CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Air Separation for Coal Gasification
Module Author: Jason Keith
Author Affiliation: Michigan Technological University
Course: Separations
Text Reference: Geankoplis (4th edition), Section 13.4, Wankat (2nd edition), Section
16.3.
Concepts: Determine the size of a membrane separation unit to purify air to be used in
hydrogen production via coal gasification.
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 with oxygen to
produce electricity (Figure 1). Fundamental to a hydrogen economy with fuel cell
vehicles is an understanding of current hydrogen generation technologies.
Consider the schematic of a compressed hydrogen tank (2000 psi, regulated to 10 psi)
feeding a proton exchange membrane fuel cell, as seen in Figure 2 below. Hydrogen can
currently be obtained from steam reforming of natural gas or other hydrocarbons or from
coal gasification. In this module, we will focus on analysis of an air separation unit to be
used in a process to obtain hydrogen via coal gasification.
-
-
e
e
H2
H2O
O2
H+
H2
H2O
H2
O2
O2
H+
H2
Computer
(Electric
Load)
Pressure
regulator
H2 feed line
Air in
H2
H2
H2
H2
H2
H2O
H2O
H+
H+
O2
Anode
Cathode
Electrolyte
Figure 1. Reactions in the PEMFC
H2 out
H2 tank
Fuel Cell
Air / H2O out
Figure 2. Diagram for fueling a laptop.
1st Draft
J.M. Keith
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September 15, 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 ).
Background
Coal gasification has been proposed as a short-term source of hydrogen for fuel cell
vehicles. In this process, coal is combined with steam and oxygen to form a mixture
composing primarily of carbon monoxide, carbon dioxide, and hydrogen. After
separation, pure hydrogen is obtained for use on the fuel cell. Thus, in this module we
will size a membrane to concentrate oxygen from the air, which will lead to a smaller
coal gasification reactor.
The simplest (but not most effective) membrane design is called the complete mixing
model, seen in Figure 3 below. A feed enters a well-mixed chamber. The gases in the
feed can either pass through the membrane (and become the permeate stream) or exit
with the reject stream.
Permeate: Vp, yp, pl
Feed:
Lf, xf, ph
Reject: Lf – Vp, xo, ph
Figure 3. Complete mixing model for membrane separator. The streams are
labeled along with the flow rate, oxygen mole fraction, and pressure.
The air feed enters at flow rate Lf and oxygen mole fraction xf = 0.209 at a high pressure
ph. The bottom mixing chamber has an oxygen mole fraction xo. The reject exits at a flow
rate Lo and an oxygen mole fraction xo and at a high pressure ph. The permeate exits at a
flow rate Vp and oxygen mole fraction yp.
A detailed analysis of this system is given in the reference texts. The resulting equations
are summarized below for your use in this module.
1st Draft
J.M. Keith
Page 2
September 15, 2008
In the case when the feed mole fraction xf, fraction permeated  = Vp/Lf, oxygen / nitrogen
permeability ratio *, and pressure ratio pl/ph are given, the unknowns are the permeate
oxygen mole fraction yp, the reject oxygen mole fraction xo, and membrane area Am. The
analysis shows that the permeate oxygen mole fraction is given by the quadratic equation:
yp 
 b  b 2  4ac
2a
(1)
where
a  
pl
p
p
p
 l    *   * l   * l 
ph ph
ph
ph
b  1  x f 
(2)
pl
p
p
p
 l    *   * l   * l    * x f
ph ph
ph
ph
(3)
c   * x f
(4)
Also,
xo 
x f  yp
(5)
(1   )
Finally,
Am 
 Lf yp
(6)
( PA' / t )( ph xo  pl y p )
where PA' is the oxygen permeability and t is the membrane thickness.
The following example problem is modeled after example 13-4.2 of Geankoplis.
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J.M. Keith
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September 15, 2008
Example Problem Statement: For a mid-size scale coal gasification reactor (enough to
power about 6000 hydrogen cars daily) a membrane separation unit has a volumetric flow
of air of 1 x 108 cm3/s. The following information is available:






The oxygen permeability is 500 x 10-10 (cm3(STP)-cm)/(s-cm2-cm Hg)
The feed (and reject) pressure is 380 cm Hg
The nitrogen permeability is 35 x 10-10 (cm3(STP)-cm)/(s-cm2-cm Hg)
The permeate pressure is 20 cm Hg.
You want to collect 10% of the feed as the permeate.
The membrane thickness is 2.54 x 10-3 cm (1 mil).
Perform the following calculations:
a) Determine the oxygen mole fraction in the permeate.
b) Determine the oxygen mole fraction in the reject.
c) Determine the membrane area in m2.
Example Problem Solution:
Part a)
Step 1) The permeability ratio * is equal to 500/35 = 14.3. The fraction permeated is  =
0.1. The pressure ratio is ph/pl = 380/20 = 19.
Step 2) We will use the quadratic equation given in Equations 1-4. Thus, we find that:
a  
pl
p
p
p
 l    *   * l   * l 
ph ph
ph
ph
1 1
1
1
 (0.1)  (14.3)(0.1)  (14.3)  (14.3) (0.1)
19 19
19
19
a  1.96
a  0.1 
(7)
Step 3) Furthermore,
b  1  x f 
pl
p
p
p
 l    *   * l   * l    * x f
ph ph
ph
ph
b  1  0.1  0.209 
1
1
1
1
 (0.1)  (14.3)(0.1)  (14.3)  (14.3) (0.1)  (14.3)(0.209)
19 19
19
19
b  5.74
(8)
Step 4) Also,
c   * x f  14.3(0.209)  2.99
(9)
Step 5) Finally, inserting into Equation 1 gives the oxygen mole fraction in the permeate:
1st Draft
J.M. Keith
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September 15, 2008
2
 b  b 2  4ac  5.74  (5.74)  4(1.96)( 2.99)
yp 

2a
2(1.96)
y p  0.68
(10)
Part b)
We can use Equation 5 to solve for the oxygen mole fraction in the reject:
xo 
x f  y p
(1   )

0.209  (0.1)(0.68)
 0.16
(1  0.1)
(11)
Part c)
Inserting into the design equation (Equation 6) gives the membrane area as:
Am 
L f y p
( P / t )( p h xo  pl y p )
'
A

cm 3
(0.1)(1  10
)(0.68)
s
Am 

 500  10 10
cm 3  cm 

380 cmHg(0.16) - 20 cmHg(0.68) 
3
2
 2.54  10 cm s  cm  cmHg 
8
(12)
Am  7.3  10 9 cm 2  7.3  10 5 m 2
Analysis: Such a large area is typically accomplished with a large number of hollow
tubes with a very thin wall. The area can be reduced with a thinner membrane or by
operating with a cross-flow, co-current, or counter-current flow.
1st Draft
J.M. Keith
Page 5
September 15, 2008
Home Problem Statement: You wish to design a two-stage membrane separation unit as
shown in the figure below.
Permeate
Feed
Reject
Permeate
Feed
Reject
Figure 4. Complete mixing model for multistage membrane
separator.
For a mid-size scale coal gasification reactor (enough to power about 6000 hydrogen cars
daily) a membrane separation unit has a volumetric flow of air of 1 x 108 cm3/s. The
following information is available:






The oxygen permeability is 500 x 10-10 (cm3(STP)-cm)/(s-cm2-cm Hg) in each
unit
The feed (and reject) pressure is 380 cm Hg in each unit
The nitrogen permeability is 35 x 10-10 (cm3(STP)-cm)/(s-cm2-cm Hg) in each
unit
The permeate pressure is 20 cm Hg in each unit
You want to collect 30% of the feed as the permeate from each unit.
The membrane thickness is 2.54 x 10-3 cm (1 mil).
Perform the following calculations:
a)
b)
c)
d)
1st Draft
Determine the oxygen mole fraction in the permeate from the first unit.
Determine the oxygen mole fraction in the reject from the first unit.
Determine the oxygen mole fraction in the permeate from the second unit.
Determine the oxygen mole fraction in the reject from the second unit.
J.M. Keith
Page 6
September 15, 2008