Ind. Eng. Chem. Res. 2010, 49, 5725–5733
5725
Feasible Input Manipulation for a Nonisothermal Direct Methanol Fuel Cell System W. Wu* and Y.-T. Lin Department of Chemical and Materials Engineering, National Yunlin UniVersity of Science and Technology, Douliou, Yunlin 64002, Taiwan, R.O.C.
Through a semiexperimental dynamic modeling of a direct methanol fuel cell (DMFC) system, transient behaviors on describing cell temperature, methanol crossover, water crossover, and electric power are presented. By different operating manners, the new input determinations are addressed. To investigate the performance of DMFC systems, the presented fuel and exergetic efficiencies for the specific probe of energy utilization are evaluated. Through both steady-state and unsteady state analyses, the manipulation of cathode pressure and anode inlet flow temperature is a feasible approach for temperature regulation, but methanol crossover is restricted. 1. Introduction The direct methanol fuel cell (DMFC) with proton exchange membrane (PEM) as electrolyte and aqueous methanol solution as fuel becomes quite attractive for portable and stationary power applications. The characteristics of this type of fuel cell include low emissions, low operating temperature, and rapid response to changes in load, etc. In fact, there are some drawbacks in the development of DMFC systems including the low kinetics of methanol electro-oxidation on the anode catalyst, crossover of unreacted methanol from the anode to the cathode, water flooding at the cathode, and thermal management. To study cell characteristics and operating manners, the mathematical modeling of a class of DMFC systems has been significantly developed. Through experiments and model-based analysis, Zhou et al.1 and Sundmacher et al.2 first used a set of differential-algebraic equations (DAEs) to describe an isothermal dynamic model. Moreover, Schultz et al.3 showed an extension of mathematical modeling of DMFCs with different anode and cathode reaction mechanisms, and Oliveiraa et al.4 provided a comparative study on the DMFC model by using analytical, semiempirical, and mechanistic models. Recently, two-dimensional dynamic modeling and numerical simulation for a specified DMFC system was verified via experiments.5 For most DMFC systems, the performance enhancement usually depended on high methanol feed concentration. Chen and Zhao6 showed that the temperature of DMFC strongly depended on the thermal effect owing to an exothermic reaction between permeated methanol and oxygen at the cathode. Wang et al.7 found that the cell dynamic response was significantly changed by the cell temperature and oxygen flow rate. By nonisothermal dynamic modeling of DMFC, Ko et al.8 indicated that anode feed concentration had a significant larger impact on methanol crossover, temperature, and cell voltage than anode and cathode flow rates. In addition, Jeong et al.9 used the dynamic optimization algorithm to minimize methanol consumption of an isothermal DMFC system, and Xu et al.10 used the numerical algorithm to decide the optimal feed concentration for the DMFC model in regard to the highest power density output. Moreover, they showed that the cell performance was usually evaluated by utilizing the dynamic modeling and numerical simulation. * To whom correspondence should be addressed. Fax: +886-55312071. E-mail:
[email protected].
Recently, the exergy analysis has become a popular approach to identifying process efficiency and accounts for impact on the surroundings.11 Based on the second law of thermodynamics, exergy or available energy is the fraction of energy that can do useful work. It is a measure to evaluate chemical, thermal, and electrical energy such that the quality of exergy loss can be found, providing a clue for improving the conversion system. Cownden et al.12 first used the exergy analysis procedure to conduct the irreversibilities of each utility of fuel cell systems and construct the rate of exergy destruction; Wang and Wang13 used exergy analysis to determine the efficiencies, exergy loss, and performance for proton exchange membrane fuel cells (PEMFC); Rashidi et al.14 studied the effect of pressure, temperature, and current density for a hybrid molten carbonate fuel cell (MCFC) to evaluate the prescribed exergy destruction of this power system; and Li et al.15 demonstrated the exergetic efficiencies for the steady-state DMFC model under consideration of methanol crossover. Since methanol transport across the membrane of DMFC may cause depolarization losses at the cathode and conversion losses in terms of lost fuel leading to poor performance, the methanol solution at the anode is usually very dilute. However, a large amount of water may increase the risk of water flooding at the cathode and degrade the stability of the overall system operation. In general, permeated methanol catalytically oxidized at the cathode could release a lot of heat to warm up the cell to the working temperature. Since operating conditions of methanol and water crossover may directly affect overall system performance, the feasible input manipulation is an alternative approach on the performance improvement of fuel cell systems. In this article, the state-space representation of a DMFC system can describe its dynamic characteristics and establish the specific input-to-output relationship. Through both steady-state and unsteady state analyses, the preliminary operating manner is based on how to reduce methanol crossover or water crossover. By exergy analysis, the defined fuel and exergetic efficiencies provide a new assessment technique on energy utilization. Moreover, the feasible input manipulation is developed and successfully verified by the simulation tests of an illustrative DMFC model.
