Theory for Enhanced Thermal Concentrators: Thermal Conductivity Coupling

Abstract

In this chapter, we propose the theory of conductivity coupling to solve the problem that the concentrating efficiency of a thermal concentrator is restricted by its geometric configuration. We first discuss a monolayer scheme with an isotropic thermal conductivity, which can break the upper limit but is still restricted by the geometric structure. We further explore another degree of freedom by considering the monolayer scheme with an anisotropic thermal conductivity or adding the second shell with an isotropic thermal conductivity, thereby freeing the concentrating efficiency from the geometric configuration. Finite-element simulations are performed to confirm the theoretical predictions, and experimental suggestions are also provided to improve feasibility. These results may have potential applications for thermal camouflage and provide insights into other diffusive systems such as static magnetic fields and DC fields for achieving similar behaviors.

1 Opening Remarks

The theory of transformation thermotics [1, 2] has promoted an advanced control of heat transfer based on thermal metamaterials [3, 4]. As a representative example, a thermal concentrator [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24] can increase its interior temperature gradient without distorting its exterior one. So far, many schemes have been proposed to design thermal concentrators. The initial explorations are based on the theory of transformation thermotics [5,6,7,8,9,10,11,12,13,14] which is a bridge linking space transformations and material transformations. Therefore, the effect of thermal concentrating can be achieved by coating a region (i.e., the core) with a designed shell (i.e., the thermal concentrator). This scheme has three features: (I) the thermal conductivities inside and outside the shell are identical; (II) the shell has an anisotropic thermal conductivity that is commonly realized by a layered structure [15,16,17,18,19]; and (III) both temperature gradient and heat flux are enhanced in the core. An alternative scheme is based on the effective medium theory [20,21,22] with also three features: (I) the thermal conductivity inside the shell should be smaller than that outside the shell; (II) the shell requires only a homogeneous and isotropic thermal conductivity; and (III) temperature gradient increases but heat flux decreases in the core. Recently, topology optimization has also become a powerful tool for designing thermal concentrators [23, 24], which largely reduces the requirements for materials and structures [25,26,27,28].

Despite varieties of schemes, the concentrating efficiency of a thermal concentrator, commonly reflected in the ratio of its interior to exterior temperature gradients, has an upper limit. Specifically, when a circular concentrator with inner radius $r_c$ and outer radius $r_s$ is designed, the upper limit for the concentrating efficiency is $\eta =r_s/r_c$ [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24], indicating that the isotherms in the shell are completely compressed to the core. To reach the upper limit, the theory of transformation thermotics requires a shell with an extremely anisotropic thermal conductivity [29,30,31,32], and the effective medium theory needs to fabricate a core with a near-zero thermal conductivity [20,21,22]. However, breaking the upper limit for concentration efficiency is still challenging.

To solve the problem, we investigate a monolayer scheme and two extended schemes with the coupling of thermal conductivities. These three schemes feature the simultaneous concentrating of heat flux and temperature gradient with only homogeneous materials. More importantly, they contribute to much higher efficiency than existing schemes. Nevertheless, apparent negative thermal conductivities are required, which can be effectively realized with external heat energy and have been applied to design thermal metamaterials [33,34,35,36].

2 Monolayer Scheme with Isotropic Thermal Conductivity

We first discuss a monolayer scheme in the Cartesian coordinate system $x_i$ ($i=1,\,2,\,3$ for three dimensions and $i=1,\,2$ for two dimensions). A confocal core-shell structure is embedded in a background (Fig. 8.1a). The semi-axis of the core (or shell) along the $x_i$ axis is denoted as $r_{ci}$ (or $r_{si}$). The thermal conductivities of the core, shell, and background are denoted as $\kappa _c$, $\kappa _s$, and $\kappa _b$, respectively. The conversion between the Cartesian coordinates $x_i$ and elliptical (or ellipsoidal) coordinates $\rho _j$ is

$$\begin{aligned} \sum _i\frac{x_i^2}{\rho _j+r_{ci}^2}=1, \end{aligned}$$

(8.1)

with parameters of $j=1,\,2,\,3$ for three dimensions and $j=1,\,2$ for two dimensions. The coordinate $\rho _1$ $\left( >-r_{ci}^2\right)$ denotes an elliptical (or ellipsoidal) boundary. For example, the inner and outer boundaries of the shell can be denoted as $\rho _1=\rho _c$ $\left( =0\right)$ and $\rho _1=\rho _s$, respectively. In the presence of a uniform thermal field along the $x_i$ axis, thermal conduction equation can be expressed in the elliptical (or ellipsoidal) coordinate system as [38]

$$\begin{aligned} \frac{\partial }{\partial \rho _1}\left[ g\left( \rho _1\right) \frac{\partial T}{\partial \rho _1}\right] +\frac{g\left( \rho _1\right) }{\rho _1+r_{ci}^2} \frac{\partial T}{\partial \rho _1}=0, \end{aligned}$$

