Padulles Dynamic Model II

Version 0.9

The Padulles dynamic model can predict the transient response of cell voltage, temperature of the cell, hydrogen/oxygen out flow rates and cathode and anode channel temperatures/pressures under sudden change in load current. Hence, a dynamic fuel cell simulation is developed in this model, which incorporates the dynamics of flow and pressure in the anode and cathode channels and mass/ heat transfer transient features in the fuel cell body.
This model is based on several assumptions:

  1. The stack is fed with hydrogen and air
  2. Cell temperature is stable at all times
  3. The ratio of pressures between the interior and exterior of the electrode channels is large
  4. The channels that transport gases along the electrodes have a fixed volume
  5. Only source of voltage loss is ohmic polarization
  6. Nernst equation can be applied too

In this model, Nernst and fuel cell potential were modeled as a function of water, oxygen and hydrogen gases partial pressure that can be calculated from independent variables or constants. The partial pressure of gases is proportional to the molar flow of each gas.

Fig1. Padulles-2 Dynamic Model Block Diagram

Nernst Voltage

$$E_{Nernst}=N_0\times [E_0+\frac{RT}{2F}ln(\frac{P_{H_2}\times \sqrt{P_{O_2}}}{P_{H_2O}})]$$

$$P_{H_2}=\frac{\frac{1}{K_{H_2}}}{1+\tau_{H_2}^{(s)}}[(q_{H_2}^{(inlet)}-(2\times K_r \times i)]$$

$$P_{O_2}=\frac{\frac{1}{K_{O_2}}}{1+\tau_{O_2}^{(s)}}[(q_{O_2}^{(inlet)}-(K_r \times i)]$$

$$P_{H_2O}=\frac{\frac{1}{K_{H_2O}}}{1+\tau_{H_2O}^{(s)}}[(q_{H_2O}^{(inlet)}-(2\times K_r \times i)]$$

$$K_r=\frac{N_0}{4F}$$

$$q_{O_2}^{(inlet)}=\frac{q_{H_2}^{(inlet)}}{r_{h-o}}$$

$$q_{H_2O}^{(inlet)}=q_{H_2}^{(inlet)}$$

In [1]:
from opem.Dynamic.Padulles2 import Kr_Calc,qO2_Calc,PH2_Calc,PO2_Calc,PH2O_Calc,Enernst_Calc
In [2]:
Kr=Kr_Calc(N0=5)
Kr
Out[2]:
1.2955435375179044e-08
In [3]:
qO2=qO2_Calc(qH2=0.0004,rho=1.168)
qO2
Out[3]:
0.0003424657534246576
In [4]:
PH2=PH2_Calc(KH2=0.0004,tH2=3.37,Kr=Kr,I=1,qH2=0.0004)
PH2
Out[4]:
0.22881812879247693
In [5]:
PO2=PO2_Calc(KO2=0.0000211,tO2=6.74,Kr=Kr,I=1,qO2=qO2)
PO2
Out[5]:
2.0968979878594753
In [6]:
PH2O=PH2O_Calc(KH2O=0.000007716,tH2O=18.418,Kr=Kr,I=1,qH2O=0.0004)
PH2O
Out[6]:
2.669532068584946
In [7]:
Enernst=Enernst_Calc(E0=0.6,N0=5,T=343, PH2=PH2, PO2=PO2,PH2O=PH2O)
Enernst
Out[7]:
2.845819522254369

FC Voltage

$$V_{Fuelcell}=E_{Nernst}-Bln(Ci)-(R^{int}\times i)$$

In [8]:
from opem.Dynamic.Padulles2 import Vcell_Calc
FC_Voltage=Vcell_Calc(Enernst=Enernst, B=0.04777,C=0.0136,I=1,Rint=0.00303)
FC_Voltage
Out[8]:
3.0480899579320604

Power Of PEMFC

$$P=V_{cell}\times i$$

$$P_{Thermal}=i\times(N \times E_{th}-V_{Stack})$$

$$E_{th}=\frac{-\Delta H}{nF}=1.23V$$

In [9]:
from opem.Dynamic.Padulles2 import Power_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=FC_Voltage,i=2)
Power
Out[9]:
6.096179915864121
In [10]:
Power_Thermal_Calc(VStack=FC_Voltage,N=5,i=2)
Out[10]:
6.20382008413588

