have designed many thermal devices based on thermal effective medium theory [28,29,30,31,32,33,34] and numerical calculation methods. For thermal effective medium theory, the most representative one is scattering cancellation [35, 36], which uses double-layer isotropic materials to achieve a perfect thermal cloak. After that, the researchers also implemented a thermal concentrator using this bilayer structure [37]. Additionally, the thermal effective medium theory can not only design the functions predicted by the transformation theory, but also the functions predicted by the non-transformation theory, which greatly enriches the design ideas of thermal metamaterials. Since the core of the thermal effective medium theory is to calculate the equivalent thermal conductivity, it is a little weak to control the transient heat transfer process [8, 38, 39]. Therefore, some researchers also developed the transient heat transfer theory to regulate the wave-like temperature field [40,41,42,43], which also established a unified understanding of the diffusion system and the fluctuation system. Moreover, with the diversification of device functions, we have more precise and complex requirements for the spatial distribution of thermal conductivity. Therefore, numerical calculation methods have been developed to fabricate metamaterials with desired thermal conductivity. Printable freeform thermal metamaterials can directly create functional cells with certain thermal conductivity using transformation thermotics and topology optimization [44, 45]. These functional cells are local microstructures composed of die steel and polydimethylsiloxane. This approach solves challenges such as the limited shape adaptability of thermal metadevices and the need for prior knowledge of the background temperatures. Numerical methods have now become advanced and powerful for designing feasible thermal metamaterials with various applications. These days, inspired by the optical conformal mapping theory, thermal conformal mapping theory has been proposed to simplify the preparation of transformation thermotics and eliminate anisotropy [46]. All the theories mentioned above constitute the new research direction of theoretical thermodynamics [47, 48], which is regarded as an important part of the field of thermodynamics together with thermodynamic statistical physics and heat transfer [49, 50].

Based on these theories and methods, researchers design various functional thermal metamaterials to regulate thermal conduction at first [3]. Further, they also use thermal metamaterials to realize the coordinated regulation of heat convection and heat conduction [3, 51,52,53,54,55]. In 2020, Xu et al. unify thermal radiation into the framework of the theory of transformational thermotics [56], which can simultaneously deal with the three basic heat transfer modes of heat conduction, heat convection, and heat radiation. Based on this, they further design an omnithermal metamaterial to realize the functions of thermal cloaking, thermal concentration and thermal rotation. For the simultaneous regulation of three heat transfer modes using effective medium theory, Wang et al. fabricate omnithermal restructurable metasurfaces which show both illusion viewed in infrared light and in visible light by considering the three basic modes of heat transfer in theoretical designs [30]. Yang et al. propose an effective medium theory to handle conductive, radiative, and convective processes simultaneously, which is based on the Fourier law, the Rosseland diffusion approximation, and the Darcy law [29]. In addition to these unified paradigms, researchers have also utilized many artificially designed structures possessing three thermal transfer modes, such as radiative cooling and engineered cellular solids [57]. In this chapter, we will introduce the transformation omnithermotics in Sect. 13.2 and effective medium theory for omnithermotics in Sect. 13.3. At last, we will introduce radiative cooling and engineered cellular solids in Sect. 13.4.

2 Transformation Omnithermotics

Transformation Omnithermotics, as introduced by Xu et al. in their work [56], provides a comprehensive framework for understanding and manipulating the three basic modes of heat transfer: conduction, convection, and radiation simultaneously. They suppose that fluids are incompressible and laminar with a low speed and neglect the momentum conservation of fluids when handling thermal convection. Therefore, the transient process of heat transfer with conduction, convection, and radiation is dominated by

$$\begin{aligned} \begin{gathered} \rho _f C_f \partial T / \partial t+\nabla \cdot \left( -\boldsymbol{\kappa }_f \cdot \nabla T+\rho _f C_f \boldsymbol{v}_f T-\boldsymbol{\alpha }_f T^3 \cdot \nabla T\right) =0, \\ \partial \rho _f / \partial t+\nabla \cdot \left( \rho _f \boldsymbol{v}_f\right) =0, \end{gathered} \end{aligned}$$
(13.1)

