Padulles-Amphlett 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

This model is an integration of Padulles-Hauer dynamic model with Amphlett static model. The advantage of this dynamic model is using Amphlett equation for simulating the polarization values. Amphlett model as the most complicated and preferable static model, but the most precise. Based on this model, the obtained polarization voltage is identical to the experimental results.

Fig1. Padulles-Amphlett 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_Amphlett 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

PEM Losses Model

Activation

$$\eta_{activation}=\xi_{1}+\xi_{2}T+\xi_{3}T[ln(C_{O_{2}})]+\xi_{4}T[ln(i)]$$

$$\xi_{1}=-0.948$$

$$\xi_{2}=0.00286+0.0002\times ln(A)+(4.3\times10^{-5})[ln(C_{H_{2}})]$$

$$\xi_{3}=7.6\times10^{-5}$$

$$\xi_{4}=-1.93\times10^{-4}$$

$$C_{H_{2}}=\frac{P_{H_2}}{1.09\times10^{6}\times exp(\frac{77}{T})}$$

$$C_{O_{2}}=\frac{P_{O_2}}{5.08\times10^{6}\times exp(\frac{-498}{T})}$$

In [9]:
from opem.Dynamic.Padulles_Amphlett import Eta_Act_Calc
Eta_Act=Eta_Act_Calc(T=343,PO2=PO2 , PH2=PH2, i=2, A=50.6)
Eta_Act
Out[9]:
0.38390602046791616

Ohmic

$$\eta_{ohmic}=i(R_{electronic}+R_{Proton})$$

$$R_{Proton}=\frac{\rho_m\times I}{A}$$

$$\rho_m=\frac{181.6[1+0.03(\frac{i}{A})+0.062(\frac{T}{303})^2(\frac{i}{A})^{2.5}]}{[\lambda-0.634-3(\frac{i}{A})]exp[4.18(\frac{T-303}{T})]}$$

  • Relectronic Should be approximately constant over the relatively narrow temperature range of PEM fuel cell operation. Therefore, the parameter Relectronic can be taken as a constant, but is generally difficult to predict and, therefore, is initially an unknown.
In [10]:
from opem.Dynamic.Padulles_Amphlett import Eta_Ohmic_Calc
Eta_Ohmic=Eta_Ohmic_Calc(i=2, l=0.0178, A=50.6, T=343, lambda_param=23, R_elec=0)
Eta_Ohmic
Out[10]:
0.003531492225469087

Concentration

$$\eta_{Concentration}=-B\times ln(1-\frac{J}{J_{Max}})$$

$$J=\frac{i}{A}$$

$$J_{Max}=\frac{i_L}{A}$$

In [11]:
from opem.Dynamic.Padulles_Amphlett import Eta_Conc_Calc
Eta_Conc=Eta_Conc_Calc(i=2, A=50.6, B=0.016, JMax=1.5)
Eta_Conc
Out[11]:
0.0004272617031588504

FC Voltage

$$Loss=\eta_{Activation}+\eta_{Ohmic}+\eta_{Concentration}$$

$$V_{Fuelcell}=E_{Nernst}-N_0\times Loss$$

In [12]:
from opem.Dynamic.Padulles_Amphlett import Loss_Calc,Vcell_Calc
Loss=Loss_Calc(Eta_Act=Eta_Act,Eta_Conc=Eta_Conc,Eta_Ohmic=Eta_Ohmic)
Loss
Out[12]:
0.38786477439654415
In [13]:
FC_Voltage=Vcell_Calc(Enernst=Enernst, Loss=Loss, N=5)
FC_Voltage
Out[13]:
0.9840777868416166

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 [14]:
from opem.Dynamic.Padulles_Amphlett import Power_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=FC_Voltage,i=2)
Power
Out[14]:
1.9681555736832332
In [15]:
Power_Thermal_Calc(VStack=FC_Voltage,N=5,i=2)
Out[15]:
10.331844426316767

Efficiency of PEMFC

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

In [16]:
from opem.Dynamic.Padulles_Amphlett import Efficiency_Calc
Efficiency_Calc(Vcell=FC_Voltage,N=5)
Out[16]:
0.1261638188258483

