CACHE Modules on Energy in the Curriculum
Fuel Cells
Module 5 (Final Draft): Equation of State for Hydrogen Fuel
Module Author: Jason Keith
Author Affiliation: Michigan Technological University
Module Secondary Author: Michael Gross
Author Affiliation: Bucknell University
Course: Thermodynamics
Text Reference: Sandler (2nd ed.), Section 4.4
Smith, Van Ness, and Abbott (7th ed.), Section 3.5
Concepts: Gas law, equation of state
Problem Motivation:
Fuel cells are a promising alternative energy conversion technology. One type of fuel
cell, a proton exchange membrane fuel cell (PEMFC) reacts hydrogen fuel with oxygen
to produce electricity. Fundamental to the design of fuel cells is an understanding of
nonideal equations of state, as hydrogen fuel is often stored at high pressure. We are
interested in determining the impact of the nonideal behavior of the compressed hydrogen
has on fuel cell performance. After completing this module, you will be able to size
compressed fuel tanks for fuel cell applications. One example of a fuel cell application
would be powering a laptop as shown below.
Computer
(Electric Load)
e-
eH2
H2
H2O
H2
O2
O2
H+
H2
e
H2O
O2
H+
e-
-
H2
H2
H2
H2
H2
Air in
H2O
H2 feed line
H2O
H+
H+
O2
Anode
Gas
Chamber
Anode
Cathode
Electrolyte
Cathode
Gas
Chamber
Figure 1. Reactions with PEMFC
H2 out
Fuel Cell
H2 tank
Air / H2O out
Figure 2. Diagram for Fueling a Laptop
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J.M. Keith
J.M. Keith
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June 5, 2007
April 23, 2008
The PEMFC reactions are:
Anode:
Cathode:
Overall:
H2
→ 2H+ + 2e½ O2 + 2H+ + 2e- → H2O
H2 + ½ O2
→ H2O
For each mole of hydrogen consumed, two moles of electrons are passed through the
electric load. To convert electron flow (moles of electrons/s) to electrical current
(coulombs/s or amps), one would use Faraday’s constant: F  96,485 coulombs / mole of
electrons. The primary objective of a fuel cell is to deliver energy to the electric load. To
calculate the energy delivery rate (also know as power) one would multiply the current
times the cell voltage: Power = Current · Voltage. (Recall the unit conversions:
Coulomb  Volt  Joule and Joule / s  Watt ).
Problem Information
Example Problem Statement: A gas cartridge contains hydrogen gas at room
temperature and 500 psi pressure. This cartridge has been proposed as a fuel tank for a
fuel cell powered laptop computer. Note that the volume of the cylinder is 2 L.
a. Determine the number of moles of hydrogen in the tank utilizing the PengRobinson equation of state and compare this result with the ideal gas law result,
which is 2.78 moles of hydrogen.
b. Calculate and compare the operating time of a laptop running on this cartridge of
H2 using the Peng-Robinson equation of state and the ideal gas law. The laptop
requires 35 W of power at 0.7 V.
To solve this problem, note that the Peng-Robinson equation of state is given by:
P
RT
a 2

1  b 1  2b  b 2  2
where P is the pressure,  is the density, R is the gas constant, T is the absolute
temperature, and a and b are coefficients for the Peng-Robinson equation of state. They
can be calculated knowing the critical temperature Tc and pressure Pc of the gas under
pressure, and are given by the following relationships (see, for example, Section 4.7 in
Sandler):
a = ac
ac = 0.45724 R2Tc2/Pc
 = [1+(1–Tr1/2)]2
Tr = T/Tc
b = 0.07780 RTc/Pc
 = 0.37464 + 1.54226  – 0.26993 2.
Note that is the acentric factor and relates to the compressibility of the gas (for
example, see Table 4.6-1 in Sandler for values of Tc, Pc, and ).
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J.M. Keith
June 5, 2007
Revision
J.M. Keith
April 23, 2008
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The critical properties of hydrogen gas are obtained from Table 4.6-1 of Sandler as Tc =
33.2 K, Pc = 1.297 MPa,  = -0.22.
Example Problem Solution:
Part a) We will use the Peng-Robinson equation of state to determine the gas density.
Step 1) For simplicity all terms in the Peng-Robinson equation of state will be tracked in
terms of atmospheres. The tank pressure is thus
500 psi / (14.696 psi/atm) = 34.0 atm
Since the pressure and temperature are known we will solve for the density. Note that the
Peng-Robinson equation of state is a cubic equation of state. This means there are three
answers for the gas density. We may have to decide which answer is appropriate to use.
Step 2) We will now determine the various parameters (, Tr, , ac, a, and b parameters)
in the model for use in the equation of state. In our solution, we will keep two significant
figures in the solution (not counting zeroes).
For the given value of we calculate  = 0.37464 + 1.54226(–0.22) – 0.26993(–0.22)2 or
 = 0.022.
The reduced temperature is:
T/Tc = 298.15/33.2 = 8.98.
Thus, we can calculate the parameter  = [1+(1–Tr1/2)]2 = [1 + 0.022 (1 – 8.981/2)2 or
0.91.
To determine the value of a in the Peng-Robinson equation of state, we need to calculate
ac according to:
2
L2 atm
0.45724  0.08206 L atm  33.2 2 K 2 0.101325 MPa


