# Selected Topics in Heat and Mass Transfer

of 196 ## Content

Selected Topics in heat and Mass
Transport
..A compilation of selected presentations

2011

Jundika Candra Kurnia
Agus Pulung Sasmito
Sachin Vinayak Jangam
Hee Joo Poh

PREFACE
This e-book consists of selected topics to be covered as part of the postgraduate
course entitled ME6203 Mass Transport in the Mechanical Engineering Department of
the National University of Singapore, given by Professor A. S. Mujumdar.

For the benefit of wider audience interested in the themes covered, this e-book is
being offered freely. It contains handouts of the PowerPoint presentations made by the
authors. We hope that this compilation will be useful to research students as well as
researchers in academia and industrial R&D. Professor Mujumdar recommended and
provided guidance to the authors of various chapters included in this e-book.
The authors would be happy to hear from readers about any related matter they
wish to discuss or seek clarification on.
Jundika Candra Kurnia
Agus Pulung Sasmito
Sachin Vinayak jangam
Hee Joo Poh
Singapore

Index

Presentation Title / Author names
No
01
02
03
04
05
06

Heat Transfer in Square duct
Jundika C. Kurnia
Computational Study of Energy-Efficient Thermal Drying Using
Intermittent Impinging Jets
Jundika C. Kurnia
Mass transport in a micro-channel T-Junction with coiled-base
channel design
Agus P. Sasmito
Mass Transport Considerations in PEM Fuel Cell Modeling
Hee Joo Poh
Heat Transfer in Fluidized Beds-An Overview
Sachin V. Jangam
Mass Transfer in Fluidized Beds - An Overview
Sachin V. Jangam

Heat Transfer in
Square duct

ME 6204: Convective Heat Transfer

Heat Transfer in Square Duct

Prof. Arun S. Mujumdar
[email protected]

Mathematical formulation
• Governing equations:
– Conservation of mass
– Conservation of momentum
– Conservation of energy

∇ ⋅ ( ρ u) = 0

(

)

Τ
∇ ⋅ ( ρ u ⊗ u) = −∇P + ∇ ⋅ ⎡⎢ μ ∇u + ( ∇u ) ⎤⎥

ρ c p u ⋅∇T = k ∇ 2T

• Constitutive equation (air)

Density
Dynamic viscosity
Thermal conductivity
Specific heat

ρ=
k=

Pabs
RspecificT

−6
, μ = 2.67 ×10

R
15 R ⎡ 4 c p M 1 ⎤
+ ⎥ , cp = .
μ⎢
M
4 M ⎣15 R
3⎦

• For water, properties are set
as constant
Nomenclature:

ρ = fluid density
μ = fluid viscosity
u = fluid velocity
T = fluid temperature
kt = fluid thermal conductivity

MT

σΩ (T )

Pabs = Absolute pressure
Rspecific= Specific gas constant
cp = specific heat
σ = Collision diameter
Ω = Collision integral
M = molecular weight

,

Mathematical Formulation
• Turbulent model used in this simulation is k-ε
model
⎡⎛
ρμ ⎞ ⎤
∂k
+ ∇ ⋅ ( ρ uk ) = ∇ ⋅ ⎢⎜ μ + t ⎟ ∇k ⎥ + ρμt G − ρε ,
∂t
σ k ⎠ ⎦⎥
⎣⎢⎝
⎡⎛
ρμ ⎞ ⎤ C ρμ Gε
∂ε
ε2
+ ∇ ⋅ ( ρ uε ) = ∇ ⋅ ⎢⎜ μ + t ⎟ ∇ε ⎥ + 1ε t
− C2 ε ρ ,
∂t
σ k ⎠ ⎥⎦
k
k
⎢⎣⎝
⎛ ⎡ ∂u ⎤ 2 ⎡ ∂v ⎤ 2 ⎡ ∂w ⎤ 2 ⎞ ⎛ ∂u ∂v ⎞ 2 ⎛ ∂u ∂w ⎞ 2 ⎛ ∂w ∂v ⎞ 2
+
+
G = 2⎜ ⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥ ⎟ + ⎜ + ⎟ + ⎜ +
,
⎜ ⎣ ∂x ⎦ ⎣ ∂y ⎦ ⎣ ∂z ⎦ ⎟ ⎝ ∂y ∂x ⎠ ⎝ ∂z ∂x ⎠⎟ ⎝⎜ ∂y ∂z ⎠⎟

k2
μt = Cμ ,

ε

Nomenclature:
u, v, w = component velocity
μt= turbulent viscosity
k = turbulent kinetic energy
ε = turbulent dissipation
G = turbulent generation rate

C1ε = 1.44
C2ε = 1.92
Cμ = 0.09
σk = 1.0
σε = 1.0

3

Mathematical Formulation
• Nusselt number calculation
Tmean =
V=
h=

1
Ac

1
VAc

∫ TudA ,
c

Ac

∫ udA ,
c

Ac

Q&
,
Tsurface − Tmean

Nu =

hD
k

Nomenclature:

Tmean = mixed mean temperature

Tsurface
Ac

h
Q&

= surface temperature
= cross-section area
= convective heat transfer
= heat flux

Nu = Nusselt number
D = Hydraulic diameter
k = Conductive heat transfer
V = mixed mean velocity

4

Geometry
• The flow configuration considered is full tube
flow inside square duct, as illustrated in figure

Schematic representation of flow in a) square duct and b) development
of a momentum boundary layer

5

Numerics
• Finite-volume based solver: Fluent 6.3.
• Mesh independence study ~10000 cells.
• Pressure velocity coupling: SIMPLE (Semi-Implicit
• Second-order upwind discretization.
• Algebraic Multi-grid Method (AMG).
• Relative residual ~10-6.
• It took around one minute to converge in Quad-core 2.83
GHz with 8 GB RAM.
• CFD analysis was carried out by Agus Pulung Sasmito
and Jundika Candra Kurnia (ME, NUS)

6

LAMINAR FLOW
7

Boundary condition
Air

Water

• Case 1: Constant heat flux at
wall

• Case 3: Constant heat flux at
wall

– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; Q = 30 watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

• Case 2: Constant wall
temperature
– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

– Inlet: water velocity = 0.1 m/s;
T water = 25 °C.
– Wall: no-slip; Q = 2870
watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

• Case 4: Constant wall
temperature
– Inlet: water velocity = 0.1 m/s;
T water = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

8

Velocity in the middle channel

Air (Cases 1 and 2)

Water (Cases 3 and 4)

9

Temperature in the middle of the channel

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

10

Temperature in the wall of the channel

Air (Case 1)

Water (Case 3)

11

Heat flux at the wall

Air (Case 2)

Water (Case 4)

12

Nusselt Number

Air (Cases 1 and 2)

Water (Cases 3 and 4)

Nusselt number asymptotic
Case 1: 2.8
Case 2: 2.7
Case 3: 2.7
Case 4: 2.2

13

Wall and Mean temperature

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

14

TURBULENT FLOW WITH
RE=20000

15

Boundary condition
Air

Water

• Case 1: Constant heat flux at
wall

• Case 3: Constant heat flux at
wall

– Inlet: air velocity = 30 m/s; T
air = 25 °C.
– Wall: no-slip; Q = 497 watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 20000.

• Case 2: Constant wall
temperature
– Inlet: air velocity = 30 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 20000.

– Inlet: water velocity = 2 m/s; T
water = 25 °C.
– Wall: no-slip; Q = 170370
watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 20000.

• Case 4: Constant wall
temperature
– Inlet: water velocity = 2 m/s; T
water = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 20000.

16

Velocity in the middle channel

Air (Cases 1 and 2)

Water (Cases 3 and 4)

17

Temperature in the middle of the channel

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

18

Temperature in the wall of the channel

Air (Cases 1)

Water (Cases 3)

19

Heat flux at the wall

Air (Cases 2)

Water (Cases 4)

20

Nusselt Number

Air (Cases 1 and 2)

Water (Cases 3 and 4)

Nusselt number asymptotic
Case 1: 45
Case 2: 34
Case 3: 623
Case 4: 127

21

Wall and Mean temperature

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

22

TURBULENT FLOW
WITH Re=60000

23

Boundary condition
Air

Water

• Case 1: Constant heat flux at
wall

• Case 3: Constant heat flux at
wall

– Inlet: air velocity = 80 m/s; T
air = 25 °C.
– Wall: no-slip; Q = 1272
watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 60000.

• Case 2: Constant wall
temperature
– Inlet: air velocity = 80 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 60000.

– Inlet: water velocity = 6 m/s; T
water = 25 °C.
– Wall: no-slip; Q = 433879
watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 60000.

• Case 4: Constant wall
temperature
– Inlet: water velocity = 6 m/s; T
water = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 60000.

24

Velocity in the middle channel

Air (Cases 1 and 2)

Water (Cases 3 and 4)

25

Temperature in the middle of the channel

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

26

Temperature in the wall of the channel

Air (Cases 1)

Water (Cases 3)

27

Heat flux at the wall

Air (Cases 2)

Water (Cases 4)

28

Nusselt Number

Air (Cases 1 and 2)

Water (Cases 3 and 4)

Nusselt number asymptotic
Case 1: 105
Case 2: 67
Case 3: 1700
Case 4: 314

29

Wall and Mean temperature

Air (Case 1)

Air (Case 2)

Water (Case 3)

Water (Case 4)

30

Summary of heat transfer calculation

Air (Laminar)

Water(Laminar)

Air (Turbulent Re 20000)

Water (Turbulent Re 20000)

31

LAMINAR FLOW WITH VARIABLE T
and Q BOUNDARY condition

32

Boundary conditions-Variable T,Q
Air
• Constant heat flux at wall
– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; Q = 30 watt/m2.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

• Constant wall temperature
– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

33

Temperature distribution

Distance from entrance: 5 cm

Distance from entrance: 25 cm

34

Temperature distribution

Distance from entrance: 50 cm

Distance from entrance: 100 cm

35

LAMINAR FLOW WITH
HEATING/COOLING

36

Boundary condition
• Case 1: Heating

Inlet: air velocity = 1.6 m/s; T air = 25 °C.
Wall: no-slip; T wall = 50 °C, 100 °C, 200 °C.
Outlet: Pout = 1 atm; ∇⋅Q=0.
Re ≈ 1000.

• Case 2: Cooling
– Inlet: air velocity = 1.6 m/s; T air = 50 °C, 100 °C,
200 °C.
– Wall: no-slip; T wall = 25 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.
37

Nusselt number distribution in entry length

Heating

Cooling

38

LAMINAR FLOW WITH BOUYANCY
39

Boundary condition
Laminar flow (Air)
• Boundary conditions
– Inlet: air velocity = 0.15 m/s;
T air = 25 °C.
– Wall: no-slip; T wall = 200 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 100.

Case 1: No gravity force
Case 2: Horizontal placement
Case 3: Tilted 45°
Case 4: Vertical placement
Gravity force is applied for
case 2, 3 and 4.

40

Velocity at entry region (z=2.5 cm)

Case 1

Case 3

Case 2

Case 4

41

Temperature at entry region (z=2.5 cm)

Case 1

Case 2

Case 3

Case 4

42

Velocity at z= first 10 cm

Case 1

Case 2

Case 3

Case 4

43

Temperature at z= first 10 cm

Case 1

Case 2

Case 3

Case 4

44

Nusselt number

45

TAPERED DUCT

46

Boundary condition
Laminar flow (Air)
Case 1: Divergent duct

Inlet: air velocity = 16
m/s;
T air = 25 °C.
Wall: no-slip; T wall = 50
°C.
Outlet: Pout = 1 atm;
∇⋅Q=0.
Re ≈ 1000.

Case 2: Convergent duct

Inlet: air velocity = 0.16
m/s; T air = 25 °C.
Wall: no-slip; T wall = 50
°C.
Outlet: Pout = 1 atm;
∇⋅Q=0.
Re ≈ 1000.

47

Velocity in the middle channel

Divergent duct (Case1)

Convergent duct (Case 2)

48

Temperature in the middle channel

Divergent duct (Case1)

Convergent duct (Case 2)

49

Nusselt number

50

POWER LAW FLUID

51

Boundary condition
Power law fluid
⎛ ∂u ⎞

⎝ ∂y ⎠

n

τ =K⎜
where:

K = flow consistency index (Pa s n )
∂u
= shear rate or the velocity gradient (s −1 )
∂y
n = flow behaviour index
⎛ ∂u ⎞

⎝ ∂y ⎠

n −1

μeff = K ⎜

μeff = apparent or effective viscosity (Pa s )
n

Type of fluid

<1

Pseudoplastic

1

Newtonian

>1

Dilatant

Laminar flow (Air)
• Case 1 (Pseudoplastic n=0.5)
– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.

• Case 2 (Dilatant n=1.5)
– Inlet: air velocity = 1.6 m/s; T
air = 25 °C.
– Wall: no-slip; T wall = 50 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.

52

Velocity in the middle channel

Pseudoplastic (Case1)

Dilatant (Case 2)

53

Temperature in the middle channel

Pseudoplastic (Case1)

Dilatant (Case 2)

54

Nusselt number

55

PULSATING FLOW

56

Boundary condition
• Case 1 (Frequency = 5 Hz)
– Inlet: air velocity = Vin; T air =
25 °C.
– Wall: no-slip; T wall = 200 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

• Case 2 (Frequency = 10 Hz)
– Inlet: air velocity = Vin; T air =
25 °C.
– Wall: no-slip; T wall = 200 °C.
– Outlet: Pout = 1 atm; ∇⋅Q=0.
– Re ≈ 1000.

57

Velocity in the middle channel

0s

0.05 s

0.1 s

0.15 s
Case 1 (frequency 5 Hz)

58

Temperature in the middle channel

0s

0.05 s

0.1 s

0.15 s

59

Case 1 (frequency 5 Hz)

Velocity in the middle channel

0s

0.01 s

0.02 s

0.03 s
Case 2 (frequency 20 Hz)

60

Velocity in the middle channel

0.04 s

0.05 s

61

Case 2 (frequency 20 Hz)

Temperature in the middle channel

0s

0.01 s

0.02 s

0.03 s
Case 2 (frequency 20 Hz)

62

Temperature in the middle channel

0.04 s

0.05 s

63

Case 2 (frequency 20 Hz)

Average Nusselt number

64

Summary
• Flow inside square duct has been simulated for variety of
BCs and for buoyancy effect.
• Several cases - laminar and turbulent flow are
considered
• Cooling/heating, buoyancy effect, power law fluid,
tapered duct and pulsating inlet flow have also been
simulated; no analytical solution possible
• Heat transfer distributions are calculated

65

References
• W. Kays, M. Crawford, B. Weigand, Convective
heat and mass transfer 4th Edition, McGraw-Hill,
2005.
• S. Kakac and Y. Yener, Convective Heat
Transfer, Hemisphere Pub, 1982.
• A. Bejan, Convection heat transfer, Wiley, 2004.
• F. P. Incropera and D. P. Dewitt, Fundamentals
of Heat and Mass Transfer, 5th Edition, Wiley,
2001.
• J. H. Leinhard IV and J. H. Leinhard V, A Heat
Transfer Textbook, 3rd edition, 1980.
66

Computational Study of
Energy-Efficient Thermal
Drying Using Intermittent
Impinging Jets

ME 6203 Mass Transport

Computational Study of Energy-Efficient
Thermal Drying Using Intermittent
Impinging Jets
Prof. Arun S. Mujumdar
Email: [email protected]
Tel: +65-6516-4623, Fax: +65-6777-6235
2011

Guest lecturer
Jundika Candra Kurnia
Department of Mechanical Engineering
National University of Singapore

Outline
• Overview of drying
• Physical model
– Problem description
– Key assumptions

• Numerical methodology
• Selected results

Case study
Velocity
y contours
Temperature contours
Drying kinetics

• Summary
• Q&A

2

1

Overview
• Drying
– Widely known as the most common way to preserve food
– Essential operation in chemical, agricultural, biotechnology, food,
polymer ceramics,
polymer,
ceramics pharmaceutical,
pharmaceutical pulp and paper
paper, mineral
processing and wood processing industries
– Involves simultaneous transport process

• Transport processes in drying:
– Mass transfer
– Heat transfer
– Flow
Occur simultaneously both internally and externally

• Induces deformation:
– Shrinkage
– Cracking
(Not modeled here)

3

Impinging jet drying
• Various drying methods are available for
different material
p g g jjet drying
y g
• One of this method is impinging
– Offer high transport rate (mass and energy)
– Effective for drying of continuous sheets (paper
drying); discrete flat/curved objects.

• Impinging jets, however, have several
drawbacks
– Non-uniform drying
– High energy consumption compare to parallel flow

• Further study in impinging jet drying is required
– Pulsating jet is proposed to speed up drying kinetics
– Not commonly used yet

4

2

Physical Model
An orifice nozzle is used in this study- Axisymmetric case

Substrate dried is a potato chip. It is placed in a drying chamber under one
impinging jet. Pulsating and intermittent flow is applied on the inlet

5

Physical model
The aforementioned condition can be brought into computational domain as follows

Drying chamber

Inlet

L  0.4 m

Tin  45 C

z  0.02 m

Substrate dimension (chip)
Ls  30mm

Vin  fixed pulsating and intermitent H s  0.5 and 5mm
RH in  17.5%

6

3

What phenomena occur during drying?
Transport mechanism of mass? Energy?

7

Physical model
Basic mechanisms
• moisture diffusion from the inner drying
substrate towards its surface,
surface where it
evaporates
• conductive heat transfer within the drying
substrate
• evaporation and convection of the vapor
from the surface of the drying substrate
into the drying air
• convection heat transfer from drying air to
8
the surface of the drying substrate

4

Why do we need assumptions?
What are the key assumptions?
Comment on validityy of assumptions?
p

9

Assumptions
In developing the mathematical model, several assumptions are
• The drying substrate is compact and homogeneous with uniform
initial temperature and moisture content.
• Within the drying substrate, the diffusivity of water vapor is 100
times larger than the diffusivity of liquid water.
• The thermophysiscal properties of the drying substrate are
temperature and moisture content-dependent and isotropic (equal in
all directions).
• Variations in dependent variables in span wise direction are
negligible since width of the drying substrate is much larger than its
negligible,
height (reduction in dimensionality from three to two dimensions).
• The shape of the drying substrate remains constant. No shrinkage
or deformation is accounted for.
• Newtonian fluid
10

5

• How to translate these physical phenomena into
mathematical model?
• What conservation equations do we need to model
th physical
h i l phenomena?
h
?
the

11

Conservation equations
For drying substrate (chip)
• Conservation of
mass:
– Liquid water
– Water vapor

Transient term (time dependent)
Diffusion

cl
     Dlbcl    Kcl ,
t
cv
     Dvb cv   Kcl ,
t
Evaporation
Diffusion
Transient term (time dependent)

• Conservation of
energy Transient term (time dependence; heat capacity)
Conduction

T
b c pb
     kbT    q
t

Cooling due to
evaporation

12

6

Conservation equation
For drying air (Impinging jet) unsteady case

  u  0,

• Mass

Inertia/net rate

Viscous

 u

 u  u   p   u -  a u ' ,
 t

• Momentum
• Energy

For incompressible fluid

a 

 T

 uT     ka T   a c pa u ' T ' ,
 t

 a c pa 

• Mass
M
off
water vapor

Conduction

Turbulent
T
b l t
heating

cv
     Dva cv   u cv ,
t
Convection

Transient term (storage)

Diffusion

13

What is a turbulence model?
Why do we need one?
What is Reynolds averaging?
Basic /popular turbulence models

14

7

Turbulence model
• A turbulence model is a model which is used to approximates the
physical behavior of turbulent flows*
• Turbulence model are necessary in numerical simulation due to
impracticality in computing all scales of turbulent motion. Therefore,
approximate
i t methods
th d (turbulence
(t b l
models)
d l ) are introduced
i t d
d to
t simplify
i lif
and reduce computational cost.
• Reynolds averaging refers to the process of averaging a variable or
an equation in time. For example, if we have time dependent
variable , we can decompose this variable into an average part
and fluctuating part in the following way:



1
 (t )dt ,  '    .

TT

where T has to be long enough to phase out fluctuation part on (t).
• Aside from time averaging, Reynolds averaging also deals with
space averaging and Ensemble averaging**
*J.J. Bertin, J. Periaux, J. Ballmann, 1992, Advances in hypersonics v2: Modeling hypersonic flows, Birkhauser, Cambridge
**J. Sodja, 2007, Turbulence model in CFD, Ljubljana, Slovenia (http://www-f1.ijs.si/~rudi/sola/Turbulence-models-in-CFD.pdf)

15

Turbulence model
• Various turbulence models have been developed. They
can be categorized as*:

Algebraic/Zero-equation models
One equation models
Two equation models
Second-order closure models

• Among these models,
k-, k-, LES (Large Eddy Simulation), RSM (Reynolds
Stress Model) are among the most popular turbulence
model used in computational fluid dynamics.
• In this study, Reynolds Stress Model (RSM) is used as it
has been shown to be superior to k-ε and k-ω turbulence
16
*D. C. Wilcox, 2006, Turbulence modeling for CFD, DCW Industries, La Canada, Clifornia
** P. Xu, B. Yu, S. Qiu, H. J. Poh and A. S. Mujumdar. Turbulent impinging jet heat transfer enhancement due to intermittent pulsation. International Journal of
Thermal Science, 49 (7): 1247-1252, 2010.

