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PEM Fuel Cell Stack: Polarization, Efficiency & Water

Size a PEM fuel cell stack or check an existing one. Plot polarization and power, calculate HHV/LHV efficiency, hydrogen and air demand, and cathode water balance.

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Published 30. October 2026 · version 2 of 6. November 2026

#efficiency #hydrogen #PEM fuel cells #polarization curve #stack sizing #water balance

This PEM fuel cell stack calculator connects a polarization curve to practical stack sizing, hydrogen consumption, electrical efficiency and cathode water balance. Engineers and students can compare an existing series stack with a proposed design, or estimate the active cell area needed for a target net DC output.

Choose Size stack to enter the required useful power and minimum loaded stack voltage. The tool rounds the series cell count up, then calculates active area per cell at your chosen current density. In Evaluate stack, enter its series cell count and active area instead. The area is the electrochemically active surface of one cell; total active area adds all cells and is not the stack footprint.

The default polarization curve uses Kim and colleagues’ published hydrogen/air fit at 70 °C and 1 atm. Its operating and plotting window is restricted to 0.02–0.70 A/cm², a conservative interval inside Fig. 5’s measured 1 atm air data, which end near 0.72 A/cm². This window excludes the near-zero-current region and does not establish a continuous-duty rating. Choose Custom fit to reproduce your own measured curve at its operating conditions. Alternatively, enter a measured or rated loaded cell voltage for a single operating point. In that mode the fitted-curve charts are blank; the sizing and balance results still work. The fitted equation uses a base-10 logarithm and excludes zero current. Its intercept is not an open-circuit voltage.

Select a continuous operating current density within the measured data. The voltage and power plots reveal the tradeoff between compact active area and lower voltage at higher load. Curves hold the selected cell count and area fixed: they do not resize the stack at every plotted current. Subtract the actual compressor, pump, fan and control power to obtain net DC power. Overall hydrogen utilization includes fresh-fuel purge losses. The efficiency table distinguishes gross output on reacted hydrogen from gross and net output on supplied hydrogen, with both HHV and LHV net efficiency.

For the water balance, enter cathode absolute pressure, gas temperature, inlet relative humidity and air stoichiometry. Reaction water follows Faraday’s law. An ideal-gas equilibrium balance compares inlet vapor plus generated water with the saturated exhaust capacity after oxygen consumption. It reports outlet vapor, liquid discharge and unused vapor capacity. Unused capacity means the exhaust is unsaturated; it does not measure membrane drying. Liquid discharge is an equilibrium estimate, not a flooding diagnosis or guaranteed water recovery.

This tool does not model membrane hydration, electro-osmotic drag, anode water, local current distribution, heat rejection, aging, transients, pressure losses or compressor performance. Changing gas conditions does not automatically correct the fitted voltage. Cell thickness, seals, cooling plates and manifolds are required to turn active area into physical stack dimensions or mass. Check startup and open-circuit voltage against the converter separately. Installation work, including hydrogen handling and electrical connections, belongs to a qualified person.

How it is calculated

The polarization model is Kim et al. (1995), Eq. (5), PDF p. 3 (printed p. 2672): $V = E_0-b\log_{10}(j/j_{ref})-Rj-m\exp(nj)$ . Here $j_{ref}=0.001$ A/cm², preserving the paper's mA/cm² logarithm convention. The default Table II row is H₂/air, 70 °C, 1 atm: E₀ = 0.943 V, b = 0.062 V per decade, R = 0.238 Ω·cm², m = 0.00375 V and n = 6.38 cm²/A. These coefficients are condition-specific. Published-mode operation and plotting are restricted to 0.02–0.70 A/cm², conservatively inside the experimental 1 atm air points in Fig. 5 (PDF p. 3), whose final point is near 0.72 A/cm². The bounds are tool-use limits chosen from the plot, not numerical limits stated by the authors. Rated-voltage mode uses the entered loaded voltage without predicting a curve.

