A proton exchange membrane fuel cell-compound thermoelectric system: Bidirectional modeling and energy conversion potentials

Energy Conversion and Management 207 (2020) 112517

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A proton exchange membrane fuel cell-compound thermoelectric system: Bidirectional modeling and energy conversion potentials Yang Caia,b,c, Wei-Wei Wanga,b,c, Lei Wanga,b,c, Di Liud, Fu-Yun Zhaoa,b,c,

T

⁎

a

Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan, Hubei Province, China Shenzhen Research Institute, Wuhan University, Shenzhen, Guangdong Province, China c School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei Province, China d College of Pipeline and Civil Engineering, China University of Petroleum, Qingdao, Shandong Province, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Fuel cell-thermoelectric hybrid system Thermodynamic performance Parametric comparison Thermoelectric conversion conditions Operating modes

Thermoelectric device may appear as thermoelectric cooling (TEC) mode or thermoelectric generation (TEG) mode when it is generally applied to recover the waste heat produced from proton exchange membrane fuel cell (PEMFC), typically operating in the range of 60–80 °C. Although PEMFC integrated thermoelectric cooler or generator has been investigated separately in the past years, researches regarding their simultaneous TEC and TEG modes are still not reported so far. In the present work, a comprehensive thermodynamic performance analysis of the fuel cell-thermoelectric hybrid (FC-TEH) system considering TEC and TEG models simultaneously is conducted to exploit the energy conversion potential of the electrochemical and thermoelectric coupling processes. Irreversible characteristics and exergoeconomic performance of the hybrid system are thoroughly analyzed through combining finite time thermodynamics and thermodynamic economics. Subsequently, parametric comparisons between the fuel cell-thermoelectric cooling (FC-TEC) and the fuel cell-thermoelectric generation (FC-TEG) models are sensitively identified in terms of the decision targets, such as power output, energy efficiency, exergy efficiency and unit exergy cost. In addition, operating regimes of thermoelectric models in FC-TEH system are further determined to reveal thermoelectric conversion conditions and ensure efficient operation of the thermoelectric device (TED). Present results further demonstrate that FC-TEH system firstly behaves as FC-TEC in the current density range of 0–1.2 A/cm2, then FC-TEG and ultimately FC-TEC mode; where, only the TEG mode has the positive influence on the power output of the hybrid system. In addition, effective ranges of current density for the FC-TEG mode and minimum unit exergy cost are also confirmed. Present research may be significant for fully enhancing the energy and exergy performance of electrochemical – thermoelectric process.

1. Introduction Disadvantageous environmental influence of using fossil fuels and growing energy demand have prompted an extensive interest and an urgent desire to seek out cleaner energy alternatives and more efficient energy conversion systems. Among various systems, proton exchange membrane fuel cell (PEMFC) has received more attention and become a promising solution in recent years to replace conventional power generators, owing to their superior aspects: high efficiency, low emissions, low operating temperature, and simple maintenance requirement [1]. PEMFC can use hydrogen as fuel to directly transform the chemical potential into electrical power without the mechanical movement, and the exhaust product is an almost water stream, which is harmless to the environment. For these reasons, PEMFC has great potential in the

⁎

representative areas of automobiles, portable devices, and buildings, etc. as power generation devices or thermodynamic hybrid systems for specific requirements [2]. Relative, one important consideration of their commercialization stage is that waste heat produced from the electrochemical process should be immediately removed, so that the fuel cell can be maintained a specified operation temperature range (60–80 °C) [3]. The waste heat generated from the PEMFC can be recycled and reused in some application fields, especially in buildings where electricity and heat could be simultaneously desired [4]. Therefore, there is an urgent need that feasible cooling method must be adopted in the operation of the PEMFC to reduce energy consumption and further enhance system performance [5]. In recent times, with the evolution of advanced materials and material processing technologies, thermoelectric systems have been

Corresponding author at: School of Power and Mechanical Engineering, Wuhan University, 430072 Wuhan, Hubei Province, China. E-mail address: [email protected] (F.-Y. Zhao).

https://doi.org/10.1016/j.enconman.2020.112517 Received 15 July 2019; Received in revised form 16 January 2020; Accepted 17 January 2020 0196-8904/ © 2020 Elsevier Ltd. All rights reserved.

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Nomenclature A c d Ex Exd f F i I K L Lh m n N PFC PH2 PO2 Q r R Rgc S Sgen t T T0 U Z

ZT

dimensionless figure of merit

Greek symbols

area (m2) unit exergy cost ($/kWh) interest rate exergy (W) destroyed exergy (W) Exergy destruction efficiency Faraday constant (C/mol) PEMFC's current density TED's Operating current (A) thermal conductance (W/K) length (m) annual operating time mass flow number of thermoelectric devices number of cells FC's output power (W) hydrogen partial pressure (atm) oxygen partial pressure (atm) thermal capacity (W) exergy destruction factor electrical resistance (Ω) universal gas constant (J/mol K) Seebeck coefficient (V/K) entropy generation rate (W/K) system life temperature (K) ambient temperature (K) voltage (V) capital investment cost ($)

ζ η ψ

thermal loss coefficient energy efficiency exergy efficiency

Subscripts ab c ch d dis et f FC FC-TEH h HE ie in ire out ph re TE TEC TED TEG

absorb cold side of TE chemical destruction dissipate external fluid fuel cell Fuel cell-thermoelectric hybrid hot side of TE heat exchanger internal input irreversible output physical reversible thermoelectric thermoelectric cooling model thermoelectric device thermoelectric generation model

influences and optimization of the thermoelectric cooler to maximize cooling exergy output and minimize entropy generation simultaneously. Cai et al. [14,15] carried out the thermodynamic model jointly with energy balance, finite time thermodynamics and cost analyses on the multiple thermoelectric coolers for fully evaluating system performance. Manikandan et al. [16] developed an irreversible thermodynamic model to predict the overall performance of an annular thermoelectric cooling system. Nemati et al. [17] investigated the effects of significant variables, e.g. thermocouple length, cross section area ratios, applied current on the exergetic efficiency and cost. It is clear from these studies that researchers often depend on various decisive objectives derived from the above thermodynamic analyzes to attain the optimal system performance. Different from the TAC system, the TSC system mainly converts waste heat into electrical energy via Seebeck effect when the temperature difference exists on both sides of thermoelectric device, and thereby results in a temperature drop of heat source. Generally, the TSC system could be thought as a thermoelectric generation (TEG) system, which uses waste heat from the hot equipment to realize the aim of cooling. Researches on TSC systems have increased observably in recent times due to no need of additional energy consumption in equipment cooling [18]. As much, the design and optimization have been developed to improve the whole system performance based on the concepts of energy, exergy, entropy generation, exergoeconomics or their combined forms [19–22]. An important factor that needs to be considered could be whether the TEG power can meet the pumping power under normal flow rate. Kiflemariam et al. [23] reported that the net power could achieve positive value as the fluid flow velocity was not greater than 0.5 l/min. Despite a wide variety of studies on TAC and TSC systems respectively, there are few published reports regarding the simultaneous TAC and TSC systems, particularly in the thermodynamic coupling analysis aspect.

recommended as a promising option for fuel cells cooling, mainly owing to their superior features of environment-friendly, portable, compact architecture and no moving parts [6]. According to the reciprocal conversion principle of thermoelectric process, the thermoelectric cooling systems can be divided into thermoelectric active cooling (TAC) and thermoelectric self cooling (TSC) systems [7]. As a solid-state cooling device, thermoelectric system is successfully incorporated with various heat sinks (e.g. air heat sink, water heat sink and heat pipe) to eliminate the heat generated from fuel cell during operation. To carry out the waste heat and improve power output, the fuel cell can be organically integrated with the thermoelectric devices to form a fuel cell-thermoelectric hybrid (FC-TEH) system [8]. Therefore, how to enhance the electrical and thermal performance and explore the energy conversion potential of the electrochemical - thermoelectric coupling process is a conspicuous challenge in the widespread applications of the FC-TEH system. 1.1. Review of thermoelectric cooling systems As previously reported by Cai et al. [9], the thermoelectric cooling systems in broad sense can be categorized into TAC and TSC systems in terms of thermal management aspect. By using TAC or TSC system, the waste heat from specific heat sources can be dissipated due to the thermoelectric effects, which causes the noteworthy temperature drop of the heat reservoirs. Currently, many efforts by researchers have been made on the performance improvement of the TAC systems through various thermodynamic analysis methods. For example, Liu et al. [10,11] theoretically and experimentally investigated the cooling performance of the TAC system coupled with heat pipes via energy analysis. Zhu et al. [12] used entropy generation analysis to optimize the heat sink of thermoelectric cooler in terms of the entropy generation number and exergy efficiency. Tan et al. [13] discussed the parametric 2

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1.2. Thermodynamic analysis of proton exchange membrane fuel cell

