Performance Analysis of a Metamaterial-Based Near-Field Thermophotovoltaic System Considering Cooling System Energy Consumption

Abstract

A mathematical physical model of a near-field thermophotovoltaic (TPV) system containing a metamaterial emitter comprised of W-nanowire arrays embedded in a SiC host is constructed. It is notable that this model incorporates a cooling system. On this basis, the influence of the emitter temperature and filling ratio, the cell and emitter thicknesses, and the emitter cell vacuum gap on the TPV system output performance is analyzed. It is found that the cooling device energy consumption increases by two orders of magnitude with decreased emitter cell gap size in the near-field; the highest possible value is 267.74 W·cm⁻². However, the maximum net system efficiency is only 11.79 %. The emitter radiation capability can be enhanced by increasing the emitter temperature, but the cooling system energy consumption remains a significant problem. When the emitter temperature increases to 2200 K, the net system efficiency and net output density reach maximum values of only 6.22 % and 36.28 W·cm⁻², respectively. Further investigation demonstrates that a large emitter thickness can induce a surface disturbance phenomenon, resulting in rapid decreases in the net system output power density and net system efficiency to − 251.85 W·cm⁻² and − 12.86 %, respectively. However, when the cell thickness increases beyond 1000 nm, the net system efficiency and output power density are stable at 2.13 % and 24.12 W·cm⁻², respectively. Finally, the emitter filling ratio should be increased as much as possible to maintain good system performance.

Appendix

Calculation method of energy transmission coefficient $\xi (\omega ,\beta )$ was reported in papers [3, 13, 23,24,25,26,27,28,29,30]. In particular, papers [3, 13, 24, 28,29,30] have described the calculation method of energy transmission coefficient of multilayer planar structure. A calculation method of energy transmission coefficient between two suspended metamaterial films was reported in papers [23, 27, 31], at the same time. Because in this work, we have studied a five-layer parallel-plate near-field thermophotovoltaic system, combining all the literature above with each other; the expression of energy transmission coefficient $\xi (\omega ,\beta )$ between metamaterial emitter and thermophotovoltaic cell for both propagating waves ($\kappa < \omega /c$) and evanescent waves ($\kappa > \omega /c$) can be expressed as:

$$ \xi (\omega ,\beta ) = \left\{ {\begin{array}{*{20}l} {\frac{{\left( {1 - \left| {R_{2}^{s} } \right|^{2} - \left| {T_{2}^{s} } \right|^{2} } \right)\left( {1 - \left| {R_{4}^{s} } \right|^{2} - \left| {T_{4}^{s} } \right|^{2} } \right)}}{{\left| {1 - R_{2}^{s} R_{4}^{s} e^{{2i\gamma_{3} d}} } \right|^{2} }} + \frac{{\left( {1 - \left| {R_{2}^{p} } \right|^{2} - \left| {T_{2}^{p} } \right|^{2} } \right)\left( {1 - \left| {R_{4}^{p} } \right|^{2} - \left| {T_{4}^{p} } \right|^{2} } \right)}}{{\left| {1 - R_{2}^{p} R_{4}^{p} e^{{2i\gamma_{3} d}} } \right|^{2} }},} \hfill & {\quad \kappa < \omega /c} \hfill \\ {\frac{{4IM(R_{2}^{s} )IM(R_{4}^{s} )e^{{ - 2\text{Im} (\gamma_{3} )d}} }}{{\left| {1 - R_{2}^{s} R_{4}^{s} e^{{2i\gamma_{3} d}} } \right|^{2} }} + \frac{{4IM(R_{2}^{p} )IM(R_{4}^{p} )e^{{ - 2\text{Im} (\gamma_{3} )d}} }}{{\left| {1 - R_{2}^{p} R_{4}^{p} e^{{2i\gamma_{3} d}} } \right|^{2} }},} \hfill & {\quad \kappa > \omega /c} \hfill \\ \end{array} } \right. $$

(13)

Where reflection coefficients R and transmission coefficients T are given by:

$$ R_{l}^{\nu } = \frac{{r_{l - 1,l}^{\nu } + r_{l,l + 1}^{\nu } e^{{2i\gamma_{l} t_{l} }} }}{{1 + r_{l - 1,l}^{\nu } + r_{l,l + 1}^{\nu } e^{{2i\gamma_{l} t_{l} }} }} $$

(14)

$$ T_{l}^{\nu } = \frac{{t_{l - 1,l}^{\nu } + t_{l,l + 1}^{\nu } e^{{i\gamma_{l} t_{l} }} }}{{1 + r_{l - 1,l}^{\nu } + r_{l,l + 1}^{\nu } e^{{2i\gamma_{l} t_{l} }} }} $$

(15)

