Padulles Dynamic Model I

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 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-1 Dynamic Model Block Diagram

Nernst Voltage

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

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

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

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

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

In [1]:
from opem.Dynamic.Padulles1 import Kr_Calc,qO2_Calc,PH2_Calc,PO2_Calc,Enernst_Calc
In [2]:
Kr=Kr_Calc(N0=88)
Kr
Out[2]:
2.2801566260315118e-07
In [3]:
qO2=qO2_Calc(qH2=0.0004,rho=1.168)
qO2
Out[3]:
0.0003424657534246576
In [4]:
PH2=PH2_Calc(KH2=0.0000422,tH2=3.37,Kr=Kr,I=1,qH2=0.0004)
PH2
Out[4]:
2.166559852694447
In [5]:
PO2=PO2_Calc(KO2=0.0000211,tO2=6.74,Kr=Kr,I=1,qO2=qO2)
PO2
Out[5]:
2.0955811367185566
In [6]:
Enernst=Enernst_Calc(E0=0.6,N0=88,T=343, PH2=PH2, PO2=PO2)
Enernst
Out[6]:
54.28658891467849

FC Voltage

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

In [7]:
from opem.Dynamic.Padulles1 import Vcell_Calc
FC_Voltage=Vcell_Calc(Enernst=Enernst, B=0.04777,C=0.0136,I=1,Rint=0.00303)
FC_Voltage
Out[7]:
54.48885935035618

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 [8]:
from opem.Dynamic.Padulles1 import Power_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=FC_Voltage,i=2)
Power
Out[8]:
108.97771870071236
In [9]:
Power_Thermal_Calc(VStack=FC_Voltage,N=88,i=2)
Out[9]:
107.50228129928763

Efficiency of PEMFC

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

In [10]:
from opem.Dynamic.Padulles1 import Efficiency_Calc
Efficiency_Calc(Vcell=FC_Voltage,N=88)
Out[10]:
0.39691768174793257