where \(\rho _f\), \(C_f\), \({\kappa }_f\), and \(\boldsymbol{v}_f\) are the density, heat capacity, thermal conductivity, and velocity of fluids, respectively. \(\boldsymbol{\alpha }_f=16{\boldsymbol{\beta }_f}^{-1}{\boldsymbol{n}_f}^{2}\sigma /3\) can be treated as radiative coefficient where \(\boldsymbol{\beta }_f\), \(\boldsymbol{n}_f\), and \(\sigma (=5.67\times 10^{-8}\,\mathrm {W/(m^2K^4)})\) are the Rosseland mean attenuation coefficient, relative refractive index, and the Stefan-Boltzmann constant, respectively.

Equation (13.1) can keep its form-invariance after a space transformation if the materials’ parameters obey the following transformation rules:

$$\begin{aligned} \begin{gathered} \rho _f^{\prime }=\rho _f / {\text {det}} \Lambda \\ C_f^{\prime }=C_f \\ \boldsymbol{\kappa }_f^{\prime }=\Lambda \boldsymbol{\kappa }_f \Lambda ^\tau / {\text {det}} \Lambda \\ \boldsymbol{v}_f^{\prime }=\Lambda \boldsymbol{v}_f \\ \boldsymbol{\alpha }_f^{\prime }=\Lambda \boldsymbol{\alpha }_f \Lambda ^\tau / {\text {det}} \Lambda \end{gathered} \end{aligned}$$
(13.2)

where \(\Lambda \) is the Jacobian transformation matrix and \(\Lambda ^\tau \) is the transpose of \(\Lambda \). As is shown in Eq. (13.2), for thermal convection, they have to transform flow velocity directly. It is extremely difficult in experiments. Therefore, they resort to porous media to control flow velocity and ensure feasibility. With the calculated \(\Lambda \) according to the expected function, they design three devices including omnithermal cloaking, concentrating, and rotating as model applications. Finite-element simulations are also provided to confirm these applications.

This work not only unifies the three basic modes of heat transfer within the theoretical framework of transformation omnithermotics, but also provides novel hints and potential applications to thermal management.

3 Effective Medium Theory for Omnithermotics

3.1 Omnithermal Restructurable Metasurfaces

In addition to manipulating the three basic modes of heat transfer with omnithermal metamaterials, Wang et al. design omnithermal restructurable metasurfaces which achieve both infrared-light illusion and visible-light similarity [30]. By considering the three basic modes of heat transfer (omnithermotics) in theoretical designs, the authors delve into the tuning of surface apparent temperature and emissivity, which can be executed synergistically. In the experimental demonstration, they use radiation-cavity effects to modulate the full-wavelength emissivity. They use a cylindrical cavity structure as it is easy to manufacture, demonstrated in Fig. 13.1. The whole metasurface consists of an array of those cylindrical cavity unit cells. The effective emissivity of an isolated cylindrical cavity \(\varepsilon _e\) depends on its area ratio of the mouth and the intra-cavity, which can be expressed as:

$$\begin{aligned} \varepsilon _e=\left[ 1+\frac{S_0}{S_1}\left( \frac{1}{\varepsilon _b}-1\right) \right] ^{-1} \end{aligned}$$
(13.3)

where \(S_0\) and \(S_1\) are the areas of the mouth and the inwall, respectively, and \(\varepsilon _{sur}\) is the intrinsic surface emissivity. Through adjusting the unit cells’ emissivity, they can tailor the effective emissivity of the surface so as to form a specific apparent-temperature distribution in infrared imaging.

