Research Article An Improved Empirical Fuel Cell Polarization...

Research ArticleAn Improved Empirical Fuel Cell Polarization Curve ModelBased on Review Analysis

Dong Hao12 Jianping Shen3 Yongping Hou12 Yi Zhou12 and Hong Wang4

1Lab of Clean Energy Automotive Engineering Center Tongji University Shanghai 201804 China2School of Automotive Studies Tongji University Shanghai 201804 China3Shanghai Motor Vehicle Inspection Center Shanghai 201805 China4China National Institute of Standardization Beijing 100088 China

Correspondence should be addressed to Yongping Hou yphoutongjieducn

Received 20 November 2015 Revised 9 March 2016 Accepted 18 April 2016

Academic Editor Iftekhar A Karimi

Copyright copy 2016 Dong Hao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on a review analysis of empirical fuel cell polarization curvemodels in the literature an improvedmodel that can predict fuelcell performance with only measured current-voltage data is developed The fitting characteristics of this new model are validatedby fitting bench test data and road test data In the case of bench test data a comparison of the new model and two representativemodels is conducted and the results show that the newmodel presents the best fitting effects over a whole range of current densitiesMoreover the fitted ohmic resistances derived from the new model show good agreement with the measured values obtainedthrough a current interruption test In the case of using road test data the new model also presents excellent fitting characteristicsand convenience for application It is the authorrsquos belief that the new model is beneficial for the application-oriented research offuel cells due to its prominent features such as conciseness flexibility and high accuracy

1 Introduction

A polymer electrolyte membrane fuel cell (PEMFC) is anelectrochemical device that converts chemical energy storedin hydrogen directly to electricity and heat with water as theonly byproduct of the reaction With the prominent featuresof zero emissions low operating temperature quick startupand high efficiency PEMFCs have been broadly consideredthe best electrical energy sources for automotive stationaryand portable power devices [1] In the area of PEMFCsresearch mathematical fuel cell models including mecha-nism and empirical models are widely applied in the designcontrol and optimization of PEMFCs [2ndash7] Compared withmechanism models empirical PEMFC polarization curvemodels are generally used in the application-oriented inves-tigations and industrial research with advantages such asflexibility simplicity and acceptable accuracy

For PEMFC the polarization curve which describes therelationship between output voltage and current densityis the most important characteristic of performance Thus

a variety of empirical fuel cell polarization curve modelshave been developed in the past decades to reproduce themeasured polarization curves containing a series of current-voltage data points Furthermore the values of fitting param-eters derived from the empirical models are also valuablereferences for the investigation of PEMFCs

This work began with a review on existing empiricalfuel cell polarization curve models Based on this a newempirical model for the entire range of current densities wasthen presented Validation of the new model was executedthrough fitting bench test data and road test data Regardingbench tests a polarization curve test and current interruptiontest were carried out on a fuel cell stack consisting of 57cells Taking the measured current-voltage data as referencevalues the fitting accuracy of the newmodel was investigatedand compared with two representative models Finally thevalidity of the new model was discussed by comparingthe fitted ohmic resistances with the experimental valuesRegarding road tests some discrete current-voltage data of a90-cell stack which were sampled from a demonstrating fuel

Hindawi Publishing CorporationInternational Journal of Chemical EngineeringVolume 2016 Article ID 4109204 10 pageshttpdxdoiorg10115520164109204

2 International Journal of Chemical Engineering

cell sightseeing vehicle were used to analyze the applicationeffects of different models

2 Review Analysis of Existing PolarizationCurve Models

With the aim of improving fitting accuracy of fuel cellperformance throughout the range of operation a num-ber of researchers have developed numerous empirical andsemiempirical fuel cell polarization curve models since theearly 1990s Ten polarization curve models are reviewed inthis section To distinguish fitting parameters from otherparameters or constants fitting parameters are written inboldface throughout this work

The first empirical fuel cell polarization curve modelwith five fitting parameters was presented by Kim et al [8](denoted by model K)

119864cell = EO minus b log (119894) minus R119894 minusm exp (n119894) (1)

The parameter EO can be described as

EO = 119864rev + b log (119894119900) (2)

where b log(119894) represents activation loss R119894 dedicates ohmicloss and m exp(n119894) is an empirical term that approximatesmass transfer loss It is noteworthy that EO is simply a fittingparameter rather than the open circuit voltage (OCV) or thereversible voltage of the fuel cell Fitting (1) has been citedby many studies and publications because of its advantagesthat is it fits the current-voltage data over the entire rangeof current densities under different temperatures pressuresgas compositions and so on However the measured valueof OCV the corresponding voltage of 0A cmminus2 cannot beused during the fitting process because of the log(119894) termIn addition although this equation provides excellent fittingcharacteristics with both medium and high current densitiesthere is a minor deficiency in fitting accuracy with smallcurrent densities

Based on model K several improved models aiming atmodifying the term ofmass transfer loss have been proposedThe model suggested by Lee et al [9] is expressed as

119864cell = EO minus b log (119894) minus R119894 minusm exp (n119894)

minus b log( 119901

119901O2

)

(3)

where the last term containing the pressure ratio logarithmis a form of the Nernst equation which is introduced primar-ily to describe potential changes in the cathode This term isan empirical constant parameter that has no relationshipwithcurrent density In addition the pressure of oxygen must bemeasured before using this model

The model of Squadrito et al [10] is

119864cell = EO minus b log (119894) minus R119894 + 120572S119894k ln (1 minus 120573S119894) (4)

where 120572S119894k accounts for an ldquoamplification termrdquo of the log-

arithmic term both 120572S and k are fitting parameters without

any physical meaning 120573 is a fitting parameter representingthe inverse of the limiting current density This modelimproves the fitting characteristic of the mass transfer lossterm by increasing a fitting parameter

The model of Chu et al [11] can be described as

119864cell = EO minus b log (119894) minus R119894 minus 119894119898m exp (n119894119898) (5)

119894119898= 119894 minus 119894

119889(for 119894 gt 119894

119889) (6)

