Thickness optimization of the output power and effective thermoelectric figure of merit of thin thermoelectric generator

Thermoelectric generators directly convert waste thermal energy into electrical power. Electric currents can be generated from temperature gradients by the thermoelectric effect; the energy conversion efficiency is known to be characterized by the dimensionless thermoelectric figure of merit, given by Refs. 1,2

$$\begin{eqnarray}&&{{zT}}_{{\rm{b}}}=\displaystyle \frac{{\alpha }^{2}{T}_{{\rm{m}}}}{\rho \kappa },\end{eqnarray} \tag{ 1 }$$

where T_m is the mean temperature between the cold side and the hot side, α is the Seebeck coefficient, ρ is the electrical resistivity, and κ is the thermal conductivity. The thermoelectric figure of merit zT_b provides a measure of the quality of thermoelectric materials and can be used as a guide to compare the thermoelectric properties of materials.

The recent demand for thermoelectric generators for the Internet of Things (IoT) and wearable devices has prompted the development of thin film thermoelectric devices, which are flexible and could operate around room temperature. ^3–9) Such devices should work under limited temperature gradients and require sufficient energy conversion efficiency. Conventionally, the thermoelectric figure of merit and the power factor have been used to relate the material properties such as the electrical resistivity and the thermal conductivity with the maximum energy conversion efficiency and the maximum output power, respectively. However, for thin film thermoelectric devices, the electrical contact resistivity and the heat transfer on the surface facing the environments need to be considered because of the large areal size compared to the film thickness. ^4,6,7,9) Therefore, the energy conversion efficiency could be influenced by the film dimensions in addition to the thermoelectric figure of merit zT_b of the bulk material. To consider the influence of film dimensions on the energy conversion efficiency, we define below zT_eff(d), where d indicates the film thickness.

We first introduce the effective electrical resistivity using the intrinsic material electrical resistivity ρ Ω m and the contact electrical resistivity ρ_c Ω m². Since they are connected electrically in series, the effective resistivity can be expressed as ρ_eff = ρ + ρ_c/d. When the electrical contact resistivity of one side is given by ρ_c0 and that of the other side is given by ρ_cd , the total electrical contact resistivity is expressed as ρ_c = ρ_c0 + ρ_cd . Similarly, we define the effective thermal conductivity κ_eff by the intrinsic material thermal conductivity κ (W m⁻¹ K⁻¹) and the heat transfer coefficient h (W m^{− 2} K⁻¹) as ${\kappa }_{\mathrm{eff}}={\left(1/\kappa +1/({hd})\right)}^{-1}$, where the heat transfer coefficient is inverse of the contact thermal resistivity; the effective thermal conductivity is given by the inverse of the sum of the intrinsic material thermal resistivity and the contact thermal resistivity. When the heat transfer coefficient of one side is given by h₀ and that of the other side is given by h_d , the total heat transfer coefficient is expressed as $h={\left(1/{h}_{0}+1/{h}_{d}\right)}^{-1}$. Previously, the effective thermal conductivity was given by the sum of the intrinsic material thermal conductivity and the thermal interface conductance, which indicates parallel thermal flows of these components. ⁵⁾

We introduce the figure of merit by considering the maximum of energy conversion efficiency by taking into account the film dimensions. For thick bulk materials, the maximum conversion efficiency is known to be an increasing function of zT_b at the mean temperature of the cold and the hot side. Therefore, the maximum conversion efficiency is higher as zT_b increases. We need to define zT_eff(d) for films in such a way that the maximum conversion efficiency increases by increasing zT_eff(d).

We consider the case that the film of thickness d and the surface area of S is located between x = 0 and d in the x coordinate as shown in Fig. 1. The ambient temperature around x = 0 is T_h and the ambient temperature around x = d is T_c; we assume T_h > T_c. We also assume that systems operate under a constant temperature difference rather than under constant heat flux. ¹⁰⁾ The energy conversion efficiency can be expressed as Ref. 11

$$\begin{eqnarray}&&\eta =\displaystyle \frac{{VI}}{\alpha T(0)I-{{RI}}^{2}/2+K\left[T(0)-T(d)\right]},\end{eqnarray} \tag{ 2 }$$