1.3. Performance enhancement of fuel cell-thermoelectric hybrid system

Recently, the PEMFC has attracted much research attention for both combined heat and power and portable power applications, which motivates researchers to investigate its performance by utilizing the numerical and experimental approaches. In the aspect of numerical analysis, Change et al. [24] presented the analysis model and performance estimation of a high temperature PEMFC-based CCHP system. Authayanun et al. [25] studied the energy and exergy analyses of a HTPEMFC based trigeneration system to achieve both the optimum electrical exergy and system exergy efficiencies. Chang et al. [26] designed a novel combined cooling heating and power (CCHP) system integrated PEMFC and the effect of six organic working fluids on energy and exergy performance were thoroughly analyzed. Romdhane et al. [27] proposed a comprehensive modeling study on a CCHP system based on PEMFC and LiBr-H2O vapor absorption for residential application. Chen et al. [28] applied a novel multi-optimization approach in the PEMFC parametric optimization, aiming at improving the system power output and efficiency. In the aspect of experimental analysis, Laurencelle et al. [29] proposed an experimental investigation of a PEMFC stack composed of 36 cells and the coefficient values of their model expressions were further derived at different FC’s temperatures. Xie et al. [30] put forward to add a thermal energy conversion to PEMFC, generating more electricity for the overall system. Also, a lab-scale test prototype was fabricated to experimentally explore thermal potential in electricity generation. Özgür et al. [31] investigated the energy and exergy performance of a 1 kW PEMFC through a laboratory test platform. Gimba et al. [32] carefully investigated the intrinsic relationship between the exergy performance and operating parameters and presented the comparisons of the energy and exergy performance under different variables. Despite few experiment studies of PEMFC relative to that of numerical analysis, these testing data could be very significant as a benchmark to validate and compare for various thermodynamic modeling. Additionally, according to all above researches, one of the key challenges in the mass commercialization is its high capitalized cost. For the sake of fully evaluating the effectiveness of present energy conversion system, exergoeconomic analysis is introduced by Bejan [33] and Tsatsaronis [34] as a cost-performance approach, in which the costs of each system component can be calculated on the basis of cost balance expressions. Barbir et al. [35] found that there was an intimate connection between the efficiency and economics of PEMFC and the cost of generating electricity could be less than 0.08 $/kWh at the hydrogen cost of 10 $/GJ. In Sarabchi et al.’s work [36], the exergoeconomic analysis incorporated with exergy analysis was considered in PEMFC hybrid cogeneration system and discovered that the total product unit cost as well as the carbon dioxide mass specific emission can decrease by up to 17.72% and 16.3%. The thermodynamic and exergoeconomic performance of the PEMFC integrated a proton exchange membrane electrolyzer was evaluated at steady state operation condition, proving that increasing the current density would enhance the cost rate of power generation [37]. A PEM fuel cell-based micro-CHP system with a lithium-ion battery energy storage system has better economic advantages in cases of a long lifespan [38]. Besides, the thermal management during operating process is another critical issue in PEMFC for its widely applications. As a result, a series of cooling systems or concepts have been presented for PEMFC’s cooling implement in recent years, including but not limited, air cooling [39], liquid cooling [40], heat pipe cooling [41] and phase change cooling [42], etc. In their studies on PEMFC, a comprehensive thermodynamic evaluation approach combined with exergoeconomic analysis could be indispensable and this is a forceful need to introduce an effective cooling system into PEMFC for significantly reducing energy consumption.

Thermoelectric devices (TED) are eco-friendly energy convertors which can be used to cool fuel cells. Many new thermodynamic models or analyzes on the FC-TEH systems were witnessed in past years to estimate this coupled system performance and identify parameters characteristics under various operating conditions. For instance, Zhang et al. [43] introduced a thermoelectric generator-driven thermoelectric cooler to efficiently residual heat from solid oxide fuel cells (SOFC) and indicated that the power density and efficiency achieved a slight increase of 2.3% and 4.6% over the single SOFC. Saufi et al. [44] and Hasani et al. [45] experimentally conducted the performance of a combined FC-TEH system, proving that the TED could be a competitive choice to harvest the waste heat from the PEMFC in its narrow temperature range. Gao et al. [46] proposed the structure optimization of FC-TEH subsystem, and concluded that the subsystem power output can be enhanced by 12.9%. Referring to these studies, energy and exergy performance is a major concern of the present FC-TEH system and thereby has received a lot of attention in different fields. It is well known that exergoeconomic analysis is a cost-performance analysis, which should also be considered as a supplement to traditional energy and exergy analysis in thermodynamic systems. Although using exergoeconomic analysis to FC system has been conducted extensively, few studies have been done on the FC-TEH system by exergoeconomic analysis. In terms of application areas, FC-TEH systems are considered as excellent candidates for combined heat and power (CHP) because the simultaneous thermal and electrical energy can be directly produced during the chemical reaction. Studies, by Kwan et al. [47,48], have shown that these is a slight increment in thermodynamic efficiency for FC-TEH system as compared to that of traditional CHP system. In addition to the CHP system, through the combined adsorption chiller the FC-TEH system has significant potential to compete with traditional CCHP system if the hydrogen price goes down below 2 US$/kg [49]. Thermoelectric cooler can be flexibly adopted instead of adsorption chiller to form a cascading thermoelectric generator and cooler for CCHP purposes [50]. These results indicate that in a FC-TEH system the thermal and electrical performance is of equal importance to the researchers. Therefore, the research of improving thermal and electrical efficiency is also one of the critical issues in the field of energy conversion. Generally, using TED in FC system can be mainly categorized in two groups: (1) fuel cell-thermoelectric cooling (FC-TEC) system, and (2) fuel cell-thermoelectric generation (FC-TEG) system. This theory is based primarily on the fact that the thermoelectric device may behaves as TEC mode for cooling or TEG mode for generation power in FC, especially in PEMFC. Previously, the simultaneous TEC and TEG models have been established in the fields of electronic cooling [51], photovoltaic power generation [52] or energy harvesting [53], aiming to efficiently achieve energy conversion capability. Referring to the analysis results in Ref. [9], as the operating current increases continuously, the TED mode is transformed from TEG to TEC at the surface temperature range of 60–100 °C. In addition, the possible TEC - TEG operating mode shift may occur heavily depending on the working current and hot–cold side temperature difference [54]. To clearly identify the operating mode for TED in a specific hybrid system, the positive power production or input and absorbed power can be uniformly employed as determined objectives and then the system performance will be characterized through the present model [55,56]. Kwan et al. [57] preliminarily conducted the bi-directional characteristics of the TED in FCTEH system via theoretical and experimental approaches. Results revealed that the switching processes between the TEC and TEG modes have significant influences on the system performance and parametric optimization. Form the aforementioned literatures, one could observe that few researches have paid their attentions on the systematical investigation 3

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of the TEC - TEG operating mode shift and switched conditions in FCTEH system, especially in the case of PEMFC-TEH system under various decisive parameters. Moreover, internal and external irreversibilities of FC-TEH system have never been identified and quantified. Therefore, this research seeks to fill in these gaps and its key purposes will be to, (1) explore the energy conversion potential of the electrochemical thermoelectric coupling process; (2) describe and estimate the overall system performance on the basis of energy, exergoeconomic and irreversible analysis; (3) investigate the effects of decisive variables on system performance; (4) reveal the switched conditions of TED in FCTEH system; (5) provide a recommendation on application potential and limits of FC-TEH system. The remaining sections of this paper are organized as follows; First of all, thermodynamic modeling jointly with energy, exergoeconomic and irreversible analysis is fully established for PEMFC-TEH system in Section 2. Building on the numerical results from the theoretical analysis, validation and comparison of the PEMFC model is further elaborated in Section 3. Then the effects of several related parameters, such as current density, number of thermoelectric modules and thermal losses on system performance are analyzed and further discussed in Section 4. Furthermore, the operating mode of TED in PEMFC-TEH system is sensitively confirmed according to the current density and hot–cold side temperature difference of TED. Finally, some valuable conclusions are summarized in Section 5.

input and output energy of the TED. Noted that the thermoelectric device could have a positive or negative influence on the overall generation power, highly depending to what model it is operated. Typically, the FC-TEH model can be separated into two categories: FC-TEC model in which the TED operated in TEC model, and FC-TEG model in which the thermoelectric device works as TEG model. On the other words, when the TED is operated in the TEC model, the electric power consumption value is negative which represents the power input for the TED. Inversely the electric power consumption value is positive which denotes the power output for the thermoelectric device. The detailed thermodynamic modeling for both two models is presented in the later subsection to reveal the universal performance features of the FC-TEH system. The thermodynamic modeling of the present system is established with regard to the following assumptions.

2. Thermodynamic modeling of the fuel cell-thermoelectric hybrid system

(8)

Fig. 1 depicts a simple schematic overview of a PEMFC-TEH system in which the TED has been operated as the TEC and TEG models respectively to absorb heat produced from the PEMFC stack. As presented in Fig. 1, the proposed PEMFC-TEH system is mainly composed of a PEMFC stack, a series of TEDs and heat sinks. In this hybrid system, the PEMFC stack can be used as the prime mover of electricity production continuously if the hydrogen is adequately fed to the anode of FC. During the electrochemical reaction process, the FC would generate electric production, water and heat synchronously, and according to the research report the amount of waste heat generated is almost equal to that of electric production. Here, the waste heat is fully taken up by a series of TEDs, which is sandwiched between the FC and heat sinks. The heat sinks are installed at the dissipated side of the TEDs to remove the extra thermal energy for possible heat demands (space heating, hot water). The arrows hint possible fuel or energy inputs and outputs. PFC, Qab,TE and Qdis,TE are respectively generation power of the FC, total

2.1. Energy and exergy analysis

(1) (2) (3) (4) (5) (6) (7)

The FC-TEH system is simulated under steady-state conditions. Stable operating pressure and temperature are assumed in the FC. Current leakage of the FC is assumed to be negligible. For the thermoelectric device, Thomson effect and the joining thermal and electric resistances are neglected. Thermal losses for thermoelectric device and heat exchanger are neglected. The performance of each TED is assumed to be consistent. The pressure loss in the heat exchanger is not considered in all cases. The fuels fed to the FC are regarded as steady, incompressible and the chemical reaction is also adequately completed.

