Padulles-Hauer Dynamic Model

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

Padulles-Hauer Dynamic Model is a dynamic electrochemical simulation model of a grid independent proton exchange membrane (PEM) fuel cell. This model includes a methanol reformer to generate hydrogen from methanol and the PEM stack. The model is used to predict the output voltage and power of a PEMFC. It has to be noted that the reformer model is a second order transfer function.

Fig1. Padulles-Hauer Dynamic Model Block Diagram

Nernst Voltage

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

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

$$\frac{q_{H_2}^{(inlet)}}{q_{methanol}}=\frac{CV}{\tau_{1}^{(s)}+(\tau_{2}^{(s)})^2+(\tau_{1}+\tau_{2})^{(s)}+1}$$

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

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

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

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

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

In [1]:
from opem.Dynamic.Padulles_Hauer import qH2_Calc,Kr_Calc,qO2_Calc,PH2_Calc,PO2_Calc,PH2O_Calc,Enernst_Calc
In [2]:
qH2=qH2_Calc(qMethanol=0.0002,CV=2,t1=2,t2=2)
qH2
Out[2]:
3.6363636363636364e-05
In [3]:
Kr=Kr_Calc(N0=5)
Kr
Out[3]:
1.2955435375179044e-08
In [4]:
qO2=qO2_Calc(qH2=qH2,rho=1.168)
qO2
Out[4]:
3.113325031133251e-05
In [5]:
PH2=PH2_Calc(KH2=0.0000422,tH2=3.37,Kr=Kr,I=1,qH2=qH2)
PH2
Out[5]:
0.19704428889827239
In [6]:
PO2=PO2_Calc(KO2=0.0000211,tO2=6.74,Kr=Kr,I=1,qO2=qO2)
PO2
Out[6]:
0.1905549730945132
In [7]:
PH2O=PH2O_Calc(KH2O=0.000007716,tH2O=18.418,Kr=Kr,I=1,qH2O=qH2)
PH2O
Out[7]:
0.2425275190047356
In [8]:
Enernst=Enernst_Calc(E0=0.6,N0=5,T=343, PH2=PH2, PO2=PO2,PH2O=PH2O)
Enernst
Out[8]:
2.9234016588243374

FC Voltage

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

In [9]:
from opem.Dynamic.Padulles_Hauer import Vcell_Calc
FC_Voltage=Vcell_Calc(Enernst=Enernst, B=0.04777,C=0.0136,I=1,Rint=0.00303)
FC_Voltage
Out[9]:
3.1256720945020287

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 [10]:
from opem.Dynamic.Padulles_Hauer import Power_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=FC_Voltage,i=2)
Power
Out[10]:
6.251344189004057
In [11]:
Power_Thermal_Calc(VStack=FC_Voltage,N=5,i=2)
Out[11]:
6.048655810995943

Efficiency of PEMFC

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

In [12]:
from opem.Dynamic.Padulles_Hauer import Efficiency_Calc
Efficiency_Calc(Vcell=FC_Voltage,N=5)
Out[12]:
0.40072719160282416