where V and I denote the voltage across the film and the electrical current passing through the film, respectively; R and K denote the electrical resistance and the thermal conductance, respectively. The temperature at x = 0 and x = d is denoted by T(0) and T(d), respectively; T(d) differs from the environmental temperature denoted by T_c if the heat transfer coefficient is low. We calculate T(0) and T(d) when the heat transfer coefficient at the hot side (x = 0) is h_h and that at the cold side (x = d) is h_c. T(0) reduces to T_h in the limit of h_h → ∞. As shown in the Appendix, we obtain

$$\begin{eqnarray}&&T(0)-T(d)={\rm{\Delta }}T/(1+\kappa /({hd})),\end{eqnarray} \tag{ 3 }$$

where ΔT = T_h − T_c and $h={\left(1/{h}_{0}+1/{h}_{d}\right)}^{-1}$. We also find

$$\begin{eqnarray}&&{T}_{{\rm{m}}}=(T(0)+T(d))/2=({T}_{{\rm{h}}}+{T}_{{\rm{c}}})/2.\end{eqnarray} \tag{ 4 }$$

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When the thermoelectric power is generated, the external load with the resistance R_ext is imposed. We introduce R_ext = yR_eff, where R_eff = (ρ + ρ_c/d)d/S for the film with the thickness d and the surface area of S facing the environments. ^1,12–14) We have I = α[T(0) − T(d)]/[(1 + y)R_eff] and V = α[T(0) − T(d)] − IR_eff = α[T(0) − T(d)]y/(1 + y). The power can be expressed as

$$\begin{eqnarray}&&P={IV}=\displaystyle \frac{{\left({\alpha }_{\mathrm{eff}}{\rm{\Delta }}T\right)}^{2}}{{R}_{\mathrm{eff}}}\displaystyle \frac{y}{{\left(1+y\right)}^{2}},\end{eqnarray} \tag{ 5 }$$

where α_eff is defined by

$$\begin{eqnarray}&&{\alpha }_{\mathrm{eff}}=\displaystyle \frac{\alpha }{1+\kappa /({hd})}.\end{eqnarray} \tag{ 6 }$$

By introducing α_eff, the thermoelectric power can be expressed in the conventional form as shown in Eq. (5).

Using y = R_ext/R_eff, the energy conversion efficiency given by (2) can be expressed for films as

$$\begin{eqnarray}&&\eta (y)=\displaystyle \frac{y{\rm{\Delta }}T}{(1+y){T}_{{\rm{m}}}+{\left(1+y\right)}^{2}/{Z}_{\mathrm{eff}}+y{\rm{\Delta }}T/2},\end{eqnarray} \tag{ 7 }$$

where Z_eff is defined by

$$\begin{eqnarray}&&{Z}_{\mathrm{eff}}=\displaystyle \frac{{\alpha }_{\mathrm{eff}}^{2}}{{R}_{\mathrm{eff}}{K}_{\mathrm{eff}}},\end{eqnarray} \tag{ 8 }$$

and we have R_eff = (ρ + ρ_c/d)d/S and ${K}_{\mathrm{eff}}\,={\left({\kappa }^{-1}+1/({hd})\right)}^{-1}S/d$. Using d η(y)/dy = 0, ¹⁵⁾ we find y at the maximum conversion efficiency as ${y}_{\max }=\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}};$ the maximum conversion efficiency is obtained as

$$\begin{eqnarray}\begin{array}{rcl}{\eta }_{\max } & = & \displaystyle \frac{{\rm{\Delta }}T\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}}{2(1+\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}})/{Z}_{\mathrm{eff}}+2{T}_{{\rm{m}}}+{T}_{{\rm{h}}}\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}}\\ & = & \displaystyle \frac{{\rm{\Delta }}T\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}(\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}-1)}{(2{T}_{{\rm{m}}}-{T}_{{\rm{h}}})\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}+{T}_{{\rm{h}}}(1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}})}\\ & = & \left(\displaystyle \frac{{\rm{\Delta }}T}{{T}_{{\rm{h}}}}\right)\displaystyle \frac{\left(\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}-1\right)}{\sqrt{1+{Z}_{\mathrm{eff}}{T}_{{\rm{m}}}}+({T}_{{\rm{c}}}/{T}_{{\rm{h}}})},\end{array}\end{eqnarray} \tag{ 9 }$$

where ΔT/T_h is the Carnot efficiency; the maximum efficiency approaches the Carnot efficiency by increasing Z_eff. By introducing α_eff defined by Eq. (6), the conventional form of the maximum conversion efficiency given by Eq. (9) is obtained, where Z = zT_b given by Eq. (1) is redefined by Eq. (8); it is sufficient to study Z_eff in the place of Z = α²/(RK) with R = ρ d/S and K = κ S/d for films. The maximum conversion efficiency is higher for the higher value of Z_eff.

