Larminie-Dicks Static Model

Version 0.9

Larminie-Dicks model is obtained for large variation of the load parameters. In this model, the fuel cell is represented by means of its voltage–current characteristic obtained in static operating mode. In fact, Larminie-Dicks static model presents the fuel cell voltage as a function of the current magnitude. The obtained polarization curve is composed of three main regions corresponding to the predominance of electrochemical activation phenomena (region I), a linear part (region II) where the voltage drop is mainly due to electronic and ionic internal resistances and the last region where the diffusion kinetics of gases through the electrodes becomes the limiting factor (region III). This last zone is characterized by a rapid voltage fall. It has to be noted that this curve is common to all fuel cells and consequently no units are depicted on it.

Fig1. Graphical Abstract of Static Models

Vcell

$$V_{cell}=E_0-A\times ln(\frac{i+i_n}{i_0})-R_m\times (i+i_n)+B\times ln(1-\frac{i+i_n}{i_L})$$

In [1]:
from opem.Static.Larminie_Dicks import Vcell_Calc
Vcell=Vcell_Calc(E0=1.178, i=1,i_0=0.00654,i_n=0.23 ,i_L=100.0 ,R_M=0.0018,A=0.0587,B=0.0517)
Vcell
Out[1]:
0.8677440917797067
  • Notice : from opem.Larminie_Dicks in version (0.3>)

VStack

$$V_{Stack}=N\times V_{cell}$$

In [2]:
from opem.Static.Larminie_Dicks import VStack_Calc
VStack=VStack_Calc(Vcell=Vcell,N=23)
VStack
Out[2]:
19.958114110933252

Power Of PEMFC

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

$$P_{Stack}=N\times P$$

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

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

In [3]:
from opem.Static.Larminie_Dicks import Power_Calc,PowerStack_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=Vcell,i=2)
Power
Out[3]:
1.7354881835594134
In [4]:
PowerStack_Calc(Power=Power,N=23)
Out[4]:
39.916228221866504
In [5]:
Power_Thermal_Calc(VStack=VStack,N=23,i=2)
Out[5]:
16.663771778133494

Efficiency of PEMFC

$$\eta=\frac{\mu_F\times V_{cell}}{HHV}$$

In [6]:
from opem.Static.Larminie_Dicks import Efficiency_Calc
Efficiency_Calc(Vcell=Vcell)
Out[6]:
0.5562462126792992