Where the subscript number l = 1, 2, 3, 4 or 5 is the layer index shown in Fig. 1a in the main text; ν = s or p represents the polarization states; γ is the vertical-component wavevector, $\beta$ is the parallel-component wavevector. According to paper [29], if the layer 5 is not a vacuum, the energy associated with propagating waves will be absorbed when it cannot transmit into the vacuum substrate. In this case, $\left( {1 - \left| {R_{l}^{\nu } } \right|^{2} - \left| {T_{l}^{\nu } } \right|^{2} } \right)$ will be replaced by $\left( {1 - \left| {R_{l}^{\nu } } \right|^{2} } \right)$ in Eq. 13; $r_{l - 1 ,l}^{\text{s}}$ and $r_{l - 1 ,l}^{\text{p}}$ are the Fresnel reflection coefficients for s and p polarizations at the interface of layers l − 1 and l, which can be written as $r_{l - 1,l}^{s} = (\gamma_{l - 1} - \gamma_{l} )/(\gamma_{l - 1} + \gamma_{l} )$ and $r_{l - 1,l}^{p} = (\varepsilon_{l\parallel } \gamma_{l - 1} - \varepsilon_{l - 1,\parallel } \gamma_{l} )/(\varepsilon_{l\parallel } \gamma_{l - 1} + \varepsilon_{l - 1,\parallel } \gamma_{l} )$ [3, 27, 31], respectively; the metamaterial emitter in this work is an inhomogeneous medium made of tungsten and SiC. When it comes to the calculation of the reflection coefficient of layer 2, $R_{2}^{\nu }$, the wavelength-dependent dielectric function for the metamaterial emitter is needed, so that according to papers [2, 3, 32, 33], effective medium theory (EMT) is employed here to obtain effective dielectric functions of the emitter by approximating it as a homogeneous medium. In this work, Maxwell–Garnett method (MGM) is applied, which can be expressed as:

$$ \varepsilon_{{\parallel ,{\text{eff}}}} = \varepsilon_{\text{d}} \frac{{(f + 1)\varepsilon_{\text{m}} + (1 - f)\varepsilon_{\text{d}} }}{{(1 - f)\varepsilon_{\text{m}} + (f + 1)\varepsilon_{\text{d}} }} $$

(16)

and

$$ \varepsilon_{{ \bot ,{\text{eff}}}} = f\varepsilon_{\text{m}} + (1 - f)\varepsilon_{\text{d}} $$

(17)

With the literature research, we know that the permittivity of material can be well described by Drude model and Lorentz model [2, 24, 25, 34, 35], so that in this work, we employed the Drude model to describe the wavelength-dependent dielectric function of tungsten $\varepsilon_{\text{d}} (\omega )$ as:

$$ \varepsilon_{\text{d}} (\omega ) = \varepsilon_{\infty } + \frac{{\omega_{\text{p}}^{2} }}{{\omega_{0}^{2} - i\omega \gamma - \omega^{2} }} $$

(18)

The parameters needed for calculation are obtained from Ref. [36].

However, because SiC is polar material, wavelength-dependent dielectric function $\varepsilon_{\text{m}} (\omega )$ is more suitable to describe in Lorentz model [25]:

$$ \varepsilon_{\text{m}} (\omega ) = \varepsilon_{\infty } - \frac{{\omega_{\text{p}}^{2} }}{\omega (\omega + i\gamma )} $$

(19)

The parameters needed for calculation are obtained from Ref. [25].

In this work, we have also employed an accurate model for the calculation of dark current in this work as follows.

Refer to Ref. [14], the dark current can be calculated by:

$$ J_{0} = e\left( {\frac{{n_{\text{in}}^{2} D_{\text{h}} }}{{N_{\text{D}} \sqrt {\tau_{\text{h}} } }} + \frac{{n_{\text{in}}^{2} D_{\text{e}} }}{{N_{\text{A}} \sqrt {\tau_{\text{e}} } }}} \right) $$

(20)

When the temperature of cell is 300 K, the intrinsic carrier concentration $n_{\text{in}}^{{}}$ is 1.4 × 1012 cm⁻³ [37]; $\tau_{\text{e}}$, $\tau_{\text{h}}$ are the relaxation time of electron and hole, respectively; diffusion coefficient $D_{\text{h}}$ and $D_{\text{e}}$ can be calculated by Einstein’s relation: $D_{n} = \left( {\frac{kT}{q}} \right)\mu_{n}$ [17], where $\mu_{n}$ is carrier mobility [38] and mobility of electron and hole can be, respectively, calculated by:

$$ \mu_{\text{e}} = \mu_{{\hbox{min} ,{\text{e}}}} + \frac{{\mu_{{\hbox{max} ,{\text{e}}}} \left( {\frac{300}{T}} \right)^{{\theta_{{1,{\text{e}}}} }} - \mu_{{\hbox{min} ,{\text{e}}}} }}{{1 + \left( {\frac{{N_{\text{e}} }}{{N_{{{\text{ref}},{\text{e}}}} \left( {\frac{T}{300}} \right)^{{\theta_{{2,{\text{e}}}} }} }}} \right)^{{\alpha_{\text{e}} }} }} $$

(21)

$$ \mu_{\text{h}} = \mu_{\text{min,h}} + \frac{{\mu_{\text{max,h}} \left( {\frac{300}{T}} \right)^{{\theta_{{ 1 , {\text{h}}}} }} - \mu_{\text{min,h}} }}{{1 + \left( {\frac{{N_{\text{h}} }}{{N_{\text{ref,h}} \left( {\frac{T}{300}} \right)^{{\theta_{{ 2 , {\text{h}}}} }} }}} \right)^{{\alpha_{h} }} }} $$

(22)

The parameters needed for calculation are obtained from Ref. [15, 39, 40].