8

Turbulence model for impinging jet drying
Reynolds stress model (5 equations)

Rij
t

 Cij  Pij  Dij   ij   ij  ij

• Accumulation
A
l ti
• Convective

Rij

 

 ui'u 'j

t
t
Cij     a ui' u 'j U

• Production

U j

U i 
 R jm
Pij    Rim

xm
xm 

• Rotation

ij  2k u 'j um' eikm  ui' um' e jkm



k2
Dij     t Rij  with  t  C , C  0.09 and  k  1.0

 k

2
 ij   ij
• Dissipation
3

2
2

C1  Rij  k ij   C2  Pij  P ij 
ij
• Pressure –strain
3
3
k

interaction
with C1  1.8 and C2  0.6

• Diffusion

To solve this model, k- turbulence model is required

17

Governing equations
• k- turbulence model


k
     uk         t
t
k


 
 k   t G   ,
 

   C  G

2
     u        t     1 t
 C2   ,
t
 k  
k
k

  u  2  v  2  w  2   u v 2  u w 2  w v 2

,
G  2                
  x   y   z    y x   z x   y z 

k2
t  C ,

Nomenclature:
u, v, w = component velocity
t= turbulent viscosity
k = turbulent kinetic energy
 = turbulent dissipation
G = turbulent generation rate

C1 = 1.44
C2 = 1.92
C = 0.09
k = 1.0
 = 1.0

18

9

Constitutive relations
• Density air

 air  1.076 105 Tair2  1.039 102 Tair  3.326

• Dynamic viscosity of air

air  5.211015 Tair3  4.077 1011Tair2  7.039 108 Tair  9.19 107

• Conductivity of air

kair  4.084 1010 Tair3  4.519 107 Tair2  2.35 104 Tair  0.0147

• Specific heat of air

c p , air  4.647 106 Tair3  4.837 103 Tair2  1.599Tair  1175

• Heat evaporation

h fg  1000  2.394(T  273.15)  2502.1

• Density of substrate

b 

b ,ref 1  X 

1  SbX

0.049
0
049
47
1
1   0.611
0 611X
X
• Conductivity
C d ti it off substrate
b t t kb 
exp  

 
3 
1 X
 8.3143 10  Ts  273.15 335.15   1  X
• Specific heat of substrate c  1750  2345  X 
p ,b

 1 X 
• Diffusivity of water vapor and
 2044 
 0.0725 
Dvb  Dlb  1.29 106 exp 

 exp 
X

 Ts  273.15 
liquid water inside substrate
19

Constitutive relations
• Heat of wetting (heat to evaporate

H w  8.207 106 X 4  4.000 106 X 3  6.161105 X 2
2.368 104 X  1163 for 0.01  X  0.2

bound water)

hevap  h fg  H w

• Total heat of evaporation
• Moisture content

X

mass of water
 l
mass of dry product  s

W

l

X
mass of water

 l 
mass of wet product  s  l b 1  X

 Dry basis
 Wet basis

• Equilibrium moisture content
(GAB model)
• Free moisture content

Xe 

X mCKAw

1  KAw   KAw  CKAw 

,

X m  0.0209, K  0.976, C  4.416

X free  X  X e

(free to be removed)

• Cooling rate due to evaporation q  hevap M l Kcl

• Rate of water evaporation
• Diffusivity of water vapor in air

 Ea

K  K 0 e RT

20

Dva  2.775 106  4.479 108 T  1.656 1010 T 2

10

Constitutive relations

Relation of moisture content to concentration of water inside substrate
 w  W  b ,
w 

b ,ref 1  X 
X

,
1 X
1  SbX

M w cw 

b ,ref  X  X 2 

1  SbX  X  SbX 2

SbX 2   Sb  1 X  1 

,

b,ref

M w cw

 X  X ,
2

b,ref  2 
b ,ref 

 Sb 
 X   Sb  1 
 X  1  0,
M w cw 
M wcw 

can be solved analytically for X and by neglecting wrong root the solution is
b  b 2  4ac
,
2a
where
X

b ,ref 

a   Sb 
,
M wcw 

b,ref 

b   Sb  1 
,
M w cw 

c  1.

21

Correlations
Calculation of h, Nu, Nu distributions in impinging jets
• Local Nusselt number

Nu ( x, t ) 

hx D jet

along the target surface
• Local heat transfer
coefficient
• Local heat transfer flux
• Time
Ti
averaged
d llocall
Nusselt number

hx 

k fluid

qx
T
 jet  Twall 

qx  k fluid

T ( x)
y
t

Nuavg ( x) 

1
Nu ( x, t )dt
t 0
x

• Time averaged Nusselt
number

y 0

Nuavg 

t

1 1
Nu ( x, t )dtdx
x 0 t 0
22

11

Nomenclature
cl

concentration of liquid water [mol m-3]

p

Pressure [Pa]

cv

concentration of water vapor [mol m-3]

Dva

diffusivity of vapor on the drying air [m2 s]

Dlb

diffusivity of liquid inside the drying substrate [m2 s]

μ

dynamic viscosity of the drying air [Pa s]

Dvb

diffusivity of vapor inside the drying substrate [m2 s]

ρa

density of the drying air [kg m-3]

T

temperature [K]

cpa

specific heat of the drying air [J kg-1 K-1]

q

cooling rate due to evaporation [W m-3]

ka

thermal conductivity of the drying air [W m-2 K-1]

K

production of water vapor mass per unit volume

Ea

activation energy [kJ mol-1]

-3

ρb

density of the drying substrate [kg m ]

R

universal gas constant [J K−1 mol−1]

cpb

specific heat of the drying substrate [J kg-1 K-1]

Ml

molecular weight of water [kg kmol-1]

kb

thermal
h
l conductivity
d i i off the
h drying
d i substrate
b
[W m-22 K-11]

Δhhevap

totall heat
h off evaporation
i [J kg
k -11]

-1

u

mean velocity [m s ]

X

moisture content (dry basis) [kg kg-1]

u’

fluctuate velocity [m s-1]

W

moisture content (wet basis) [kg kg-1]

23

Initial and boundary conditions
Initial conditions:

where

• Substrate
T  T0 , cl  cl 0,b , cv  cv 0,b ,

cl 0.b 

• Drying air

cv 0,b  0,
0

T  T0 , cl  cl 0, a , cv  c v 0, a , u  v  0,

Boundary conditions:

W b ,0
Ml

,

cl 0.a  0,
cv 0,a  1000

• Drying chamber inlet

RH  a ,0

1  RH  M l

.

u  0, v  vin , T  Tin , RH  RH in , cv  cv 0,,a .
• Drying chamber outlet

p  pout , n    Dcv   0, n   k T   0.
• Drying chamber wall

u  v  0, n    Dcv  cv u   0, n  (k T )  0.

24

12

Boundary conditions and parameters
• Parameters needed to solve
the model are

Inlet velocity for various cases

b ,ref  1420 kg m 3 ,

– vin = 2 m s-1

• Pulsating
g laminar jjet
– vin = 1+1sin(2ft) m s-1

Tin

 45C ,

Ml

 0.018 kg mol 1 ,

X0

 4.6,

• Intermitent laminar jet
– vin= 2 m s-1(on),0 m s-1(off)

 a ,45C  1.110 kg m 3 ,

a ,45C  1.934 105 kg m 1 s 1

– vin = 20 m s-1

• Pulsating turbulent jet
– vin = 10+10sin(2ft)
10+10 i (2 ft) m s-11

• Intermitent laminar jet
– vin= 2 m s-1(on),0 m s-1(off)

R

 8.314 J K 1 mol 1 ,

Ea
Sb

 48.7 kJ mol 1 ,
 1.4

f

1
Hz
120
25

Numerics
• Gambit: creating geometry, meshing,
labeling boundary condition
• Fluent: solver based on finite volume
method
– Domain is discretized onto a finite set of
control volumes (or cells).
– General conservation (transport)equations
for mass, momentum, energy, species, etc.
are solved on this set of control volumes.

 dV   V  dA     dA   S dV
t V
A
A
 


  V
 

Convection

Diffusion

Generation

– Partial differential equations are discretized
into a system of algebraic equations.
– All algebraic equations are then solved
numerically to render the solution field.
26

13

Numerics
Flow chart of computational fluid dynamics (CFD)*

27
*http://progdata.umflint.edu/MAZUMDER/Fluent/Intro%20Training/L-1%20Introduction%20to%20CFD.pdf

Numerics
• User Defined Scalars: solving for water liquid and vapor
• User Defined Functions Macros
– DEFINE_SOURCE, DEFINE_DIFUSIVITY, DEFINE_FLUX,
DEFINE_PROFILE,
PROFILE ETC

• Three different mesh sizes of 2000, 4000, 8000 elements
were implemented and compared in terms of velocity,
temperature and moisture content to ensure a meshindependent. We found that the result from the mesh
size 2000 deviates 5% and 4000 deviates 1% from 8000
elements Therefore
elements.
Therefore, a mesh of around 4000 was
chosen.
• Relative residual 10-6 for all dependent variable.
• It took around 30-50 min to converge in Quadcore 1.8
GHz with 8 GB RAM for 5 to 8 h drying time
28

14

SELECTED RESULTS
29

Contours of velocity

30

15

Contours of temperature

31

Temperature contours in substrate

32

16

Moisture profile in substrate

33

Drying kinetics (0.5 mm substrate)

• Impinging jet is not recomended
especially when drying cost is
considered (energy cost).
• Effect of the pulsating and
intermittent flow can be seen on this
case
• Frequency and velocity have no
effect on this case, most likely due
to the thin substrate

34

17

Drying kinetics (5 mm substrate)

• Velocity slightly affect drying
kinetics. It is clearer compare to
that for thin substrate.
• Effect of the pulsating and
intermittent flow can be seen in this
case
• Frequency has no effect
35

Conclusion
• Simple physical model to show how one
can use a math model consisting of
conservation equations and relevant
boundary conditions
• For gas-side, we use continuity,
momentum, energy and species equations
• On drying material side we consider
simple diffusion model for both water and
vapor. For low temperature only liquid

36

18

References
 M. V. De Bonis, G. Ruocco, 2008, A Generalized Conjugate Model for
Forced Convection Drying Based on An Evaporative Kinetics, Journal of
Food Engineering, Vol. 89, pp: 232-240
[] M. R. Islam,
s a , J.
J C
C. Ho,
o, A. S
S. Mujumdar,
uju da , 2003,
003, Convective
Co ect e Drying
y g with
t Timee
Varying Heat Input, Drying Technology, Vol. 21(7), pp: 1333-1356
 J. Srikiatden a, J. S. Roberts, 2008, Predicting moisture profiles in potato
and carrot during convective hot air drying using isothermally measured
effective diffusivity, Journal of Food Engineering, Vol. 84, pp: 516-525
 W. Kays, M. Crawford, B. Weigand, 2005, Convective Heat and Mass
Transfer 4th ed., McGraw Hill, Singapore
 F. P. Incropera and D. P. Dewitt , 2001, Fundamentals of Heat and Mass
Transfer 5th Edition
Transfer,
Edition, Wiley
 P. Xu, B. Yu, S. Qiu, H. J. Poh, A. S. Mujumdar, 2010, Turbulent Impinging
Jet Heat Transfer Enhancement Due to Intermittent Pulsation, International
Journal of Thermal Sciences. doi:10.1016/j.ijthermalsci.2010.01.020
 H. J. Poh, K. Kumar, A. S. Mujumdar, 2005, Heat transfer from a pulsed
laminar impinging jet, International Communications in Heat and Mass
Transfer, Vol. 32, pp:1317–1324
37

For Self-Study
• How would you model
– Case where jet temperature is 200 C?
– The jet is superheated steam at atmospheric
pressure and 200 C?
– The drying chamber is at very low (but finite)
pressure and dried by superheated steam
– Will drying time be reduced if the slab is
flipped after some time? Why?

38

19

Mass transport in a microchannel T-Junction with
coiled-base channel design

ME 6203 Mass Transport Guest Lecture

Mass transport in a micro-channel
T-Junction with coiled-base channel design
Prof. Arun S. MUJUMDAR
Email: [email protected]
Tel: +65-6516-4623, Fax: +65-6777-6235

Guest lecturer
Agus Pulung SASMITO
Minerals Metals Materials Technology Center
National University of Singapore

2011

Outline
• Overview of micro-channel T-Junction
• Physical model
– Problem description
– Key assumptions

Numerical methodology
Selected results
Mass transport enhancement
Concluding remarks
2

Overview
• Micro-channel T-Junction
– Widely used in industry, especially pharmaceutical, for mixing
and reaction processes
– Relatively easy to control the reactions, especially for highly
exothermic reaction
– Involves simultaneous transport process

• Transport process in T-Junction:
– Momentum transfer
– Mass transfer
– Heat transfer
Occur simultaneously

• Main phenomena:
– Mixing
– Surface reactions
3

Micro-Channel T-Junction
• Passive mixing for various chemical reaction; it
does not require additional energy for mixing
processes
• Micro-channel T-Junction, however, has several
drawbacks
– Poor mixing, especially at short channel and high
Reynolds number
– High pressure drop due to impingement effect

• Various innovative designs are proposed to
improve mixing and reactions
– Coiled channel
– Channel with fins
– Impinging jet channel

4

Typical geometry
Example case of mixing and reaction of
methane oxidation in platinum surface

A typical micro-channel Tjunction is used to mix methane
and air; the channel surface is
coated with catalyst (platinum)
Inlet

O2
N2
T = 300K

Tin = 300 K
Pt surface; T 1290K

h
Pt surface; T 1290K

Vin ≈ Re 500
Micro-channel T-junction
CO
CO2
H2O

L = 120 mm
h = 1 mm
Gas species: CH 4 , O 2 , H 2 , H 2 O,

CO 2 , HO 2 , N 2
Surface species: Pt(s), H(s), O(s),

CH4
H2
T = 300K

OH(s), H 2 O(s), H3 (s),
CH 2 (s), CH(s), C(s),
CO(s), CO 2 (s)

L

Solid species: Pt(b)

5

Innovative coiled-base design
• Coiled-base channel design is
proposed to enhance heat and
mass transfer.
• Coiled-base channel design
has been widely used in
industrial applications due to
compact structure, ease of
manufacture, higher heat and
mass transfer.
• The presence of secondary
flow induced by coil curvature
and complex temperature and
concentration profiles caused
by curvature-induced torsion
are among significant
phenomena observed in
coiled-base channel.
• Length is kept constant for
comparison purpose.

Typical straight T-junction

Conical T-junction

Helical T-junction
In-plane spiral T-junction
6

Physical model
Basic mechanisms

Convective heat and mass transfer
Mixing in the opposing jet at T junction
Surface reactions at the channel wall which include
– mass consumptions and generation,
– heat release due surface reaction,
– multi-step surface reactions including adsorption
reaction, surface reaction, and desorption reaction.
Surface species are calculated from site balance
equations
Surface reactions create sources of bulk phase, which
determines its deposition rate on a surface.

7

Assumptions
In developing the mathematical model, several assumptions are
taken:
• The flow is steady-state, laminar, newtonian flow and
species mixture is follows ideal gas law.
• There are three types of species: gas, surface (site) and
solid species. The model treats chemical species
deposited on surfaces as distinct from the same
chemical species in the gas
• Thermo-physical properties of species mixture follows
mixing law of ideal gas with temperature dependent
effect.
• Gas phase reaction are closely coupled with surface
reactions.
8

• How to translate this physical phenomena into a
mathematical model?
• What conservation equations do we need to interpret
the physical phenomena?

9

Conservation equations
∇ ⋅ ρu = 0

• Mass

Compressible
flow

• Momentum

viscous

(

)

Effect of volume
dillatation

2
T

∇ ⋅ ρ u ⊗ u = −∇p + ∇ ⋅ ⎜ ⎡ μ ∇u + ( ∇u ) ⎤ − μ ( ∇ ⋅ u ) I ⎟
⎦ 3

⎝⎣
Pressure
Inertia/net rate

• Species

∇ ⋅ ( ρ uωi ) = −∇ ⋅ ( ρ Di∇ωi ) + Ri
convective

• Energy

diffusive

reaction

i: CH 4 , O 2 , H 2 , H 2 O, CO 2 , HO 2

∇ ⋅ ( ρ cpuT ) = ∇ ⋅ ( keff ∇T ) + S temp
convective

conductive

Heat due to
reactions

Detailed reactions
• Theory: consider the rth wall surface reaction written in general forms
Ng

Nb

Ns

Ng

Nb

Ns

i =1

i =1

i =1

Kr
⎯⎯
→ ∑ gi'',r Gi + ∑ bi'', r Bi + ∑ si'',r Gi

∑ g Gi + ∑ b Bi + ∑ s Gi ←⎯
i =1

'
i ,r

'
i ,r

i =1

i =1

'
i ,r

where Gi, Bi, and Si represents the gas phase species, the solid species, and the
surface-adsorbed (or site) species, respectively. g’, b’, s’ are the stoichiometric
coefficients for each reactant species; g’’, b’’, and s’’ are the stoichiometric coefficients
for each product species; and Kr is the overallNreaction rate constant.
g

i ,r
i ,r
ℜr = k f ,r ∏ [Gi ]wall
[ Si ]wall

• The rate of rth reaction is

g'

s'

i =1

• The net molar rate of production or consumption of each species i is
N
given by
Rˆ =
g '' − g ' ℜ
i = 1, 2,3,..., N

∑(
rxn

i ,gas

i ,r

i ,r

)

r

g

Rˆi ,bulk = ∑ ( bi'',r − bi',r )ℜr

i = 1, 2,3,..., N b

r =1

N rxn

r =1

N rxn

Rˆi ,site = ∑ ( si'',r − si' ,r )ℜr

i = 1, 2,3,..., N s

r =1

• Reaction rate constant is computed using Arrhenius expresion

k f , r = Ar T β r e − Er / RT

Wall surface reaction boundary conditions
• It is assumed that, on the reacting surface, the mass flux of each gas
species is balanced with its rate of production/consumption

ρ wall Di

∂ωi ,wall
∂n

− m depωi,wall = M w,i Rˆi,gas
∂ [ Si ]wall
∂t

= Rˆi,site

i = 1, 2,3,..., N g
i = 1, 2,3,..., N s

• The mass fraction at the wall is related to concentration by

[Gi ]wall =

ρ wallωi,wall
M w ,i

• mdep is the net rate of mass deposition or etching as a result of surface
reaction
Nb
m dep = ∑ M w,i Rˆi ,bulk
i =1

• [Si]wall is the site species concentration at the wall, and defined as
where

[ Si ]wall = ρsite zi
ρsite is the site density and
zi is the site coverage of species i

Multi-step reaction mechanism
Ar

βr

1

H2 + 2Pt(s) => 2H(s)

4.36e7

0.5

0

2

2H(s) => H2 + 2Pt(s)

3.7e20

0

6.74e7

No

Reaction

Er (J/kmol)

3

O2 + 2Pt(s) => 2O(s)

1.8e17

-0.5

0

4

O2 + 2PT(s) => 2O(s)

2.01e14

0.5

0

5

2O(s) => O2 + 2Pt(s)

3.7e20

0

2.13e8

6

H2O + Pt(s) => H2O(s)

2.37e8

0.5

0

7

H2O(s) => H2O + Pt(s)

1e13

0

4.03e7

8

OH + Pt(s) => OH(s)

3.25e8

0.5

0

9

OH(s) => OH + Pt(s)

1e13

0

1.93e8

10

H(s) + O(s) => OH(s) + Pt(s)

3.7e20

0

1.15e7

11

H(s) + OH(s) => H2O(s) + Pt(s)

3.7e20

0

1.74e7

12

OH(s) + OH(s) => H2O(s) + O(s)

3.7e20

0

4.82e7

13

CO + Pt(s) => CO(s)

7.85e15

0.5

0

14

CO(s) => CO + Pt(s)

1e13

0

1.25e8

15

CO2(s) => CO2 + Pt(s)

1e13

0

2.05e7

16

CO(s) + O(s) => CO2(s) + Pt(s)

3.7e20

0

1.05e8

17

CH4 + 2Pt(s) => CH3(s) + H(s)

2.3e16

0.5

0

18

CH3(s) + Pt(s) => CH2(s) + H(s)

3.7e20

0

2e7

19

CH2(s) + Pt(s) => CH(s) + H(s)

3.7e20

0

2e7

20

CH(s) + Pt(s) => C(s) + H(s)

3.7e20

0

2e7

21

C(s) + O(s) => CO(s) + Pt(s)

3.7e20

0

6.28e7

22

CO(s) + Pt(s) => C(s) + O(s)

1e17

0

1.84e8

23

OH(s) + Pt(s) => H(s) + O(s)

1.56e18

0

1.15e7

24

H2O(s) + Pt(s) => H(s) + OH(s)

1.88e18

0

1.74e7

25

H2O(s) + O(s) => OH(s) + OH(s)

4.45e20

0

4.82e7

Constitutive relations
• Mixture density
ρ = pM / RT
• Mean molecular mass

(

M = ωCH4 / M CH4 + ωH2 / M H2 + ωO2 / M O2 + ωH2O / M H2O + ωOH / M OH + ωCO2 / M CO2 + ωCO / M CO + ωN2 / M N2

• Mixture viscosity
μ =∑
α

xα μα
∑ xα Φ α,β

with α , β = CH 4 , H 2 , O 2 , H 2 O, OH, CO, CO 2 , N 2

β

Φ α,β

1 ⎛ Mα
=
⎜1 +
8 ⎜⎝ M β

⎟⎟

−1/ 2

1
1

(g) 2
⎢1 + ⎛ μα ⎞ ⎛ M β ⎞ 4 ⎥
⎢ ⎜⎜ μ (g) ⎟⎟ ⎜ M ⎟ ⎥
⎝ β ⎠ ⎝ α⎠ ⎥
⎣⎢

2

• Effective thermal conductivity; heat capacity
keff = ∑ ki ωi
cp = ∑ ωi cp,i
i

)

−1

Constitutive relations (Cont’d)
• Figure of Merit is used to evaluate the effectiveness of the
mixing and reaction rate in the micro-channel T-Junction. It
defined as reactant conversion rate over pumping power
required. Since the mass flow rate is kept constant; hence

ωin − ωout

FoM =
V=

• Mean velocity

• Mixed-mean temperature

• Mixed-mean mass fraction

Δp

1
Ac

∫ udA ,
c

Ac

Tmean =

1
VAc

ωi ,mean =

∫ TudA ,
c

Ac

1
VAc

∫ ω udA
i

c

Ac

Boundary conditions
• At the air inlet

Uin = 5 m/s (Re~500)
Tin = 300K
ωO2 = 0.21
ωN2 = 1- ωO2

• At the methane
inlet

Uin = 5 m/s (Re~500)
Tin = 300K
ωCH4 = 0.9
ωH2 = 0.1

• At the walls
– No-slip condition
– No species flux
– Twall = 1290 K

• At the outlet
– Pout = 101325
– ∇T = ∇ωi = 0

Numerics
• Gambit for meshing and labeling boundary conditions:
– fine structured mesh near wall to resolve boundary layer;
increasingly coarser mesh to the middle of the channel to reduce
computational cost
– Mesh independence test were carried out for three different
mesh size—coarse, medium, fine—in terms of velocity, pressure,
temperature and species.

