Transport phenomena in electrochemical systems Charge and mass

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 1

• Dr Bradley P Ladewig, presenting instead of Francois Lapicque • Ph. D 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!) 2

1 - Various physical phenomena in electrolyte solution H H O Men+ O H Metal ions (and also anions) are highly solvated. H Solvatation Existing forces and hindrance to motion Ionic atmosphere (negative charge) Electric field Men+ Fion NB: these effects are rarely accounted for in models Fionic atm. 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 3

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

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 Specific flux of species i Ci vi Flux for diffusion and migration Theory of irreversible processes µie: electrochemical potential Ion activity Elec. potential 5

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 Diffusion term (Fick) Migration : Motion of ions (zi) under the electric field NB: This equation is not rigorous in most cases, however, it is often used because of its simplicity Other expressions available from theory of irreversible processes (Stefan maxwell, Onsager …) 6

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 7

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: which leads to the Stokes Einstein’s relation only in dilute media For more concentrated media, various laws…. Dµ 0. 7/T = Constant * Transference number: fraction of the current transported by species i 8

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 with Transference numbers The salt behaves like an non-dissociated species, with the overall diffusivity D being compromise between D+ and D- Expression for the current density NB: Although extensively used, 9 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 anoth * Turbulent, where the fluid is divided into aggregates. The velocity of the aggregate possesse 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 10

3 - Flow fields in electrochemical cells Two dimensionless numbers allow the flow to be defined in the considered system Friction factor Tangent. stress/kinetic energy Reynolds number 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 11

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 12

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 Gas 1 Liquids 1000 Pr = n/a 1 10 Le = a/D 1 13

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 Did 2 Ci/dx 2 = 0 NB: the velocity profile is Sc 1/3 thicker than the concentration profile. i. e. 10 or so Linear profile of the concentration Only the diffusion term 14

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

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…) 16

4 - Mass transfer to electrode surfaces (C’td) Examples L=1 m dp=0. 005 m n=10 -6 m 2/s D=10 -9 m 2/s k. L = A <u>n Laminar flow 1/3 < n < 0. 5 Turbulent flow 0. 6 < n < 0. 8 17

4 - Mass transfer to electrode surfaces How to determine them? * 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 Poorly accurate!! Access to overall data, only Reliability of the data? Is your system so close? * Measure the limiting current at electrode surfaces • Access to local rates with microelectrodes • Find the right electrochemical system (solution, electroactive species • Do measurement with the academic system • Deduce estimate for k. L in the real case using dimensionless analysis 18