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Parametric optimum designs of a vacuum thermionic solar cell Tianjun Liao Abstract:In this paper, a vacuum thermionic solar cells consisting of a series of vacuum thermionic power generator (VTPG), a heat sink, and a concentrator is proposed. The arithmetic expressions for the power output and overall efficiency of the VTPG solar cell are derived. The Parametric optimum designs are presented and discussed. The effects of the concentrating ratio and work function of the anode on the performance characteristics are investigated. The obtained results can provide important guiding significance for optimization of the actual VTPG solar cell and performance improvement. Key words:Parametric optimum designs; Vacuum thermionic power generator; Solar cell 1. Introduction 2. Model description The VTPG solar cell composed of a concentrator, a collector, a heat sink and VTPGS connected in series exchange heat with the collector and heat sink, as shown in Fig. 1 (a). The collector acts as the high-temperature heat reservoir of the VTPG solar cell for a further production of electric energy. The wasted heat flow is transferred from the heat sink to the environment. Fig. 1 (b) shows a VTPG consisting of two electrodes placed near one another in a vacuum gap, one electrode is normally called the cathode, and the other is called the anode. Ordinarily, electrons in the cathode are prevented from escaping from the surface by a potential-energy barrier. We call the minimum energy required to allow an electron to be liberated from material as its work function. When the cathode at high temperature, the numbers of electron have sufficient thermal energy to overcome the materials work function and escape into vacuum to produce thermionic current. After collected by the anode, these electrons return to the cathode through the load resistance RL . TC and TA are the temperatures of the cathode and the anode, TL and T0 are the temperatures of heat sink and ambient, in this model, TC TA TL T0 holds, C is the concentrator ratio, G is the solar irradiation, 0 and 1 are the emissivity factor of the collector and electrodes, 0 is the collector optical efficiency, Ar is the effective surface area of the cathode and anode, U is the heat transfer coefficient per unit area between the TPGM and the heat sink, is the Stefan-Boltzmann constant, CGAr0 and 0 Ar TC4 T04 are the heat flow into the each VTPG and radiation loss from the surface of the cathode to the environment, 1 TC4 TA4 is thermal radiant energy from the cathode to the anode per unit area, J C and J A are the current densities emitted from the cathode and anode, UAr TA TL is the heat flow from the anode to the heat sink. Fig.1. (a) A schematic diagram of the VTPG solar cell and (b) structural schematic diagram of the VTPG. 3. Parametric optimum designs In order to investigate the performance of the VTPG, some assumptions are given by: (i) the heat transfer between the collector and the cathode isn’t taken into account. It is assumed that the temperature of the collector is equal to the cathode. (ii) It is assumed that no space charge or other phenomena limit the current density and the current density from the cathode to be in the positive direction. (iii) It is assumed that the cathode and anode of the TPG have the same surface area Ar . Considering the radiation loss on the surface of the collector and the heat transfer between the VTPG and heat sink, the rate of heat Qin entering the VPTG and the rate of heat Qout leaving from the heat sink are given by, respectively [2]: Qin CGAc0 0 Ac TC4 T04 , (1) Qout UAc TA TL . (2) Thermal efficiency of the solar collector is given by s 0 TC4 T04 Qin 1 , CGAc0 CG0 (3) The Richardson-Dushman equation describes the current density emitted from a heated metal surface. To derive the equation, a metal in thermal equilibrium is considered. The spectral density of the electrons is given by the Fermi-Dirac distribution: f FD v me 4 3 3 me 4 3 3 1 1 exp me v 2 2 kT 1 , (4) 1 exp me v 2 kT 2 where v is the velocity, k the Boltzmann constant, me the mass of a electron. , and T are, respectively, the electrochemical potential, work function and temperature of the metal materials To calculate the current density J nqv flowing away from the surface, the contributions from all electrons with a positive velocity normal to the metal surface have to be integrated. If the x-direction is chosen as the direction of the current, it follows: 0 J dvx dvy dvz vx f FD v A0T 2e where q the charge of an electron, A0 kT , (5) qme k 2 the Richardson constant. 2 2 3 According to the following Richardson-Dushman equations [1, 16], the emitted net current density J J C J A determined by the two electrodes’ work functions and temperatures can be expressed as J C A0TC2 exp C kTC , (6) J A A0TA2 exp A kTA , (7) where C and A denote the work function of the cathode and anode. Internally, there exist two kinds of heat flows, one is the heat transport between the electrodes due to the emission of electrons and another is the radiation loss of the two electrodes. According to the first law of thermodynamics, the first kind of heat flow, the rate of heat Q1 leaving the cathode and the rate of heat Q2 entering the anode are given by [2] Q1 Ar J C C 2kTC J A C 2kTA q , (8) Q2 Ar J C A 2kTC J A A 2kTA q . (9) It is assumed that the two electrodes are perfect thermal radiators, thus, the second kind of heat flow, the thermal radiation from the anode to the cathode can be given by [2] QR 1 Ar TC4 TA4 , (10) where 1 is the effective emissivity of the electrodes. For practical TPGM, 1 usually lies in the range of 0.1~0.2 [24]. According to the theory of non-equilibrium thermodynamics, combining Eqs. (7)-(9), one can obtain the rate ( QC ) of total leaving the cathode and the rate ( QA ) of total heat flow entering the anode [2] (11) (12) QC =Q1 QR =Ar J C C 2kTC J A C 2kTA q +1 TC4 TA4 , QA =Q2 QR =Ar J C A 2kTC J A A 2kTA q +1 TC4 TA4 . By using the law of conservation of energy, we can obtain Qin ns QC , (13) Qout ns QA , (14) where ns is the number of VTPGS connected in series. Combining Eqs. (1), (2), (10)-(13), the thermal balance equations can be derived: Ac CG0 0 TC4 T04 ns Ar J C C 2kTC J A C 2kTA q +1 TC4 TA4 , (15) Ac U TA TL ns Ar ns Ar J C A 2kTC J A A 2kTA q +1 TC4 TA4 , (16) The power output PVTPG and efficiency VTPG of TPGM are given by: PVTPG Qin Qout VJns Ar , VTPG PVTPG VJns Ar , Qin CGAc0 0 Ac TC4 T04 (17) (18) where V C A q is operating voltage of a per TPG unit. The over thermal efficiency is product of the thermal efficiency of the collector and the thermal efficiency of the VPTG solar cell sVTPG VJns Ar CGAc0 (19) Theoretically, combining Eq. (15) with Eq. (16), the TC and TA can be solved out under different voltage V and Ac ns Ar . Then, substituting the values of TC and TA into Eqs. (17) and (19), the PVTPG and can be determined. 4. Conclusion References [1] Hatsopoulos G N, Kaye J. Measured thermal efficiencies of a diode configuration of a thermo electron engine. J. Appl. Phys. 1958; 29: 1124–1125. [2] Houston J M. 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