119894119898= 0 (for 119894 le 119894

119889) (7)

where 119894119889is the smallest current density that causes the voltage

to deviate from linearityWhen the current density is less than119894119889 the term of mass transfer loss in (5) is equal to 0 That

is mass transfer loss occurs only in the high current densityregion nevertheless in fact it occurs over the entire rangeof current densities With the foregoing considerations animproved model was proposed by Xia and Chan [12]


119894119898= 119894 minus 119894

119889(for 119894 gt 119894

119889) (9)

119894119898= 0 (for 119894 le 119894

119889) (10)

In this model first 119894119898in (5) is substituted by parameter

119894 As a result when the current density is less than 119894119889 the

mass transfer loss in (8) is equal to 119894m which means that therelationship between mass transfer loss and current densityis linear in the low and moderate current density range Thecoefficient m indicates that the mass transfer phenomenonoccurs over the entire range of current densities The nonlin-ear exp(n119894

119898) term indicates that the phenomenon occurs at

current densities that are higher than 119894119889

Similarly some modifications on the mass transfer lossterm were executed by Pisani et al [13]

119864cell = EO minus b log (119894) minus R119894 + aP ln(1 minus119894

ilSminus120583(1minus119894il)) (11)

where S is a fitting parameter describing the flooding phe-nomenon and 120583 is an empirical constant The derivation ofthismodel is based on the observation that the strongest non-linear contributions to the cell voltage drop at high currentdensities arise from interface phenomenon happening in thecathode reactive region One of the limitations of this modelis the fact that there are three fitting parameters and oneempirical constant to be included in the mass transfer lossterm

In summary (3)ndash(5) (8) and (11) are obtained by modi-fying the term of mass transfer loss in model K Each of thesemodels consists of a constant voltage fitting parameter a loga-rithmic term approximating the activation loss a linear termrepresenting the ohmic loss and one or two terms describingthe mass transfer loss As with model K the parameters ofEO in these models are merely fitting parameters without anytheoretical meanings Moreover owing to the existence of theterm log(119894) which tends towards infinity when the currentdensity decreases to zero it is difficult to precisely predict theOCV and the performance at small current densities

International Journal of Chemical Engineering 3

Considering that the models mentioned above cannotaccurately fit no-load operation and small current densitiesFraser and Hacker [14] executed some modifications to (1)and presented a model (denoted by model F) as follows

119864cell = 119864rev minus b log(119894 + iloss

io) minus R119894 minusm exp (n119894) (12)

The reversible voltage which is a constant term can beestimated with known fuel cell operation temperature andpartial pressures of oxygen and hydrogen Additional fittingparameters iloss and io are introduced into the logarithmicterm Consequently this equation provides accurate fittingcharacteristics with small current densities and OCV It isworth noting that the reversible voltage here is merely anapproximate value because it is difficult to measure the actualvalues of the operation temperature and partial pressures ofthe reactants

In the case of a hydrogenair fuel cell the thermodynamicreversible voltage 119864rev is calculated from the modifiedNernst equation which considers both temperature andpressure changes [9]

119864rev = 1198640

+Δ

119899119890119865(119879 minus 119879

0) minus

119877119892119879

119899119865ln(

119886H2O

119886H2

11988612

O2

) (13)

where 1198640 is the standard-state reversible voltage (1229V)and Δ is the entropy of the reaction (assuming that it isindependent of temperature) which is minus16328 Jmolminus1 Kminus1at temperature 119879

0 119886119909is the activity of 119909 For an ideal gas

119886119909= 1199011199091199010 where 119901

119909is the partial pressure of gas 119909 and

1199010 is the standard-state pressure (1 atm) For a hydrogenair

fuel cell operating at a low temperature with liquid water as aproduct the activity of water is 1

By substituting the known parameters mentioned aboveinto (13) the reversible voltage can be described as a functionof temperature and pressure as follows

119864rev = 1229 minus 846 times 10minus4

(119879 minus 29815) minus 431

times 10minus5

119879 ln( 1

119901H2

11990112

O2

)

(14)

As shown by (14) the operation temperature of thefuel cell and partial pressures of oxygen and hydrogen areindispensable for calculating the value of the reversiblevoltage To reduce the number of indispensable measuredparameters in the model a fitting term of reversible voltagewas used in (15) by Weydahl et al [15] and in (16) by Pohet al [16]

119864cell = Erev minus b log(119894 + iloss

io) minus R (119894 + iloss)

+ c ln(1 minus 119894 + ilossilim

)

(15)


io) minus R119894 minus 119894m exp (n119894) (16)

Either (15) or (16) contains the maximum number of fit-ting parameters among these ten existing models Comparedwith model K two additional fitting parameters are intro-duced To regard the reversible voltage as a fitting parameterit is difficult to obtain a unique reasonable solution of ErevAs a consequence there are many solutions for other fittingparameters

Equation (17) developed by Haji [17] also regards thereversible voltage as a constant term and presents outstandingfitting accuracy over the entire region of current densitiesConsider

119864cell = 119864rev minus [a + b ln (119894 + 119894loss)] minus R119894

minus c ln( 119894lim119894lim minus (119894 + 119894loss)

)

(17)

Although there are only four fitting parameters in thismodel the limiting current density and current loss areobtained through measurements As a result the range ofapplication of this model is limited

In addition an analysis technique to evaluate six sourcesof polarization losses in hydrogenair proton exchange mem-brane fuel cells was developed by Williams et al [18] Theoverall polarization curve was described by two separateequations one for current densities smaller than 119894

119887and one

for current densities greater than 119894119887 At

119894 lt 119894119887

119864cell = 119864rev + 119887 log (119894o) minus 119887 log (119894) minus 119894119877electrode

minus 119894119877nonelectrode

(18)

At119894 ge 119894119887

119864cell = 119864rev + 1198871015840 log (1198941015840o) minus 119887

1015840 log[ 119894

(1 minus 119894119894lim)]

minus 119894119877electrode minus 119894119877nonelectrode

(19)

The approach in their work is very clear and effectivefor evaluating different sources of polarization losses underlaboratory conditions Based on determinations of someparameters including reversible voltage nonelectrode ohmicresistance cathode electrode ohmic resistance limited cur-rent density Tafel slope and exchange current density theoutput voltage of fuel cells is calculated The developmentof this model is aimed at distinguishing different polariza-tion losses in fuel cells rather than reproducing the wholepolarization curve based on voltagecurrent data Howeverit provided useful insights for analyzing the polarizationcurve to future works on developing a model without anyparameters that lack physical significance