In this subsection, energy and exergy based analysis is investigated for the PEMFC and thermoelectric system respectively, deriving a series of related equations representing the theoretical model. 2.1.1. The PEMFC-representative one component of this hybrid system For the PEMFC subsystem, the practical cell potential (Ucell) is clearly lower than its reversible cell potential (Ure) due to the existing irreversibility during the electrochemical reaction. The empirical equations are adopted for describing the performance of PEMFC stack with the assumption that each single cell in the stack is coincident at steady state. The actual output voltage of a single fuel cell can be determined as below:

Ucell = Ure − Uire

Fig. 1. Schematic diagram of a PEMFC-TEH system incorporated with the description of the bidirectional model. 4

(1)

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Y. Cai, et al.

transfer. Heat loss ratio ζ is introduced to quantifiably evaluate and characterize the rate of removal heat. Based on this, the absorbed heat and lost heat can be respectively expressed by the following Eqs. (14) and (15).

The reversible voltage (Ure) can be calculated in a certain temperature and pressure conditions through the use of the Nernst equation, which is defined as:

Ure = -

- Δgf0 Rgc TFC ΔG Δs = − (TFC − 298.15) + (ln PH2 Po2 ) 2F 2F 2F 2F

(2)

where ΔG, Δg0f and Δs denote the Gibbs free energy change, standard molar Gibbs free energy change and entropy change respectively. F and Rgc stand for the Faraday’s constant and universal gas constant respectively, TFC denotes the operating temperature of the FC. Here, PH2 and PO2 are the partial pressures of hydrogen and oxygen, which are calculated by [49]:

Qab, TE = - Δ H − PFC − Qloss

(14)

Qloss = ξQ heat = Kloss (TFC − T0)

(15)

By combining Eqs. (1, 10, 13–15), the absorbed heat represented by Eq. (14) can be given as:

Qab, TE = (1 − ξ )[ - Δh

N iA i − N iA (Ure − bln( )− ir − mexp(ni))] 2F i0 (16)

PH2

⎤ ⎡ 1 ⎥ ⎢ = 0.5PH2 O ⎢ − 1⎥ 1.653i ⎛ ⎞ ⎥ ⎢ exp T1.334 X H2 O ⎝ FC ⎠ ⎦ ⎣

For the sake of describing the exergy performance of this FC-TEH model, the total exergy of the reactants should be determined on the basis of the chemical and physical specific exergy rate. Actually, the air fed in the cathode could be considered as free fuel from the environment. For a perfect gas, the general calculations of the physical and chemical exergy are presented in detail in Refs. [32]. The total exergy content of the hydrogen can be determined through the following equation:

(3)

0.291i PO2 = 1 − X H2 O − X Nchannel exp ⎛⎜ 0.832 ⎞⎟ 2 ⎝ TFC ⎠

(4)

where

log(PHsat2 O ) = −2.1749 + 0.0295(TFC − 273) − 9.1837 ∗ 10−5 273)2

10−7 (TFC

273)3

E xin,H2 = m H2 (ex ch + exph) H2

(17)

(5)

In this equation, parameter mH2 represents the mass flow of the hydrogen, which can be confirmed by Faraday’s Law as:

(6)

m H2 = MH2

)

(7)

X N2, in = 0.79(1 − X H2 O )

(8)

In which MH2 is the molar mass of the hydrogen. Finally, energy and exergy efficiency of the stand alone PEMFC can be respectively defined as the ratios of electric production to the amounts of possible energy (−ΔH) and exergy input (Exin,H2) as:

(TFC −

X H2 O

+ 1.4454 ∗

−

PH2 O = Pop

X Nchannel 2

=

X N2,out =

X N2, in − X N2, out ln

(

X N2, in X N2, out

1 − X H2 O 1+

(

λ air − 1 λ air

)( ) 0.21 0.79

i Uire = Uact + Uohm + Ucon = bln ⎛ ⎞ + ir + mexp(ni) i ⎝ 0⎠ ⎜

PFC - ΔH

(19)

ψFC =

PFC Exin,H2

(20)

2.1.2. The thermoelectric system-representative secondary component of this hybrid system As described in Fig. 1, a bidirectional model for the TED is proposed according to the thermoelectric theory, the first one, called “TEC model”, denotes the thermoelectric refrigerator; the second model denotes the thermoelectric generator, which is called “TEG model”. It is emphasized that for the thermoelectric device in either TEC or TEG model the temperature side connected to the fuel cell is referred to as the absorbed side of the TED and the other side is called the dissipated side. Meantime, the heat transfer coefficients in the absorbed and dissipated sides are expressed as KFC,TE and Kdis and the higher value is, the better heat transfer potential is. For the TEC model, expressions for the energy balance associated with absorbed load Qc,TEC, dissipated load Qh,TEC and power input PTEC are respectively written as:

⎟

(10)

where coefficients b, r, and m are the functions of fuel cell temperature, as shown in Ref. [29]. Parameters i and i0 represent the current density and exchange current density. Coefficient n is the growth rate factor, which is considered to be 8 cm2/A in this study. For the PEMFC stack with a number of cells (N), the total output voltage and the corresponding power are given by:

UFC = NUcell

(11)

PFC = UFC iA

(12)

Then, the total thermal energy of the PEMFC stack can be obtained

Qc, TEC = STEC Tc, TEC ITEC −

1 2 ITEC RTEC − KTEC (Th, TEC − Tc, TEC ) 2

(21)

Qh, TEC = STEC Th, TEC ITEC +

1 2 ITEC RTEC − KTEC (Th, TEC − Tc, TEC ) 2

(22)

as:

Q heat = - Δ H− PFC

N iA = - Δh − UFC iA 2F

(18)

ηFC = (9)

In which parameters PH2O, XH2O and XN2 stand for the water saturation pressure, molar fractions of water and nitrogen respectively, λair stands for the air stoichiometric ratio. Taking into consideration the all overpotentials including activation overpotential Uact ohmic overpotential Uohm and concentration overpotential Ucon, the irreversible voltage (Uire) can be expressed as:

NiA 2F

(13)

PTEC = Qh, TEC − Qc, TEC = STEC ITEC (Th, TEC − Tc, TEC ) + ITEC 2RTEC

where ΔH and Δh represent the total possible energy and molar enthalpy change. N is the number of the cells, A is effective polar area of the PEMFC. In this research, the thermal energy produced by the FC is mostly taken away through the TEDs and the rest of the heat is assumed to be lost to the ambient environment via convective or conductive heat

(23)

where S, K, and R represent the Seebeck coefficient, thermal conductance and electrical resistance respectively. I, T represent the operating current and temperature. Subscripts “TEC, h,TEC and c,TEC” denote the TEC model and hot and cold ends of the TEC model respectively. 5

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The absorbed load and dissipated loads for the TEDs are respectively given as:

Qab, TE = nTEC Qc, TEC = KFC , TE (TFC − Tc, TEC )

(24)

Qdis, TE = nTEC Qh, TEC = K dis (Th, TEC − T0)

(25)

( + K( =

Tc, TEC

K dis n

1 I 2R 2 TEC

(

K dis n

(K + SI

TEC

Th, TEC =

) T)

1

+ K − SITEC TFC + 2 ITEC 2R +

K dis n

KFC n

K dis n

+ K − SITEC

)

0

+ K − SITEC +

(

)(K + SI

)T − ( c

K

TEC

1 I 2R 2 TEC

+

+

KFC n

KFC T n FC

)−K

2

(26)

) (27)

It is observed from the above equations that once the fuel cell’s and ambient temperatures (TFC, T0) and the thermal conductance in the absorbed and dissipated sides are obtained, the hot and cold end temperatures of this TED can be calculated according to the iterative approach. Then the performance and operating mode of the TED can be further confirmed. To compare with the stand alone FC model, the thermal energy/ exergy has been considered in the analytical expressions for the energy and exergy efficiencies of this FC-TEH system. Because of the negative influence of the TED on the total power generation, the total electric production and efficiency of this hybrid system can be respectively expressed as:

PFC

- TEC

= PFC - nTEC PTEC

- TEC

=

ψFC

- TEC

=

PFC PFC

- TEC

+ nTEC Q h,TEC - ΔH

+ Ex out , TEC Exin,H2

(29)

- TEC

(30)

In which nTEC represents the number of TEDs, Exout,TEC represents the useful thermal exergy rate obtained from the heat exchanger, which is calculated in the subsection 2.2. For the TED in the TEG model, the thermoelectric device is regarded as the solid-state heat engine and the total electric production of the FCTEH system is larger than that in the TEC model due to the additional power generation from the TED. The general expressions for heating capacity Qh,TEG, cooling capacity Qc,TEG and power output PTEG are respectively written as:

Based on the Eqs. (21)–(25), the hot and cold end temperatures of the TEC model are solved by the following equations: KFC n

ηFC

Qh, TEG = STEG Th, TEG ITEG −

1 ITEG 2RTEG + KTEG (Th, TEG − Tc, TEG ) 2

(31)

Qc, TEG = STEG Tc, TEG ITEG +

1 ITEG 2RTEG + KTEG (Th, TEG − Tc, TEG ) 2

(32)