10.1021/ie901829a 2010 American Chemical Society Published on Web 04/29/2010
5726
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
Figure 1. The run schematic of the DMFC system.
2. Nonlinear Dynamic Model The dynamic modeling of the DMFC system has been well developed recently; the one-dimensional, rigorous, mathematical model is formulated in the appendix (Figure 1). Referring to the nonlinear behavior and specific characteristics of the nonisothermal DMFC system, the systematic approach by using the state-space representation is introduced, ξ˙ ) f(ξ, Θ, χ) 0 ) g(ξ, Θ) y ) h(ξ)
(1)
out CL CL T where ξ ) [xout a,CH3OH,xa,CO2,xa,CH3OH,xa,CO2,ηa,ηc,T] represents the state vector of the compositions at the anode compartment and the anode catalyst layer, charges at the anode and cathode, and in T cell temperature. χ ) [Pc,xina,CH3OH,n˙in ˙ in a ,n c ,Ta ] represents the input vector of the system, and the output vector function is specified Xover Xover T by h ) [nCH 3OH,nH2O ,Vcell,T] . The unknown surface fraction Θ ) [Θ1, · · · ,Θ5]T is determined from the nonlinear algebra equation by eqs A12-A16. Remark 1. Under certain conditions and assumptions,2 the illustrative DMFC model is treated as a class of nonlinear differential-algebraic-equation (DAE) systems.16 Its representation can describe the main characteristics of the DMFC system and specify the input-to-output relationship. The mathematical equations of DMFC shown in the appendix section are mainly referenced by Sundmacher et al.2 and Ko et al.8 Remark 2. If the output functions are related to stack performance, efficiency, and lifetime of DMFC, the corresponding multiple input manipulations need to be specified. With regards to the parameter constraints in eq 1, the original DAE system is reduced as a nonlinear ordinary-differential-equation (ODE) system where the analytic solution of Θ should exist, that is, the nonlinear inversion of Θ ) g-1(ξ) can hold. If model errors and parameter uncertainties are considered, the original DAE system would be described by a singularly perturbed system17 as follows:
ˆ , χ) ξ˙ ) f(ξ, Θ ˙ ˆ ) g˜(ξ, Θ ˆ) εΘ
(2)
y ) h(ξ) where ε > 0 is a small parameter. The nonlinear function g˜ involves the uncertain dynamics, parametric uncertainties, and the slight variation of system parameters. 3. System Analysis The steady-state and unsteady-state analyses of dynamic characteristics of DMFC are turned out to determine feasible
Figure 2. Steady-state profiles with different methanol concentration: (a) cell voltage vs current density; (b) power density vs current density; (c) methanol crossover vs current density.
operating manners. Through changeable input manipulation, the negative effects of methanol crossover and water flooding at the cathode (water crossover) are addressed as follows. 3.1. Steady-State Analysis. Under the steady-state condition, eq 1 is reduced into the algebraic equation 0 ) F(ξss, Θ, χ) y ) h(ξss)
(3)
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
5727
Figure 4. Unsteady-state profiles: (a) methanol feed concentration vs cell temperature; (b) methanol feed concentration vs water crossover.