• Fluent for discretization and solving dependent variables
– Based on finite volume discretization method
– Pressure-velocity coupling is solved by well-known SIMPLE
method
– Overall, it requires ~300 MB memory and 2 h solving time on
workstation with Quadcore 2.63 GHz processor for convergence
criteria 10-6 for all dependent variables.

• ChemKIN for reaction kinetics
– To set up details multi-step reaction mechanism and thermophysical properties of gas species.

Flow chart for numerical solver

Results and discussion
Velocity profiles at channel length 50mm

m/s

• Fully developed
flow exists in
the straight
channel
• Secondary flow
is developed in
the coiled base
channel.
• Higher velocity
intensity exists
in the outer wall
of the coiled
base channel

Oxygen mass fraction at channel length 50mm
• Straight channel has
higher oxygen mass
fraction means that
lesser oxygen is
consumed for
reactions.
• Among the coiled
base channel design,
helical coil gives
better conversion
rate.
• Secondary flow
enhance mass
transport in the
surface reaction

Methane mass fraction at channel length 50 mm
• At straight channel, the
methane concentration
is higher at the right
side of the wall, it
means that the
methane is not mixed
well with air.
• Helical coil yields the
best mixing and
reaction among others
• The presence of
secondary flow
improve the mixing rate
of reactant species

Oxygen mass fraction along channel
0.18
helical
straight
in plane spiral
conical

Oxygen mass fraction

0.16
0.14
0.12
0.1
0.08
0.06
0

20

40

60

80

100

120

Length / mm

• Helical coil gives the best conversion rate among other designs.
• Straight T-junction yields the lowest conversion rate due to poor
mixing.

Mixed mean temperature along channel
1300
1200
1100
Temperature / K

1000
900
800
700

conical
in plane spiral
helical
straight

600
500
400
300
0

20

40

60
Length / mm

80

100

120

• Coiled base channel also gives higher heat transfer rate as
compared to straight channel
• Coiled base channel is suitable for highly exothermic/endothermic
reaction to control the desired environment/temperature.

Effect of Reynolds number
Re 100
Re 500
Re 1000

0.17

Oxygen mass fraction

0.15

Re increasing

0.13
0.11
0.09
0.07
0.05
0.03
0

20

40

60

80

100

Length / mm

• Lower mass flow rate performs better conversion rate
• This is due to longer residence time of the species

120

Effect of coil diameter
0.18
r4
r3
r5

Oxygen mass fraction

0.16
0.14
diameter increasing
0.12
0.1
0.08
0.06
0

20

40

60

80

100

120

Length / mm

• Smaller coil diameter produces slightly better conversion rate.
• This is due to higher secondary flow produced in smaller coil
diameter.

Pressure drop
20000
straight
conical
in-plane
helical

18000
16000
14000
p / pa

12000
10000
8000
6000
4000
2000
0
0

200

400

600

800

1000

Reynolds

• Straight channel requires the lowest pressure drop;
whereas, the helical coil has the highest pressure drop
• Pressure drop increses as the mass flow increasing

Figure of Merit
3.50E-04
straight
conical
in-plane
helical

3.00E-04

Figure of Merit

2.50E-04
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
100

500
Reynolds

1000

• Straight channel has the highest figure of merit among other
designs; however, for industrial application where space is limited
and pumping power is not an issue, such as in pharmaceutical
industry, coiled base channel design can be a desirable choice

Concluding remarks

• Coiled base channel design can improve heat
and mass transfer as compare to straight Tjunction channel.
• This improvement is due to the presence of
secondary flow.
• However, higher pressure drop is required for
coiled base channel design.
• For industrial application where space is limited,
conversion rate is the most important, and
pumping power is not an issue, coiled base
channel design can be a desirable choice

Nomenclature
ρ = density, kgm −3

ℜ = rate of rth reaction

u = velocity, ms −1
p = pressure, pa

k f ,r = reaction rate constant using Arrhenius expression
Ar = pre-exponential factor

μ = dynamic viscosity, Pas
ωi = mass fraction of species i
Di = diffusivity of species i, ms

β r = temperature exponent
Er = activation energy for the reaction, Jkgmol
−2

Ri = reaction rate of species i, kgm −3
c p = specific heat, Jkg −1K −1

R = universal gas constant, Jkg -1mol-1K -1
M = mean molecular mass
m dep = net rate of mass deposition, kg
x = mol fraction

T = temperature, K
keff = effective thermal conductivity, WmK −1
S temp = heat release due to reactions, Wm -3
Gi = gas species, mol
Bi = bulk/solid species, mol
gi' , gi'' = stoichiometric coefficient for gas reactant, and product
bi' , bi'' = stoichiometric coefficient for bulk reactant, and product
si' , si'' = stoichiometric coefficient for site reactant, and product

References
 O. Deutschmann, L.i. Maier, U. Riedel, A.H. Stroemman, R.W. Dibble,
Hydrogen Assisted Catalytic Combustion of Methane on Platinum,
Catalysis Today 59,141--150 (2000).
 V. Kumar, M. Paraschivoiu, K.D.P. Nigam, Single phase fluid flow and
mixing in microchannel, Chemical Engineering Science, 2011, in press.
 S. Vatisth, V. Kumar, K.D.P. Nigam, A review on the potential application
of curved geometries in process industry, Industrial Engineering Chemistry
Research 47, 3291-3337 (2008).
 J.C. Kurnia, A.P. Sasmito, A.S. Mujumdar, Evaluation of heat transfer
performance of helical coils of non-circular tubes, J. Zhejiang University
Science: A, 2011, in press.
 J.C. Kurnia, A.P. Sasmito, A.S. Mujumdar, Laminar convective heat
transfer in coils of non-circular cross-section tube: a computational fluid
dynamics study, Thermal Science, 2011, accepted.
 J.C. Kurnia, A.P. Sasmito, A.S. Mujumdar, Numerical investigation of
laminar heat transfer performance of various cooling channel designs,
Applied Thermal Engineering, 2011, in press.
 Fluent user guide documentation, http://www.fluent.com
30

For Self Study

• How can one reduce pressure drop in the coiled
base channel design?
• What happens if the channel is not in square
cross-section, e.g. circular, triangle, star-shape
etc?
• Will reaction and mixing rate improve if we add
fins inside the channel?
• What if the reactant species are in different
phase, e.g. gas and liquid? Is the model
presented still valid?

Mass Transport
Considerations in PEM
Fuel Cell Modeling

ME6203 Mass Transport
Mass Transport Considerations
in PEM Fuel Cell Modeling

Prof. Arun S. Mujumdar, ME, NUS
Dr. Poh Hee Joo, IHPC

March 2010
1

Outline - Part 1 (Handout)

Fuel Cell Introduction
Fuel cell Mass Transport

Diffusive transport in electrode
Convective transport in flow structures

Analytical Modeling with MATLAB
Summary

2

1

Why PEMFC Modeling in Mass
Transport course?
• Fuel cells are becoming important in academic
commercialized. Much more development is
needed to enhance performance cost-effectively
• Excellent industrial illustration of a case where
math modeling of transport phenomenaincluding mass transport is critically important
• It is an excellent illustration of how very complex
transport processes can be modeled and what
are the different levels of math models which are
possible
3

Why PEMFC model?
• In ME6203 one objective is to look at advanced mass
transport problems of real interest and examine how a
model can be developed based on fundamentals
• It is also an example of complex interaction between
various transport phenomena. Illustrates need for
significant information needed for such a model
• Due to time limitation, different levels of modeling e.g.
phase models etc are not discussed. Model is only as
good as assumptions made- they must be realistic.
4

2

Mass Transport and PEMFC
• Whenever there is species movement causing
concentration changes –there is mass transport
• Mechanisms are: diffusion, convection, electro-osmosis
etc
• PEMFC includes flow in channels, flow in porous media
• Involves proton and electron transfer, catalytic chemical
reactions, heat transfer etc
• An excellent-but complex-example for study of transport
phenomena
• Here, please focus on the technique of math modeling
rather than the complex details which are beyond the
scope of this course.
• Suitable for Term Paper Projects e.g. 1D analytical
modeling of different types of fuel cells
5

Preamble
• With this preamble , let us proceed to fuel cells..
• Numerous resources are available on the web for
self-study
• Advanced models are being worked on at hundreds
of labs around the world-useful for innovation!
• Several excellent textbooks available as well
• No need to go beyond what is in this PPT-except for
those who choose to work on term papers on this
subject.
• Poh Hee Joo will be happy to provide relevant
resources and ideas to those interested
• Caution: Some aspects are complex and are
included only for completeness of coverage. You do
not need to get into those details.
6

3

What is a Fuel Cell ?
A fuel cell is a device that generates electricity by a chemical
reaction taking place at two electrodes (anode and cathode)

Backing layer

Backing layer

7

Physical Description of a PEMFC
PEMFC – Proton Exchange Membrane Fuel Cell

Anode bipolar plate
Anode gas channel
Anode gas diffusion layer
Anode catalyst layer

MEA Membrane
Electrode
Assembly

Membrane

Cathode catalyst layer
Cathode gas diffusion layer
Cathode gas channel
Cathode bipolar plate
8

4

Basic Fuel Cell Operation
1. Reactant transport
•Efficient delivery of reactants by using flow field plates in
combination with porous electrode structures.
2. Electrochemical reaction
•Choosing right catalyst and carefully designing reaction
zones
3. Ionic (and Electronic) Conduction
•Thin electrolyte for ionic conduction, without fuel cross over
4. Product Removal
•“Flooding” by product water can be major issue in PEMFC
9

Applications of fuel cells

• Transportation
• Stationary Power Generation
• Residential
• Portable Power Generation
• Space and Defense

10

5

Fuel Cells: Classification
PEM fuel cell

Solid oxide fuel cell

Polymer membrane

Ceramic membrane

800C

At Anode

Characterized by
•Electrolyte materials
•Operating temperature
•Fuel used

600 – 10000C

PEM Fuel Cell
Solid Oxide Fuel Cell
Direct Methanol Fuel Cell

2H2 → 4H+ + 4e

At Cathode
O2 + 4e + 4H+ → 2H2O
½ O2 + 2e → O2-

H2 + O2-→CO2 + H2O + 2e
CH3OH + H2O →CO2 + 6H+ + 6e

3/2O2 + 6H+ + 6e →3H2O

11

Replacement for IC Engines in
transportation

Higher energy efficiency
Zero or ultra-low emission

Replacement for batteries in portable
electronics

Higher energy density
Nearly zero recharge time
Independent scaling between power
(determined by fuel cell size) and capacity
(determined by fuel cell reservoir)
12

6

Fuel Cells: Some Limitations

High cost of fuel cell
Low volumetric power density comparing to I.C.
engines and batteries
Safety, Availability, Storage and Distribution of
pure hydrogen fuel
Alternative fuels (e.g. methanol, gasoline)
difficult to use directly and require reforming
Susceptibility to environmental poisons
Operational temperature compatibility concerns
13

Challenges to Fuel Cell Commercialization
• Simple question, but difficult answer
– Prototype developed by SERC-PEMFC for
2W portable battery charger fuel cell of
NOKIA mobile phone cost about \$300

14

7

Fuel cells: Interdisciplinary field of
science and engineering:

Thermodynamics
Electrochemistry
Chemistry and Chemical Engineering
Fluid Mechanics
Heat and Mass Transfer
Material Science (metallurgy) and materials engineering
Polymer Science and specifically ionomer chemistry
Design, manufacturing and engineering optimization
Solid mechanics and mechanical engineering
Electromagnetism and electrical engineering
Etc etc
15

PEMFC (Interdisciplinary!)
Membrane Science

High temperature
cation membrane
Reduce CO poisoning
of catalyst)

Catalysis
and
Electrochemistry
Alternative catalyst
(reduce cost)
High catalyst utilization

Membrane for DMFC
Improved performance
(prevent methanol crossover)
(electro-oxidation/Reduction)
Anion membrane
(use of low cost catalyst)

Measurement and
Characterization
(relate performance to
electrochemical processes)

Thermofluids
and
Component Design

System Integration

Transport phenomena System design & configuration
(molecular diffusion, (reduce cost, improved efficient)
ion migration,
convection)
Interconnection
(increase power o/p)
Multi-phase physics
(water management) Heat and water management
(operational stability)
Heat transfer
(performance stability)
Fluid dynamics
(flow channel design)
Courtesy of SERC Fuel Cell Project 16

8

Schematic of a cross sectional view of PEM
Fuel Cell unit

Cathode
Catalyst
Cathode
Bipolar Plate

Air
channel

Cathode
Electrode
(GDL)

O2

½O2 + 2H+ + 2e → H2O

e-

Anode
Catalyst

Membrane

H+

Anode
Anode
Electrode Bipolar Plate
(GDL)

H2

H2
channel

H2 → 2H+ + 2e

17

Role of Each Component
1.

2.

3.

4.

5.

Cathode/Anode Bipolar plate
Electronic Conduction
Heat Transport
Air/H2 channel
Reactant Transport & Product Removal (Mass Transfer)
Heat Transport
Cathode/Anode GDL
Ionic and Electronic Conduction
Reactant Transport & Product Removal (Mass Transfer)
Heat Transport
Cathode/Anode Catalyst
Electrochemical reaction (Mass Transfer – reactant consumption and
product generation)
Ionic and Electronic Conduction
Membrane
Ionic conduction
Water transport
18

9

Water Management in PEMFC
Cathode

Electrolyte
membrane

Anode

Water produced
within cathode
Water is dragged from
anode to cathode sides by
protons moving through
electrolyte (electro-osmotic
drag)

Water is back diffused from
cathode to anode, if cathode side
holds more water

Water is supplied by
externally humidifying
air/O2 supply

Water is supplied by
externally humidifying
hydrogen supply

Water is removed by
O2 depleted air
leaving the fuel cell

Water is removed by
circulating hydrogen
19

Why Mass Transfer is Important in PEMFC
Component

Mass Transport Implication

Where mass transport
limitation exists

Air/H2 channel

To provide homogenous distribution
of reactants across an electrode
surface while minimizing pressure
drop and maximizing water removal
capability

Reactant depletion for
downstream channel
Impurity contamination, e.g.
N2

Cathode/Anode GDL

Porous electrode support to
reinforce catalyst, allow easy gas
enhances electrical conductivity

Liquid water flooding block the
pores for gas diffusion into
catalyst layer

Cathode/Anode Catalyst

Electrochemical reaction takes
place at the catalyst layer,
consume reactant (H2 and O2) and
generate product (H2O)

Poor total reaction surface
optimal electrochemical
performance

Membrane

To separate the air and H2 while
allowing liquid water and ionic
transport across membrane

Membrane dry-out at high
temperature, and loss of its
proton conducting capability
20

10

Fuel Cell : Mass Transport

To produce electricity, fuel cell must be continually
supplied with fuel and oxidant. At the same time,
products must be continuously removed so as to
avoid “strangling” the cell. The process of
supplying reactants and removing products is
termed fuel cell mass transport.

Why is it important? Poor mass transport can lead
to significant fuel cell performance loss, as the
reactant depletion and/or product accumulation
within catalyst layer (not at the fuel cell inlet) will
adversely affect performance. This is called
concentration or mass transport loss, and can be
minimized by careful optimization of mass
transport in the fuel cell electrodes and fuel cell
flow structures
21

Transport in Electrode vs. Flow
Structure

Difference between mass transport in fuel cell electrode and fuel
cell flow structures in one of length scale, and this lead to
difference in transport mechanism

For fuel cell flow structures, dimensions are generally on the
millimeter or centimeters scale. Flow pattern typically consisted of
well-defined channel arrays. Gas transport in the channel is
dominated by fluid flow and convection.

For fuel cell electrodes, it exhibit structure and porosity on the
micrometer and nanometer length scale. The tortuous geometry
of electrodes insulated gas molecules from convective forces
present in flow channel. Gas transport within electrodes is
dominated by diffusion.

Velocity scale could also affect transport
mechanism
*

22

11

Transport in Electrodes: Diffusive
Transport
Flow
Structure

Anode
Electrode

Reactants (R) In

Catalyst
Layer

Electrolyte

JR
jrxn

Products (P) Out

Concentration

An electrochemical
reaction on catalyst layer
side of an electrode and
convective mixing on the
other flow channel side of
the electrode set up
transport across the
electrode.

Flow
channel

JP

c Ro

c *P

c Po

c R*

Reaction in catalyst
layer consumes R,
generates P

Schematic of mass transport situation within
typical fuel cell electrode
23

From Faraday’s Law : current i evolved by an

electrochemical reaction is a direct measure of
the rate of electrochemical reaction i = dQ = nF dN
dt

n is the number of electrons transferred,
F is Faradays constant, 96,485 C/mol,
dN
is the rate of electrochemical reaction, mol/s.
dt

*The

dt

current density

j=

i
 1 dN 
= nF 
 = nFJ
A
 A dt 

J is the molar flux, mol/cm2s.
24

12

Transport in Electrode: Diffusive Transport

At steady state, the diffusion flux of reactants and
products down the concentration gradient across the
electrode (diffusion layer) will exactly match the
consumption/production rate of reactants and products
at the catalyst layer.
Diffusion flux (kmol/m2s) of reactants to the catalyst
layer may be described by
dc J = − D eff c *R − c Ro
J diff = − D
diff
δ
dx

*
o
eff c R − c R
j
=

nFD
δ
Using the flux balance equation, one can solve for
reactant concentration in the catalyst layer

*
0

cR = cR −

25

nFD eff

Transport in Electrode: Diffusive
Transport

jL = nFD

cR0

Limiting Current Density, jL
δ
Limiting current density of fuel cell will be encountered
when reactant concentration in the catalyst layer drops
all the way to zero.
Fuel cell mass transport design strategies focus on
increasing the limiting current density by:
1.
2.

eff

Ensuring a high reactant concentration at flow channel by
designing good flow structures that even distribute reactants
Ensuring that effective diffusivity is large and diffusion layer
thickness is small by carefully optimizing fuel cell operating
conditions, electrode structure, and diffusion layer thickness.

Theoretical typical jL are on the order of 1-10A/cm2
26

13

Question 1
1. Discuss the factors that determine jL,
limiting current density. List three ways to
increase jL.

