Chamberlin-Kim static model is an empirical equation which was developed to fit the experimental cell potential (E) vs. current density (J) data for proton exchange membrane fuel cells (PEMFCs), at several temperatures, pressures, and oxygen compositions in the cathode gas mixture. The exponential term compensates for the mass-transport regions of the V vs. i plot; i.e., the increase in slope of the pseudolinear region and the subsequent rapid fall-off of the cell potential with increasing current density. The terms E0 and b yield the electrode kinetic parameters for oxygen reduction in the PEMFC and R represents the resistance, predominantly ohmic and, to a small extent, the charge transfer resistance of the electro-oxidation of hydrogen. The exponential term characterizes the mass-transport region of the V vs. i plot. The parameter n has more pronounced effects than the parameter m in this region. In Chamberline Kim’s model, the values of the parameters(five parameters: E0, b, R, m, n) vary depending on many variables, including the composition of the Membrane Electrode Assemblies(MEA), the fuel and oxidant used, besides the local temperature, pressure, and humidity of the MEA. They also depend on the stack itself, so that it can not be transposed to another fuel cell without new parameter identification.
Fig1. Graphical Abstract of Static Models
from opem.Static.Chamberline_Kim import *
Vcell=Vcell_Calc(E0=0.982,b=0.0689,R=0.328,m=0.000125,n=9.45,i=1,A=50.0)
Vcell
1.244827379954939
VStack=VStack_Calc(Vcell=Vcell,N=1)
VStack
1.244827379954939
Power=Power_Calc(Vcell=Vcell,i=2)
Power
2.489654759909878
PowerStack_Calc(Power=Power,N=2)
4.979309519819756
Efficiency_Calc(Vcell=Vcell)
0.797966269201884
i
=0 to i
=4 with step
=0.1Test_Vector={"A":50.0,"E0":0.982,"b":0.0689,"R":0.328,"m":0.000125,"n":9.45,"N":1,"i-start":1,"i-stop":4,"i-step":0.1,"Name":"Test"}
Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
{'I': [1, 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': [1.244827379954939, 1.3613617901715511, 1.4771371289340156, 1.5922057745841485, 1.706612017960278, 1.820393802022961, 1.933583995558465, 2.0462113472500323, 2.158301214515197, 2.2698761299021477, 2.380956247931119, 2.491559702356007, 2.601702895230229, 2.7114007333123014, 2.820666823283489, 2.929513634375034, 3.0379526349346824, 3.145994407952373, 3.2536487494474096, 3.360924752781809, 3.4678308813294754, 3.5743750314441862, 3.6805645872928157, 3.786406468826155, 3.891907173928067, 3.9970728155998123, 4.101909154889422, 4.206421630157481, 4.310615383174736, 4.414495282468661], 'V': [1.244827379954939, 1.2376016274286827, 1.230947607445013, 1.2247736727570373, 1.2190085842573415, 1.2135958680153074, 1.2084899972240406, 1.2036537336764896, 1.1990562302862204, 1.1946716473169199, 1.1904781239655595, 1.1864570011219082, 1.1825922251046495, 1.1788698840488268, 1.1752778430347872, 1.1718054537500135, 1.168443321128724, 1.1651831140564344, 1.162017410516932, 1.1589395699247618, 1.1559436271098251, 1.153024203691673, 1.1501764335290048, 1.1473958996442895, 1.1446785805670785, 1.1420208044570892, 1.1394192096915061, 1.136870710853373, 1.1343724692565096, 1.1319218672996567]}
Data=Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=True,ReportMode=True)
########### Chamberline-Kim-Model Simulation ########### Analyzing . . . I : 1 PEM Efficiency : 0.797966269201884 Power : 1.244827379954939 W Power-Stack : 1.244827379954939 W VStack : 1.244827379954939 V Vcell : 1.244827379954939 V ########### I : 1.1 PEM Efficiency : 0.7933343765568479 Power : 1.3613617901715511 W Power-Stack : 1.3613617901715511 W VStack : 1.2376016274286827 V Vcell : 1.2376016274286827 V ########### I : 1.2 PEM Efficiency : 0.7890689791314186 Power : 1.4771371289340156 W Power-Stack : 1.4771371289340156 W VStack : 1.230947607445013 V Vcell : 1.230947607445013 V ########### I : 1.3 PEM Efficiency : 0.7851113286904084 Power : 1.5922057745841485 W Power-Stack : 1.5922057745841485 W VStack : 1.2247736727570373 V Vcell : 1.2247736727570373 V ########### I : 1.4 PEM Efficiency : 0.7814157591393215 Power : 1.706612017960278 W Power-Stack : 1.706612017960278 W VStack : 1.2190085842573415 V Vcell : 1.2190085842573415 V ########### I : 1.5 PEM Efficiency : 0.7779460692405816 Power : 1.820393802022961 W Power-Stack : 1.820393802022961 W VStack : 1.2135958680153074 V Vcell : 1.2135958680153074 V ########### I : 1.6 PEM Efficiency : 0.7746730751436158 Power : 1.933583995558465 W Power-Stack : 1.933583995558465 W VStack : 1.2084899972240406 V Vcell : 1.2084899972240406 V ########### I : 1.7 PEM Efficiency : 0.771572906202878 Power : 2.0462113472500323 W Power-Stack : 2.0462113472500323 