(8.2)

with a definition of $g\left( \rho _1\right) =\prod \limits _i\left( \rho _1+r_{ci}^2\right) ^{1/2}$. For three dimensions, $4\pi g\left( \rho _1\right) /3$ represents the volume of an ellipsoid. For two dimensions, $\pi g\left( \rho _1\right)$ denotes the area of an ellipse. The temperature distributions along the $x_i$ axis in the core $T_{ci}$, shell $T_{si}$, and background $T_{bi}$ can be expressed as [38]

$$\begin{aligned} \left\{ \begin{aligned} T_{ci}&=A_{ci}x_i,\\ T_{si}&=\left[ A_{si}+B_{si}\phi _i\left( \rho _1\right) \right] x_i,\\ T_{bi}&=\left[ A_{bi}+B_{bi}\phi _i\left( \rho _1\right) \right] x_i, \end{aligned} \right. \end{aligned}$$

(8.3)

with a definition of $\phi _i\left( \rho _1\right) =\int _{\rho _c}^{\rho _1}\left[ \left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) \right] ^{-1}d\rho _1$. $A_{ci}$, $A_{si}$, $B_{si}$, and $B_{bi}$ can be determined by the continuities of temperature and normal heat flux. Since the temperature distribution in the background should be undistorted, we take $B_{bi}=0$ and then obtain

$$\begin{aligned} \kappa _b=\frac{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s+\left( 1-L_{si}\right) \left( \kappa _c-\kappa _s\right) f}{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s-L_{si}\left( \kappa _c-\kappa _s\right) f}\kappa _s, \end{aligned}$$

(8.4)

with a definition of $f=g\left( \rho _c\right) /g\left( \rho _s\right) =\prod \limits _ir_{ci}/r_{si}$. The shape factor $L_{c1}$ (or $L_{s1}$) reflects the flattening degree of the ellipse, and the larger the shape factor is, the more flattening the ellipse is.

Fig. 8.1

Adapted from Ref. [37]

a Monolayer scheme with an isotropic thermal conductivity. b Concentrating efficiency $\eta$ as a function of geometric configuration $r_{s1}/r_{c1}$. Lines and points denote theoretical results and simulation results, respectively.

Then, the concentrating efficiency of a thermal concentrator can then be defined as the ratio of its interior and exterior temperature gradients (taking $B_{bi}=0$),

$$\begin{aligned} \eta =\frac{\nabla T_{ci}}{\nabla T_{bi}}=\frac{A_{ci}}{A_{bi}}=\frac{\kappa _s}{L_{ci} \kappa _c+\left( 1-L_{ci}\right) \kappa _s-L_{si}\left( \kappa _c-\kappa _s\right) f}. \end{aligned}$$

(8.5)

For a two-dimensional circular case with $L_{ci}=L_{si}=1/2$, Eq. (8.5) can be reduced to

$$\begin{aligned} \eta =\frac{2\kappa _s}{\kappa _c+\kappa _s-\left( \kappa _c-\kappa _s\right) f}. \end{aligned}$$

(8.6)

For a three-dimensional spherical case with $L_{ci}=L_{si}=1/3$, Eq. (8.5) can be reduced to

$$\begin{aligned} \eta =\frac{3\kappa _s}{\kappa _c+2\kappa _s-\left( \kappa _c-\kappa _s\right) f}. \end{aligned}$$

(8.7)

We also consider the same thermal conductivities inside and outside the shell and then obtain two coupling conditions to satisfy $\kappa _c=\kappa _b$,

$$\begin{aligned} \kappa _s=\kappa _c, \end{aligned}$$

(8.8)

$$\begin{aligned} -\frac{1-L_{ci}-\left( 1-L_{si}\right) f}{L_{ci}-L_{si}f}\kappa _s=\kappa _c. \end{aligned}$$

(8.9)

Equation (8.8) leads to a trivial case with $\kappa _c=\kappa _s=\kappa _b$ and $\eta =1$. However, if we apply the coupling condition described by Eq. (8.9), the concentrating efficiency largely increases,

$$\begin{aligned} \eta =f^{-1}=\prod _ir_{si}/r_{ci}. \end{aligned}$$

(8.10)

Clearly, the concentrating efficiency exceeds the upper limit for existing thermal concentrators $\eta =r_{s1}/r_{c1}$, and a smaller $L_{c1}$ yields a larger $\eta$ (Fig. 8.1b). However, the geometric configuration still restricts the concentrating efficiency, so we further consider the following two schemes by adding another degree of freedom.

3 Monolayer Scheme with Anisotropic Thermal Conductivity

We further consider a shell with an anisotropic thermal conductivity. Since it is not convenient to unify two and three dimensions, we discuss them independently. Nevertheless, the conclusion of three dimensions is similar to that of two dimensions. We first discuss a two-dimensional circular shell with inner and outer radii of $r_c$ and $r_s$, respectively (Fig. 8.2a). Thermal conduction equation can be written in the cylindrical coordinate system $\left( r,\,\theta \right)$ as [29]

$$\begin{aligned} \frac{1}{r} \frac{\partial }{\partial r}\left( r \kappa _{srr} \frac{\partial T}{\partial r}\right) +\frac{1}{r} \frac{\partial }{\partial \theta }\left( \kappa _{s\theta \theta } \frac{\partial T}{r\partial \theta }\right) =0, \end{aligned}$$