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 [7]:
Test_Vector={"A":0.0587,"E0":1.178,"B":0.0517,"RM":0.0018,"i_0":0.00654,"i_L":100.0,"i_n":0.23,"N":23,"i-start":0.1,"i-stop":4,"i-step":0.1,"Name":"Test"}
  • Notice : "Name", new in version 0.5
In [8]:
from opem.Static.Larminie_Dicks import Static_Analysis
Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
Out[8]:
{'EFF': [0.6070918465825977,
  0.596983288796134,
  0.5889668542476844,
  0.5823143704480743,
  0.576622053314997,
  0.5716425142657562,
  0.5672131517182866,
  0.5632213484876261,
  0.5595858755325545,
  0.5562462126792992,
  0.5531560572234717,
  0.5502791922121564,
  0.5475867544887774,
  0.5450553693735705,
  0.5426658416714328,
  0.5404022150580957,
  0.5382510820424253,
  0.536201068437617,
  0.5342424419218519,
  0.532366810486187,
  0.5305668870842597,
  0.5288363037744284,
  0.527169463367337,
  0.5255614198480028,
  0.5240077811244296,
  0.5225046292799733,
  0.52104845467996,
  0.5196361011410657,
  0.5182647200068938,
  0.5169317314482642,
  0.515634791665824,
  0.514371764946645,
  0.5131406997374682,
  0.5119398080610783,
  0.5107674477304784,
  0.5096221069165698,
  0.5085023907052405,
  0.5074070093438323,
  0.5063347679284521],
 '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': -0.8085109813421083,
 'P': [2.178245545538361,
  4.2839520804010585,
  6.339639219122075,
  8.357375844670763,
  10.344599636471047,
  12.3063200471132,
  14.246125518556482,
  16.16670558698882,
  18.070147092697248,
  19.958114110933252,
  21.831963266495983,
  23.692820899886605,
  25.541636576374536,
  27.379221314373194,
  29.206275598756516,
  31.02341036205516,
  32.83116300025978,
  34.63000980397505,
  36.420375750696486,
  38.20264232048878,
  39.97715380802481,
  41.74422247473829,
  43.504132794926115,
  45.257144985951214,
  47.00349796686133,
  48.74341185627015,
  50.47709009557581,
  52.20472126503602,
  53.926480646157316,
  55.64253157309116,
  57.35302660740627,
  59.05810856411401,
  60.75791141171519,
  62.452561064987066,
  64.14217608599348,
  65.82686830619949,
  67.50674338046491,
  69.18190128197547,
  70.85243674576417],
 'Ph': [0.650754454461639,
  1.3740479195989417,
  2.1473607808779245,
  2.958624155329237,
  3.8004003635289525,
  4.6676799528867985,
  5.556874481443514,
  6.465294413011179,
  7.390852907302754,
  8.331885889066747,
  9.287036733504017,
  10.255179100113395,
  11.235363423625465,
  12.226778685626805,
  13.228724401243484,
  14.24058963794484,
  15.261836999740217,
  16.29199019602495,
  17.330624249303508,
  18.37735767951122,
  19.431846191975197,
  20.49377752526172,
  21.562867205073875,
  22.638855014048787,
  23.721502033138655,
  24.810588143729856,
  25.905909904424192,
  27.007278734963975,
  28.11451935384268,
  29.22746842690884,
  30.345973392593724,
  31.469891435885998,
  32.5990885882848,
  33.73343893501293,
  34.872823914006524,
  36.017131693800515,
  37.16625661953508,
  38.32009871802452,
  39.47856325423582],
 'Status': True,
 'V': [21.78245545538361,
  21.41976040200529,
  21.132130730406917,
  20.893439611676907,
  20.689199272942094,
  20.510533411855334,
  20.35160788365212,
  20.208381983736025,
  20.07794121410805,
  19.958114110933252,
  19.847239333178166,
  19.74401741657217,
  19.647412751057335,
  19.55658665312371,
  19.47085039917101,
  19.389631476284475,
  19.312448823682224,
  19.238894335541694,
  19.168618816156048,
  19.10132116024439,
  19.03673990858324,
  18.97464657942649,
  18.914840345620053,
  18.857143744146338,
  18.801399186744536,
  18.74746609856544,
  18.695218553916966,
  18.644543308941437,
  18.59533815384735,
  18.54751052436372,
  18.500976324969766,
  18.455658926285626,
  18.411488306580363,
  18.36840031323149,
  18.326336024569564,
  18.285241196166524,
  18.24506577850403,
  18.205763495256704,
  18.167291473272865],
 'V0': 20.957577180240715}
  • Notice : "Status", "V0", "K" and "EFF" , new in version 0.8
In [9]:
Data=Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
###########
Larminie-Dicks-Model Simulation
###########
Analyzing . . .
I : 0.1
PEM Efficiency : 0.6070918465825977 
Power : 0.09470632806688527 W
Power-Stack : 2.178245545538361 W
Power-Thermal : 0.650754454461639 W
