Equivalent power output and parametric optimum design of a PEM fuel cell-based hybrid system

Electrical Power and Energy Systems 63 (2014) 429–433

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Equivalent power output and parametric optimum design of a PEM fuel cell-based hybrid system Xiaohang Chen, Yuan Wang, Yinghui Zhou ⇑ Department of Physics, Xiamen University, Xiamen 361005, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 13 August 2013 Received in revised form 1 June 2014 Accepted 3 June 2014

Keywords: Hybrid system PEM fuel cell Three-heat-source heat pump Performance evaluation Parametric optimization

a b s t r a c t A new hybrid system composed of a proton exchange membrane (PEM) fuel cell, an irreversible threeheat-source heat pump, and a regenerator is originally established, so that the waste heat produced in the PEM fuel cell may be efficiently utilized. With the help of the current models of PEM fuel cells and three-heat-source cycles, expressions for the equivalent power output and efficiency of the hybrid system are analytically derived. The curves of the equivalent power output and efficiency of the hybrid system varying with the electric current density and the equivalent power output versus equivalent efficiency curves are represented through numerical calculation. The maximum equivalent power output of the hybrid system is determined. Some of the key parameters such as the equivalent efficiency of the hybrid system, the electric current density of the fuel cell, and the heat transfer areas of the three-heat-source heat pump are optimally designed. The performance between the hybrid system and the single PEM fuel cell is compared, and consequently, the advantages of the hybrid system are expounded. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction Because polymer electrolyte membrane (PEM) fuel cell has the advantages [1] of high power density, low operating temperature, short start-up time, and zero-pollution, it has been considered to be the best candidate to replace the combustion power generator in vehicular [2–4] and attracted many researchers to continuously discuss the performance of PEM fuel cells [5–10] and deeply analyze the influence of the flow orientation of water, fuel cell temperature, gas diffusion layer, cathode catalyst layer thickness, and so on, on the performance of PEM fuel cells [11–14]. In the design of improved PEM fuel cells, other aspects have also been investigated such as the combined use of PEM fuel cell/ultracapacitor or PEM fuel cell/ultracapacitor/battery [15–19] and the control strategy for a hybrid fuel cell/capacitor power source in electric vehicles to avoid fuel starvation problems [20–22]. However, the problems how to effectively utilize the waste heat produced in PEM fuel cells are often ignored when one tries to improve the performance of PEM fuel cells. Unlike the solid oxide fuel cell [23–26], the PEM fuel cell does not have a high working temperature so that the waste heat produced in the PEM fuel cell cannot be effectively used to drive a heat engine. Thus, one possible way of the effective uses for the waste heat produced in the

⇑ Corresponding author. E-mail address: [email protected] (Y. Zhou). http://dx.doi.org/10.1016/j.ijepes.2014.06.014 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

PEM fuel cell is to combine a three-heat-source heat pump with the PEM fuel cell, so that the hybrid system may provide not only a power output but also an additional pumping-heat. In the present paper, the novel model of a hybrid system composed of a PEM fuel cell, a three-heat-source heat pump, and a regenerator is originally set up, from which the equivalent power output and efficiency of the hybrid system are analytically derived. The general performance characteristics of the hybrid system are discussed and some main parameters are optimized. Some significant results are obtained.

The hybrid system composed of a PEM fuel cell and a heat pump We consider a new hybrid system composed of a PEM fuel cell, a three-heat-source heat pump, and a regenerator, as shown in Fig. 1, where the PEM fuel cell operated at temperature T acts as the hightemperature heat reservoir of the heat pump for a further utilization of waste heat, qH is the heat flow from the fuel cell to the heat pump, q0 is the rate of heat exchange between the cycle and the environment at temperature T0, qP is the rate of heat pumping from the cycle to the heated space at temperature TP, qL is the heat leak rate directly from the fuel cell to the environment, and Pc is the power output of the fuel cell. The role of the regenerator in the hybrid system is to preheat the incoming fuel and air by means of relative high-temperature exhaust water in the PEM fuel cell so that the export temperature of the inlet reactants is ensured

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T

Table 1 Parameters used in the model of a PEM fuel cell.

Pc

PEM fuel cell qH

Regenerator

Heat pump

qP TP

qL

q0

T0

Fig. 1. The schematic diagram of a hybrid system composed of a PEM fuel cell, a three-heat-source heat pump, and a regenerator.

to attain the working temperature T of the fuel cell. When the fuel cell works stably, the working temperature T will be kept to be constant. By using such a hybrid system, the waste heat produced in the PEM fuel cell can be efficiently utilized and the performance of the system can be heightened. Below, we will first analyze the performance of the PEM fuel cell and three-heat-source heat pump, respectively, and then synthetically discuss the performance of the hybrid system.