(8.11)

where $\kappa _{srr}$ and $\kappa _{s\theta \theta }$ are the radial and tangential thermal conductivities of the shell, respectively. The temperature distributions of the core $T_c$, shell $T_s$, and background $T_b$ can be written as [29]

$$\begin{aligned} \left\{ \begin{aligned} T_c&=A_c r\cos \theta ,\\ T_s&=\left( A_sr^{d_{s1}}+B_sr^{d_{s2}}\right) \cos \theta ,\\ T_b&=\left( A_b r+B_br^{-1}\right) \cos \theta , \end{aligned} \right. \end{aligned}$$

(8.12)

with definitions of $d_{s1}=\sqrt{\kappa _{s\theta \theta }/\kappa _{srr}}$ and $d_{s2}=-\sqrt{\kappa _{s\theta \theta }/\kappa _{srr}}$. $A_c$, $A_s$, $B_s$, and $B_b$ are four constants to be determined by the boundary conditions. By taking $B_b=0$, we can further derive

$$\begin{aligned} \kappa _b=\frac{d_{s1}\left( \kappa _c-d_{s2}\kappa _{srr}\right) -d_{s2}\left( \kappa _c-d_{s1}\kappa _{srr}\right) f^{\left( d_{s1}-d_{s2}\right) /2}}{\kappa _c-d_{s2}\kappa _{srr}-\left( \kappa _c-d_{s1}\kappa _{srr}\right) f^{\left( d_{s1}-d_{s2}\right) /2}}\kappa _{srr}, \end{aligned}$$

(8.13)

with a definition of $f=r_c^2/r_s^2$. We also define the concentrating efficiency as

$$\begin{aligned} \eta =\frac{A_c}{A_b}=\frac{\left( d_{s1}-d_{s2}\right) \kappa _{srr}f^{\left( d_{s1}-1\right) /2}}{\kappa _c-d_{s2}\kappa _{srr}-\left( \kappa _c-d_{s1}\kappa _{srr}\right) f^{\left( d_{s1}-d_{s2}\right) /2}}. \end{aligned}$$

(8.14)

For an isotropic case with $d_{s1}=-d_{s2}=1$, Eq. (8.14) can be simplified as

$$\begin{aligned} \eta =\frac{2\kappa _{srr}}{\kappa _c+\kappa _{srr}-\left( \kappa _c-\kappa _{srr}\right) f}, \end{aligned}$$

(8.15)

which has the same form as Eq. (8.6) in Sect. 8.2.

Fig. 8.2

Adapted from Ref. [37]

a Monolayer scheme with an anisotropic thermal conductivity. b $\kappa _{s\theta \theta }/\kappa _c$ and $\eta$ as a function of $\kappa _{srr}/\kappa _c$ when $r_s/r_c=2$. Lines and points denote theoretical results and simulation results, respectively.

We also obtain two coupling conditions for $\kappa _c=\kappa _b$,

$$\begin{aligned} d_{s1}\kappa _{srr}=\kappa _c, \end{aligned}$$

(8.16)

$$\begin{aligned} d_{s2}\kappa _{srr}=\kappa _c. \end{aligned}$$

(8.17)

Equations (8.16) and (8.17) can be unified as

$$\begin{aligned} \kappa _{srr}\kappa _{s\theta \theta }=\kappa _c^2, \end{aligned}$$

(8.18)

which is plotted with the dotted line in Fig. 8.2b.

When the thermal conductivities of the core and shell satisfy Eq. (8.18), Eq. (8.14) becomes

$$\begin{aligned} \eta =f^{-\left( 1-\kappa _c/\kappa _{srr}\right) /2}=\left( r_s/r_c\right) ^{1-\kappa _c/\kappa _{srr}}, \end{aligned}$$

(8.19)

which is plotted with the dashed-dotted line in Fig. 8.2b. Obviously, the minimum value $\eta \rightarrow 0$ appears when $\kappa _{srr}/\kappa _c\rightarrow 0^+$, and the maximum value $\eta \rightarrow \infty$ appears when $\kappa _{srr}/\kappa _c\rightarrow 0^-$. Moreover, we can observe $\eta \rightarrow r_s/r_c$ when $\kappa _{srr}/\kappa _c\rightarrow \pm \infty$, which is just the upper limit for existing concentrating efficiency (see the solid line in Fig. 8.2b). If the thermal conductivity of the shell is isotropic and nontrivial $\kappa _{srr}/\kappa _c=1/d_{s2}=-1$, the concentrating efficiency also exceeds the upper limit and becomes $\eta =r_s^2/r_c^2$, which is in accordance with the two-dimensional conclusion in Sect. 8.2. Therefore, the concentrating efficiency can exceed the upper limit and even approach infinity when $\kappa _{srr}/\kappa _c\rightarrow 0^-$.

Fig. 8.3

Adapted from Ref. [37]

a Bilayer scheme with isotropic thermal conductivities. b $\kappa _s/\kappa _c$ and $\eta$ as a function of $\kappa _t/\kappa _c$ when $r_{s1}/r_{c1}=1.2$, $r_{t1}/r_{c1}=1.4$, and $L_{c1}=1/3$. Lines and points denote theoretical results and simulation results, respectively.