27

c
The limiting current density is given by j = nFD δ
Factors determining the jL are reactant concentration at flow
channel, effective diffusivity and diffusion layer thickness. We
could increase jL by
– Ensuring a high reactant concentration at flow
channel by designing good flow structures that
even distribute reactants
– Ensuring effective diffusivity is large;
– Ensuring diffusion layer thickness is small
• by carefully optimizing fuel cell operating
conditions, electrode structure, and diffusion
layer thickness
eff

0
R

L

28

14

Typical process of reactant
transport to reactant sites

If considering the convection mass transfer across electrode surface,
how is the limiting current density being derived?
x=H

c

o
R

Convection, hm

Diffusion

Diffusion and
reaction

c Rs

Reactant molar
flux, JR

Flow channel

x

c *R

HE

x=0
Gas Diffusion
Layer (porous)

Catalyst Layer
(porous)

29

Process of reactant transport to reactant sites
Convection mass transfer at the electrode surface

(

J = hm c Ro − c Rs

)

(1)

Diffusion mass transport through the Gas Diffusion Layer

 c s − c R*
J = D eff  R
 HE



(2)

Combining Equation 1 & 2

J=

c Ro − c R*
∑ Rm

∑R

m

=

H
1
+ effE
hm D

(3)

From Faraday’s Law, current density is proportional to the rate of
electrochemical reaction
 1
H 
i
 1 dN 
+ effE (C Ro − C R* ) (4)
j = nF 
j = = nF 
 = nFJ
h
A
 m D 
 A dt 
Limiting current density

 1 H
jL = nF  + effE
 hm D

−1

 o
 CR

(5)

30

15

Transport in Electrode: Diffusive
Transport

Concentration affects fuel cell performance through reaction
kinetics.
This is because reaction kinetics also depend on the reactant and
product concentration at the reaction sites.
Reactant depletion/product accumulation in the catalyst layers
lead to fuel cell performance loss.
This is called fuel cell concentration (or mass transport) loss.

ηconc =

RT  1  jL
1 + 
nF  α  jL − j

ηconc - Voltage loss due to reactant depletion in the catalyst layer

Increasing jL can greatly extend a fuel cell’s potential operating
range; therefore mass transport design is an active area of
current fuel cell research.

31

Question 2
2.

Using the limiting current density equation, calculate
the limiting current density for a fuel cell cathode
running on air at 1 Atm and 25°C. Assume only O2 and
N2 and ignore the presence of water vapor. Mass
fraction of O2 in air is 0.23. Assume the diffusion layer
is 500µm and has a porosity of 40%.

Hint* : Using Chapman-Enskog theory (Chp 5, Cussler)
to find the binary diffusion coefficient, and
Bruggemann correction to account for the effective
diffusivity in porous structure. Molar concentration for
O2 can be obtained by mole fraction of O2 multiply by
the total molar concentration for the air mixture. n is
the number of electrons consumed per mole of the
reactant consumed. Molecular weight for N2 and O2
are 28 and 32, respectively
32

16

In the H2-O2 fuel cell, the electrochemical reaction at
the cathode is given by O2 + 4H+ + 4e → H2O. Hence
n = 4. F is the Faraday constant, 96,485 C/mol. The
binary diffusion coefficient is given by ChapmanEnskog theory.σO2 = 3.467. σN2 = 3.798. σO2-N2 =
3.6325. From the necessary calculation, Ω = 0.9186.
Therefore, Dij = 2x10-5 m2/s. From Bruggemann
correction Dij,eff = 5.06 x 10-6 m2/s. From the ideal gas
equation, total molar concentration for the mixture is =
40.9 mol/m3. Molecular weight for the mixture O2-N2 is
= 28.92. Mole faction of O2 = = 0.20786. Therefore,
molar concentration of O2 = 8.5015 mol/m3. Limiting
current density = 33,204 A/m2 = 3.32A/cm2
33

Transport in Flow Structure:
Convective Transport
• Fuel cell flow structures are designed to distribute
reactants across fuel cell
• One could possibly use single-chamber structure, and
encapsulate the entire fuel cell collector in a single
compartment. Unfortunately, this would make reactants
tend to stagnant inside the chamber, leading to poor
reactant distribution and high mass transport losses;
hence poor fuel cell performance
• Conversely, employing intricate flow structure containing
many small flow channels keeps the reactants constantly
flowing across fuel cell, encouraging uniform convection,
mixing and homogenous reactant distribution.
34

17

Convective Transport Contd
• Analyzing convective gas transport in the
complex real world flow structures is only really
possible with numerical methods. A common
technique is to use CFD modeling
• However, basic analysis of simple flow scenarios
is still possible with the principle of fluid
mechanics, which can still yield great insight into
fuel cell mass transport and flow structure
design
35

Transport in Flow Structure:
Convective Transport
Inlet

Outlet
u

x
y

Dh
JC
Convection transfer at surface
JD
Diffusion

Electrode

Membrane

Pressure difference between inlet and outlet drives the fluid flow.
Although gas flowing in stream-wise direction along flow channel,
convective mass transport can also occur in transverse direction from
flow channel into (or out of) electrode. This happens when
concentration of species i is different at the electrode surface versus
the flow channel bulk.
36

18

Transport in Flow Structure:
Convective Transport
• Mass flux (kg/m2s) due to convective mass transfer
may be estimated by

(

J C ,i = hm ρi , s − ρi

)

• Mass transfer convection coefficient, hm, is dependent
on the channel geometry, the physical properties of
species i and j, and the wall conditions. It can be found
from the nondimensional Sherwood number
hm = Sh

Dij
Dh
37

Transport in Flow Structure:
Convective Transport
• Gas is depleted along flow channel
• As hydrogen or air is consumed continuously along
a flow channel, the reactants tend to become
depleted, especially near the outlet. Depletion poses
adverse effect on fuel cell performance, since
concentration losses increase as reactant
concentrations decrease.
• A simple 2D mass transport model for fuel cell
cathode is developed. This is to determine how the
oxygen concentration decreases along flow channel
using macro-scale mass flux balance.
*
• Refer to Note 1 for O2 mass concentration profile
along cathode catalyst layer.
* important

38

19

Transport in Flow Structure: Convective
Transport
Gas is depleted along flow channel

y

To find oxygen concentration profile along the catalyst
layer
Electrolyte
C

J O2

RXN

J O2

E

uin

ρO

J O2
2

y =C

Cathode catalyst layer

DIFF
y=E

Gas diffusion layer

HE

CONV
y=E

ρO
Cathode flow channel

2

HC
x

Schematic of a 2D fuel cell transport model including diffusion and convection
39

Convective Transport in Flow Structure :
Assumptions
1.
2.
3.
4.
5.

Flow channel has a square cross section.
The catalyst layer is infinitely thin.
Water exists only in vapor form.
Diffusive mass transport dominates in
the diffusion layer. Furthermore, only ydirection diffusion is considered.
6. Convection mass transport dominates in
the flow channel
40

20

Transport in Flow Structure:
Convective Transport

From Faraday’s Law, if fuel cell is producing a current density at location X,
then the O2 mass flux (kg/cm2s) that is consuming is given by
^

J O2

rxn

x = X , y =C

= M O2

j( X )
4F

(1)

The O2 flux consumed by the electrochemical reaction must be provided by
diffusion in the gas diffusion layer, described by Fick’s law
diff

^

J O2

x= X , y=E

= − DOeff2

ρO

2

x = X , y =C

− ρ O2

x= X , y=E

(2)

HE

O2 mass flux due to mass transport through the gas diffusion layer is provided
by convective mass transport between the flow channel and gas diffusion layer
^

J O2

conv

x= X , y=E

= − hm  ρ O2

x= X , y=E

− ρ O2

x = X , y = channel



(3)
41

Mass flux balance between convective transport in flow
structure and diffusion transport in GDL

1. O2 mass flux consumed by the
electrochemical reaction at the catalyst
layer
2. O2 mass flux due to diffusion mass
transport through the gas diffusion layer
3. O2 mass flux provided by convective
mass transport between the flow channel
and gas diffusion layer surface.
Mass flux 1 = Mass flux 2 = Mass flux 342

21

Transport in Flow Structure:
Convective Transport
To maintain the flux balance, O2 mass flux in equations 1, 2 & 3 must be same
rxn

^

J O2

diff

^

= J O2

x = X , y =C

^

x= X , y= E

conv

= J O2

(4)

x= X , y= E

The following relations can be derived
diff

^

J O2

ρO

x= X , y=E

x = X , y =C

2

ρO

= M O2
= ρO2

x= X , y=E

2

j(X )
4F

x= X , y=E

= ρ O2

(5)

− M O2

x = X , y = channel

j( X ) H E
4 F DOeff2

− M O2

(6)

j(X ) 1
4 F hm

(7)
43

Transport in Flow Structure:
Convective Transport
Couple y direction O2 mass transport in the diffusion layer to the x direction O2
mass transport in the flow channel by considering the overall flux balance in
the control volume (dotted box)

u in H C ρ O2

x = 0 , y = channel

− u in H C ρ O2

X

x = X , y = channel

^

= ∫ J O2
0

conv

dx (8)
y=E

O2 leaving out of the top of the control volume can be related to the current
density produced by fuel cell.

^

X

J O2

0

conv

y=E

dx = ∫ M O2
X

0

j (x )
dx
4F

(9)

Combining equations 6, 7, 8 & 9
ρO

2

x = X , y =C

= ρ O2

x = X , y = channel

M O2  j ( X ) H E j ( X ) X j (x )

+
+∫
dx  (10)
eff

0
4 F  hm
u in H C 
DO2

44

22

Transport in Flow Structure:
Convective Transport
Assume current density is constant along the x direction
ρO

2

= ρ O2

x = X , y =C

x = X , y = channel

− M O2

H
j  1
X
+ E +
4 F  hm DOeff2 u in H C

(11)

hm can be determined based on constant-flux Sherwood number
hm =

ShF DO2
(12)

HC

Final expression for oxygen concentration profile along the catalyst layer
ρO

2

x = X , y =C

= ρ O2

x = X , y = channel

− M O2

H
j  H C
X
+ E +
4 F  ShF DO2 DOeff2 u in H C

(13)

Linear profile

45

Transport in Flow Structure:
Convective Transport
Three terms that affect O2 concentration profile at the
reaction site for fuel cell are
1.Inlet flow velocity, uin

Supplying more O2 improves mass transport, thus
increasing O2 concentration at the catalyst layer

2.Diffusion layer thickness, HE

Decreasing diffusion layer thickness also increases the
O2 concentration at the catalyst layer.

3.Channel size, HC

A little tricky as HC appears in both numerator of the
first term and denominator of third term in the
parentheses. However, with constant volume flow rate,
uinHC is constant. Therefore, decreasing channel size
will increase the O2 concentration.
46

23

Transport in Flow Structure:
Convective Transport

Flow Structure Pattern
Flow plate typically contain dozen or even
hundreds of fine channels (or groves) to
homogenously distribute gas flow over the fuel
cell surface. The shape, size and pattern of
flow channels can significantly affect fuel cell
performance.
In PEMFCs, flow field design effort often focus
on the water removal capability of cathode
side.

47

1.

Transport in Flow Structure:
Convective Transport
Three basic flow structure patterns are
Outlet

Parallel flow

2.

Serpentine flow

3.

Low overall pressure drop between gas inlet and
outlet
However, when the width of the flow field is relatively
large, flow distribution in each channel may not be
uniform

Outlet

Excellent water removal capability, as only one flow
path exists in the pattern and liquid water is forced to
exit the channel
However, in large area cell, serpentine design leads
to large pressure drop

Interdigitated flow

Inlet

Promotes forced convection of the reactant gases
through the gas diffusion layer.
Far better water management, leading to improved
mass transport
Significant pressure drop, but possible to be
overcome by employing extremely small rib spacing.

Inlet
Outlet

Inlet

48

24

Analytical Modelling with MATLAB
• Motivation
• Shorter turnaround time
• Accessibility
• Working with basics: enhancing understanding
Issues:
• Oversimplification
• Disregard for physical factors
• Incapable of complex reality simulations
Way out:
• Compare results with numerical methods
49

Analytical Modelling with MATLAB
• Working directly with equations :
parametric study
• Basics:

Conservation of Momentum
Conservation of Mass
Conservation of Energy
Conservation of Species
Conservation of Charge
50

25

Modelling different components
• Cathode gas flow field
– Cathode: rate determining reaction
– Falls first in the flow of events taking place in
the fuel cell
– Can define the inlet conditions and assess the
resulting effects without considering other
components
– Single channel flow field for simplicity
51

Cathode Gas flow channel

(b)

Charge Conductor
(a)

Gas Diffusion Layer

(c)

Flow Channel
Figure 6 (a): PEM Fuel Cell Structure; (b): GDL and Current Conductor assembly;
(c): Flow Channel dimensions – w=width, d=depth, L=length

52

26

Limiting current density
• theoretical maximum value of the current density
• calculated by assuming zero concentration of oxygen at
the GDL/cathode catalyst layer interface
i L = nFhm [

(C m ,in − C m ,out )
]
C m ,in
ln(
)
C m ,out

hm = Sh

Di , j
Dh

C m,out
C m ,in

= exp

− hm L
vm d

Dh = [

4⋅d ⋅w
]
2( d + w)

53

Assumptions

The fuel cell system is assumed to be operating at a constant temperature
Fuel cell operating pressure is assumed to be constant
Velocity in the gas flow channel is assumed to be constant
The cell is in steady state operation
The current density is assumed to be constant on the electrode surface
Chamber gases are assumed to be of uniform composition
Ideal gas properties are observed by all gases
Incompressible and laminar flow in flow channel
Isotropic and homogeneous membrane and electrode structures
Negligible ohmic potential drop in solid components
The mass and energy transport is modeled from macro-perspective using volumeaveraged conservation equations

Any model is only as good as the assumptions it is based upon are valid – Franco Barbir

54

27

(Analytical) Procedure
• Eqn in question: i L = nFhm [

(C m ,in − C m ,out )
]
C m ,in
ln(
)
C m ,out

• Two cases: Square and Rectangular cross section
• Range: L – 10cm-30cm; w – 1mm-5mm; d – 1mm-5mm
• ‘for’ loop and data point for iL for every 10μm of channel
depth/width, and for every 100μmm of channel length
for i=100:500
d(i) = i/100000;

*Refer

for i=100:500
w(i) = i/100000;

for i=1000:3000
channel_length(i) = i/10000;

to Appendix for MATLAB code
55

Square cross section

56

28

Square cross section
10
9.5

85

9

80

8.5

75

8
70
7.5
65
7
60

6.5

55

6

50

5.5

45
0

0.05

0.1

0.15

0.2

0.25

0.3

5
0.35

Channel Length (L) / m

Maximum Limiting Current Density
(iLmax) / Acm-2

Channel Lenght/Channel Depth (L/d)

90

L/d
(iLmax)

57

Rectangular cross section: iL v. d
w=1mm

w=3mm

w=5mm

58

29

14

80

13

70

12

60

11
50
10
40
9
30
8
20

7

10

6
5

0
0

0.05

0.1

0.15

0.2

0.25

0.3

Channel Lenght/Channel Depth (L/d)

Maximum Limiting Current Density
(iLmax) / Acm-2

Rectangular cross section

iLmax (w=3mm)
iLmax (w=5mm)
L/d (w=3mm)
L/d (w=5mm)

Channel Length (L)/ m

59

Rectangular cross section: iL v. w
d=1.5mm

d=3mm

d=5mm

60

30

Rectangular cross section: iL v. L
d=1 mm

d=3mm

d=5mm

61

Analytical Approach - Discussion

Square cross section: There exist optimum channel depths (widths)

Rectangular cross section:
o channel length

iL

o channel width

iL

for lower depths (d< ~ 1-2.5mm depending upon L),

channel width

iL

for higher channel depths

o channel depth

iL

for a very narrow channel,

There exist optimum channel depths for channels wider than ~ 1.5mm

62

31

Closing Remarks
• Mass transport is critical for enhanced fuel
cell performance – solution could be
obtained through simple analytical or more
complicated computational fuel cell
dynamics simulation
• All fuel cell model necessarily incorporates
assumptions - accuracy is strongly
dependent on validity of assumptions
63

Analytical Approach - Conclusion

Establish the relation between geometric parameters and corresponding
performance of the fuel cell (in terms of current density)

Analytical and numerical studies did not coincide, mainly due to the
assumptions with which the analytical study is carried out

Numerical trend for change in values for current density v. channel depth
was similar to the analytical one – a big positive!

64

32

Acknowledgements
• Erik Birgersson, Department of Chemical
and Bio-molecular Engineering, NUS
• Rina Lum, IHPC
• Agus Pulung Sasmito, NUS
• Narissara Bussayajarn, SIMTech
• Xing Xiuqing & Wu Yanling, IHPC
• Gaurav Pundir, NUS
65

33

APPNDIX B: MATLAB CODE [SQ CS – LCD v CD]
% Square cross section, limiting current density v channel depth for diff lengths
T=25;
% Fuel Cell operating temperature in degrees
C
air_velocity = 2;
% Air velocity (m/s)
temperature = T+273;
% Temperature in K
reference_temperature = 273;
% Reference Temperature
meu = 15.89e-6;
% Kinematic viscosity (m2/s)
D_O2_N2_at_reference = 1.84e-5; % Diffusion coefficient
Sh=4.86;
% Sherwood no (uniform surface concentration)
X_O2 = 0.21;
% Mole fraction of
Oxygen
R=8.314;
% Ideal Gas Constant
P=101.325;
% Pressure (kPa)
n=4;
% Number of electrons transferred per mol of
reactant consumed
F=96487;
% Define different channel lengths
channel_length1=0.1;
channel_length2=0.11;
channel_length3=0.12;
channel_length4=0.13;
channel_length5=0.14;
channel_length6=0.15;
channel_length7=0.16;
channel_length8=0.17;
channel_length9=0.18;
channel_length10=0.19;
channel_length11=0.20;
channel_length12=0.21;
channel_length13=0.22;
channel_length14=0.23;
channel_length15=0.24;
channel_length16=0.25;
channel_length17=0.26;
channel_length18=0.27;
channel_length19=0.28;
channel_length20=0.29;
channel_length21=0.30;
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=100:500
d(i) = i/100000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k(i) = d(i);
% Calculate Reynold's number
RE(i) = air_velocity*D_k(i)/meu;
% Binary diffusivity coefficient
h_m(i) = Sh*D_o2_n2/D_k(i);
% Channel outlet O2 concentration and limiting current density for different
channel lengths
C_O2_out1(i) = C_O2_in*exp(-h_m(i)*channel_length1/(d(i)*air_velocity));

limitng_current_density1(i) = n*F*h_m(i)*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
C_O2_out2(i) = C_O2_in*exp(-h_m(i)*channel_length2/(d(i)*air_velocity));
limitng_current_density2(i) = n*F*h_m(i)*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
C_O2_out3(i) = C_O2_in*exp(-h_m(i)*channel_length3/(d(i)*air_velocity));
limitng_current_density3(i) = n*F*h_m(i)*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
C_O2_out4(i) = C_O2_in*exp(-h_m(i)*channel_length4/(d(i)*air_velocity));
limitng_current_density4(i) = n*F*h_m(i)*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
C_O2_out5(i) = C_O2_in*exp(-h_m(i)*channel_length5/(d(i)*air_velocity));
limitng_current_density5(i) = n*F*h_m(i)*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
C_O2_out6(i) = C_O2_in*exp(-h_m(i)*channel_length6/(d(i)*air_velocity));
limitng_current_density6(i) = n*F*h_m(i)*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
C_O2_out7(i) = C_O2_in*exp(-h_m(i)*channel_length7/(d(i)*air_velocity));
limitng_current_density7(i) = n*F*h_m(i)*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
C_O2_out8(i) = C_O2_in*exp(-h_m(i)*channel_length8/(d(i)*air_velocity));
limitng_current_density8(i) = n*F*h_m(i)*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
C_O2_out9(i) = C_O2_in*exp(-h_m(i)*channel_length9/(d(i)*air_velocity));
limitng_current_density9(i) = n*F*h_m(i)*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
C_O2_out10(i) = C_O2_in*exp(-h_m(i)*channel_length10/(d(i)*air_velocity));
limitng_current_density10(i) = n*F*h_m(i)*(C_O2_in C_O2_out10(i))/log(C_O2_in/C_O2_out10(i))/10000;
C_O2_out11(i) = C_O2_in*exp(-h_m(i)*channel_length11/(d(i)*air_velocity));
limitng_current_density11(i) = n*F*h_m(i)*(C_O2_in C_O2_out11(i))/log(C_O2_in/C_O2_out11(i))/10000;
C_O2_out12(i) = C_O2_in*exp(-h_m(i)*channel_length12/(d(i)*air_velocity));
limitng_current_density12(i) = n*F*h_m(i)*(C_O2_in C_O2_out12(i))/log(C_O2_in/C_O2_out12(i))/10000;
C_O2_out13(i) = C_O2_in*exp(-h_m(i)*channel_length13/(d(i)*air_velocity));
limitng_current_density13(i) = n*F*h_m(i)*(C_O2_in C_O2_out13(i))/log(C_O2_in/C_O2_out13(i))/10000;
C_O2_out14(i) = C_O2_in*exp(-h_m(i)*channel_length14/(d(i)*air_velocity));
limitng_current_density14(i) = n*F*h_m(i)*(C_O2_in C_O2_out14(i))/log(C_O2_in/C_O2_out14(i))/10000;
C_O2_out15(i) = C_O2_in*exp(-h_m(i)*channel_length15/(d(i)*air_velocity));
limitng_current_density15(i) = n*F*h_m(i)*(C_O2_in C_O2_out15(i))/log(C_O2_in/C_O2_out15(i))/10000;
C_O2_out16(i) = C_O2_in*exp(-h_m(i)*channel_length16/(d(i)*air_velocity));
limitng_current_density16(i) = n*F*h_m(i)*(C_O2_in C_O2_out16(i))/log(C_O2_in/C_O2_out16(i))/10000;