The temperature difference at both ends of the material is given by T(0) − T(d) which is related to the ambient temperature difference ΔT by Eq. (3). The factor 1 + κ/(hd) in Eq. (3) is absorbed in α_eff in Eq. (6). Using α_eff, Z_eff is defined and the maximum conversion efficiency is a monotonically increasing function of Z_eff for any thickness of the thermoelectric material. If we use α instead of α_eff in Z_eff, the correlation to the maximum conversion efficiency is lost. Therefore, it is essential to define Z_eff in terms of α_eff, when the heat transfer coefficient is low for thin film (hd < κ).

By taking into account the contact electrical resistivity and the heat transfer coefficient, the effective thermoelectric figure of merit for film conductors [Eq. (8)] can be expressed as zT_eff(d) = Z_eff T_m and zT_eff(d) which is explicitly expressed as

$$\begin{eqnarray}&&{{zT}}_{{\rm{eff}}}(d)=\displaystyle \frac{{\alpha }_{{\rm{eff}}}^{2}{T}_{{\rm{m}}}}{{\rho }_{{\rm{eff}}}{\kappa }_{{\rm{eff}}}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\,\,\,\,\,\,\,=\,{{zT}}_{{\rm{b}}}{S}_{{zT}}(d),\end{eqnarray} \tag{ 11 }$$

where zT_b is the figure of merit of bulk given by Eq. (1) and the size factor for the effective figure of merit can be expressed as

$$\begin{eqnarray}&&{S}_{{zT}}(d)=\displaystyle \frac{1}{[1+{\rho }_{{\rm{c}}}/(\rho d)][1+\kappa /({hd})]}.\end{eqnarray} \tag{ 12 }$$

The thickness dependence of zT_eff(d) = Z_eff T_m can be studied using S _zT (d); the size factor of the effective figure of merit [Eq. (12)] is a monotonically increasing function of the thickness d of the thermoelectric generating layer.

It should be stressed that the conventional figure of merit defined only by the material parameters [zT_b in Eq. (1)] is not a good index of the maximum conversion efficiency for thin film conductors; the appropriate index is given by Eq. (11), where the size factor given by Eq. (12) is taken into account. The equivalent expression of Eq. (12) was defined to capture the interface effects in thermoelectric microrefrigerators. ⁶⁾

We also study the power factor of thin film by maximizing the electrical output power. The maximum power is obtained for y = 1 in Eq. (5) and is given by

$$\begin{eqnarray}&&{P}_{\max }=\displaystyle \frac{{\left({\alpha }_{\mathrm{eff}}{\rm{\Delta }}T\right)}^{2}}{4{R}_{\mathrm{eff}}}.\end{eqnarray} \tag{ 13 }$$

The condition y = 1 leads to R_ext = R_eff, where R_eff = (ρ + ρ_c/d)d/S involves the contact resistivity. In general, the external load resistance should match the effective resistance involving both the contact resistivity and the intrinsic material resistivity. Equation (13) indicates that the thickness dependence of the maximum electric output power can be studied by ${\alpha }_{\mathrm{eff}}^{2}/{R}_{\mathrm{eff}}$. Conventionally, the bulk power factor is defined by

$$\begin{eqnarray}&&{{Pf}}_{{\rm{b}}}=\displaystyle \frac{{\alpha }^{2}}{\rho },\end{eqnarray} \tag{ 14 }$$