4 Bilayer Scheme with Isotropic Thermal Conductivity

We then consider the second shell whose isotropic thermal conductivity and semi-axis along the $x_i$ axis are denoted as $\kappa _t$ and $r_{ti}$, respectively (Fig. 8.3a). With the conclusion of the monolayer scheme (Eq. (8.4)), the effective thermal conductivity of the core and the first shell $\kappa _{cs}$ can be calculated by

$$\begin{aligned} \kappa _{cs}=\frac{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s+\left( 1-L_{si}\right) \left( \kappa _c-\kappa _s\right) f}{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s-L_{si}\left( \kappa _c-\kappa _s\right) f}\kappa _s. \end{aligned}$$

(8.20)

We then treat the core and the first shell as an effective core with an effective thermal conductivity of $\kappa _{cs}$, so we can further derive

$$\begin{aligned} \kappa _b=\frac{L_{si}\kappa _{cs}+\left( 1-L_{si}\right) \kappa _t+\left( 1-L_{ti}\right) \left( \kappa _{cs}-\kappa _t\right) p}{L_{si}\kappa _{cs}+\left( 1-L_{si}\right) \kappa _t-L_{ti}\left( \kappa _{cs}-\kappa _t\right) p}\kappa _t, \end{aligned}$$

(8.21)

with a definition of $p=g\left( \rho _s\right) /g\left( \rho _t\right) =\prod \limits _ir_{si}/r_{ti}$. $\rho _t$ denotes the outer boundary of the second shell. $L_{ti}$ is the shape factor of the second shell along the $x_i$ axis,

$$\begin{aligned} L_{ti}=\frac{g\left( \rho _t\right) }{2} \int \limits _{\rho _t}^{\infty }\left[ \left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) \right] ^{-1}d\rho _1. \end{aligned}$$

(8.22)

We can also express the concentrating efficiency as

$$\begin{aligned} \eta =\frac{A_{ci}}{A_{bi}}=\frac{\kappa _s\kappa _t}{\lambda _1+\lambda _2+\lambda _3}, \end{aligned}$$

(8.23)

where $\lambda _1$, $\lambda _2$, and $\lambda _3$ take the form of

$$\begin{aligned} \left\{ \begin{array}{l} \lambda _1=\left[ L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s\right] \left[ L_{si}\kappa _s+\left( 1-L_{si}\right) \kappa _t-L_{ti}\left( \kappa _s-\kappa _t\right) p\right] ,\\ \lambda _2=-L_{ti}\left( \kappa _c-\kappa _s\right) \left[ \left( 1-L_{si}\right) \kappa _s+L_{si}\kappa _t\right] fp,\\ \lambda _3=L_{si}\left( 1-L_{si}\right) \left( \kappa _c-\kappa _s\right) \left( \kappa _s-\kappa _t\right) f. \end{array} \right. \end{aligned}$$

(8.24)

As a more general model, the bilayer scheme can also be reduced to the monolayer scheme in Sect. 8.2 at two certain conditions. When $\kappa _c=\kappa _s$, Eq. (8.23) can be reduced to

$$\begin{aligned} \eta =\frac{\kappa _t}{L_{si} \kappa _s+\left( 1-L_{si}\right) \kappa _t-L_{ti}\left( \kappa _s-\kappa _t\right) p}. \end{aligned}$$

(8.25)

When $\kappa _s=\kappa _t$, Eq. (8.23) becomes

$$\begin{aligned} \eta =\frac{\kappa _t}{L_{ci} \kappa _c+\left( 1-L_{ci}\right) \kappa _t-L_{ti}\left( \kappa _c-\kappa _t\right) fp}. \end{aligned}$$

(8.26)

Obviously, Eqs. (8.25) and (8.26) have similar forms as Eq. (8.5) in Sect. 8.2.

We can also derive two coupling conditions for $\kappa _c=\kappa _b$,

$$\begin{aligned} M\left( \kappa _s,\,\kappa _t\right) =\kappa _c, \end{aligned}$$

(8.27)

$$\begin{aligned} N\left( \kappa _s,\,\kappa _t\right) =\kappa _c. \end{aligned}$$

(8.28)

M and N are two analytical functions. Therefore, one $\kappa _t$ corresponds to two $\kappa _s$ for satisfying $\kappa _c=\kappa _b$, i.e., $\kappa _s=m\left( \kappa _t\right)$ being a continuous function (see the dotted line in the upper inset of Fig. 8.3b) and $\kappa _s=n\left( \kappa _t\right)$ being a quasi-hyperbolic function (see the dotted line in the lower inset of Fig. 8.3b). We do not express the concrete forms of m and n because they are too complicated.