In conclusion all of themodels reviewed above have theirmerits and drawbacks and their numbers of fitting parame-ters range from 4 to 7 The shortages and inconveniences ofthese models are listed in Table 1

After reviewing existing models of polarization curves itcan be concluded that model K [8] and model F [14] are the


Table 1 Shortages and inconveniences of existing empirical fuel cell polarization curve models

Features Equation number(1) (3) (4) (5) (8) (11) (12) (15) (16) (17)

Number of fitting parameters 5 5 6 5 5 6 6 7 7 4Unavailable to fit OCV radic radic radic radic radic radic

Low fitting accuracy of small current densities radic radic radic radic radic radic

Requiring estimating 119864rev or low fitting accuracy of Erev radic radic radic radic

Requiring measured parameters such as temperature and gas partial pressure radic radic radic

Requiring estimating 119894119889

radic radic

Mass transfer loss occurring only in high current density region radic

Requiring other empirical constants radic

Requiring measured 119894lim and 119894loss radic

twomost representativemodels Hence these twomodels arechosen as the references for comparison with the new modelgiven in the following part of this work

3 Improved Empirical PolarizationCurve Model

In this work a series of modifications is performed on thesimplified version of theoretical fuel cell polarization curvemodels and an improved empirical model is then obtainedThe processes of modifications are elaborated as follows

The fuel cell output voltage can be expressed by thefollowing equation

119864cell = 119864rev minus Δ119881act minus Δ119881ohmic minus Δ119881conc (20)

The activation loss Δ119881act is described by the Butler-Volmer equation and is associated with sluggish electrodekinetics The Butler-Volmer equation can be reduced to theTafel equation

Δ119881act = 119886 + 119887 log (119894) (21)

where

119886 = minus23

119877119892119879

120572119865log (119894119900)

119887 = 23

119877119892119879

120572119865

(22)

Term 119887 is called the Tafel slope Hence the activation losscan be given in a simplified form

Δ119881act = 119887 log(119894

119894119900

) (23)

The ohmic loss Δ119881ohmic which obeys Ohmrsquos law occursbecause of resistance to the flow of ions in the electrolyteand resistance to the flow of electrons through the electricallyconductive fuel cell components Consider

Δ119881ohmic = 119894119877 (24)

The mass transfer loss Δ119881conc occurs when reactants arerapidly consumed at the electrode by the electrochemical

reaction so that concentration gradients are establishedThe voltage loss caused by the mass transport limitation isexpressed by the following equation

Δ119881conc =119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

) (25)

In addition to the activation ohmic and mass transferlosses current losses due to hydrogen permeation andelectron crossover also exist in fuel cells Neglecting the effectof current losses on ohmic loss and mass transfer loss asimplified version of the theoretical fuel cell polarizationcurve model that consists of numerous parameters can bedescribed as [19]

119864cell = 119864rev minus 119887 log(119894 + 119894loss119894119900

) minus 119877119894

minus

119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

)

(26)

When external current density 119894 equals zero the fuel celloperates as an open circuit so (26) can be expressed as

119864OCV = 119864rev minus 119887 log(119894loss119894119900

) (27)

The following equation can then result from (27)

119864rev = 119864OCV + 119887 log(119894loss119894119900

) (28)

In a fuel cell as the cell current becomes high which indi-cates that the electrochemical reaction rate on the electrodesurface is fast the mass transfer rate of the reactants is notsufficiently fast to provide enough reactants to the electrodesurface Depletion of reactants at the electrode surface leadsto a drop in cell voltage The calculation of the cell voltagedrop in this part is difficult in some conditions without alloperational parameters so the empirical equation suggestedby Kim et al [8] is usually used to estimate the mass transferdrop

Δ119881conc = m exp (n119894) (29)


The term of mass transfer loss m exp(n119894) is proposedbased on experimental data with two zero physical significantparameters However for practical purposes the adoption ofthese parameters is meaningful

When a fuel cell is under no-load operation the value of(29) ism which does not conform to zero for the theoreticalmass transfer loss Hence to set the empirical term of masstransfer loss equal to zero under the open circuit condition(29) is modified into the following form

Δ119881conc = m [exp (n119894) minus 1] (30)

Combined with (26) (28) and (30) a new empirical fuelcell polarization curve model can be written as

119864cell = 119864OCV minus b log(119894 + ilossiloss

) minus R119894

minusm [exp (n119894) minus 1] (31)

where 119864OCV is known as a measured value five fittingparameters b iloss Rm and n are adopted

Neglecting the mass transfer loss the new model can beexpressed as (32)The neglect of themass transport limitationresults in a simple model that is generally used to representthe low and moderate current density regions of a fuel cellpolarization curve in which only slight mass transfer lossappears Consider


) minus R119894 (32)

In this simplified formation there are three fitting param-eters b iloss and R

The new model (denoted model N) can be applied torepresent measured polarization curves with only current-voltage data Meanwhile the application of this model doesnot require reversible voltage and complicated operationalparameters such as the fuel cell temperature and partialpressures of reactants As a result of the modifications of thelogarithmic term and mass transfer loss term the newmodelis available to accurately fit very small current densities andthe fitted polarization curve passes through the test OCVabsolutely

The assumptions approximations and limitations of theproposed model are illustrated as follows

(i) Assume that the anode losses of fuel cells are negligi-ble compared with the cathode losses

(ii) Neglect the effect of current losses on ohmic loss andmass transfer loss

(iii) The activation polarization is described by the Tafelequation

(iv) The ohmic loss is expressed by Ohmrsquos law and theohmic resistance remains constant over the wholerange of current density

(v) Assume that the mass transfer loss approximatesm[exp(n119894) minus 1]

Table 2 Operational conditions of the fuel cell stack (bench test)

Parameters ValuesRelative humidity of hydrogen Without humidificationRelative humidity of air 80Stoichiometry of hydrogen 12Stoichiometry of air 25Hydrogen inlet temperature 60∘CAir inlet temperature 60∘CHydrogen inlet pressure 151 atmAir inlet pressure Without pressurizationCoolant outlet temperature Approximately 60∘C