where ${P}_{\max }$ is multiplied by d to eliminate the trivial thickness dependence originating from R = ρ d/S, which remains even for ρ_c = 0. As a result, the bulk power factor is expressed only by material parameters such as ρ and α. Using Eq. (13), we obtain the power factor of thin film with thickness d as ${{Pf}}_{{\rm{eff}}}(d)={\alpha }_{{\rm{eff}}}^{2}/{\rho }_{{\rm{eff}}}$. However, it is more appropriate to define an effective power factor using ${{Pf}}_{{\rm{g}}}(d)={\alpha }_{{\rm{eff}}}^{2}/{R}_{{\rm{eff}}}$ to absorb all terms associated with d for studying the thickness dependence. We rewrite Pf_g(d) = Pf_eff(d)/d as

$$\begin{eqnarray}&&{{Pf}}_{{\rm{g}}}(d)={{Pf}}_{{\rm{b}}}{S}_{\mathrm{Pw}}(d),\end{eqnarray} \tag{ 15 }$$

where the size factor for the effective power factor is given by

$$\begin{eqnarray}&&{S}_{\mathrm{Pw}}(d)=\displaystyle \frac{1}{d[1+{\rho }_{{\rm{c}}}/(\rho d)]{\left[1+\kappa /({hd})\right]}^{2}}.\end{eqnarray} \tag{ 16 }$$

The equivalent expression of Eq. (13) was introduced previously by noticing the temperature difference between the actual device temperature and the ambient temperature. ⁷⁾

For a given thickness d, both ρ_c < ρ d (small enough contact resistance) and h > κ/d (high enough heat transfer coefficient) should be satisfied to associate the maximum conversion efficiency and the maximum power with the bulk dimensionless thermoelectric figure of merit [Eq. (1)] and the bulk power factor [Eq. (14)], respectively. For the maximum conversion efficiency and the maximum power, the effect of finite thickness should be considered either when d < ρ_c/ρ or d < κ/h holds.

Contrary to the size factor of the effective figure of merit [Eq. (12)], which is a monotonically increasing function of the thickness d, the size factor of the effective power factor [Eq. (16)] has a maximum at the thickness of the thermoelectric generating layer given by

$$\begin{eqnarray}&&{d}_{\max }=\displaystyle \frac{\kappa /h+\sqrt{\kappa /h\left(\kappa /h+8{\rho }_{{\rm{c}}}/\rho \right)}}{2}.\end{eqnarray} \tag{ 17 }$$

When κ/h > 8ρ_c/ρ is satisfied, the above equation is simplified to

$$\begin{eqnarray}&&{d}_{\max }\approx \kappa /h.\end{eqnarray} \tag{ 18 }$$

The appearance of the maximum in the output power can be understood in the following way. When the film thickness is small, the actual temperature difference at the both ends of the film is smaller than the ambient temperature difference as shown in Eq. (3). As a result, the Seebeck power is reduced. By increasing the film thickness, the actual temperature difference increases and the Seebeck power also increases. Therefore, the output power increases up to a certain film thickness. When the film thickness exceeds ${d}_{\max }$, the electric power output decreases with the increasing film thickness because the electrical resistance [R_eff = (ρ + ρ_c/d)d/S] increases with d.

We study the size factor of the effective figure of merit [Eq. (12)] and the size factor of the effective power factor [Eq. (16)] for the benchmark organic/inorganic thermoelectric materials. Specifically, we use the physical properties of inorganic materials (Bi₂Te₃) and organic materials (PEDOT) which are summarized in Table I. The thickness is varied from mm to cm range according to Ref. 7. The maxima calculated from Eq. (17) lie in this length scale by substituting the material properties listed in Table I.