Figure 3. Steady-state profiles with different cell temperature: (a) cell voltage vs current density; (b) power density vs current density.
where F ) [f,g]T is a full vector function. On the basis of the given input values of χ, the steady-state profile of the output function is obtained though the numerical approach. If the methanol feed concentration increases, the voltage drop, shown in Figure 2a, is improved at high current density and the corresponding power density, shown in Figure 2b, appears at the maximum power. However, the methanol crossover, shown in Figure 2c, becomes obvious when the current density increases. In addition, Figure 3a shows that the voltage drop can be improved by high cell temperature, and Figure 3b shows that the higher cell temperature induces the higher power output at the same current density. Remark 3. By steady-state analysis, methanol feed concentration strongly affect cell temperature, methanol crossover, and power density. In general, the heat radiation produced by permeated methanol is used to warm up the power system, but attention should be paid to the operating temperature of DMFC because of the different rates of methanol crossover. 3.2. Unsteady-State Analysis. The steady-state analysis shows that the operating temperature of the DMFC system could affect power or fuel (methanol) efficiency, and output variables of DMFC may be controlled by manipulating input variables. Therefore, the interaction between input and output, that is, χ vs y, needs to be investigated by following the unsteady-state analysis.
Open-Loop Tests. The selected input variables including in methanol feed concentration (xa,CH 3OH), anode inlet flow rate in (n˙a ), cathode inlet flow rate (n˙in c ), and cathode pressure (Pc) are individually manipulated, respectively. The open-loop simulation results are as follows. (i) Figure 4 panels a and b show that methanol feed concentration can individually regulate cell temperature and water crossover, respectively. However, water crossover cannot be reduced by increasing methanol feed concentration. (ii) Figure 5 panels a and b show that anode inlet flow rate can individually regulate cell temperature and water crossover, respectively. Similarly, water crossover cannot be reduced by increasing anode inlet flow rate. (iii) Figure 6 panels a and b show that cathode inlet flow rate can individually regulate cell temperature and water crossover, respectively. Water crossover can be reduced by increasing cathode inlet flow rate, but the large air flow would reduce the cell temperature. (iv) Figure 7 panels a and b show that cathode pressure can individually regulate cell voltage and methanol crossover, respectively. Since the cathode pressure may change the diffusion rate of fuel in the membrane,18 methanol crossover can be reduced by increasing cathode pressure, which has been studied through the experimental validation.19,20 Remark 4. Since water crossover may induce water flooding, and methanol crossover may cause voltage drop and fuel loss, both water and methanol crossover fluxes can be treated as physical constraints of the DMFC system operation. By in unsteady-state analysis, three manipulated inputs, xa,CH ˙ in a, 3OH, n
5728
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
Figure 5. Unsteady-state profiles: (a) anode inlet flow rate vs cell temperature; (b) anode inlet flow rate vs water crossover. in
and n˙c , cannot improve the power performance due to undesired water or methanol crossover flux. Through open-loop tests, the feasible input condition is verified by their transient behaviors under step changes of prescribed inputs. Similarly, the stability condition in regard to output multiplicity can be examined by the algebraic equation. In our approach, the input-output linearization technique21 is used to ensure the unique input function according to the set point of the output j,yjsp χi ) Rj(ξss, Θ, ysp j )
(4)
in in Xover Xover where i ∈ [Pc,xa,CH ˙ in ˙ in a ,n c ,Ta ] and j ∈ [nCH3OH,nH2O ,Vcell,T]. 3OH,n To rapidly eliminate inadequate manipulating variables, the nonlinear inverse function Rj should be obtained by the following algebraic equation
Lrf hj[ξss, Θ, χi] ) 0
(5)
where Lfhj ≡ ∂hj/∂ξ f and Lfrhj ≡ ∂/∂ξ (Lfr-1hj). The system relative degree r (g1) is determined by the selected input i and the output j. Remark 5. The input-output linearization technique is implemented to develop the static feedforward control if the nonlinear inversion of eq 5 exists. For instance, the steady-state input condition of cathode inlet flow rate with respect to prescribed water crossover of a DMFC system is determined [ξss,Θ,χn˙inc ] ) 0. Similarly, by the nonlinear inversion of L2f hnHXover 2O if the nonlinear inversion of LfhnCH Xover [ξss,Θ,χP ] ) 0 can hold, c OH 3
Figure 6. Unsteady-state profiles: (a) cathode inlet flow rate vs cell temperature; (b) cathode inlet flow rate vs water crossover.