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 [13]:
Test_Vector={"T":343,"E0":0.6,"N0":5,"KO2":0.0000211,"KH2":0.0000422,"KH2O":0.000007716,"tH2":3.37,"tO2":6.74,"t1":2,"t2":2,"tH2O":18.418,"B":0.04777,"C":0.0136,"Rint":0.00303,"rho":1.168,"qMethanol":0.0002,"CV":2,"i-start":0.1,"i-stop":4,"i-step":0.1,"Name":"Test"}
  • Notice : "Name", new in version 0.5
In [14]:
from opem.Dynamic.Padulles_Hauer import Dynamic_Analysis
Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
Out[14]:
{'EFF': [0.41518043908246366,
  0.4108963136482338,
  0.4083740564879825,
  0.4065731449109761,
  0.4051674903968853,
  0.4040118444230801,
  0.4030287270042349,
  0.4021718894938075,
  0.4014115005665013,
  0.40072719160282416,
  0.40010443449551725,
  0.39953250222749515,
  0.3990032485837277,
  0.3985103413824903,
  0.3980487608857143,
  0.39761446042126253,
  0.3972041300730298,
  0.39681502801851076,
  0.3964448575287326,
  0.3960916755547374,
  0.39575382364054146,
  0.3954298749226794,
  0.39511859292081414,
  0.39481889910524637,
  0.39452984708947947,
  0.39425060188740335,
  0.3939804230873111,
  0.3937186510874208,
  0.3934646957478549,
  0.39321802696722546,
  0.39297816680494896,
  0.39274468285467545,
  0.3925171826377131,
  0.39229530883366054,
  0.39207873520256487,
  0.3918671630816706,
  0.3916603183622587,
  0.3914579488697281,
  0.3912598220840501],
 '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': -0.03643090556526363,
 'P': [0.3238407424843217,
  0.6409982492912448,
  0.955595292181879,
  1.2685082121222457,
  1.5801532125478528,
  1.8907754319000147,
  2.2005368494431226,
  2.509552590441359,
  2.81790873397684,
  3.1256720945020287,
  3.4328960479715387,
  3.7396242208493544,
  4.045892940639,
  4.351732927896794,
  4.657170502362857,
  4.962228466057358,
  5.2669267647683755,
  5.571282993379892,
  5.875312788575817,
  6.1790301386539035,
  6.482447631232071,
  6.78557665367318,
  7.088427556999405,
  7.391009791250212,
  7.693332018244849,
  7.995402206276542,
  8.297227710218774,
  8.59881533974927,
  8.900171417816479,
  9.201301831033076,
  9.502212073343667,
  9.8029072840527,
  10.103392281094736,
  10.403671590268678,
  10.703749471030022,
  11.003629939333312,
  11.303316787934786,
  11.602813604498742,
  11.902123787796803],
 'PH2': [0.19717074233280188,
  0.1971566919511875,
  0.19714264156957312,
  0.19712859118795872,
  0.19711454080634436,
  0.19710049042472996,
  0.19708644004311557,
  0.19707238966150117,
  0.19705833927988675,
  0.19704428889827239,
  0.197030238516658,
  0.1970161881350436,
  0.19700213775342923,
  0.19698808737181484,
  0.19697403699020044,
  0.19695998660858605,
  0.19694593622697168,
  0.19693188584535729,
  0.1969178354637429,
  0.19690378508212852,
  0.19688973470051413,
  0.19687568431889974,
  0.19686163393728537,
  0.19684758355567097,
  0.19683353317405658,
  0.19681948279244216,
  0.19680543241082776,
  0.1967913820292134,
  0.196777331647599,
  0.1967632812659846,
  0.19674923088437024,
  0.19673518050275585,
  0.19672113012114145,
  0.19670707973952706,
  0.1966930293579127,
  0.1966789789762983,
  0.1966649285946839,
  0.19665087821306954,
  0.19663682783145514],
 'PH2O': [0.2426831613626925,
  0.24266586776736396,
  0.24264857417203542,
  0.24263128057670688,
  0.24261398698137834,
  0.2425966933860498,
  0.24257939979072127,
  0.24256210619539273,
  0.24254481260006414,
  0.2425275190047356,
  0.24251022540940706,
  0.24249293181407852,
  0.24247563821874998,
  0.24245834462342142,
  0.24244105102809288,
  0.24242375743276434,
  0.2424064638374358,
  0.24238917024210727,
  0.24237187664677873,
  0.2423545830514502,
  0.24233728945612165,
  0.2423199958607931,
  0.24230270226546458,
  0.242285408670136,
  0.24226811507480747,
  0.2422508214794789,
  0.24223352788415034,
  0.2422162342888218,
  0.24219894069349326,
  0.24218164709816473,
  0.2421643535028362,
  0.24214705990750765,
  0.2421297663121791,
  0.24211247271685057,
  0.24209517912152204,
  0.2420778855261935,
  0.24206059193086493,
  0.2420432983355364,
  0.24202600474020786],
 'PO2': [0.1906263686382979,
  0.1906184358000996,
  0.1906105029619013,
  0.19060257012370302,
  0.1905946372855047,
  0.1905867044473064,
