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

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!)

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

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

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

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 …)

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

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

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

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

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

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 = 0.023 Re-0.2 for 104 < Re < 2 106 (the pressure drop varies with <u> 1.8

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 1 1 1 Liquids 1000 10

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

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

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…)

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 0.6 < n < 0.8

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