then the steady-state response of methanol crossover depends on the manipulation of cathode pressure with nonlinear function sp χPc ) RnCH Xover (ξss,Θ,ynXover ). The steady-state responses of water CH3OH 3OH and methanol crossover have been depicted in Figures 6b and 7b, respectively. By virtue of above steady-state and unsteady-state analyses, the manipulation of cathode pressure can suppress methanol crossover and improve the power density. However, these analyses on energy efficiency are unclear. A new perspective in energy utilization would be carried out by the following approach. 4. Energy and Exergy To address the feasible operating manner from the overall energy concept, the specification of fuel and exergy efficiencies is expected to become an effective approach. Under the steadystate condition and no heat loss assumption, the temperature equation by eq A20 is rewritten as
∑ n˙ x (h - h ∑h r A - ∑h
0)
in in a a,i
in a,i
out a,i )
+
i
out s a,i a
i
in in in c xc,j(hc,j
- hout c,j ) -
j
out s c,j rcA
j
∑ n˙
Xover Xover - QCH n As - icellVcell (6) 3OH CH3OH
Notably, eq 6 represents the overall energy balance for a DMFC system. Whereas exergy is defined as the maximum work that can be extracted from the substance as it undergoes a reversible
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
εout c,j )
∫
T
298.15
(
Cjp,c(T) 1 -
5729
)
To dT + εjch - δj(T - 298.15) + T RTo ln xin c,j (9b)
where εich and εjch represent the chemical exergy and δi and δj represent the temperature correction coefficient. Thermodynamic parameters have been shown in Li et al.15 and Ishihara et al.22 Because the permeated methanol may poison the catalyst on the cathode, the excessive oxygen is used to burn permeated methanol. Moreover, the thermal exergy generated from the methanol crossover oxidation is represented by
(
Xover εtherm ) 1-
)
To Xover Xover s Q n A T CH3OH CH3OH
(10)
Remark 6. According to the second law of thermodynamics and the definition of exergy, the physical exergy is expressed by ε1 ) ∆H - To∆S
(11)
In a reversible process, ∆S ) ∆H/T. If the chemical exergy derives from a compositional imbalance between a substance and its environment, then a material flow including chemical exergy can be written as ε2 ) ∆H - To∆S -
∑ ∆µ x
(12)
i i
i
Figure 7. Unsteady-state profiles: (a) cathode pressure vs cell voltage; (b) cathode pressure vs methanol crossover.
process from a given state to the environment state, exergy should take into account the quality of the energy as well as the quantity due to a consideration of entropy. Basically, entropy increases in the irreversible process that causes exergy destruction, so exergy destruction εdest in regard to the exergy balance is expressed by εdest )
∑ n˙
in in in a xa,i(εa,i
- εout a,i ) +
i
∑ n˙
in in in c xc,j(εc,j
- εout c,j ) -
ηtotal )
j
Xover εtherm
- Welec (7)
where the electric power Welec ) icellVcell, the inlet molar flow exergy of each species at the anode and cathode are shown by εin a,i
)
εin c,j )
∫
T
Ci (T) 298.15 K p,a
∫
T
298.15 K
)
(8a)
(
To dT T
)
(8b)
Cjp,c(T) 1 -
and the outlet molar flow exergy of each species at the anode and cathode are expressed as εout a,i )
∫
T
(
Ci (T) 1 298.15 p,a
)
Xover εtherm + Welec
∑ n˙
in in in a xa,i(εa,i
To dT + εich - δi(T - 298.15) + T RTo ln xin a,i (9a)
- εout a,i ) +
i
∑ n˙
in in in c xc,j(εc,j
- εout c,j )
(13)
j
Second, the fuel efficiency is formulated by ηfuel )
(
To 1dT T
where µi is the chemical potential of species i in the flow at normal temperature To and atmospheric pressure Po. The exergy equations for outlet material flows at the anode and cathode, respectively, are shown in eqs 9a and 9b. On the basis of the prescribed exergy balance, the irreversibility by chemical reaction, heat transfer, pressure drop, and mixing proceedure in the electrochemical process would cause exergy destruction. Notably, exergy balance of the DMFC system is depicted in Figure 8. To define the DMFC system efficiencies, Figure 8 indicates that the chemical exergy of fuel is mainly converted into electricity and heat, a portion of thermal exergy including methanol crossover oxidation is used to heat the stack, and the other portion becomes the exergy loss. First, the total exergetic efficiency is described by
iell Xover iell + 6FnCH 3OH
(14)
Remark 7. Exergy analysis involves different effects between fuel, electricity, heat, and exergy loss. Both fuel and exergetic efficiencies provide a new assessment manner on energy utilization. The total exergetic efficiency is treated as a measure of exergy destruction, and the fuel efficiency is described as a measure between the electric power and power loss. In the above formulations, fuel efficiency is directly affected by methanol crossover but total exergetic efficiency indirectly depends on the thermal exergy due to methanol crossover oxidation and operating temperature. In our opinion, the large thermal exergy or high operating temperature may enhance the total exergetic
5730
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
Figure 8. Exergy balance of the DMFC system.
