Omnithermal Metamaterials: Mastering Diverse Heat Transfer Modes

Abstract

Transformation omnithermotics has emerged as a revolutionary theory within heat transfer, seeking to cohesively control the three primary heat transfer modes: conduction, convection, and radiation. Historically, orchestrating these modes in unison has posed immense challenges due to their unique properties and operational dynamics. This chapter delves deep into the core principles of transformation omnithermotics and its prowess in seamlessly integrating the complexities of these varied modes. We present an exhaustive exploration of cutting-edge devices birthed from this concept, such as the omnithermal cloaking, concentrating, and rotating apparatuses. As underscored by finite-element simulations and bolstered by experimental insights, these devices signify a pioneering horizon in thermal management. In this chapter, we endeavor not only to clarify the integration of these fundamental heat transfer modes within transformation omnithermotics but also to spotlight the vast potential it unlocks for avant-garde thermal management techniques.

1 Opening Remarks

Heat transfer operates through three fundamental modes: conduction, convection, and radiation. Each possesses unique attributes and mechanisms, yet they often function synergistically. Take, for instance, solar vapor generation as example. In this context, radiation is pivotal, associated with the capture of solar energy. Convection is manifest in the evaporation of seawater, while conduction is ubiquitously involved in nearly every phase. In the study of heat transfer within aerogels, it’s imperative to consider all three modes. At its core, proficient thermal management hinges on the harmonious interaction of conduction, convection, and radiation.

The effective control or manipulation of heat flow is an evolving research field combining transformation thermodynamics and metamaterials. Transformation theory, as a general and powerful methodology, has already been extended from wave systems (e.g., electromagnetic propagation) to diffusion systems (e.g., heat conduction [1], Noise Shielding, thus yielding the rapid development of thermal metamaterials or metadevices, i.e. thermal cloaks, concentrators, rotators, camouflaging, invisible sensors [2] and thermal transparency, have been proposed [3, 4]. Moreover, some novel thermal metamaterials are also involved, such as digital thermal metasurfaces, thermal chameleon-like metashells, energy-free thermostat devices, thermal dipoles, thermal crystals, phonon transport, and printed circuit boards containing various thermal metamaterials for heat dissipation of electronic devices. In addition to the typical thermal convection metamaterials, we also summarize the phenomena of thermal wave [5], zero-index cloak, thermal anti-parity-time symmetry, thermal convection–conduction crystal, negative thermal transport, and so on.

Transformation-based thermal metadevices manipulate heat flows through tailored constitutive parameters. However, these thermal metamaterials take only conductive effects into consideration [6]. Recent studies have begun to incorporate convective effects into their exploration of thermal metamaterials, leading to the emergence of innovative applications. However, it’s worth noting that these studies have often focused on only one or two of the fundamental conduction, convection [7, 8], and radiation modes. This selective consideration allows researchers to delve deeper into specific aspects while potentially paving the way for a more comprehensive understanding of thermal metamaterial behaviors and their practical uses. For example, Xu and Huang designed circular/elliptical core-shell structured metamaterials with functions like thermal transparency [9], cloaking (the shell or cloak can be single-layered or bilayered) and expanding within such a conduction-radiation system [10].

Within the prevailing research landscape, there’s a conspicuous gap in a holistic transformation theory that encompasses conduction, convection, and radiation concurrently. Each mechanism, given its distinct attributes, presents challenges when pursued for amalgamation into a singular theoretical construct. Nevertheless, considering the routine simultaneous manifestations of these processes in numerous systems, both the academic and industrial spheres are voicing an escalating need for a transformation theory that can integrally address all three facets.

Classical thermodynamics often helps to passively describe macroscopic heat phenomena of natural systems, which means people almost cannot change the heat phenomena, but understand them according to the four thermodynamic laws. In contrast, thermal metamaterials, together with the governing theories, make it possible to actively manipulate macroscopic heat phenomena of artificial systems, which enables people to change the heat phenomena at will.