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 [11]:
Test_Vector={"T":343,"E0":0.6,"N0":88,"KO2":0.0000211,"KH2":0.0000422,"tH2":3.37,"tO2":6.74,"B":0.04777,"C":0.0136,"Rint":0.00303,"rho":1.168,"qH2":0.0004,"i-start":0,"i-stop":4,"i-step":0.1,"Name":"Test"}
  • Notice : "Name", new in version 0.5
In [12]:
from opem.Dynamic.Padulles1 import Dynamic_Analysis
Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
[Error] Vcell Calculation Error (Enernst:54.28850557413407, B:0.04777, C:0.0136, I:0, Rint:0.00303)
[Error] PEM Efficiency Calculation Failed (Vcell:None, N:88)
[Error] Power Calculation Error (Vcell:None, i:0)
Out[12]:
{'EFF': [None,
  0.3977513543332525,
  0.39750655365258875,
  0.39736185898012566,
  0.3972581497084044,
  0.3971768980805572,
  0.3971098513395853,
  0.3970526072135646,
  0.39700253793845613,
  0.39695794855256433,
  0.39691768174793257,
  0.39688091206389015,
  0.3968470300107991,
  0.3968155727355017,
  0.3967861804549145,
  0.3967585679566162,
  0.39673250531602267,
  0.3967078044696878,
  0.3966843096322109,
  0.3966618903075967,
  0.39664043609530514,
  0.3966198527648516,
  0.3966000592444172,
  0.39658098527940633,
  0.3965625695896912,
  0.3965447584032809,
  0.3965275042777518,
  0.39651076514422556,
  0.3964945035252972,
  0.39647868589026086,
  0.39646328211968823,
  0.3964482650578333,
  0.39643361013612227,
  0.39641929505459833,
  0.3964052995109346,
  0.3963916049687393,
  0.39637819445850625,
  0.39636505240584513,
  0.3963521644826228,
  0.39633951747744234],
 'I': [0,
  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': -0.03833515487668971,
 'P': [None,
  5.4603305922868905,
  10.913939937085477,
  16.364950800237498,
  21.814239516787904,
  27.262222284249443,
  32.709144235138965,
  38.155167342794705,
  43.60040672655301,
  49.044948459566434,
  54.48885935035618,
  59.93219276894392,
  65.37499233585899,
  70.81729437266857,
  76.25912959399092,
  81.7005243136264,
  87.14150132765376,
  92.58208057591786,
  98.02227964735785,
  103.46211417271105,
  108.90159813432699,
  114.34074411387355,
  119.77956349276191,
  125.21806661606087,
  130.65626292785475,
  136.09416108400603,
  141.5317690468494,
  146.9690941652981,
  152.40614324306785,
  157.84292259714354,
  163.2794381081724,
  168.71569526413202,
  174.15169919835796,
  179.58745472281433,
  185.02296635732776,
  190.45823835537988,
  195.89327472694944,
  201.32807925881536,
  206.76265553266293,
  212.1970069412828],
 'PH2': [2.169032719858579,
  2.168785433142166,
  2.1685381464257527,
  2.168290859709339,
  2.1680435729929264,
  2.1677962862765128,
  2.1675489995600996,
  2.1673017128436864,
  2.1670544261272733,
  2.16680713941086,
  2.166559852694447,
  2.1663125659780333,
  2.16606527926162,
  2.165817992545207,
  2.165570705828794,
  2.1653234191123807,
  2.1650761323959675,
  2.164828845679554,
  2.164581558963141,
  2.1643342722467276,
  2.1640869855303144,
  2.1638396988139013,
  2.1635924120974876,
  2.163345125381075,
  2.1630978386646613,
  2.162850551948248,
  2.162603265231835,
  2.162355978515422,
  2.1621086917990087,
  2.1618614050825955,
  2.161614118366182,
  2.1613668316497687,
  2.1611195449333556,
  2.1608722582169424,
  2.1606249715005292,
  2.160377684784116,
  2.1601303980677025,
  2.1598831113512893,
  2.159635824634876,