Efficiency of PEMFC

$$\eta=\frac{\mu_F\times V_{Fuelcell}}{N_0\times HHV}$$

In [11]:
from opem.Dynamic.Padulles2 import Efficiency_Calc
Efficiency_Calc(Vcell=FC_Voltage,N=5)
Out[11]:
0.3907807638374436

Linear Approximation

Sometimes quick calculations regarding fuel cell efficiency–power-size relationships need to be made. Linear approximation is a good method to find a rough estimate of the value of polarization function at a particular point. A linear polarization curve has the following form: $$V_{cell}=V_0-kI$$ where V0 is the intercept (actual open circuit voltage is always higher) and k is the slope of the curve.

$$Parameter$$ $$Description$$ $$Unit$$
$$V_0$$ Intercept of the curve obtained by linear approximation $$V$$
$$k$$ Slope of the curve obtained by linear approximation $$A^{-1}$$
$$P_{max}$$ Maximum power obtained by linear approximation $$W$$
$$V_{FC}|P_{max}$$ Cell voltage at maximum power obtained by linear approximation $$V$$
  • Notice : These parameters are only available in HTML report

Overall Parameters

$$Parameter$$ $$Description$$ $$Unit$$
$$\eta|P_{Max}$$ Cell efficiency at maximum power $$--$$
$$P_{Max}$$ Maximum power $$W$$
$$P_{Elec} $$ Total electrical power $$W$$
$$P_{Thermal} $$ Total thermal power $$W$$
$$V_{FC}|P_{Max}$$ Cell voltage at maximum power $$V$$
  • Notice : P(Thermal) & P(Elec) calculated by Simpson's Rule