Within the vast domain of “Physics” as a foundational discipline, thermal metamaterials have emerged as an innovative thermal system [11, 12]. They shed light on distinctive thermodynamic behaviors and enrich our understanding of thermodynamics and statistical physics [13,14,15,16,17,18]. Turning to the primary discipline of “Power Engineering and Engineering Thermophysics”, there lies a fresh approach to heat transfer regulation. Theoretical thermodynamics focuses on the proactive manipulation of macroscopic thermal phenomena in engineered systems. This approach is anchored in the theoretical architecture of transformation thermodynamics and its ancillary theories. In essence, transformation thermodynamics employs a technique encompassing coordinate shifts between two separate spaces, enabling a meticulous integration of spatial geometric parameters with thermal attributes such as thermal conductivity.

The theories previously highlighted stand as the pinnacle of recent progress in theoretical thermodynamics. In conjunction with statistical thermophysics and heat transfer, they constitute an essential pillar in the expansive domain of thermodynamics.

In addressing the previously discussed challenges, a comprehensive study by Xu et al. [19] delves into the formulation of the transformation omnithermotics theory. This theory incorporates all three fundamental heat transfer modes simultaneously. Drawing on this foundation, the team introduced three prototypical devices: omnithermal cloaking, concentrating, and rotating. Respectively, these devices serve to shield central objects from detection, enhance local heat flux, and control the direction of local heat flux. Finite-element simulations validate the efficacy of these devices, showing impressive results in both steady-state and transient scenarios. Moreover, the researchers employed porous media to modulate flow velocity, ensuring the practical viability of their approach. With the present theory, Yang et al. [20] further design an omnithermal metamaterial switchable between transparency and cloaking, which results from the nonlinear properties of radiation and convection. Moreover, Wang et al. [21] design an omnithermal restructurable thermal metasurface for infrared illusion. The following sections will delve deeper into the intricacies of this groundbreaking theory.

2 Omnithermal Metamaterials Based on Transformation Theory

2.1 Theory of Transformation Omnithermotics

Recently, several researchers introduced the notion of transformation omnithermotics, a holistic model designed to manipulate conduction, convection, and radiation concurrently. Initially, these scholars examined the heat transfer equation—a continuity equation representing total heat flux—and devised an array of devices, encompassing cloaking, concentrating, and rotating functionalities. Within this context, the radiative flux is characterized using the Rosseland diffusion approximation, transforming the heat transfer equation into a convection-diffusion equation. Here, the combined temperature-responsive conductivity integrates both conductive and convective elements. Notably, both conductive/radiative thermal conductivities and velocity undergo direct transformations in line with the transformation theory.

Subsequently, the focus shifted to a collection of equations that govern heat and mass transfer within porous media. This collection encompasses the Darcy law, the continuity equation for fluid motion, and the aforementioned heat transfer equation. Given the transformation convection theory highlighted earlier, the design of pertinent thermal devices involves altering both the permeability of porous media (or the dynamic viscosity of fluids in the case of convection) and the collective thermal conductivity for conduction-radiation. Intriguingly, porous media serve as a representative model that encapsulates all three primary heat transfer mechanisms, especially at elevated temperatures. Consequently, the researchers forecasted the emergence of omnithermal metamaterials capable of orchestrating heat conduction, convection, and radiation in unison. Detailed explorations are as followed.

Fluids and gases play a crucial role in convective heat transfer. Regardless of their varied forms, they obey the same heat transfer equation. Xu et al. studied the temporary heat transfer process in pure fluids or gases, considering conduction, convection, and radiation. This process can be described by