C_O2_out17(i) = C_O2_in*exp(-h_m(i)*channel_length17/(d(i)*air_velocity));
limitng_current_density17(i) = n*F*h_m(i)*(C_O2_in C_O2_out17(i))/log(C_O2_in/C_O2_out17(i))/10000;
C_O2_out18(i) = C_O2_in*exp(-h_m(i)*channel_length18/(d(i)*air_velocity));
limitng_current_density18(i) = n*F*h_m(i)*(C_O2_in C_O2_out18(i))/log(C_O2_in/C_O2_out18(i))/10000;
C_O2_out19(i) = C_O2_in*exp(-h_m(i)*channel_length19/(d(i)*air_velocity));
limitng_current_density19(i) = n*F*h_m(i)*(C_O2_in C_O2_out19(i))/log(C_O2_in/C_O2_out19(i))/10000;
C_O2_out20(i) = C_O2_in*exp(-h_m(i)*channel_length20/(d(i)*air_velocity));
limitng_current_density20(i) = n*F*h_m(i)*(C_O2_in C_O2_out20(i))/log(C_O2_in/C_O2_out20(i))/10000;
C_O2_out21(i) = C_O2_in*exp(-h_m(i)*channel_length21/(d(i)*air_velocity));
limitng_current_density21(i) = n*F*h_m(i)*(C_O2_in C_O2_out21(i))/log(C_O2_in/C_O2_out21(i))/10000;
end
% Plot limiting current density v channel depth
plot(d(100:500), limitng_current_density1(100:500), d(100:500),
limitng_current_density2(100:500), d(100:500), limitng_current_density3(100:500),
d(100:500), limitng_current_density4(100:500), d(100:500),
limitng_current_density5(100:500), d(100:500), limitng_current_density6(100:500),
d(100:500), limitng_current_density7(100:500), d(100:500),
limitng_current_density8(100:500), ':', d(100:500),
limitng_current_density9(100:500), ':', d(100:500),
limitng_current_density10(100:500), ':', d(100:500),
limitng_current_density11(100:500), ':', d(100:500),
limitng_current_density12(100:500), ':', d(100:500),
limitng_current_density13(100:500), ':', d(100:500),
limitng_current_density14(100:500), ':', d(100:500),
limitng_current_density15(100:500), '--', d(100:500),
limitng_current_density16(100:500), '--', d(100:500),
limitng_current_density17(100:500), '--', d(100:500),
limitng_current_density18(100:500), '--', d(100:500),
limitng_current_density19(100:500), '--', d(100:500),
limitng_current_density20(100:500), '--', d(100:500),
limitng_current_density21(100:500), '--')

APPNDIX C: MATLAB CODE [RCT CS – LCD v CD (W = ?mm)]
% Rectangular x-section, limiting current density v channel depth for diff lengths
% Replace highlighted ? with relevant channel width (in mm) before running the
code
T=25;
degrees C
air_velocity = 2;
temperature = T+273;
reference_temperature = 273;

% Fuel Cell operating temperature in
% Air velocity (m/s)
% Temperature in K
% Reference Temperature

meu = 15.89e-6;
D_O2_N2_at_reference = 1.84e-5;
Sh=4.86;
(uniform surface concentration)
X_O2 = 0.21;
Oxygen
R=8.314;
P=101.325;
w=?e-3;
n=4;
F=96487;

% Kinematic viscosity (m2/s)
% Diffusion coefficient
% Sherwood number
% Mole fraction of
% Ideal Gas Constant
% Pressure (kPa)
% Channel width (m)
% Number of electrons transferred per mol of
reactant consumed

% Define different channel lengths
channel_length1=0.1;
channel_length2=0.11;
channel_length3=0.12;
channel_length4=0.13;
channel_length5=0.14;
channel_length6=0.15;
channel_length7=0.16;
channel_length8=0.17;
channel_length9=0.18;
channel_length10=0.19;
channel_length11=0.20;
channel_length12=0.21;
channel_length13=0.22;
channel_length14=0.23;
channel_length15=0.24;
channel_length16=0.25;
channel_length17=0.26;
channel_length18=0.27;
channel_length19=0.28;
channel_length20=0.29;
channel_length21=0.30;
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=100:500
d(i) = i/100000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k(i) = 2*w*d(i)/(w+d(i));
% Calculate Reynold's number
RE(i) = air_velocity*D_k(i)/meu;
% Binary diffusivity coefficient
h_m(i) = Sh*D_o2_n2/D_k(i);
% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m(i)*channel_length1/(d(i)*air_velocity));
limitng_current_density1(i) = n*F*h_m(i)*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
C_O2_out2(i) = C_O2_in*exp(-h_m(i)*channel_length2/(d(i)*air_velocity));
limitng_current_density2(i) = n*F*h_m(i)*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;

C_O2_out3(i) = C_O2_in*exp(-h_m(i)*channel_length3/(d(i)*air_velocity));
limitng_current_density3(i) = n*F*h_m(i)*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
C_O2_out4(i) = C_O2_in*exp(-h_m(i)*channel_length4/(d(i)*air_velocity));
limitng_current_density4(i) = n*F*h_m(i)*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
C_O2_out5(i) = C_O2_in*exp(-h_m(i)*channel_length5/(d(i)*air_velocity));
limitng_current_density5(i) = n*F*h_m(i)*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
C_O2_out6(i) = C_O2_in*exp(-h_m(i)*channel_length6/(d(i)*air_velocity));
limitng_current_density6(i) = n*F*h_m(i)*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
C_O2_out7(i) = C_O2_in*exp(-h_m(i)*channel_length7/(d(i)*air_velocity));
limitng_current_density7(i) = n*F*h_m(i)*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
C_O2_out8(i) = C_O2_in*exp(-h_m(i)*channel_length8/(d(i)*air_velocity));
limitng_current_density8(i) = n*F*h_m(i)*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
C_O2_out9(i) = C_O2_in*exp(-h_m(i)*channel_length9/(d(i)*air_velocity));
limitng_current_density9(i) = n*F*h_m(i)*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
C_O2_out10(i) = C_O2_in*exp(-h_m(i)*channel_length10/(d(i)*air_velocity));
limitng_current_density10(i) = n*F*h_m(i)*(C_O2_in C_O2_out10(i))/log(C_O2_in/C_O2_out10(i))/10000;
C_O2_out11(i) = C_O2_in*exp(-h_m(i)*channel_length11/(d(i)*air_velocity));
limitng_current_density11(i) = n*F*h_m(i)*(C_O2_in C_O2_out11(i))/log(C_O2_in/C_O2_out11(i))/10000;
C_O2_out12(i) = C_O2_in*exp(-h_m(i)*channel_length12/(d(i)*air_velocity));
limitng_current_density12(i) = n*F*h_m(i)*(C_O2_in C_O2_out12(i))/log(C_O2_in/C_O2_out12(i))/10000;
C_O2_out13(i) = C_O2_in*exp(-h_m(i)*channel_length13/(d(i)*air_velocity));
limitng_current_density13(i) = n*F*h_m(i)*(C_O2_in C_O2_out13(i))/log(C_O2_in/C_O2_out13(i))/10000;
C_O2_out14(i) = C_O2_in*exp(-h_m(i)*channel_length14/(d(i)*air_velocity));
limitng_current_density14(i) = n*F*h_m(i)*(C_O2_in C_O2_out14(i))/log(C_O2_in/C_O2_out14(i))/10000;
C_O2_out15(i) = C_O2_in*exp(-h_m(i)*channel_length15/(d(i)*air_velocity));
limitng_current_density15(i) = n*F*h_m(i)*(C_O2_in C_O2_out15(i))/log(C_O2_in/C_O2_out15(i))/10000;
C_O2_out16(i) = C_O2_in*exp(-h_m(i)*channel_length16/(d(i)*air_velocity));
limitng_current_density16(i) = n*F*h_m(i)*(C_O2_in C_O2_out16(i))/log(C_O2_in/C_O2_out16(i))/10000;
C_O2_out17(i) = C_O2_in*exp(-h_m(i)*channel_length17/(d(i)*air_velocity));
limitng_current_density17(i) = n*F*h_m(i)*(C_O2_in C_O2_out17(i))/log(C_O2_in/C_O2_out17(i))/10000;
C_O2_out18(i) = C_O2_in*exp(-h_m(i)*channel_length18/(d(i)*air_velocity));

limitng_current_density18(i) = n*F*h_m(i)*(C_O2_in C_O2_out18(i))/log(C_O2_in/C_O2_out18(i))/10000;
C_O2_out19(i) = C_O2_in*exp(-h_m(i)*channel_length19/(d(i)*air_velocity));
limitng_current_density19(i) = n*F*h_m(i)*(C_O2_in C_O2_out19(i))/log(C_O2_in/C_O2_out19(i))/10000;
C_O2_out20(i) = C_O2_in*exp(-h_m(i)*channel_length20/(d(i)*air_velocity));
limitng_current_density20(i) = n*F*h_m(i)*(C_O2_in C_O2_out20(i))/log(C_O2_in/C_O2_out20(i))/10000;
C_O2_out21(i) = C_O2_in*exp(-h_m(i)*channel_length21/(d(i)*air_velocity));
limitng_current_density21(i) = n*F*h_m(i)*(C_O2_in C_O2_out21(i))/log(C_O2_in/C_O2_out21(i))/10000;
end
% Plot limiting current density v channel depth
plot(d(100:500), limitng_current_density1(100:500), d(100:500),
limitng_current_density2(100:500), d(100:500), limitng_current_density3(100:500),
d(100:500), limitng_current_density4(100:500), d(100:500),
limitng_current_density5(100:500), d(100:500), limitng_current_density6(100:500),
d(100:500), limitng_current_density7(100:500), d(100:500),
limitng_current_density8(100:500), ':', d(100:500),
limitng_current_density9(100:500), ':', d(100:500),
limitng_current_density10(100:500), ':', d(100:500),
limitng_current_density11(100:500), ':', d(100:500),
limitng_current_density12(100:500), ':', d(100:500),
limitng_current_density13(100:500), ':', d(100:500),
limitng_current_density14(100:500), ':', d(100:500),
limitng_current_density15(100:500), '--', d(100:500),
limitng_current_density16(100:500), '--', d(100:500),
limitng_current_density17(100:500), '--', d(100:500),
limitng_current_density18(100:500), '--', d(100:500),
limitng_current_density19(100:500), '--', d(100:500),
limitng_current_density20(100:500), '--', d(100:500),
limitng_current_density21(100:500), '--')

APPNDIX D: MATLAB CODE [RCT CS – LCD v CD (L = ?m)]
%Rectangular x-section, limiting current density v channel depth for diff widths
% Replace highlighted ? with relevant channel length (in m) before running the
code
T=25;
% Fuel Cell operating
temperature in degrees C
air_velocity = 2;
% Air velocity (m/s)
temperature = T+273;
% Temperature in K
reference_temperature = 273;
% Reference Temperature
meu = 15.89e-6;
% Kinematic viscosity (m2/s)
D_O2_N2_at_reference = 1.84e-5; % Diffusion coefficient
Sh=4.86;
% Sherwood number
(uniform surface concentration)
X_O2 = 0.21;
% Mole fraction of
Oxygen
R=8.314;
% Ideal Gas Constant
P=101.325;
% Pressure (kPa)

n=4;
F=96487;
channel_length=?;

% Number of electrons transferred per mol of
reactant consumed
% Channel Length (m)

% Define different channel widths
w1=1e-3;
w2=1.5e-3;
w3=2e-3;
w4=2.5e-3;
w5=3e-3;
w6=3.5e-3;
w7=4e-3;
w8=4.5e-3;
w9=5e-3
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=100:500
d(i) = i/100000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k1(i) = 2*w1*d(i)/(w1+d(i));
% Calculate Reynold's number
RE1(i) = air_velocity*D_k1(i)/meu;
% Binary diffusivity coefficient
h_m1(i) = Sh*D_o2_n2/D_k1(i);
% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m1(i)*channel_length/(d(i)*air_velocity));
limitng_current_density1(i) = n*F*h_m1(i)*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
D_k2(i) = 2*w2*d(i)/(w2+d(i));
RE2(i) = air_velocity*D_k2(i)/meu;
h_m2(i) = Sh*D_o2_n2/D_k2(i);
C_O2_out2(i) = C_O2_in*exp(-h_m2(i)*channel_length/(d(i)*air_velocity));
limitng_current_density2(i) = n*F*h_m2(i)*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
D_k3(i) = 2*w3*d(i)/(w3+d(i));
RE3(i) = air_velocity*D_k3(i)/meu;
h_m3(i) = Sh*D_o2_n2/D_k3(i);
C_O2_out3(i) = C_O2_in*exp(-h_m3(i)*channel_length/(d(i)*air_velocity));
limitng_current_density3(i) = n*F*h_m3(i)*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
D_k4(i) = 2*w4*d(i)/(w4+d(i));
RE4(i) = air_velocity*D_k4(i)/meu;
h_m4(i) = Sh*D_o2_n2/D_k4(i);
C_O2_out4(i) = C_O2_in*exp(-h_m4(i)*channel_length/(d(i)*air_velocity));
limitng_current_density4(i) = n*F*h_m4(i)*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
D_k5(i) = 2*w5*d(i)/(w5+d(i));
RE5(i) = air_velocity*D_k5(i)/meu;

h_m5(i) = Sh*D_o2_n2/D_k5(i);
C_O2_out5(i) = C_O2_in*exp(-h_m5(i)*channel_length/(d(i)*air_velocity));
limitng_current_density5(i) = n*F*h_m5(i)*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
D_k6(i) = 2*w6*d(i)/(w6+d(i));
RE6(i) = air_velocity*D_k6(i)/meu;
h_m6(i) = Sh*D_o2_n2/D_k6(i);
C_O2_out6(i) = C_O2_in*exp(-h_m6(i)*channel_length/(d(i)*air_velocity));
limitng_current_density6(i) = n*F*h_m6(i)*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
D_k7(i) = 2*w7*d(i)/(w7+d(i));
RE7(i) = air_velocity*D_k7(i)/meu;
h_m7(i) = Sh*D_o2_n2/D_k7(i);
C_O2_out7(i) = C_O2_in*exp(-h_m7(i)*channel_length/(d(i)*air_velocity));
limitng_current_density7(i) = n*F*h_m7(i)*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
D_k8(i) = 2*w8*d(i)/(w8+d(i));
RE8(i) = air_velocity*D_k8(i)/meu;
h_m8(i) = Sh*D_o2_n2/D_k8(i);
C_O2_out8(i) = C_O2_in*exp(-h_m8(i)*channel_length/(d(i)*air_velocity));
limitng_current_density8(i) = n*F*h_m8(i)*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
D_k9(i) = 2*w9*d(i)/(w9+d(i));
RE9(i) = air_velocity*D_k9(i)/meu;
h_m9(i) = Sh*D_o2_n2/D_k9(i);
C_O2_out9(i) = C_O2_in*exp(-h_m9(i)*channel_length/(d(i)*air_velocity));
limitng_current_density9(i) = n*F*h_m9(i)*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
end
% Plot limiting current density v channel depth
plot(d(100:500), limitng_current_density1(100:500), d(100:500),
limitng_current_density2(100:500), d(100:500), limitng_current_density3(100:500),
d(100:500), limitng_current_density4(100:500), d(100:500),
limitng_current_density5(100:500), d(100:500), limitng_current_density6(100:500),
d(100:500), limitng_current_density7(100:500), d(100:500),
limitng_current_density8(100:500), '--', d(100:500),
limitng_current_density9(100:500), '--')

APPNDIX E: MATLAB CODE [RCT CS – LCD v CW (D = ?mm)]
%Rectangular x-section, limiting current density v channel width for diff lengths
% Replace highlighted ? with relevant channel depth (in mm) before running the
code
T=25;
% Fuel Cell operating temperature in degrees
C
air_velocity = 2;
% Air velocity (m/s)
temperature = T+273;
% Temperature in K
reference_temperature = 273;
% Reference Temperature
meu = 15.89e-6;
% Kinematic viscosity (m2/s)
D_O2_N2_at_reference = 1.84e-5; % Diffusion coefficient
Sh=4.86;
% Sherwood number
(uniform surface concentration)
X_O2 = 0.21;
% Mole fraction of
Oxygen
R=8.314;
% Ideal Gas Constant
P=101.325;
% Pressure (kPa)
d=?e-3;
% Channel depth (m)
n=4;
% Number of electrons transferred per mol of
reactant consumed
F=96487;
% Define different channel lengths
channel_length1=0.1;
channel_length2=0.11;
channel_length3=0.12;
channel_length4=0.13;

channel_length5=0.14;
channel_length6=0.15;
channel_length7=0.16;
channel_length8=0.17;
channel_length9=0.18;
channel_length10=0.19;
channel_length11=0.20;
channel_length12=0.21;
channel_length13=0.22;
channel_length14=0.23;
channel_length15=0.24;
channel_length16=0.25;
channel_length17=0.26;
channel_length18=0.27;
channel_length19=0.28;
channel_length20=0.29;
channel_length21=0.30;
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=100:500
w(i) = i/100000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k(i) = 2*w(i)*d/(w(i)+d);
% Calculate Reynold's number
RE(i) = air_velocity*D_k(i)/meu;
% Binary diffusivity coefficient
h_m(i) = Sh*D_o2_n2/D_k(i);
% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m(i)*channel_length1/(d*air_velocity));
limitng_current_density1(i) = n*F*h_m(i)*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
C_O2_out2(i) = C_O2_in*exp(-h_m(i)*channel_length2/(d*air_velocity));
limitng_current_density2(i) = n*F*h_m(i)*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
C_O2_out3(i) = C_O2_in*exp(-h_m(i)*channel_length3/(d*air_velocity));
limitng_current_density3(i) = n*F*h_m(i)*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
C_O2_out4(i) = C_O2_in*exp(-h_m(i)*channel_length4/(d*air_velocity));
limitng_current_density4(i) = n*F*h_m(i)*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
C_O2_out5(i) = C_O2_in*exp(-h_m(i)*channel_length5/(d*air_velocity));
limitng_current_density5(i) = n*F*h_m(i)*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
C_O2_out6(i) = C_O2_in*exp(-h_m(i)*channel_length6/(d*air_velocity));
limitng_current_density6(i) = n*F*h_m(i)*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
C_O2_out7(i) = C_O2_in*exp(-h_m(i)*channel_length7/(d*air_velocity));

limitng_current_density7(i) = n*F*h_m(i)*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
C_O2_out8(i) = C_O2_in*exp(-h_m(i)*channel_length8/(d*air_velocity));
limitng_current_density8(i) = n*F*h_m(i)*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
C_O2_out9(i) = C_O2_in*exp(-h_m(i)*channel_length9/(d*air_velocity));
limitng_current_density9(i) = n*F*h_m(i)*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
C_O2_out10(i) = C_O2_in*exp(-h_m(i)*channel_length10/(d*air_velocity));
limitng_current_density10(i) = n*F*h_m(i)*(C_O2_in C_O2_out10(i))/log(C_O2_in/C_O2_out10(i))/10000;
C_O2_out11(i) = C_O2_in*exp(-h_m(i)*channel_length11/(d*air_velocity));
limitng_current_density11(i) = n*F*h_m(i)*(C_O2_in C_O2_out11(i))/log(C_O2_in/C_O2_out11(i))/10000;
C_O2_out12(i) = C_O2_in*exp(-h_m(i)*channel_length12/(d*air_velocity));
limitng_current_density12(i) = n*F*h_m(i)*(C_O2_in C_O2_out12(i))/log(C_O2_in/C_O2_out12(i))/10000;
C_O2_out13(i) = C_O2_in*exp(-h_m(i)*channel_length13/(d*air_velocity));
limitng_current_density13(i) = n*F*h_m(i)*(C_O2_in C_O2_out13(i))/log(C_O2_in/C_O2_out13(i))/10000;
C_O2_out14(i) = C_O2_in*exp(-h_m(i)*channel_length14/(d*air_velocity));
limitng_current_density14(i) = n*F*h_m(i)*(C_O2_in C_O2_out14(i))/log(C_O2_in/C_O2_out14(i))/10000;
C_O2_out15(i) = C_O2_in*exp(-h_m(i)*channel_length15/(d*air_velocity));
limitng_current_density15(i) = n*F*h_m(i)*(C_O2_in C_O2_out15(i))/log(C_O2_in/C_O2_out15(i))/10000;
C_O2_out16(i) = C_O2_in*exp(-h_m(i)*channel_length16/(d*air_velocity));
limitng_current_density16(i) = n*F*h_m(i)*(C_O2_in C_O2_out16(i))/log(C_O2_in/C_O2_out16(i))/10000;
C_O2_out17(i) = C_O2_in*exp(-h_m(i)*channel_length17/(d*air_velocity));
limitng_current_density17(i) = n*F*h_m(i)*(C_O2_in C_O2_out17(i))/log(C_O2_in/C_O2_out17(i))/10000;
C_O2_out18(i) = C_O2_in*exp(-h_m(i)*channel_length18/(d*air_velocity));
limitng_current_density18(i) = n*F*h_m(i)*(C_O2_in C_O2_out18(i))/log(C_O2_in/C_O2_out18(i))/10000;
C_O2_out19(i) = C_O2_in*exp(-h_m(i)*channel_length19/(d*air_velocity));
limitng_current_density19(i) = n*F*h_m(i)*(C_O2_in C_O2_out19(i))/log(C_O2_in/C_O2_out19(i))/10000;
C_O2_out20(i) = C_O2_in*exp(-h_m(i)*channel_length20/(d*air_velocity));
limitng_current_density20(i) = n*F*h_m(i)*(C_O2_in C_O2_out20(i))/log(C_O2_in/C_O2_out20(i))/10000;
C_O2_out21(i) = C_O2_in*exp(-h_m(i)*channel_length21/(d*air_velocity));
limitng_current_density21(i) = n*F*h_m(i)*(C_O2_in C_O2_out21(i))/log(C_O2_in/C_O2_out21(i))/10000;
end
% Plot limiting current density v channel width

plot(w(100:500), limitng_current_density1(100:500), w(100:500),
limitng_current_density2(100:500), w(100:500), limitng_current_density3(100:500),
w(100:500), limitng_current_density4(100:500), w(100:500),
limitng_current_density5(100:500), w(100:500), limitng_current_density6(100:500),
w(100:500), limitng_current_density7(100:500), w(100:500),
limitng_current_density8(100:500), ':', w(100:500),
limitng_current_density9(100:500), ':', w(100:500),
limitng_current_density10(100:500), ':', w(100:500),
limitng_current_density11(100:500), ':', w(100:500),
limitng_current_density12(100:500), ':', w(100:500),
limitng_current_density13(100:500), ':', w(100:500),
limitng_current_density14(100:500), ':', w(100:500),
limitng_current_density15(100:500), '--', w(100:500),
limitng_current_density16(100:500), '--', w(100:500),
limitng_current_density17(100:500), '--', w(100:500),
limitng_current_density18(100:500), '--', w(100:500),
limitng_current_density19(100:500), '--', w(100:500),
limitng_current_density20(100:500), '--', w(100:500),
limitng_current_density21(100:500), '--')