PTEG = Qh, TEG − Qc, TEG = STEG ITEG (Th, TEG − Tc, TEG ) − ITEG 2RTEG

(33)

In which subscripts “TEG, h,TEG and c,TEG” denote the TEG model and hot and cold ends of the TEG model respectively. Similarly, the overall absorbed and dissipated loads can be respectively calculated as:

Qab, TE = nTEG Qh, TEG = KFC , TE (TFC − Th, TEG )

(34)

Qdis, TE = nTEG Qc, TEG = K dis (Tc, TEC − T0)

(35)

By equating Eqs. (31)–(35) and then solving for Th,TEG and Tc,TEG, the expressions for Th,TEG and Tc,TEG are given as:

(28)

Fig. 2. Step-by-step connection of system components in a PEMFC-TEH system. 6

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( + K( = KFC n

Th, TEG

K dis n

1 I 2R 2 TEG

(

K dis n

(K + SI

TEG

Tc, TEG =

) T)

1

+ K − SITEG TFC + 2 ITEG 2R +

K dis n

+ K − SITEG

)

Qdis, TE Qab, TE ⎞ − Ex d, TE = T0 Sgen,TE = T0 ⎛⎜ ⎟ T Tab, TE ⎠ , dis TE ⎝

K dis n 0

+ K − SITEG

+

(

KFC n

)(K + SI

)T − ( h

TEG

1 I 2R 2 TEG

+

+

)−K

KFC n

KFC T n FC

2

In the above equation, subscripts “ dis,TE” and “ab,TE” indicate the dissipated side and absorbed side of the TED. In other words, for the TED in TEC model, the dissipated side and absorbed side represent the hot and cold sides of this TED respectively, as opposed to that in TEG model. The exergy flow rate in the exhausted heat taken away by the heat exchanger can be calculated as [9]:

(36)

) (37)

K

After calculating Th,TEG and Tc,TEG, the total electric production and efficiency of the FC-TEH system in the TEG model can be respectively expressed as:

PFC

- TEG

= PFC + nTEG PTEG

ηFC

- TEG

=

ψFC

- TEG

=

(39)

+ Ex out , TEG Exin,H2

(40)

PFC

Qdis, TE ⎤ T Ex d, HE = T0 Sgen,HE = T0 ⎡mf cf ln out − ⎢ T Tdis, TE ⎥ 0 ⎣ ⎦

(38)

PFC, TEG + nTEG Qc,TEG - ΔH

T Ex out , TE = mf cf ⎡ (Tout − T0) − T0 ln ⎛ out ⎞ ⎤ ⎢ ⎝ T0 ⎠ ⎥ ⎣ ⎦ ⎜

2.2. Irreversible characteristic analysis

Ex d, FC

rFC = rFC

rFC

= Ex d, FC + Ex dis, TE + Ex d, HE

Pre − PFC 1 1 ⎞ ⎛1 − 1 ⎞ + ( 1− ξ ) Q heat ⎜⎛ − ⎟ + ξQ heat TFC TFC ⎠ TFC ⎠ ⎝ T0 ⎝ Tab, TE ⎟

(42) where Pre, Sgen,it and Sgen,et are the reversible power output, internal entropy generation rate and external entropy generation rate, respectively. It is noteworthy that, the first term on the right hand side of Eq. (42) denotes the internal entropy generation rate due to the overpotential losses in FC module. The second and third terms denote the entropy generation rates, resulting from the thermal transfer between the FC and TED or environment. Substituting Eqs. (1, 11, 12, 14, 15) into Eq. (42), the expression for Sgen,FC is rearranged as: ⎜

(47)

Ex d, FC PFC

- TEC

=

- TEG

=

(48)

Ex d, FC - TEC + Ex out , TEC

(49)

Ex d, FC - TEG + Ex out , TEG

(50)

PFC

- TEC

PFC

- TEG

Actually, the reference value of this factor should be as small as possible, which means to generate more desired exergy production while minimize the exergy destruction content. It can be seen that the exergy destruction could be larger than the exergy production when the value of r is more than 1. In this case, this hybrid system may be low efficient in terms of the irreversible analysis. In addition, anther irreversible index associate with the exergy destruction and exergy input of this hybrid system is defined as the proportion of the exergy loss to overall exergy input, which can be proposed as:

Sgen, FC = Sgen,it + Sgen,et

NiAUire 1 1 ⎞ ⎛1 − 1 ⎞ + Qab, TE ⎜⎛ − ⎟ + Q loss TFC TFC ⎠ TFC ⎠ ⎝ T0 ⎝ Tab, TE

(46)

Exergy destruction is a significant index for the FC-TEH system, which indicates the decrement of the potential work loss due to its existing irreversibility. However, only exergy loss amount of the thermodynamic system cannot directly reflect the decremental margin of irreversibility because the exergy input and output should be simultaneously taken into consideration during the FC-TEH operation. In order to quantitatively evaluate and characterize the inefficiency of the hybrid system, two new benign ratios, including exergy destruction factor r and exergy destruction efficiency f, are introduced similar to the system efficiency, and, but with a slightly various implication. Exergy destruction factor of the FC-TEH system can be assumed as the ratio of the exergy loss to overall exergy output. In this regard, the exergy destruction factors for the stand alone FC model, FC-TEC model and FCTEG models can be respectively expressed as:

The irreversible entropy generation rate caused by FC’s operating at the steady state is given as below:

Sgen,FC =

- TE

(41)

⎜

⎟

Finally, the overall exergy loss of this FC-TEH system can be calculated through the following equation:

According to the irreversible thermodynamics, the irreversible losses occurring because of the irreversibility in the entire energy system are essentially dependent on the entropy generation of this system and the environment temperature [9,33]. Entropy generation is a significant index, which hints the potential work lost owing to the finite potential difference. Therefore, by knowing the entropy generations for the each component in the system one can calculate the system exergy loss and further identify the inefficiency in view of the entropy generation analysis. For an irreversible FC-TEH system, the total irreversibility can be divided into three constituent parts, which results from the corresponding components of FC, TED and heat exchanger (HE), as exhibited in Fig. 2. Referring to this figure, the overall hybrid system consists of three parts A, B and C, and the product of the previous system is also the input of the next system. The reversible system component is one which does not possess irreversibility, that is, the entropy generation for this component is numerically zero. For instance, if the component TED is reversible and the hybrid system is in the FC-TEG model, one has the socalled the case of “reversible for TED in FC-TEG model” in this research. A general expression of the exergy loss for the component A is given by:

=

(45)

where mf, cf, Tout are the coolant flow rate, specific heat capacity of the coolant and the exit temperature of the coolant in the heat exchanger. The exergy flow rate in the exhausted heat taken away by the heat exchanger can be calculated as [40]:

- TEG

Ex d, FC = T0 Sgen,FC

(44)

fFC =

⎟

(43)

Ex d, FC Ex in

(51)

fFC

- TEC

=

Ex d, FC - TEC Ex in

(52)

fFC

- TEG

=

Ex d, FC - TEG Ex in

(53)

where fFC, fFC-TEC, and fFC-TEG stand for the exergy destruction

The exergy destruction in a TED can be written as follows: 7

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efficiencies for the stand alone FC model, FC-TEC model and FC-TEG model respectively. Once the exergy destruction rates related to the corresponding models are obtained by using a numerical approach, the exergy destruction efficiency can be determined according to the Eqs. (51)–(53). In this research, the exergy destruction factor and efficiency are the main criterions from the viewpoint of irreversible thermodynamics.

Cfuel = ch ∗ Exin, H 2

2.3. Exergoeconomic analysis

cFC =

(59)

where ch represents the hydrogen price (0.07 $/kWh), Exin,H2 represents the input exergy of the hydrogen. Accordingly, the unit exergy cost of the stand alone FC model, FCTEC model and FC-TEG model are obtained using the following equations: ·

The exergoeconomic concept has been proposed and extended by Bejan’s work [33] on the basis of the combination of the irreversible thermodynamics with economic analysis to evaluate the effectiveness of energy conversion systems. The main goal of using the exergoeconomic analysis is to calculate the exergy cost of the defined fuel and product and quantify the capital cost of each component in this hybrid system. Actually, the exergoeconomic analysis, as a supplement to the energy and exergy analysis, can be utilized for simultaneous economic and thermodynamic considerations. In the present work, the specific exergy costing (SPECO) approach is adopted, in which the product of the entire system is defined and then the unit exergy cost for the product is calculated by using the exergy cost balance equations associated with the capital cost and fuel cost. As mentioned above, the general exergy-based cost equation for an energy conversion system is given as: ·

.

.

∑ Ce,k + Cw,k = Cq,k

.

+

e

i

.

(54)

PFC

(62)

The simulation model jointly with energy, irreversible and exergoeconomic analysis has been carried out for the FC-TEH system, employing MATLAB program environment. Since the analytical expressions for hot and cold end temperatures of the TED are derived, as proposed in Eqs. (26), (27), (36) and (37), the performance of the FCTEH system in the FC-TEC or FC-TEG model can be numerically calculated by iterative method. First of all, the hot and cold end temperatures of the TED are hypothetically given as the initial parameters and then a MATLAB program can be further developed to solve Eqs. (13), (21)–(25) and (31)–(35) by using Gauss-Seidel iterative method, where the convergent criterion is a small specified number (10 −3). Once these temperatures are numerically obtained, the values of power output, energy efficiency, exergy efficiency and unit exergy cost can be subsequently calculated to describe the FC-TEH system performance and reveal the irreversibility characteristics. Parameters used in predicting system performance in this study have been summarized in Tables 1 and 2 respectively, which are obtained from the relative Refs. [6,15,29,33,35].