VStack : 21.78245545538361 V
Vcell : 0.9470632806688526 V
###########
I : 0.2
PEM Efficiency : 0.596983288796134 
Power : 0.18625878610439384 W
Power-Stack : 4.2839520804010585 W
Power-Thermal : 1.3740479195989417 W
VStack : 21.41976040200529 V
Vcell : 0.9312939305219692 V
###########
I : 0.3
PEM Efficiency : 0.5889668542476844 
Power : 0.27563648778791633 W
Power-Stack : 6.339639219122075 W
Power-Thermal : 2.1473607808779245 W
VStack : 21.132130730406917 V
Vcell : 0.9187882926263877 V
###########
I : 0.4
PEM Efficiency : 0.5823143704480743 
Power : 0.3633641671595984 W
Power-Stack : 8.357375844670763 W
Power-Thermal : 2.958624155329237 W
VStack : 20.893439611676907 V
Vcell : 0.908410417898996 V
###########
I : 0.5
PEM Efficiency : 0.576622053314997 
Power : 0.4497652015856977 W
Power-Stack : 10.344599636471047 W
Power-Thermal : 3.8004003635289525 W
VStack : 20.689199272942094 V
Vcell : 0.8995304031713954 V
###########
I : 0.6
PEM Efficiency : 0.5716425142657562 
Power : 0.5350573933527478 W
Power-Stack : 12.3063200471132 W
Power-Thermal : 4.6676799528867985 W
VStack : 20.510533411855334 V
Vcell : 0.8917623222545797 V
###########
I : 0.7
PEM Efficiency : 0.5672131517182866 
Power : 0.6193967616763688 W
Power-Stack : 14.246125518556482 W
Power-Thermal : 5.556874481443514 W
VStack : 20.35160788365212 V
Vcell : 0.884852516680527 V
###########
I : 0.8
PEM Efficiency : 0.5632213484876261 
Power : 0.7029002429125574 W
Power-Stack : 16.16670558698882 W
Power-Thermal : 6.465294413011179 W
VStack : 20.208381983736025 V
Vcell : 0.8786253036406967 V
###########
I : 0.9
PEM Efficiency : 0.5595858755325545 
Power : 0.7856585692477064 W
Power-Stack : 18.070147092697248 W
Power-Thermal : 7.390852907302754 W
VStack : 20.07794121410805 V
Vcell : 0.8729539658307849 V
###########
I : 1.0
PEM Efficiency : 0.5562462126792992 
Power : 0.8677440917797067 W
Power-Stack : 19.958114110933252 W
Power-Thermal : 8.331885889066747 W
VStack : 19.958114110933252 V
Vcell : 0.8677440917797067 V
###########
I : 1.1
PEM Efficiency : 0.5531560572234717 
Power : 0.9492157941954775 W
Power-Stack : 21.831963266495983 W
Power-Thermal : 9.287036733504017 W
VStack : 19.847239333178166 V
Vcell : 0.8629234492686159 V
###########
I : 1.2
PEM Efficiency : 0.5502791922121564 
Power : 1.0301226478211567 W
Power-Stack : 23.692820899886605 W
Power-Thermal : 10.255179100113395 W
VStack : 19.74401741657217 V
Vcell : 0.858435539850964 V
###########
I : 1.3
PEM Efficiency : 0.5475867544887774 
Power : 1.1105059381032407 W
Power-Stack : 25.541636576374536 W
Power-Thermal : 11.235363423625465 W
VStack : 19.647412751057335 V
Vcell : 0.8542353370024928 V
###########
I : 1.4
PEM Efficiency : 0.5450553693735705 
Power : 1.190400926711878 W
Power-Stack : 27.379221314373194 W
Power-Thermal : 12.226778685626805 W
VStack : 19.55658665312371 V
Vcell : 0.85028637622277 V
###########
I : 1.5
PEM Efficiency : 0.5426658416714328 
Power : 1.2698380695111529 W
Power-Stack : 29.206275598756516 W
Power-Thermal : 13.228724401243484 W
VStack : 19.47085039917101 V
Vcell : 0.8465587130074352 V
###########
I : 1.6
PEM Efficiency : 0.5404022150580957 
Power : 1.348843928785007 W
Power-Stack : 31.02341036205516 W
Power-Thermal : 14.24058963794484 W
VStack : 19.389631476284475 V
Vcell : 0.8430274554906293 V
###########
I : 1.7
PEM Efficiency : 0.5382510820424253 
Power : 1.427441869576512 W
Power-Stack : 32.83116300025978 W
Power-Thermal : 15.261836999740217 W
VStack : 19.312448823682224 V
Vcell : 0.8396716879861836 V
###########
I : 1.8
PEM Efficiency : 0.536201068437617 
Power : 1.5056526001728283 W
Power-Stack : 34.63000980397505 W
Power-Thermal : 16.29199019602495 W
VStack : 19.238894335541694 V
Vcell : 0.8364736667626824 V
###########
I : 1.9
PEM Efficiency : 0.5342424419218519 
Power : 1.583494597856369 W
Power-Stack : 36.420375750696486 W
Power-Thermal : 17.330624249303508 W
VStack : 19.168618816156048 V
Vcell : 0.833418209398089 V
###########
I : 2.0
PEM Efficiency : 0.532366810486187 
Power : 1.6609844487169034 W
Power-Stack : 38.20264232048878 W
Power-Thermal : 18.37735767951122 W
VStack : 19.10132116024439 V
Vcell : 0.8304922243584517 V
###########
I : 2.1
PEM Efficiency : 0.5305668870842597 
Power : 1.738137122088035 W
Power-Stack : 39.97715380802481 W