The power output and efficiency of a PEM fuel cell The PEM fuel cell is an electrochemical system that converts the chemical energy of a reaction such as hydrogen and oxygen directly into the electrical and thermal energy. The overall electrochemical reaction is [27,28] H2 ðgasÞ þ 12 O2 ðgasÞ ! H2 OðliquidÞþ heat þ electricity. As described in Refs. [29–33], there always exist some irreversible losses in the PEM fuel cell, which mainly result from three overpotential losses such as the activation overpotential (Vact), ohmic overpotential (Vohm), and concentration overpotential (Vconc). The power output and efficiency of the fuel cell are, respectively, given by [29,30,33]:

" Pc ¼ iAc V 0 

   b # RT i t mem i 2 i ln  i b1 kne F i0 im rmem

ð1Þ

Parameter

Value

Number of electrons, ne Faraday constant, F (C mol1) Membrane thickness, tmem (cm) Universal gas constant, R (J mol1 K1) Charge transfer coefficient of the electrodes, k Limiting current density, im (A cm2) Concentration overpotential constant, b2 Environment temperature, T0 (K) Operating temperature, T (K) Molar enthalpy change, Dh (kJ mol1) Partial pressure of H2, pH2 (atm) Partial pressure of O2, pO2 (atm) Membrane conductivity, rmem (cm X1) Parameter, b1 Exchange current density, i0 (A cm2)

2 96485 0.018 8.314 0.5 2.2 2.0 298 363 284.0 2.305 0.4068 0.05520 0.3298 3.902  108

The coefficient of performance of an irreversible three-heat-source heat pump The heat pump in the hybrid system is operated among three heat sources and may be referred to as the three-heat-source heat pump [34,35]. Such a heat pump is directly driven by heat rather than work [34,36]. It has been proved that when the finite-rate heat transfer between the three-heat-source heat pump and the heat reservoirs and the internal irreversibility [37] of the working fluid in the cycle are considered, the coefficient of performance of the three-heat-source heat pump for a given rate of heat input qH may be expressed as [35,37]:

w¼

 1 1 1 2 IR T P T  T 0 2 xþ x þ ; 2 4 T qH b

 pffiffiffiffi2 1 0 where x ¼ 1  TT0  IR TqP T , b ¼ AU 1 þ IR > 0, IR P 1 is the interHb nal irreversibility of the working fluid in the cycle, U is the heat transfer coefficient between the working substance in the cycle and the three heat reservoirs, A = AH + AP + A0 is the total effective heat transfer area of the heat pump, and AH, AP, and A0 are, respectively, the heat transfer areas between the cycle and the three heat reservoirs at temperatures T, TP, and T0. The three heat transfer areas may be, respectively, determined by:

AH ¼

A T0 pffiffiffiffi ; 1 þ IR T 0 þ ðw  1ÞT

ð4Þ

AP ¼

pffiffiffiffi A IR pffiffiffiffi ; 1 þ IR

ð5Þ

A ðw  1ÞT pffiffiffiffi : 1 þ IR T 0 þ ðw  1ÞT

ð6Þ

and

"

   b # Pc ne F RT i tmem i 2 ; i gc ¼ ¼ ln  i b1 V0  Dh kne F i0 im rmem DH_

ð2Þ

where  pffiffiffiffiffiffiffi V 0 ¼ 1:229  8:5  104 ðT  T 0 Þ þ 4:3085  105 T ln PH2 PO2 , ne is the number of electrons, F is Faraday’s constant, R is the universal gas constant, pH2 and pO2 are the partial pressures of reactants H2 and O2, respectively, DH_ ¼ DhiAc =ðne FÞ, Dh is the molar enthalpy change, k is the charge transfer coefficient of the electrodes, i is the electric current density, Ac is the effective area of electrodes of the PEM fuel cell, i0 is the exchange current density in the electrodes of the PEM fuel cell, tmem is the membrane thickness, rmem is the membrane conductivity, im is the limiting current density, b1 is a parameter depending on pO2 and T , and b2 is a constant. In the following discussion, the material parameters of the PEM fuel cell, such as the membrane conductivity and the charge transfer coefficient of the electrodes, are considered to be some constants which are independent of the working temperature and listed in Table 1. Such a model is ideal, but it is a good approximation to capture the physical properties of the investigating problems.