When Eq. (8.27) is satisfied, the upper limit of $\eta =r_{t1}/r_{c1}$ can be broken, but the concentrating efficiency can still not tend to infinity (see the dashed-dotted line in the upper inset of Fig. 8.3b). Moreover, Eq. (8.27) contains two special cases that can be reduced to the conclusion in Sect. 8.2. One features a concentrating efficiency of $\eta =f^{-1}$ with the same thermal conductivities of the second shell and core,

$$\begin{aligned} -\frac{1-L_{ci}-\left( 1-L_{si}\right) f}{L_{ci}-L_{si}f}\kappa _s=\kappa _t=\kappa _c. \end{aligned}$$

(8.29)

The other features a concentrating efficiency of $\eta =p^{-1}$ with the same thermal conductivities of the first shell and core,

$$\begin{aligned} \kappa _s=-\frac{1-L_{si}-\left( 1-L_{ti}\right) p}{L_{si}-L_{ti}p}\kappa _t=\kappa _c. \end{aligned}$$

(8.30)

Fortunately, Eq. (8.28) can lead to an infinite efficiency. $\kappa _t/\kappa _c\rightarrow 0^-$ and $\kappa _t/\kappa _c\rightarrow 0^+$, respectively, yield $\eta \rightarrow \infty$ and $\eta \rightarrow -\infty$, and the thermal conductivity of the first shell satisfies

$$\begin{aligned} -\frac{1-L_{ci}-\left( 1-L_{si}\right) f}{L_{ci}+\left( 1-L_{si}\right) f}\kappa _s\approx \kappa _c. \end{aligned}$$

(8.31)

Meanwhile, $\kappa _t/\kappa _c\rightarrow \mp \infty$ can also lead to $\eta \rightarrow \pm \infty$, and the thermal conductivity of the first shell satisfies

$$\begin{aligned} -\frac{1-L_{ci}+L_{si}f}{L_{ci}-L_{si}f}\kappa _s\approx \kappa _c. \end{aligned}$$

(8.32)

Moreover, Eq. (8.28) also contains a special case that can be reduced to the conclusion in Sect. 8.2. That is, the concentrating efficiency of $\eta =\left( fp\right) ^{-1}$ occurs when the two shells have the same thermal conductivities,

$$\begin{aligned} \begin{aligned} \kappa _c=-\frac{1-L_{ci}-\left( 1-L_{ti}\right) fp}{L_{ci}-L_{ti}fp}\kappa _s=-\frac{1-L_{ci}-\left( 1-L_{ti}\right) fp}{L_{ci}-L_{ti}fp}\kappa _t. \end{aligned} \end{aligned}$$

(8.33)

There is another case for $\eta =\left( fp\right) ^{-1}$ if the thermal conductivities of the two shells satisfy

$$\begin{aligned} \begin{aligned} \kappa _c=-\frac{1-L_{ci}-\left( 1-L_{si}\right) f}{L_{ci}-L_{si}f}\kappa _s=-\frac{1-L_{si}-\left( 1-L_{ti}\right) p}{L_{si}-L_{ti}p}\kappa _t. \end{aligned} \end{aligned}$$

(8.34)

Conductivity coupling occurs layer by layer in this case. The core is coupled with the first shell described by Eq. (8.9). Then, they are treated as an effective core with an effective thermal conductivity of $\kappa _c$. The effective core is then coupled with the second shell described by the similar form of Eq. (8.9).

Another unique feature of Eq. (8.28) is the concentrating efficiency of $\eta <0$ when the thermal conductivity of the second shell satisfies

$$\begin{aligned} \kappa _t>\frac{1-L_{si}+L_{ti}p}{1-L_{si}-\left( 1-L_{ti}\right) p}\kappa _c, \end{aligned}$$

(8.35)

or

$$\begin{aligned} 0<\kappa _t<\frac{L_{si}-L_{ti}p}{L_{si}+\left( 1-L_{ti}\right) p}\kappa _c, \end{aligned}$$

(8.36)

indicating that the temperature gradient in the core changes its direction.

Then we can draw a brief conclusion for these three schemes. The monolayer scheme with an isotropic thermal conductivity can break the upper limit but is still restricted by its geometric configuration. To be free from geometric configurations, we further consider the monolayer scheme with an anisotropic thermal conductivity and the bilayer scheme with isotropic thermal conductivities. For the former, the efficiency can tend to infinity with $\kappa _{srr}/\kappa _c\rightarrow 0^-$. For the latter, the efficiency can also reach infinity when $\kappa _t/\kappa _c\rightarrow 0^-$ or $\kappa _t/\kappa _c\rightarrow -\infty$. Moreover, the latter features $\eta <0$ if the coupling condition is appropriately chosen.

Fig. 8.4

Adapted from Ref. [37]

a–c Simulations of the monolayer scheme with an isotropic thermal conductivity. d $T^*$ as a function of $x^*$. Parameters: a $L_{c1}=0.4$ and $\kappa _s/\kappa _c=-0.58$; b $L_{c1}=0.5$ and $\kappa _s/\kappa _c=-1$; c $L_{c1}=0.6$ and $\kappa _s/\kappa _c=-1.87$; and a–c $r_{s1}/r_{c1}=1.5$ and $\kappa _c=\kappa _b$.