  • Notice : These parameters are only available in HTML report

Full Run

  • Run from i=0 to i=4 with step=0.1
In [12]:
Test_Vector={"T":343,"E0":0.6,"N0":5,"KO2":0.0000211,"KH2":0.0000422,"KH2O":0.000007716,"tH2":3.37,"tO2":6.74,"tH2O":18.418,"B":0.04777,"C":0.0136,"Rint":0.00303,"rho":1.168,"qH2":0.0004,"i-start":0.1,"i-stop":4,"i-step":0.1,"Name":"Test"}
  • Notice : "Name", new in version 0.5
In [13]:
from opem.Dynamic.Padulles2 import Dynamic_Analysis
Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
Out[13]:
{'EFF': [0.4265389961269225,
  0.42225504989786394,
  0.4197329719509197,
  0.41793223959535725,
  0.41652676431084773,
  0.4153712975747617,
  0.4143883594017745,
  0.4135317011453444,
  0.41277149148017567,
  0.4120873617867766,
  0.4114647839578894,
  0.41089303097642904,
  0.4103639566273662,
  0.40987122872897686,
  0.409409827543193,
  0.40897570639787845,
  0.4085655553769282,
  0.40817663265783805,
  0.4078066415116356,
  0.40745363888936376,
  0.40711596633503955,
  0.4067921969851981,
  0.4064810943595029,
  0.40618157992825576,
  0.40589270730496035,
  0.40561364150350754,
  0.405343642112191,
  0.4050820495292294,
  0.404828273614746,
  0.40458178426735336,
  0.4043421035464688,
  0.40410879904574315,
  0.40388147828648513,
  0.403659783948294,
  0.4034433897912177,
  0.4032319971525013,
  0.40302533192342654,
  0.402823141929393,
  0.4026251946503726],
 'I': [0.1,
  0.2,
  0.3,
  0.4,
  0.5,
  0.6,
  0.7,
  0.8,
  0.9,
  1.0,
  1.1,
  1.2,
  1.3,
  1.4,
  1.5,
  1.6,
  1.7,
  1.8,
  1.9,
  2.0,
  2.1,
  2.2,
  2.3,
  2.4,
  2.5,
  2.6,
  2.7,
  2.8,
  2.9,
  3.0,
  3.1,
  3.2,
  3.3,
  3.4,
  3.5,
  3.6,
  3.7,
  3.8,
  3.9],
 'K': -0.03641691581104169,
 'P': [0.33270041697899955,
  0.6587178778406679,
  0.9821751543651522,
  1.3039485875375147,
  1.6244543808123062,
  1.943937672649885,
  2.2625604423336885,
  2.580437815146949,
  2.897655870190834,
  3.2142814219368576,
  3.5303678463586916,
  3.845958769939376,
  4.161090520201494,
  4.475793817720427,
  4.790094982255358,
  5.104016815845523,
  5.417579264298068,
  5.730799922516047,
  6.04369442720244,
  6.356276766674075,
  6.668559528567949,
  6.980554100266,
  7.292270832809483,
  7.603719176256948,
  7.914907792446728,
  8.225844649691133,
  8.536537102882745,
  8.84699196171837,
  9.157215549165555,
  9.467213751856068,
  9.776992063753617,
  10.08655562418175,
  10.395909251094128,
  10.705057470308757,
  11.014004541300244,
  11.322754480042237,
  11.631311079310091,
  11.939677926787208,
  12.247858421264336],
 'PH2': [2.169018669476965,
  2.1690046190953503,
  2.1689905687137356,
  2.1689765183321215,
  2.1689624679505073,
  2.1689484175688927,
  2.1689343671872785,
  2.168920316805664,
  2.1689062664240497,
  2.168892216042435,
  2.168878165660821,
  2.1688641152792063,
  2.168850064897592,
  2.168836014515978,
  2.1688219641343633,
  2.1688079137527487,
  2.1687938633711346,
  2.16877981298952,
  2.1687657626079058,
  2.168751712226291,
  2.168737661844677,
  2.168723611463063,
  2.168709561081448,
  2.168695510699834,
  2.1686814603182194,
  2.168667409936605,
  2.1686533595549906,
  2.168639309173376,
  2.168625258791762,
  2.1686112084101477,
  2.168597158028533,
  2.168583107646919,
  2.1685690572653042,
  2.16855500688369,
  2.1685409565020755,
  2.168526906120461,
  2.1685128557388467,
  2.168498805357232,
  2.168484754975618],
 'PH2O': [2.669687710942903,
  2.6696704173475743,
  2.669653123752246,
  2.6696358301569174,
  2.669618536561589,
  2.6696012429662606,
  2.669583949370932,
