CACHE / AIChE Modules on Energy in the Curriculum
Module Title: Battery / Fuel Cell Vehicle Range
Module Author: Jason Keith
Author Affiliation: Michigan Technological University
Concepts: Energy usage for transportation
Problem Motivation: The availability of energy has become an important part of our society. In
this and related problems, we will discuss issues of energy consumption, energy reserves, and
energy related emissions. Furthermore, we will analyze conventional and alternative energy
systems. A particular emphasis will be placed on the generation of hydrogen for use in fuel cells
for transportation and/or stationary applications.
Problem Information
Example Problem Statement: Consider a fuel cell / battery hybrid system. The fuel cell has 400
cells and the electric motor operates at 300 V.
a) The vehicle consumes 2.1 kW power from the battery alone at 20 miles per hour. If the
battery has 1.1 kW-hr of energy storage, determine the driving range in miles.
b) The vehicle consumes 15 kW power from the fuel cell alone at 60 miles per hour. If the
vehicle carries 4 kg hydrogen, determine the driving range in miles.
Example Problem Solution:
a. Battery energy (E, in kW-hr) is equal to the product of operation time (t, hr) and power
consumption rate (P, kW):
E  Pt
(1)
As such, the time of operation is given as:
t
1.1 kW-hr
 0.52 hr
2.1 kW
The driving range (R, mi) is equal to the product of the speed (v, mph) and the operation
time (t, hr):
R  vt
From which the driving range can be calculated as:
R  (20 mph)(0.52 hr) = 10.5 mi
(2)
b. For a fuel cell, we first need to relate the power consumption and the voltage to the
current flow. The current flow will then be related to the hydrogen consumption rate.
Once this is known, the vehicle range can be determined.
The current (I, A) is equal to the power (P, W) divided by the voltage (V, V):
I
P
V
(3)
Thus,
I
15 kW 1000 W
 50 A
1 kW 300 V
For a hydrogen fuel cell, the current can be used to determine the hydrogen
consumption rate (H2, mol/s), the number of cells (N), and Faraday’s constant (F =
96485 C/mol e–):
 H2 
IN
2F
(4)
Note that in this equation, 2 represents that for every mole of hydrogen fuel (H2) there
are 2 moles of electrons produced (from the anode reaction H2 → 2 H+ + 2e–).
Thus,
H2 
mol H 2
50 A 400 cells C/s
 0.104

2 mol e 96485C A
s

mol
e
mol H 2
The molecular weight (M = 2 g/mol) can be used to give the hydrogen consumption rate
as:
mH2  M  H2  0.207
g H2
s
Next, the total hydrogen supply (Mtank, g) can be used to determine the total operation
time for the fuel cell (tfuelcell, s).
tfuelcell 
M tank
mH2
(5)
Since the problem statement gives the hydrogen supply as 4 kg, we must first convert to
4000 g to have consistent units. Thus,
tfuelcell 
4000 g
 19300 s  5.36 hr
g
0.207
s
Finally, we can use equation 2 to determine the range:
R  (60 mph)(5.36 hr) = 320 mi
Home Problem Statement: Consider a fuel cell / battery hybrid system. The fuel cell has 375
cells and the electric motor operates at 300 V.
a) The vehicle consumes 3 kW power from the battery alone at 35 miles per hour, traveling
50 miles. If the total energy in the battery is 16 kW-hr (as it is in the Chevy Volt),
determine the fraction of the available energy in the battery that is consumed. Note
that it is typical that only a fraction of the battery energy is used to allow for thousands
of charge/discharge cycles.
b) The vehicle consumes 10 kW power from the fuel cell alone for 4 hours at 40 miles per
hour. Determine the fuel economy of the vehicle in miles per kg hydrogen.