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

(14.1)

where $\rho _f$, $C_f$, $\boldsymbol{\kappa }_f$, and $\boldsymbol{v}_f$ are the density, heat capacity, thermal conductivity, and velocity of fluid, respectively $\boldsymbol{\alpha }_f(= 16\beta _f^{-1}\boldsymbol{n}_f^2\sigma /3)$ can be regarded as radiative coefficient where $\beta _f$, $\boldsymbol{n}_f$, and $\sigma (= 5.67 \times 10^{-8}W m^{-2}K^{-4}$ are the Rosseland mean attenuation coefficient, relative refractive index, and the Stefan–Boltzmann constant, respectively. T and t denote temperature and time, respectively. Physically speaking, the two equations in Eq. (1) describe the conservation of heat flux and mass flux, respectively. Conductive flux $\boldsymbol{J}_1$is determined by the Fourier law$\boldsymbol{J}_1 = -\kappa _f\cdot \boldsymbol{\nabla }T$; convective flux $\boldsymbol{J}_2$ is given by $\boldsymbol{J}_2 = \rho _f C_f \boldsymbol{v}_f T$; radiative flux $\boldsymbol{J}_3$ is dealt with the Rosseland diffusion approximation $\boldsymbol{J}_3 = -\alpha _f T^3\cdot \boldsymbol{\nabla }T$; and total flux $\boldsymbol{J}_T$ is the summation of conductive, convective, and radiative fluxes $\boldsymbol{J}_T = \boldsymbol{J}_1 +\boldsymbol{J}_2 +\boldsymbol{J}_3$.

Equation (14.1) is form-invariant under the space transformation from a curvilinear space S to a physical space S’, which is determined by the Jacobian transformation matrix $\boldsymbol{\nabla }$. To prove the form-invariance, Xu et al. wrote down the component form of Eq. (14.1). In the curvilinear space with a set of contravariant basis ($\boldsymbol{g}^i$,$\boldsymbol{g}^j$,$\boldsymbol{g}^k$) and corresponding contravariant components ($s^i$,$s^j$,$s^k$), Eq. (14.1) can be rewritten as

$$\begin{aligned} \begin{gathered} \sqrt{g}\rho _f C_f\partial _t T+\partial _i(\sqrt{g}(-\kappa ^{ij}_f\partial _j T+\rho _f C_f v_f^i T-\alpha _f^{ij} T^3\partial _j T))=0,\\ \sqrt{g}\partial _t\rho _f+\partial _i(\sqrt{g}\rho _f v^i_f)=0, \end{gathered} \end{aligned}$$

(14.2)

where g is the determinant of the matrix $\boldsymbol{g}_i\cdot \boldsymbol{g}_j$, and ($\boldsymbol{g}_i$, $\boldsymbol{g}_j$, $\boldsymbol{g}_k$) is a set of covariant basis. Equation (14.2) is written in the curvilinear space, which should be rewritten in the physical space with the Cartesian coordinates ($s^{i'}$,$s^{j'}$,$s^{k'}$),

$$\begin{aligned} \begin{gathered} \sqrt{g}\rho _f C_f\partial _t T+\partial _{i'}\frac{\partial s^{i'}}{\partial s^i}\left( \sqrt{g}\left( -\kappa ^{ij}_f\frac{\partial s^{j'}}{\partial s^j}\partial _{j'} T+\rho _f C_f v_f^i T-\alpha _f^{ij} T^3\frac{\partial s^{j'}}{\partial s^j}\partial _{j'} T\right) \right) =0,\\ \sqrt{g}\partial _t\rho _f+\partial _{i'}\frac{\partial s^{i'}}{\partial s^i}(\sqrt{g}\rho _f v^i_f)=0, \end{gathered} \end{aligned}$$

(14.3)

where $\partial s^{i'}/\partial s^i$ and $\partial s^{j'}/\partial s^j$ are just the components of the Jacobian transformation matrix $\Lambda$, and $\sqrt{g}=1/det\boldsymbol{\Lambda }$. The key to transformation theory is to turn space transformation into material transformation. For this purpose, Eq. (14.3) can be rewritten as