(55)

.

(56)

d )t

d (1 + (1 + d )t − 1

- TEG

Z ch + + Ex out , TEG ψFC - TEG

(61)

2.4. Implementation of the mathematical model of the system

here, the levelized capital investment cost rate is a function of operating time (Lh), system life (t), interest rate (d) and capital recovery factor (CRF), which can be transformed to the cost rate as:

CRF =

- TEC

It should be mentioned that for the stand alone FC model, the electric production is regarded as the exclusively useful output while for the FC-TEC model and FC-TEG model the thermal and electric exergy is simultaneously considered as the output exergy, aiming at exploring the exergoeconomic-based potential of the FC-TEH system. The detailed results and discussions for the unit exergy cost of this system have been proposed in the Section 4 with the consideration of depth analysis and explanation.

.

Zk = Zk ·CRF ·φ /(3600Lh )

PFC

Z ch + + Ex out , TEC ψFC - TEC ·

cFC − TEG =

In which i and e stand for the entry and exit exergy streams for the . part k in this system. Zk stands for the capitalized cost rate of the component k in this system. C stands for the exergy cost rate of each exergy stream. It should be mentioned that for this hybrid system the total exergy output is the product and the hydrogen used in the chemical reaction is the fuel. Eq. (54) indicates that the exergy cost rate of the product can be calculated if the exergy cost associated with the fuel and the investment cost rate is given according to the appropriate references. For a specific component or an entire thermodynamic system, the exergy cost of the product is defined as the product of the unit exergy cost and the exergy stream, which is expressed as:

Cp = cp·Ex p

(60) ·

cFC − TEC =

.

∑ Ci,k + Zk

Z c + h PFC ψFC

Table 1 Parameters used in the parametric study for PEMFC model [24,29].

(57)

In which the annual operating time Lh and system life t are respectively considered to be 8760 h and 5 years. It is assumed that there is no maintenance for the entire system (φ = 1). The total system cost Zk is the summation of the costs of the FC (ZFC), TED (ZTE) and heat exchanger (ZHE), which can be determined as:

Z = ZFC + ZTE + ZHE = Zfix +NZcell + (Ev L + Ear ) ATE Far + EHE K dis (58) where Zfix, Zcell, Ev, Ear, and EHE stand for the FC stack fixed cost, fuel cell cost, volumetric manufacturing cost of the TED, areal manufacturing cost of the TED and heat exchanger cost, respectively. The specific values of these important parameters with respect to the exergoeconomic analysis involve above are given in this study, which are derived from the data available in the literatures [15,33,35]. The hydrogen fuel is considered as the entering stream in this study. According to the Ref. [30], the fuel cost can be calculated as: 8

Parameter

Symbol

Value

Nominal power output Amount of cells Active area of the cell Operating temperature Operating pressure Standard molar Gibbs free energy change Standard molar entropy change Faraday constant Universal gas constant Air stoichiometric ratio exchange current density Coefficient of concentration loss Chemical exergy of hydrogen Fluid flow rate Standard temperature Standard pressure

– N A TFC Pop Δgf0 Δs F Rgc λair i0 n exCH mf T0 P0

5 kW 35 232 cm2 333–353 K 3 atm −237.3 kJ/mol −163.4 kJ/mol 96,485 C/mol 8.314 J/mol K 2 0.04 mA/cm2 8 cm2/A 159,138 KJ/kg 0.5 kg/s 298.15 K 1 atm

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parameters are presented and discussed to carefully illustrate the FCTEH system performance. In the meantime, parametric comparisons between the fuel cell-thermoelectric cooling (FC-TEC) and the fuel cellthermoelectric generation (FC-TEG) models are sensitively identified in terms of the decision targets, such as power output, energy efficiency, exergy efficiency and unit exergy cost. In addition, irreversible characteristics of this FC-TEH system are quantitatively evaluated with respect to the exergy destruction factor and efficiency.

Table 2 Design parameters used in predicting thermoelectric system performance and exergoeconomic cost [6,15,33,35]. Parameters

Values

Thermoelectric element length (mm) Thermoelectric element width (mm) Thermoelectric element thickness (mm) Thickness of thermocouples L (mm) Area ratio of semiconductor columns to TE module (Far) Absorbed/dissipated side thermal conductance Kab,TE (W/K) Thermal conductance K (W/K) Electrical resistance R (Ω) Figure of merit Z (K−1) Absorbed end heat conductance (W/K) Dissipater end heat conductance (W/K) Stack fixed cost Zfix ($) Cell cost Zcell ($/cell) Volumetric manufacturing cost Ev ($/cm3) Areal manufacturing cost Ear ($/cm2) Heat exchanger cost EHE ($/(W/K)) Hydrogen cost ch ($/kWh) System operating years t System operating hours Lh (h) Maintenance factor φ Interest rate d

40 40 3.9 1.4 0.3 1120 0.5186 1.9591 0.00253 20 20 1000 500 0.89 0.017 1 0.07 5 8760 1.0 10%

4.1. Analysis of the decision parameters The decision parameters, such as PEMFC current density, number of thermoelectric devices and thermal losses not only influence the energy, exergy performance of the FC-TEH system, but also the irreversible and exergoeconomic characteristics, therefore the impacts of theses factor on the FC-TEH performance should be studied. 4.1.1. Effect of PEMFC current density The PEMFC current density is a significant factor for the FC-TEHE system, therefore the impact of this variable is first investigated. Fig. 5 displays the evolution of the power output with current density i at various cell temperatures for the stand-alone FC model, FC-TEC model and FC-TEG model respectively. It can be seen from Fig. 5 that for the stand alone FC model, as the current density increases, the power output first increases to a maximum value then decreases, and the greater the value of TFC is, the higher the generated electricity is. This is because, the proton transfer process in the membrane and catalyst activity are strengthened at the elevated temperature. For the FC-TEH system in the current density range of 0–1.2 A/cm2, this hybrid system first behaves as the FC-TEC model, then FC-TEG model and finally FCTEC model again. Thus, there exist effective ranges of current density for the FC-TEG model, which are respectively confirmed to be 0.615–0.855 A/cm2, 0.77–1.037 A/cm2, and 0.92–1.17 A/cm2 at TFC = 333 K, 343 K and 353 K. Also, the effective range of current density for the FC-TEC model is more widespread than that for the FCTEG model. By comparing the three models, the FC-TEG model produces the highest power output among them, while FC-TEC’s the lowest. It can be explained in that the thermoelectric device has a positive influence in the power production for the FC-TEC model as depicted in Eq. (28) while the opposite is true for the FC-TEG model as shown in Eq. (38). For instance, at the operating temperature of 343 K, the maximized power outputs of the FC, FC-TEC and FC-TEG models are respectively 4658.62 W, 4738.75 W and 4525.3 W for the current density of 0.955 A/cm2, 0.95 A/cm2, 1.037 A/cm2. Figs. 6 and 7 show the respective variations of the energy efficiency

3. Validation and comparison of the PEMFC model Based on the experimental results reported by Laurencelle et al. [29], Chang et al. [38], and the numerical results reported by Ebrahimi et al. [49], the numerical modeling proposed in the present work is verified and compared. Firstly, the comparative polarization curves of the simulated and experimental results with various operating temperature of 312 K, 329 K, and 345 K are presented for PEMFC model, as shown in Fig. 3. As seen, the variation tendency of the current results is basically consistent with the experimental results and there is a relative close result between curves. Although the simulated results in all operating temperatures are slight higher that of the experimental results, the results between the simulated and experimental cases are in good accordance to a certain degree. This may attribute to the fact that the thermal losses and current leakage occur in actual experiments while these factors are ignored in the simulation process. In order to further judging the difference between the simulated and experimental results, the values of correlation coefficient (CC) [32] and the root-mean-square error (RMSE) [14] are respectively calculated in this comparison. For the operating temperatures of 312 K, 329 K, and 345 K, the values of CC and RMSE are approximately 0.997/0.031, 0.998/0.068 and 0.998/ 0.005 respectively, which indicate the theoretical modeling for the PEMFC is reliable and applicable. A comparative analysis of the simulation results and experimental data from the Ref. [38] was performed and the average error was approximately 4.84%. Furthermore, the results comparison of the predicted output power and cell voltage between the present theoretical model and the analysis model from Ref. [49] are proposed, as depicted in Fig. 4. It can be seen from this figure that these two situations for the output power and cell voltage show the similar trends. Despite the identical number of cells considered in FC model, the results of the output power and cell voltage are clearly higher than that in Ref. [49]. The reason could be that the irreversible losses of the FC in the present work are calculated according to the experimental parameters while in reference these losses are identified from the theoretical empirical formulas, causing in result difference in these both situations. By comparing current results with the above study, one can find that the FC stack can provide sufficient output power demand for some specific applications. 4. Results and discussion

Fig. 3. Comparative polarization curves of the simulated and experimental results for PEMFC [29,38].

In this section, the numerical results under different operating 9

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Fig. 4. Results comparison of the predicted output power, cell voltage between the present theoretical model and Ref. [49].