Figure 10. Unsteady-state profiles: (a) load demand vs methanol crossover; (b) fuel efficiency vs total efficiency.
Figure 9. Steady-state profiles: (a) total efficiency with different cathode pressure; (b) total efficiency with different inlet flow temperature at the anode.
efficiency, but it induces the undesired fuel efficiency. It is a kind of dilemma approach between process operation and energy utilization. By previous analysis, the manipulation of cathode pressure can cope with methanol crossover. Figure 9a shows that the total (exergetic) efficiency of the DMFC system is improved by the increase of cathode pressure. To enhance the same total efficiency, Figure 9b shows that the manipulation of inlet flow temperature at the anode is an alternative approach. From the
aspect of energy utilization, both Pc and Tin a are denoted as feasible inputs. If the current load is increased stepwise at constant methanol feed concentration ()125 mol/m3) and the methanol crossover is decreased stepwise under feasible input manipulation, as shown in Figure 10a, the fuel efficiency can be increased stepwise and the total exergetic efficiency can be kept above 50% (as seen in Figure 10b). Similarly, when responses of methanol crossover at constant methanol inlet concentrations, 250 mol/m3 and 500 mol/m3, shown in Figure 11a, are given, the corresponding fuel efficiencies, shown in Figure 11b, are strongly affected by methanol crossover at different methanol feed concentration and the corresponding total efficiencies, shown in Figures 11c, are kept at approximately 50%. The feasible input manipulation is successfully verified by steady-state analysis and unsteady-state analysis. The manipulation of cathode pressure can suppress the effect of methanol crossover but the total exergetic efficiency can be kept at the desired percentage under the manipulation of the inlet flow temperature at the anode. Obviously, the cell temperature
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
5731
basis of the constrained operation for methanol crossover, fuel efficiency can be improved. Following the presented operation analysis, the control technique becomes a further effective approach to solve the problem between energy density and fuel efficiency while a feasible input manipulation is taken in consideration. Appendix As shown in Figure 1, methanol/water as a fuel and air is supplied to a DMFC to generate electricity where heat and water are generated as byproduct. The electrochemical reaction scheme including methanol is oxidized to carbon dioxide at the anode, and oxygen is reduced to water at the cathode: anode: CH3OH + H2O f CO2 + 6H+ + 6e-
(A1)
3 6H+ + O2 + 6e- f 3H2O 2
(A2)
3 CH3OH + O2 f CO2 + 2H2O 2
(A3)
cathode:
overall:
According to the experimental-based DMFC modeling,2 the isothermal dynamics are composed of the following: (i) mass balances for the anode compartment out dxa,CH 3OH
dt
1 in out ) (xa,CH - xa,CH )3OH 3OH τ kLSAS out CL (x - xa,CH ) (A4) 3OH Va a,CH3OH
out dxa,CO 2
Figure 11. Unsteady-state profiles with different methanol feed concentration: (a) methanol crossover; (b) fuel efficiency; (c) total exergetic efficiency.
regulation by additional input variable could take over the thermal effect by burning permeated methanol and meanwhile release the duty of cathode pressure on high pressure operation. Consequently, both fuel and total exergetic efficiencies could be improved by both input manipulations.