For a series stack, $I=jA$ , $V_s=NV$ and $P_g=V_s I$ . Sizing uses $N=\lceil V_{min}/V\rceil$ , $P_g=P_{target}+P_{aux}$ and $A=P_g/(NVj)$ . This meets the loaded minimum voltage with an integer cell count; it does not select a maximum voltage.

Faraday's law gives $\dot n_{H_2}=NI/(2F)$ and $\dot n_{O_2}=NI/(4F)$ . Fresh H₂ is reacted H₂ divided by overall utilization. Dry air supply is $\lambda_{air}\dot n_{O_2}/0.21$ . This follows DOE Fuel Cell Handbook, seventh edition (2004), section 9.1.1, PDF pp. 352–354. Hydrogen HHV is 285.8 kJ/mol (DOE PDF p. 61); LHV is 241.826 kJ/mol (NIST gas-phase water formation enthalpy). Efficiency divides the selected electrical output by the corresponding reacted or supplied fuel energy rate.

Water saturation pressure uses the NIST WebBook Stull (1947) Antoine fit, $\log_{10}(p_{sat}/bar)=4.6543-1435.264/(T_K-64.848)$ , within its published 255.9–373 K range. Inlet vapor per mole of dry air is $RH\,p_{sat}/(p-RH\,p_{sat})$ . Outlet dry gas is inlet dry air minus reacted O₂. Saturated vapor capacity is $\dot n_{dry,out}p_{sat}/(p-p_{sat})$ . Water above that capacity leaves as liquid; otherwise all available water leaves as vapor. This balance is derived here from species conservation and Dalton's law.

Assumptions

Identical cells in one series string, uniform steady current density and 100% Faradaic efficiency. Pure hydrogen, dry air with 21% O₂ by mole, and DOE's 28.85 g/mol dry-air convention. Overall hydrogen utilization refers to fresh supply; recirculation flow is not calculated. Auxiliaries are a fixed user-entered load at the operating point.

The cathode inlet and outlet have one gas temperature and absolute pressure. All electrochemical product water is assigned to the cathode, with zero net membrane water transfer and no liquid entering. Exhaust is in vapor/liquid equilibrium, with no entrainment or incomplete evaporation effects. The Antoine fit is an engineering approximation. HHV and LHV use 25 °C standard fuel reference enthalpies, independent of gas operating temperature.

Where it applies

Use the published polarization fit only for its stated cell and conditions, or supply coefficients fitted to your actual measurements. Changing temperature, pressure, humidity, stoichiometry or utilization does not refit the curve. Published mode blocks operating current densities outside 0.02–0.70 A/cm² and plot endpoints above 0.70 A/cm². These conservative bounds stay inside Fig. 5's measurements; they do not certify continuous operation. A custom fit must stay within your own measured range. A rated voltage specifies one operating point only. Positive fitted voltage does not establish an acceptable continuous operating limit.

The water balance is a lumped cathode exhaust estimate. It cannot predict membrane hydration, flooding, anode humidification demand or recoverable condensate after an external cooler. No mechanical stack envelope, heat exchanger size, hydrogen storage, protection system or installation approval is calculated. Account for degradation margins, minimum individual-cell voltage and the full converter voltage window separately.

References

Kim, J., Lee, S.-M., Srinivasan, S. and Chamberlin, C. E. (1995). Modeling of Proton Exchange Membrane Fuel Cell Performance with an Empirical Equation. Journal of The Electrochemical Society 142(8), 2670–2674. DOI: https://doi.org/10.1149/1.2050072. Open author/institutional copy: https://schatzcenter.org/docs/Modeling_Proton_Exchange_Membrane_Fuel_Cell_Performance_with_Empirical_Equation.pdf
EG&G Technical Services / U.S. DOE NETL (2004). Fuel Cell Handbook, Seventh Edition, sections 2.2 and 9.1.1; worked Examples 9-1 and 9-2. https://netl.doe.gov/sites/default/files/netl-file/FCHandbook7.pdf
NIST Chemistry WebBook, SRD 69. Water: Antoine equation, Stull (1947), 255.9–373 K; pressure in bar and temperature in K. https://webbook.nist.gov/cgi/cbook.cgi?ID=C7732185&Mask=4&Type=ANTOINE
NIST Chemistry WebBook, SRD 69. Water gas-phase standard enthalpy of formation, CODATA review value −241.826 kJ/mol. https://webbook.nist.gov/cgi/cbook.cgi?ID=C7732185&Mask=1

