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
from opem.Static.Larminie_Dicks import *
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
0.8677440917797067
VStack=VStack_Calc(Vcell=Vcell,N=23)
VStack
19.958114110933252
Power=Power_Calc(Vcell=Vcell,i=2)
Power
1.7354881835594134
PowerStack_Calc(Power=Power,N=23)
39.916228221866504
Efficiency_Calc(Vcell=Vcell)
0.5562462126792992
i
=0 to i
=4 with step
=0.1Test_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"}
Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
{'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], '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], '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]}
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 VStack : 21.78245545538361 V Vcell : 0.9470632806688526 V ########### I : 0.2 PEM Efficiency : 0.596983288796134 Power : 0.18625878610439384 W Power-Stack : 4.2839520804010585 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 VStack : 21.132130730406917 V Vcell : 0.9187882926263877 V ########### I : 0.4 PEM Efficiency : 0.5823143704480743 Power : 0.3633641671595984 W Power-Stack : 8.357375844670763 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 VStack : 20.689199272942094 V Vcell : 0.8995304031713954 V ########### I : 0.6 PEM Efficiency : 0.5716425142657562 Power : 0.5350573933527478 W Power-Stack : 12.3063200471132 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 VStack : 20.35160788365212 V Vcell : 0.884852516680527 V ########### I : 0.8 PEM Efficiency : 0.5632213484876261 Power : 0.7029002429125574 W Power-Stack : 16.16670558698882 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 VStack : 20.07794121410805 V Vcell : 0.8729539658307849 V ########### I : 1.0 PEM Efficiency : 0.5562462126792992 Power : 0.8677440917797067 W Power-Stack : 19.958114110933252 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 VStack : 19.847239333178166 V Vcell : 0.8629234492686159 V ########### I : 1.2 PEM Efficiency : 0.5502791922121564 Power : 1.0301226478211567 W Power-Stack : 23.692820899886605 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 VStack : 19.647412751057335 V Vcell : 0.8542353370024928 V ########### I : 1.4 PEM Efficiency : 0.5450553693735705 Power : 1.190400926711878 W Power-Stack : 27.379221314373194 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 VStack : 19.47085039917101 V Vcell : 0.8465587130074352 V ########### I : 1.6 PEM Efficiency : 0.5404022150580957 Power : 1.348843928785007 W Power-Stack : 31.02341036205516 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 VStack : 19.312448823682224 V Vcell : 0.8396716879861836 V ########### I : 1.8 PEM Efficiency : 0.536201068437617 Power : 1.5056526001728283 W Power-Stack : 34.63000980397505 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 VStack : 19.168618816156048 V Vcell : 0.833418209398089 V ########### I : 2.0 PEM Efficiency : 0.532366810486187 Power : 1.6609844487169034 W Power-Stack : 38.20264232048878 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 VStack : 19.03673990858324 V Vcell : 0.8276843438514452 V ########### I : 2.2 PEM Efficiency : 0.5288363037744284 Power : 1.8149661945538385 W Power-Stack : 41.74422247473829 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 VStack : 18.914840345620053 V Vcell : 0.8223843628530457 V ########### I : 2.4 PEM Efficiency : 0.5255614198480028 Power : 1.9677019559109223 W Power-Stack : 45.257144985951214 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 VStack : 18.801399186744536 V Vcell : 0.8174521385541103 V ########### I : 2.6 PEM Efficiency : 0.5225046292799733 Power : 2.1192787763595717 W Power-Stack : 48.74341185627015 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 VStack : 18.695218553916966 V Vcell : 0.8128355893007376 V ########### I : 2.8 PEM Efficiency : 0.5196361011410657 Power : 2.269770489784175 W Power-Stack : 52.20472126503602 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 VStack : 18.59533815384735 V Vcell : 0.8084929632107544 V ########### I : 3.0 PEM Efficiency : 0.5169317314482642 Power : 2.4192405031778765 W Power-Stack : 55.64253157309116 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 VStack : 18.500976324969766 V Vcell : 0.8043902749986854 V ########### I : 3.2 PEM Efficiency : 0.514371764946645 Power : 2.5677438506136525 W Power-Stack : 59.05810856411401 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 VStack : 18.411488306580363 V Vcell : 0.8004994915904505 V ########### I : 3.4 PEM Efficiency : 0.5119398080610783 Power : 2.7153287419559593 W Power-Stack : 62.452561064987066 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 VStack : 18.326336024569564 V Vcell : 0.7967972184595463 V ########### I : 3.6 PEM Efficiency : 0.5096221069165698 Power : 2.862037752443456 W Power-Stack : 65.82686830619949 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 VStack : 18.24506577850403 V Vcell : 0.7932637295001752 V ########### I : 3.8 PEM Efficiency : 0.5074070093438323 Power : 3.007908751390238 W Power-Stack : 69.18190128197547 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 VStack : 18.167291473272865 V Vcell : 0.7898822379683854 V ########### Done!
Inputs, Constants & Middle Values
$$Parameter$$ | $$Description$$ | $$Unit$$ | $$Value$$ |
$$E_0$$ | 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 | $$\Omega$$ | $$User$$ |
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.