Power-Thermal : 19.431846191975197 W
VStack : 19.03673990858324 V
Vcell : 0.8276843438514452 V
###########
I : 2.2
PEM Efficiency : 0.5288363037744284 
Power : 1.8149661945538385 W
Power-Stack : 41.74422247473829 W
Power-Thermal : 20.49377752526172 W
VStack : 18.97464657942649 V
Vcell : 0.8249846338881083 V
###########
I : 2.3
PEM Efficiency : 0.527169463367337 
Power : 1.891484034562005 W
Power-Stack : 43.504132794926115 W
Power-Thermal : 21.562867205073875 W
VStack : 18.914840345620053 V
Vcell : 0.8223843628530457 V
###########
I : 2.4
PEM Efficiency : 0.5255614198480028 
Power : 1.9677019559109223 W
Power-Stack : 45.257144985951214 W
Power-Thermal : 22.638855014048787 W
VStack : 18.857143744146338 V
Vcell : 0.8198758149628843 V
###########
I : 2.5
PEM Efficiency : 0.5240077811244296 
Power : 2.0436303463852754 W
Power-Stack : 47.00349796686133 W
Power-Thermal : 23.721502033138655 W
VStack : 18.801399186744536 V
Vcell : 0.8174521385541103 V
###########
I : 2.6
PEM Efficiency : 0.5225046292799733 
Power : 2.1192787763595717 W
Power-Stack : 48.74341185627015 W
Power-Thermal : 24.810588143729856 W
VStack : 18.74746609856544 V
Vcell : 0.8151072216767583 V
###########
I : 2.7
PEM Efficiency : 0.52104845467996 
Power : 2.194656091111992 W
Power-Stack : 50.47709009557581 W
Power-Thermal : 25.905909904424192 W
VStack : 18.695218553916966 V
Vcell : 0.8128355893007376 V
###########
I : 2.8
PEM Efficiency : 0.5196361011410657 
Power : 2.269770489784175 W
Power-Stack : 52.20472126503602 W
Power-Thermal : 27.007278734963975 W
VStack : 18.644543308941437 V
Vcell : 0.8106323177800625 V
###########
I : 2.9
PEM Efficiency : 0.5182647200068938 
Power : 2.3446295933111876 W
Power-Stack : 53.926480646157316 W
Power-Thermal : 28.11451935384268 W
VStack : 18.59533815384735 V
Vcell : 0.8084929632107544 V
###########
I : 3.0
PEM Efficiency : 0.5169317314482642 
Power : 2.4192405031778765 W
Power-Stack : 55.64253157309116 W
Power-Thermal : 29.22746842690884 W
VStack : 18.54751052436372 V
Vcell : 0.8064135010592922 V
###########
I : 3.1
PEM Efficiency : 0.515634791665824 
Power : 2.4936098524959247 W
Power-Stack : 57.35302660740627 W
Power-Thermal : 30.345973392593724 W
VStack : 18.500976324969766 V
Vcell : 0.8043902749986854 V
###########
I : 3.2
PEM Efficiency : 0.514371764946645 
Power : 2.5677438506136525 W
Power-Stack : 59.05810856411401 W
Power-Thermal : 31.469891435885998 W
VStack : 18.455658926285626 V
Vcell : 0.8024199533167663 V
###########
I : 3.3
PEM Efficiency : 0.5131406997374682 
Power : 2.6416483222484866 W
Power-Stack : 60.75791141171519 W
Power-Thermal : 32.5990885882848 W
VStack : 18.411488306580363 V
Vcell : 0.8004994915904505 V
###########
I : 3.4
PEM Efficiency : 0.5119398080610783 
Power : 2.7153287419559593 W
Power-Stack : 62.452561064987066 W
Power-Thermal : 33.73343893501293 W
VStack : 18.36840031323149 V
Vcell : 0.7986261005752822 V
###########
I : 3.5
PEM Efficiency : 0.5107674477304784 
Power : 2.788790264608412 W
Power-Stack : 64.14217608599348 W
Power-Thermal : 34.872823914006524 W
VStack : 18.326336024569564 V
Vcell : 0.7967972184595463 V
###########
I : 3.6
PEM Efficiency : 0.5096221069165698 
Power : 2.862037752443456 W
Power-Stack : 65.82686830619949 W
Power-Thermal : 36.017131693800515 W
VStack : 18.285241196166524 V
Vcell : 0.7950104867898489 V
###########
I : 3.7
PEM Efficiency : 0.5085023907052405 
Power : 2.935075799150648 W
Power-Stack : 67.50674338046491 W
Power-Thermal : 37.16625661953508 W
VStack : 18.24506577850403 V
Vcell : 0.7932637295001752 V
###########
I : 3.8
PEM Efficiency : 0.5074070093438323 
Power : 3.007908751390238 W
Power-Stack : 69.18190128197547 W
Power-Thermal : 38.32009871802452 W
VStack : 18.205763495256704 V
Vcell : 0.7915549345763785 V
###########
I : 3.9
PEM Efficiency : 0.5063347679284521 
Power : 3.080540728076703 W
Power-Stack : 70.85243674576417 W
Power-Thermal : 39.47856325423582 W
VStack : 18.167291473272865 V
Vcell : 0.7898822379683854 V
###########
Done!
  • Notice : "PrintMode" & "ReportMode" , new in version 0.5
In [10]:
Static_Analysis(InputMethod={},TestMode=True,PrintMode=False,ReportMode=True)
Out[10]:
{'Message': '[Error] Larminie-Dicks Simulation Failed!(Check Your Inputs)',
 'Status': False}