  2.159388537918463],
 'PO2': [2.0969773162414582,
  2.096837698289168,
  2.096698080336878,
  2.096558462384588,
  2.0964188444322978,
  2.0962792264800076,
  2.0961396085277175,
  2.0959999905754274,
  2.0958603726231373,
  2.0957207546708467,
  2.0955811367185566,
  2.095441518766267,
  2.0953019008139764,
  2.0951622828616863,
  2.095022664909396,
  2.094883046957106,
  2.094743429004816,
  2.094603811052526,
  2.0944641931002357,
  2.094324575147945,
  2.0941849571956554,
  2.094045339243365,
  2.0939057212910748,
  2.0937661033387847,
  2.0936264853864945,
  2.0934868674342044,
  2.0933472494819143,
  2.093207631529624,
  2.093068013577334,
  2.092928395625044,
  2.0927887776727534,
  2.0926491597204633,
  2.0925095417681736,
  2.092369923815883,
  2.092230305863593,
  2.092090687911303,
  2.0919510699590127,
  2.0918114520067226,
  2.0916718340544325,
  2.0915322161021423],
 'Ph': [None,
  5.36366940771311,
  10.734060062914523,
  16.1070491997625,
  21.4817604832121,
  26.857777715750554,
  32.23485576486103,
  37.61283265720528,
  42.991593273446995,
  48.37105154043357,
  53.751140649643816,
  59.13180723105608,
  64.51300766414099,
  69.89470562733142,
  75.27687040600905,
  80.65947568637358,
  86.04249867234626,
  91.42591942408212,
  96.80972035264215,
  102.19388582728892,
  107.578401865673,
  112.96325588612645,
  118.34843650723809,
  123.7339333839391,
  129.11973707214523,
  134.50583891599396,
  139.8922309531506,
  145.27890583470193,
  150.66585675693213,
  156.05307740285645,
  161.4405618918276,
  166.828304735868,
  172.21630080164203,
  177.60454527718562,
  182.9930336426722,
  188.38176164462013,
  193.77072527305054,
  199.15992074118464,
  204.54934446733702,
  209.9389930587172],
 'Status': True,
 'V': [None,
  54.6033059228689,
  54.56969968542738,
  54.54983600079166,
  54.535598791969754,
  54.524444568498886,
  54.515240391898274,
  54.507381918278156,
  54.50050840819126,
  54.49438717729603,
  54.48885935035618,
  54.483811608130836,
  54.4791602798825,
  54.47484182512967,
  54.470806852850664,
  54.46701620908427,
  54.46343832978359,
  54.46004739759874,
  54.456822026309915,
  54.45374430142687,
  54.450799067163494,
  54.44797338755883,
  54.44525613307359,
  54.4426376591569,
  54.44010955327281,
  54.43766443360241,
  54.435295787249764,
  54.432997838999285,
  54.4307654439528,
  54.428593999015014,
  54.4264793693908,
  54.42441782713936,
  54.42240599948686,
  54.420440825095255,
  54.41851951686111,
  54.41663953010853,
  54.414798535263735,
  54.41299439427442,
  54.41122514017446,
  54.40948895930328],
 'V0': 54.539964013494824}
  • Notice : "Status", "V0", "K" and "EFF" , new in version 0.8
In [13]:
Data=Dynamic_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
###########
Padulles-I-Model Simulation
###########
Analyzing . . .
[Error] Vcell Calculation Error (Enernst:54.28850557413407, B:0.04777, C:0.0136, I:0, Rint:0.00303)
[Error] PEM Efficiency Calculation Failed (Vcell:None, N:88)
[Error] Power Calculation Error (Vcell:None, i:0)
I : 0
E : 54.28850557413407 V
FC Efficiency : None 
FC Power : None W
FC Voltage : None V
PH2 : 2.169032719858579 atm
PO2 : 2.0969773162414582 atm
Power-Thermal : None W
###########
I : 0.1
E : 54.288313997298886 V
FC Efficiency : 0.3977513543332525 
FC Power : 5.4603305922868905 W