dt
1 in kLSAS out out CL ) (xa,CO - xa,CO )(x - xa,CO ) 2 2 2 τ Va a,CO2
(A5) (ii) mass balances for the anode catalyst layer CL dxa,CH 3OH
dt
)
kLSAS out AS Xover CL (xa,CH3OH - xa,CH )nCH3OH CL 3OH Va CaVCL a AS ra (A6) CaVCL a
5. Conclusions In this article, the input-to-output effect of a dynamic DMFC system is completely investigated via steady-state and unsteadystate analyses. On the basis of a well-developed mathematical model for a nonisothermal DMFC, the main dynamics of cell temperature and power output can be straightforwardly obtained. According to specific constraints on methanol crossover and water flooding, the systematic analysis provides an alternative algorithm to determine feasible inputs. Exergy analysis shows that the fuel efficiency is affected by methanol crossover, and the exergetic efficiency depends on the working temperature of fuel cell. Through analyses and observations, both cathode pressure and inlet flow temperature at the anode are recommended as feasible input manipulations. It is concluded that if the cell temperature could be regulated, then the total exergetic efficiency can be kept around the desired percentage. On the
CL dxa,CO 2
dt
)
kLSAS out AS CL (x x ) + ra a,CO a,CO 2 2 VCL CaVCL a a
(A7)
(iii) charge balances anode: dηa 1 ) (icell - 6Fra) dt ca
(A8)
cathode: dηc 1 Xover ) (-icell - 6F(rc + nCH )) (A9) 3OH dt cc In eqs A6-A9, ra and rb represent the rate equations of primary electrode reactions at the anode and cathode, respectively, described by
5732
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010
( ){
ra ) ka exp
βaF η RT a
( ){ βcF η RT c
rc ) kc exp
}
1 F exp - ηa Θ2 K1 RT (A10)
CL CaΘ13xCH 3OH
(
1 - exp -
(
)
)(
)}
PO2 F ηc RT Pc
3/2
(A11)
where the surface fractions Θ1 and Θ2 are determined by solving the following algebraic equations: Θ4 -
Θ5 F exp - ηa ) 0 K2 RT
(
Θ2Θ52 -
Θ2Θ5 -
)
(A12)
Θ2Θ12Θ42 )0 K3
CL ΘΘ CaxCO 2 1 4
K4
(A13)
)0
(A14) (A15)
Θ4 + Θ5 ) 1
(A16)
Xover In eqs A6 and A9, the methanol crossover nCH 3OH is expressed by
Xover nCH ) 3OH
Pe exp(Pe) exp(Pe) - 1
d
(A17)
where the Peclet number Pe is defined as Pe ≡
υdM M DCH 3OH
(A18)
M and the diffusion coefficient of methanol DMeOH is described as a function of cell temperature,
( (
M M DCH ) DCH exp 3OH 3OH,ref
∆E 1 1 R Tref T
))
(A19)
Moreover, the overall cell voltage is described by Vcell ) Vocell - ηa + ηc -
dM icell κM
(A20)
In eqs A4-A7, the effective mass transfer coefficient kLS is affected by the methanol feed concentration. In eq A18, the convective flow velocity υ is affected by the cell current density, temperature, and cathode pressure. The diffusion coefficient of M methanol DMeOH is affected by the cell temperature. Therefore, the reaction rate and methanol crossover strongly depend on cell temperature. The nonisothermal model for DMFC by overall energy balance is shown as follows: (iv) energy balance
∑x C + n ∑x ) + ∑ n˙ x (h - h
dT ) (mcellCp,cell + na dt (
∑ n˙ x
in in in a a,i(ha,i
i
out i a,i p,a
i
in in c c,j
out c,j rcA
j
out j -1 c,j Cp,c)
j
out - ha,i
∑h
c
j s
in c,j
( )
nHXover ) (-5.27 + 0.027T) 2O
Θ1 + Θ2 + Θ3 ) 1
M CaDCH 3OH CL xa,CH3OH M