Inputs

Input	Symbol	Unit	Default
Calculation Size stack for a target net DC power and minimum loaded voltage, or evaluate an existing series stack. Active area is per cell.	`task`		Size stack
Cell voltage source Kim et al. (1995), Table II, H₂/air at 70 °C and 1 atm. Published operation and plotting are restricted to 0.02–0.70 A/cm², conservatively inside Fig. 5 data. Custom fit uses your coefficients. Measured voltage defines one point; both curve charts are blank.	`voltage_mode`		Kim: 70 °C, 1 atm, H₂/air
Operating current density Published Kim fit: use 0.02–0.70 A/cm², inside Fig. 5 measurements. With your own fit or measured voltage, choose a supported continuous operating point.	`j_op`	A/cm^2	0.4
Measured cell voltage Mean loaded voltage of one cell at the entered current density.	`V_rated`	V	0.7
Target net DC power Power remaining after the auxiliary load entered below.	`P_target`	kW	5
Minimum loaded stack voltage Selects the number of series cells by rounding up. Check open-circuit and converter voltage limits separately.	`V_min`	V	48
Series cells One series string, equal current through every cell.	`N_given`		80
Active area per cell Electrochemically active area, excluding seals and manifolds.	`A_given`	cm^2	200
Auxiliary DC power Total compressor, fan, pump and control load at this operating point. DC conversion losses must be included here if required.	`P_aux`	W	250
Overall hydrogen utilization Reacted hydrogen divided by fresh hydrogen supplied, including purge losses. For a recirculated anode, use overall utilization, not single-pass utilization.	`U_H2`		0.95
Cathode air stoichiometry Oxygen supplied divided by oxygen reacted. Dry air is taken as 21% oxygen by mole.	`lambda_air`		2
Cathode gas temperature Both inlet humidity and outlet equilibrium use this temperature. It changes the water balance; it does not refit the polarization curve.	`T_gas`	°C	70
Cathode absolute pressure Same absolute pressure at inlet and outlet; pressure drop is neglected. This does not correct the fitted voltage.	`p_abs`	kPa	101.325
Inlet relative humidity 0 = dry air; 1 = saturated air, at the entered cathode gas temperature.	`RH_in`		0.5
Fit intercept E₀ Kim equation intercept referenced to j = 1 mA/cm². This is not the open-circuit voltage.	`E_custom`	V	0.943
Tafel slope b Volts per decade of current density; the equation uses log₁₀.	`b_custom`	V	0.062
Area specific resistance The ohmic term is R × j, with j in A/cm².	`R_custom`	ohm*cm^2	0.238
Mass transport coefficient m The transport voltage loss is m × exp(n × j). Fit at the actual temperature, pressure, humidity and flows.	`m_custom`	V	0.00375
Mass transport coefficient n Reciprocal current density. Published 0.00638 cm²/mA becomes 6.38 cm²/A.	`n_custom`	cm^2/A	6.38
Curve upper current density Published Kim fit: set above 0.02 and no higher than 0.70 A/cm², and at least as high as the operating point. For a custom fit, keep within your measured data and above zero volts.	`j_end`	A/cm^2	0.7

Results

With the default inputs above, the calc gives:

Result	Value
Net DC power	5 kW
Loaded stack voltage	48.51 V
Stack current	108.21 A
Series cells	76
Active area per cell	270.5 cm²
Net LHV efficiency (%)	46.09
Net HHV efficiency (%)	39
Fresh hydrogen supply	0.3256 kg/h
Dry air supply	21.078 kg/h
Reaction water generated	2.764 kg/h
Equilibrium cathode liquid discharge	0 kg/h
Total active electrode area	2.056 m²

Checked against sources

Each case below was calculated by this calc and agrees with the value from its source (within the tolerance where one is shown).