$$\begin{aligned} {\begin{gathered} \frac{\rho _f}{det \boldsymbol{\Lambda }} C_f\partial _t T+\partial _{i'} \left( -\frac{\frac{\partial s^{i'}}{\partial s^i}\kappa ^{ij}_f\frac{\partial s^{j'}}{\partial s^j}}{det \boldsymbol{\Lambda }}\partial _{j'}T + \frac{\rho _f}{det \boldsymbol{\Lambda }}C_f \frac{\partial s^{i'}}{\partial s^i} v^i_f T-\frac{\frac{\partial s^{i'}}{\partial s^i}\alpha ^{ij}_f \frac{\partial s^{j'}}{\partial s^j}}{det \boldsymbol{\Lambda }}T^3\partial _{j'}T\right) =0,\\ \partial _t\frac{\rho _f}{det\boldsymbol{\Lambda }}+\partial _{i'}\left( \frac{\rho _f}{det\boldsymbol{\Lambda }}\frac{\partial s^{i'}}{\partial s^i}v^i_f\right) =0. \end{gathered}} \end{aligned}$$

(14.4)

Thus, transformation rules can be derived as

$$\begin{aligned} \left\{ \begin{aligned} \rho '_f &= \rho _f/det \boldsymbol{\Lambda },\\ C'_f &= C_f,\\ \boldsymbol{\kappa }'_f &= \boldsymbol{\Lambda }\boldsymbol{\kappa }_f\boldsymbol{\Lambda ^\tau }/det\Lambda ,\\ \boldsymbol{v}'_f &= \boldsymbol{\Lambda v}_f,\\ \boldsymbol{\alpha }'_f &= \boldsymbol{\Lambda }\boldsymbol{\alpha }_f\boldsymbol{\Lambda }^\tau /det\boldsymbol{\Lambda }. \end{aligned}\right. \end{aligned}$$

(14.5)

where $\boldsymbol{\Lambda }^\tau$ is the transpose of $\boldsymbol{\Lambda }$.

The radiative coefficient is defined by two primary parameters: the Rosseland mean attenuation coefficient and the relative refractive index. Given that natural materials exhibit a limited range of relative refractive indices, there’s no need to alter them, which means $\boldsymbol{n}'_f = \boldsymbol{n}_f$. Consequently, the transformation of the radiative coefficient $\boldsymbol{\alpha }_f$ is given by $\boldsymbol{\beta }'_f = \boldsymbol{\Lambda }^{-\tau }\boldsymbol{\beta }_f\boldsymbol{\Lambda }^{-1}det\boldsymbol{\Lambda }$. For simplicity, relative refractive indices were also left untransformed in the finite-element simulations.

2.2 Applications of Omnithermal Metamaterials Based on Transformation Theory

The space transformation of cloaking is

$$\begin{aligned} \begin{aligned} r' &= \frac{(R_2-R_1)r}{R_2}+R_1\\ \theta '&=\theta \end{aligned} \end{aligned}$$

(14.6)

where $R_1$ and $R_2$ are the inner and outer radii. respectively.

The space transformation of concentrating is

$$\begin{aligned} \begin{aligned} r' &= \frac{R_1 r}{R_m}, (r<R_m)\\ r' &= \frac{(R_1-R_m)R_2 + (R_2-R_1)r}{R_2-R_m}, (R_m<r<R_2)\\ \end{aligned} \end{aligned}$$

(14.7)

where $R_m$ is a medium radius determining the concentrating ratio.

The space transformation of rotating is

$$\begin{aligned} {\begin{aligned} r' &= r,\\ \theta ' &= \theta + \theta _0, (r<R_1)\\ \theta ' &= \theta + \frac{(r-R_2)\theta _0}{(R_1-R_2)}, (R_1 <r<R_2) \end{aligned}} \end{aligned}$$

(14.8)

where $\theta _0$ is a rotation degree. Then, the Jacobian transformation matrix of Eqs. (14.6)-(14.8) can be calculated by

$$\begin{aligned} \left( \begin{array}{cc} \frac{\partial r'}{\partial r} &{} \frac{\partial r'}{r\partial \theta } \\ \frac{r'\partial \theta '}{\partial r} &{} \frac{r'\partial \theta '}{r\partial \theta } \end{array}\right) =\Lambda \end{aligned}$$

(14.9)

Fig. 14.1

(from Ref. [19])