where i ) CH3OH, H2O(l), CO2, j ) O2, N2, H2O(g). Assuming that the compositions at the cathode are assumed to be constants, the inlet flow rate at the anode/cathode is also assumed to be equal to the corresponding outlet flow rate, and the system is Xover adiabatic. In eq A20, the methanol oxidation heat, QCH 3OH, is affected by cell temperature, the heat capacities of anode and cathode, Cip,a and Cip,c, are affected by cell temperature, the in enthalpies of the inlet stream at the anode/cathode, hin a,i and hc,i, are affected by the cell temperature and the inlet flow temperature at the anode/cathode, respectively, and the enthalpies of out the output stream at the anode/cathode, hout a,i and hc,j , are also affected by cell temperature. All detailed formulations of thermodynamic functions have been shown in Ko et al.8 Regarding water crossover through membrane, the experimental-based model is expressed by
out c,j )
-
×
∑h
s out a,i raA
-
i
Xover Xover - QCH n As - icellVcell) (A21) 3OH CH3OH
icell kp Fwater - (Pc - Pa) F µ Mwater (A22)
All thermodynamic parameters in eqs A4-A22 have been found in the literature.2,8,22,23 Notably, the presented dynamic model is a reduced, one-dimensional system under prescribed assumptions. For some DMFC modeling, some kinetic parameters and heat and mass transfers are still unclear at this time. Notations As ) cross-sectional active electrode area, 9 × 10-4 m2 Ca ) total concentration at the anode, mol/m3 ca ) double layer capacity of anode, F/m2 Cc ) total concentration at the cathode, mol/m3 cc ) double layer capacity of cathode, F/m2 Cp,cell ) heat capacity of cell, J/(kg/K) Cip,a ) heat capacity of outlet stream of component i from the anode, J/(mol/K) j Cp,c ) heat capacity of outlet stream of component j from the cathode, J/(mol/K) dM ) PEM thickness, 200 µm M DM CH3OH ) diffusion coefficient of methanol in membrane (DCH3OH,ref -10 2 ) 2.9 × 10 ), m /s F ) Faraday constant, 96485 C/mol in ha,i ) inlet enthalpy of component i into the anode, J/mol in hc,j ) inlet enthalpy of component j into the cathode, J/mol out ha,i ) outlet enthalpy of component i from the anode, J/mol out hc,j ) outlet enthalpy of component j from the cathode, J/mol ka ) rate constant of the reaction at the anode compartment, 1.0 × 10-7 mol/(m2/s) kc ) rate constant of the reaction at the cathode compartment, 7.0 × 10-5 mol/(m2/s) LS k ) mass transfer coefficient, m/s kp ) hydraulic permeability of membrane, 1.57 × 10-18 m2 Ki ) equilibrium constant of reaction i (K1 ) 1, K2 ) 0.02, K3 ) 1, K4 ) 0.02) Mwater ) molecular weight of water, kg/mol mcell ) cell mass, kg na,nc ) mole amount at the anode/cathode, mol Pa ) anode pressure, 0.1 MPa Pc ) cathode pressure, 0.3 MPa PO2 ) partial pressure of oxygen, Pa Xover QCH 3OH ) reaction heat due to methanol crossover, J/mol R ) universal gas constant, 8.314 J/(mol/K) Va ) volume of the anode compartment, 2.0 × 10-6 m3 VaCL ) volume of the anode catalyst layer, m3 o Vcell ) open circuit voltage of cell, V
Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010 xi ) mole fraction of component i in catalyst layer in xa,i ) inlet mole fraction of component i into the anode in xc,j ) inlet mole fraction of component j into the cathode out xa,i ) outlet mole fraction of component i from the anode out xc,j ) outlet mole fraction of component j from the cathode CL
Greek Symbols ηa,ηb ) anode/cathode electrode overpotential, V τ ) mean residence time, s Θi ) surface fraction covered by component i κM ) conductivity of membrane, 17 Ω-1 m-1 υ ) convective flow velocity in membrane, m/s µ ) pore fluid viscosity in membrane, 3.353 × 10-4 kg/(m/s) Fwater ) water density, kg/m3 βa,βc ) charge transfer coefficient at the anode/cathode