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 [17]:
Test_Vector={"A":50.6,"l":0.0178,"lambda":23,"JMax":1.5,"T":343,"E0":1.229,"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":0.016,"R":0,"rho":1.168,"qMethanol":0.0002,"CV":2,"i-start":0.1,"i-stop":4,"i-step":0.1,"Name":"Test"}
In [18]:
from opem.Dynamic.Padulles_Amphlett import Dynamic_Analysis
Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
Out[18]:
{'EFF': [0.6589193654570529,
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 'I': [0.1,
  0.2,
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  1.0,
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 'K': -0.24141814874622033,
 'P': [0.5139571050565013,
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  2.3014338158615835,
  2.72491286817189,
  3.142649276570101,
  3.555440520606415,
  3.9638828203358005,
  4.368438911998416,
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 'PH2': [0.19717074233280188,
  0.1971566919511875,
  0.19714264156957312,
  0.19712859118795872,
  0.19711454080634436,
  0.19710049042472996,
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  0.19707238966150117,
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  0.1969178354637429,
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  0.19687568431889974,
  0.19686163393728537,
  0.19684758355567097,
  0.19683353317405658,
  0.19681948279244216,
  0.19680543241082776,
  0.1967913820292134,
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  0.1966930293579127,
  0.1966789789762983,
  0.1966649285946839,
  0.19665087821306954,
  0.19663682783145514],
 'PH2O': [0.2426831613626925,
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  0.24264857417203542,
  0.24263128057670688,
  0.24261398698137834,
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 'PO2': [0.1906263686382979,
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  0.1905946372855047,
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 'Ph': [0.10104289494349877,
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 'Status': True,
 'V': [5.139571050565013,
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 'V0': 4.698328442367755}
  • Notice : "Status", "V0", "K" and "EFF" , new in version 0.8
In [19]:
Data=Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
###########
Padulles-Amphlett-Model Simulation
###########
Analyzing . . .
I : 0.1
E : 6.0684154992732005 V
Eta Activation : 0.18557231242539243 V
Eta Concentration : 2.109426805213159e-05 V
Eta Ohmic : 0.00017548304819292376 V
FC Efficiency : 0.6589193654570529 
FC Power : 0.5139571050565013 W
FC Voltage : 5.139571050565013 V
Loss : 0.1857688897416375 V
PH2 : 0.19717074233280188 atm
PH2O : 0.2426831613626925 atm
PO2 : 0.1906263686382979 atm
Power-Thermal : 0.10104289494349877 W
###########
I : 0.2
E : 6.068413961701556 V
Eta Activation : 0.23146009851376736 V
Eta Concentration : 4.221638333089875e-05 V
Eta Ohmic : 0.0003510800160998837 V
FC Efficiency : 0.629377818863534 
FC Power : 0.9818293974271133 W
FC Voltage : 4.909146987135566 V
Loss : 0.23185339491319815 V
PH2 : 0.1971566919511875 atm
PH2O : 0.24266586776736396 atm
PO2 : 0.1906184358000996 atm
Power-Thermal : 0.2481706025728869 W
###########
I : 0.3
E : 6.068412424065923 V
Eta Activation : 0.2583036192079603 V
Eta Concentration : 6.336641945755048e-05 V
Eta Ohmic : 0.0005267910327125488 V
FC Efficiency : 0.6120440436878553 
FC Power : 1.4321830622295815 W