DOE Example 9-1: hydrogen reacted per ampere

Source: DOE Fuel Cell Handbook, seventh edition (2004), Example 9-1, printed pp. 9-1 to 9-2 (PDF pp. 352–353). Published factor: 0.037605 kg H₂/(h·kA). One cell, 1 A isolates the Faraday balance; no polarization data are inferred.

Inputs: Calculation = Evaluate stack; Cell voltage source = Measured voltage; Series cells = 1; Active area per cell = 100 cm²; Operating current density = 0.01 A/cm²; Auxiliary DC power = 0 W; Overall hydrogen utilization = 1; the others as above.

Result	Source	This calc
Hydrogen reacted	3.761 × 10⁻⁵ kg/h	3.761 × 10⁻⁵ kg/h

DOE Example 9-2: 1 MW at 0.7 V per cell

Source: DOE Fuel Cell Handbook, seventh edition (2004), Example 9-2, PDF pp. 353–354. Published 118.4 lb/h reacted H₂, 148.0 lb/h supply and 16,142 lb/h dry air. Converted to kg/h; tolerances allow the example's rounded current and intermediate values. Current density is arbitrary here: area is sized to reproduce its power and voltage.

Inputs: Cell voltage source = Measured voltage; Target net DC power = 1,000 kW; Minimum loaded stack voltage = 350 V; Operating current density = 0.5 A/cm²; Auxiliary DC power = 0 W; Overall hydrogen utilization = 0.8; Cathode air stoichiometry = 4; the others as above.

Result	Source	This calc
Series cells	500	500
Loaded stack voltage	350 V	350 V
Stack current	2,858 A	2,857 A
Hydrogen reacted	53.707 kg/h	53.723 kg/h
Fresh hydrogen supply	67.134 ± 0.07 kg/h	67.154 kg/h
Dry air supply	7,322 ± 10 kg/h	7,323 kg/h

Kim Fig. 5: measured 1 atm air point near 200 mA/cm²

Source: Kim et al. (1995), DOI 10.1149/1.2050072, Fig. 5, PDF p. 3 / printed p. 2672. Independent graphical reading of the circular experimental H₂/air markers at 70 °C, 1 atm: about 0.74 V at 200 mA/cm². ±0.02 V allows plot resolution and rounded fit coefficients; this is a plot-reading check, not a new precision measurement.

Inputs: Operating current density = 0.2 A/cm²; the others as above.

Result	Source	This calc
Selected loaded cell voltage	0.74 ± 0.02 V	0.7393 V

Kim Fig. 5: measured 1 atm air point near 600 mA/cm²

Source: Kim et al. (1995), DOI 10.1149/1.2050072, Fig. 5, PDF p. 3 / printed p. 2672. Independent graphical reading of the circular experimental H₂/air markers at 70 °C, 1 atm: about 0.45 V at 600 mA/cm². ±0.02 V allows plot resolution and rounded fit coefficients.

Inputs: Operating current density = 0.6 A/cm²; the others as above.

Result	Source	This calc
Selected loaded cell voltage	0.45 ± 0.02 V	0.4556 V

DOE HHV efficiency at 0.7 V and complete utilization

Source: DOE Fuel Cell Handbook, seventh edition (2004), Eq. (2-19), PDF p. 62 / printed p. 2-9 gives η = 0.675 × Vcell. At 0.7 V this is 0.4725. Tolerance accounts for the handbook's rounded factor; the calculator uses 285.8 kJ/mol directly.

Inputs: Calculation = Evaluate stack; Cell voltage source = Measured voltage; Series cells = 100; Auxiliary DC power = 0 W; Overall hydrogen utilization = 1; the others as above.