APPNDIX F: MATLAB CODE [RCT CS – LCD v CW (L = ?m)]
%Rectangular x-section, limiting current density v channel width for diff depths
% Replace highlighted ? with relevant channel length (in m) before running the
code
T=25;
C
air_velocity = 2;
temperature = T+273;
reference_temperature = 273;
meu = 15.89e-6;
D_O2_N2_at_reference = 1.84e-5;
Sh=4.86;
(uniform surface concentration)
X_O2 = 0.21;
Oxygen
R=8.314;
P=101.325;
n=4;
F=96487;
channel_length=?;

% Fuel Cell operating temperature in degrees
% Air velocity (m/s)
% Temperature in K
% Reference Temperature
% Kinematic viscosity (m2/s)
% Diffusion coefficient
% Sherwood number
% Mole fraction of
% Ideal Gas Constant
% Pressure (kPa)
% Number of electrons transferred per mol of
reactant consumed
% Channel Length (m)

% Define different channel depths
d1=1e-3;
d2=1.5e-3;
d3=2e-3;
d4=2.5e-3;
d5=3e-3;
d6=3.5e-3;
d7=4e-3;
d8=4.5e-3;
d9=5e-3
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;

% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=100:500
w(i) = i/100000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k1(i) = 2*w(i)*d1/(w(i)+d1);
% Calculate Reynold's number
RE1(i) = air_velocity*D_k1(i)/meu;
% Binary diffusivity coefficient
h_m1(i) = Sh*D_o2_n2/D_k1(i);
% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m1(i)*channel_length/(d1*air_velocity));
limitng_current_density1(i) = n*F*h_m1(i)*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
D_k2(i) = 2*w(i)*d2/(w(i)+d2);
RE2(i) = air_velocity*D_k2(i)/meu;
h_m2(i) = Sh*D_o2_n2/D_k2(i);
C_O2_out2(i) = C_O2_in*exp(-h_m2(i)*channel_length/(d2*air_velocity));
limitng_current_density2(i) = n*F*h_m2(i)*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
D_k3(i) = 2*w(i)*d3/(w(i)+d3);
RE1(i) = air_velocity*D_k3(i)/meu;
h_m3(i) = Sh*D_o2_n2/D_k3(i);
C_O2_out3(i) = C_O2_in*exp(-h_m3(i)*channel_length/(d3*air_velocity));
limitng_current_density3(i) = n*F*h_m3(i)*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
D_k4(i) = 2*w(i)*d4/(w(i)+d4);
RE4(i) = air_velocity*D_k4(i)/meu;
h_m4(i) = Sh*D_o2_n2/D_k4(i);
C_O2_out4(i) = C_O2_in*exp(-h_m4(i)*channel_length/(d4*air_velocity));
limitng_current_density4(i) = n*F*h_m4(i)*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
D_k5(i) = 2*w(i)*d5/(w(i)+d5);
RE5(i) = air_velocity*D_k5(i)/meu;
h_m5(i) = Sh*D_o2_n2/D_k5(i);
C_O2_out5(i) = C_O2_in*exp(-h_m5(i)*channel_length/(d5*air_velocity));
limitng_current_density5(i) = n*F*h_m5(i)*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
D_k6(i) = 2*w(i)*d6/(w(i)+d6);
RE6(i) = air_velocity*D_k6(i)/meu;
h_m6(i) = Sh*D_o2_n2/D_k6(i);
C_O2_out6(i) = C_O2_in*exp(-h_m6(i)*channel_length/(d6*air_velocity));
limitng_current_density6(i) = n*F*h_m6(i)*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
D_k7(i) = 2*w(i)*d7/(w(i)+d7);
RE7(i) = air_velocity*D_k7(i)/meu;
h_m7(i) = Sh*D_o2_n2/D_k7(i);
C_O2_out7(i) = C_O2_in*exp(-h_m7(i)*channel_length/(d7*air_velocity));
limitng_current_density7(i) = n*F*h_m7(i)*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;

D_k8(i) = 2*w(i)*d8/(w(i)+d8);
RE8(i) = air_velocity*D_k8(i)/meu;
h_m8(i) = Sh*D_o2_n2/D_k8(i);
C_O2_out8(i) = C_O2_in*exp(-h_m8(i)*channel_length/(d8*air_velocity));
limitng_current_density8(i) = n*F*h_m8(i)*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
D_k9(i) = 2*w(i)*d9/(w(i)+d9);
RE9(i) = air_velocity*D_k9(i)/meu;
h_m9(i) = Sh*D_o2_n2/D_k9(i);
C_O2_out9(i) = C_O2_in*exp(-h_m9(i)*channel_length/(d9*air_velocity));
limitng_current_density9(i) = n*F*h_m9(i)*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
end
% Plot limiting current density v channel width
plot(w(100:500), limitng_current_density1(100:500), w(100:500),
limitng_current_density2(100:500), w(100:500), limitng_current_density3(100:500),
w(100:500), limitng_current_density4(100:500), w(100:500),
limitng_current_density5(100:500), w(100:500), limitng_current_density6(100:500),
w(100:500), limitng_current_density7(100:500), w(100:500),
limitng_current_density8(100:500), '--', w(100:500),
limitng_current_density9(100:500), '--')

APPNDIX G: MATLAB CODE [RCT CS – LCD v CL (d = ?m)]
%Rectangular x-section, limiting current density v channel length for diff widths
% Replace highlighted ? with relevant channel depth (in mm) before running the
code

T=25;
% Fuel Cell operating temperature in
degrees C
air_velocity = 2;
% Air velocity (m/s)
temperature = T+273;
% Temperature in K
reference_temperature = 273;
% Reference Temperature
meu = 15.89e-6;
% Kinematic viscosity (m2/s)
D_O2_N2_at_reference = 1.84e-5; % Diffusion coefficient
Sh=4.86;
% Sherwood number
(uniform surface concentration)
X_O2 = 0.21;
% Mole fraction of
Oxygen
R=8.314;
% Ideal Gas Constant
P=101.325;
% Pressure (kPa)
d=?e-3;
% Channel width (m)
n=4;
% Number of electrons transferred per mol of
reactant consumed
F=96487;
% Define different channel width
w1=1e-3;
w2=1.5e-3;
w3=2e-3;
w4=2.5e-3;
w5=3e-3;
w6=3.5e-3;
w7=4e-3;
w8=4.5e-3;
w9=5e-3;
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=1000:3000
channel_length(i) = i/10000;
% Hydraullic diameter equal to 2LW/(L+W)

D_k1 = 2*w1*d/(w1+d);
% Calculate Reynold's number
RE1 = air_velocity*D_k1/meu;
% Binary diffusivity coefficient
h_m1 = Sh*D_o2_n2/D_k1;
% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m1*channel_length(i)/(d*air_velocity));
limitng_current_density1(i) = n*F*h_m1*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
D_k2 = 2*w2*d/(w2+d);
RE2 = air_velocity*D_k2/meu;
h_m2 = Sh*D_o2_n2/D_k2;
C_O2_out2(i) = C_O2_in*exp(-h_m2*channel_length(i)/(d*air_velocity));
limitng_current_density2(i) = n*F*h_m2*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
D_k3 = 2*w3*d/(w3+d);
RE3 = air_velocity*D_k3/meu;
h_m3 = Sh*D_o2_n2/D_k3;
C_O2_out3(i) = C_O2_in*exp(-h_m3*channel_length(i)/(d*air_velocity));
limitng_current_density3(i) = n*F*h_m3*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
D_k4 = 2*w4*d/(w4+d);
RE4 = air_velocity*D_k4/meu;
h_m4 = Sh*D_o2_n2/D_k4;
C_O2_out4(i) = C_O2_in*exp(-h_m4*channel_length(i)/(d*air_velocity));
limitng_current_density4(i) = n*F*h_m4*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
D_k5 = 2*w5*d/(w5+d);
RE5 = air_velocity*D_k5/meu;
h_m5 = Sh*D_o2_n2/D_k5;
C_O2_out5(i) = C_O2_in*exp(-h_m5*channel_length(i)/(d*air_velocity));
limitng_current_density5(i) = n*F*h_m5*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
D_k6 = 2*w6*d/(w6+d);
RE6 = air_velocity*D_k6/meu;
h_m6 = Sh*D_o2_n2/D_k6;
C_O2_out6(i) = C_O2_in*exp(-h_m6*channel_length(i)/(d*air_velocity));
limitng_current_density6(i) = n*F*h_m6*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
D_k7 = 2*w7*d/(w7+d);
RE7 = air_velocity*D_k7/meu;
h_m7 = Sh*D_o2_n2/D_k7;
C_O2_out7(i) = C_O2_in*exp(-h_m7*channel_length(i)/(d*air_velocity));
limitng_current_density7(i) = n*F*h_m7*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
D_k8 = 2*w8*d/(w8+d);
RE8 = air_velocity*D_k8/meu;
h_m8 = Sh*D_o2_n2/D_k8;
C_O2_out8(i) = C_O2_in*exp(-h_m8*channel_length(i)/(d*air_velocity));

limitng_current_density8(i) = n*F*h_m8*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
D_k9 = 2*w9*d/(w9+d);
RE9 = air_velocity*D_k9/meu;
h_m9 = Sh*D_o2_n2/D_k9;
C_O2_out9(i) = C_O2_in*exp(-h_m9*channel_length(i)/(d*air_velocity));
limitng_current_density9(i) = n*F*h_m9*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
end
% Plot limiting current density v channel length
plot(channel_length(1000:3000), limitng_current_density1(1000:3000),
channel_length(1000:3000), limitng_current_density2(1000:3000),
channel_length(1000:3000), limitng_current_density3(1000:3000),
channel_length(1000:3000), limitng_current_density4(1000:3000),
channel_length(1000:3000), limitng_current_density5(1000:3000),
channel_length(1000:3000), limitng_current_density6(1000:3000),
channel_length(1000:3000), limitng_current_density7(1000:3000),
channel_length(1000:3000), limitng_current_density8(1000:3000), ':',
channel_length(1000:3000), limitng_current_density9(1000:3000), ':')

APPNDIX H: MATLAB CODE [RCT CS – LCD v CL (w = ?m)]
%Rectangular x-section, limiting current density v channel length for diff depths
% Replace highlighted ? with relevant channel width (in mm) before running the
code

T=25;
% Fuel Cell operating temperature in
degrees C
air_velocity = 2;
% Air velocity (m/s)
temperature = T+273;
% Temperature in K
reference_temperature = 273;
% Reference Temperature
meu = 15.89e-6;
% Kinematic viscosity (m2/s)
D_O2_N2_at_reference = 1.84e-5; % Diffusion coefficient
Sh=4.86;
% Sherwood number
(uniform surface concentration)
X_O2 = 0.21;
% Mole fraction of
Oxygen
R=8.314;
% Ideal Gas Constant
P=101.325;
% Pressure (kPa)
w=?e-3;
% Channel width (m)
n=4;
% Number of electrons transferred per mol of reactant
consumed
F=96487;
% Define different channel depth
d1=1e-3;
d2=1.5e-3;
d3=2e-3;
d4=2.5e-3;
d5=3e-3;
d6=3.5e-3;
d7=4e-3;
d8=4.5e-3;
d9=5e-3;
% Convective Mass Transfer Coefficient
D_o2_n2 = D_O2_N2_at_reference*(temperature/reference_temperature)^1.5;
% The concentration of oxygen at channel inlet
C_O2_in = 1000*X_O2*P/(R*temperature);
for i=1000:3000
channel_length(i) = i/10000;
% Hydraullic diameter equal to 2LW/(L+W)
D_k1 = 2*w*d1/(w+d1);
% Calculate Reynold's number
RE1 = air_velocity*D_k1/meu;
% Binary diffusivity coefficient
h_m1 = Sh*D_o2_n2/D_k1;

% The concentration of oxygen at channel outlet
C_O2_out1(i) = C_O2_in*exp(-h_m1*channel_length(i)/(d1*air_velocity));
limitng_current_density1(i) = n*F*h_m1*(C_O2_in C_O2_out1(i))/log(C_O2_in/C_O2_out1(i))/10000;
D_k2 = 2*w*d2/(w+d2);
RE2 = air_velocity*D_k2/meu;
h_m2 = Sh*D_o2_n2/D_k2;
C_O2_out2(i) = C_O2_in*exp(-h_m2*channel_length(i)/(d2*air_velocity));
limitng_current_density2(i) = n*F*h_m2*(C_O2_in C_O2_out2(i))/log(C_O2_in/C_O2_out2(i))/10000;
D_k3 = 2*w*d3/(w+d3);
RE3 = air_velocity*D_k3/meu;
h_m3 = Sh*D_o2_n2/D_k3;
C_O2_out3(i) = C_O2_in*exp(-h_m3*channel_length(i)/(d3*air_velocity));
limitng_current_density3(i) = n*F*h_m3*(C_O2_in C_O2_out3(i))/log(C_O2_in/C_O2_out3(i))/10000;
D_k4 = 2*w*d4/(w+d4);
RE4 = air_velocity*D_k4/meu;
h_m4 = Sh*D_o2_n2/D_k4;
C_O2_out4(i) = C_O2_in*exp(-h_m4*channel_length(i)/(d4*air_velocity));
limitng_current_density4(i) = n*F*h_m4*(C_O2_in C_O2_out4(i))/log(C_O2_in/C_O2_out4(i))/10000;
D_k5 = 2*w*d5/(w+d5);
RE5 = air_velocity*D_k5/meu;
h_m5 = Sh*D_o2_n2/D_k5;
C_O2_out5(i) = C_O2_in*exp(-h_m5*channel_length(i)/(d5*air_velocity));
limitng_current_density5(i) = n*F*h_m5*(C_O2_in C_O2_out5(i))/log(C_O2_in/C_O2_out5(i))/10000;
D_k6 = 2*w*d6/(w+d6);
RE6 = air_velocity*D_k6/meu;
h_m6 = Sh*D_o2_n2/D_k6;
C_O2_out6(i) = C_O2_in*exp(-h_m6*channel_length(i)/(d6*air_velocity));
limitng_current_density6(i) = n*F*h_m6*(C_O2_in C_O2_out6(i))/log(C_O2_in/C_O2_out6(i))/10000;
D_k7 = 2*w*d7/(w+d7);
RE7 = air_velocity*D_k7/meu;
h_m7 = Sh*D_o2_n2/D_k7;
C_O2_out7(i) = C_O2_in*exp(-h_m7*channel_length(i)/(d7*air_velocity));
limitng_current_density7(i) = n*F*h_m7*(C_O2_in C_O2_out7(i))/log(C_O2_in/C_O2_out7(i))/10000;
D_k8 = 2*w*d8/(w+d8);
RE8 = air_velocity*D_k8/meu;
h_m8 = Sh*D_o2_n2/D_k8;
C_O2_out8(i) = C_O2_in*exp(-h_m8*channel_length(i)/(d8*air_velocity));
limitng_current_density8(i) = n*F*h_m8*(C_O2_in C_O2_out8(i))/log(C_O2_in/C_O2_out8(i))/10000;
D_k9 = 2*w*d9/(w+d9);
RE9 = air_velocity*D_k9/meu;
h_m9 = Sh*D_o2_n2/D_k9;
C_O2_out9(i) = C_O2_in*exp(-h_m9*channel_length(i)/(d9*air_velocity));

limitng_current_density9(i) = n*F*h_m9*(C_O2_in C_O2_out9(i))/log(C_O2_in/C_O2_out9(i))/10000;
end
% Plot limiting current density v channel length
plot(channel_length(1000:3000), limitng_current_density1(1000:3000),
channel_length(1000:3000), limitng_current_density2(1000:3000),
channel_length(1000:3000), limitng_current_density3(1000:3000),
channel_length(1000:3000), limitng_current_density4(1000:3000),
channel_length(1000:3000), limitng_current_density5(1000:3000),
channel_length(1000:3000), limitng_current_density6(1000:3000),
channel_length(1000:3000), limitng_current_density7(1000:3000),
channel_length(1000:3000), limitng_current_density8(1000:3000), ':',
channel_length(1000:3000), limitng_current_density9(1000:3000), ':')

Heat Transfer in Fluidized
Beds-An Overview

1/13/2011

Special Topic in ME6204 Convective Heat
Transfer:
Heat Transfer in Fluidized Beds-An Overview

Professor A. S. Mujumdar
ME Department, NUS, 2011

Contents
Introduction
Particle Characterization
Flow past spheres, non-spheres
Heat transfer from spheres under large Re range
Fluid Beds – Concept and modifications
Geldart’s Classification
Heat Transfer
Bed to Immersed Surface Heat Transfer – Correlations
Sample Calculations
FB Drying – Plug flow vs Well-mixed
Fluid be combustion (Simple Model)
Example of Fluid Bed Calculation
Closing Remarks

1

1/11/2011

Introduction
• fluidized bed describes a finely granulated layer of solid material (referred to as “the
mass”) that is loosened by fluid flowing through to such an extent that the particles
of solid material are free to move to a certain degree
• It is called “fluidized” because the solid material takes on properties similar to those
of a fluid (liquid)
• Fluidized beds are used widely in engineering for applications such as combustion,
reactors, drying, gasification of coal/biomass, thermal treatments of metal and
powder coating, granulation, heat transfer etc ……
• Gas-solid fluidized systems are characterized by temperature uniformity and high
heat transfer coefficients, due to the intense mixture of the solid material by the
presence of gas bubbles
• Liquid- solid fluidized beds have also been used for wide applications
• Heat transfer in fluid beds can be from Fluid – Particle, Particle – Fluid, Wall – Fluid
and to immersed surfaces, if any- care needed in calculation method used!