  2.6695666557756033,
  2.6695493621802746,
  2.669532068584946,
  2.6695147749896178,
  2.669497481394289,
  2.669480187798961,
  2.6694628942036323,
  2.6694456006083036,
  2.669428307012975,
  2.6694110134176463,
  2.6693937198223177,
  2.669376426226989,
  2.669359132631661,
  2.669341839036332,
  2.669324545441004,
  2.6693072518456753,
  2.6692899582503467,
  2.669272664655018,
  2.6692553710596894,
  2.6692380774643607,
  2.6692207838690325,
  2.669203490273704,
  2.6691861966783756,
  2.669168903083047,
  2.6691516094877183,
  2.6691343158923897,
  2.669117022297061,
  2.6690997287017324,
  2.669082435106404,
  2.6690651415110755,
  2.669047847915747,
  2.6690305543204187],
 'PO2': [2.09696938340326,
  2.0969614505650616,
  2.0969535177268637,
  2.0969455848886653,
  2.096937652050467,
  2.0969297192122687,
  2.0969217863740703,
  2.096913853535872,
  2.0969059206976737,
  2.0968979878594753,
  2.096890055021277,
  2.0968821221830787,
  2.0968741893448803,
  2.096866256506682,
  2.0968583236684837,
  2.096850390830286,
  2.0968424579920875,
  2.096834525153889,
  2.096826592315691,
  2.0968186594774925,
  2.096810726639294,
  2.096802793801096,
  2.0967948609628975,
  2.096786928124699,
  2.096778995286501,
  2.0967710624483025,
  2.096763129610104,
  2.096755196771906,
  2.0967472639337075,
  2.0967393310955096,
  2.0967313982573113,
  2.096723465419113,
  2.0967155325809146,
  2.0967075997427163,
  2.096699666904518,
  2.0966917340663196,
  2.0966838012281213,
  2.096675868389923,
  2.0966679355517246],
 'Ph': [0.2822995830210005,
  0.5712821221593323,
  0.8628248456348478,
  1.1560514124624854,
  1.450545619187694,
  1.7460623273501152,
  2.0424395576663112,
  2.3395621848530515,
  2.6373441298091667,
  2.9357185780631427,
  3.234632153641309,
  3.5340412300606245,
  3.833909479798507,
  4.1342061822795735,
  4.434905017744642,
  4.7359831841544775,
  5.037420735701932,
  5.339200077483954,
  5.6413055727975605,
  5.943723233325926,
  6.246440471432053,
  6.5494458997340015,
  6.8527291671905175,
  7.156280823743052,
  7.460092207553273,
  7.764155350308868,
  8.068462897117257,
  8.37300803828163,
  8.677784450834444,
  8.982786248143931,
  9.288007936246384,
  9.593444375818251,
  9.899090748905872,
  10.204942529691243,
  10.510995458699757,
  10.817245519957764,
  11.123688920689911,
  11.43032207321279,
  11.737141578735665],
 'Status': True,
 'V': [3.3270041697899955,
  3.293589389203339,
  3.273917181217174,
  3.259871468843787,
  3.2489087616246124,
  3.2398961210831416,
  3.232229203333841,
  3.225547268933686,
  3.219617633545371,
  3.2142814219368576,
  3.2094253148715377,
  3.2049656416161465,
  3.2008388616934567,
  3.1969955840860194,
  3.1933966548369055,
  3.190010509903452,
  3.18681133194004,
  3.183777734731137,
  3.180891803790758,
  3.1781383833370374,
  3.1755045374133086,
  3.172979136484545,
  3.170552536004123,
  3.168216323440395,
  3.1659631169786913,
  3.163786403727359,
  3.1616804084750902,
  3.1596399863279894,
  3.1576605341950192,
  3.1557379172853564,
  3.1538684076624572,
  3.152048632556797,
  3.1502755306345844,
  3.1485463147966937,
  3.1468584403714983,
  3.1452095777895104,
  3.1435975890027272,
  3.1420205070492657,
  3.140476518272907],
 'V0': 3.263468110052509}
  • Notice : "Status", "V0", "K" and "EFF" , new in version 0.8
In [14]:
Data=Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
###########
Padulles-II-Model Simulation
###########
Analyzing . . .
I : 0.1
E : 3.0120122442199793 V
FC Efficiency : 0.4265389961269225 