FC Voltage : 54.6033059228689 V
PH2 : 2.168785433142166 atm
PO2 : 2.096837698289168 atm
Power-Thermal : 5.36366940771311 W
###########
I : 0.2
E : 54.28812240067271 V
FC Efficiency : 0.39750655365258875 
FC Power : 10.913939937085477 W
FC Voltage : 54.56969968542738 V
PH2 : 2.1685381464257527 atm
PO2 : 2.096698080336878 atm
Power-Thermal : 10.734060062914523 W
###########
I : 0.3
E : 54.287930784251316 V
FC Efficiency : 0.39736185898012566 
FC Power : 16.364950800237498 W
FC Voltage : 54.54983600079166 V
PH2 : 2.168290859709339 atm
PO2 : 2.096558462384588 atm
Power-Thermal : 16.1070491997625 W
###########
I : 0.4
E : 54.28773914803044 V
FC Efficiency : 0.3972581497084044 
FC Power : 21.814239516787904 W
FC Voltage : 54.535598791969754 V
PH2 : 2.1680435729929264 atm
PO2 : 2.0964188444322978 atm
Power-Thermal : 21.4817604832121 W
###########
I : 0.5
E : 54.28754749200584 V
FC Efficiency : 0.3971768980805572 
FC Power : 27.262222284249443 W
FC Voltage : 54.524444568498886 V
PH2 : 2.1677962862765128 atm
PO2 : 2.0962792264800076 atm
Power-Thermal : 26.857777715750554 W
###########
I : 0.6
E : 54.28735581617328 V
FC Efficiency : 0.3971098513395853 
FC Power : 32.709144235138965 W
FC Voltage : 54.515240391898274 V
PH2 : 2.1675489995600996 atm
PO2 : 2.0961396085277175 atm
Power-Thermal : 32.23485576486103 W
###########
I : 0.7
E : 54.287164120528516 V
FC Efficiency : 0.3970526072135646 
FC Power : 38.155167342794705 W
FC Voltage : 54.507381918278156 V
PH2 : 2.1673017128436864 atm
PO2 : 2.0959999905754274 atm
Power-Thermal : 37.61283265720528 W
###########
I : 0.8
E : 54.28697240506729 V
FC Efficiency : 0.39700253793845613 
FC Power : 43.60040672655301 W
FC Voltage : 54.50050840819126 V
PH2 : 2.1670544261272733 atm
PO2 : 2.0958603726231373 atm
Power-Thermal : 42.991593273446995 W
###########
I : 0.9
E : 54.286780669785365 V
FC Efficiency : 0.39695794855256433 
FC Power : 49.044948459566434 W
FC Voltage : 54.49438717729603 V
PH2 : 2.16680713941086 atm
PO2 : 2.0957207546708467 atm
Power-Thermal : 48.37105154043357 W
###########
I : 1.0
E : 54.28658891467849 V
FC Efficiency : 0.39691768174793257 
FC Power : 54.48885935035618 W
FC Voltage : 54.48885935035618 V
PH2 : 2.166559852694447 atm
PO2 : 2.0955811367185566 atm
Power-Thermal : 53.751140649643816 W
###########
I : 1.1
E : 54.2863971397424 V
FC Efficiency : 0.39688091206389015 
FC Power : 59.93219276894392 W
FC Voltage : 54.483811608130836 V
PH2 : 2.1663125659780333 atm
PO2 : 2.095441518766267 atm
Power-Thermal : 59.13180723105608 W
###########
I : 1.2
E : 54.286205344972856 V
FC Efficiency : 0.3968470300107991 
FC Power : 65.37499233585899 W
FC Voltage : 54.4791602798825 V
PH2 : 2.16606527926162 atm
PO2 : 2.0953019008139764 atm
Power-Thermal : 64.51300766414099 W
###########
I : 1.3
E : 54.286013530365594 V
FC Efficiency : 0.3968155727355017 
FC Power : 70.81729437266857 W
FC Voltage : 54.47484182512967 V
PH2 : 2.165817992545207 atm
PO2 : 2.0951622828616863 atm
Power-Thermal : 69.89470562733142 W
###########
I : 1.4
E : 54.28582169591637 V
FC Efficiency : 0.3967861804549145 
FC Power : 76.25912959399092 W
FC Voltage : 54.470806852850664 V
PH2 : 2.165570705828794 atm
PO2 : 2.095022664909396 atm
Power-Thermal : 75.27687040600905 W
###########
I : 1.5
E : 54.28562984162091 V
FC Efficiency : 0.3967585679566162 
FC Power : 81.7005243136264 W