FC Voltage : 4.773943540765272 V
Loss : 0.25889377666013036 V
PH2 : 0.19714264156957312 atm
PH2O : 0.24264857417203542 atm
PO2 : 0.1906105029619013 atm
Power-Thermal : 0.41281693777041856 W
###########
I : 0.4
E : 6.068410886366294 V
Eta Activation : 0.27735002084480426 V
Eta Concentration : 8.454445034568427e-05 V
Eta Ohmic : 0.0007026162388380664 V
FC Efficiency : 0.5997083306020967 
FC Power : 1.871089991478542 W
FC Voltage : 4.677724978696355 V
Loss : 0.278137181533988 V
PH2 : 0.19712859118795872 atm
PH2O : 0.24263128057670688 atm
PO2 : 0.19060257012370302 atm
Power-Thermal : 0.5889100085214584 W
###########
I : 0.5
E : 6.068409348602667 V
Eta Activation : 0.2921240370409447 V
Eta Concentration : 0.00010575055020278165 V
Eta Ohmic : 0.0008785557847524419 V
FC Efficiency : 0.5901112348363035 
FC Power : 2.3014338158615835 W
FC Voltage : 4.602867631723167 V
Loss : 0.2931083433758999 V
PH2 : 0.19711454080634436 atm
PH2O : 0.24261398698137834 atm
PO2 : 0.1905946372855047 atm
Power-Thermal : 0.7735661841384167 W
###########
I : 0.6
E : 6.0684078107750326 V
Eta Activation : 0.3041956781419353 V
Eta Concentration : 0.00012698479353178086 V
Eta Ohmic : 0.0010546098289093816 V
FC Efficiency : 0.582246339352968 
FC Power : 2.72491286817189 W
FC Voltage : 4.54152144695315 V
Loss : 0.3053772727643765 V
PH2 : 0.19710049042472996 atm
PH2O : 0.2425966933860498 atm
PO2 : 0.1905867044473064 atm
Power-Thermal : 0.9650871318281099 W
###########
I : 0.7
E : 6.068406272883388 V
Eta Activation : 0.31440243547871893 V
Eta Concentration : 0.000148247255132642 V
Eta Ohmic : 0.0012307785370829418 V
FC Efficiency : 0.5755767905806046 
FC Power : 3.142649276570101 W
FC Voltage : 4.489498966528716 V
Loss : 0.3157814612709345 V
PH2 : 0.19708644004311557 atm
PH2O : 0.24257939979072127 atm
PO2 : 0.19057877160910808 atm
Power-Thermal : 1.162350723429899 W
###########
I : 0.8
E : 6.068404734927729 V
Eta Activation : 0.3232442167420945 V
Eta Concentration : 0.00016953801010392253 V
Eta Ohmic : 0.0014070620817435461 V
FC Efficiency : 0.5697821347125663 
FC Power : 3.555440520606415 W
FC Voltage : 4.444300650758018 V
Loss : 0.32482081683394204 V
PH2 : 0.19707238966150117 atm
PH2O : 0.24256210619539273 atm
PO2 : 0.1905708387709098 atm
Power-Thermal : 1.3645594793935858 W
###########
I : 0.9
E : 6.068403196908046 V
Eta Activation : 0.3310434726426763 V
Eta Concentration : 0.00019085713384438152 V
Eta Ohmic : 0.0015834606415773538 V
FC Efficiency : 0.564655672412507 
FC Power : 3.9638828203358005 W
FC Voltage : 4.404314244817556 V
Loss : 0.332817790418098 V
PH2 : 0.19705833927988675 atm
PH2O : 0.24254481260006414 atm
PO2 : 0.19056290593271147 atm
Power-Thermal : 1.5711171796642 W
###########
I : 1.0
E : 6.068401658824337 V
Eta Activation : 0.33802037026202836 V
Eta Concentration : 0.00021220470205456714 V
Eta Ohmic : 0.0017599744011013664 V
FC Efficiency : 0.5600562707690275 
FC Power : 4.368438911998416 W
FC Voltage : 4.368438911998416 V
Loss : 0.3399925493651843 V
PH2 : 0.19704428889827239 atm
PH2O : 0.2425275190047356 atm
PO2 : 0.1905549730945132 atm
Power-Thermal : 1.7815610880015846 W
###########
I : 1.1
E : 6.068400120676597 V
Eta Activation : 0.3443319458183834 V
Eta Concentration : 0.0002335807907384422 V
Eta Ohmic : 0.0019366035503462617 V
FC Efficiency : 0.5558832653691355 
FC Power : 4.769478416867183 W
FC Voltage : 4.335889469879257 V
Loss : 0.34650213015946807 V