3.2 Omnithermal Metamaterials with Switchable Function

As illustrated in Sect. 13.2, Xu et al. manipulate the three basic modes of heat transfer with transformation omnithermotics. In contrast, Yang et al. propose an effective medium theory to handle conductive, radiative, and convective processes simultaneously, which is based on the Fourier law, the Rosseland diffusion approximation, and the Darcy law [29]. Moreover, They also discuss to what extent the three different heat transfer modes dominate the heat transfer process at different temperature intervals. In this circumstance, they design an omnithermal metamaterial switchable between transparency and cloaking, which results from the nonlinear properties of radiation and convection.

Fig. 13.1
An illustration of the principle of effective emissivity. It depicts a cavity structure with an emissivity of 0.2 (upper panel), which is equivalent to a flat surface with an emissivity of approximately 0.6 (lower panel).

(from Ref. [30])

Cavity structure (upper panel) and effective emissivity principle. The effective emissivity of a flat surface with a cavity (upper panel) is equivalent to \(\varepsilon _sur\) of another flat surface (lower panel), which is quantitatively expressed in Eq. (13.3).

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Theory—If we consider all methods of heat transfer in a porous medium, the conductive heat flux \(\boldsymbol{J_1}\), radiative flux \(\boldsymbol{J_2}\), and convective flux \(\boldsymbol{J_3}\), are given by the Fourier law, the Rosseland diffusion approximation, and the Darcy law:

$$\begin{aligned} \begin{aligned} &\boldsymbol{J_1}=-\kappa \nabla T\\ &\boldsymbol{J_2}=-\gamma T^3 \nabla T\\ &\boldsymbol{J_3}=\rho _f c_f \boldsymbol{v} T \end{aligned} \end{aligned}$$
(13.4)

where \(\rho _f\) and \(c_f\) are the density and heat capacity of the fluid. \(\boldsymbol{v}\) represents the velocity of the fluid, which is given by the Darcy law \(\boldsymbol{v}=-\varsigma / \eta _f\nabla P\), where \(\varsigma \) is the permeability of the porous medium, \(\eta _f\) is the viscosity of the fluid, and P denotes pressure. \(\kappa \) is the thermal conductivity of the porous medium. \(\gamma =16\beta ^{-1}n^{2}\sigma /3\) is the radiative coefficient, where \(\beta \) is the Rosseland mean extinction coefficient, n is the relative refractive index, which is set to 1 for brevity, and \(\sigma \) is the Stefan–Boltzmann constant. The total heat flux is \(\boldsymbol{J}=\boldsymbol{J_1}+\boldsymbol{J_2}+\boldsymbol{J_3}\) which satisfy the Laplace equation:

$$\begin{aligned} \nabla \cdot \boldsymbol{J}=\nabla \cdot (\boldsymbol{J_1}+\boldsymbol{J_2}+\boldsymbol{J_3}) \end{aligned}$$
(13.5)

Then, two boundary conditions are considered: (I) the temperature field and pressure field are parallel and (II) the temperature field and pressure field are perpendicular. Yang et al. confirm that the effective medium theory is applicable in both conditions.

Design—For omnithermal transparency (Fig. 13.2a), the radius of the core is \(r_c\) with the thermal conductivity \(\kappa _c\), the Rosseland mean extinction coefficient \(\beta _c\), and the permeability \(\varsigma _c\). The core is coated by a shell with the corresponding parameters \(r_s\), \(\kappa _s\), \(\beta _s\), and \(\varsigma _s\). Based on the effective medium theory, three effective values can be obtained:

$$\begin{aligned} \begin{aligned} \kappa _e & =\kappa _s \frac{\kappa _c+\kappa _s+\left( \kappa _c-\kappa _s\right) p}{\kappa _c+\kappa _s-\left( \kappa _c-\kappa _s\right) p}\\ \beta _e^{-1} & =\beta _s^{-1} \frac{\beta _c^{-1}+\beta _s^{-1}+\left( \beta _c^{-1}-\beta _s^{-1}\right) p}{\beta _c^{-1}+\beta _s^{-1}-\left( \beta _c^{-1}-\beta _s^{-1}\right) p} \\ \varsigma _e & =\varsigma _s \frac{\varsigma _c+\varsigma _s+\left( \varsigma _c-\varsigma _s\right) p}{\varsigma _c+\varsigma _s-\left( \varsigma _c-\varsigma _s\right) p} \end{aligned} \end{aligned}$$
(13.6)

where \(\kappa _e\), \({\beta _e}_{-1}\), and \(\varsigma _e\) represent the effective thermal conductivity, the effective reciprocal of the Rosseland mean extinction coefficient and the effective permeability of the core-shell structure. \(p=r_c/r_s\) is the area fraction.

