Chamberline-KimStatic Model

Version 0.9

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

Vcell

$$V_{cell}=E_0-b\times ln(J)-(R\times J)-m\times exp(nJ)$$

$$J=\frac{i}{A}$$

In [1]:
from opem.Static.Chamberline_Kim import Vcell_Calc
Vcell=Vcell_Calc(E0=0.982,b=0.0689,R=0.328,m=0.000125,n=9.45,i=1,A=50.0)
Vcell
Out[1]:
1.244827379954939
  • Notice : from opem.Chamberline_Kim in version (0.3>)

VStack

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

In [2]:
from opem.Static.Chamberline_Kim import VStack_Calc
VStack=VStack_Calc(Vcell=Vcell,N=1)
VStack
Out[2]:
1.244827379954939

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.Chamberline_Kim import Power_Calc,PowerStack_Calc,Power_Thermal_Calc
Power=Power_Calc(Vcell=Vcell,i=2)
Power
Out[3]:
2.489654759909878
In [4]:
PowerStack_Calc(Power=Power,N=2)
Out[4]:
4.979309519819756
In [5]:
Power_Thermal_Calc(VStack=VStack,N=2,i=2)
Out[5]:
2.430345240090122

Efficiency of PEMFC

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

In [6]:
from opem.Static.Chamberline_Kim import Efficiency_Calc
Efficiency_Calc(Vcell=Vcell)
Out[6]:
0.797966269201884

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":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"}
  • Notice : "Name", new in version 0.5
In [8]:
from opem.Static.Chamberline_Kim import Static_Analysis
Static_Analysis(InputMethod=Test_Vector,TestMode=True,PrintMode=False,ReportMode=True)
Out[8]:
{'EFF': [0.797966269201884,
  0.7933343765568479,
  0.7890689791314186,
  0.7851113286904084,
  0.7814157591393215,
  0.7779460692405816,
  0.7746730751436158,
  0.771572906202878,
  0.7686257886450131,
  0.7658151585364871,
  0.7631270025420254,
  0.7605493596935309,
  0.7580719391696471,
  0.7556858231082223,
  0.753383232714607,
  0.7511573421474446,
  0.7490021289286691,
  0.7469122526002784,
  0.7448829554595718,
  0.7429099807210011,
  0.7409895045575802,
  0.7391180792895339,
  0.7372925855955158,
  0.7355101920796727,
  0.7337683208763324,
  0.7320646182417239,
  0.7303969292894269,
  0.7287632761880597,
  0.7271618392669932,
  0.7255909405767029],
 '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],
 'K': -0.0372516118425709,
 '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],
 'Ph': [-0.01482737995493899,
  -0.008361790171551031,
  -0.0011371289340155854,
  0.006794225415851485,
  0.015387982039721892,
  0.02460619797703889,
  0.03441600444153501,
  0.04478865274996757,
  0.055698785484803184,
  0.06712387009785219,
  0.07904375206888092,
  0.09144029764399271,
  0.10429710476977118,
  0.11759926668769842,
  0.13133317671651062,
  0.14548636562496609,
  0.16004736506531766,
  0.17500559204762714,
  0.1903512505525901,
  0.20607524721819082,
  0.22216911867052458,
  0.23862496855581392,
  0.2554354127071847,
  0.2725935311738447,
  0.29009282607193304,
  0.30792718440018774,
  0.3260908451105779,
  0.3445783698425195,
  0.36338461682526346,
  0.38250471753133874],
 'Status': True,
 '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],
 'V0': 1.2696835857181188}
  • 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)
###########
Chamberline-Kim-Model Simulation
###########
Analyzing . . .
I : 1
PEM Efficiency : 0.797966269201884 
Power : 1.244827379954939 W
Power-Stack : 1.244827379954939 W
Power-Thermal : -0.01482737995493899 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
Power-Thermal : -0.008361790171551031 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
Power-Thermal : -0.0011371289340155854 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
Power-Thermal : 0.006794225415851485 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
Power-Thermal : 0.015387982039721892 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
Power-Thermal : 0.02460619797703889 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
Power-Thermal : 0.03441600444153501 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
Power-Thermal : 0.04478865274996757 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
Power-Thermal : 0.055698785484803184 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
Power-Thermal : 0.06712387009785219 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
Power-Thermal : 0.07904375206888092 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
Power-Thermal : 0.09144029764399271 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
Power-Thermal : 0.10429710476977118 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
Power-Thermal : 0.11759926668769842 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
Power-Thermal : 0.13133317671651062 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
Power-Thermal : 0.14548636562496609 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
Power-Thermal : 0.16004736506531766 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
Power-Thermal : 0.17500559204762714 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
Power-Thermal : 0.1903512505525901 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
Power-Thermal : 0.20607524721819082 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
Power-Thermal : 0.22216911867052458 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
Power-Thermal : 0.23862496855581392 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
Power-Thermal : 0.2554354127071847 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
Power-Thermal : 0.2725935311738447 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
Power-Thermal : 0.29009282607193304 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
Power-Thermal : 0.30792718440018774 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
Power-Thermal : 0.3260908451105779 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
Power-Thermal : 0.3445783698425195 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
Power-Thermal : 0.36338461682526346 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
Power-Thermal : 0.38250471753133874 W
VStack : 1.1319218672996567 V
Vcell : 1.1319218672996567 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] Chamberline-Kim Simulation Failed!(Check Your Inputs)',
 'Status': False}

Parameters

Inputs, Constants & Middle Values

  1. User : User Input
  2. System : Simulator Calculation (Middle Value)
$$Parameter$$ $$Description$$ $$Unit$$ $$Value$$
$$E_0$$ 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 $$cm^2$$ $$User$$
$$N$$ 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 $$A$$ $$User$$
$$J$$ Actual current density of the cell $$Acm^{-2}$$ $$System$$
$$\mu_F$$ The fuel utilization $$--$$ $$0.95$$
$$HHV$$ Higher Heating Value Potential $$V$$ $$1.482$$
$$E_{th}$$ Theoretical Potential $$V$$ $$1.23$$

Reference

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.