PH2 : 0.197030238516658 atm
PH2O : 0.24251022540940706 atm
PO2 : 0.19054704025631486 atm
Power-Thermal : 1.9955215831328177 W
###########
I : 1.2
E : 6.068398582464819 V
Eta Activation : 0.35009414904739194 V
Eta Concentration : 0.00025498547620500224 V
Eta Ohmic : 0.002113348284589288 V
FC Efficiency : 0.5520623292851136 
FC Power : 5.167303402108664 W
FC Voltage : 4.306086168423887 V
Loss : 0.35246248280818626 V
PH2 : 0.1970161881350436 atm
PH2O : 0.24249293181407852 atm
PO2 : 0.19053910741811658 atm
Power-Thermal : 2.212696597891336 W
###########
I : 1.3
E : 6.068397044188998 V
Eta Activation : 0.35539503345654255 V
Eta Concentration : 0.0002764188350699048 V
Eta Ohmic : 0.0022902088041253615 V
FC Efficiency : 0.548537017783373 
FC Power : 5.562165360323403 W
FC Voltage : 4.27858873871031 V
Loss : 0.3579616610957378 V
PH2 : 0.19700213775342923 atm
PH2O : 0.24247563821874998 atm
PO2 : 0.19053117457991825 atm
Power-Thermal : 2.4328346396765976 W
###########
I : 1.4
E : 6.06839550584913 V
Eta Activation : 0.36030304442922906 V
Eta Concentration : 0.00029788094425712723 V
Eta Ohmic : 0.0024671853140681515 V
FC Efficiency : 0.5452634554373536 
FC Power : 5.954276933375901 W
FC Voltage : 4.253054952411358 V
Loss : 0.36306811068755435 V
PH2 : 0.19698808737181484 atm
PH2O : 0.24245834462342142 atm
PO2 : 0.19052324174171997 atm
Power-Thermal : 2.6557230666240987 W
###########
I : 1.5
E : 6.068393967445208 V
Eta Activation : 0.3648724409731032 V
Eta Concentration : 0.00031937188100060893 V
Eta Ohmic : 0.002644278024175193 V
FC Efficiency : 0.5422068606479247 
FC Power : 6.343820269580719 W
FC Voltage : 4.229213513053812 V
Loss : 0.367836090878279 V
PH2 : 0.19697403699020044 atm
PH2O : 0.24244105102809288 atm
PO2 : 0.19051530890352164 atm
Power-Thermal : 2.881179730419282 W
###########
I : 1.6
E : 6.068392428977227 V
Eta Activation : 0.36914696409844006 V
Eta Concentration : 0.0003408917228459314 V
Eta Ohmic : 0.0028214871486926026 V
FC Efficiency : 0.539339194118889 
FC Power : 6.7309531426037355 W
FC Voltage : 4.2068457141273345 V
Loss : 0.3723093429699786 V
PH2 : 0.19695998660858605 atm
PH2O : 0.24242375743276434 atm
PO2 : 0.19050737606532336 atm
Power-Thermal : 3.1090468573962653 W
###########
I : 1.7
E : 6.068390890445182 V
Eta Activation : 0.3731623911228729 V
Eta Concentration : 0.00036244054765199196 V
Eta Ohmic : 0.0029988129062160497 V
FC Efficiency : 0.5366375214822406 
FC Power : 7.115813534854511 W
FC Voltage : 4.185772667561477 V
Loss : 0.376523644576741 V
PH2 : 0.19694593622697168 atm
PH2O : 0.2424064638374358 atm
PO2 : 0.19049944322712503 atm
Power-Thermal : 3.339186465145489 W
###########
I : 1.8
E : 6.068389351849069 V
Eta Activation : 0.3769483587657406 V
Eta Concentration : 0.00038401843359268825 V
Eta Ohmic : 0.003176255519565377 V
FC Efficiency : 0.5340828446480225 
FC Power : 7.4985231388582365 W
FC Voltage : 4.165846188254576 V
Loss : 0.3805086327188987 V
PH2 : 0.19693188584535729 atm
PH2O : 0.24238917024210727 atm
PO2 : 0.19049151038892673 atm
Power-Thermal : 3.5714768611417647 W
###########
I : 1.9
E : 6.068387813188879 V
Eta Activation : 0.38052969267197334 V
Eta Concentration : 0.00040562545915863245 V
Eta Ohmic : 0.0033538152156708046 V
FC Efficiency : 0.5316592495454955 
FC Power : 7.8791900782642434 W
FC Voltage : 4.146942146454865 V
Loss : 0.38428913334680276 V
PH2 : 0.1969178354637429 atm
PH2O : 0.24237187664677873 atm
PO2 : 0.19048357755072845 atm