ð3Þ

and

A0 ¼

When the working substance of the cycle is reversible, IR = 1 and the three-heat-source heat pump is an endoreversible cycle. In such a case, we can obtain [35]:

AP ¼ AH þ A0 ¼ A=2:

ð7Þ

The equivalent power output and efficiency of the hybrid system As shown in Fig. 1, one part of the waste heat produced in the PEM fuel cell is directly released to the environment, which is called the heat leak qL, and the other part is transferred to the three-heat-source heat pump in the hybrid system. The heat leak may be expressed as [37,38]:

qL ¼ K l Al ðT  T 0 Þ;

ð8Þ

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X. Chen et al. / Electrical Power and Energy Systems 63 (2014) 429–433

where Kl is the convective and/or conductive heat leak coefficient and Al is the effective heat transfer area. According to the first law of thermodynamics and Fig. 1, one can derive the rate of heat input from the PEM fuel cell to the three-heat-source heat pump as:

ð9Þ

Substituting Eqs. (2) and (8) into Eq. (9) yields

ð10Þ

where C ¼  KAl Ac lDnhe F . Using Eqs. (3) and (10), we obtain

"

w¼

#12

1 1 2 IR T P T  T0 Xþ X þ ; pffiffiffiffi 2 2 4 Bð1 þ IR Þ T ð1  gc Þi  CðT  T 0 Þ

where X ¼ 1  TT0 

1 p ffiffiffi 2

Bð1þ

IR Þ

IR T P T 0 ð1gc ÞiCðTT 0 Þ

ð11Þ

and B ¼ ðDhAc Þ=ðne FUAÞ.

Using the relation w ¼ qP =qH and Eqs. 1, 2, 10, and (11), we can derive the equivalent power output and efficiency of the hybrid system as:

  T0 P ¼ Pc þ qP 1  TP     Ac Dh T0 ¼ ½ð1  gc Þi  CðT  T 0 Þw igc þ 1  ne F TP

ð12Þ

and

g¼



Pc þ qP 1  TT 0P DH_

  T0 C 1  gc  ðT  T 0 Þ w: ¼ gc þ 1  TP i

i 6 iP :

ð14Þ

It shows that iP is one important parameter of the hybrid system because it determines the upper bound of the optimized current density. When the current density is operated in the optimal region determined by Eq. (14), the equivalent efficiency of the hybrid system will increase as the equivalent power output density is decreased, and vice versa, as shown in Fig. 4 which may be directly generated by using the data in Figs. 2 and 3, where gP is the equivalent efficiency at the maximum equivalent power output density. Fig. 4 shows clearly that the optimally operating region of the equivalent efficiency of the hybrid systems should be determined by g P gP . It indicates that Pmax and gP are also two important parameters of the hybrid systems, where Pmax gives the upper bound of the equivalent power output density, while gP determines the lower bounds of the optimized values of the equivalent efficiency. 0.5

ð13Þ

It should be pointed out that the output products of the PEM fuel cell and three-heat-source heat pump in the hybrid system are, respectively, Pc and qP, as shown in Fig. 1. However, the sum of Pc and qP is unmeaning in the calculative process, because the former is the power output and the latter is the heat flow. Thus, it is necessary to introduce a Carnot efficiency (1  T0/TP) in Eqs. (12) and (13) so that qP(1  T0/TP) is equivalent to a power output. By using such a simple and reasonable conversion method, both Pc and qP(1  T0/TP) can be directly added, and consequently, the overall performance of the hybrid system can be discussed. Using Eqs. 1, 2, 11, 12, and (13), we can evaluate the performance of the hybrid system, optimize the main performance parameters of the hybrid system, and reveal the advantages of the hybrid system. General performance characteristics and parametric optimum designs Eqs. 1, 2, 11, 12, and (13) show clearly that the performance of the hybrid system depends on a series of electrochemical and thermodynamic parameters such as the operating temperature T and working current density i of the PEM fuel cell, the internal irreversibility IR of the heat pump, and the synthesis parameters B and C of the hybrid system. It can be proved that the equivalent power output density and efficiency of the hybrid system are of monotonically decreasing functions of IR, B, and C. For the given values of IR, B, and C, one can discuss the performance of the hybrid system by using Eqs. 1, 2, 10, 11, 12, and (13). Below, numerical calculations are carried out, based on the parameters summarized in Table 1, which are derived from the data available in literature [27–29,39–43]. These parameters are kept constant unless mentioned specifically. Using Eqs. 1, 2, 11, 12, and (13), one can obtain the curves of the equivalent power output density P ¼ P=Ac and efficiency g of the hybrid system varying with the current density, as shown in Figs. 2 and 3, respectively. It is seen from Fig. 2 that there are a maximum

P*max 0.4

P*

2

Ac Dh ½ð1  gc Þi  CðT  T 0 Þ; ne F

P* ( W/cm )

qH ¼

0.3

( Pc *) max

0.2

0.1

ic 0.4

P * c

iP

0.8

1.2

1.6

i ( A/cm ) 2

Fig. 2. The equivalent power output density versus current density curves of the hybrid system, where the dash curve represents the power output density versus current density curve of the PEM fuel cell, iP is the current density at the maximum equivalent power output density P max of the hybrid system, ic is the current density at the maximum power output density ðP c Þmax of the PEM fuel cell, and the parameters B = 0.1, C = 0.001, and IR = 1.05 are chosen.