For omnithermal cloaking (Fig. 13.2b), the radius of the core is \(r_c\) with thermal conductivity \(\kappa _c\), reciprocal of the Rosseland mean extinction coefficient \({\beta _c}_{-1}\), and permeability \(\varsigma _c\). The core is coated by an inner shell with parameters \(r_{s1}\), \(\kappa _{s1}\), \({\beta _{s1}}_{-1}\), and \(\varsigma _{s1}\) and an outer shell with parameters \(r_{s2}\), \(\kappa _{s2}\), \({\beta _{s2}}_{-1}\), and \(\varsigma _{s2}\). Bilayer cloaking requires the inner shell to be insulated; i.e., \(\kappa _{s1}\rightarrow 0\), \({\beta _{s1}}_{-1}\rightarrow 0\), and \(\varsigma _{s1}\rightarrow 0\). Then, the effective thermal conductivity, the effective reciprocal of the Rosseland mean extinction coefficient, and the effective permeability of the core-shell-shell structure can be derived:

$$\begin{aligned} \begin{aligned} \kappa _e & =\kappa _{s 2} \frac{1-p}{1+p}\\ \beta _e^{-1} & =\beta _{s 2}^{-1} \frac{1-p}{1+p} \\ \varsigma _e & =\varsigma _{s 2} \frac{1-p}{1+p} \end{aligned} \end{aligned}$$
(13.7)

where \(p=(r_{s1}/r_{s2})^2\) is the air fraction.

Fig. 13.2
3 sets of 3 heat maps for omni thermal transparency with 3 concentric circles, cloaking with 3 concentric circles and reference core structures with 3 concentric circles. Temperature distribution is plotted using a color gradient scale for different velocities.

(from Ref. [29])

Simulation results of (a1)–(a3) transparency, (b1)–(b3) cloaking, and (c1)–(c3) references with uniform fields.

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Ominthermal transparency and cloak—Fig. 13.2 shows the simulation results with boundary condition (I). The simulation results for omnithermal transparency, omnithermal cloaking, and references are presented in Fig. 13.2a1–a3, b1–b3, and c1–c3, respectively. The temperatures of the right and left boundaries are 413 and 313 K for Fig. 13.2a1–c1, 913 and 813 K for Fig. 13.2a2–c2, and 1413 and 1313 K for Fig. 13.2a3–c3. The pressures of the right and left boundaries in Fig. 13.2a1–c3 are 1000 and 0 Pa, respectively. For omnithermal transparency, the temperature distribution in the background should be the same as the reference, as if there was no core-shell structure in the center. To ensure that omnithermal transparency works, setting \(\kappa _b=\kappa _e\), \(\beta _b=\beta _e\) and \(\varsigma _b=\varsigma _e\) based on Eq. (13.6). For bilayer cloaking, the cloaking region should be at a constant temperature, and the temperature distribution in the background should not be disturbed. Thus \(\kappa _{s1}\), \({\beta _{s1}}^{-1}\) and \(\varsigma _{s1}\) should be set as very small values. Based on Eq. (13.7), the properties of the background should be equal to the effective properties of the bilayer cloak: \(\kappa _b=\kappa _e\), \(\beta _b=\beta _e\) and \(\varsigma _b=\varsigma _e\). The results of omnithermal transparency and cloaking conditions confirm the theory.

Fig. 13.3
3 sets of 3 heat maps for omni thermal transparency with 3 concentric circles, cloaking with 3 concentric circles and reference core structures with 3 concentric circles. Temperature distribution is plotted using a color gradient scale for different velocities.