Power-Thermal : 3.8058099217357566 W
###########
I : 2.0
E : 6.0683862744646095 V
Eta Activation : 0.3839273955127959 V
Eta Concentration : 0.0004272617031588504 V
Eta Ohmic : 0.003531492225469087 V
FC Efficiency : 0.5293532727253193 
FC Power : 8.257911054514981 W
FC Voltage : 4.1289555272574905 V
Loss : 0.3878861494414239 V
PH2 : 0.19690378508212852 atm
PH2O : 0.2423545830514502 atm
PO2 : 0.19047564471253012 atm
Power-Thermal : 4.04208894548502 W
###########
I : 2.1
E : 6.068384735676256 V
Eta Activation : 0.38715939375662295 V
Eta Concentration : 0.00044892724472251814 V
Eta Ohmic : 0.0037092867838082735 V
FC Efficiency : 0.5271534226603188 
FC Power : 8.634773063176024 W
FC Voltage : 4.1117966967504875 V
Loss : 0.39131760778515373 V
PH2 : 0.19688973470051413 atm
PH2O : 0.24233728945612165 atm
PO2 : 0.19046771187433184 atm
Power-Thermal : 4.280226936823977 W
###########
I : 2.2
E : 6.068383196823811 V
Eta Activation : 0.39024111055794025 V
Eta Concentration : 0.00047062216330069346 V
Eta Ohmic : 0.0038871991293599716 V
FC Efficiency : 0.5250498125090778 
FC Power : 9.009854782655777 W
FC Voltage : 4.095388537570807 V
Loss : 0.3945989318506009 V
PH2 : 0.19687568431889974 atm
PH2O : 0.2423199958607931 atm
PO2 : 0.1904597790361335 atm
Power-Thermal : 4.520145217344226 W
###########
I : 2.3
E : 6.068381657907269 V
Eta Activation : 0.39318591119501267 V
Eta Concentration : 0.0004923465386680586 V
Eta Ohmic : 0.004065229504538212 V
FC Efficiency : 0.5230338745789967 
FC Power : 9.383227709947201 W
FC Voltage : 4.0796642217161745 V
Loss : 0.3977434872382189 V
PH2 : 0.19686163393728537 atm
PH2O : 0.24230270226546458 atm
PO2 : 0.19045184619793523 atm
Power-Thermal : 4.761772290052799 W
###########
I : 2.4
E : 6.068380118926627 V
Eta Activation : 0.3960054536369255 V
Eta Concentration : 0.0005141004509246927 V
Eta Ohmic : 0.004243378155424144 V
FC Efficiency : 0.5210981356038789 
FC Power : 9.754957098504613 W
FC Voltage : 4.0645654577102555 V
Loss : 0.40076293224327436 V
PH2 : 0.19684758355567097 atm
PH2O : 0.242285408670136 atm
PO2 : 0.1904439133597369 atm
Power-Thermal : 5.005042901495387 W
###########
I : 2.5
E : 6.068378579881878 V
Eta Activation : 0.39870996749954657 V
Eta Concentration : 0.0005358839804978295 V
Eta Ohmic : 0.00442164533169592 V
FC Efficiency : 0.5192360379260482 
FC Power : 10.12510273955794 W
FC Voltage : 4.050041095823176 V
Loss : 0.4036674968117403 V
PH2 : 0.19683353317405658 atm
PH2O : 0.24226811507480747 atm
PO2 : 0.19043598052153862 atm
Power-Thermal : 5.24989726044206 W
###########
I : 2.6
E : 6.068377040773017 V
Eta Activation : 0.40130847825734167 V
Eta Concentration : 0.0005576972081436541 V
Eta Ohmic : 0.004600031286563196 V
FC Efficiency : 0.5174417957708685 
FC Power : 10.493719618233213 W
FC Voltage : 4.036046007012774 V
Loss : 0.40646620675204853 V
PH2 : 0.19681948279244216 atm
PH2O : 0.2422508214794789 atm
PO2 : 0.1904280476833403 atm
Power-Thermal : 5.4962803817667885 W
###########
I : 2.7
E : 6.068375501600038 V
Eta Activation : 0.4038089891176398 V
Eta Concentration : 0.0005795402149490941 V
Eta Ohmic : 0.004778536276705824 V
FC Efficiency : 0.5157102786607133 
FC Power : 10.860858468594623 W
FC Voltage : 4.022540173553564 V
Loss : 0.4091670656092947 V
PH2 : 0.19680543241082776 atm
PH2O : 0.24223352788415034 atm
PO2 : 0.190420114845142 atm
Power-Thermal : 5.744141531405379 W
###########
I : 2.8
E : 6.068373962362936 V