Fig. 8.5

Adapted from Ref. [37]

a–c Simulations of the monolayer scheme with an anisotropic thermal conductivity. d Temperature distribution of the existing scheme based on transformation theory close to the upper limit of concentrating efficiency. e $T^*$ as a function of $x^*$. Parameters: a $\kappa _{srr}/\kappa _c=-0.5$; b $\kappa _{srr}/\kappa _c=-1$; c $\kappa _{srr}/\kappa _c=-2$; d $\kappa _{srr}/\kappa _c=(r+100)/r$; and a–d $r_s/r_c=2$ and $\kappa _c=\kappa _b$.

5 Finite-Element Simulation

We also perform finite-element simulations to confirm the theories with COMSOL Multiphysics. From a practical perspective, although interfacial thermal resistance exists widely, its effect at the macroscopic scale is not dominant, so it is reasonable to ignore it. Without loss of generality, we consider a two-dimensional case with size $10\times 10$ cm$^2$ and set the core and background thermal conductivities as 1 W m$^{-1}$ K$^{-1}$. The left boundaries are set at 313 K, the right boundaries are set at 273 K, and the upper and lower boundaries are adiabatic. To compare the concentrating efficiency of different thermal concentrators, we introduce a dimensionless temperature of $T^*=100(T-T_0)/T_0$ and a dimensionless position of $x^*=x/w$, where $T_0$ and w denote the central temperature and half-length of the system, respectively.

Thermal concentrating aims to increase the temperature gradient in the core without distorting that in the background. In order to confirm Eqs. (8.5), (8.8), and (8.9) and demonstrate the expected case shown in Fig. 8.1, we design three structures with different shape factors, and the corresponding results are presented in Figs. 8.4a–c. The temperature profiles outside the shells are undistorted as if there were no core-shell structures in the center. Meanwhile, the isotherms in the cores are concentrated as expected. According to the Fourier law $\boldsymbol{J}=-\kappa \boldsymbol{\nabla }T$, heat fluxes are also enhanced in the cores due to the larger temperature gradients. The dimensionless temperatures are plotted as a function of dimensionless position in Fig. 8.4d.

By considering the monolayer scheme with an anisotropic thermal conductivity, we confirm the theoretical prediction of Eqs. (8.14) and (8.17). Then, we design three structures with different thermal conductivities of the shells (Fig. 8.5a–c). Similar to Fig. 8.4, the temperature profiles in Fig. 8.5a–c prove the effect of thermal concentrating. Also, we draw the temperature distribution of the thermal concentrator (Fig. 8.5d) designed by transformation theory for comparison. Figure 8.5e displays the temperature distribution along the central horizontal axis. As presented in Fig. 8.2b, the temperature gradient in the core increases with the increment of $\kappa _{srr}/\kappa _c$, leading to the improvement of concentrating efficiency. Thus, we can control $\kappa _{srr}/\kappa _c\rightarrow 0^-$ for an extreme concentrating efficiency.

Fig. 8.6

Adapted from Ref. [37]

a–e Simulations of the bilayer scheme with isotropic thermal conductivities. f $T^*$ as a function of $x^*$. Parameters: a $\kappa _{s}/\kappa _{c}=0.0826$ and $\kappa _{t}/\kappa _{c}=-0.05$; b $\kappa _{s}/\kappa _{c}=-1.14$ and $\kappa _{t}/\kappa _{c}=-0.05$; c $\kappa _{s}/\kappa _{c}=-0.175$ and $\kappa _{t}/\kappa _{c}=-10$; d $\kappa _{s}/\kappa _{c}=-0.122$ and $\kappa _{t}/\kappa _{c}=15$; e $\kappa _{s}/\kappa _{c}=-2.83$ and $\kappa _{t}/\kappa _{c}=0.05$; and a–e $r_{s1}/r_{c1}=1.2$, $r_{t1}/r_{c1}=1.4$, $L_{c1}=1/3$, and $\kappa _{c}=\kappa _{b}$.

For the bilayer scheme with isotropic thermal conductivities, two coupling conditions (Eqs. (8.27) and (8.28)) are available. Similar to the structures in Figs. 8.4 and 8.5, those in Fig. 8.6 also ensure that isotherms outside the shells are straight and those in the cores are denser, thereby realizing the effect of thermal concentrating. With the coupling condition of Eq. (8.27), the efficiency changes continuously with $\kappa _t/\kappa _c$ (Fig. 8.3b). We further design a structure to display the concentrating efficiency when $\kappa _t/\kappa _c\rightarrow 0^-$ (Fig. 8.6a). The coupling condition of Eq. (8.28) can lead to an infinite efficiency. That is, $\eta \rightarrow +\infty$ when $\kappa _t/\kappa _c\rightarrow 0^-$ (Fig. 8.6b) or $\kappa _t/\kappa _c\rightarrow -\infty$ (Fig. 8.6c), and $\eta \rightarrow -\infty$ for $\kappa _t/\kappa _c \rightarrow 0^+$ (Fig. 8.6d) or $\kappa _t/\kappa _c \rightarrow +\infty$ (Fig. 8.6e). As shown in Fig. 8.6f, the effect of thermal concentrating can be quantitatively observed.