(from Ref. [29])

Simulation results of (a1)–(a3) transparency, (b1)–(b3) cloaking, and (c1)–(c3) references with uniform fields. The pressures of the bottom and top boundaries are 1000 and 0 Pa, and the left and right boundaries are insulated. The other parameters are the same as those for Fig. 13.2.

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For boundary condition (II), the pressures of the bottom and top boundaries are at 1000 and 0 Pa, respectively. The other conditions are kept unchanged. Finite-element simulations are performed again with the new boundary conditions and the corresponding results are shown in Fig. 13.3. Obviously, this change does not influence the effects of omnithermal transparency and cloaking. Moreover, this scheme is robust when applying a nonuniform thermal field. The results are stable; see Fig. 13.4. In Fig. 13.4a1–c3, there is an elliptical source with high temperature and pressure at the bottom of the simulation box. Then fix the upper boundary at 0 Pa and the source boundary at 1000 Pa. The other boundaries are insulated. The temperature settings are 413–313 K for Fig. 13.4a1–c1, 913–813 K for Fig. 13.4a2–c2, and 1413–1313 K for Fig. 13.4a3–c3. The other parameters are same as those for Fig. 13.2.

Fig. 13.4
3 sets of 3 heat maps for omni thermal transparency with 3 concentric circles, cloaking with 3 concentric circles and reference core structures with 3 concentric circles. Temperature distribution is plotted using a color gradient scale for different velocities.

(from Ref. [29])

Simulation results of (a1)–(a3) transparency, (b1)–(b3) cloaking, and (c1)–(c3) references with nonuniform fields.

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Function switching between transparency and cloaking—The Rosseland diffusion approximation suggests that the radiative flux \(\boldsymbol{J_2}\) is proportional to \(T_3\). The convective flux \(\boldsymbol{J_3}\) is proportional to T, and the conductive flux \(\boldsymbol{J_1}\) is independent of T. The total flux \(\boldsymbol{J}\) are compared in Fig. 13.2a1–a3 and find that \(\boldsymbol{J}\) increases with increasing concrete temperature (the temperature difference remains unchanged). In other words, the radiative and convective effects increase with increasing temperature. These nonlinear properties induce a switchable omnithermal metamaterial; see Fig. 13.5. When the device functions with a high-temperature interval (Fig. 13.5b), bilayer cloaking is presented. When the device functions with a normal temperature interval (Fig. 13.5c), it switches to transparency. Different from the previous work, Yang et al. introduce radiation and convection to achieve such an intelligent metamaterial that can adaptively switch its functions according to the external temperature. Actually, the switch between the different functions results from the competition between three mechanisms of heat transfer. Generally, thermal conduction is dominant at normal temperatures, and thermal convection and radiation are dominant at high temperatures. Therefore, Yang et al. design the parameters related to thermal conduction to meet the requirements of transparency and design the parameters related to thermal convection and radiation to meet the requirements of bilayer cloaking. Then, the device can exhibit the cloaking function at high temperatures and the transparency function at normal temperatures.

Fig. 13.5
A schematic of a switchable onmithermal metamaterial. It presents the transition between a high-temperature bilayer cloak and normal-temperature transparency.

(from Ref. [29])

Switchable omnithermal metamaterial. a Schematic diagram. b Bilayer cloak with a high-temperature interval. c Transparency with a normal temperature interval. The pressures of the right and left boundaries are 1000 and 0 Pa, and the other boundaries are insulated.