FC Voltage : 54.46701620908427 V
PH2 : 2.1653234191123807 atm
PO2 : 2.094883046957106 atm
Power-Thermal : 80.65947568637358 W
###########
I : 1.6
E : 54.28543796747497 V
FC Efficiency : 0.39673250531602267 
FC Power : 87.14150132765376 W
FC Voltage : 54.46343832978359 V
PH2 : 2.1650761323959675 atm
PO2 : 2.094743429004816 atm
Power-Thermal : 86.04249867234626 W
###########
I : 1.7
E : 54.28524607347429 V
FC Efficiency : 0.3967078044696878 
FC Power : 92.58208057591786 W
FC Voltage : 54.46004739759874 V
PH2 : 2.164828845679554 atm
PO2 : 2.094603811052526 atm
Power-Thermal : 91.42591942408212 W
###########
I : 1.8
E : 54.285054159614596 V
FC Efficiency : 0.3966843096322109 
FC Power : 98.02227964735785 W
FC Voltage : 54.456822026309915 V
PH2 : 2.164581558963141 atm
PO2 : 2.0944641931002357 atm
Power-Thermal : 96.80972035264215 W
###########
I : 1.9
E : 54.284862225891644 V
FC Efficiency : 0.3966618903075967 
FC Power : 103.46211417271105 W
FC Voltage : 54.45374430142687 V
PH2 : 2.1643342722467276 atm
PO2 : 2.094324575147945 atm
Power-Thermal : 102.19388582728892 W
###########
I : 2.0
E : 54.28467027230115 V
FC Efficiency : 0.39664043609530514 
FC Power : 108.90159813432699 W
FC Voltage : 54.450799067163494 V
PH2 : 2.1640869855303144 atm
PO2 : 2.0941849571956554 atm
Power-Thermal : 107.578401865673 W
###########
I : 2.1
E : 54.28447829883886 V
FC Efficiency : 0.3966198527648516 
FC Power : 114.34074411387355 W
FC Voltage : 54.44797338755883 V
PH2 : 2.1638396988139013 atm
PO2 : 2.094045339243365 atm
Power-Thermal : 112.96325588612645 W
###########
I : 2.2
E : 54.2842863055005 V
FC Efficiency : 0.3966000592444172 
FC Power : 119.77956349276191 W
FC Voltage : 54.44525613307359 V
PH2 : 2.1635924120974876 atm
PO2 : 2.0939057212910748 atm
Power-Thermal : 118.34843650723809 W
###########
I : 2.3
E : 54.284094292281814 V
FC Efficiency : 0.39658098527940633 
FC Power : 125.21806661606087 W
FC Voltage : 54.4426376591569 V
PH2 : 2.163345125381075 atm
PO2 : 2.0937661033387847 atm
Power-Thermal : 123.7339333839391 W
###########
I : 2.4
E : 54.28390225917852 V
FC Efficiency : 0.3965625695896912 
FC Power : 130.65626292785475 W
FC Voltage : 54.44010955327281 V
PH2 : 2.1630978386646613 atm
PO2 : 2.0936264853864945 atm
Power-Thermal : 129.11973707214523 W
###########
I : 2.5
E : 54.28371020618635 V
FC Efficiency : 0.3965447584032809 
FC Power : 136.09416108400603 W
FC Voltage : 54.43766443360241 V
PH2 : 2.162850551948248 atm
PO2 : 2.0934868674342044 atm
Power-Thermal : 134.50583891599396 W
###########
I : 2.6
E : 54.28351813330103 V
FC Efficiency : 0.3965275042777518 
FC Power : 141.5317690468494 W
FC Voltage : 54.435295787249764 V
PH2 : 2.162603265231835 atm
PO2 : 2.0933472494819143 atm
Power-Thermal : 139.8922309531506 W
###########
I : 2.7
E : 54.2833260405183 V
FC Efficiency : 0.39651076514422556 
FC Power : 146.9690941652981 W
FC Voltage : 54.432997838999285 V
PH2 : 2.162355978515422 atm
PO2 : 2.093207631529624 atm
Power-Thermal : 145.27890583470193 W
###########
I : 2.8
E : 54.28313392783386 V
FC Efficiency : 0.3964945035252972 
FC Power : 152.40614324306785 W
FC Voltage : 54.4307654439528 V
PH2 : 2.1621086917990087 atm
PO2 : 2.093068013577334 atm
Power-Thermal : 150.66585675693213 W
###########
I : 2.9
E : 54.28294179524345 V
FC Efficiency : 0.39647868589026086 