(vi) The model can be used under all conditions (includ-ing different operational conditions and componentmaterials) as long as the voltage-current data ofpolarization curves are measured

4 Results and Discussion

41 Validation Using Bench Test Data The comparison ofthe measured and fitted fuel cell performance is carried outwith a polarization curve derived from a fresh commercial8 kW fuel cell stack The stack consists of 57 cells whichare assembled in series The active area per single cell is312 cm2 The fuel cell stack test station for this study is G500designed by Greenlight Innovation Canada [20] The G500is a complete fuel cell testing solution that includes precisegas flow control using mass flow controllers accurate gashumidification stack coolant proprietary pressure controlexpandable cell voltage monitoring capability and a water-cooled load bank with a maximum power of 30 kW Theoperational conditions of the fuel cell stack are shown inTable 2

The polarization curve test of fuel cell stack was con-ducted under a constant gas stoichiometry condition Thecurrent-voltage characteristics were obtained by varying thecurrent density from 0 to the maximum value at a suitableinterval and maintaining each current density until theaverage voltage of a single cell was stabilized within plusmn5mVfor 3min [21] The measured polarization curve of a fuel cellstack that has been normalized to the average of a single cellis shown by the black line in Figure 1 The measured OCV ofthe fuel cell stack is 09560V which plays a significant role inmodel N

In addition the ohmic resistances of the fuel cell stackare obtained through a current interruption test The averageohmic resistances of single cells corresponding to differentcurrent densities are acquired by dividing the ohmic resis-tances of the fuel cell stack by the cell number of 57 andthe range from 01062Ω cm2 to 01086Ω cm2 with an averagevalue of 01071Ω cm2 as depicted in Figure 1

411 Validation of New Model To validate the new modelthe fitting characteristics are investigated and compared withthose of model K and model F Model N and model K canfit the test data directly whereas model F requires the value


Measured polarization curve of fuel cell stackMeasured ohmic resistance of fuel cell stack

02 04 06 08 10 12 1400Current density (A cmminus2)

04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )

Figure 1 Measured polarization curve and ohmic resistances of thefuel cell stack

04

05

06

07

08

09

10

Volta

ge (V

)


MeasuredFitted by model K

Fitted by model FFitted by model N

Figure 2 Comparison of measured and fitted polarization curves(bench test)

of reversible voltage As shown in Table 2 the hydrogenand oxygen partial pressures are 151 atm and 021 atm (1 times21 atm) respectively The operational temperature of thefuel cell stack is approximately 33315 K Consequently areversible voltage of 11940V is obtained by substituting theabovementioned values into (14)

The three fitted polarization curves and measuredcurrent-voltage data are depicted in Figure 2 It is clear thatthese three models exert excellent fitting characteristics withboth medium and high current densities However whereasthe current density declines to zero the voltage fitted bymodel K increases dramatically and tends towards infinityThe region of small current densities in Figure 2 is presentedin Figure 3 It can be observed that the curve fitted by modelK deviates from the other two fitting curves significantlyin a very small current density region (approximately 0ndash001 Acm2)

080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)

002 004 006 008 010 012000Current density (A cmminus2)



Figure 3 Comparison of measured and fitted polarization curves(small current densities) (bench test)

To evaluate the prediction accuracy of these three modelsin terms of quantity over the whole region of currentdensities the fitted voltages of each model are comparedwith the measured values at different current densities (25test points in total) For model K because the fitted voltagecorresponding to 0Acm2 cannot be obtained the fittedvoltage at 1mA cmminus2 is used here as a substitution of OCV

The absolute deviation between fitted voltage and mea-sured voltage is defined as

119889fit (119894119901) =10038161003816100381610038161003816119864fit (119894119901) minus 119864test (119894119901)

10038161003816100381610038161003816 (33)

where 119894119901is the current density of test point 119901 and 119864fit(119894119901)

and 119864test(119894119901) are the fitted voltage and measured voltage at119894119901 respectively All absolute deviations of the three models

are less than 5mV which meets the requirement for voltagestabilization in the polarization curve test as depicted inFigure 4 It is obvious that only the OCV fitted by model Nhas no deviation from the measured valueThat is comparedwith model K and model F the polarization curve fitted bymodel N completely passes through the measured OCV Forthe region of 0ndash05 A cmminus2 the fitting accuracy of model Kis slightly lower than those of the other two models For theregion of 05ndash14 A cmminus2 the three models present almost thesame fitting accuracy

The average of the absolute deviations between fittedvoltage and measured voltage is defined as (34) and theresults are given in Table 3 Consider

119889fit =sum25

119901=1119889fit (119894119901)

25 (34)

As illustrated in Table 3 the average of the absolutedeviations derived from model N is the minimum amongthe three values As a result model N exhibits the best fittingaccuracy in the entire region of current densities Comparedwith model F data that are difficult to measure and estimate


Table 3 Average and maximum values of absolute deviationsderived from three models

Model K Model F Model NAverage of absolute deviations (mV) 15919 14883 14642Maximum absolute deviation (mV) 38224 31011 34242

Absolute deviations derived with model KAbsolute deviations derived with model FAbsolute deviations derived with model N

0

1

2

3

4

5

Volta

ge (m

V)


Figure 4 Absolute deviations of fitted voltage andmeasured voltagederived from the three models

are nonessential to the application of model N In additionthe fitting parameter for exchange current density in model Fis eliminated so that the structure of model N is much morecompact

The values of fitting parameters obtained from the threemodels are listed in Table 4 As seen in the table the fittedvalues of b R m and n derived from each of the threemodels exhibit slight differences The value of EO derivedfrom model K is unreasonable and much smaller than theother two values The application of the log(119894) term in modelK is the main reason for this result Model F predicts theOCV approximately whereas model N predicts the OCVperfectly Regarding the ohmic resistance R and Tafel slopeb the results obtained from model N approximate those ofmodel F and obviously deviate from those of model K Aslight distinction is detected between the internal currentdensities derived with model F and model N and both ofthem are reasonable [19]