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4 Other Artificially Designed Structures

4.1 Radiative Cooling

Theory—Radiative cooling is a prevalent phenomenon in our daily lives. For instance, the cool sensation we experience during nighttime arises because the sun’s radiation is minimal, allowing the Earth to emit heat into outer space [58]. Figure 13.6 illustrates the principle of radiative cooling. Picture an object exposed to sunlight: it releases heat via radiation while simultaneously absorbing heat from the direct sun rays. Furthermore, natural convection and thermal conduction play roles in determining the object’s temperature. To effectively cool the object, it’s essential to encase it in a specially designed layer that amplifies its radiation and reflects sunlight. The net cooling power can be described as \(P_{cool}(T)=P_{rad}(T)-P_{atm}(T)-P_{sun}(T)-P_{cond+conv}(T)\), where \(P_{rad}(T)\) is the radiative power, \(P_{atm}(T)\) is the absorbed power due to incident atmospheric thermal radiation, \(P_{sun}(T)\) is the incident solar power absorbed by the object, and \(P_{cond+conv}(T)\) is the power loss or gain due to convection and conduction. For the best cooling performance, the radiative cooler should emit powerfully and selectively within the atmospheric transparency window to amplify \(P_{rad}(T)\). Simultaneously, it should reflect all other light, especially in the visible and near-infrared spectrum, to reduce \(P_{atm+sun}(T)\). It’s notable that in every application of radiative cooling, heat conduction and convection also significantly contribute. In other words, the combined impact of heat radiation, heat conduction, and heat convection produces the targeted cooling in specific areas.

Design—In 2013, Fan’s team developed a metal-dielectric photonic structure that, through numerical simulations, minimized solar absorption and selectively emitted within the atmospheric transparency window [59]. This photonic crystal, boasting a multilayer micro-nano structure with periodic perforations, achieved a net cooling power surpassing 100 \(\mathrm {W/m^2}\). They later refined this multilayer design, crafting a photonic crystal from \(\mathrm {SiO_2}\), \(\mathrm {HfO_2}\), and Silver [60]. This new design managed to reach temperatures 5 degrees Celsius below ambient under direct sunlight. However, the complexity and expense of its fabrication posed challenges to its broad-scale adoption.

Fig. 13.6
A schematic of the full-heat transfer modes. It highlights the earth’s surface, host, outer space, metamaterial for thermal radiation cooling, thermal radiation from the atmospheric layer, thermal radiation through atmospheric windows, thermal radiation from the sun, and natural thermal convection and conduction.

Adapted from Ref. [3]

Schematic diagram showing the full heat-transfer modes associated with radiative cooling.

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Addressing these challenges, other researchers introduced a novel randomized glass-polymer hybrid metamaterial crafted from polymethyl-pentene [61]. This design embedded tiny \(\mathrm {SiO_2}\) spheres, averaging eight microns in diameter, on one side of the material, while the opposite side was silver-coated. These 8-micron glass spheres emitted heat from beneath the material in the 8-14 \(\mathrm {{\mu }m}\) infrared range, aligning with the atmospheric transparency window. Concurrently, the polymethyl-pentene and silver layers reflected a majority of incoming visible light. Demonstrating a radiative cooling power of 93 watts per square meter under direct sunlight, this metamaterial not only proved efficient but was also cost-effective and scalable for mass production. Consequently, such metamaterial films have been successfully mass-produced. In 2018, a group of researchers developed hierarchically porous poly coatings made from vinylidene fluoride-co-hexafluoropropene [62]. These coatings exhibited superior passive daytime radiative cooling properties. To create them, poly and water were combined in acetone and then air-dried. The evaporation process of both acetone and water led to the formation of micro-nano air cavities within the poly. Consequently, this coating boasted a high reflectance of 96% in the visible light spectrum and delivered a cooling power of 96 watts per square meter. This polymer method, being more user-friendly and simpler to produce than previous techniques, emerges as a promising contender in the field of passive radiative cooling materials.

The radiative cooling experiments described earlier are primarily effective in arid regions. Liu and his team delved into the impact of atmospheric water vapor on the efficiency of radiative cooling [63]. Both their theoretical analyses and experimental findings indicated that as the column of water vapor increases, the cooling power diminishes, albeit at a decreasing rate. This decline in cooling power is attributed to the enhanced absorption of atmospheric radiation by the radiative cooler due to the increase in water vapor. Their research offers valuable insights for the real-world implementation of radiative cooling across various climatic conditions.