FC Power : 157.84292259714354 W
FC Voltage : 54.428593999015014 V
PH2 : 2.1618614050825955 atm
PO2 : 2.092928395625044 atm
Power-Thermal : 156.05307740285645 W
###########
I : 3.0
E : 54.282749642742786 V
FC Efficiency : 0.39646328211968823 
FC Power : 163.2794381081724 W
FC Voltage : 54.4264793693908 V
PH2 : 2.161614118366182 atm
PO2 : 2.0927887776727534 atm
Power-Thermal : 161.4405618918276 W
###########
I : 3.1
E : 54.282557470327596 V
FC Efficiency : 0.3964482650578333 
FC Power : 168.71569526413202 W
FC Voltage : 54.42441782713936 V
PH2 : 2.1613668316497687 atm
PO2 : 2.0926491597204633 atm
Power-Thermal : 166.828304735868 W
###########
I : 3.2
E : 54.282365277993584 V
FC Efficiency : 0.39643361013612227 
FC Power : 174.15169919835796 W
FC Voltage : 54.42240599948686 V
PH2 : 2.1611195449333556 atm
PO2 : 2.0925095417681736 atm
Power-Thermal : 172.21630080164203 W
###########
I : 3.3
E : 54.282173065736494 V
FC Efficiency : 0.39641929505459833 
FC Power : 179.58745472281433 W
FC Voltage : 54.420440825095255 V
PH2 : 2.1608722582169424 atm
PO2 : 2.092369923815883 atm
Power-Thermal : 177.60454527718562 W
###########
I : 3.4
E : 54.28198083355201 V
FC Efficiency : 0.3964052995109346 
FC Power : 185.02296635732776 W
FC Voltage : 54.41851951686111 V
PH2 : 2.1606249715005292 atm
PO2 : 2.092230305863593 atm
Power-Thermal : 182.9930336426722 W
###########
I : 3.5
E : 54.28178858143586 V
FC Efficiency : 0.3963916049687393 
FC Power : 190.45823835537988 W
FC Voltage : 54.41663953010853 V
PH2 : 2.160377684784116 atm
PO2 : 2.092090687911303 atm
Power-Thermal : 188.38176164462013 W
###########
I : 3.6
E : 54.281596309383765 V
FC Efficiency : 0.39637819445850625 
FC Power : 195.89327472694944 W
FC Voltage : 54.414798535263735 V
PH2 : 2.1601303980677025 atm
PO2 : 2.0919510699590127 atm
Power-Thermal : 193.77072527305054 W
###########
I : 3.7
E : 54.281404017391424 V
FC Efficiency : 0.39636505240584513 
FC Power : 201.32807925881536 W
FC Voltage : 54.41299439427442 V
PH2 : 2.1598831113512893 atm
PO2 : 2.0918114520067226 atm
Power-Thermal : 199.15992074118464 W
###########
I : 3.8
E : 54.28121170545457 V
FC Efficiency : 0.3963521644826228 
FC Power : 206.76265553266293 W
FC Voltage : 54.41122514017446 V
PH2 : 2.159635824634876 atm
PO2 : 2.0916718340544325 atm
Power-Thermal : 204.54934446733702 W
###########
I : 3.9
E : 54.28101937356888 V
FC Efficiency : 0.39633951747744234 
FC Power : 212.1970069412828 W
FC Voltage : 54.40948895930328 V
PH2 : 2.159388537918463 atm
PO2 : 2.0915322161021423 atm
Power-Thermal : 209.9389930587172 W
###########
Warning : There are errors in the simulations in some of I amounts; please refer to the .opem file for review. If you are confident about this parameters, ignore this error.
Done!
  • Notice : "PrintMode" & "ReportMode" , new in version 0.5
In [14]:
Dynamic_Analysis(InputMethod={},TestMode=True,PrintMode=False,ReportMode=True)
Out[14]:
{'Message': '[Error] Padulles-I 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 $$kmol.s^{-1}.atm^{-1}$$ $$User$$
$$K_{O_2}$$ Oxygen valve constant $$kmol.s^{-1}.atm^{-1}$$ $$User$$
$$\tau_{H_2}^{(s)}$$ Hydrogen 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 $$atm$$ $$System$$
$$P_{O_2}$$ 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 $$V$$ $$1.23$$

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.