412 Analysis of Fitted Ohmic Resistance During the polar-ization curve test current-voltage data of every single cell inthe stack have also been acquired The polarization curves of57 single cells are fitted by model N such that the values offitting parameters can be investigated further

The ohmic resistances of 57 single cells derived frommodel N are depicted in Figure 5 The bars represent theohmic resistances fitted from 57 single cell polarizationcurves and the dashed line shows their average with a

Table 4Values of fitting parameters obtained from the threemodels(bench test)

Model K Model F Model N119864rev (V) NA 11940 NA

119864OCV (V) 07703 (EO)09559 (at 1mA cmminus2) 09557 09560

b (Vdecade) 006362 006642 006677R (Ω cm2) 01121 01083 01073iloss (A cm2) NA 0001225 0001241io (10minus7 A cm2) NA 37980 NAm (V) 0005165 0005248 0005339n (cm2 Aminus1) 22389 22421 22353

5 10 15 20 25 30 35 40 45 50 550Serial number of single cell

000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )

Average of fitted ohmic resistances of numbers 1ndash57 cellsFitted ohmic resistances of numbers 1ndash57 cells

Figure 5 Fitted ohmic resistances of 57 single cells

Fitted value of average ohmic resistance of single cellReference value of average ohmic resistance of single cell

R3

R2

R1

002 004 006 008 010 012000Ohmic resistance (Ω cm2)

Average of fitted ohmic resistances of numbers 1ndash57 cells

Figure 6 Comparison of ohmic resistances obtained frommeasure-ment and fits

value of 01109Ω cm2 The fitted ohmic resistances of 57 cellsremain in a reasonable range [19] with a maximum valueof 01282Ω cm2 (cell number 57) and a minimum value of00996Ω cm2 (cell number 24)

Because the ohmic resistance of the fuel cell stack hasbeen obtained through the current interruption test theaverage value of the measured results that is 01071Ω cm2is taken as the reference value to the fitted values in whatfollows As seen in Figure 6119877

1and119877

2represent the reference

and fitted values of the average ohmic resistance of a single


cell respectively1198773represents the average of the fitted ohmic

resistances of cells numbers 1ndash57 which corresponds to thedashed line in Figure 5 The fitted average ohmic resistanceof a single cell is 01073Ω cm2 larger than the referencevalue by 019 The average of 57 fitted ohmic resistances is01109Ω cm2 which is 355 larger than the reference value

From the foregoing analysis in addition to the excellentfitting characteristics of current-voltage data the fittingresults of the ohmic resistance derived with model N arein good agreement with the measured values and can be areference for research on polarization curves

The model in this work is developed based on a reviewof other existing models The fitting accuracy of similarempirical models under different operational conditions andcomponent materials has been corroborated by many otherworks such as [8 10ndash13 16 17] To be different from theprevious works the resistances were measured through acurrent interruption test and the residuals of the voltage atdifferent current densities were calculated Because of theforgiving nature of the model it is difficult to obtain a uniquesolution of a set of test data The adoption of the least squaremethod during the fitting process can ensure an optimalsolution of parameters in an empirical model Inevitably thefitting values cannot be completely equal to the theoreticalvalue but they can be used as references to understand thevoltage losses in fuel cells Although some parameters in theproposed model cannot be measured a comparison of fittingresults between our results and those of other works was con-ducted As shown in Table 4 the fitting values of parametersare similar to those obtained from two representativemodelsThis corroboration method was also adopted in [12 14]

42 Validation Using Road Test Data To investigate theperformance and durability of fuel cell stacks for automotiveapplications road tests or demonstrations of fuel cell vehiclesare significant and indispensable Under a real road environ-ment a vehicular fuel cell stack experiencesmuchmore com-plicated and harsher operational conditions than those underbench tests such as dynamic cycling and changes of ambienttemperature pressure and relative humidity Because of thefrequent load change the performance data of the fuel cellstack sampled from the road test are massive and discretein total contrast to those obtained from the bench test As aconsequence the empirical fuel cell polarization curvemodelis an effective tool to extract the whole polarization curvefrom the complex data

In this work some experimental data which are obtainedfrom a demonstration fuel cell sightseeing vehicle with asampling frequency of 1Hz are used to compare the appli-cation effects of the three models that is model K model Fand model N The fuel cell stack adopted in this sightseeingvehicle has a designed output power of 58 kW and consists of90 cells assembled in series The electrochemical active areaper cell is 250 cm2 Under a real road environment the knownoperational conditions of the stack are listed in Table 5

Similarly model N and model K can fit the discrete datadirectly whereas model F requires the value of reversiblevoltage According to the information in Table 5 hydrogenand oxygen partial pressures are approximately 16 atm and

Table 5 Operational conditions of the fuel cell stack (road test)

Parameters ValuesHydrogen inlet pressure Approximately 16 atmAir inlet pressure Approximately 148 atmCoolant outlet temperature Approximately 55∘C

05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600

Current density (A cmminus2)

Raw dataFitted by model K


Figure 7 Comparison of measured and fitted polarization curves(road test)

03108 atm (148 times 21 atm) respectively The operationaltemperature of the stack is approximately 32815 K Thus anestimated reversible voltage of 12020V is gained by substitut-ing the known data into (14) Compared with bench test it ismuchmore difficult to accuratelymaintain steady operationalconditions of the fuel cell stack during the road test owing tothe complicated operation of the vehicle As a consequenceit is difficult to obtain an exact value of the reversible voltage

The raw data depicted in Figure 7 show that all outputcurrent densities of the fuel cell stack are less than 05 A cmminus2In addition the shape of the scatterplot also does not presentan apparent characteristic of the mass transport limitationAs a result simple formations of three models similar to (32)which neglect the mass transfer loss terms containingm andn are adopted to fit the scattered data [8 14] Another pointto consider is that the value of119864OCV inmodel N is the averageof sampled OCVs in raw data that is 09576V

As plotted in Figure 7 these three models present almostthe same fitting results in thewhole range of current densitieswhereas the voltage fitted by model K increases dramaticallyand tends towards infinity when the current density approxi-mates zero As shown in Figure 8 which enlarges the smallcurrent density region in the region of approximately 0ndash001 Acm2 the curve fitted by model K deviates from theother two fitting curves expressively