Fig. 5. Effect of current density on the power output with various cell temperatures for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

Fig. 6. Effect of current density on the energy efficiency with various cell temperatures for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

η, and exergy efficiency ψ with the current density for difference operating temperatures. As can be observed in Fig. 6, increasing the current density continuously decreases the energy efficiency for the stand alone FC model as well as the FC-TEC and FC-TEG models, which means that a greater amount of energy content is input to the system as compared to the generated power. Specifically, the energy efficiencies of the FC-TEH system decline from 0.94 to 0.796, 0.94 to 0.836 and 0.94 to 0.856 for TFC = 333 K, 343 K and 353 K respectively. Referring to Fig. 7, opposite to the energy efficiency, the exergy efficiency first shifts to augment and then reduces for the FC-TEH system with a rise of current density. It is mentioned that, the exergy efficiency for the FCTEG model is greater than that for the alone FC due to simultaneous consideration of the thermal and electric exergy in the output product. The results also show that with the increment of current density the exergy efficiency value at the case of TFC = 333 K changes from the maximal to the minimum among of all temperature cases. The reason for this variation is due to the fact that the TED in the TEC model exhibits better performance at a lower temperature while poor performance in the TEG model, which is also suggested in Fig. 5. Maximal

exergy efficiencies for TFC = 333 K, 343 K and 353 K are respectively obtained to be 0.438, 0.424 and 0.413 respectively. Fig. 8 illustrates the unit exergy cost variation of the alone FC model, FC-TEC model and FC-TEG model with current density for various values of TFC. As can be observed in Fig. 8 the unit exergy cost of the FC-TEC or FC-TEG model is greater than the single FC in the current range of 0–1.2 A/cm2 for the same operating temperature. This is mainly because that the TEDs in the hybrid system increase the total capital investment cost, thereby augmenting the unit exergy cost. It can also be seen from Fig. 8 that compared with the FC-TEC system the FCTEG model possesses the lower unit exergy cost due to its larger exergy output. The minimum unit exergy costs are approximately 0.33 $/kWh, 0.311 $/kWh, and 0.3 $/kWh at TFC = 333 K, 343 K and 353 K respectively, which are increased by 4.1%, 3.67% and 5.26% compared to that of single FC (0.317 $/kWh, 0.3 $/kWh and 0.285 $/kWh). The results prove that despite the slight improvement of unit exergy cost for the FC-TEG model this hybrid system has better thermodynamic characteristics over the stand alone FC owing to its higher power output and thermodynamic efficiency. 10

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attributed to the model transition of the TED from TEC to TEG model in the FC-TEH system. Figs. 10 and 11 show the effect of number of thermoelectric devices in the energy and exergy efficiencies for the stand-alone FC model, FCTEC model and FC-TEG model respectively. It shows that the energy efficiency values for the FC-TEH system almost keep constant at 0.873 and 0.882 for i = 1.037 A/cm2 and 0.95 A/cm2 respectively. This is due to the fact that although augmenting n within certain limits causes a rise in the electric efficiency, the thermal efficiency shifts to decline, and if the current density is given in the performance calculation, the total absorbed load from the FC to the TEDs remains constant. Referring to the Fig. 11, the evolution of exergy efficiency follows the same trend of the power output as proposed in Fig. 9 and the amount of n switched from TEC to TEG model is consistent for these two indexes. Furthermore, at the n range of 100–300, the exergy efficiency ranges are about 0.207–0.325 and 0.325–0.331 respectively for i = 1.037 A/cm2. The variation in unit exergy cost due to the growth in the n is presented in Fig. 12. It is evident from Fig. 12 that the unit exergy cost shows a drop with increase in the n, then achieves the lower value, and ultimately a slow rise for the FC-TEH system. This means that more TEDs installed allow a slight augment in the power output and thermodynamic efficiency for the FC-TEG model, further causing a reduction in the unit exergy cost.

Fig. 7. Effect of current density on the exergy efficiency with various cell temperatures for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

4.1.3. Effect of thermal loss In FC-TEH system, both for system electric power output and for irreversibility, the thermal loss between the FC and the environment plays a critical role in their performance characteristics. As showed in Eq. (15), the thermal loss coefficient ζ is used to represent the thermal loss degree for the FC-TEH system, which gives rise to an inevitable reduction in energy and exergy amounts. Fig. 13 describes the relationship of the power output and thermal loss coefficient for the current density of 1.037 A/cm2 and 0.95 A/cm2 respectively. As seen for the case of i = 1.037 A/cm2, varying ζ from 0 to 30% would persistently improve the power output from 4089.55 W to 4601.33 W. The reason behind this variation is the fact that the thermal loss coefficient highly influences the absorbed load from the FC to the TED, which accounts for the hot and cold side temperatures of the TED thus, an elevated ζ results in a growth in power output. By comparing the stand alone FC, FC-TEC and FC-TEG models, one can observe that the power output for the single FC model is higher than the FC-TEC model’s case within certain limit of ζ and afterwards the power output for the FC model is lower than that for the FC-TEG model. Figs. 14 and 15 illustrate the effect of thermal loss coefficient on the energy η and exergy efficiencies ψ for the stand-alone FC model, FC-TEC

Fig. 8. Effect of current density on the unit exergy cost with various cell temperatures for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

4.1.2. Effect of number of thermoelectric devices The number of thermoelectric devices n not only directly affects the absorbed load of the TEDs, but also significantly affects the investment cost of system components. Therefore, the effect of number of thermoelectric devices on the important performance parameters for the FC-TEC and FC-TEG models should be investigated. Since altering the number of thermoelectric devices does not interface with any of the single FC’s calculations, the alone FC performance remains unchanged, considered as a case for reference in this study. Fig. 9 demonstrates the effect of altering number of thermoelectric devices on the power output for the stand-alone FC model, FC-TEC model and FC-TEG model respectively. In order to obtain the representative characteristics, the current density leading to the maximal power output for the FC-TEH system is chosen for simulation in this case. It is obvious from the Fig. 9 that the power output for the FC-TEH system has a large upward tendency initially and then a slight decline, as the value of n continuously increases from 100 to 300. Thus, these exist maximum values of power outputs at about 4623 W and 4744.78 W when the number of thermoelectric devices are respectively 267 and 216 for the current density of 1.037 A/cm2, and 0.95 A/cm2. For the case of i = 1.037 A/cm2, the generated electricity for this FC-TEH system is lower than that for the FC at the n range of 100–200, after that, the FC-TEH’s greater, which is

Fig. 9. Effect of number of thermoelectric devices on the power output for the stand-alone FC model, FC-TEC model and FC-TEG model respectively. 11

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Fig. 10. Effect of number of thermoelectric devices on the energy efficiency for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

Fig. 13. Effect of thermal loss coefficient on the power output for the standalone FC model, FC-TEC model and FC-TEG model respectively.

Fig. 11. Effect of number of thermoelectric devices on the exergy efficiency for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

Fig. 14. Effect of thermal loss coefficient on the energy efficiency for the standalone FC model, FC-TEC model and FC-TEG model respectively.

Fig. 12. Effect of number of thermoelectric devices on the unit exergy cost for the stand-alone FC model, FC-TEC model and FC-TEG model respectively.

Fig 15. Effect of thermal loss coefficient on the exergy efficiency for the standalone FC model, FC-TEC model and FC-TEG model respectively.

model and FC-TEG model respectively. Augmenting the thermal loss coefficient from 0 to 30% causes a linear lessening in the energy efficiency of the FC-TEH system from 0.999 to 0.809 and from 0.999 to

0.823 for i = 1.037 A/cm2 and 0.95 A/cm2 respectively. However, opposite to the energy efficiency, the exergy efficiency for the FC-TEH system shows an increasing tendency with a rise of ζ. This variation of ψ can be justified with the fact that although there is a reduction of the 12

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absorbed load as the ζ value increases, the reduction of the output thermal exergy would be compensated by the increased power generation, causing an increase in the ψ value, as suggested in Eqs. (30) and (40). Fig. 16 depicts the variation in unit exergy cost of these three models with changing the ζ value from 0 to 30%. Referring to this figure, the minimum unit exergy costs for the FC-TEH system are about 0.348 $/kWh and 0.32 $/kWh at i = 1.037 A/cm2 and 0.95 A/cm2 respectively, which are slightly higher 2.35% and 2.56% over the single FC’s case (0.34 $/kWh and 0.312 $/kWh). Therefore, if we operate the thermal loss between the FC and the environment in the FC-TEH system through suitable thermal insulation one can have high thermodynamic performance and low unit exergy cost simultaneously.