ac 
 0.27
mol K
atm
mol 2

 1.297 MPa
from which we can calculate
a = ac = 0.27 (0.91) = 0.25 L2atm/mol2.
Furthermore, we can calculate
0.0778  0.08206 L atm  33.2 K 0.101325 MPa
L


b
 0.017
mol K
atm
mol

 1.297 MPa
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J.M. Keith
J.M. Keith
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June 5, 2007
April 23, 2008
Step 3) The a and b parameters can be inserted into the Peng-Robinson equation to solve
for the density. This can be done by graphing
RT
a 2

P
1  b 1  2b  b 2  2
as a function of the density . A single real root occurs when  ~ 1.38 mol/L, as seen in
the figure below.
Alternatively one can use a root finding program using the ideal gas density as an initial
guess.
Step 4) Multiplying the gas density by the tank volume gives approximately 2.76 mol.
The relative error when compared with the ideal gas law is given as 100(2.78 – 2.76)/2.78
< 1%. Note that if the pressures were higher (say 2000 psi which is common in a
compressed cylinder in the laboratory) the relative error is larger.
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Part b)
Step 1) Given the power requirement of 35 W at 0.7 V, the required current (I) can be
calculated with P =VI.
I
P 35W
C

 50 A  50
V 0.7V
s
Step 2) With Faraday’s constant and knowing that two moles of electrons are produced
for every mole of H2 reacted the laptop operating time can be calculated using the PengRobinson equation of state.
 2.76 mol H 2  2 mol e -  96485 C  1 s  1 min 




  178 min  2h 58min
- 

 1 mol H 2  1 mol e  50 C  60 s 
Step 3) For the ideal gas law the operating time is given as:
 2.78 mol H 2  2 mol e -  96485 C  1 s  1 min 




  179 min  2h 59min
- 

 1 mol H 2  1 mol e  50 C  60 s 
Summary. In this module we have used the Peng-Robinson equation of state to
determine the number of moles of hydrogen in a small tank. We then used this to
determine the operating time for a fuel cell for a laptop computer.
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J.M. Keith
J.M. Keith
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Home Problem Statement:
A gas cartridge contains hydrogen gas at room temperature and 1000 psi pressure. This
cartridge has been proposed as a fuel tank for a fuel cell powered laptop computer. Note
that the volume of the cylinder is 2 L.
a. Determine the number of moles of hydrogen in the tank utilizing the Van der
Waals equation of state and compare this result with the ideal gas law result,
which is 5.56 moles of hydrogen.
b. An electronics company wants their laptops to operate continuously for 5 hours at
a voltage of 0.7 V before having to recharge the fuel cartridge described above.
Calculate the maximum possible power rating of the laptop (in watts) that would
meet these specifications.
To solve this problem, note that the Van der Waals equation of state is given by:
P
RT
 a 2
1  b
where P is the pressure,  is the density, R is the gas constant, T is the absolute
temperature, and a and b are coefficients for the Van der Waal’s equation of state. They
can be calculated knowing the critical temperature Tc and pressure Pc of the gas under
pressure, and are given by the following relationships (see, for example, Section 4.6 in
Sandler):
a = 27/64 R2Tc2/Pc
b = 0.125 RTc/Pc
where Tc is the critical temperature and Pc is the critical pressure.
The critical properties of hydrogen gas are obtained from Table 4.6-1 of Sandler as Tc =
33.2 K, Pc = 1.297 MPa.
1st draft
Revision
J.M. Keith
J.M. Keith
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June 5, 2007
April 23, 2008