Mobility and Coalescence of
Water Droplet Formed in Fuel Cells
Overview




Introduction
Objectives
Mathematical Modeling
Sample Simulations
Introduction



A unit cell of PEMFC is
composed of a
membrane electrode
assembly, two diffusion
media, and bipolar
plates.
The diffusion medium
needs to be optimized to
enhance the cell
performance.
In cathode diffusion
medium, the product
water flows towards the
channel through gasphase diffusion or liquidphase motion.
Porous Cathode
H2O
H2
H+
Porous Anode
O2
At high current densities, the liquid flow rate
increases due to the increased condensation, and
when the channel is at the local vapor saturation
condition, liquid water flows out of diffusion
medium and surface droplet are formed.
Fiber Screens
Stacked fiber screens, and the same exposed
to water-vapor saturated atmosphere.
Objective



Characterization of the mobility of micro-drops in the
diffusion medium.
Growth of the micro drops through coalescence.
Consider effect of:







Initial droplet number density;
Initial droplet size;
Vapor diffusion rates;
Rate of droplet onset;
Growth rate of drops;
Surface contact angles;
Thickness of the membrane.
Proposed Porous Model
Mathematical Model

Basic assumptions


Governing equations





Laminar flow ; incompressible and Newtonian fluid
 
 V  0
Continuity






  
Momentum   FV
    FVV  P  .  F V  Fb
t
f  
 V .f  0
Free-surface tracking
t



Modified flow equations in presence of solid walls
Finite volume method ; structured mesh ;
Youngs algorithm ; volume fraction for walls

Energy Equation- Enthalpy Method

Original Energy Equation
h
1
 (V  )h 
  ( k T )
t


Final Energy Equation
h
1
1
 (V  )h 
  (  h ) 
2
t


h  0:

ks
,  0
cs
0  h  H f :   0,
h  Hf :

 0
H k
kl
,   f l
cl
cl
h
interphase
cl
Hf
Tm
solid
cs
T
liquid
Free Surface Tracking

Step 1. Specify liquid domain using VOF method

in a cell full of liquid
 f 1

in a cell with no liquid
Define a function f  f  0
 0  f  1 for a cell partly filled with liquid


Represent actual liquid domain by corresponding f values
liquid
Actual liquid region
0
0
.10
.38
.48
0
.25
.87
1
1
.12
.91
1
1
1
Volume of Fluid representation
Modeling Solid Walls

Step 1: Define a Volume Fraction
 1
for a cell in the liquid
0
for a cell in the wall
0    1 for a cell partially
occupied by the wall


Step 2: Use  instead of a
stair-step model
Step 3: Modify fluid flow
equations
Liquid
0.2
0.8
1
0
0.05
0.7
0
0
0.1
Wall
Modified Fluid Equations

Continuity, Momentum and VOF Equations


  V  0




 

  F V
    F VV  P  .  F V  Fb
t




f
 (V .) f  0
t



Sample Simulations
Impaction of a water Drop on a fiber
Droplet diameter: 2 mm
3.18 mm tube; 1 m/s
offset 1.55 mm
two perpendicular
tubes (0.5 mm)
no offset
Code Validation
Droplet : 2 mm, 1 m/s; Tube: 3.18 mm (0.125 in); Offset: 1.55 mm
Code validation (continued)
Droplet : 2 mm, 1 m/s; Tube: 3.18 mm (0.125 in); Offset: 1.55 mm
Droplet impact with heat transfer
Droplet flow through a modeled Porous Medium

Radial Capillary

A staggered array
of perforated
plates
Draw back and coalescence:Drop Spacing 42 m ;
Droplets coalesce
Contact Angle 45o
Contact Angle 120o
Drop Spacing 44 m ; Contact Angle 120o
Droplets do not coalesce
Droplet Collisions


Head- On Collisions

Permanent Coalescence

Reflexive Separation
or
Off-axis Collisions

Permanent Coalescence

Stretching Separation then
Permanent Coalescence

Stretching Separation

Tearing
or
Droplet Coalescence
Drop Collisions: Experiments vs. Simulations
Experimental
Numerical
Coalescence collision of two drops.
Off-Axis Collisions
Tearing (Water, We=56, Re=2392, X=0.73 and =0.5)
Thank you
Summary

Our 3D free surface code can simulate the
detailes of the droplet dynamics in porous
systems.
Continuity and Momentum

Two-step time discretization of momentum equation:

~ n
 n 1
n  n 1  n
V V
n
   VV  n    F V  G  n Fb
(i )
t
F
F

 n 1 ~
V V
1
  n p n 1
(ii )
t
F
 




Step (i): evaluate convective, viscous and surface tension
effects explicitly
Step (ii): combine with continuity equation to obtain the Poisson
pressure equation:

~
 1
 V
n 1 
    n p 
t
 F

Free Surface Tracking

Step 2. Use function f to advect the free surface to
a new location at each time step
ut
f  
 V .f  0
t

u
Step 3. Reconstruct the free surface shape at the
new location using Youngs algorithm
0
n̂
.10


Obtain normal nˆ  f / f
.05 .20
.87
1
and use values of f : get free surface cutting planes
Youngs’ 3D-VOF
Interfaces - piecewise “planes”
Interface plane is fitted within
a single cell
Interface slope and fluid
position are determined from
inspection of neighboring cells
Pentagonal Section
Youngs’ 3D - Cases
Triangle
Quadrilateral - A
Pentagon
Quadrilateral - B
Hexagon
Youngs’ VOF
Implementation
Surface Reconstruction
Using f-field determined cell “normal” (i.e. n  f )
f
Determine “case” using normal
Position plane with known slope based upon volume
fraction
Compute plane area and vertices
Fluid Advection,
Compute flux across cell side (case dependent)
Operator Split (i.e. do for x, y and z sweeps)
 f 
 V  f  0
t
General Solution Procedure
1. Specify initial surface geometry and velocities
2. Begin Cycle, increment time and repeat 2-6 until
done

n
~
3. Explicitly update V  V
convective terms
viscous terms
surface tension terms
4. Implicitly calculateP n1 using ICCG Method

5. UpdateV n1 with P n1 and apply BCs
6. Advect f-field using Youngs VOF and re-apply BCs
Nondimensional Parameters
Impact Parameter
ds
V
s


Vr
Weber Number
X
2b sin   
We 
dl  d s
ds Vr

b
Relative Velocity Vr  Vl 2  Vs2  2VlVs cos 

V
dl
X
Vr
Reflexive Separation

(X=0)

Fragmentat ion 


Stretching Separation
(X>0)
Size Ratio

ds
dl
Head-on Collisions
Separation (Water, We=58, Re=1127)
0
1
6
14
2
7
15
16
3
8
17
18
4
9
19
10
20
21
5
11
12
13
22
23
24
Off-axis Collisions
Coalescence (Water, We=25, Re=740, X=0.7)
0
11
1
12
2
13
3
14
4
15
5
16
6
7
17
8
18
9
10
19
20