It can be seen from Fig. 20 that increasing current density from 0 to 1.2 A/cm2 leads to augment in absorbed power from 0 to 9421.59 W for TFC = 343 K and the reason is that a higher current density causes a greater energy input of the FC (see Eq. (13). According to Fig. 21, the power of the TED is regarded as the power input for the hybrid system in the FC-TEC model while the power output in the FC-TEG model. In the case of TFC = 343 K, with a rise of current density, the TED operated first in the TEC model, then TEG model and finally TEC model again and the effective range of current density for the FC-TEG model is about 0.77–1.037 A/cm2. The reason behind this change is the fact that as current density augments the absorb load increases considerably as given by Fig. 20, thereby causes the change of thermoelectric working current I, which has a significant influence on the thermoelectric operating models (see Ref. (52)). Moreover, the order of the current density range for the FC-TEG model from small to large is: TFC = 333 K < TFC = 343 K < TFC = 353 K, which means that the FC-TEG model is more pronounced at higher TFC. It also shows that the maximum generated electricity of the TEDs in the TEG model are confirmed to be 50.42 W, 83.37 W and 124.48 W for TFC = 333 K, 343 K, and 353 K respectively while the lower values of power input of the TEDs in the TEC model are zero. The hot and cold side temperature difference trends of the TED in the FC-TEH system are illustrated in Fig. 22 for various FC’s operating temperatures. For the case of TFC = 343 K, the hot and cold side temperature difference monotonously decreases with the increment of i for the FC-TEG model while the trend is reversed for the FC-TEC model. A interesting phenomenon which can be seen from the Fig. 22 is that the absolute values of the hot and cold side temperature difference (called switched temperature difference) in FC-TEC and TEG models are the same when the current density values leading to PTE = 0 are attained. This is because the hot and cold ends in TEC and TEG models are opposite under the same external conditions as shown in Fig. 1. In this typical current density, the switched temperature differences are obtained as approximately 29.38 K/26.09 K, 37.91 K/33.62 K and 46.29 K/40.99 K respectively for the various values of TFC at 333 K, 343 K and 353 K. According to correlative results in Figs. (20)–(22), the current density has the different effects on the thermoelectric operating model in the FC-TEH system.

4.2. Comparative results of irreversible modeling In this section, the cases of the reversible and irreversible thermoelectric system, including the TED and HE, are paid close attention to and investigated through the previous analysis in Section 2.2. Comparative results of these irreversible modeling and exergy destruction indexes r and f have been plotted respectively in Figs. 17–19. The results in Fig. 17 show clearly that if the TED is assumed to be reversible, the FC-TEH system is operated in the FC-TEG model for the entire range of the current density from 0 to 1.2 A/cm2, which indicates that in this current density range the hot end temperature of this TED is always higher than the cold side temperature of the TED. Similar to the results in Fig. 7, the order of the exergy efficiency for the various models from small to large is: FC-TEC < FC < FC-TEG, which is attributed to the positive contribution of the TEG in the power output for the overall hybrid system. For a typical current density of 0.95 A/cm2, the values of exergy efficiency for the cases IV, VII, Ⅱ and Ⅴ are respectively 0.37, 0.371, 0.409, and 0.413, which are increased by 1.93%, 2.2%, 12.67% and 13.77% over the stand alone FC model (0.363). The changes of exergy destruction factor and exergy destruction efficiency with the current density for the irreversible cases which correspond to different irreversible models are respectively described in Figs. 18 and 19. According to the Fig. 18, the effect of current density has a smaller impact on the exergy destruction factor in cases Ⅱ and Ⅴ compared with other cases because in these two cases the values of r increase slower from 0.011 to 0.149 and from 0.011 to 0.142. It is apparent that if the component HE is reversible in the hybrid system (i.e. cases III and IV), as the current density increases within a certain range this system would behave as the FC-TEC model initially, then FCTEG model and ultimately FC-TEC model again, following the same tendency of the results in cases VI and VII. Also in these two conditions, the minimum values of exergy destruction factor can be obtained at 0.192 and 0.905 respectively and the corresponding i values are 0.85 A/ cm2 and 0.64 A/cm2. As we know in Eqs. (48)–(50), the situation of r < 1 indicates that the amount of exergy destruction is less than the exergy output amount, which could be beneficial for the overall system. Referring to the Fig. 19, the exergy destruction efficiency follows almost the same variation trend of the r shown in Fig. 18 and there exist low values of f at approximately 0.383 and 0.075 in cases VI, VII, III and IV. Besides, this figure also describes that the exergy destruction efficiency profiles are ranked by VI < Ⅱ < IV < I < VII at the current density of 0.95 A/cm2.

4.4. Application potential and limits As is well-known, the system efficiency of the TED highly depends on the property of semiconductor materials, which can be quantitatively characterized by a dimensionless figure of merit ZT. In recent years, exploring high-performance thermoelectric materials and novel

4.3. Operating mode of the thermoelectric device As presented in Sections 4.1 and 4.2, the thermodynamic performance and irreversible characteristics of the FC-TEH system have been thoroughly analyzed and compared in terms of these important criterions. For the purpose of further identifying the operating state of the TED in this FC-TEH, the results of the absorbing power, power consumption and hot–cold side temperature difference of the TED with current density are given in detail as demonstrated in Figs. (20)–(22) respectively.

Fig. 16. Effect of thermal loss coefficient on the unit exergy cost for the standalone FC model, FC-TEC model and FC-TEG model respectively. 13

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Fig. 20. Variation of absorbed power of thermoelectric device with current density at various cell temperatures.

Fig. 17. Variation of exergy efficiency with current density for various irreversible processes of the FC-THE system.

Fig. 21. Variation of power input or output with current density at various cell temperatures for the FC-TEC model and FC-TEG model respectively.

Fig. 18. Variation of exergy destruction factor with current density for various irreversible processes of the FC-THE system.

Fig. 22. Variation of temperature differences between both sides of thermoelectric device with current density at various cell temperatures for the FC-TEC model and FC-TEG model respectively.

Fig. 19. Variation of exergy destruction efficiency with current density for various irreversible processes of the FC-THE system.

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the FC-TEC and the FC-TEG models is sensitively conducted with respect to the decision targets such as power output, energy efficiency, exergy efficiency, unit exergy cost. Furthermore, the operating regions of the TEC and TEG models in the FC-TEH system are simultaneously determined to reveal thermoelectric conversion conditions and ensure efficient operation of the TED, though these two models have been severally investigated by some authors to improve their thermodynamic efficiencies. The main significant conclusions are drawn as follows:

material processing methods to attain a higher ZT value are always fascinating though challenging. As reported in Ref. [7], a higher ZT value of 3 can be reached through effective measures under laboratory conditions, which significantly improves the energy conversion efficiency for the TED. Generally, the FC-TEH system is a combined system which mainly includes the FC and the TEDs, thereby its performance is subject to the operation of component TED. To explore and characterize energy conversion potential of the TED, the effect of parameter ZT on the system efficiency should be quantitatively investigated. For the case of UA→∞, the maximum exergy efficiency of this TED in the TEC and TEG models can be expressed respectively as follows:

ψTEC =

T0 − Tc Th − Tc

ψTEG =

Th − Tc Th − T0

1 + ZTm −

Th

a) For the FC-TEH system in the current density range of 0–1.2 A/cm2, it first behaves as the FC-TEC model, then FC-TEG model and finally FC-TEC model again. There exist effective ranges of current density for the FC-TEG model, which are respectively confirmed to be 0.615–0.855 A/cm2, 0.77–1.037 A/cm2, and 0.92–1.17 A/cm2 at TFC = 333 K, 343 K and 353 K. At the operating temperature of 343 K, the maximized power outputs of the FC, FC-TEC and FC-TEG models are respectively 4658.62 W, 4738.75 W and 4525.3 W for the current density of 0.955 A/cm2, 0.95 A/cm2, and 1.037 A/cm2. b) Increasing the current density continuously decreases the energy efficiency for the FC-TEH system from 0.94 to 0.796, 0.94 to 0.836 and 0.94 to 0.856 for the various operating temperatures respectively. Opposite to the energy efficiency, the exergy efficiency first shifts to augment and then reduces for the FC-TEH system with a rise of current density. The minimum unit exergy costs are approximately 0.33 $/kWh, 0.311 $/kWh, and 0.3 $/kWh at TFC = 333 K, 343 K and 353 K respectively, which are increased by 4.1%, 3.67% and 5.26% compared to that of single FC (0.317 $/kWh, 0.3 $/kWh and 0.285 $/kWh). c) These exist maximum values of power outputs at about 46.23 W and 4744.78 W when the numbers of thermoelectric devices are respectively 267 and 216 for the current density of 1.037 A/cm2, and 0.95 A/cm2. For the case of i = 1.037 A/cm2, the power output for the FC-TEH system is lower than that for the FC at the n range of 100–200, after that, the FC-TEH’s greater. At the n range of 100–300, the exergy efficiency ranges are about 0.207–0.325 and 0.325–0.331 respectively for i = 1.037 A/cm2. d) The power output for the single FC model is higher than the FC-TEC model’s case within certain limit of ζ and afterwards the power output for the FC model is lower than that for the FC-TEG model. Augmenting the thermal loss coefficient from 0 to 30% causes a linear lessening in the energy efficiency of the FC-TEH system from 0.999 to 0.809 and from 0.999 to 0.823 for i = 1.037 A/cm2 and 0.95 A/cm2 respectively. e) The values of exergy efficiency for the cases IV, VII, II and Ⅴ are respectively 0.37, 0.371, 0.409, and 0.413, which are increased by

Tc

1 + ZTm + 1

(63a)

1 + ZTm − 1 1 + ZTm +

Tc

Th

(63b)