0.5

0.4

η

qH ¼ DH_  Pc  qL :

equivalent power output density P max for the hybrid system and a corresponding current density iP. It is also seen from Figs. 2 and 3 that in the region of i > iP, the equivalent power output density and efficiency of the hybrid system will decrease as the current density i is increased. Thus, the region of i > iP is not the optimally working region of the hybrid system. Obviously, the optimally working region of the current density is not arbitrary and should be:

0.3

η

0.2

ηc 0.1

0.4

0.8

2

1.2

1.6

i ( A/cm ) Fig. 3. The equivalent efficiency versus current density curves of the hybrid system, where the dash curve represents the efficiency versus current density curve of the PEM fuel cell. The values of relevant parameters are the same as those used in Fig. 2.

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X. Chen et al. / Electrical Power and Energy Systems 63 (2014) 429–433

0.6

0.5

P*max

Aj /A

2

P* ( W/cm )

AP /A

0.5

0.4

0.3

0.4

AH/A 0.3

0.2

0.2

0.1 0.0

0.1

0.2

η P0.3

η

0.4

0.1

0.5

A0 /A 0.4

0.8

iP 1.2

1.6

2

i ( A/cm )

Fig. 4. The equivalent power output density versus equivalent efficiency curve of the hybrid system, where gP is the equivalent efficiency at the maximum equivalent power output density P max . The values of relevant parameters are the same as those used in Fig. 2.

Figs. 2 and 3 clearly show that the performance of the hybrid system is much better than that of a single PEM fuel cell. Using Eqs. (12) and (13), one can easily prove:

  T0 P=Pc ¼ g=gc ¼ 1 þ 1  ½1  gc  CðT  T 0 Þ=iw=gc  r; TP

ð15Þ

which may be conveniently used to plot the r–i curves, as shown in 5. It is seen from 5 and Eq. (14) that r increases with the increase of i, but the current density is not allowed to exceed the upper bound iP of the optimized current density, because the equivalent power output density and efficiency of the hybrid system in the region of i > iP will decrease with the increase of i. When i = iP, the equivalent power output and efficiency of the hybrid system are about 1.46 times of those of a single PEM fuel cell. It shows clearly that using such a hybrid system established above, one may effectively improve the performance of the PEM fuel cell and increase not only the power output but also the efficiency of the system. Using Eqs. 4, 5, 6, and (11), one can plot the Aj =A  i curves, as shown in Fig. 6. It is seen from Eq. (5) or Fig. 6 that AP/A is independent of the current density i. According to Fig. 6 and Eq. (14), one can determine the optimal regions of AH/A and A0/A of the heat pump as:

AH =A 6 ðAH =AÞP

Fig. 6. The Aj =A  i ðj ¼ H; P; 0Þ curves of a three-heat-source heat pump. The values of relevant parameters are the same as those used in Fig. 2.

It should be pointed out that although the model mentioned above is ideal, the results obtained are meaning. The fundamental characteristics of the hybrid system may be revealed. The optimally working regions of some of the key parameters determined by Figs. 2–6 can be used to provide a good guidance for the design and operation of practical hybrid systems. Conclusions We have originally set up the model of the hybrid system composed of a PEM fuel cell, an irreversible three-heat-source heat pump, and a regenerator by utilizing the existing models of PEM fuel cells and three-heat-source cycles. Expressions for the equivalent power output and efficiency of the hybrid system are derived and used to discuss the optimal performance of the hybrid system. The optimally working regions of some of important parameters including the efficiency, current density, and heat transfer areas are determined. It is important to find that not only the power output but also the efficiency of the hybrid system is much better than those of a single PEM fuel cell. The results obtained here may provide some theoretical bases for the optimal design and operation of practical PEM fuel cell/three-heat-source cycle hybrid systems.

ð16Þ Acknowledgments

and

A0 =A P ðA0 =AÞP ;

ð17Þ

r

where (AH/A)P and (A0/A)P are , respectively, the values of (AH/A) and (A0/A) at the maximum equivalent power output of the hybrid system. 3.0 2.8 2.6 2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0

0.4

0.8

i p 1.2

1.6

2

i (A/cm ) Fig. 5. The r–i curve, where r ¼ P=P c ¼ g=gc . The values of relevant parameters are the same as those used in Fig. 2.

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