Problem Set 03
Course: ENERGY 201C - Energy Storage & Conversion
Due Date: Thursday, 5 June 2025
1. [20 points] Ballard Power Systems is a hydrogen fuel-cell company whose headquarters are located in Burnaby, British Columbia, Canada near Joseph’s hometown. Joseph is starting as an intern at Ballard and one of the first tasks that his manager gives him after he finishes his onboarding is to characterize the power-current relationship of one of Ballard’s fuel cells: the Ballard HD6 75 kW Fuel Cell (Fig. 1).
Figure 1: Ballard HD6 75 kW Fuel Cell.
Specifically, Joseph’s manager hands him the following plot with data (Fig. 2) and asks him to determine an empirical relationship between the power and current.
Fuel Cell Power (kW)
Figure 2: Current vs. power data for a fuel cell plot given to Joseph by his manager.
This empirical relationship is expected to be used in some internal modeling tools. His manager indicates that the currents shown in the plot are the expected magnitudes of the currents that the fuel cell will experience during operation (i.e., the operating currents would not exceed what is shown in the data).
(a) Determine the empirical relationship between the fuel cell power as a function of fuel cell current from the data given. Summarize the characteristics of this relationship (monotonic or non-monotonic, increasing or decreasing) that you observe. Hint: is a linear fit is sufficient to capture the generic characteristics we learned in class for the power vs. current relationship in fuel cells?
(b) Joseph’s manager is particularly concerned that this particular cell may “starve” during operation. Given the data and empirical relationship you have derived, do you think that the manager’s concern is valid? Hint: What characteristic(s) of the power vs. current relationship would be signature of starvation?
2. [50 points] The purpose of this problem is to familiarize with the calculations of the polarization curve for a PEM fuel cell and understand its dependency on operating and physical parameters. Using the model of the polarization curve described in the supplemental notes (please check Canvas), write a script. to evaluate the following:
(a) Open-circuit voltage (Nernst potential): Calculate and represent on a table the open cir-cuit voltage for T = 20, 50, 80 ◦C and p = 1, 1.5, 2, 2.5, 3 bar. For the calculation, assume the same pressure existing in the cathode and in the anode.
(b) Activation overvoltage: evaluate and graph the activation overvoltage as a function of current density i varying between 0 and 1.8 A/cm2
for following two cases:
i. T = 80 ◦C and p = 1, 1.5, 2, 2.5, 3 bar (5 curves on same graph);
ii. T = 20, 50, 80 ◦C and p = 2 bar (3 curves on same graph).
Provide observations and comments on the results.
(c) Ohmic loss: evaluate and graph the Ohmic losses as a function of current density i varying between 0 and 1.8 A/cm2
, in the following cases:
i. tm = 100, 150 µm, T = 80 ◦C and λm = 14 (i.e., RHm = 100%) (2 curves on same graph);
ii. tm = 125 µm, T = 80 ◦C, and RHm = 20, 50, 70, 100% (4 curves on same graph);
iii. tm = 125 µm, T = 20, 50, 80 ◦C, and RHm = 7% (3 curves on same graph);
Provide observations and comments on trends.
(d) Concentration Loss:
i. Evaluate and graph the concentration loss as a function of current density i varying between 0 and 1.8 A/cm2
for following cases:
a. T = 80 ◦C and p = 1, 1.5, 2, 2.5, 3 bar (5 curves on same graph);
b. T = 20, 50, 80 ◦C and p = 2 bar (3 curves on same graph);
Provide observations and comments on your results.
ii. Plot the polarization curves for current density i varying between 0 and 1.8 A/cm2
, considering tm = 125 µm, T = 80 ◦C, RHm = 70%, p = 1, 2, 3 bar (3 curves on same graph).
iii. Plot the power density (W/cm2
) corresponding to the polarization curves in Part 2 as a function of current density i.