  0.19057877160910808,
  0.1905708387709098,
  0.19056290593271147,
  0.1905549730945132,
  0.19054704025631486,
  0.19053910741811658,
  0.19053117457991825,
  0.19052324174171997,
  0.19051530890352164,
  0.19050737606532336,
  0.19049944322712503,
  0.19049151038892673,
  0.19048357755072845,
  0.19047564471253012,
  0.19046771187433184,
  0.1904597790361335,
  0.19045184619793523,
  0.1904439133597369,
  0.19043598052153862,
  0.1904280476833403,
  0.190420114845142,
  0.19041218200694368,
  0.1904042491687454,
  0.19039631633054707,
  0.1903883834923488,
  0.19038045065415046,
  0.19037251781595219,
  0.19036458497775385,
  0.19035665213955555,
  0.19034871930135727,
  0.19034078646315894,
  0.19033285362496066,
  0.19032492078676233],
 'Ph': [0.2911592575156784,
  0.5890017507087553,
  0.889404707818121,
  1.1914917878777547,
  1.4948467874521474,
  1.7992245680999854,
  2.1044631505568776,
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  3.0243279054979717,
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  11.13637006066669,
  11.451683212065214,
  11.767186395501259,
  12.082876212203196],
 'Status': True,
 'V': [3.2384074248432166,
  3.204991246456224,
  3.1853176406062635,
  3.171270530305614,
  3.1603064250957056,
  3.1512923865000246,
  3.1436240706330323,
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  3.0690246268332686,
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  3.0565638720370307,
  3.054950483225618,
  3.0533720011838796,
  3.051826612255591],
 'V0': 3.1748727715256186}
  • Notice : "Status", "V0", "K" and "EFF" , new in version 0.8
In [15]:
Data=Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
###########
Padulles-Hauer-Model Simulation
###########
Analyzing . . .
I : 0.1
E : 2.9234154992732004 V
FC Efficiency : 0.41518043908246366 
FC Power : 0.3238407424843217 W
FC Voltage : 3.2384074248432166 V
PH2 : 0.19717074233280188 atm
PH2O : 0.2426831613626925 atm
PO2 : 0.1906263686382979 atm
Power-Thermal : 0.2911592575156784 W
###########
I : 0.2
E : 2.9234139617015558 V
FC Efficiency : 0.4108963136482338 
FC Power : 0.6409982492912448 W
FC Voltage : 3.204991246456224 V
PH2 : 0.1971566919511875 atm
PH2O : 0.24266586776736396 atm
PO2 : 0.1906184358000996 atm
Power-Thermal : 0.5890017507087553 W
###########
I : 0.3
E : 2.9234124240659227 V
FC Efficiency : 0.4083740564879825 
FC Power : 0.955595292181879 W
FC Voltage : 3.1853176406062635 V
PH2 : 0.19714264156957312 atm
PH2O : 0.24264857417203542 atm
PO2 : 0.1906105029619013 atm
Power-Thermal : 0.889404707818121 W
###########
I : 0.4
E : 2.9234108863662946 V
FC Efficiency : 0.4065731449109761 
FC Power : 1.2685082121222457 W
FC Voltage : 3.171270530305614 V
PH2 : 0.19712859118795872 atm
PH2O : 0.24263128057670688 atm
PO2 : 0.19060257012370302 atm
Power-Thermal : 1.1914917878777547 W
###########
I : 0.5
E : 2.9234093486026658 V
FC Efficiency : 0.4051674903968853 
FC Power : 1.5801532125478528 W
FC Voltage : 3.1603064250957056 V
PH2 : 0.19711454080634436 atm
PH2O : 0.24261398698137834 atm
PO2 : 0.1905946372855047 atm
Power-Thermal : 1.4948467874521474 W
###########
I : 0.6
E : 2.923407810775032 V
FC Efficiency : 0.4040118444230801 
FC Power : 1.8907754319000147 W
FC Voltage : 3.1512923865000246 V
PH2 : 0.19710049042472996 atm
PH2O : 0.2425966933860498 atm
PO2 : 0.1905867044473064 atm
Power-Thermal : 1.7992245680999854 W
###########
I : 0.7
E : 2.923406272883388 V
FC Efficiency : 0.4030287270042349 
FC Power : 2.2005368494431226 W
FC Voltage : 3.1436240706330323 V
PH2 : 0.19708644004311557 atm
PH2O : 0.24257939979072127 atm
PO2 : 0.19057877160910808 atm
Power-Thermal : 2.1044631505568776 W
###########
I : 0.8
E : 2.9234047349277277 V
FC Efficiency : 0.4021718894938075 
FC Power : 2.509552590441359 W
FC Voltage : 3.1369407380516985 V
PH2 : 0.19707238966150117 atm
PH2O : 0.24256210619539273 atm
PO2 : 0.1905708387709098 atm
Power-Thermal : 2.4104474095586417 W
###########
I : 0.9
E : 2.9234031969080454 V
FC Efficiency : 0.4014115005665013 
FC Power : 2.81790873397684 W
FC Voltage : 3.1310097044187106 V
PH2 : 0.19705833927988675 atm
PH2O : 0.24254481260006414 atm
PO2 : 0.19056290593271147 atm
Power-Thermal : 2.7170912660231608 W
###########