4.2 Engineered Cellular Solids

Owing to the recent progress in 3D printing techniques and the capability to produce multifunctional materials with intricate nano/microstructures, rationally-designed lattices that form metamaterials and meta-structures have garnered significant interest [57]. Engineered cellular solids, mirroring the properties of their natural counterparts found in entities like wood, cork, bone, and coral, represent a notable category of these lightweight, high-performance materials. Beyond their mechanical attributes, the hydrodynamic and thermal characteristics of designed lattices, especially when evaluating various heat transfer modalities under diverse boundary scenarios, are crucial. Given the challenges in formulating a precise analytical model for integrated convective and radiative heat transfer, a computational approach leveraging the finite volume method (FVM) or finite element method (FEM) becomes imperative. Among the various lattice designs, the mechanical attributes of BCC lattices, representative of cubic cellular structures, have been extensively studied [57]. This is attributed to their straightforward production via additive manufacturing or molding and their applicability in fields such as energy engineering, infrastructure, transportation, and aerospace. In 2022, Shahrzadi et al. introduce the architecture of BCC lattices and a comprehensive study that includes contributions of all three modes of heat transfer for the BCC lattices has been conducted in this work. It’s imperative to assess the thermal characteristics of lattices both with and without the influence of radiation, as this factor is often omitted to streamline the thermal performance analysis of designed lattices. Shahrzadi et al. ascertain situations where radiation heat transfer cannot be overlooked in the thermal analysis of BCC lattices, by comparing the results in the absence and the presence of radiation heat transfer. They also investigated the effect of different porosity and Reynolds numbers on the heat transfer in the BCC lattice.

Design—The design of the three-dimensional unit cell draws inspiration from BCC crystals within the cubic crystal system family. By adjusting the diameter of the cylindrical struts in the BCC unit cell, its porosity changes while the cell size remains constant; see Fig. 1(a) in Ref. [57]. Porosity is characterized as the proportion of the vacant space in the unit cell relative to its overall volume. The equivalent pore diameter refers to the size of the largest sphere that can be enclosed by the unit cell’s struts. As shown in Fig. 1(b) in Ref. [57], the equivalent pore diameter can be calculated by:

$$\begin{aligned} \frac{d_p}{2}=\frac{L \sqrt{2}}{2} \sin \theta -\frac{d_s}{2} \end{aligned}$$
(13.8)

Governing equations—The governing equations for the simulation of fluid flow and heat transfer in porous media are presented below, which are simplified based on steady-state, incompressible flow, and temperature-independent material property assumptions.

$$\begin{aligned} \begin{aligned} \nabla \cdot (\rho \vec {v})&=0 \\ \rho (\vec {v} \cdot \nabla \vec {v})&=-\nabla P+\mu \nabla ^2 \vec {v} \\ \rho C_p(\vec {v} \cdot \nabla T)&=\nabla \cdot (k \nabla T)+\dot{q} \\ \nabla \cdot (I(\vec {r}, \vec {s}) \vec {s})+\left( a+\sigma _s\right) I(\vec {r}, \vec {s})&=a n^2 \frac{\sigma T^4}{\pi } +\frac{\sigma _s}{4 \pi } \int _0^{4 \pi } I\left( \vec {r}, \vec {s^{\prime }}\right) \Phi \left( \vec {s} \cdot \vec {s^{\prime }}\right) d \Omega ^{\prime } \end{aligned} \end{aligned}$$
(13.9)

which are corresponding continuity, momentum, and energy conservation equations along with the Discrete Ordinates equation for radiation heat transfer accordingly. \(\vec {v}\) is velocity vector, \(\rho \) is density, P is pressure, \(\mu \) is dynamic viscosity, \(C_p\) is specific heat, T is temperature, \(\kappa \) is thermal conductivity, q is the heat generation, I is radiation intensity, \(\vec {r}\)is position vector, \(\vec {s}\)is direction vector, \(\vec {s^{\prime }}\) is scattering direction vector, a is absorption coefficient, \(\sigma \) is scattering coefficient, n is reflective index, \(\sigma _s\) is the Stefan-Boltzmann constant, \(\Phi \) is the phase function, and \(\Omega ^{\prime }\) represents solid angle. All mentioned equations are used for the fluid phase, and the energy conservation equation is used for both fluid and solid phases. For the solid phase, the left side of the energy conservation equation is zero. The ANSYS meshing tool is used to generate computational unstructured meshes for the simulations.