Literature Cited (1) Zhou, Su.; Schultz, T.; Peglow, M.; Sundmacher, K. Analysis of the nonlinear dynamics of a direct methanol fuel cell. Phys. Chem. Chem. Phys. 2001, 3, 347. (2) Sundmacher, K.; Schultz, T.; Zhou, S.; Scott, K.; Ginkel, M.; Gilles, E. D. Dynamics of the direct methanol fuel cell (DMFC): Experiments and model-based analysis. Chem. Eng. Sci. 2001, 56, 333. (3) Schultz, T.; Krewer, U.; Sundmacher, K. Impact of electrode kinetics on the dynamic response of a DMFC to change of methanol feed concentration. J. Power Sources 2007, 165, 138. (4) Oliveiraa, V. B.; Falca˜o, D. S.; Rangel, C. M.; Pinto, A. M. F. R. A comparative study of approaches to direct methanol fuel cells modeling. Int. J. Hydrogen Energy 2007, 32, 415. (5) Basri, S.; Kamarudin, S. K.; Daud, W. R. W.; Yaakub, Z.; Ahmad, M. M.; Hashim, N.; Hasran, U. A. Unsteady-state modelling for a passive liquid-feed DMFC. Int. J. Hydrogen Energy 2009, 34, 5759. (6) Chen, R.; Zhao, T. S. Mathematical modeling of a passive-feed DMFC with heat transfer effect. J. Power Sources 2005, 152, 122. (7) Wang, M.; Guo, H.; Ma, C. Dynamic characteristics of a direct methanol fuel cell. J. Fuel Cell Sci. Technol. 2006, 3, 202. (8) Ko, D. H.; Lee, M. J.; Jang, W. H.; Krewer, U. Non-isothermal dynamic modelling and optimization of a direct methanol fuel cell. J. Power Sources 2008, 180, 71.
5733
(9) Jeong, I.; Kim, J.; Pak, S.; Nam, S. W.; Moon, I. Optimum operating strategies for liquid-fed direct methanol fuel cells. J. Power Sources 2008, 185, 828. (10) Xu, C.; Follmann, P. M.; Biegler, L. T.; Jhon, M. S. Numerical simulation and optimization of a direct methanol fuel cell. Comput. Chem. Eng. 2005, 29, 1849. (11) Bakshi, B. R.; Fiksel, J. The quest for sustainability: Challenges for process systems engineering. AIChE J. 2003, 49, 1350. (12) Cownden, R.; Nahon, R.; Rosen, M. A. Exergy analysis of a fuel cell power system for transportation applications. Exergy Int. J. 2001, 1, 112. (13) Wang, S.; Wang, S. Exergy analysis and optimization of methanol generating hydrogen system for PEMFC. Int. J. Hydrogen Energy 2006, 31, 1747. (14) Rashide, R.; Dincer, I.; Berg, P. Energy and exergy analysis of a hybrid molten carbonate fuel cell system. J. Power Sources 2008, 185, 1107. (15) Li, X.; He, Y.; Yin, B.; Miao, Z.; Li, X. Exergy flow and energy utilization of direct methanol fuel cells based on a mathematic model. J. Power Sources 2008, 178, 344. (16) Liu, X. P.; Rohani, S.; Jutan, A. Tracking control of general nonlinear differential-algebraic equation systems. AIChE J. 2003, 49, 1743. (17) Khalil, H. K. Nonlinear Systems, 2nd ed.; Macmillan: New York, 1996. (18) Lu, G. Q.; Liu, F. Q.; Wang, C. Y. Water transport through Nafion 112 membrane in DMFCs. Electrochem Solid-State Lett. 2005, 8, A1. (19) Boyer, C.; Gamburzev, S.; Appleby, A. J. Evaluation of methods to increase the oxygen partial pressure in PEM fuel cells. J. Appl. Electrochem. 1999, 29, 1095. (20) Scott, K.; Taama, W. M.; Argyropoulos, P.; Sundmacher, K. The impact of mass transport and methanol crossover on the direct methanol fuel cell. J. Power Sources 1999, 83, 204. (21) Wu, W. Approximate output regulation of a class of nonlinear systems. J. Process Control 2001, 11, 69. (22) Ishihara, A.; Mitsushima, S.; Kamiya, N.; Ota, K. Exergy analysis of polymer electrolyte fuel cell systems using methanol. J. Power Sources 2004, 126, 34. (23) Scott, K.; Taama, W.; Cruickshank, J. Performance and modelling of a direct methanol solid polymer electrolyte fuel cell. J. Power Sources 1997, 65, 159.
ReceiVed for reView November 18, 2009 ReVised manuscript receiVed April 9, 2010 Accepted April 13, 2010 IE901829A