Fig. 8.7

Adapted from Ref. [37]

Experimental suggestions. a Schematic diagram for realizing apparent negative conductivity. b Without temperature control. c Continuous temperature control. d Discrete point heat sources whose temperatures are shown in Tables 8.1 and 8.2. The core and background in d are a brass plate (109 W m$^{-1}$ K$^{-1}$) drilled with 2116 air circles with a radius of 0.1 cm. The shell is drilled with 282 air ellipses with a major (or minor) semi-axis of 0.35 cm (or 0.02 cm). Other parameters: b and d $\kappa _s=\textrm{diag}(23,\,92)$ W m$^{-1}$ K$^{-1}$; c $\kappa _s=\textrm{diag}(-23,\,-92)$ W m$^{-1}$ K$^{-1}$; and b–d $r_c=1$ cm, $r_s=2$ cm, $\kappa _c=\kappa _b=46$ W m$^{-1}$ K$^{-1}$. The black lines and blue arrows in b–d denote isotherms and heat fluxes, respectively.

Table 8.1 Temperatures of point heat sources at the outer boundary of the shell in Fig. 8.7d.

Table 8.2 Temperatures of point heat sources at the inner boundary of the shell in Fig. 8.7d.

6 Experimental Suggestion

The coupling conditions require apparent negative thermal conductivities [33,34,35,36], which cannot happen spontaneously in experiments. To achieve the equivalent effect, we can resort to external heat sources (Fig. 8.7a). According to the thermal uniqueness theorem [39, 40], as long as we realize the same boundary temperature distributions by adding external heat sources at the inner and outer boundaries of the shell, we can obtain the same temperature profiles. Since the central temperature gradient and heat flux in Fig. 8.7c are almost the same as those in Fig. 8.7b, we prove that an apparent negative thermal conductivity can be effectively achieved using external heat sources. Then we design a structure as a feasible experimental suggestion (Fig. 8.7d). We add a series of point heat sources at the inner and outer boundaries of the shell (Fig. 8.7d). The precise temperatures are presented in Tables 8.1 and 8.2, which can be experimentally controlled by adjusting the voltages of heaters and coolers according to Eqs. (1) and (2) in Ref. [39]. The required thermal conductivity can be realized by punching air holes on a brass plate (109 W m$^{-1}$ K$^{-1}$), whose left and right edges are put into hot (313 K) and cold (273 K) sinks, respectively. To achieve the thermal conductivities of the core and background in Fig. 8.7b, 2116 air circles are drilled on the brass, leading to an effective thermal conductivity of 46 W m$^{-1}$ K$^{-1}$ (calculated by Eq. (11) in Ref. [41]). The shell region is composed of 282 air ellipses, leading to an effective thermal conductivity of $\textrm{diag}(23,\,92)$ W m$^{-1}$ K$^{-1}$ (calculated by Eq. (11) in Ref. [41]). By comparing the temperature distributions in Fig. 8.7b–d, we can confirm that the scheme in Fig. 8.7d can realize the effect of Fig. 8.7b in experiments.

7 Conclusion

We break the upper limit for the concentrating efficiency of existing thermal concentrators by coupling thermal conductivities. We first explore a monolayer scheme with an isotropic thermal conductivity, which can break the upper limit but is still restricted by its geometric configuration. Then, we consider a shell with an anisotropic thermal conductivity or add the second shell with an isotropic thermal conductivity as another degree of freedom, which renders the concentrating efficiency free from geometric configurations. Apparent negative thermal conductivities are required in these three schemes, which can be effectively realized by external energy or thermoelectric materials. Since negative permeability [42,43,44] and negative electric conductivity [45] have been, respectively, revealed in static magnetic fields and DC fields, it is promising to extend our results to these diffusive fields due to the similar equation forms (i.e., the Laplace equation). Moreover, the present theory is applicable not only for thermal concentrators with $\eta >1$ but also for thermal invisible sensors with $\eta =1$ [46, 47] and thermal cloaks with $\eta =0$ (perfect cloaking) or $\eta <1$ (imperfect cloaking) [48, 49]. A typical feature for concentrating, sensing, or cloaking is the undistorted background temperature distribution, so these schemes may provide insights into thermal camouflage [50] for misleading infrared detection. It is also promising to extend the related mechanisms towards multi-function and micro/nano-scale.

8 Exercise and Solution

Exercise

1. Prove that the three-dimensional case in Sect. 8.3 is similar to two dimensions.

Solution

1. The tensorial thermal conductivity of the shell can be expressed in the spherical coordinate system $\left( r,\,\theta ,\,\phi \right)$ as $\boldsymbol{\kappa }_{s}=\textrm{diag}\left( \kappa _{srr},\,\kappa _{s\theta \theta },\,\kappa _{s\phi \phi }\right)$. For simplicity, we assume a axial symmetry with $\kappa _{s\theta \theta }=\kappa _{s\phi \phi }$. Therefore, thermal conduction is independent of $\phi$, which is dominated by

$$\begin{aligned} \frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 \kappa _{srr} \frac{\partial T}{\partial r}\right) +\frac{1}{r\sin \theta } \frac{\partial }{\partial \theta }\left( \sin \theta \kappa _{s\theta \theta } \frac{\partial T}{r \partial \theta }\right) =0. \end{aligned}$$