Fig. 23 presents the variation of exergy efficiency with the change of ZT from 0.73 to 3 for the various hot and cold end temperature differences (ΔT = 10 K, 20 K and 30 K). In this simulation, the hot side temperature of the TED in the TEC model is set as the ambient temperature and it is reversed for the TEG model’s case. According to results depicted in Fig. 23, as ZT increases continuously from 0.73 to 3, the exergy efficiencies in these two models are monotonously increased from 0.12 to 0.32 and from 0.14 to 0.34 with the change rates of 166.67% and 142.86% respectively at ΔT = 10 K. Besides, opposite to the TEC model, the growth of hot and cold side temperature difference has a positive influence on the performance improvement of the TEG model. One notable phenomenon indicated from Eqs. (63a) and (63b) is that when the ZT value tends to be infinite, the exergy efficiency value approaches 1. This is due to the fact that increasing ZT would diminish the internal irreversibility, further resulting in the reduction in the TED exergy losses. Except for the application barrier caused from the thermoelectric materials, technical limitations in the system components being considered, including costs and effectiveness of the system materials and production approaches, could be another conclusive obstacle for its widely commercialized development [34]. In this regard, the variation of the exergy destruction of the FC-TEH system operating in the TEC and TEG models for various operating temperature are investigated as described in Fig. 24. Referring to this figure, one can find that growing current density increases the exergy destruction persistently and the minimum exergy destruction values can be obtained as 740.01 W, 1137.6 W and 1591.36 W. This means that for the entire range of current density from 0 to 1.2 A/cm2, these irreversible losses could be absolutely unavoidable due to technical limitations mentioned above. From the viewpoint of the entropy generation, the minimum exergy destruction represents the product of the environment temperature and the minimum entropy generation, which is indispensable within this hybrid system under this condition. Inversely, the avoidable exergy losses can be diminished properly by optimizing system design and operating parameters that have the potential for improving system efficiency. Based on this, it is advantageous for the overall system to reduce the unavoidable exergy destruction as little as possible. 5. Conclusions This study has implemented a comprehensive thermodynamic performance analysis of the fuel cell-thermoelectric hybrid (FC-TEH) system able to identify the TEC and TEG models simultaneously and exploit the energy conversion potential of the electrochemical and thermoelectric coupling processes. Irreversible characteristics and exergoeconomic performance of the hybrid system are thoroughly analyzed through integrating finite time thermodynamics and thermodynamic economics. Following that, a parametric comparison between

Fig. 23. Variation of exergy efficiency with dimensionless figure of merit ZT for the TEC model and TEG model respectively. 15

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influence the work reported in this paper. Acknowledgements Present research has been financially supported by the China National Key R&D Program (Grant No. 2018YFB0904200, Grant No. 2018YFC0705201), National Natural Science Foundation of China (Grant No. 51778504, Grant No. U1867221, Grant No. 51208192), Joint Zhuzhou-Hunan Provincial Natural Science Foundation (Grant No. 2018JJ4064), and National Defense Research Funds for the Central Universities (Grant No. 2042018gf0031, Wuhan University). References [1] Priya K, Sathishkumar K, Rajasekar N. A comprehensive review on parameter estimation techniques for Proton Exchange Membrane fuel cell modelling. Renew Sustain Energy Rev 2018;93:121–44. [2] Daud WRW, Rosli RE, Majlan EH, Hamid SAA, Mohamed R, Husaini T. PEM fuel cell system control: a review. Renew Energy 2017;113:620–38. [3] Islam MR, Shabani B, Rosengarten G. Nanofluids to improve the performance of PEM fuel cell cooling systems: a theoretical approach. Appl Energy 2016;178:660–71. [4] Elmer T, Worall M, Wu S, Riffat SB. Fuel cell technology for domestic built environment applications: state of-the-art review. Renew Sustain Energy Rev 2015;42:913–31. [5] Rahgoshay SM, Ranjbar AA, Ramiar A, Alizadeh E. Thermal investigation of a PEM fuel cell with cooling flow field. Energy 2017;134:61–73. [6] Cai Y, Liu D, Zhao F-Y, Tang J-F. Performance analysis and assessment of thermoelectric micro cooler for electronic devices. Energy Convers Manage 2016;124:203–11. [7] Cai Y, Wang Y, Liu D, Zhao F-Y. Thermoelectric cooling technology applied in the field of electronic devices: updated review on the parametric investigations and model developments. Appl Therm Eng 2019;148:238–55. [8] Kwan TH, Zhang Y, Yao Q. A coupled 3D electrochemical and thermal numerical analysis of the hybrid fuel cell-thermoelectric device system. Int J Hydrogen Energy 2018;43(52):23450–62. [9] Cai Y, Wang W-W, Ding W-T, Yang G-B, Liu D, Zhao F-Y. Entropy generation minimization of thermoelectric systems applied for electronic cooling: parametric investigations and operation optimization. Energy Convers Manage 2019;186:401–14. [10] Liu D, Zhao F-Y, Yang H-X, Tang G-F. Thermoelectric mini cooler coupled with micro thermosiphon for CPU cooling system. Energy 2015;83:29–36. [11] Liu D, Cai Y, Zhao F-Y. Optimal design of thermoelectric cooling system integrated heat pipes for electric devices. Energy 2017;128:403–13. [12] Zhu L, Yu J. Optimization of heat sink of thermoelectric cooler using entropy generation analysis. Int J Therm Sci 2017;118:168–75. [13] Tan H, Fu H, Yu J. Evaluating optimal cooling temperature of a single-stage thermoelectric cooler using thermodynamic second law. Appl Therm Eng 2017;123:845–51. [14] Cai Y, Wang L, Ding W-T, Liu D, Zhao F-Y. Thermal performance of an active thermoelectric ventilation system applied for built space cooling: network model and finite time thermodynamic optimization. Energy 2019;170:915–30. [15] Cai Y, Mei S-J, Liu D, Zhao F-Y, Wang H-Q. Thermoelectric heat recovery units applied in the energy harvest built ventilation: parametric investigation and performance optimization. Energy Convers Manage 2018;171:1163–76. [16] Manikandan S, Kaushik SC. Energy and exergy analysis of an annular thermoelectric cooler. Energy Convers Manage 2015;106:804–14. [17] Nemati A, Nami H, Yari M, Ranjbar F. Effect of geometry and applied currents on the exergy and exergoeconomic performance of a two-stage cascaded thermoelectric cooler. Int J Refrig 2018;85:1–12. [18] Martínez A, Astrain D, Rodríguez A. Experimental and analytical study on thermoelectric self cooling of devices. Energy 2011;36(8):5250–60. [19] Li D, Xuan Y, Li Q, Hong H. Exergy and energy analysis of photovoltaic-thermoelectric hybrid systems. Energy 2017;126:343–51. [20] Luo D, Wang R, Yu W, Sun Z, Meng X. Modelling and simulation study of a converging thermoelectric generator for engine waste heat recovery. Appl Therm Eng 2019;153:837–47. [21] Manikandan S, Kaushik SC. The influence of Thomson effect in the performance optimization of a two stage thermoelectric generator. Energy 2016;100:227–37. [22] Asaadi S, Khalilarya S, Jafarmadar S. A thermodynamic and exergoeconomic numerical study of two-stage annular thermoelectric generator. Appl Therm Eng 2019;156:371–81. [23] Kiflemariam R, Lin C-X. Numerical simulation of integrated liquid cooling and thermoelectric generation for self-cooling of electronic devices. Int J Therm Sci 2015;94:193–203. [24] Chang H, Duan C, Xu X, Pei H, Shu S, Tu Z. Technical performance analysis of a micro-combined cooling, heating and power system based on solar energy and high temperature PEMFC. Int J Hydrogen Energy 2019;44(38):21080–9. [25] Authayanun S, Hacker V. Energy and exergy analyses of a stand-alone HT-PEMFC based trigeneration system for residential applications. Energy Convers Manage 2018;160:230–42.

Fig. 24. Variation of exergy destruction with current density at various cell temperatures for the FC-TEC model and FC-TEG model respectively. The figure shows the definition of specific unavoidable exergy destruction (Exd,un = Exd,min = T0Sgen,min).

1.93%, 2.2%, 12.67% and 13.77% over the stand alone FC model (0.363). The exergy destruction efficiency profiles are ranked by VI < II < IV < I < VII at the current density of 0.95 A/cm2. f) The maximum power output of the TEDs in the TEG model are confirmed to be 50.42 W, 83.37 W and 124.48 W for TFC = 333 K, 343 K, and 353 K respectively while the lower values of power input of the TEDs in the TEC model are zero. The absolute values of the hot and cold side temperature difference (called switched temperature difference) in FC-TEC and TEG models are the same when the current density values leading to PTE = 0 are attained. In this typical current density, the switched temperature differences are obtained as approximately 29.38 K/26.09 K, 37.91 K/33.62 K and 46.29 K/40.99 K respectively for the various operating temperatures of 333 K, 343 K and 353 K. g) As ZT increases continuously from 0.73 to 3, the exergy efficiencies in these two models are monotonously increased from 0.12 to 0.32 and from 0.14 to 0.34 with the change rates of 166.67% and 142.86% respectively at ΔT = 10 K. The minimum exergy destruction values can be obtained as 740.01 W, 1137.6 W and 1591.36 W and these irreversible losses could be absolutely unavoidable due to technical limitations, such as costs and effectiveness of the system materials and production approaches. Overall, this paper has performed a thermodynamic coupling analysis to the FC-TEH system operated in the TEC or TEG mode for an abundant investigation on some decisive parameters which strongly influence the system energy, irreversible and exergoeconomic performance. In the future, extensive efforts will be made to break through the constraints associated with thermoelectric materials, processing approaches as well as costs, further achieving performance improvement in the electrochemical and thermoelectric combining processes. CRediT authorship contribution statement Yang Cai: Conceptualization, Methodology, Software, Writing original draft. Wei-Wei Wang: Data curation. Lei Wang: Investigation. Di Liu: Writing - review & editing. Fu-Yun Zhao: Supervision, Writing review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to 16

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