FC Power : 0.33270041697899955 W
FC Voltage : 3.3270041697899955 V
PH2 : 2.169018669476965 atm
PH2O : 2.669687710942903 atm
PO2 : 2.09696938340326 atm
Power-Thermal : 0.2822995830210005 W
###########
I : 0.2
E : 3.012012104448671 V
FC Efficiency : 0.42225504989786394 
FC Power : 0.6587178778406679 W
FC Voltage : 3.293589389203339 V
PH2 : 2.1690046190953503 atm
PH2O : 2.6696704173475743 atm
PO2 : 2.0969614505650616 atm
Power-Thermal : 0.5712821221593323 W
###########
I : 0.3
E : 3.0120119646768333 V
FC Efficiency : 0.4197329719509197 
FC Power : 0.9821751543651522 W
FC Voltage : 3.273917181217174 V
PH2 : 2.1689905687137356 atm
PH2O : 2.669653123752246 atm
PO2 : 2.0969535177268637 atm
Power-Thermal : 0.8628248456348478 W
###########
I : 0.4
E : 3.0120118249044676 V
FC Efficiency : 0.41793223959535725 
FC Power : 1.3039485875375147 W
FC Voltage : 3.259871468843787 V
PH2 : 2.1689765183321215 atm
PH2O : 2.6696358301569174 atm
PO2 : 2.0969455848886653 atm
Power-Thermal : 1.1560514124624854 W
###########
I : 0.5
E : 3.0120116851315726 V
FC Efficiency : 0.41652676431084773 
FC Power : 1.6244543808123062 W
FC Voltage : 3.2489087616246124 V
PH2 : 2.1689624679505073 atm
PH2O : 2.669618536561589 atm
PO2 : 2.096937652050467 atm
Power-Thermal : 1.450545619187694 W
###########
I : 0.6
E : 3.012011545358149 V
FC Efficiency : 0.4153712975747617 
FC Power : 1.943937672649885 W
FC Voltage : 3.2398961210831416 V
PH2 : 2.1689484175688927 atm
PH2O : 2.6696012429662606 atm
PO2 : 2.0969297192122687 atm
Power-Thermal : 1.7460623273501152 W
###########
I : 0.7
E : 3.0120114055841967 V
FC Efficiency : 0.4143883594017745 
FC Power : 2.2625604423336885 W
FC Voltage : 3.232229203333841 V
PH2 : 2.1689343671872785 atm
PH2O : 2.669583949370932 atm
PO2 : 2.0969217863740703 atm
Power-Thermal : 2.0424395576663112 W
###########
I : 0.8
E : 3.0120112658097153 V
FC Efficiency : 0.4135317011453444 
FC Power : 2.580437815146949 W
FC Voltage : 3.225547268933686 V
PH2 : 2.168920316805664 atm
PH2O : 2.6695666557756033 atm
PO2 : 2.096913853535872 atm
Power-Thermal : 2.3395621848530515 W
###########
I : 0.9
E : 3.0120111260347056 V
FC Efficiency : 0.41277149148017567 
FC Power : 2.897655870190834 W
FC Voltage : 3.219617633545371 V
PH2 : 2.1689062664240497 atm
PH2O : 2.6695493621802746 atm
PO2 : 2.0969059206976737 atm
Power-Thermal : 2.6373441298091667 W
###########
I : 1.0
E : 3.0120109862591664 V
FC Efficiency : 0.4120873617867766 
FC Power : 3.2142814219368576 W
FC Voltage : 3.2142814219368576 V
PH2 : 2.168892216042435 atm
PH2O : 2.669532068584946 atm
PO2 : 2.0968979878594753 atm
Power-Thermal : 2.9357185780631427 W
###########
I : 1.1
E : 3.0120108464830992 V
FC Efficiency : 0.4114647839578894 
FC Power : 3.5303678463586916 W
FC Voltage : 3.2094253148715377 V
PH2 : 2.168878165660821 atm
PH2O : 2.6695147749896178 atm
PO2 : 2.096890055021277 atm
Power-Thermal : 3.234632153641309 W
###########
I : 1.2
E : 3.0120107067065027 V
FC Efficiency : 0.41089303097642904 
FC Power : 3.845958769939376 W
FC Voltage : 3.2049656416161465 V
PH2 : 2.1688641152792063 atm
PH2O : 2.669497481394289 atm
PO2 : 2.0968821221830787 atm
Power-Thermal : 3.5340412300606245 W
###########
I : 1.3
E : 3.0120105669293777 V
FC Efficiency : 0.4103639566273662 
FC Power : 4.161090520201494 W
FC Voltage : 3.2008388616934567 V
PH2 : 2.168850064897592 atm
PH2O : 2.669480187798961 atm
PO2 : 2.0968741893448803 atm
Power-Thermal : 3.833909479798507 W
###########
I : 1.4
E : 3.012010427151724 V
FC Efficiency : 0.40987122872897686 
FC Power : 4.475793817720427 W
FC Voltage : 3.1969955840860194 V