2

1/11/2011

Particle Characterization
Particle Size
6Vp 1⁄3
dv = �

π

Volume Diameter (dV)
Surface Diameter (dS)
Surface-Volume Diameter (dSV)
Sieve Diameter

S p 1 ⁄2
d S =� �
π

6V p d 3v
dSv =
=
S p d 2s

18μU t
d st =�
�ρ p −ρ s �g

Stokes Diameter (dst)
Free Falling Diameter
Drag Diameter
Projected Area Diameter
Feret Diameter
Martin Diameter ….etc

Commonly used for applications in packed and fluid bed
VP – Volume of the particle; SP – Surface area of the particle;
Ut – terminal settling velocity of the particle

Particle Characterization

contd……

Shape

Equivalent particle diameter, dp = dsv = 6/as

Sphere

dp = diameter of sphere

Cylinder with length (ly) equal to
diameter (dy)

dp = dy , the diameter of cylinder

Cylinder dy ≠ ly

6dy
dp = �

4 + 2 dy ⁄ly

Ring with outside diameter of do, and
inside diameter of di
Mixed sizes

Irregular shapes with ф = 0.5 to 0.7

dp = 1.5�do − di �

dp =

1
∑i �xi ⁄dpi �

dp = ф dv ; dv ≈ dA

Table: Suggested equivalent particle diameters for catalysts in catalytic reactor application

3

1/11/2011

Particle Characterization

contd……

Particle Shape
Sphericity (ф)
Surface area of volume equivalent sphere
ф=
Surface area of particle

Circularity (⊄)

Circumference of a circle having same
cross − sectional area as the particle
⊄=
Actual Perim eter of the cross −section

Operational Sphericity and Circularity
Heywood Shape Factor
Particle Density

Vp
k= 3
da

d a =�4A p ⁄π

da – Projected area diameter; AP – Projected area; Vp – Volume of particle

Methods for Direct Characterization
Sieve Analysis
In U.S. – Taylor sieve size and U.S. Sieve size
In Europe – British standard and German DIN sieve size
Mesh Number – number of parallel wires per inch
Imaging Techniques: Direct measurements using enlarged photographic or electronic images
of microscope
Optical microscope (1µm to 150 µm)
Scanning electron microscope (SEM, 5µm to 0.01 µm)
Transmission electron microscope (TEM , 5µm to 0.01 µm)
Gravity and Centrifugal Sedimentation
Characterization by Elutriation
Resistivity and Optical Zone Sensing Techniques
Coulter Counter

4

1/11/2011

Particle size ranges for various methods
Particle sizing method

Applicable particle
size (mm)

Measured Dimension

Sieving
Dry
Wet

>10
2 – 500

Sieve diameter

Microscopic examination
Optical
Electronic

1.0 – 100
0.01 – 500

Length, Projected Area
Statistical diameter

Zone Sensing
Resistivity
Optical

0.6 – 1200
1.0 – 800

Volume

Elutriation
Laminar Flow
Cyclone

3 – 75
8 – 50

Stokes diameter

Gravity Sedimentation
Pipette and hydrometer
Photoextinction
X-ray

1 – 100
0.5 – 100
0.1 – 130

Stokes diameter

Mechanical Properties of Particles
Hardgrove Grindability Index (HGI)
Higher the HGI higher the grindability of material

Attrition Index
Important in fluidization, can affect entrainment and elutriation

Solids impaction on plate
Abrasiveness Index
factor used to determine the effective rate of wear of the
aforementioned consumables
Erosiveness Index

5

1/11/2011

Dimensionless numbers
Dimensionless numbers of interest
Significance

For FB Heat Transfer

Flow passed over a single particle
Drag force resisting very slow steady relative motion between a
rigid particle (sphere) of diameter dP and a fluid of infinite extent of
viscosity µ is composed of two components

Total drag force resisting motion

Where ur is the relative velocity
This is known as the stokes law and is valid mainly for
This also hold true up to

with some error
ReP = Ur dP ρf / μ

6

1/11/2011

Flow passed over a single particle
Particle Drag Coefficient
Ratio of force on the particle and the fluid dynamic
pressure caused by the fluid times the projected
area of the particle
F
C D =�

�1⁄2�ρ f U 2r A P

Ur

AP

F

1
F = CD ρf Ur2 AP
2

Drag Coefficient CD is a function of particle’s Reynolds number
Three different regimes based on the magnitude of ReP
For the Stokes Regime
0.3 < (Re)P
The Intermediate Regime
Dallavalle (1948)

CD = ��ReP �

0.3 < (Re)P < 500

Schiller & Naumann 1933
The Newton’s Law Regime

CD = 0.44

(Re)P > 500

Drag coefficient for different shapes

Drag Coefficient against Reynolds Number (Re)

7

1/11/2011

Particle falling under gravity through a fluid
The general force acting on particle are gravity, buoyancy, drag
Gravity

-

Buoyancy -

Drag =

Acceleration Flow

For spherical particle (for acceleration = 0)
Fb
Now

So equation becomes

Ut is called as terminal settling velocity of the particle
On solving for CD

Particle falling under gravity through a fluid
For Stokes law Regime

Hence
In stokes law regime the terminal settling velocity is proportional to the squere of the
particle diameter
In the Newton’s Law Regime

CD = 0.44

In the Newton’s law regime the terminal settling velocity is proportional to the square
root of particle diameter and independent of viscocity

8

1/11/2011

Multiple particle system
The motion of each particle is affected by the presence of others
Simple analysis for the fluid particle interaction is not valid but can be adopted to
model the multiple particle system
For suspension of particles the stokes law is assumed to apply but an effective
suspension viscosity and effective average suspension density is assumed
Effective suspension viscosity µe = µ / f(ε)
Average suspension density = ρave = ε ρf + (1-ε) ρP
where ε is the voidage or volume fraction occupied by fluid
For suspension of particles the drag coefficient in stokes law regime becomes
where
And Ur is the relative velocity of particle

Multiple particle system
Under terminal velocity condition for a particle falling under gravity in a suspension the
force balance

Substituting µe and ρave

Urel,t is known as the particle settling velocity in presence of other particles or Hindered
Settling Velocity

9

1/11/2011

Multiple particle system
f(ε) was shown theoretically by Einstein to be

For uniform sphere forming a suspension of solid column fraction less than 0.1
Or (1-ε) < 0.1,
Richardson and Zaki have given the values of f(ε)
for stokes regime to be

For Newton’s Regime

Characterization of Multiple Particle System
Different definitions of average particle diameter
Arithmetic Mean
Surface Mean
Volume Mean
Volume Surface Mean
Weight Mean
Length Mean

���� =

∑� �� ���
∑� � �

2
∑� �� ���
��� = �
∑� ��

3 ∑ � �3
� � ��
��� = �
∑� ��

���� =

3
∑� �� ���
2
∑� �� ���

��� = � �� ��� =

��� =

2
∑� �� ���

4
∑� �� ���
3
∑� �� ���

3
�� ���

Where �� = ∑
3
� �� ���

∑� �� ���

10

1/11/2011

Fluid Beds - Basics

Fluidized beds: Particle
suspended in an upward gas
stream

11

1/11/2011

Pressure drop in fixed beds
Pressure drop through packed bed / fixed bed of uniform
sized solids is correlated by Ergun equation

It has two factors, viscous force and the kinetic energy force
At low Reynolds number only viscous losses predominate

At high Reynolds number only kinetic energy losses need to be considered

Pressure drop & Minimum Fluidization Velocity

At the onset of fluidization, the gravity force on the particles in the bed must be
balanced by the drag, buoyancy, and pressure forces.

12

1/11/2011

Pressure drop & Minimum Fluidization Velocity

The umf, the superficial velocity at minimum fluidizing condition is found by using
expression for ∆p/L

In a simplified form

For small particles (ReP < 20)

Or

Pressure drop & Minimum Fluidization Velocity
for larger particles the simplified form is
or

Wen and Yu have found for variety of systems
and
for small particles the simplified form is

For larger particles (ReP > 20)
For whole range of Reynolds number

13

1/11/2011

Geldart’s Classification

A: Aeratable (Umb > Umf) Material has significant deaeration time (FCC Catalyst)
B: Bubbles Above (Umb = Umf) 500 micron sand
C: Cohesive (Flour, Fly Ash)
D: Spoutable (wheat, 2000 micron polyethylene pellets)

Geldart’s Classification

contd…

Group

Characteristics and Properties

A

• Good fluidization quality, aeratable, easily fluidized, smooth at low
velocity and bubbling at higher velocity and slug at high velocity, bed
expands, Good solids mixing
• Small mean particle size, low density, typically 30< dp <100μm and
ρ<1400 kgm-3.

B

• Good fluidization quality, sand-like particles, vigorous bubbling, slug
at high velocity, small bed expansion, good solids mixing in bubbling
• Typically 40μm<dp<500μm, 1400kgm-3<ρ<4000kgm-3

C

D

• Bad fluidization quality, cohesive due to strong interparticle force,
severe slugging and agglomeration, may generates electrostatic
charges, poor solids mixing.
• Fine and ultra-fine particles
• Poor fluidization quality, spoutable, difficult to fluidize in deep bed
depth, large bubbles, severe channeling, relatively poor solids mixing
• Large and/or dense particles, typically dp >500μm, ρ>1400kgm-3

14

1/11/2011

Geldart’s Classification

contd…

Geldart’s Classification

contd…

15

1/11/2011

Fluidization Regimes

Velocity Increasing

Fluidization Regimes: Description
Velocity range

Regime

Fluidization Features and Appearance

0 ≤ U ≤ Umf

Fixed Bed

Particles are quiescent; gas flows through interstices

Umf ≤ U ≤ Umb

Particulate
Regime

Bed expands smoothly and homogeneously with small-scale particle
motion; bed surface is well defined

Umb ≤ U ≤ Ums

Bubbling Regime

Gas bubbles form above distributor, coalesce and grow; gas bubbles
promote solids mixing during rise to surface and breakthrough

Ums ≤ U ≤ UC

Slug flow Regime

Bubble size approaches bed cross section; bed surface rises and falls
with regular frequency with corresponding pressure fluctuation

UC ≤ U ≤ Uk

Transition to
turbulent
Fluidization

Pressure fluctuation decrease gradually until turbulent fluidization
regime is reached

Uk ≤ U ≤ Utr

Turbulent Regime

Small gas voids and particle clusters and streamers dart to and fro;
bed surface is diffused and difficult to distinguish

U ≥ Utr

Fast Fluidization

Particles are transported out of the bed and need to be replaced and
recycled; normally has a dense phase region at bottom coexisting
with a dilute phase region on top; no bed surface

U » Utr

Pneumatic
Conveying

Usually a once-through operation; all particles fed are transported
out in dilute phase with concentration varying along the column
height; no bed surface

16

1/11/2011

Fluidization Regimes: Description
Transition between regimes
Equations have been published for transition lines
between various regimes. This map can be used to
identify the type of flow regime that will exist for a
given particle under specific flow conditions.
Smooth or particulate fluidization

Bubbling or aggregative fluidization

Slugging criteria

Design of distributor in fluidized beds
 Uniform gas sparging is govern by the effective design of gas distribution
system and is very important to have uniform heat transfer
 Parameter affecting performance of Gas distribution system comprise of;

Gas Sparger geometry

Gas chamber geometry

Pressure drop across the gas distribution system

 A good distributor should;

Obtain a spatially uniform gas distribution, without stagnant zones

Prevent solids loss by leakage

Minimize solid erosion

Avoid choking of the distributor

Have a definite and non-changing (with time) pressure drop for the gas

17

1/11/2011

Design of distributor in fluidized beds
Pressure Drop across the Distributor
The pressure drop across the distributor ΔPd is used as the criterion for
design, and ΔPd, values varying from 0.01 to 1.0 times the pressure drop
across the bed ΔPb have been suggested.
Siegel (1986):
ΔPd/ΔPb = 0.14 - 0.22 [Galileo number (1 - 10,000)]
Kunii & Levenspiel (1991):
ΔPd/ΔPb = 0.1 – 0.3

Siegel, M. H., Merchuk, J.C., Schugerl, K., 1986. Air-Lift Reactor Analysis: Interrelationships between Riser,
Downcomer and Gas-Liquid Separator Behavior, including Gas Recirculation Effects. AIChE Journal 32(10),
1585-1595

Kunni, D., Levenspiel, O., 1991. Fluidization Engineering, 2nd Ed., Butterworth-Heinemann, Boston.

Design of distributor in fluidized beds
Nozzle Position (Litz, 1972)
Side entry:

H = 0.2D + 0.5D noz

H = 18D noz

, when

D noz > D 100

Dnoz < D 100

, when

Bottom entry:

H = 3(D − D noz )

H = 100D noz
Where,

, when

, when

D noz > D 36

D noz < D 36

D - diameter of gas distribution chamber
Dnoz - diameter of the nozzle
H - distance between the nozzle centerline and the
distributor plate

 Litz, W. J., 1972. Design of gas distributors. Chemical Engineering 13, 162-166.

18

1/11/2011

Different types of fluid beds / modifications

Example
A packed bed is composed of cubes 0.02 m on a side. The bulk density
of the packed bed, with air, is 980 kg/m3. The density of the solid cubes is
1500 kg/m3.
• Calculate the void fraction (ε) of the bed.
• Calculate the effective diameter (Dp) where Dp is the diameter of a sphere
having the equivalent volume.
• Determine the sphericity of the cubes.
• Estimate the minimum fluidization velocity using water at 38 C and a tower
diameter of 0.15 m.
Void Fraction
We know : Vbed = V fluid + Vsolids

and Wbed = W fluid + Wsolids

ρ bedVbed = ρ fluidV fluid + ρ solidsVsolids
ρ solidsVsolids >> ρ fluid V fluid

19

1/11/2011

Example

contd…..

∴ ρbedVbed ≅ ρ solidsVsolids

ε =

Effective diameter
3

a =

π
6

D

ε =

kg
980 3
m
= 1−
kg
1500 3
m

ρbed
ρ solids

1−

and

Vbed −

Vbed
= 0.35

3
p

(0.02)3 = π D 3p
6

ρbedVbed
ρ solids

∴ D p = 0.025m

Sphericity
6
(6 π )1 3 a =  π 1 3 = 0.81
Φs =
 
6
6
a

Mimimum Fluidization Velocity

p

− ρ f )g =

2
150(1 − ε mf )µ

ρ f umf
+ 1.75
3 
Φ s D p ε mf
 Φ s D p umf ρ f


kg
kg 
m
kg

− 994
1500
 ∗ 9.80 2 = 4959 2 2
m3
m3 
s
m s

Example

contd…..
3
Φ sε mf
=

1
∴ ε mf = 0.445
14

2
1.75 ∗ ρ f umf
3
Φ s D p ε mf

=

kg
2
∗ umf
kg
2
m3
= 9.748 ×105 2 2 ∗ umf
0.81∗ 0.025 ∗ (0.445)3
m s
1.75 ∗ 994

150 ∗ (1 − ε mf )µ ∗ umf
2

2

3
Φ s D p ε mf

= 1597

kg 
 ∗ u mf
150 ∗ (1 − 0.445) ∗ (0.693 cp ) ∗  0.001
m
s 

=
2
2
3
(0.81) ∗ (0.025 m ) ∗ (0.445)

kg
∗ umf
m2s 2

0 = 9.748 ×105

umf = 0.071

kg
kg
kg
2
∗ u mf
+ 1597 2 2 ∗ u mf − 4959 2 2
m s
m s
m2 s 2

m
s

20

1/11/2011

Heat Transfer in fixed and Fluidized beds
• Main advantage of fluidized beds is the extremely large area of solid surface
exposed to the fluidizing media
• High solid surface area greatly facilitates solid-to-gas heat transfer
• Because of the solids mixing generated within the bulk of a bubbling gas
fluidized bed, temperature gradients are reduced to negligible proportions
• High rates of heat transfer are obtainable between the fluidizing solids and
the immersed transfer surface
Particle-to-gas heat transfer
Bed-to-surface heat transfer
Use of immersed surfaces

21

1/11/2011

Heat Transfer in Fixed Beds
Heat transfer in fixed bed consists of following mechanisms
(1) conduction heat transfer between particles,
(2) convective heat transfer between particles and fluid,
(3) interaction of both (1) and (2),
(4) Radiative heat transfer between particles and the flowing gas
(5) Heat transfer between bed wall and bed particles

(1) Particle to fluid heat transfer
Heat transfer to single particle can be expressed as
Heat transfer coefficient can be evaluated as

(2) Heat transfer through wall (one dimensional model)
For homogeneous model, temperature of fluid and of bed are assumed identical

for

and

Heat Transfer in Fixed Beds
(3) Heat transfer through wall (two dimensional model)
For homogeneous model,
temperature of fluid and of bed are assumed identical

for

and

for

and

22

1/11/2011

Heat Transfer in Fluid beds
Heat transfer in a bubbling fluidized bed
• Gas to particle heat transfer coefficients are typically small, of the order of 5 - 20 W/m2K
• However, because of the very large heat transfer surface area provided by a mass of small
particles, the heat transfer between gas and particles is rarely limiting in fluid bed heat
transfer
• One of the most commonly used correlations for gas-particle heat transfer coefficient is that
of Kunii and Levenspiel (only for low particle Reynolds numbers)

for
• Gas to particle heat transfer is relevant where a hot fluidized bed is fluidized by cold gas
• While following are the correlation suggested based on experimental data*

for
for
*Chen,

J.C., Heat Transfer in handbook of fluidization and fluid systems, 2003

Analysis of Gas-particle heat transfer

The energy balance across the element gives

Integrating with boundary conditions Tg = Tg0 at L = 0

23

1/11/2011

Analysis of Gas-particle heat transfer

The distance Ln, in which the gas-to-particle temperature falls by a factor

Is given by,

The distance over which the temperature distance is reduced to half its initial value,
L0.5 is then

Analysis of Gas-particle heat transfer
A bed of 450µm particles is operating at 150˚C. The temperature and superficial velocity
of the incoming gas are 550˚C and 0.4 m/s, respectively. Approximately how far will
the incoming gas have penetrated into the bed before it is cooled to 350˚C
Gas physical properties can be estimated at average temperature over the specified
range. In this particular case the average temperature is 450˚C. Hence the physical
properties of air

Gas velocity at 450˚C for an inlet velocity of 0.4m/s at 550˚C will be 0.35m/s

24

1/11/2011

Heat Transfer in Fluid beds
Bed to surface heat transfer
In a bubbling fluidized bed the coefficient of heat transfer between bed and immersed
surfaces (vertical bed walls or tubes) can be considered to be made up of three additive
components
The particle convective component hcp, which is dependent upon heat transfer
through particle exchange between the bulk of the bed and the region adjacent to
the transfer surface (heat transfer due to the motion of packets of solids carrying heat
to and from the surface)

The interphase gas convective component hgc, by which heat transfer between
particle and surface is augmented by interphase gas convective heat transfer
The radiant component of heat transfer hr
Thus,
Approximate range
of significance

40µm

1mm

> 800 µm and at
higher static
pressure

Higher temperatures
(> 900 K) and
difference

Heat Transfer in Fluid beds
Particle convective heat transfer
On a volumetric basis the solids in the fluidized bed have about one thousand times the
heat capacity of the gas and so, since the solids are continuously circulating within the
bed, they transport the heat around the bed. For heat transfer between the bed and a
surface the limiting factor is the gas conductivity, since all the heat must be transferred
through a gas film between the particles and the surface

Heat transfer from bed particles to an immersed surface

25

1/11/2011

Heat Transfer in Fluid beds
Particle convective heat transfer
The particle-to-surface contact area is too small to allow significant heat transfer.
Factors affecting the gas film thickness or the gas conductivity will therefore influence
the heat transfer under particle convective conditions.
Decreasing particle size, for example, decreases the mean gas film thickness and so
improves hpc. However, reducing particle size into the Group C range will reduce
particle mobility and so reduce particle convective heat transfer. Increasing gas
temperature increases gas conductivity and so improves hpc.
Particle convective heat transfer is dominant in Group A and B powders. Increasing
gas velocity beyond minimum fluidization improves particle circulation and so
increases particle convective heat transfer.
The heat transfer coefficient increases with fluidizing velocity up to a broad maximum
hmax and then declines as the heat transfer surface becomes blanketed by bubbles.