Temporary cloaking simulations. The size of each simulation box is 0.1 m $\times$ 0.1 m, $R_1=0.024$ m, and $R_2=0.036$ m, where $R_1$ and $R_2$ are the inner and outer radii, respectively. The parameters of background fluid are as following: density of the fluid are as following: density of the fluid $\rho _f= 1000$ kg/m${^3}$, heat capacity of the fluid $C_f = 1000$ J kg$^{-1}$ K$^{-1}$, thermal conductivity of the fluid $\kappa _f=1$ W m$^{-1}$ K$^{-1}$, velocity of the fluids $\upsilon _f = 10^{-5}$ m/s, the Rosseland mean attenuation coefficient $\beta _f = 100$ m$^{-1}$, and relative refractive index $\boldsymbol{n}_f$=1. a–d, e–h, and i–l show the temperature evolutions over time with case I, II, and III, respectively. White lines represent isotherms, and color surfaces denote temperature distributions.

To explore the varying impacts of the three primary heat transfer methods, Xu et al. simulated three distinct scenarios. In the first scenario (I), the temperature range is between 300 and 360 K with the background velocity directed along the +x axis. Here, convection plays a predominant role. For the second scenario (II), the temperature spans from 300–1200 K and the background velocity remains on the +x axis. In this case, the effects of radiation become evident. In the third scenario (III), with the same temperature range as (II), the background velocity shifts to the -x axis. For this scenario, the overall flux diminishes due to the convective flux moving in the opposite direction compared to scenarios (I) and (II). When starting at an initial temperature of 300 K, it takes approximately 80, 30, and 50 min respectively for these scenarios to stabilize.

Figure 14.1 presents the transient simulation results of cloaking. The temperature evolutions with time under three different cases (say, I, II, and III) are demonstrated in Fig. 14.1a–d, e–h, and i-j, respectively. To avoid the problems of singular parameters, the boundary at r = $R_1$ is set with insulated and no-flow conditions. In this way, any object can be placed in the central white regions of Fig. 14.1. Moreover, the background isotherms are kept unchanged, thus confirming the cloaking effect. Xu et al. also plotted the conductive, convective, radiative, and total flux of the background in Fig. 14.1d, h, and l for quantitative analyses. The distributions of heat fluxes are presented in Fig. 14.2a–c.

Fig. 14.2

(from Ref. [19])

Heat flux distributions in background along x axis. a, b, and c present the distributions of steady heat flux with the same boundary conditions applied in the first, second, and third columns in Fig. 14.1, respectively.

Fig. 14.3

(from Ref. [19])

Temporary concentrating simulations. The background parameters and boundary conditions are the same as those for Fig. 14.1. The transformation media and flow velocity are designed according to Eq. (14.7) where $R_m=0.032$ m.

Fig. 14.4

(from Ref. [19])

Temporary rotating simulations. The background parameters and boundary conditions are the same as those for Fig. 14.1. The transformation media and flow velocity are designed according to Eq. (14.8) where $\theta = \pi /2$.

The concentrating effect can be observed in Fig. 14.3. Central isotherms in each simulation are markedly denser than those in peripheral areas, signifying enhanced heat flux concentrations. This aligns with the expected concentration effect. In Fig. 14.4, the rotation effect is evident. The central isotherms take on a horizontal orientation, with higher temperatures prominent at the base. This denotes a vertically upward direction of heat flux, signifying a successful rotation of heat fluxes by an angle of $\pi /2$ rad.

In the aforementioned discussion, the flow velocity was directly transformed, given that the velocity field had been pre-established. While this approach is mathematically sound, it poses significant challenges in practical experimentation. To address this, Xu et al. modulated flow velocity using porous media, ensuring a more feasible execution.