I : 1.0
E : 2.9234016588243374 V
FC Efficiency : 0.40072719160282416 
FC Power : 3.1256720945020287 W
FC Voltage : 3.1256720945020287 V
PH2 : 0.19704428889827239 atm
PH2O : 0.2425275190047356 atm
PO2 : 0.1905549730945132 atm
Power-Thermal : 3.0243279054979717 W
###########
I : 1.1
E : 2.9234001206765963 V
FC Efficiency : 0.40010443449551725 
FC Power : 3.4328960479715387 W
FC Voltage : 3.1208145890650347 V
PH2 : 0.197030238516658 atm
PH2O : 0.24251022540940706 atm
PO2 : 0.19054704025631486 atm
Power-Thermal : 3.3321039520284623 W
###########
I : 1.2
E : 2.9233985824648183 V
FC Efficiency : 0.39953250222749515 
FC Power : 3.7396242208493544 W
FC Voltage : 3.116353517374462 V
PH2 : 0.1970161881350436 atm
PH2O : 0.24249293181407852 atm
PO2 : 0.19053910741811658 atm
Power-Thermal : 3.640375779150646 W
###########
I : 1.3
E : 2.923397044188998 V
FC Efficiency : 0.3990032485837277 
FC Power : 4.045892940639 W
FC Voltage : 3.1122253389530767 V
PH2 : 0.19700213775342923 atm
PH2O : 0.24247563821874998 atm
PO2 : 0.19053117457991825 atm
Power-Thermal : 3.9491070593610007 W
###########
I : 1.4
E : 2.923395505849129 V
FC Efficiency : 0.3985103413824903 
FC Power : 4.351732927896794 W
FC Voltage : 3.1083806627834245 V
PH2 : 0.19698808737181484 atm
PH2O : 0.24245834462342142 atm
PO2 : 0.19052324174171997 atm
Power-Thermal : 4.258267072103206 W
###########
I : 1.5
E : 2.923393967445207 V
FC Efficiency : 0.3980487608857143 
FC Power : 4.657170502362857 W
FC Voltage : 3.1047803349085714 V
PH2 : 0.19697403699020044 atm
PH2O : 0.24244105102809288 atm
PO2 : 0.19051530890352164 atm
Power-Thermal : 4.567829497637144 W
###########
I : 1.6
E : 2.923392428977226 V
FC Efficiency : 0.39761446042126253 
FC Power : 4.962228466057358 W
FC Voltage : 3.101392791285848 V
PH2 : 0.19695998660858605 atm
PH2O : 0.24242375743276434 atm
PO2 : 0.19050737606532336 atm
Power-Thermal : 4.877771533942644 W
###########
I : 1.7
E : 2.9233908904451815 V
FC Efficiency : 0.3972041300730298 
FC Power : 5.2669267647683755 W
FC Voltage : 3.098192214569633 V
PH2 : 0.19694593622697168 atm
PH2O : 0.2424064638374358 atm
PO2 : 0.19049944322712503 atm
Power-Thermal : 5.188073235231625 W
###########
I : 1.8
E : 2.9233893518490675 V
FC Efficiency : 0.39681502801851076 
FC Power : 5.571282993379892 W
FC Voltage : 3.0951572185443843 V
PH2 : 0.19693188584535729 atm
PH2O : 0.24238917024210727 atm
PO2 : 0.19049151038892673 atm
Power-Thermal : 5.498717006620109 W
###########
I : 1.9
E : 2.9233878131888784 V
FC Efficiency : 0.3964448575287326 
FC Power : 5.875312788575817 W
FC Voltage : 3.0922698887241142 V
PH2 : 0.1969178354637429 atm
PH2O : 0.24237187664677873 atm
PO2 : 0.19048357755072845 atm
Power-Thermal : 5.809687211424183 W
###########
I : 2.0
E : 2.9233862744646095 V
FC Efficiency : 0.3960916755547374 
FC Power : 6.1790301386539035 W
FC Voltage : 3.0895150693269517 V
PH2 : 0.19690378508212852 atm
PH2O : 0.2423545830514502 atm
PO2 : 0.19047564471253012 atm
Power-Thermal : 6.120969861346097 W
###########
I : 2.1
E : 2.923384735676255 V
FC Efficiency : 0.39575382364054146 
FC Power : 6.482447631232071 W
FC Voltage : 3.086879824396224 V
PH2 : 0.19688973470051413 atm
PH2O : 0.24233728945612165 atm
PO2 : 0.19046771187433184 atm
Power-Thermal : 6.432552368767931 W
###########
I : 2.2
E : 2.92338319682381 V
FC Efficiency : 0.3954298749226794 
FC Power : 6.78557665367318 W
FC Voltage : 3.0843530243968997 V
PH2 : 0.19687568431889974 atm
PH2O : 0.2423199958607931 atm
PO2 : 0.1904597790361335 atm
Power-Thermal : 6.744423346326822 W
###########
I : 2.3
E : 2.923381657907269 V
FC Efficiency : 0.39511859292081414 
FC Power : 7.088427556999405 W
FC Voltage : 3.0819250247823504 V
PH2 : 0.19686163393728537 atm
PH2O : 0.24230270226546458 atm
PO2 : 0.19045184619793523 atm
Power-Thermal : 7.0565724430005945 W
###########
I : 2.4
E : 2.9233801189266266 V
FC Efficiency : 0.39481889910524637 
FC Power : 7.391009791250212 W
FC Voltage : 3.079587413020922 V
PH2 : 0.19684758355567097 atm
PH2O : 0.242285408670136 atm
PO2 : 0.1904439133597369 atm
Power-Thermal : 7.368990208749787 W
###########
I : 2.5
E : 2.923378579881877 V
FC Efficiency : 0.39452984708947947 
FC Power : 7.693332018244849 W
FC Voltage : 3.07733280729794 V