Parameters

Inputs, Constants & Middle Values

  1. User : User Input
  2. System : Simulator Calculation (Middle Value)
$$Parameter$$ $$Description$$ $$Unit$$ $$Value$$
$$E_0$$ Fuel Cell reversible no loss voltage $$V$$ $$User$$
$$A$$ The slope of the Tafel line $$V$$ $$User$$
$$i_n$$ Internal current $$A$$ $$User$$
$$i_0$$ Exchange current at which the overvoltage begins to move from zero $$A$$ $$User$$
$$i_L$$ Limiting current $$A$$ $$User$$
$$B$$ Constant in the mass transfer term $$V$$ $$User$$
$$R_m$$ The membrane and contact resistances $$\Omega$$ $$User$$
$$N$$ Number of single cells $$--$$ $$User$$
$$i_{start}$$ Cell operating current start point $$A$$ $$User$$
$$i_{step}$$ Cell operating current step $$A$$ $$User$$
$$i_{stop}$$ Cell operating current end point $$A$$ $$User$$
$$\mu_F$$ The fuel utilization $$--$$ $$0.95$$
$$HHV$$ Higher Heating Value Potential $$V$$ $$1.482$$
$$E_{th}$$ Theoretical Potential $$V$$ $$1.23$$

Reference

I. Sadli, P. Thounthong, J.-P. Martin, S. Rael, B. Davat. 2006. "Behaviour of a PEMFC supplying a low voltage static converter." Journal of Power Sources (Elsevier) 156: 119–125. doi:10.1016/j.jpowsour.2005.08.021.