As seen in Table 6 the value of EO derived with model Kis unreasonable and much smaller than the other two valueswhereas model F and model N predict almost the same value


Table 6 Values of fitting parameters obtained from three models(road test)


119864OCV (V) 07510 (EO)(09431 at 1mA cmminus2) 09574 09576

b (Vdecade) 006412 006532 006516R (Ω cm2) 03310 03287 03292iloss (A cm2) NA 00006616 00006508io (10minus7 A cm2) NA 11926 NA

075

080

085

090

095

100

105

Volta

ge (V

)

002 004 006000Current density (A cmminus2)



Figure 8 Comparison of measured and fitted polarization curves(small current densities) (road test)

of OCV The fitted values of b and R derived with model Npresent extremely slight differences from those of model Fbut relatively obvious deviations from those of model K Thefitted internal current densities obtained from model F andmodel N exhibit only a slight distinction

Through the foregoing analysis it can be concluded thatmodel F andmodelN showmuch better fitting characteristicsthan model K because of the minimum prediction error atOCV and small current density region Although model Nshows almost the same fitting characteristics as model F themost competitive advantage of model N is that the reactantpressures and operational temperature of the fuel cell stackare not indispensable for fitting In addition only three fittingparameters one less than those of model F are required formodel N Each fitting parameter in model N has physicalmeaning whereas EO in model K is merely a voltage fittingtermThese features of model N are beneficial for researchersto extract polarization curves of fuel cells under a road testaccurately and efficiently

43 Advantages of the NewModel Among all proposedmod-els the model in this paper presents good fitting accuracyand the best advantage of convenience collectively As shown

in Table 1 the shortcomings of the existing models can besummarized as follows

(1) Inability to fit open circuit voltage(2) Low fitting accuracy of small current densities(3) Requiring estimate of reversible voltage 119864rev or low

fitting accuracy of 119864rev(4) Requiring measured parameters such as temperature

and gas partial pressure(5) Requiring estimation of the smallest current density

that causes the voltage to deviate from linearity 119894119889

(6) Assuming that mass transfer loss occurs only in thehigh current density region

(7) Requiring other empirical constants(8) Requiring measured limited current density 119894lim and

internal current density 119894loss

Themodel proposed in this work avoids all shortcomingsand inconveniences mentioned above and it is the authorrsquosbelief that this is one of the most meaningful aspects of thiswork

5 Conclusion

The purpose of this work is to obtain a simple easy-to-useefficient and application-oriented polarization curve modelwith an acceptable fitting accuracy based on a review analysisAfter summarizing numerous existing empirical fuel cellpolarization curve models a new model has been developedin this paper

The following conclusions can be obtained

(1) The newmodel contains five fitting parameters Com-pared with most existing models it does not requiremore fitting parameters

(2) The new model has a compact formation and itsapplication only requires current-voltage data that areeasy tomeasureHence it is beneficial for application-oriented investigations and industrial research

(3) In the case of using bench test data the new modelprovides almost ideal fitting characteristics with theentire range of current densities Compared withother models its advantages can be illustrated asfollows (a) the fitting curve definitely crosses thepoint of measured OCV (b) the fitting curve presentsexcellent accuracy over the whole region of currentdensities Moreover all of the values of the fittingparameters derived from the new model appear inreasonable ranges The fitted ohmic resistances offuel cells ideally agree with the experimental dataobtained through the current interruption test As aconsequence the fitted results of the parameters canbe references for research on fuel cell polarizationcurves In addition at present under laboratoryconditions it is difficult to precisely measure someparameters such as the limited current density andinternal loss current density The development of


a model without any parameters that lack physicalsignificance is the most important research directionof our future works

(4) In the case of using road test data it is difficultto precisely measure the steady pressures of reac-tants operational temperature limited current den-sity internal loss current density and so on As aconsequence the most prominent advantage of thenew model is that excellent fitting characteristicsover the whole current density region are presentedwithout measuring any pressures or temperaturesConsequently it is suitable for extracting the fuel cellstack performance accurately and efficiently from thediscrete data sampled in a real road test

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper

Acknowledgments

The work was financially supported under the grants of theNational Nature Science Foundation of China Project no51275357

References

[1] YWang K S Chen J Mishler S C Cho and X C Adroher ldquoAreview of polymer electrolyte membrane fuel cells technologyapplications and needs on fundamental researchrdquo AppliedEnergy vol 88 no 4 pp 981ndash1007 2011

[2] N Duic M Lerer and M G Carvalho ldquoIncreasing thesupply of renewable energy sources in island energy systemsrdquoInternational Journal of Sustainable Energy vol 23 no 4 pp177ndash186 2003

[3] M A R S Al-Baghdadi ldquoA simple mathematical modelof performance for proton exchange membrane fuel cellsrdquoInternational Journal of Sustainable Energy vol 26 no 2 pp79ndash90 2007

[4] J E Choi and Y C Bae ldquoSwelling effect of a polymer electrolytemembrane on the development of a semi-empirical cell voltagemodelrdquo Journal of Applied Electrochemistry vol 39 no 9 pp1419ndash1424 2009

[5] S A Grigoriev A A Kalinnikov P Millet V I Porembsky andV N Fateev ldquoMathematical modeling of high-pressure PEMwater electrolysisrdquo Journal of Applied Electrochemistry vol 40no 5 pp 921ndash932 2010

[6] A A Kulikovsky ldquoA simple and accurate fitting equation forhalf of the faradaic impedance arc of a PEM fuel cellrdquo Journalof Electroanalytical Chemistry vol 738 pp 108ndash112 2015

[7] A A Tahmasbi AHoseini and R Roshandel ldquoA new approachto multi-objective optimisation method in PEM fuel cellrdquoInternational Journal of Sustainable Energy vol 34 no 5 pp283ndash297 2015

[8] J Kim S-M Lee S Srinivasan and C E Chamberlin ldquoModel-ing of proton exchangemembrane fuel cell performancewith anempirical equationrdquo Journal of the Electrochemical Society vol142 no 8 pp 2670ndash2674 1995