PH2 : 0.19683353317405658 atm
PH2O : 0.24226811507480747 atm
PO2 : 0.19043598052153862 atm
Power-Thermal : 7.681667981755151 W
###########
I : 2.6
E : 2.923377040773016 V
FC Efficiency : 0.39425060188740335 
FC Power : 7.995402206276542 W
FC Voltage : 3.0751546947217467 V
PH2 : 0.19681948279244216 atm
PH2O : 0.2422508214794789 atm
PO2 : 0.1904280476833403 atm
Power-Thermal : 7.99459779372346 W
###########
I : 2.7
E : 2.923375501600037 V
FC Efficiency : 0.3939804230873111 
FC Power : 8.297227710218774 W
FC Voltage : 3.073047300081027 V
PH2 : 0.19680543241082776 atm
PH2O : 0.24223352788415034 atm
PO2 : 0.190420114845142 atm
Power-Thermal : 8.307772289781228 W
###########
I : 2.8
E : 2.923373962362936 V
FC Efficiency : 0.3937186510874208 
FC Power : 8.59881533974927 W
FC Voltage : 3.0710054784818825 V
PH2 : 0.1967913820292134 atm
PH2O : 0.2422162342888218 atm
PO2 : 0.19041218200694368 atm
Power-Thermal : 8.62118466025073 W
###########
I : 2.9
E : 2.9233724230617057 V
FC Efficiency : 0.3934646957478549 
FC Power : 8.900171417816479 W
FC Voltage : 3.0690246268332686 V
PH2 : 0.196777331647599 atm
PH2O : 0.24219894069349326 atm
PO2 : 0.1904042491687454 atm
Power-Thermal : 8.934828582183522 W
###########
I : 3.0
E : 2.923370883696343 V
FC Efficiency : 0.39321802696722546 
FC Power : 9.201301831033076 W
FC Voltage : 3.0671006103443585 V
PH2 : 0.1967632812659846 atm
PH2O : 0.24218164709816473 atm
PO2 : 0.19039631633054707 atm
Power-Thermal : 9.248698168966925 W
###########
I : 3.1
E : 2.9233693442668414 V
FC Efficiency : 0.39297816680494896 
FC Power : 9.502212073343667 W
FC Voltage : 3.0652297010786023 V
PH2 : 0.19674923088437024 atm
PH2O : 0.2421643535028362 atm
PO2 : 0.1903883834923488 atm
Power-Thermal : 9.562787926656334 W
###########
I : 3.2
E : 2.9233678047731946 V
FC Efficiency : 0.39274468285467545 
FC Power : 9.8029072840527 W
FC Voltage : 3.0634085262664685 V
PH2 : 0.19673518050275585 atm
PH2O : 0.24214705990750765 atm
PO2 : 0.19038045065415046 atm
Power-Thermal : 9.877092715947303 W
###########
I : 3.3
E : 2.9233662652153996 V
FC Efficiency : 0.3925171826377131 
FC Power : 10.103392281094736 W
FC Voltage : 3.0616340245741624 V
PH2 : 0.19672113012114145 atm
PH2O : 0.2421297663121791 atm
PO2 : 0.19037251781595219 atm
Power-Thermal : 10.191607718905265 W
###########
I : 3.4
E : 2.923364725593449 V
FC Efficiency : 0.39229530883366054 
FC Power : 10.403671590268678 W
FC Voltage : 3.059903408902552 V
PH2 : 0.19670707973952706 atm
PH2O : 0.24211247271685057 atm
PO2 : 0.19036458497775385 atm
Power-Thermal : 10.506328409731324 W
###########
I : 3.5
E : 2.923363185907339 V
FC Efficiency : 0.39207873520256487 
FC Power : 10.703749471030022 W
FC Voltage : 3.0582141345800062 V
PH2 : 0.1966930293579127 atm
PH2O : 0.24209517912152204 atm
PO2 : 0.19035665213955555 atm
Power-Thermal : 10.82125052896998 W
###########
I : 3.6
E : 2.923361646157063 V
FC Efficiency : 0.3918671630816706 
FC Power : 11.003629939333312 W
FC Voltage : 3.0565638720370307 V
PH2 : 0.1966789789762983 atm
PH2O : 0.2420778855261935 atm
PO2 : 0.19034871930135727 atm
Power-Thermal : 11.13637006066669 W
###########
I : 3.7
E : 2.923360106342616 V
FC Efficiency : 0.3916603183622587 
FC Power : 11.303316787934786 W
FC Voltage : 3.054950483225618 V
PH2 : 0.1966649285946839 atm
PH2O : 0.24206059193086493 atm
PO2 : 0.19034078646315894 atm
Power-Thermal : 11.451683212065214 W
###########
I : 3.8
E : 2.9233585664639925 V
FC Efficiency : 0.3914579488697281 
FC Power : 11.602813604498742 W
FC Voltage : 3.0533720011838796 V
PH2 : 0.19665087821306954 atm
PH2O : 0.2420432983355364 atm
PO2 : 0.19033285362496066 atm
Power-Thermal : 11.767186395501259 W
###########
I : 3.9
E : 2.9233570265211877 V
FC Efficiency : 0.3912598220840501 
FC Power : 11.902123787796803 W
FC Voltage : 3.051826612255591 V
PH2 : 0.19663682783145514 atm
PH2O : 0.24202600474020786 atm
PO2 : 0.19032492078676233 atm
Power-Thermal : 12.082876212203196 W
###########
Done!
  • Notice : "PrintMode" & "ReportMode" , new in version 0.5
In [16]:
Dynamic_Analysis(InputMethod={},TestMode=True,PrintMode=False,ReportMode=True)
Out[16]:
{'Message': '[Error] Padulles-Hauer Simulation Failed!(Check Your Inputs)',
 'Status': False}