Eta Activation : 0.40621862980268425 V
Eta Concentration : 0.0006014130823336223 V
Eta Ohmic : 0.004957160562216277 V
FC Efficiency : 0.5140369160418929 
FC Power : 11.226566246354942 W
FC Voltage : 4.009487945126765 V
Loss : 0.4117772034472342 V
PH2 : 0.1967913820292134 atm
PH2O : 0.2422162342888218 atm
PO2 : 0.19041218200694368 atm
Power-Thermal : 5.993433753645058 W
###########
I : 2.9
E : 6.068372423061707 V
Eta Activation : 0.4085437792118771 V
Eta Concentration : 0.0006233158920510905 V
Eta Ohmic : 0.005135904406545483 V
FC Efficiency : 0.5124176186550433 
FC Power : 11.590886533977079 W
FC Voltage : 3.996857425509338 V
Loss : 0.4143029995104737 V
PH2 : 0.196777331647599 atm
PH2O : 0.24219894069349326 atm
PO2 : 0.1904042491687454 atm
Power-Thermal : 6.244113466022921 W
###########
I : 3.0
E : 6.0683708836963435 V
Eta Activation : 0.4107901672807063 V
Eta Concentration : 0.0006452487261915484 V
Eta Ohmic : 0.005314768076451755 V
FC Efficiency : 0.5108487132409738 
FC Power : 11.953859889838787 W
FC Voltage : 3.9846199632795956 V
Loss : 0.4167501840833496 V
PH2 : 0.1967632812659846 atm
PH2O : 0.24218164709816473 atm
PO2 : 0.19039631633054707 atm
Power-Thermal : 6.496140110161214 W
###########
I : 3.1
E : 6.068369344266841 V
Eta Activation : 0.4129629601316751 V
Eta Concentration : 0.0006672116671831024 V
Eta Ohmic : 0.0054937518419525275 V
FC Efficiency : 0.5093268879567676 
FC Power : 12.315524150794642 W
FC Voltage : 3.9727497260627875 V
Loss : 0.41912392364081075 V
PH2 : 0.19674923088437024 atm
PH2O : 0.2421643535028362 atm
PO2 : 0.1903883834923488 atm
Power-Thermal : 6.74947584920536 W
###########
I : 3.2
E : 6.068367804773196 V
Eta Activation : 0.41506683170178466 V
Eta Concentration : 0.0006892047977937692 V
Eta Ohmic : 0.005672855976278701 V
FC Efficiency : 0.5078491464607577 
FC Power : 12.675914695660513 W
FC Voltage : 3.96122334239391 V
Loss : 0.42142889247585713 V
PH2 : 0.19673518050275585 atm
PH2O : 0.24214705990750765 atm
PO2 : 0.19038045065415046 atm
Power-Thermal : 7.0040853043394895 W
###########
I : 3.3
E : 6.0683662652154 V
Eta Activation : 0.417106024344736 V
Eta Concentration : 0.0007112282011333409 V
Eta Ohmic : 0.005852080755831333 V
FC Efficiency : 0.5064127690649867 
FC Power : 13.035064675732759 W
FC Voltage : 3.9500195987068967 V
Loss : 0.42366933330170065 V
PH2 : 0.19672113012114145 atm
PH2O : 0.2421297663121791 atm
PO2 : 0.19037251781595219 atm
Power-Thermal : 7.259935324267242 W
###########
I : 3.4
E : 6.06836472559345 V
Eta Activation : 0.4190844003836543 V
Eta Concentration : 0.0007332819606552831 V
Eta Ohmic : 0.0060314264601405215 V
FC Efficiency : 0.5050152796886153 
FC Power : 13.393005217342077 W
FC Voltage : 3.9391191815711992 V
Loss : 0.42584910880445015 V
PH2 : 0.19670707973952706 atm
PH2O : 0.24211247271685057 atm
PO2 : 0.19036458497775385 atm
Power-Thermal : 7.516994782657924 W
###########
I : 3.5
E : 6.068363185907339 V
Eta Activation : 0.42100548618901656 V
Eta Concentration : 0.0007553661601586168 V
Eta Ohmic : 0.006210893371826288 V
FC Efficiency : 0.503654417602863 
FC Power : 13.74976560055816 W
FC Voltage : 3.9285044573023313 V
Loss : 0.4279717457210015 V
PH2 : 0.1966930293579127 atm
PH2O : 0.24209517912152204 atm
PO2 : 0.19035665213955555 atm
Power-Thermal : 7.775234399441842 W
###########
I : 3.6
E : 6.068361646157063 V
Eta Activation : 0.4228725100457559 V
Eta Concentration : 0.000777480883789843 V