As shown in Fig. 2, S _zT (d) (the size factor for the effective figure of merit) monotonically increases by increasing d for both inorganic materials (Bi₂Te₃) and organic materials (PEDOT). By considering anisotropy in intrinsic material electrical resistivity and intrinsic material thermal conductivity of PEDOT, we show the case that the temperature gradient is applied in the in-plane direction, and the case that the temperature gradient is applied in the through-plane direction for PEDOT, where PEDOT would tend to lie down at the surface. In the same figure, we also show S_Pw(d) (the size factor for the effective power factor). S_Pw(d) shows a maximum as a function of the film thickness for both inorganic materials (Bi₂Te₃) and organic materials (PEDOT). Judging from S_Pw(d), PEDOT with the temperature gradient in the through-plane direction (PEDOT-Through) can be optimized at a thinner film thickness compared to the other materials. The optimum value of d for S_Pw(d) roughly corresponds to the inflection point for S _zT (d), indicating that the decrease of S _zT (d) by decreasing d mainly occurs around the optimum value of d. The output power can be optimized around 2 cm for Bi₂Te₃ and could be optimized below 1 cm for PEDOT-Through. The optimum thickness of the thermoelectric generator can be estimated by plotting S_Pw(d) as a function of d or using Eq. (17) once the material parameters as shown in Table I are available. Though the effective figure of merit is a monotonically increasing function of d, the large part of the figure of merit can be reduced by reducing the thickness from 10 to 1 cm for Bi₂Te₃ and the figure of merit can be largely reduced by reducing the thickness to around 1 cm for PEDOT-Through. By employing the finite element methods, the results similar to Fig. 2 were reported, where the maximum efficiency monotonically increases by increasing d and the maximum power shows a maximum as a function of d. ¹⁶⁾ (see supplementary data available online at stacks.iop.org/JJAP/61/080903/mmedia for α_eff, κ_eff and ρ_eff against d)

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In Fig. 3, we study the film thickness denoted by ${d}_{\max }$ at the maximum in S_Pw(d) as a function of the heat transfer coefficient denoted by h. ${d}_{\max }$ decreases by increasing the heat transfer coefficient because the temperature gradient could be closer to the imposed temperature gradient applied through ambient temperature if the heat transfer is higher. In Fig. 3, ${d}_{\max }$ of PEDOT-Through is smaller than ${d}_{\max }$ of Bi₂Te₃, though the difference decreases by increasing h.

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In Fig. 4, we study the film thickness denoted by ${d}_{\max }$ at the maximum in S_Pw(d) as a function of the contact resistivity denoted by r_c. ${d}_{\max }$ increases by increasing r_c. ${d}_{\max }$ of PEDOT-Through is more than an order of magnitude smaller than ${d}_{\max }$ of Bi₂Te₃.

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In practice, thermoelectric generators may be composed of numerous series-connected semiconductors. When each unit is identical, S_Pw(d) and S _zT (d) of the N repeated units are independent of N as shown below. If the electrical resistivity, the thermal conductivity, the electrical contact resistivity, and the heat transfer coefficient of the unit are denoted by ρ_u, κ_u, ρ_c,u, and h_u, those for the electrically connected units in series are expressed as N ρ_u, N κ_u, N ρ_c,u, and Nh_u. Because S_Pw(d) and S _zT (d) of the N repeated units are expressed in terms of the ratio such as κ/h and ρ/ρ_c, they are independent of N. However, the dimensionless figure of merit for a single material should be generalized if a thermoelectric power generator consists of series-connected p-type and n-type semiconductors. We introduce the effective Seebeck coefficient α_eff,p = α_p/[1 + κ_p/(h_p d)], effective electrical resistivity ρ_eff,p = ρ_p + ρ_c,p/d, and effective thermal conductivity ${\kappa }_{\mathrm{eff},{\rm{p}}}={\left(1/{\kappa }_{{\rm{p}}}+1/({h}_{{\rm{p}}}d)\right)}^{-1}$ of the p-type semiconductor, where the subscript p indicates the property of p-type semiconductor; the material properties for the n-type semiconductor are defined similarly. When the p-type semiconductor with the area S_p and n-type semiconductor with the area S_n are connected electrically in series and thermally in parallel, while the thickness d is common for both semiconductors, ${Z}_{\mathrm{eff}}=({\alpha }_{\mathrm{eff}}^{2}{T}_{{\rm{m}}})/({R}_{\mathrm{eff}}{K}_{\mathrm{eff}})$ given by Eq. (8) still holds, where we have R_eff = ρ_eff,p d/S_p + ρ_eff,n d/S_n, K_eff = κ_eff,p S_p/d + κ_eff,n S_n/d and α_eff = α_eff,p − α_eff,n; the minus sign in front of α_eff,n is introduced to take into account the opposite direction of electron flow in n-type semiconductor against the direction of hole flow in p-type semiconductor. If R_eff K_eff is minimized by optimizing the ration S_p/S_n, we find ${{zT}}_{{\rm{eff}}}(d)={\left({\alpha }_{{\rm{eff}},{\rm{p}}}-{\alpha }_{{\rm{eff}},{\rm{n}}}\right)}^{2}{T}_{{\rm{m}}}/{\left[\sqrt{{\rho }_{{\rm{eff}},{\rm{p}}}{\kappa }_{{\rm{eff}},{\rm{p}}}}+\sqrt{{\rho }_{{\rm{eff}},{\rm{n}}}{\kappa }_{{\rm{eff}},{\rm{n}}}}\right]}^{2}$, where ${S}_{{\rm{p}}}/{S}_{{\rm{n}}}=\sqrt{{\rho }_{{\rm{p}}}{\kappa }_{{\rm{n}}}/({\rho }_{{\rm{n}}}{\kappa }_{{\rm{p}}})}$ is obtained by d(R_eff K_eff)/dX using X = S_p/S_n. In the limit of d → ∞, the known result is recovered. ¹⁾

