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 based on some assumption such; the gases are ideal, the stack is fed with hydrogen and air, temperature is stable at all times, the ratio of pressures between the interior and exterior of the channel is large, The channels that transport gases along the electrodes have a fixed volume, only source of losses is ohmic and 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
from opem.Dynamic.Padulles_Hauer import *
qH2=qH2_Calc(qMethanol=0.0002,CV=2,t1=2,t2=2)
qH2
3.6363636363636364e-05
Kr=Kr_Calc(N0=5)
Kr
1.2955435375179044e-08
qO2=qO2_Calc(qH2=qH2,rho=1.168)
qO2
3.113325031133251e-05
PH2=PH2_Calc(KH2=0.0000422,tH2=3.37,Kr=Kr,I=1,qH2=qH2)
PH2
0.19704428889827239
PO2=PO2_Calc(KO2=0.0000211,tO2=6.74,Kr=Kr,I=1,qO2=qO2)
PO2
0.1905549730945132
PH2O=PH2O_Calc(KH2O=0.000007716,tH2O=18.418,Kr=Kr,I=1,qH2O=qH2)
PH2O
0.2425275190047356
Enernst=Enernst_Calc(E0=0.6,N0=5,T=343, PH2=PH2, PO2=PO2,PH2O=PH2O)
Enernst
2.9234016588243374
FC_Voltage=Vcell_Calc(Enernst=Enernst, B=0.04777,C=0.0136,I=1,Rint=0.00303)
FC_Voltage
3.1256720945020287
Power=Power_Calc(Vcell=FC_Voltage,i=2)
Power
6.251344189004057
Efficiency_Calc(Vcell=FC_Voltage,N=5)
0.40072719160282416
i
=0 to i
=4 with step
=0.1Test_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"}
Dynamic_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': [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], 'V': [3.2384074248432166, 3.204991246456224, 3.1853176406062635, 3.171270530305614, 3.1603064250957056, 3.1512923865000246, 3.1436240706330323, 3.1369407380516985, 3.1310097044187106, 3.1256720945020287, 3.1208145890650347, 3.116353517374462, 3.1122253389530767, 3.1083806627834245, 3.1047803349085714, 3.101392791285848, 3.098192214569633, 3.0951572185443843, 3.0922698887241142, 3.0895150693269517, 3.086879824396224, 3.0843530243968997, 3.0819250247823504, 3.079587413020922, 3.07733280729794, 3.0751546947217467, 3.073047300081027, 3.0710054784818825, 3.0690246268332686, 3.0671006103443585, 3.0652297010786023, 3.0634085262664685, 3.0616340245741624, 3.059903408902552, 3.0582141345800062, 3.0565638720370307, 3.054950483225618, 3.0533720011838796, 3.051826612255591]}
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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### 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 ########### Done!
Inputs, Constants & Middle Values
$$Parameter$$ | $$Description$$ | $$Unit$$ | $$Value$$ |
$$T$$ | 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 | $$C.kmol^{-1}$$ | $$96484600$$ |
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.