To confirm the theoretical analyses, Xu et al. also performed finite-element simulations. The first row in Fig. 14.5 of Ref. [19] presents the results with transforming the permeability of porous media as $\boldsymbol{\eta }'_s$ = $\boldsymbol{\Lambda }\boldsymbol{\eta } _s\boldsymbol{\Lambda }^\tau /det\boldsymbol{\Lambda }$. The second row in Fig. 14.5 of Ref. [19] shows the results with directly transforming flow velocity as $\boldsymbol{\nu }'_f$ = $\boldsymbol{\Lambda }\boldsymbol{\nu }_f /det\boldsymbol{\Lambda }$. As a result, they confirmed the feasibility of controlling flow velocity with designed permeability by comparing the velocity distributions of two different methods.

Fig. 14.5

(from Ref. [19])

Manipulating flow velocity with porous media. Color surfaces denote velocity distributions. a, c, and e are the velocity distributions obtained from transforming the permeability of porous media. b, d, and f are the velocity distributions obtained from directly transforming flow velocity. The boundary conditions of the first row are 1000 and 0 Pa for the left and right boundaries, and no-flow conditions for the upper and lower boundaries. The permeability of porous media in the background is $10^{-12}m^2$ (which is common in nature), and the dynamic viscosity of fluids is 0.001 Pa s.

3 Omnithermal Metamaterials Based on Effective Medium Theory

In their work, Yang et al. [20] introduced an effective medium theory that simultaneously addresses conductive, radiative, and convective (termed as “omnithermal”) processes. This theory is anchored on the Fourier law, the Rosseland diffusion approximation, and the Darcy law. By considering all heat transfer modalities within a porous medium, they deduced that the passive and stable heat transfer process is governed by the Laplace equation, expressed as follows:

$$\begin{aligned} \begin{aligned} \nabla \cdot (-\kappa \nabla T-\gamma T^3\nabla T-\rho _f c_f \xi /\eta _f T\nabla P)=0 \end{aligned} \end{aligned}$$

(14.10)

where $\rho _f$ and $c_f$ are the density and heat capacity of the fluid respectively. $\xi$ is the permeability of the porous medium. $\kappa$ is the thermal conductivity of the porous medium. $\gamma$ is the radiative coefficient.

For the condition that the temperature field and the pressure field are perpendicular, Eq. (14.10) can be rewritten as

$$\begin{aligned} \begin{aligned} \nabla \cdot (-\kappa \nabla T-\gamma T^3\nabla T)&=\rho _f c_f\nabla \cdot (\xi /\eta _f T\nabla P)\\ &=\rho _f c_f \nabla \cdot (-vT)=-\rho _f c_f v\cdot \nabla T \end{aligned} \end{aligned}$$

(14.11)

Using the current theory, we have developed an omnithermal metamaterial that can switch between transparency and cloaking. This capability stems from the nonlinear properties of radiation and convection. Finite-element simulations validate the robustness of our approach across varied boundary conditions.

Fig. 14.6

(from Ref. [20])

Schematic diagrams of a transparency and b cloaking. Orange lines represent heat fluxes. $\boldsymbol{J}$ is the total heat flux and its components, i.e., conductive flux $\boldsymbol{J}_1$, radiative flux $\boldsymbol{J}_2$, and convective flux $\boldsymbol{J}_3$.

Leveraging the nonlinear properties of radiation and convection, we introduce a switchable omnithermal metamaterial. This metamaterial can adaptively alternate between transparency, as shown in Fig. 14.6a, and cloaking, depicted in Fig. 14.6b, based on temperature variations. Finite-element simulations have been conducted to demonstrate the viability of our approach.

In their research, Wang et al. [21] incorporate the three fundamental modes of heat transfer, termed omnithermotics, into theoretical designs and utilize radiation-cavity effects in experimental fabrication. Our approach enables the synergistic tuning of surface temperature and emissivity. Notably, these metasurfaces operate effectively in environments with fluctuating temperatures and during transient states. This strategy offers not just a platform for designing adaptable thermal illusions, but also demonstrates resilience against multi-frequency detection.

