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ABSTRACT
Photovoltaic cells that harness renewable solar energy have been combined with either electron tunnels or thermoelectric modules to work as efficient hybrid devices. In the present numerical analysis, we integrated a photovoltaic cell, an electron tunnel, and a thermoelectric module into a three-component conjugate assembly, and further studied the characteristics of this assembly in transient states. Mechanisms including (a) thermalization obeying Fermi–Dirac distribution in the cathode; (b) electron transport confined to tunnels; (c) irreversible exchange of electron fluxes between two electrodes; (d) Seebeck effects; (e) Joule heating; (f) transient heat conduction in the thermoelectric module; and (g) convective cooling over the bottom of the assembly are all intricately coupled. The set of highly nonlinear algebraic equations resulting from energy balances over five nodes are iteratively solved using the modified Newton–Raphson method jointly with under-relaxation. Thermal efficiencies of the proposed three-module combined assembly surpass those of two-unit hybrid models previously reported in the literature.
Nomenclature
| Ac | = | area of the system (m2) |
| c | = | speed of light, 3 × 108(m/s) |
| cv, te | = | specific heat capacity of thermoelectric material (J/K) |
| = | energy (eV) | |
| Ec | = | energy at the conduction-band minimum (eV) |
| Ef | = | equilibrium Fermi level without photon-excitation (eV) |
| Ef,A | = | quasi-Fermi level of photon-excited electrons in the anode (eV) |
| Ef,n | = | quasi-Fermi level of photon-excited electrons in the cathode (eV) |
| Eg | = | bandgap energy (eV) |
| Ev | = | energy at the valance-band maximum (eV) |
| e | = | elementary charge, 1.6 × 10−19(C) |
| G | = | rate of photon-electron-collision excitation (A/m3C) |
| h | = | Planck constant, 6.63 × 10−34 (J · s) |
| ℏ | = | reduced Planck constant (J · s) |
| hair | = | heat transfer coefficient for airflow over the bottom (W/m2K) |
| I | = | current inside thermoelectric material (A) |
| JA | = | current density, from anode to cathode (A/m2) |
| JC | = | current density, from cathode to anode (A/m2) |
| Jnet | = | net current density (A/m2) |
| kB | = | Boltzmann constant, 1.38 × 10−23 (J/K) |
| kx, ky, kz | = | wave vectors (rad/m) |
| L | = | thickness of the cathode (m) |
| m* | = | effective mass of the electron ( kg) |
| m1, m2, m3, m4 | = | mass of the nodal control volume (kg) |
| NC | = | effective densities of states in the conduction band (cm−3) |
| NV | = | effective densities of states in the valence band (cm−3) |
| np−n | = | number of p-n semiconductor pairs |
| n | = | concentration of photon-excited electrons (cm−3) |
| neq | = | equilibrium electron carrier concentration (cm−3) |
| PJ,i | = | Joule heating of the control volume of nodes (J) |
| Pnet | = | net heat flux exiting the cathode (W/m2) |
| Pn,rad | = | photon-enhanced radiative recombination energy flux (W/m2) |
| Prad | = | equilibrium radiative recombination energy flux (W/m2) |
| Psa | = | radiation emitted from solar-absorbing material (W/m2) |
| Psun | = | power of the solar flux (W/m2) |
| Pte,i | = | output power of thermoelectric material (W) |
| p | = | concentration of photon-excited holes (cm−3) |
| peq | = | equilibrium hole carrier concentration (cm−3) |
| R, ri | = | external and internal electrical resistances of thermoelectric module (Ω) |
| S | = | Seebeck coefficient (V/K) |
| T1, T2, T3 | = | temperature at nodes 1, 2, 3 (K) |
| T4 | = | temperature at node 4 or anode node (K) |
