CACHE Modules on Energy in the Curriculum
Fuel Cells
Module Title: Natural Convection Cooling of a Fuel Cell
Module Author: Zachary Edel and Abhijit Mukherjee
Author Affiliation: Michigan Technological University
Course: Heat Transfer
Text Reference: Incropera et al., 2007, Fundamentals of Heat and Mass Transfer, Sixth
Edition.
Concepts: Newton’s Law of Cooling, Natural Convection Correlations
Problem Motivation:
Fuel cells are a promising alternative energy technology. One type of fuel cell, a proton
exchange membrane fuel cell reacts hydrogen and oxygen together to produce electricity.
Fundamental to the design of fuel cells is an understanding of heat transfer mechanisms
within fuel cells. Heat removal from fuel cells is critical to their scaleup for large power
applications.
Consider the schematic of a compressed hydrogen tank feeding a proton exchange
membrane fuel cell, as seen in the figure below. The electricity generated by the fuel cell
is used here to power a laptop computer. We are interested in analyzing heat transfer
within the fuel cell, which involves the determination of heat transfer coefficients,
thermal conductivities, and temperatures at certain locations within the fuel cell.
Computer
(Electric Load)
H2 feed line
Air in
Anode
Gas
Chamber
Cathode
Gas
Chamber
Air / H2O out
H2 out
H2 tank
1st Draft
Fuel Cell
Z.J. Edel
Page 1
December 21, 2009
Problem Background:
Empirical correlations for free convection are generally of the form:
hL
n
 C  Ra L
k
NuL 
Where the Rayleigh number is given by:
g Ts  T L3
Ra L 

And L is the characteristic length of the flow geometry and depends on the orientation of
the convecting surface. All fluid properties are evaluated at the film temperature:
Tf 
Ts  T
2
The Vertical Plate:
Typically for laminar flow ( 104 ≤ RaL ≤ 109 ), C = 0.59 and n  1 4 and for turbulent
flows ( 109 ≤ RaL ≤ 1013 ), C = 0.10 and n  13 . Slightly better accuracy can be
achieved for laminar flow, however, if the following correlation is used:
hL
0.670Ra L 4
NuL 
 0.68 
4
492 16 9
k
9
1   0.Pr
1


RaL ≤ 109
The characteristic length scale for this orientation is the vertical height of the plate.
The Horizontal Plate:
The characteristic length scale for a horizontal plate is defined as
Lc 
As
P
Where As is the plate surface area and P is the plate perimeter.
The Nusselt number correlations for the horizontal plate depend upon both the orientation
of the surface in question and whether the surface is hotter or colder than the ambient
medium. For the top surface of a cold plate, the flow descends over the plate much like a
flat surface in a wind tunnel. For the bottom surface of a cold plate, the fluid circulates
due to the descending air near the plate displacing volume and drawing ambient air into
the region close to the plate. The conditions are exactly reversed for a hot plate in a cool
atmosphere; the top surface circulates air due to the rising of the heated fluid and the
bottom surface pushes air past it like the top surface of a cool plate.
Upper Surface of Hot Plate or Lower Surface of Cold Plate:
NuL 
1st Draft
1
hL
 0.54 Ra L 4
k
Z.J. Edel
Page 2
( 104 ≤ RaL ≤ 107 )
December 21, 2009
NuL 
1
hL
 0.15Ra L 3
k
( 107 ≤ RaL ≤ 1011 )
Lower Surface of Hot Plate or Upper Surface of Cold Plate:
NuL 
1
hL
 0.27 Ra L 4
k
( 105 ≤ RaL ≤ 1010 )
Problem Information
Example Problem Statement:
The temperature of a 7-Watt, single fuel cell is regulated by natural convection heat
transfer to surrounding air at 20oC. The fuel cell consists of a Membrane Electrode
Assembly (MEA) and two gas distribution plates. Each square, graphite plate has an
active area of 50 cm2; the MEA can be neglected for this problem. The fuel cell operates
at a steady-state temperature of 80oC.
A)
B)
What is the average heat transfer coefficient?
What is the total heat dissipation due to natural convection from the
distribution plates when the fuel cell is oriented vertically?
Example Problem Solution:
Fluid: Air, T∞ = 20oC, Tss = 80oC
Film Temperature:
T  T 353K  293K
Tf  s

 323K
2
2
Properties of Air at 323 K:
ν = 18.20 x 10-6 m2/s
α = 25.9 x 10-6 m2/s
Pr = 0.704
k = 28.0 x 10-3 W/m-K
β = 1/323K = 0.003096 K-1
Characteristic length for vertical plate: Lc = Aside1/2 = 7.071 cm
A)
Ra L 
g Ts  T L3
9.81 0.003096K 353K  293K 0.07071m

1
3
s2
18.20  106 m s 25.9  106 m s 

< 109 → Laminar Flow
Ra L  1.366 x 106
2
NuL  0.68 
1st Draft
m
0.670 Ra L
1  
0.492
Pr

9
1
0.6701.366 106  4
1
4

4
16
2
9
 0.68 
Z.J. Edel
Page 3
1  
0.492
0.704

9

4
16
9
December 21, 2009
NuL 
hnat
hL
 18.24
k

NuL  k 18.24 28.0 103 W mK


0.0707
L

hnat  7.224 W mK
B)


qtot  hnat Atot Ts  T   7.224 W m2K  2 50cm 100m2 cm2 353K  293K 
2
qtot  4.334W
1st Draft
Z.J. Edel
Page 4
December 21, 2009
Home Problem Statement:
The temperature of a 25-Watt, single fuel cell is regulated by natural convection heat
transfer to surrounding air at 20oC. The fuel cell consists of a Membrane Electrode
Assembly (MEA) and two gas distribution plates. Each square, graphite plate has an
active area of 150 cm2; the MEA can be neglected for this problem. The fuel cell
operates at a steady-state temperature of 80oC.
A)
What is the average heat transfer coefficient due to natural convection from
the top surface and the bottom surface of the distribution plates when the fuel
cell is oriented horizontally?
B)
Calculate the heat transfer rate in Watts for the top and bottom surfaces of the
distribution plates.
1st Draft
Z.J. Edel
Page 5
December 21, 2009