They conceptualized an omnithermal, reconfigurable thermal metasurface for infrared illusion, as depicted in Fig. 14.7. By individually customizing each block unit and arranging them in a specific sequence, distinctive infrared patterns can be realized. Taking into account the three heat transfer modes—conduction, convection, and radiation (omnithermotics)—we observe their influence on surface temperatures. Through the radiation-cavity effect, for instance, the effective emissivity’s reliance on the dimensions, forms, and ratios of surface cavities, it’s possible to achieve a distinct emissivity for each unit over a broad temperature spectrum.

Fig. 14.7

(from Ref. [21])

The proposed thermal metasurface. The units are arranged in three arrays (array I, array II, and array III), which can form three different images (specific gestures) in an infrared camera (right column). Meanwhile, they are similar when viewed in visible light (middle column).

4 Challenges and Prospects of Transformation Omnithermotics

For many standard coordinate transformations, devising an anisotropic radiative conductivity or a radiative conductivity tensor is challenging. This mirrors the complexities in transformation multithermotics. A pragmatic approach involves the use of composites crafted from bulk homogeneous isotropic materials, each possessing distinct Rosseland mean opacities or radiative conductivities. Viewing this from a microscopic angle, the anisotropy within an anisotropic diffusive radiation model for porous media typically arises from the uniquely directional scattering and absorption processes. The radiative conductivity tensor is intricately related to the extinction and scattering coefficients. In many scenarios, the Beer–Lambert law may not hold true, necessitating adjustments or extensions to the Rosseland model. Nonetheless, if one requires a radiative transport equation characterized by anisotropic radiative thermal conductivity in transformed space, the radiation process doesn’t strictly adhere to every assumption of the Rosseland diffusion approximation. Instead, a broader radiative Fourier law can encapsulate it.

For experimental demonstration, porous media can be harnessed, with tunable thermal conductivities, permeabilities, and extinction coefficients. Encouragingly, as alluded to above, experiments focusing on either conduction or convection have been executed. However, the pressing hurdle for current objectives is seamlessly integrating heat radiation into the established models.

Xu et al. [19] established the theory of transformation omnithermotics to control thermal conduction, convection and radiation simultaneously. In other words, within the theoretical framework of transformation omnithermotics, Xu et al. had achieved a unification among the three basic modes of heat transfer. They further designed three devices with functions of cloaking, concentrating, and rotating as model applications. Finite-element simulations are used to confirm the theory. To ensure the feasibility and completeness of this work, Xu et al. also applied porous media to control flow velocity. These results have potential applications, say, in solar vapor generation or aerogel insulation, where conduction, convection, and radiation must be taken into account simultaneously.

5 Conclusion

In conclusion, Xu et al. delved into the intricacies of conductive and radiation fluxes, employing the Fourier law and the Rosseland diffusion approximation respectively. While the Fourier law provides accurate predictions at the macroscopic scale, it falters at the nanoscale where phonons emerge as the primary heat transfer carriers. The Rosseland diffusion approximation holds relevance in optically dense media, denoting scenarios where the photon’s mean free path is substantially shorter than the system’s dimensions. Consequently, this approximation aligns with far-field effects. While the current theory proficiently manages far-field thermal radiation, it is not inherently equipped to address near-field thermal radiation, which is steered by the classic fluctuating electrodynamics theory. Thus, delving deeper into near-field thermal radiation remains on the horizon. For convective flux, they employed the Darcy law, harnessing porous media to regulate flow velocity. The Navier-Stokes equation presents a viable alternative for managing drag-free thermal convection. These insights harbor significant implications, particularly in contexts necessitating the concurrent consideration of all three fundamental heat transfer modes.

Furthermore, the transformation theory doesn’t confine designs to specific geometries, thus allowing for the adaptation of these omnithermal devices into intricate forms like ellipses. Importantly, this theory can also accommodate any blend of the three core heat transfer modes, as long as transformation protocols are upheld. For tangible demonstrations, one might consider the heat transfer properties exhibited by aerogels. Within aerogels, both fluids and gases can serve as the carriers of convective flux. Multilayered structures offer a platform to manifest anisotropy and non-uniformity. Additional porous materials, with adjustable porosity like ceramics, metals, and rocks, also emerge as promising candidates.