PH2 : 2.168836014515978 atm
PH2O : 2.6694628942036323 atm
PO2 : 2.096866256506682 atm
Power-Thermal : 4.1342061822795735 W
###########
I : 1.5
E : 3.012010287373541 V
FC Efficiency : 0.409409827543193 
FC Power : 4.790094982255358 W
FC Voltage : 3.1933966548369055 V
PH2 : 2.1688219641343633 atm
PH2O : 2.6694456006083036 atm
PO2 : 2.0968583236684837 atm
Power-Thermal : 4.434905017744642 W
###########
I : 1.6
E : 3.0120101475948298 V
FC Efficiency : 0.40897570639787845 
FC Power : 5.104016815845523 W
FC Voltage : 3.190010509903452 V
PH2 : 2.1688079137527487 atm
PH2O : 2.669428307012975 atm
PO2 : 2.096850390830286 atm
Power-Thermal : 4.7359831841544775 W
###########
I : 1.7
E : 3.012010007815589 V
FC Efficiency : 0.4085655553769282 
FC Power : 5.417579264298068 W
FC Voltage : 3.18681133194004 V
PH2 : 2.1687938633711346 atm
PH2O : 2.6694110134176463 atm
PO2 : 2.0968424579920875 atm
Power-Thermal : 5.037420735701932 W
###########
I : 1.8
E : 3.01200986803582 V
FC Efficiency : 0.40817663265783805 
FC Power : 5.730799922516047 W
FC Voltage : 3.183777734731137 V
PH2 : 2.16877981298952 atm
PH2O : 2.6693937198223177 atm
PO2 : 2.096834525153889 atm
Power-Thermal : 5.339200077483954 W
###########
I : 1.9
E : 3.012009728255522 V
FC Efficiency : 0.4078066415116356 
FC Power : 6.04369442720244 W
FC Voltage : 3.180891803790758 V
PH2 : 2.1687657626079058 atm
PH2O : 2.669376426226989 atm
PO2 : 2.096826592315691 atm
Power-Thermal : 5.6413055727975605 W
###########
I : 2.0
E : 3.012009588474695 V
FC Efficiency : 0.40745363888936376 
FC Power : 6.356276766674075 W
FC Voltage : 3.1781383833370374 V
PH2 : 2.168751712226291 atm
PH2O : 2.669359132631661 atm
PO2 : 2.0968186594774925 atm
Power-Thermal : 5.943723233325926 W
###########
I : 2.1
E : 3.0120094486933398 V
FC Efficiency : 0.40711596633503955 
FC Power : 6.668559528567949 W
FC Voltage : 3.1755045374133086 V
PH2 : 2.168737661844677 atm
PH2O : 2.669341839036332 atm
PO2 : 2.096810726639294 atm
Power-Thermal : 6.246440471432053 W
###########
I : 2.2
E : 3.0120093089114555 V
FC Efficiency : 0.4067921969851981 
FC Power : 6.980554100266 W
FC Voltage : 3.172979136484545 V
PH2 : 2.168723611463063 atm
PH2O : 2.669324545441004 atm
PO2 : 2.096802793801096 atm
Power-Thermal : 6.5494458997340015 W
###########
I : 2.3
E : 3.012009169129042 V
FC Efficiency : 0.4064810943595029 
FC Power : 7.292270832809483 W
FC Voltage : 3.170552536004123 V
PH2 : 2.168709561081448 atm
PH2O : 2.6693072518456753 atm
PO2 : 2.0967948609628975 atm
Power-Thermal : 6.8527291671905175 W
###########
I : 2.4
E : 3.0120090293460997 V
FC Efficiency : 0.40618157992825576 
FC Power : 7.603719176256948 W
FC Voltage : 3.168216323440395 V
PH2 : 2.168695510699834 atm
PH2O : 2.6692899582503467 atm
PO2 : 2.096786928124699 atm
Power-Thermal : 7.156280823743052 W
###########
I : 2.5
E : 3.0120088895626287 V
FC Efficiency : 0.40589270730496035 
FC Power : 7.914907792446728 W
FC Voltage : 3.1659631169786913 V
PH2 : 2.1686814603182194 atm
PH2O : 2.669272664655018 atm
PO2 : 2.096778995286501 atm
Power-Thermal : 7.460092207553273 W
###########
I : 2.6
E : 3.0120087497786283 V
FC Efficiency : 0.40561364150350754 
FC Power : 8.225844649691133 W
FC Voltage : 3.163786403727359 V
PH2 : 2.168667409936605 atm
PH2O : 2.6692553710596894 atm
PO2 : 2.0967710624483025 atm
Power-Thermal : 7.764155350308868 W
###########
I : 2.7
E : 3.0120086099941004 V
FC Efficiency : 0.405343642112191 
FC Power : 8.536537102882745 W
FC Voltage : 3.1616804084750902 V
PH2 : 2.1686533595549906 atm
PH2O : 2.6692380774643607 atm
