1. Transport phenomena in electrochemical systems:
Charge and mass transport in electrochemical cells
F. Lapicque, CNRS-ENSIC, Nancy, France
Outline
1- Various phenomena in electrolyte solutions
2- Mass transport rates and current density
3- Flow fields in electrochemical cells (a brief introduction to)
4- Mass transfer rates to electrode surfaces

2. Dr Bradley P Ladewig, presenting instead of Francois Lapicque
PhD in Chemical Engineering (Nafion Nanocomposite membranes for the Direct Methanol Fuel Cells)
Currently working as a Postdoc for Francois Lapicque at CNRS ENSIC, Nancy France
Originally from Australia (which is a long, long way from Serbia!)

3. 1- Various physical phenomena in electrolyte solution
Metal ions (and also anions)
are highly solvated.
Solvatation
Relaxation:
caused by interactions
between the cation and the
ionic atmosphere
This atmosphere is distorted
by the motion of Men+
(It is a sphere for nil electric field)
Electrophoretic effect :
Force on the ionic atmosphere.
acts as an increase in
solvent viscosity
Existing forces and hindrance to motion
Ionic atmosphere
(negative charge)
Electric field
Fionic atm.
Men+
Fion
NB: these effects are rarely accounted for in models

4. 1- Transport phenomena: introduction to migration
Electrical force on ions (charge q)
Velocity of the charged particle
Absolute mobility of the ion
Specific migration flux
(Stokes’ law)
Ion: very small particle
mol m-3

5. 2- Mass transport rates and current density
General equations of transport
Consider a fluid in motion
Species i Concentration Ci and velocity of ions vi
Defining a barycentric molar velocity
Convection flux Ci v
Specific flux of species i Ci vi
Flux for diffusion and migration
Theory of irreversible processes
µie: electrochemical potential
Ion activity Elec. potential

6. 2- Mass transport rates : the Nernst-Planck equation
From the expression for Ji and the relation between Ni and Ji:
Assuming ideal solutions (ai = Ci) leads to the Nernst Planck equation (steady state):
Convection :
Overall motion
of particles with
barycentric velocity
Migration :
Motion of ions (zi) under
the electric field
Diffusion
term
(Fick)
NB: This equation is not rigorous in most cases, however, it is often used because of its simplicity
Other expressions available from the theory of irreversible processes
(Stefan maxwell, Onsager …)

7. 2- Mass transport rates : expression of the current density
Equations in electrochemical systems
Current density
Electroneutrality equation
Without C gradients:
Medium conductivity (low C)
NB: The current density can be defined and calculated anywhere in the electrolytic medium

8. 2- Mass transport rates: some more useful relations
* Relation between diffusivity and ion mobility
For the expression of the migration flux and Nernst-Planck equation:
only in dilute
media
which leads to the Stokes Einstein’s relation
For more concentrated media, various laws…. Dµ0.7/T = Constant
* Transference number: fraction of the current transported by species i

9. 2- Mass transport rates: the trivial case of binary solutions
Binary solutions: one salt dissolved (one cation and one anion)
Assuming total dissociation of the salt leads to (general transient expression):
same for the anion
Replacement of the electrical term, and algebraic rearrangement leads to
The salt behaves like an
non-dissociated species, with the
overall diffusivity D being
compromise between D+ and D-
with
Transference numbers Expression for the current density
NB: Although extensively used,
the relation is only valid for binary solutions

10. 3- Flow fields in electrochemical cells (an introduction to)
Fluid in motion along a surface
The stress applied to the fluid has two components
- the normal component, corresponding to a pressure
- the second one, along the plane, corresponds to viscous force
The structure of the flow can be
* Laminar, for which the fluid is divided into thin layers (« laminae » that slide one each another
* Turbulent, where the fluid is divided into aggregates. The velocity of the aggregate possesses
a random component, in addition to its steady component
NB. For too short systems, with local changes in direction and cross-section, the flow is disturbed
or non-established

11. 3- Flow fields in electrochemical cells
Two dimensionless numbers allow the flow to be defined in the considered system
Friction factor
Reynolds number
Tangent. stress/kinetic energy
Inertia/viscous forces
<u> Average velocity, d charactetistic dimension
A few comments
Laminar/turbulent transition: for Re = 2300? Only in pipes
Very large systems are in turbulent flow… e.g. atmosphere, oceans
Minimum length for the flow to be established
Which characteristic length d? Gotta find the length of highest physical meaning
Jf is used for estimation of the pressure drop

12. 3- Flow fields in electrochemical cells: laminar and turbulent flows
Laminar flow (example of a pipe)
Jf = 8/Re
Parabolic velocity profile The pressure drop varies with <u>
Turbulent flow (example of a pipe)
More complex expression for the velocity, but the profiles are much flatter
One example for the expression of the friction factor: Blasius’ relation
Jf = Re for 104 < Re < (the pressure drop varies with <u> 1.8

13. 4- Analogy between the transports of various variables
Specific flux = - Diffusivity x Gradient of the extensive variable
Heat (J)
Weight (kg)
Example
Dimensionless numbers: ratio of the diffusivities and orders of magnitude
Sc = n/D Pr = n/a Le = a/D
Gas
Liquids

14. Did2Ci/dx2 = 0 4- Mass transfer to electrode surfaces
Mass balance (transient) in a fluid element near the electrode surface
Whow!
The Nernst-film model: a cool shortcut for approximate calculations of mass transfer rates
Steady-state conditions
Negligible migration Flux
Vicinity of the electrode (low u)
1-D approach
Did2Ci/dx2 = 0
NB: the velocity profile is
Sc1/3 thicker than the
concentration profile. i.e. 10 or so
Linear profile of the concentration
Only the diffusion term

15. neFkL 4- Mass transfer to electrode surfaces (C’td)
Expression for the current density
Defining the mass transfer coefficient, kL
neFkL
Limiting current density: when CAS tends to 0
Miximum value for the current density iL : Fe can be equal to 1
maximum production rate

16. 4- Mass transfer to electrode surfaces (C’td)
Two dimensionless numbers:
Re and Sh (Sherwood)
What’s the use of these dimensionless relations??
Hint: Possible change in velocity, dimensions and physicochemical properties (D, n…)

17. 4- Mass transfer to electrode surfaces (C’td) Examples
L=1m
dp=0.005 m
n=10-6 m2/s
D=10-9 m2/s
kL = A <u>n
Laminar flow 1/3 < n < 0.5 Turbulent flow < n < 0.8

18. 4- Mass transfer to electrode surfaces How to determine them?
Poorly accurate!!
Access to overall data, only
* Measure pressure drops in the system and
use energy correlations
(bridge between the dissipated energy
and the mass transfer rate)
Find the most suitable correlation in
your usual catalogue or in published works
* Measure the limiting current at electrode surfaces
Reliability of the data?
Is your system so close?
Access to local rates with microelectrodes
Find the right electrochemical system (solution, electroactive species
Do measurement with the academic system
Deduce estimate for kL in the real case using dimensionless analysis