[9] J H Lee T R Lalk and A J Appleby ldquoModeling electrochem-ical performance in large scale proton exchange membrane fuelcell stacksrdquo Journal of Power Sources vol 70 no 2 pp 258ndash2681998

[10] G Squadrito G Maggio E Passalacqua F Lufrano and APatti ldquoAn empirical equation for polymer electrolyte fuel cell(PEFC) behaviourrdquo Journal of Applied Electrochemistry vol 29no 12 pp 1449ndash1455 1999

[11] D Chu R Jiang andCWalker ldquoAnalysis of PEM fuel cell stacksusing an empirical current-voltage equationrdquo Journal of AppliedElectrochemistry vol 30 no 3 pp 365ndash370 2000

[12] Z T Xia and S H Chan ldquoAnalysis of carbon-filled gasdiffusion layer for H

2air polymer electrolyte fuel cells with

an improved empirical voltage-current modelrdquo InternationalJournal of Hydrogen Energy vol 32 no 7 pp 878ndash885 2007

[13] L Pisani G Murgia M Valentini and B D Aguanno ldquoA newsemi-empirical approach to performance curves of polymerelectrolyte fuel cellsrdquo Journal of Power Sources vol 108 no 1pp 192ndash203 2002

[14] S D Fraser and V Hacker ldquoAn empirical fuel cell polarizationcurve fitting equation for small current densities and no-loadoperationrdquo Journal of Applied Electrochemistry vol 38 no 4 pp451ndash456 2008

[15] H Weydahl M S Thomassen B T Boslashrresen and S Moslashller-Holst ldquoResponse of a proton exchange membrane fuel cell toa sinusoidal current loadrdquo Journal of Applied Electrochemistryvol 40 no 4 pp 809ndash819 2010

[16] C K Poh Z Tian N Bussayajarn et al ldquoPerformance enhance-ment of air-breathing proton exchange membrane fuel cellthrough utilization of an effective self-humidifying platinum-carbon catalystrdquo Journal of Power Sources vol 195 no 24 pp8044ndash8051 2010

[17] S Haji ldquoAnalytical modeling of PEM fuel cell i-V curverdquoRenewable Energy vol 36 no 2 pp 451ndash458 2011

[18] M VWilliams H R Kunz and JM Fenton ldquoAnalysis of polar-ization curves to evaluate polarization sources in hydrogenairPEM fuel cellsrdquo Journal of the Electrochemical Society vol 152no 3 pp A635ndashA644 2005

[19] F Barbir PEM Fuel CellsTheory and Practice Elsevier Amster-dam Netherlands 2005

[20] Greenlight Innovation Corporation (GIC) G500 Fuel Cell TestStation 2014 httpwwwgreenlightinnovationcomdatabasefileslibraryG500 Fuel Cell Test Stationpdf

[21] IEC (International Electrotechnical Commission) Fuel CellTechnologiesmdashPart 7-1 Single Cell Test Methods for PolymerElectrolyte Fuel Cell (PEFC) IEC 2010

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



cell sightseeing vehicle were used to analyze the applicationeffects of different models

2 Review Analysis of Existing PolarizationCurve Models

With the aim of improving fitting accuracy of fuel cellperformance throughout the range of operation a num-ber of researchers have developed numerous empirical andsemiempirical fuel cell polarization curve models since theearly 1990s Ten polarization curve models are reviewed inthis section To distinguish fitting parameters from otherparameters or constants fitting parameters are written inboldface throughout this work

The first empirical fuel cell polarization curve modelwith five fitting parameters was presented by Kim et al [8](denoted by model K)

119864cell = EO minus b log (119894) minus R119894 minusm exp (n119894) (1)

The parameter EO can be described as

EO = 119864rev + b log (119894119900) (2)

where b log(119894) represents activation loss R119894 dedicates ohmicloss and m exp(n119894) is an empirical term that approximatesmass transfer loss It is noteworthy that EO is simply a fittingparameter rather than the open circuit voltage (OCV) or thereversible voltage of the fuel cell Fitting (1) has been citedby many studies and publications because of its advantagesthat is it fits the current-voltage data over the entire rangeof current densities under different temperatures pressuresgas compositions and so on However the measured valueof OCV the corresponding voltage of 0A cmminus2 cannot beused during the fitting process because of the log(119894) termIn addition although this equation provides excellent fittingcharacteristics with both medium and high current densitiesthere is a minor deficiency in fitting accuracy with smallcurrent densities

Based on model K several improved models aiming atmodifying the term ofmass transfer loss have been proposedThe model suggested by Lee et al [9] is expressed as

119864cell = EO minus b log (119894) minus R119894 minusm exp (n119894)

minus b log( 119901

119901O2

)

(3)

where the last term containing the pressure ratio logarithmis a form of the Nernst equation which is introduced primar-ily to describe potential changes in the cathode This term isan empirical constant parameter that has no relationshipwithcurrent density In addition the pressure of oxygen must bemeasured before using this model

The model of Squadrito et al [10] is

119864cell = EO minus b log (119894) minus R119894 + 120572S119894k ln (1 minus 120573S119894) (4)

where 120572S119894k accounts for an ldquoamplification termrdquo of the log-

arithmic term both 120572S and k are fitting parameters without

any physical meaning 120573 is a fitting parameter representingthe inverse of the limiting current density This modelimproves the fitting characteristic of the mass transfer lossterm by increasing a fitting parameter

The model of Chu et al [11] can be described as


119894119898= 119894 minus 119894

119889(for 119894 gt 119894

119889) (6)

119894119898= 0 (for 119894 le 119894

119889) (7)

where 119894119889is the smallest current density that causes the voltage

to deviate from linearityWhen the current density is less than119894119889 the term of mass transfer loss in (5) is equal to 0 That

is mass transfer loss occurs only in the high current densityregion nevertheless in fact it occurs over the entire rangeof current densities With the foregoing considerations animproved model was proposed by Xia and Chan [12]


119894119898= 119894 minus 119894

119889(for 119894 gt 119894

119889) (9)

119894119898= 0 (for 119894 le 119894

119889) (10)