Parameters

Inputs, Constants & Middle Values

  1. User : User Input
  2. System : Simulator Calculation (Middle Value)
$$Parameter$$ $$Description$$ $$Unit$$ $$Value$$
$$T$$ FuelCell temperature $$K$$ $$User$$
$$N_0$$ Number of cells $$--$$ $$User$$
$$E_0$$ 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 $$s$$ $$User$$
$$\tau_{1}^{(s)}$$ Reformer time constant $$s$$ $$User$$
$$\tau_{2}^{(s)}$$ Reformer time constant $$s$$ $$User$$
$$CV$$ Conversion factor $$--$$ $$User$$
$$B$$ 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 $$--$$ $$User$$
$$q_{methanol}$$ Molar flow of methanol $$kmol.s^{-1}$$ $$User$$
$$i_{start}$$ 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 $$kmol.s^{-1}$$ $$System$$
$$q_{H_2O}^{(inlet)}$$ Molar flow of water $$kmol.s^{-1}$$ $$System$$
$$q_{H_2}^{(inlet)}$$ Molar flow of hydrogen $$kmol.s^{-1}$$ $$System$$
$$\mu_F$$ 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 $$V$$ $$1.23$$
  • $$q_{H_2}=q_{H_2O}$$

Reference

1- 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
2- Hauer, K.-H. 2001. "Analysis tool for fuel cell vehicle hardware and software (controls) with an application to fuel economy comparisons of alternative system designs." Ph.D. dissertation, Transportation Technology and Policy, University of California Davis.