(8.37)

The temperature distributions in the core $T_c$ , shell $T_s$, and background $T_b$ can be written as

$$\begin{aligned} \left\{ \begin{aligned} T_c&=A_c r\cos \theta , \\ T_s&=\left( A_s r^{h_{s1}}+B_sr^{h_{s2}}\right) \cos \theta , \\ T_b&=\left( A_b r+B_b r^{-2}\right) \cos \theta , \end{aligned} \right. \end{aligned}$$

(8.38)

with definitions of $h_{s1}=\left( -1+\sqrt{1+8\kappa _{s\theta \theta }/\kappa _{srr}}\right) /2$ and $h_{s2}=\left( -1-\sqrt{1+8\kappa _{s\theta \theta }/\kappa _{srr}}\right) /2$. By substituting Eq. (8.38) into the boundary conditions, we can obtain

$$\begin{aligned} \left\{ \begin{array}{l} A_cr_c=A_sr_c^{h_{s1}}+B_sr_c^{h_{s2}},\\ A_sr_s^{h_{s1}}+B_sr_s^{h_{s2}}=A_br_s+B_br_s^{-2},\\ \kappa _cA_c=\kappa _{srr}\left( h_{s1}A_sr_c^{h_{s1}-1}+h_{s2}B_sr_c^{h_{s2}-1}\right) ,\\ \kappa _{srr}\left( h_{s1}A_sr_s^{h_{s1}-1}+h_{s2}B_sr_s^{h_{s2}-1}\right) =\kappa _b\left( A_b-2B_br_s^{-3}\right) . \end{array} \right. \end{aligned}$$

(8.39)

We can calculate $A_c$, $A_s$, $B_s$, and $B_b$ with Eq. (8.39). By taking $B_b=0$, we can further derive

$$\begin{aligned} \kappa _b=\frac{h_{s1}\left( \kappa _c-h_{s2}\kappa _{srr}\right) -h_{s2}\left( \kappa _c-h_{s1}\kappa _{srr}\right) f^{\left( h_{s1}-h_{s2}\right) /3}}{\kappa _c-h_{s2}\kappa _{srr}-\left( \kappa _c-h_{s1}\kappa _{srr}\right) f^{\left( h_{s1}-h_{s2}\right) /3}}\kappa _{srr}, \end{aligned}$$

(8.40)

with a definition of $f=r_c^3/r_s^3$. The concentrating efficiency is

$$\begin{aligned} \eta =\frac{A_c}{A_b}=\frac{\left( h_{s1}-h_{s2}\right) \kappa _{srr}f^{\left( h_{s1}-1\right) /3}}{\kappa _c-h_{s2}\kappa _{srr}-\left( \kappa _c-h_{s1}\kappa _{srr}\right) f^{\left( h_{s1}-h_{s2}\right) /3}}. \end{aligned}$$

(8.41)

For an isotropic case with $h_{s1}=1$ and $h_{s2}=-2$, Eq. (8.14) can be simplified as

$$\begin{aligned} \eta =\frac{3\kappa _{srr}}{\kappa _c+2\kappa _{srr}-\left( \kappa _c-\kappa _{srr}\right) f}, \end{aligned}$$

(8.42)

which has the same form as Eq. (8.7) in Sect. 8.2.

We can also derive two coupling conditions for $\kappa _c=\kappa _b$,

$$\begin{aligned} h_{s1}\kappa _{srr}=\kappa _c, \end{aligned}$$

(8.43)

$$\begin{aligned} h_{s2}\kappa _{srr}=\kappa _c. \end{aligned}$$

(8.44)

Equations (8.43) and (8.44) can also be unified as

$$\begin{aligned} 2\kappa _{srr}\kappa _{s\theta \theta }-\kappa _c\kappa _{srr}=\kappa _c^2. \end{aligned}$$

(8.45)

When Eq. (8.45) is satisfied, Eq. (8.41) can be reduced to

$$\begin{aligned} \eta =f^{-\left( 1-\kappa _c/\kappa _{srr}\right) /3}=\left( r_s/r_c\right) ^{1-\kappa _c/\kappa _{srr}}, \end{aligned}$$

(8.46)

which has the same form as two dimensions (Eq. (8.19)). Therefore, the minimum value $\eta \rightarrow 0$ occurs with $\kappa _{srr}/\kappa _c\rightarrow 0^+$, and the maximum value $\eta \rightarrow \infty$ occurs with $\kappa _{srr}/\kappa _c\rightarrow 0^-$. Moreover, we can find $\eta \rightarrow r_s/r_c$ when $\kappa _{srr}/\kappa _c\rightarrow \pm \infty$. If we consider an isotropic and nontrivial shell with $\kappa _{srr}/\kappa _c=1/h_{s2}=-1/2$, the concentrating efficiency becomes $\eta =r_s^3/r_c^3$, which is also similar to the two-dimensional conclusion in Sect. 8.3.