Heat Transfer in Fluid beds
Bed to surface heat transfer

Range of fluidized bed-to-surface heat transfer coefficients

26

1/11/2011

Heat Transfer in Fluid beds
Bed to surface heat transfer

Effect of fluidizing gas velocity on bed – surface heat transfer coefficient

Heat Transfer in Fluid beds
Particle convective heat transfer
The maximum in hpc occurs relatively closer to Umf for Group B and D powders
since these powders give rise to bubbles at Umf and the size of these bubbles
increase with increasing gas velocity
Group A powders exhibit a non-bubbling fluidization between Umf and Umb and
achieve a maximum stable bubble size.
Zabrodsky (1966) has given correlation for hmax for group B particles

Khan (1978) has given correlation for hmax for group A particles

27

1/11/2011

Heat Transfer in Fluid beds
Gas convective heat transfer
Gas convective heat transfer is not important in Group A and B powders where the flow of
interstitial gas is laminar but becomes significant in Group D powders, which fluidize at
higher velocities and give rise to transitional or turbulent flow of interstitial gas
In gas convective heat transfer the gas specific heat capacity is important as the gas
transports the heat around.
Gas specific heat capacity increases with increasing pressure and in conditions where gas
convective heat transfer is dominant, increasing operating pressure gives rise to an improved
heat transfer coefficient hgc.
Baskakov and Suprun (1972) has given correlation for hgc

where Um is the superficial velocity corresponding to the maximum overall bed heat transfer
coefficient

Heat Transfer in Fluid beds
Gas convective heat transfer to immersed surfaces
Several approaches have been used to estimate hc
The most common approach assigns thermal resistance to a gaseous boundary layer at the
heat transfer surface, the enhancement of heat transfer is then attributed to the scouring
action of the solid particles on the gas film, decreasing the effective film thickness
Lava’s correlation (1952) for vertical surfaces, for larger particles, is

Wender and Cooper’s correlation (1958) for vertical tubes,

for
Where
Where r is the radial position of the heat transfer tube and Rb is the radius of the bed

28

1/11/2011

Heat Transfer in Fluid beds
Gas convective heat transfer to immersed surfaces
Vreedenberg’s correlation (1958) for horizontal tubes,

Changed
Equation
for
and

for
Where

Heat Transfer in Fluid beds
For temperatures beyond 600oC radiative heat transfer plays an increasing role
and must be accounted for in calculations
For rule of thumb estimate, the radiative heat transfer component can be
estimated using absolute temperatures and an adaptation of the stefen Boltzman equation in the form

where εr is the reduced emissivity to take into account the different emissivity properties of
surface εs and bed εb and is given by

An alternative correlation given by Panov et al. (1978) for approximate estimate is

29

1/11/2011

Application of Fluidized Beds
Industrial Processes
Physical

Dominating
Mechanism

Applications

Chemical

Heat and/or
mass transfer
between
gas/particles

Heat and/or mass
transfer between
particle/particle
or particle/surface

Heat transfer
between
bed/surface

Gas/gas reactions
in which solid acts
as catalyst or a
heat sink

Gas/solid reactions
in which solids are
transformed

• Solids Drying
• Absorption of
solvents
• Cooling of
fertilizer prills
• Food Freezing

• Plastic coating of
surfaces
• Coating of
pharmaceutical tablets
• Granulation
• Mixing of solids
• Dust Filtration

• Heat treatment of
textile fibres,
wires, rubber,
glass, metal
components
• Constant
temperature baths

• Oil cracking,
reforming
manufacturing of
• Acrylonitrile
• Phthalic Anhydride
• Polyethylene
• Chlorinated
hydrocarbons

• Coal combustion
• Coal gasification
• Roasting of nickel and
zinc sulphides
• Incineration of liquid
and solid waste
• Catalyst Regeneration
• Decomposition of
limestone

More about applications in Mass Transport PPT

30

Mass Transfer in Fluidized
Beds - An Overview

1/13/2011

Special topics in ME6203 MASS TRANSPORT

Mass Transfer in Fluidized Beds‐An Overview
Professor A. S. Mujumdar
ME Department, NUS
E‐mail ‐ [email protected]

Guest Lecturer
Sachin V. Jangam
Minerals, Metals and Material Technology Centre
NUS, Singapore (2011)

Contents
• Introduction‐Applications Why M T is important, reactions etc
• Approaches in mass transfer in Fluidized beds
• Homogeneous bed Approach‐limitations
• Mass transfer to single particles, fixed beds and fluidized
beds Empirical correlations
• Bubbling bed approach
• Kunii Levenspiel Model
• An example calculation
• Modeling a fluid bed dryer
• Key references (given at the end)

1/13/2011

Introduction
• Main  applications  of  fluidized  beds  –  Fluidized  catalytic  cracking;  Combustion  and
Gasification; Drying; Granulation;

• The  performance  of  fluidized  beds  in  all  the  above  processes  is  affected  by  the
interface mass transfer

• Many times the mass transfer in fluidized beds is a potential rate controlling  step in
fluidized bed reactors

• Depending on the type gas‐solid interactions, the rate controlling mechanism can be

1. Particle – gas mass transfer (Gas film diffusion) control
2. Pore diffusion control
3. Surface phenomenon control

Various Industrial Applications

Fluidized bed reactor

Fluid bed dryer

Fluid bed catalytic cracker

1/13/2011

Mass transfer in fluidized beds
There  are  two  approaches  that  can  be  used  for  prediction  of  mass
transfer rates in fluidized beds

• Homogeneous bed approach – considers fluidized bed behaving
as a fixed bed reactor and correlate the mass transfer coefficient
in fluidized bed in similar manner to that in a fixed bed based on
plug‐flow model

• Bubbling  bed  approach  –  Considers  fluidized  bed  to  consist  of
two phases, a bubble and an emulsion phase, the gas interchange
between the two phases constitutes the rate of mass transfer

Useful Dimensionless Numbers
Sherwood Number – ratio of convective to diffusive mass transport

Convective mass transfer coefficient
k L
Sh
Sh

Diffusive mass transfer coefficient

Schmidt Number – ratio of momentum to mass diffusivity

μ
Momentum diffusivity
Sc

Sc

ρ
Mass Diffusivity

Reynolds Number – Comparison of Inertial force to viscous force

Reynolds Number – Comparison of gravitational and viscous force

1/13/2011

Homogeneous Bed Approach
Transfer between single sphere and surrounding gas

• Rate  of  mass  transfer  between  well‐dispersed  sphere  and  surrounding  air  can  be

written as

Concentration of
A in the bulk gas
stream
Transfer rate of
A from particle
to gas

Mass transfer
coefficient of
single particle

Concentration of
A at particle‐gas
interphase

• The  single  particle  mass  transfer  coefficient  can  be  obtained  from  well  established
correlation (Froessling, 1938)

•                                                               Eq (2)
where particle Reynolds number and Schmidt number can be defined as

The mass transfer coefficient is proportional to diffusion coefficient of gas and inversly
proportional to diameter of particle

Homogeneous Bed Approach
Transfer between single sphere and surrounding gas

For non‐spherical particle the sieve diameter (dp) will replace the diameter of
the particle and the equation becomes

where particle Reynolds number is

1/13/2011

Homogeneous Bed Approach
Transfer between fixed bed particles and flowing gas

Rate of mass transfer between fixed bed of particles and surrounding gas can be written
in the same fashion as the one for single particle

Combined mass
transfer rate
from all the
particles

Average mass
transfer
coefficient of
particles

Total exterior
surface of all
individual
particles

Concentration of
A at particle‐gas
interphase

Concentration of
A in the bulk gas
stream

Homogeneous Bed Approach
Transfer between fixed bed particles and flowing gas

The correlation for the average mass transfer coefficient kg,bed was given by Ranz (1952)

Volume of bes
Total particle exterior surfaces Sex,particles
Bed voidage
segment

Method of Kunni Levenspiel
Particle surface to
particle volume ratio
Hence the equation for total exterior particle
surfaces becomes
Sphericity of bed
particles

1/13/2011

Homogeneous Bed Approach
Transfer between fluidized bed particles and fluidizing gas

The rate of particle to gas mass transfer in the differential segment of fluidized bed can

be written as

Average mass
transfer coefficient
associated with
fluidizing particles

Total exterior
surfaces of the
fluidized particles in
the segment of bed

Kg,bed for fluidized beds is always higher than that for fixed beds

Homogeneous Bed Approach
Average mass transfer coefficient for fluidizing particles

The mass transfer coefficient for fluidized bed particles can be lower or higher than that

of  single particles

Generally for low Reynolds numbers the mass transfer coefficient for single particles is
higher  than  for  fluidizing  particles  however,  reverse  is  true  for  higher  particle
Reynolds numbers (>80)

Resnick  and  White  (1949)  reported  the  average  mass  transfer  coefficient  for  the
fluidizing particles (for air system with Sc = 2.35)

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Homogeneous Bed Approach
Average mass transfer coefficient for fluidizing particles

Homogeneous Bed Approach
Particle – gas mass transfer coefficient for fluidized bed particles

Using Sh equation for
single particle Eq (2)

particle‐gas mass transfer coefficient for
fluidized bed particles

Using Sh equation for
single particle Eq (2)

Comparison of mass transfer coefficient for
fluidized bed and fixed bed

• It should be noted that the same correlations can be used for fluidized beds and fixed beds…
• “Sh” for same group of particles continues to increase with the particle Reynolds number even
during the transition from fixed bed to fluidized bed operations

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Homogeneous Bed Approach
Limitations of Homogeneous bed approach
The experimentally measured coefficient values for the bed particles under fixed bed or

fluidized  bed  conditions  can  be  lower  or  higher  than  the  theoretically  estimated
values
For  fine  particles  the  mass  transfer  coefficients  were  found  to  be  well  below  the
estimated values from correlations
The measured mass transfer coefficient by this approach should be treated as empirical
in nature
Using Sh equation
for  single particle
Eq (2)

Summary  of  particle‐gas  mass
transfer coefficient

Bubbling Bed Approach
Takes  in  to  account  existence  of  a  two  phases:  bubble  phase  and
emulsion phase

Bubble phase – considered as spherical bubbles surrounded by spherical clouds

Bubble

Cloud

There are three different models available based on this approach
Kunni and Levenspiel (bubble – emulsion transfer)
Partridge and Rowe (cloud – emulsion transfer)
Chavarie and Grace (Empirical correlation for bubble‐ emulsion transfer)

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Model of Kunni and Levenspiel
Model for vaporization or sublimation of A from all particles in bed

Assumptions:

Fresh gas enters the bed only as bubbles

Equilibrium  is  established  rapidly  between  CA  at  gas‐particle  interphase  and

its surroundings

These  assumptions  lead  to  the  following  equation  in  terms  of  bubble‐emulsion  mass
transfer coefficient (KGB)as follows

In this approach both cloud and the emulsion phases are assumed perfectly mixed

Model of Kunni and Levenspiel
Relation between KGB and Kg,bed

The  equation  reported  in  last  slide  for  mass  transfer  by  Kunni  and  Levenspiel  can  be
derived using the mass transfer rate equation for homogeneous bed approach

With the assumption that fresh gas enters the
bed  only  as  bubbles  the  above  equation
becomes

Where  CA,b  is  the  concentration  of  A  in
bubble phase

1/13/2011

Model of Kunni and Levenspiel
Relation between KGB and Kg,bed

The solutions of the previous equations finaly results in the following relation between
KGB and kg,bed

The definition of Sherwoods Number becomes

KGB for Nonporous and Nonadsorbing Particles
The  particles  dispersed  in  bubble  phase  will  not  contribute  to  any  additional  mass
transfer and hence; hence transfer across the bubble‐cloud boundary therefore is the
only source of mass transfer

Hence

Where Kbc is the bubble‐cloud interchange coefficient derived by Davidson and Harrison
as

For single particle
So the KGB equation becomes

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1/13/2011

For highly adsorbing particles, both particles dispersed in bubble and the bubble‐cloud
gas interchange can contribute to the particle‐gas mass transfer and the expression
for KGB becomes

For single particle

So the KGB equation becomes

Hence the Sherwood number for bed becomes

KGB for Porous or Partially Adsorbing Particles
For porous or partially adsorbing particles, Kunni and Levenspiel derived the following
equation

Where,

And

m is the adsorption equilibrium constant defined as

CAs is the concentration of tracer A within the particle in equilibrium with the
concentration CiA  of tracer gas at the gas‐particle interphase.

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1/13/2011

Model of Partidge and Rowe
The cloud surrounding the bubble is considered as the primary mass transfer boundary
and the bubble and the clod phases are considered as perfectly mixed single phase
as shown in the following figure

The mass transfer equation is

Volume of gas
in the bubble‐
cloud phase

Cloud
emulsion mass Cloud exterior
surface
transfer
coefficient

Model of Partidge and Rowe
Partidge  and  Rowe  have  given  a  correlation  for  mass  transfer  coefficient  in  terms  of
Sherwood number as follows

where Sc is the Schmidt number as defined earlier while Rec is defined as

Diameter of
Relative velocity

sphere with
between rising

the same
cloud and
emulsion

volume as the
cloud

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1/13/2011

Model of Chavarie and Grace
Measurement  of  mass  transfer  rates  for  bubble  containing  ozone  injected  into  an  air‐
fluidized two dimensional bed and proposed following empirical equation,

Table shows the models for kgc given by various researchers

Model of Chavarie and Grace
Table showing  some more models for kgc given by various researchers

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1/13/2011

Relation between mass and heat transfer coefficient
The correlations for particle to gas mass and heat transfer coefficient are closely related
when  the  Sherwood  number  (Sh)  is  equivalent  to  Nusselt  number  (Nu)  and  the
Schmidt number (Sc) is equivalent to Prandtl number (Pr)

For example, the correlation for Nusselt number for particles in a fixed bed is expressed
as

The corresponding correlation equation for the Sherwood number has the form

Hence the correlations obtained for Nu (heat transfer) can be converted to Sherwood
number to obtain mass transfer coefficient, provided ‐‐‐ Sh   Nu and Sc  Pr

Some Concluding Remarks on Mass Transfer in FB
Fluidized beds are very important for solid‐fluid contact for various important
industrial applications

Heat as well as mass transfer is important for design considerations

When kinetic processes such as mass transfer are carried out with fluidized
beds the particles dispersed in bubble phase should be taken in to account

Mass transfer coefficient measured for the bed as a whole (kgbed) is model
dependent

For large particles (cloudless bubble bed) the plug flow model closely matches
the bed conditions in the bed and the mass transfer coefficient for bed should
be equal to single particle mass transfer coefficient

For fine particles kgbed << kgsingle

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1/13/2011

Application of Fluidized Beds
Industrial Processes
Physical

Dominating
Mechanism

Heat and/or
mass transfer
between
gas/particles

Applications  • Solids Drying

• Absorption of
solvents
• Cooling of
fertilizer prills
• Food Freezing

Chemical

Heat and/or mass
transfer between
particle/particle
or particle/surface

Heat transfer
between
bed/surface

Gas/gas reactions
in which solid acts
as catalyst or a
heat sink

Gas/solid reactions
in which solids are
transformed

• Plastic coating of
surfaces
• Coating of
pharmaceutical tablets
• Granulation
• Mixing of solids
• Dust Filtration

• Heat treatment of
textile fibres,
wires, rubber,
glass, metal
components
• Constant
temperature baths

• Oil cracking,
reforming
manufacturing of
• Acrylonitrile
• Phthalic Anhydride
• Polyethylene
• Chlorinated
hydrocarbons

• Coal combustion
• Coal gasification
• Roasting of nickel and
zinc sulphides
• Incineration of liquid
and solid waste
• Catalyst Regeneration
• Decomposition of
limestone

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1/13/2011

Applications
Fluidized bed drying
Fluidizing  with  hot  air  is  an  attractive  means  for  drying  many  moist  powders  and
granular products

The  technique  has  been  used  industrially  for  drying  crushed  minerals,  sand,
polymers,  fertilizers,  pharmaceuticals,  crystalline  materials  and  many  other
products.

The main reason for its popularity is

Efficient  gas‐solids  contacting  leads  to  compact  units  and  relatively  low  capital  cost
combined with high thermal efficiency

Very high heat and mass transfer and hence reduced drying times

The  absense  of  moving  parts,  low  maintenance  cost  and  possibility  of  using
continuous mode

The  main  limitation  is  the  material  to  be  dried  should  be  fluidizable  and  should
have narrow particle size distribution

Fluid Bed Dryer

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1/13/2011

Fluidized Bed Drying
Different designs of fluid bed dryers

Fluidized Bed Drying
Well mixed fluid bed dryer

common FBD used in industry.
bed  temperature  uniform,  equal  to  the
product and exhaust gas temperatures.
particle  residence  time  distribution  is
wide
wide range of product moisture content.
feed  is  continuously  charged  into  FB  of
relatively  dry  particles,  this  enhances
fluidization quality.
a  series  of  well‐mixed  continuous  dryers
may  be  used  with  variable  operating
parameters.

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1/13/2011

Fluidized Bed Drying
Plug flow fluid bed dryer

vertical  baffles  are  inserted  to
create a narrow particle flow path.
narrow  particle  residence  time
distribution.
nearly equal residence time for all
particles regardless of their size
uniform product moisture content.
length‐to‐width  ratio  from  5:1  to
30:1.
inlet  region  may  be  agitated  or
apply  back‐mixing,  or  use  a  flash
dryer  to  remove  the  surface
moisture.

Modeling ‐ Fluid Bed Dryer
Diffusion model
This model assumes that the drying of single particle in the fluidized bed is controlled
by the diffusion of moisture inside the particle

Empirical model
• In this approach the drying process is divided into different periods where drying
mechanisms in each drying period are different;
• The solution of Fick’s law of diffusion expresses the moisture content in terms of
drying time by exponential function;
• Experimental  data  obtained  from  fluid  bed  drying  experiments  can  be  correlated
using exponential function

Kinetic model

Single phase model ‐ Explained later

Two‐phase model ‐ Explained later

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1/13/2011

Modeling ‐ Fluid Bed Dryer
Single phase model

The fluidized bed is regarded essentially as a continuum

Heat and mass balances are applied over the fluidized bed

Assumption ‐ particles in the bed are perfectly mixed

Mass balance

Energy balance

Modeling ‐ Fluid Bed Dryer
Two phase model

Two‐phase model of fluidized bed drying treats the fluidized bed to be composed of a
bubble phase (dilute phase) and an emulsion phase (dense phase)

gas in excess of minimum fluidization velocity, umf, flows through the bed as bubbles
whereas the emulsion phase stays stagnant at the minimum fluidization conditions
Mass balance

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1/13/2011

Modeling ‐ Fluid Bed Dryer
Two phase model
Mass balance of liquid in the bubble phase

M ass balance of liquid in the interstitial gas in the dense phase gives the following ng
equation

Mass balance of liquid in the dense‐phase particles

The coupled mass and energy balance in dense phase that consists of particles and
interstitial gas phases

Fluidized bed combustion boilers
• Fluidized bed combustion systems use a heated bed of sand‐like material suspended (fluidized)
within a rising column of air to burn many types and classes of fuel.

• This technique results in a vast improvement in combustion efficiency of high moisture content
fuels, and is adaptable to a variety of "waste type fuels.

• The scrubbing action of the bed material on the fuel particle enhances the combustion process
by stripping away the carbon dioxide and char layers that normally form around the fuel
particle.

• This allows oxygen to reach the combustible material much more readily and increases the rate
and efficiency of the combustion process

• Particles (e.g. sand) are suspended in high velocity air stream: bubbling fluidized bed

• Combustion at 840° – 950°C

• Capacity range 0,5 T/hr to 100 T/hr

• Fuels: coal, washery rejects, rice husk, bagasse and agricultural wastes

• Benefits: compactness, fuel flexibility, higher combustion efficiency, reduced SOx & NOx

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1/13/2011

Fluidized bed combustion boilers
Atmospheric Fluidized Bed Combustion (AFBC) Boiler
• Most common FBC boiler that uses preheated atmospheric air as fluidization and
combustion air

Fluidized bed combustion boilers
Pressurized Fluidized Bed Combustion (PFBC) Boiler
• Compressor supplies the forced draft and combustor is a pressure vessel
• Used for cogeneration or combined cycle power generation

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1/13/2011

Fluidized bed combustion boilers
Circulating Fluidized Bed Combustion units

A  “Circulating  Fluidized  Bed  Boiler”  commonly
abbreviated  as  CFB  is  a  device  for  generating
steam  by  burning  fossil  fuels  (coal)  in  a  furnace
operated  under  a  special  hydrodynamics
conditions.

Bed  material  is  heated  upto  the  ignition
temperature  of  the  coal  with  the  help  of  natural
gas burner. Coal and limestone are injected at the
bottom of the combustor.

Total  air  required  for  combustion  is  split  in  to
primary air and secondary air.

Circulating solids are transported in the combustor
at  a  velocity  exceeding  the  terminal  velocity  of
average particles.

Recirculation  of  solids  creating  uniformity  in  the
temperature  makes  the  combustor  as  an  efficient
combustion system

Visit us at
http://serve.me.nus.edu.sg/arun/
http://www.mujumdar.net78.net/

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1/13/2011

Key References
• Yang, W.C.; Handbook of Fluidization and Fluid –Particle Systems, Marcel Dekker, USA,
2003

• Kunni, D.; Levenspiel, O. Fluidization Engineering, 1969

• Mujumdar, A.S. Handbook of Industrial Drying, 3rd Ed; CRC Press: Boca Raton, FL,
2006

Some questions for self study
For drying of certain product, the ambient air of 80% relative humidity is heated
to 70°C before entering the fluid bed dryer of uniform particle size.
• Discuss qualitatively, what will be the effect if the air is cooled down to 5°C
and reheated back to same temperature of 70°C before entering the dryer.
During which drying rate period will this be more useful
• Discuss what will be the effect on performance if the bed particles have wide
size distribution, Suggest improvements to tackle this situation.
• What will be the effect on drying if heated inert spherical particles of same
size are added in a bed of particles to be dried?
• How can one enhance the drying of a non‐spherical particles in a fluidized
bed?

For combustion of a single particle
• Discuss what would be the combustion behavior of a single coal particle of
same mass with different shapes such as Sphere, Cylinder (D=L), cube
• Discuss qualitatively the difference in combustion of a coal particle one with
20% porosity and another with 60% porosity

23

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