We studied the case that the intrinsic electrical resistivity, thermal conductivity (regardless of phonon thermal conductivity or electronic thermal conductivity), and Seebeck coefficient are material properties independent of the dimensions. On this basis, we focused to illustrate the genuine contributions of heat transfer coefficient and contact resistivity to the maximum energy conversion efficiency and thermoelectric power.

In conclusion, the contact resistivity and heat transfer coefficient at the interface between the thermoelectric material and the environment influence the energy conversion efficiency and the electric output power if thermoelectric materials are thin and have a large surface area. The conventional thermoelectric figure of merit and the power factor are not sufficient as a measure of the thin film quality of thermoelectric materials. The effective thermoelectric figure of merit of thin film was introduced in such a way that the maximum conversion efficiency is a monotonically increasing function of the effective thermoelectric figure of merit regardless of the thickness of the thermoelectric material. The effective thermoelectric figure of merit thus defined can be expressed as a product of the conventional thermoelectric figure of merit and the size factor. The size factor of the effective thermoelectric figure of merit is shown to be an increasing function of the film thickness. Similarly, we introduced the effective power factor and the corresponding size factor; we showed the existence of the optimal film thickness. We studied the thickness dependence of the size factor of the effective thermoelectric figure of merit and the size factor of the effective power factor using the physical properties of inorganic materials (Bi₂Te₃) and organic materials (PEDOT). We showed that PEDOT with the temperature gradient in the through-plane direction is advantageous over Bi₂Te₃ as regards to the thinner optimal film thickness.

: Appendix: Derivation of Eq. (3)

The boundary condition at x = 0 is given by

$$\begin{eqnarray}&&{\left.\kappa \displaystyle \frac{\partial T}{\partial x}\right|}_{x=0}={h}_{0}\left[T(0)-{T}_{{\rm{h}}}\right],\end{eqnarray} \tag{ A·1 }$$

where the heat flux flowing into the film is considered. The boundary condition at x = d is given by

$$\begin{eqnarray}&&{\left.-\kappa \displaystyle \frac{\partial T}{\partial x}\right|}_{x=d}={h}_{d}\left[T(d)-{T}_{{\rm{c}}}\right],\end{eqnarray} \tag{ A·2 }$$

where the heat flux flowing out from the film is considered. The steady state solution of the heat equation ∇² T(x) = 0 can be expressed as T(x) = C₁ + C₂ x. The two unknown constants C₁ and C₂ can be determined from the two boundary conditions [Eqs. (A·1) and (A·2)] and we obtain

$$\begin{eqnarray}&&T(0)=\displaystyle \frac{{T}_{{\rm{h}}}\left[1+\kappa /({h}_{d}d)\right]+{T}_{{\rm{c}}}\kappa /({h}_{0}d)}{1+\kappa /({h}_{d}d)+\kappa /({h}_{0}d)},\end{eqnarray} \tag{ A·3 }$$

$$\begin{eqnarray}&&T(d)=\displaystyle \frac{{T}_{{\rm{c}}}\kappa /({h}_{d}d)+{T}_{{\rm{c}}}\left[1+\kappa /({h}_{0}d)\right]}{1+\kappa /({h}_{d}d)+\kappa /({h}_{0}d)}.\end{eqnarray} \tag{ A·4 }$$

We find Eq. (3) from (A·3) and (A·4).