PO2 : 2.096763129610104 atm
Power-Thermal : 8.068462897117257 W
###########
I : 2.8
E : 3.0120084702090426 V
FC Efficiency : 0.4050820495292294 
FC Power : 8.84699196171837 W
FC Voltage : 3.1596399863279894 V
PH2 : 2.168639309173376 atm
PH2O : 2.6692207838690325 atm
PO2 : 2.096755196771906 atm
Power-Thermal : 8.37300803828163 W
###########
I : 2.9
E : 3.0120083304234564 V
FC Efficiency : 0.404828273614746 
FC Power : 9.157215549165555 W
FC Voltage : 3.1576605341950192 V
PH2 : 2.168625258791762 atm
PH2O : 2.669203490273704 atm
PO2 : 2.0967472639337075 atm
Power-Thermal : 8.677784450834444 W
###########
I : 3.0
E : 3.012008190637341 V
FC Efficiency : 0.40458178426735336 
FC Power : 9.467213751856068 W
FC Voltage : 3.1557379172853564 V
PH2 : 2.1686112084101477 atm
PH2O : 2.6691861966783756 atm
PO2 : 2.0967393310955096 atm
Power-Thermal : 8.982786248143931 W
###########
I : 3.1
E : 3.0120080508506963 V
FC Efficiency : 0.4043421035464688 
FC Power : 9.776992063753617 W
FC Voltage : 3.1538684076624572 V
PH2 : 2.168597158028533 atm
PH2O : 2.669168903083047 atm
PO2 : 2.0967313982573113 atm
Power-Thermal : 9.288007936246384 W
###########
I : 3.2
E : 3.0120079110635234 V
FC Efficiency : 0.40410879904574315 
FC Power : 10.08655562418175 W
FC Voltage : 3.152048632556797 V
PH2 : 2.168583107646919 atm
PH2O : 2.6691516094877183 atm
PO2 : 2.096723465419113 atm
Power-Thermal : 9.593444375818251 W
###########
I : 3.3
E : 3.0120077712758215 V
FC Efficiency : 0.40388147828648513 
FC Power : 10.395909251094128 W
FC Voltage : 3.1502755306345844 V
PH2 : 2.1685690572653042 atm
PH2O : 2.6691343158923897 atm
PO2 : 2.0967155325809146 atm
Power-Thermal : 9.899090748905872 W
###########
I : 3.4
E : 3.0120076314875908 V
FC Efficiency : 0.403659783948294 
FC Power : 10.705057470308757 W
FC Voltage : 3.1485463147966937 V
PH2 : 2.16855500688369 atm
PH2O : 2.669117022297061 atm
PO2 : 2.0967075997427163 atm
Power-Thermal : 10.204942529691243 W
###########
I : 3.5
E : 3.012007491698831 V
FC Efficiency : 0.4034433897912177 
FC Power : 11.014004541300244 W
FC Voltage : 3.1468584403714983 V
PH2 : 2.1685409565020755 atm
PH2O : 2.6690997287017324 atm
PO2 : 2.096699666904518 atm
Power-Thermal : 10.510995458699757 W
###########
I : 3.6
E : 3.0120073519095425 V
FC Efficiency : 0.4032319971525013 
FC Power : 11.322754480042237 W
FC Voltage : 3.1452095777895104 V
PH2 : 2.168526906120461 atm
PH2O : 2.669082435106404 atm
PO2 : 2.0966917340663196 atm
Power-Thermal : 10.817245519957764 W
###########
I : 3.7
E : 3.012007212119725 V
FC Efficiency : 0.40302533192342654 
FC Power : 11.631311079310091 W
FC Voltage : 3.1435975890027272 V
PH2 : 2.1685128557388467 atm
PH2O : 2.6690651415110755 atm
PO2 : 2.0966838012281213 atm
Power-Thermal : 11.123688920689911 W
###########
I : 3.8
E : 3.0120070723293786 V
FC Efficiency : 0.402823141929393 
FC Power : 11.939677926787208 W
FC Voltage : 3.1420205070492657 V
PH2 : 2.168498805357232 atm
PH2O : 2.669047847915747 atm
PO2 : 2.096675868389923 atm
Power-Thermal : 11.43032207321279 W
###########
I : 3.9
E : 3.0120069325385037 V
FC Efficiency : 0.4026251946503726 
FC Power : 12.247858421264336 W
FC Voltage : 3.140476518272907 V
PH2 : 2.168484754975618 atm
PH2O : 2.6690305543204187 atm
PO2 : 2.0966679355517246 atm
Power-Thermal : 11.737141578735665 W
###########
Done!
  • Notice : "PrintMode" & "ReportMode" , new in version 0.5
In [15]:
Dynamic_Analysis(InputMethod={},TestMode=True,PrintMode=False,ReportMode=True)
Out[15]:
{'Message': '[Error] Padulles-II Simulation Failed!(Check Your Inputs)',
 'Status': False}