Eta Ohmic : 0.006390481776561363 V
FC Efficiency : 0.5023281131572472 
FC Power : 14.105373417455501 W
FC Voltage : 3.918159282626528 V
Loss : 0.43004047270610707 V
PH2 : 0.1966789789762983 atm
PH2O : 0.2420778855261935 atm
PO2 : 0.19034871930135727 atm
Power-Thermal : 8.034626582544501 W
###########
I : 3.7
E : 6.068360106342617 V
Eta Activation : 0.4246884348310017 V
Eta Concentration : 0.0007996262160448594 V
Eta Ohmic : 0.00657019196303564 V
FC Efficiency : 0.501034466832334 
FC Power : 14.459854712781162 W
FC Voltage : 3.9080688412922058 V
Loss : 0.4320582530100822 V
PH2 : 0.1966649285946839 atm
PH2O : 0.24206059193086493 atm
PO2 : 0.19034078646315894 atm
Power-Thermal : 8.29514528721884 W
###########
I : 3.8
E : 6.068358566463993 V
Eta Activation : 0.4264559863331208 V
Eta Concentration : 0.0008218022417708932 V
Eta Ohmic : 0.006750024222922298 V
FC Efficiency : 0.4997717310865285 
FC Power : 14.813234109404705 W
FC Voltage : 3.8982195024749227 V
Loss : 0.43402781279781405 V
PH2 : 0.19665087821306954 atm
PH2O : 0.2420432983355364 atm
PO2 : 0.19033285362496066 atm
Power-Thermal : 8.556765890595296 W
###########
I : 3.9
E : 6.068357026521189 V
Eta Activation : 0.42817767789163225 V
Eta Concentration : 0.0008440090461684635 V
Eta Ohmic : 0.006929978850845375 V
FC Efficiency : 0.49853829456127663 
FC Power : 15.165534920554036 W
FC Voltage : 3.888598697577958 V
Loss : 0.4359516657886461 V
PH2 : 0.19663682783145514 atm
PH2O : 0.24202600474020786 atm
PO2 : 0.19032492078676233 atm
Power-Thermal : 8.819465079445965 W
###########
Done!
  • Notice : "PrintMode" & "ReportMode" , new in version 0.5
In [20]:
Dynamic_Analysis(InputMethod={},TestMode=True,PrintMode=False,ReportMode=True)
Out[20]:
{'Message': '[Error] Padulles-Amphlett 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 $$s$$ $$User$$
$$l$$ Membrane Thickness $$cm$$ $$User$$
$$A$$ Active area $$cm^2$$ $$User$$
$$\tau_{1}^{(s)}$$ Reformer time constant $$s$$ $$User$$
$$\tau_{2}^{(s)}$$ Reformer time constant $$s$$ $$User$$
$$CV$$ Conversion factor $$--$$ $$User$$
$$B$$ An empirical constant depending on the cell and its operation state $$V$$ $$User$$
$$R_{electronic}$$ R-Electronic $$\Omega$$ $$User$$
$$\lambda$$ An adjustable parameter with a possible minimum value of 14 and a maximum value of 23 $$--$$ $$User$$
$$J_{Max}$$ Maximum current density of the cell $$Acm^{-2}$$ $$User$$
$$r_{h-o}$$ 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 $$kmol.s^{-1}$$ $$System$$
$$J$$ Actual current density of the cell $$Acm^{-2}$$ $$System$$
$$C_{O_2}$$ Concentration of oxygen in the catalytic interface of the cathode $$molcm^{-3}$$ $$System$$
$$C_{H_2}$$ Concentration of hydrogen in the catalytic interface of the anode $$molcm^{-3}$$ $$System$$
$$R_{Proton}$$ Resistance to proton flow $$\Omega$$ $$System$$
$$\xi_2$$ Parametric coefficients for cell model $$--$$ $$System$$
$$\xi_1$$ Parametric coefficients for cell model $$--$$ $$-0.948$$
$$\xi_3$$ Parametric coefficients for cell model $$--$$ $$7.6\times10^{-5}$$
$$\xi_4$$ Parametric coefficients for cell model $$--$$ $$-1.93\times10^{-4}$$
$$\mu_F$$ 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.
3- J. C. Amphlett, R. M. Baumert, R. F. Mann, B. A. Peppley, and P. R. Roberge. 1995. "Performance Modeling of the Ballard Mark IV Solid Polymer Electrolyte Fuel Cell." J. Electrochem. Soc. (The Electrochemical Society, Inc.) 142 (1): 9-15. doi: 10.1149/1.2043959.