Conductive thermal conductivity—By ignoring the thermal radiation effect, a certain temperature gradient is applied to the two opposite faces on the left and right of the BCC lattice unit cell. They calculate the conductive thermal conductivity (\(\kappa _{ec}\)); see Fig. 7 in Ref. [57]. The conductive thermal conductivity increases when porosity decreases. This is because the amount of the solid phase in the unit cell increases and the contribution of solid thermal conductivity is higher than that of fluid. Moreover, the parallel and series models specify the upper and lower bounds. The calculated conductive thermal conductivity agrees well with the experimental results for \(\varphi >\) 0.9.

Heat transfer with fluid flow—They also investigate the heat transfer and fluid flow in the BCC lattices with alternative porosities for various Reynolds numbers based on the pore diameter (\(R e_p\)). When the porosity decreases, the ability of convective heat transfer to cool down the unit cell also decreases. As the Reynolds number increases, the cooling capacity of the fluid increases, thereby reducing the temperature of the unit cell.

Radiation effects—The article also considers the radiation mechanism on the heat transfer in BCC lattices. A certain thermal gradient is applied to the opposing faces of the unit cell, then radiative thermal conductivity (\(\kappa _{er}\)) and effective thermal conductivity (\(\kappa _{eff}\)) are calculated by:

$$\begin{aligned} q=q_{\text{ cond } }+q_{\text{ rad } }=-\left( k_{e c}+k_{e r}\right) \frac{\partial T}{\partial z}=-k_{e f f} \frac{\partial T}{\partial z} \end{aligned}$$
(13.10)

It is observed that the radiative thermal conductivity increases with increasing porosity. When the applied temperature is relatively high, the enhancement of porosity to radiative heat transfer is more obvious, as shown in Fig. 11a in Ref. [57]. At lower porosities, thermal conduction plays an important role in effective thermal conductivity, while at higher porosities, radiative heat transfer plays an important role, as shown in Fig. 11c in Ref. [57].

5 Conclusion and Application

The domain of thermal metamaterials stands at the cusp of a transformative era, with the establishment of the theory of transformation omnithermotics serving as a linchpin. This theory not only unifies the three primary modes of heat transfer—conduction, convection, and radiation—but also paves the way for innovative applications. Model applications, as demonstrated in this chapter, include omnithermal cloaking, concentrating, and rotating devices, all of which have been validated through finite-element simulations. The incorporation of porous media to modulate flow velocity further underscores the comprehensive nature of this work.

Furthermore, the development of an effective medium theory in omnithermotics has enabled the simultaneous manipulation of conduction, radiation, and convection in porous media. This has led to the design of transparency, cloaking, and switchable omnithermal metamaterials. Such designs have vast potential in applications like thermal camouflage, thermal rectification, and other intelligent metamaterials. And the proposed restructurable metasurface, which exhibits both illusions in infrared light and similarity in visible light, holds promise for real-world applications, especially when structured meticulously.

Other artificially designed structures about the radiative cooling. It has been used in many practical applications, including those for making woods stronger and cooler. Some researchers also demonstrated the excellent effect of radiative cooling on daytime building cooling. Engineered cellular solids, inspired by their natural counterparts found in materials like wood and bone, have emerged as advanced lightweight materials with unparalleled properties. Their unique characteristics, such as low density and high permeability, make them indispensable in a plethora of applications. These include heat exchangers, thermal insulators, medical implants, solar energy systems, and aerospace sandwich panels [57].