Parameters

Inputs, Constants & Middle Values

  1. User : User Input
  2. System : Simulator Calculation (Middle Value)
$$Parameter$$ $$Description$$ $$Unit$$ $$Value$$
$$E_0$$ No load voltage $$V$$ $$User$$
$$K_{H_2}$$ Hydrogen valve constant $$kmol.s^{-1}.atm^{-1}$$ $$User$$
$$K_{H_2O}$$ Water valve constant $$kmol.s^{-1}.atm^{-1}$$ $$User$$
$$K_{O_2}$$ Oxygen valve constant $$kmol.s^{-1}.atm^{-1}$$ $$User$$
$$\tau_{H_2}^{(s)}$$ Hydrogen time constant $$s$$ $$User$$
$$\tau_{H_2O}^{(s)}$$ Water time constant $$s$$ $$User$$
$$\tau_{O_2}^{(s)}$$ Oxygen time constant $$s$$ $$User$$
$$B$$ Activation voltage constant $$V$$ $$User$$
$$C$$ Activation constant parameter $$A^{-1}$$ $$User$$
$$R^{int}$$ FuelCell internal resistance $$\Omega$$ $$User$$
$$r_{h-o}$$ Hydrogen-oxygen flow ratio $$--$$ $$User$$
$$q_{H_2}^{(inlet)}$$ Molar flow of hydrogen $$kmol.s^{-1}$$ $$User$$
$$i_{start}$$ Cell operating current start point $$A$$ $$User$$
$$i_{step}$$ Cell operating current step $$A$$ $$User$$
$$i_{stop}$$ Cell operating current end point $$A$$ $$User$$
$$P_{H_2}$$ Hydrogen partial pressure $$atm$$ $$System$$
$$P_{H_2O}$$ Water partial pressure $$atm$$ $$System$$
$$P_{O_2}$$ Oxygen partial pressure $$atm$$ $$System$$
$$K_r$$ Modeling constant $$kmol.s^{-1}.A^{-1}$$ $$System$$
$$q_{O_2}^{(inlet)}$$ Molar flow of oxygen $$kmol.s^{-1}$$ $$System$$
$$\mu_F$$ The fuel utilization $$--$$ $$0.95$$
$$HHV$$ Higher Heating Value Potential $$V$$ $$1.482$$
$$R$$ Universal gas constant $$J.kmol^{-1}.K^{-1}$$ $$8314.47$$
$$F$$ Faraday’s constant $$C.kmol^{-1}$$ $$96484600$$
$$E_{th}$$ Theoretical Potential $$V$$ $$1.23$$
  • Notice : $$q_{H_2}=q_{H_2O}$$

Reference

J. Padulles, G.W. Ault, J.R. McDonald. 2000. "An integrated SOFC plant dynamic model for power systems simulation." Journal of Power Sources (Elsevier) 86 (1-2): 495-500. doi:10.1016/S0378-7753(99)00430-9