W VStack : 1.2036537336764896 V Vcell : 1.2036537336764896 V ########### I : 1.8 PEM Efficiency : 0.7686257886450131 Power : 2.158301214515197 W Power-Stack : 2.158301214515197 W VStack : 1.1990562302862204 V Vcell : 1.1990562302862204 V ########### I : 1.9 PEM Efficiency : 0.7658151585364871 Power : 2.2698761299021477 W Power-Stack : 2.2698761299021477 W VStack : 1.1946716473169199 V Vcell : 1.1946716473169199 V ########### I : 2.0 PEM Efficiency : 0.7631270025420254 Power : 2.380956247931119 W Power-Stack : 2.380956247931119 W VStack : 1.1904781239655595 V Vcell : 1.1904781239655595 V ########### I : 2.1 PEM Efficiency : 0.7605493596935309 Power : 2.491559702356007 W Power-Stack : 2.491559702356007 W VStack : 1.1864570011219082 V Vcell : 1.1864570011219082 V ########### I : 2.2 PEM Efficiency : 0.7580719391696471 Power : 2.601702895230229 W Power-Stack : 2.601702895230229 W VStack : 1.1825922251046495 V Vcell : 1.1825922251046495 V ########### I : 2.3 PEM Efficiency : 0.7556858231082223 Power : 2.7114007333123014 W Power-Stack : 2.7114007333123014 W VStack : 1.1788698840488268 V Vcell : 1.1788698840488268 V ########### I : 2.4 PEM Efficiency : 0.753383232714607 Power : 2.820666823283489 W Power-Stack : 2.820666823283489 W VStack : 1.1752778430347872 V Vcell : 1.1752778430347872 V ########### I : 2.5 PEM Efficiency : 0.7511573421474446 Power : 2.929513634375034 W Power-Stack : 2.929513634375034 W VStack : 1.1718054537500135 V Vcell : 1.1718054537500135 V ########### I : 2.6 PEM Efficiency : 0.7490021289286691 Power : 3.0379526349346824 W Power-Stack : 3.0379526349346824 W VStack : 1.168443321128724 V Vcell : 1.168443321128724 V ########### I : 2.7 PEM Efficiency : 0.7469122526002784 Power : 3.145994407952373 W Power-Stack : 3.145994407952373 W VStack : 1.1651831140564344 V Vcell : 1.1651831140564344 V ########### I : 2.8 PEM Efficiency : 0.7448829554595718 Power : 3.2536487494474096 W Power-Stack : 3.2536487494474096 W VStack : 1.162017410516932 V Vcell : 1.162017410516932 V ########### I : 2.9 PEM Efficiency : 0.7429099807210011 Power : 3.360924752781809 W Power-Stack : 3.360924752781809 W VStack : 1.1589395699247618 V Vcell : 1.1589395699247618 V ########### I : 3.0 PEM Efficiency : 0.7409895045575802 Power : 3.4678308813294754 W Power-Stack : 3.4678308813294754 W VStack : 1.1559436271098251 V Vcell : 1.1559436271098251 V ########### I : 3.1 PEM Efficiency : 0.7391180792895339 Power : 3.5743750314441862 W Power-Stack : 3.5743750314441862 W VStack : 1.153024203691673 V Vcell : 1.153024203691673 V ########### I : 3.2 PEM Efficiency : 0.7372925855955158 Power : 3.6805645872928157 W Power-Stack : 3.6805645872928157 W VStack : 1.1501764335290048 V Vcell : 1.1501764335290048 V ########### I : 3.3 PEM Efficiency : 0.7355101920796727 Power : 3.786406468826155 W Power-Stack : 3.786406468826155 W VStack : 1.1473958996442895 V Vcell : 1.1473958996442895 V ########### I : 3.4 PEM Efficiency : 0.7337683208763324 Power : 3.891907173928067 W Power-Stack : 3.891907173928067 W VStack : 1.1446785805670785 V Vcell : 1.1446785805670785 V ########### I : 3.5 PEM Efficiency : 0.7320646182417239 Power : 3.9970728155998123 W Power-Stack : 3.9970728155998123 W VStack : 1.1420208044570892 V Vcell : 1.1420208044570892 V ########### I : 3.6 PEM Efficiency : 0.7303969292894269 Power : 4.101909154889422 W Power-Stack : 4.101909154889422 W VStack : 1.1394192096915061 V Vcell : 1.1394192096915061 V ########### I : 3.7 PEM Efficiency : 0.7287632761880597 Power : 4.206421630157481 W Power-Stack : 4.206421630157481 W VStack : 1.136870710853373 V Vcell : 1.136870710853373 V ########### I : 3.8 PEM Efficiency : 0.7271618392669932 Power : 4.310615383174736 W Power-Stack : 4.310615383174736 W VStack : 1.1343724692565096 V Vcell : 1.1343724692565096 V ########### I : 3.9 PEM Efficiency : 0.7255909405767029 Power : 4.414495282468661 W Power-Stack : 4.414495282468661 W VStack : 1.1319218672996567 V Vcell : 1.1319218672996567 V ########### Done!
Inputs, Constants & Middle Values
$$Parameter$$ | $$Description$$ | $$Unit$$ | $$Value$$ |
$$E_0$$ | Open circuit voltage | $$V$$ | $$User$$ |
$$b$$ | Tafel’s parameter for the oxygen reduction | $$V$$ | $$User$$ |
$$R$$ | Resistance | $$\Omega cm^2$$ | $$User$$ |
$$m$$ | Diffusion’s parameters | $$V$$ | $$User$$ |
$$n$$ | Diffusion’s parameters | $$A^{-1}cm^2$$ | $$User$$ |
$$A$$ | Active area | $$cm^2$$ | $$User$$ |
Junbom Kim, Seong-Min Lee, Supramaniam Srinivasan, Charles E. Chamberlin. 1995. "Modeling of Proton Exchange Membrane Fuel Cell Performance with an Empirical Equation." Journal of The Electrochemical Society (The Electrochemical Society) 142 (8): 2670-2674. doi:10.1149/1.2050072.