In this model first 119894119898in (5) is substituted by parameter

119894 As a result when the current density is less than 119894119889 the

mass transfer loss in (8) is equal to 119894m which means that therelationship between mass transfer loss and current densityis linear in the low and moderate current density range Thecoefficient m indicates that the mass transfer phenomenonoccurs over the entire range of current densities The nonlin-ear exp(n119894

119898) term indicates that the phenomenon occurs at

current densities that are higher than 119894119889

Similarly some modifications on the mass transfer lossterm were executed by Pisani et al [13]

119864cell = EO minus b log (119894) minus R119894 + aP ln(1 minus119894

ilSminus120583(1minus119894il)) (11)

where S is a fitting parameter describing the flooding phe-nomenon and 120583 is an empirical constant The derivation ofthismodel is based on the observation that the strongest non-linear contributions to the cell voltage drop at high currentdensities arise from interface phenomenon happening in thecathode reactive region One of the limitations of this modelis the fact that there are three fitting parameters and oneempirical constant to be included in the mass transfer lossterm

In summary (3)ndash(5) (8) and (11) are obtained by modi-fying the term of mass transfer loss in model K Each of thesemodels consists of a constant voltage fitting parameter a loga-rithmic term approximating the activation loss a linear termrepresenting the ohmic loss and one or two terms describingthe mass transfer loss As with model K the parameters ofEO in these models are merely fitting parameters without anytheoretical meanings Moreover owing to the existence of theterm log(119894) which tends towards infinity when the currentdensity decreases to zero it is difficult to precisely predict theOCV and the performance at small current densities







119864rev = 1198640

+Δ

119899119890119865(119879 minus 119879

0) minus

119877119892119879

119899119865ln(

119886H2O

119886H2

11988612

O2

) (13)



119886119909= 1199011199091199010 where 119901






(119879 minus 29815) minus 431

times 10minus5

119879 ln( 1

119901H2

11990112

O2

)

(14)





)

(15)







)

(17)



119887and one


119894 lt 119894119887



(18)

At119894 ge 119894119887


1015840 log[ 119894

(1 minus 119894119894lim)]


(19)












radic radic










Δ119881act = 119886 + 119887 log (119894) (21)

where

119886 = minus23

119877119892119879

120572119865log (119894119900)

119887 = 23

119877119892119879

120572119865

(22)


Δ119881act = 119887 log(119894

119894119900

) (23)


Δ119881ohmic = 119894119877 (24)



Δ119881conc =119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

) (25)



) minus 119877119894

minus

119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

)

(26)



) (27)


119864rev = 119864OCV + 119887 log(119894loss119894119900

) (28)


Δ119881conc = m exp (n119894) (29)







) minus R119894





) minus R119894 (32)




















04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )


04

05

06

07

08

09

10

Volta

ge (V

)







080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)








10038161003816100381610038161003816 (33)





119889fit =sum25

119901=1119889fit (119894119901)

25 (34)






0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119864rev = 1198640

+Δ

119899119890119865(119879 minus 119879

0) minus

119877119892119879

119899119865ln(

119886H2O

119886H2

11988612

O2

) (13)



119886119909= 1199011199091199010 where 119901






(119879 minus 29815) minus 431

times 10minus5

119879 ln( 1

119901H2

11990112

O2

)

(14)





)

(15)







)

(17)



119887and one


119894 lt 119894119887



(18)

At119894 ge 119894119887


1015840 log[ 119894

(1 minus 119894119894lim)]


(19)












radic radic










Δ119881act = 119886 + 119887 log (119894) (21)

where

119886 = minus23

119877119892119879

120572119865log (119894119900)

119887 = 23

119877119892119879

120572119865

(22)


Δ119881act = 119887 log(119894

119894119900

) (23)


Δ119881ohmic = 119894119877 (24)



Δ119881conc =119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

) (25)



) minus 119877119894

minus

119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

)

(26)



) (27)


119864rev = 119864OCV + 119887 log(119894loss119894119900

) (28)


Δ119881conc = m exp (n119894) (29)







) minus R119894





) minus R119894 (32)




















04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )


04

05

06

07

08

09

10

Volta

ge (V

)







080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)








10038161003816100381610038161003816 (33)





119889fit =sum25

119901=1119889fit (119894119901)

25 (34)






0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















radic radic










Δ119881act = 119886 + 119887 log (119894) (21)

where

119886 = minus23

119877119892119879

120572119865log (119894119900)

119887 = 23

119877119892119879

120572119865

(22)


Δ119881act = 119887 log(119894

119894119900

) (23)


Δ119881ohmic = 119894119877 (24)



Δ119881conc =119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

) (25)



) minus 119877119894

minus

119877119892119879

119899119890119865

ln( 119894lim119894lim minus 119894

)

(26)



) (27)


119864rev = 119864OCV + 119887 log(119894loss119894119900

) (28)


Δ119881conc = m exp (n119894) (29)







) minus R119894





) minus R119894 (32)




















04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )


04

05

06

07

08

09

10

Volta

ge (V

)







080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)








10038161003816100381610038161003816 (33)





119889fit =sum25

119901=1119889fit (119894119901)

25 (34)






0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















) minus R119894





) minus R119894 (32)




















04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )


04

05

06

07

08

09

10

Volta

ge (V

)







080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)








10038161003816100381610038161003816 (33)





119889fit =sum25

119901=1119889fit (119894119901)

25 (34)






0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












04

05

06

07

08

09

10

Volta

ge (V

)

006

007

008

009

010

011

012

Ohm

ic re

sista

nce (

Ωcm

2 )


04

05

06

07

08

09

10

Volta

ge (V

)







080

082

084

086

088

090

092

094

096

098

Volta

ge (V

)








10038161003816100381610038161003816 (33)





119889fit =sum25

119901=1119889fit (119894119901)

25 (34)






0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0

1

2

3

4

5

Volta

ge (m

V)












000002004006008010012014016018

Ohm

ic re

sista

nce (

Ωcm

2 )




R3

R2

R1






1and119877













05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















05

06

07

08

09

10

Volta

ge (V

)01 02 03 04 05 0600














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














075

080

085

090

095

100

105

Volta

ge (V

)

















5 Conclusion









Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Competing Interests


Acknowledgments


References


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



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Research Article An Improved Empirical Fuel Cell Polarization...

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