| T5 | = | temperature at node 5 or cathode node (K) |
| = | previous nodal temperatures at nodes 1, 2, 3, 4 (K) | |
| Δt | = | time step (s) |
| V | = | voltage of photovoltaic module (V) |
| Vte,i | = | voltage of thermoelectric material (V) |
| Δy | = | thickness of control volume in thermoelectric module (m) |
| ζ(kx) | = | transmission probability |
| η | = | efficiency of the proposed assembly |
| ηet | = | efficiency of the electron-tunneling module |
| ηte | = | efficiency of the thermoelectric module |
| μ | = | chemical potential (eV) |
| ν | = | frequency (s−1) |
| υ | = | velocity (m/s) |
Nomenclature
| Ac | = | area of the system (m2) |
| c | = | speed of light, 3 × 108(m/s) |
| cv, te | = | specific heat capacity of thermoelectric material (J/K) |
| = | energy (eV) | |
| Ec | = | energy at the conduction-band minimum (eV) |
| Ef | = | equilibrium Fermi level without photon-excitation (eV) |
| Ef,A | = | quasi-Fermi level of photon-excited electrons in the anode (eV) |
| Ef,n | = | quasi-Fermi level of photon-excited electrons in the cathode (eV) |
| Eg | = | bandgap energy (eV) |
| Ev | = | energy at the valance-band maximum (eV) |
| e | = | elementary charge, 1.6 × 10−19(C) |
| G | = | rate of photon-electron-collision excitation (A/m3C) |
| h | = | Planck constant, 6.63 × 10−34 (J · s) |
| ℏ | = | reduced Planck constant (J · s) |
| hair | = | heat transfer coefficient for airflow over the bottom (W/m2K) |
| I | = | current inside thermoelectric material (A) |
| JA | = | current density, from anode to cathode (A/m2) |
| JC | = | current density, from cathode to anode (A/m2) |
| Jnet | = | net current density (A/m2) |
| kB | = | Boltzmann constant, 1.38 × 10−23 (J/K) |
| kx, ky, kz | = | wave vectors (rad/m) |
| L | = | thickness of the cathode (m) |
| m* | = | effective mass of the electron ( kg) |
| m1, m2, m3, m4 | = | mass of the nodal control volume (kg) |
| NC | = | effective densities of states in the conduction band (cm−3) |
| NV | = | effective densities of states in the valence band (cm−3) |
| np−n | = | number of p-n semiconductor pairs |
| n | = | concentration of photon-excited electrons (cm−3) |
| neq | = | equilibrium electron carrier concentration (cm−3) |
| PJ,i | = | Joule heating of the control volume of nodes (J) |
| Pnet | = | net heat flux exiting the cathode (W/m2) |
| Pn,rad | = | photon-enhanced radiative recombination energy flux (W/m2) |
| Prad | = | equilibrium radiative recombination energy flux (W/m2) |
| Psa | = | radiation emitted from solar-absorbing material (W/m2) |
| Psun | = | power of the solar flux (W/m2) |
| Pte,i | = | output power of thermoelectric material (W) |
| p | = | concentration of photon-excited holes (cm−3) |
| peq | = | equilibrium hole carrier concentration (cm−3) |
| R, ri | = | external and internal electrical resistances of thermoelectric module (Ω) |
| S | = | Seebeck coefficient (V/K) |
| T1, T2, T3 | = | temperature at nodes 1, 2, 3 (K) |
| T4 | = | temperature at node 4 or anode node (K) |
| T5 | = | temperature at node 5 or cathode node (K) |
| = | previous nodal temperatures at nodes 1, 2, 3, 4 (K) | |
| Δt | = | time step (s) |
| V | = | voltage of photovoltaic module (V) |
| Vte,i | = | voltage of thermoelectric material (V) |
| Δy | = | thickness of control volume in thermoelectric module (m) |
| ζ(kx) | = | transmission probability |
| η | = | efficiency of the proposed assembly |
| ηet | = | efficiency of the electron-tunneling module |
| ηte | = | efficiency of the thermoelectric module |
| μ | = | chemical potential (eV) |
| ν | = | frequency (s−1) |
| υ | = | velocity (m/s) |